X-ray Spectroscopy of Molecules Driven by Strong Infrared Fields
Transcrição
X-ray Spectroscopy of Molecules Driven by Strong Infrared Fields
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540 Probe 538 8 3.5 Probe B PT 6 4 ∆ε B 542 d 531.0 Probe Od core ionized state Ω (eV) 2 − Energy (eV) Energy (eV) 534.0 Ω (eV) ¡&¬9x,¡& Áxz9x{jÚ|0¦Ú~'¢Rxëyxx{~'½Ówy© vT ÅAËÊË(9ËóÛܬAxwyw©bÀÆ~0¡&¬AìVÅAwxë¾wyª¡&¬Ax,ñ ¦{x~d¡&x{w©¾½Æ~9z0¡&¬Ax,¦x{~d¡&xw©½ÆÅwyi¥qסØ|Ù©¾½Ü¡&¬Ax,ÄÉx¦¡ywyAËó/ ~9¦{x¡y¬Ax,yÄRx{¦5¡&wyÂÄAw©ìVx yx~Ayס&×¥xD¡y©È¡y¬AxDñ'ÄÉ©y¡y ©~fd©~Ax<¦{~,xÃ9ÄRx{¦5¡Ü§wx{ ¾¡&©~Ay¬9 ÄÝ¢Éx5¡ØÀÂxx{~ë¡&¬9xD¡&wyZ Yx{¦5¡&©wJ|Á©½?¡&¬9x ÀÂi¥xÈÄA¦«ªx5¡`~Azë¡y¬AxN¦{x{~d¡yx{w1©½TÅw&Ú¥q¡Ø|멽P¡&¬9x`ãw&Ú|}¢A©wyÄ/¡& ©~?Ë Ionization probability Ground state A 2 0 1 2 Pump 3 4 vTJËÂË(9áf `A ס«¾¡y¥xÁ AJ¡&wy¾¡&©~Ù©¾½1¡y¬Axëê −ãw&Ú|0ÄA9ÏÄ/ÄAwy©¢Éx,ÄÉx¦¡ywy©¦{©Äd|ËßÛܬAxë x5½Y¡ ~Azßwy Ŭ£¡§ÄV~Ax §wyx¾mwyxyÄRx{¦¡y¥x{|m¡&¬AxëðÁ÷Ú«ø ãØw&Ú|ߢA©wyÄ9¡y ©~Ù©½Ü¡&¬9xë ð>Á© x¦{Ax~9z ðÝ÷JÚ&øÊãØw&Ú|ëÄ9¬A©¡&©qx{x{¦5¡&wy©~ÄÉx¦¡ywyA ©½P¡&¬9xNÀ¾¡yx{wÆzA Áx{wË r (a.u.) r(O-H) (a.u.) 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ÛܬA1x x¦¡&w©~/~q9¦{ x{wÀÂi¥xÈÄA¦«ªx5¡<©¢Rx|qÆ¡&¬AxÈ/¦«¬9wyèzA~AÅxwÜx{Þ£V¾¡y ©~ 0 ı ÔDxwyx¾ E0 ∂ + Γ̂ Ψ(t) = H(t)Ψ, ∂t c H(t) = H − V (t) ÷ /Ë(dø Ç¡y¬AxÏ¡y©¡«Á©x{¦{9jwÇÔ<¾Ï¡&©~Aj~ Γ̂ Ç¡y¬AxϾ¡ywy×Ã]©½ÜwyxjÒÃ9¾¡y ©~÷Óyx{x¢Éx ©iÀDø C~Az E = U (R(c) ) wx,Ï~A Ïx~Ax{wÅ x©½Æ¡&¬AxÅwy©9~Az'~AzÖ¦©wyxØx5Ã/¦¡&xz = U (R ) =l H 0 0 c c 0 ml ®s7 bn ²s¢²³6²Z ¬ ¡Z°±µo?«°¡p®yA\¯¯ rq Ê > ¬ \|²¸6°¡à ¬ y\t ¬ u3\µ·²¸¡¸«µ·²¸´ « ¬ª« ²³?BA;´|6y¯® ÌòÔL¹ «µ¶´²¸´ 6°sA ÄR©¡&x~£¡yj U (R) ~Az U (R) AwyxyÄRx{¦5¡&¥x{×| R ~£A¦{xÚw¦©/©wyzA~V¾¡&xN÷ù¢R©~Azëz9 ¡&~A¦x` ~¡y¬Ax ¦Ú¾yx ©½z9j¾¡y©0ϦÆÏ©x{¦Ac x«ø E $¡y¬Ax zAy©/¦j¾¡y¥xD Ïס U (∞) ½?¡y¬Ax ¦{©wyx5x5Ã/¦{ס&xzë¡«Ò¡&x c zAyy©q¦{ ¾¡&×¥x¾ËmÛܬAx<~£A¦ x{w$ÀÂÚ¥x`c ÄV¦«ª¾x¡yÜ ~Åwy©A~9z]÷ Φ øÊ~Az, xÃ/¦{¡yx{z÷ Φ øÊJ¡«¾¡yx{Ây¾¡&½Y|Ý¡&¬9x 0 c ½Ó© ©iÀÆ ~Aŧ¦{©AÄ9 x{z}xÞqA¾¡&©~A l ml ı ∂ + Γ̂ Φ = HΦ, ∂t Φ= ÀÆ¡y¬ë¡&¬AxN ~9¡& R¦{©~Az9¡&©~ Φ0 Φc , H= Φ(t = 0) = Γ̂ = 0 0 0 Γ h0 + VL00 VX0c VXc0 hc + VLcc |0i 0 ÷¿qË ø ÷¿qË dø ÀƬAxwyx |0i Ç¡&¬Axü{x{w©¾ØÄR©~£¡È¥q ¢Awy¾¡&©~VC¡&¾¡&x©½Ü¡&¬AxÅwy©9~AzÖx{x{¦¡ywy©~9 ¦x¥x{¿Ë]Ûܬ9x,zAx{¦{Ú| Ͼ¡&w×à Γ̂ ¡«ª¾x{` ~d¡y©ë¦{¦©A~d¡`©~9|}¡&¬Ax§zAx{¦{i|!w&¾¡yx©½C¦©wyxxÃ/¦{¡yx{z¡&¾¡&x Γ ¾~Az] Å~9©wyxD¡&¬9x zAx¦ÚÚ|Áw&¾¡yxD©½f¥/¢Aw&Ò¡& ©~VAx¥x Ê ~Ý¡&¬Ax1Åw©A~AzÏ¡«Ò¡&x Γ q¢Rx{¦Ú¾Ayx Γ Γ ËÛܬAxDz/|/~AϦ{Ê©½ ¡&¬9xNÀÂÚ¥xÇÄV¦«ª¾x¡&D ÜzAx5ìV~Axz!¢d|,¡&¬9xÇ~£A¦ x{w1Ô<Á ס&©~A0 ~AÊ©½Åwy©9~A0 zÙ÷ h ø~Azx5Ã/¦{ס&xz0÷ h ø 0 c ¡&¾¡&x ÷ /Ë ø hi = T + Ui (R) − Ei , i = 0, c. ÔDxwyx T 1¡&¬Ax§©ÄÉxw&¾¡y©w<©¾½Cªq~Ax¡y ¦Èx{~AxwyÅ|©½C¡&¬Ax§~q9¦{ x¿Ë Ûܬ9x ~d¡&xw&¦5¡&©~ V ii = (ψ |V |ψ ) i L i L ©½P¡&¬AxN~£A¦ x{?ÀÆס&¬ë¡&¬9xÇê ìAx{ z VLii = −(dii · EL (t)) cos(ωL t + ϕL ), dij ≡ dij (R) = Z ψi∗ (r, R)dψj∗ (r, R)dr ÷ /Ë dø CzAÉ©fxwyx~£¡Ê~ÏÅw©A~AzÏ~AzÁ¦{©wyx5x5Ã/¦{ס&xz¡&¾¡&x$z9Ax1¡y©Ç¡y¬AxÆzAª©Éx{wx{~A¦xD©¾½f¡&¬9x1Á©x{¦{9jwmz9 ÄR© x Á©Ïx~d¡& d ~Az d ËÛܬ9x R−zAx{ÄRx{~9zAx{~A¦x]©½ ¡y¬Ax{x!zAÄÉ©x}Ï©Áx{~d¡yÁϦwyA¦jܽөw¡y¬Ax ~d¡yx{w&¾¦¡&©~00ÀÆ¡y¬Ï¡&¬9xDccê)ìVx z?£¢Rx{¦{Ayx1¥q ¢Awy¾¡&©~V9¡yw&~9y¡y ©~9CwyxD¢Ax{~d¡×½ d (R) = const Ë ¡`DÀ$©w¡y¬£ÀƬ9 xÈ¡&©ë~A©¡yx§¡&¬A¾¡<ÀƬAx~ d (R) = const f¡y¬AxÞ£VzAwAÄR© x~d¡&x{wy¦ii¡y ©~ÀÆ¡&¬¡&¬Ax ê+ìAx{ z]¢Rx{¦©Áx{` ÁÄR©w¡&~d¡ÚË ½ñÙx¡&wxÚii¾¡<¡&¬AxzAÄÉ© xÇ ~d¡&xw&¦5¡& ©~!ÀÆ¡y¬ÀÂx{ª!ÄAw©¢RxÈãØw&Ú| ìVx z} ~ë¡&¬9xN½ÓwyÁxÀ$©wyª,©¾½¡y¬AxNwy©¡&¾¡&~AŧÀÂi¥xÄ9ÄAwy©ÚÃ/ Ͼ¡&©~ 1 VXc0 = VX0c ∗ = − (EX (t) · dc0 )Φ0 e−ı(Ωt+ϕX ) 2 ~Az ~d¡ywy©qzAA¦{x ¡y¬AxNzAx¡yA~A~AÅ Ω = ωX − ωc0 ÷¿qË dø ÷¿/Ë×idø ©½P¡&¬AxN¦{wyw x{w½Ówx{Þ£Ax~A¦| ω ©½P¡&¬Ax ãØwyÚ|,ìVx{z}wyx{ ¾¡&×¥x ¡y©Ý¡y¬Ax ω = E − E Ë ÛܬAx}¦{©9ÄA xzû/¦«¬AwyèzA ~AXÅx{wxÞqA¾¡&©~A÷¿qË øNÀÆ¡&¬0¡y¬Ax,~A¡yjCc0¦{©~AzA¡yc ©~÷¿0/Ë dø§wx,y©¥xz ~£AÁx{wy¦Ú|Ý~}/x{¦¾¦Ë AË×* 9ËÔ1©iÀÂx5¥x{wV¡&¬AÂ~£AÁx{wy¦ÚRy¦«¬9x{Áx` wy¾¡&¬Axwx5Ã/ÄRx{~A¥xV¢Rx{¦Ú¾Ayx ¡y© 6@?(²m´¹7²¸¡à|=vt ¬ Ê > A°wA¹ ¯µ·´|²N«²³µ·°sA\²¸0A²Iµ¶° ¬ª« ²|S¯²±ÌòÔc¹««µ·´|² y Åx5¡Ê¡y¬AxÜyÄRx{¦5¡&wAÀÂx1~Axx{zÝ¡&©Çy©×¥xÜx{Þ£VÒ¡& ©~A½Ó©wxÚ¦«¬Ý¥¾ 9xÜ©½É¡y¬AxÆÄA¬A©¡y©~ݽÓwyx{Þ£Ax~A¦| ω ©w ÄA¬A©¾¡&©qx{ x¦¡ywy©~x~Ax{wÅ|Ë`¨D y¬9©bÀÆ~~/x{¦¾Ë;/Ë /R¡&¬Ax~£AÁx{w ¦{¦«¬Ax{Áx Dx{yx{~d¡yj|yÏXÄAx{w ÀƬAx~¡&¬9xÇê w&zA ¾¡&©~,~£¡yx{wy¦¡y1©~A|ÏÀÆ¡&¬}Á© x¦{Ax{Ü~,¡&¬AxNÅw©A~Azx{x{¦¡ywy©~9 ¦`¡&¾¡&x¾Ë .x" y{z 7AB= D7 È8 E È8 q79G I v| B*9} @| ~')( ÛܬAxN ~9¡«¾~£¡&~Ax©A1ÄAwy©¢A¢A ¡Ø|Á©½Ã£Øw&Ú|}¾¢Ay©wÄ9¡&©~ P (t, Ω) = 2 =m hΦc |Vxc0 |Φ0 i = EX (t)=m hΦ0 |d0c (R)|Φc i eı(Ωt+ϕX ) ÷¿/Ë×bø Ü~9©¡&¬A~AÅÝx{yx ¡&¬A~¡y¬AxNw&¾¡yx`©½ÄÉ©Ä9Aj¾¡y ©~멾½¡y¬AxN¦{©wx5ØxÃ/¦{¡yx{z¡&¾¡&xN¢d|,¡&¬9x`ãw&Ú|ëÄAAyx ( ÷ /Ë×bø ∂ + 2Γ)ρcc (t) = P (t, Ω). ∂t ÛܬAx,z9Aw&¾¡y ©~Ù©½ÜJ¡«~AzAwyzÙãw&Ú|ßÁxÚ¾yAwx{Áx{~d¡&È §©~AÅxwÇ¡y¬V~ß¡y¬AxÄ9A xzAAwy¾¡&©~?Ë}ÛܬA Á©¡&×¥Ò¡&x{ÊA¡y©Ayx ¡y¬Ax ~£¡yx{ÅwyRÄAwy©¢A¢A ¡Ø|ÀƬA¦«¬ëÅ¥x{¡y¬Ax yÄRx{¦5¡&wAô©½;¡y¬Ax ÄAw©¢Rx yÅ~V P (Ω) = Z ∞ P (t, Ω) dt = 2Γ −∞ Z ∞ ρcc (t) dt. ÷¿qË*ø −∞ Ûܬ£A{D¦¦{©wzA ~AÅÙ¡&©0çÞ_Ë`÷ /Ë×bø5¡y¬AxãØwyÚ|Ä9wy©¢RxyÄRx{¦5¡&wA ¦{~¢Rx!¦{©ÏÄA/¡&x{z)Ay~AÅß¡&¬9x ~9¡«¾~£¡&~Ax©A,ÄAw©¢V¢9 סØ| ÷¿/Ë×bøÝ©w¡y¬AxÄR©ÄAA ¾¡&©~)©½`¡y¬Ax¦©wyx]xÃ9¦¡yx{z¡&¾¡&x¾ ρ (t) = cc Ë hΦc |Φc i ñßx1ÀÆ Ayxx<¢Rx{ ©iÀ+÷Óyxx /x¦Ë AAË £ø¡y¬V¾¡C¡&¬9xDÄA¬AyxD©½f¡y¬Ax<¦©¬Ax{wx{~d¡êwyzAjÒ¡& ©~ ϕ ©fx¦¡y ¡ywy©~9Å|Ï¡&¬Ax`ãØw&Ú|}¾¢Ay©wÄ9¡&©~}©wÜÄA¬A©¾¡&©©~A ü{¾¡&©~ÀƬAx~¡y¬AxNzAAw&Ò¡& ©~©¾½T¡&¬9x`ãw&Ú|ëLÄAAyxN y¬9©w¡yx{wN¡&¬V~ó¡&¬AxÁÄRx{wy©/zó©½$¥/¢Aw&Ò¡& ©~AÁ÷ ÄÉxw§JøË,ÛܬA ǬVÄ9ÄÉx~AÈ¢Rx{¦{AxÏ¡&¬9xÏê¶ ~9zAA¦{xz ñz9|q~VÁ ¦{zAx{ÄRx{~9zAC¡&w©~AÅ×|Ç©~ ϕ ËTÛܬAxƾ¢Ay©wÄ9¡&©~©½V¡y¬AxÄAw©¢RxÂìVx{zzA©qx{m~9©¡zAxÄÉx~Az ©~ë¡&¬9xÇÄ9¬VyxÇ©½P¡&¬Ax ãØwyÚ|ëÄAAyx ϕ xLÃ9¦x{Ä9¡< ~,¡y¬AxN¦ÚxÈ©½T¡&ÏxØzAxÄÉx~AzAx{~d¡1ÄA¬V¾yx ϕ (t) Ë X £¦v¤£ +z?U1;) 5E-;)BC%E+</:Q X !4%(' xy+<9V -:/*2A57) 8*)269V#)8 +<)2A9@%74|{#) k %7? %('3)h)j -:/*2A57) ÛܬAx1¡&¬Ax©w|Ý©½fÄAAÁÄ/ØÄ9wy©¢RxÆyÄRx{¦¡ywy©¦{©Äd|zAÁ¡yÅwyx{¾¡Êy ÁÄAìV¦{¾¡&©~ÀƬAx{~Á¡&¬AxÜãØwyi|ÝÄ9A x N¾¢Ay©w¢Éxz¢£|Á©x{¦{9 x{ ¾½Y¡&xw<¡&¬AxÝêíÄ9A xÁ¬A` x5½Y¡`¡&¬AxJ|/J¡&xËØ~¡&¬A`¦Úx¡&¬AxÝÄAw©¢Éx yÄRx{¦5¡&wA ÷ ÄÉxwÆyJø P (Ω) = hφc (Ω)|φc (Ω)i, |φc (Ω)i = Z∞ −∞ dt e−ıΩt EX (t) |φc (t)i ÷¿qË=dø ®s7 ²s¢²³6²Z ¬ ¡Z°±µo?«°¡p®yA\¯¯ rq Ê > 6°sA ¬ \|²¸6°¡à ¬ y\t ¬ u3\µ·²¸¡¸«µ·²¸´ « ¬ª« ²³?BA;´|6y¯® ÌòÔL¹ «µ¶´²¸´ ÁyÏÄ9|Ö¡&¬9x!~A©w ©½ ¡&¬9xñ¶ ~û¡&¬Ax!½Ówx{Þ£Ax~A¦|zA©Ï ~f |φ (Ω)i Ë ÔDx{wx ~£A¦{xÚwÜÀÂÚ¥xÇÄV¦«ªx5¡D¡y¬V¾¡Æx5¥©×¥x{D ~릩wyxØx5Ã/¦¡&xzÄR©¡yx{~d¡& ?x{~9x{cwyž|}yAwJ½ù¦{x¾ |φc (t)i = eıhc t ζ|φ(t)i, |φc (t)i Á¡y¬Ax ÷¿/Ë× ø 1 ζ = (dc0 ·eX ), 2 ~Azf e ¡&¬AxNÄR©j¾wy ü{¾¡&©~Ï¥x{¦5¡&©wÆ©½PÃqw&Ú|,Ä9¬A©¡&©~?Ë Ð©~dX¡&wyw|ë¡y© Φ (t) ≡ Φ (ω , t) ÷¿/Ëø$¡&¬9xÇÀÂÚ¥x§ÄA¦«ªx5¡ φ (t) ÷¿qË øÜz9©/x1~9©¡1zAx{ÄRx{~Az!©~ c ¡&¬9x,ÄA¬A©¡y©~0½ÓwyxÞ£Ax{c~A¦5| ω Ë ÌDXAx¡&©]A¦«¬û~0xyyx~d¡&j¾Üz9¥¾c~d¡«¾ÅxÀ$xëAyx φ (t) ~AJ¡&x{zÖ©½ X c Á ~ & ¡ A ¬ x £ ~ A Á { x w { ¦ É 9 j ¾ y ¡ © A ~ C © ; ½ y ¡ A ¬ x A Ä w © R ¢ < x y Å V ~ _ Æ À y ¡ ¬ & ¡ 9 ¬ x A z { x Ú | { x ë z 9 Ä y w © ¢Rx ÄAAyx¾ËÛܬAx Φc (t) ¦Ú¾ ¦{9j¾¡y ©~È©½ φ (t) mÄRx{wJ½Ó©wyÁx{zݾ ©~AÅD¡&¬Axܦ©~d¡&©Aw1÷ùvT Å9Ë/Ë×bøPÀƬ9x{wyxÜ¡y¬AxÜÅwy©A~AzÝ¡&¾¡&xÀÂÚ¥x ÄV¦«ª¾x¡ φ(t) x¥©c¥x{D½Ów©ö¡y¬AxNÏ©Ïx~£¡ÜÁAÄë¡y© t ı ÛܬAx!ãØwyÚ|ìVx zÄAw©Á©¡&x φ(t) ¾¡,¡y¬Ax~A¡&~d¡ t ¡&©0¡&¬9x x5Ã/¦¡&xz¡&¾¡&xmÀƬAx{wx¡&¬Ax~9xÀÆ|ȽөwyÁx{zÈÀÚ¥xÜÄA¦«ªx5¡ φ (t) x¥©¥x{D ~}¡&¬AxN ~d¥x{wyx§zAwyx¦¡&©~ë½Ówy© Á©Áx{~d¡ t AÄ}¡&©c 9Ë ÛܬAxÝz9|q~VÁ ¦` ~]¡&¬9xÁ~d¥x{wyxÝzAwyx{¦5¡&©~] `x{x{~ó½Ówy©Ñ¡&¬Ax ½ù¦5¡Â¡&¬A¾¡$¡y¬AxDÀÂÚ¥x`ÄV¾¦«ªx¡ φ (t) ≡ φ (0, t) $¡y¬Ax<© 9¡y ©~ c ©½P¡&¬AxÈ/¦«¬9wyèzA~AÅxw1xÞ£V¾¡&©~ c ½Ówy© Ë ζφ(t) ∂ ı φc (τ, t) = hc φc (τ, t) ∂τ τ = t ÷¿/Ë×idø ∂ φ(t) = [h0 − (d00 · EL (t)) cos(ωL t + ϕL )] φ ∂t ¡y ÜÙÀÆ¡y¬¡y¬Ax!~A¡yj¦©~AzAס&©~ ÷¿/Ë×bø φc (t, t) = core−excited state φ (t) c 0 φ( t) t ground state Tv JËb/Ë×áÛÜ Áxm©/©Ä`9yx{zÈ ~`¡&¬9x ÀÂÚ¥x§ÄA¦«ªx5¡<¦{ ¦Aj¾¡y ©~fË ^._>`acbTdfe ³ JYacH §q&P\H}i eOJ pkml\eJQcJ e isrlth [ }iIG ln_ H DÔ xwyx¾ÀÂxÆ©9¡y ~Ax$ÏÄ9 x{Áx{~d¡&¾¡&©~È©¾½R¡y¬AwyxxÜ~£AÏxwy¦Úq¡&x{¦«¬9~A Þ£AxÊ9yx{zÝ~©Awm¦©ÁÄA9¡«Ò¡& ©~V yA ¾¡&©~A{áÛܬAxDyx¦{©~Az©wyzAxw$z9ª©Éx{wyx~A¦{~AÅ÷¿Að`Ì`øÏx5¡&¬A©qz?d¡&¬9x<¨ÜzVÁJج9½Ó©w¡y¬÷Ó¨ Üæ!ø Áx¡&¬9©/z?A~Az}¡&¬AxNy¬9©w¡Æס&x{wy¾¡&×¥xNï~A¦ü{©È÷ /ïPøÜÄ9wy©ÄV¾Åd¾¡&©~ëÁx¡&¬9©/z?Ë v6u ¥ )B?U=38 ?@+<83),+ 8*4 §),+<)=3B4A= 57B'*)Q ) ÜÛ ¬Axyx¦{©~Az©wyz9x{wÝzAª©Éx{wx{~A¦ ~AÅ)÷¿9ðNÌ ø¦«¬Ax{Áx ¡&¬Ax} ÁÄA x¡ÀÂi|Ö¡y©Ù©¥x~£AÁx{w ¦{ ×| ìVw¡N©wzAx{w z9ª©Éx{wyx~d¡&j¾Tx{Þ£VÒ¡& ©~A{ËÇ¡N¦{©~9y J¡&`~!¡&¬Axz9 y¦wyx5¡& ü{¾¡&©~©½C¡y¬Ax¡&ÏxÈzAx{ÄRx{~Az9x{~d¡ /¦«¬Awè/z9 ~AÅxwÆx{Þ£V¾¡y ©~?/ÀƬAx{wxÇ¡y¬AxNzAx{w¥¾¾¡&×¥x ÀÆס&¬wyx{ÄÉx¦¡Æ¡&©Ý¡&Ïx< ÜwyxÄAj¾¦{x{z¢d|ë¡&¬Ax`ìV~9¡&x zAÉ ©fxwyx{~9¦{xNx5Ã/ÄAwx{y ©~ ¾ Å¥x{~}¢d| ÷ /Ëbø 2ı∆t H(t)|ψ(r, t)i + O[(∆t)2 ]. |ψ(r, t + ∆t)i = |ψ(r, t − ∆t)i − ~ ÔDxwyx¾ H(t) $¡y¬Ax ÔDÏ¡y©~Aj¾~¾¡ywy Ãf |ψ(r, t + ∆t)i $¡y¬AxDñ' ~Á¡&¬AxD¦{©q©wyz9 ~V¾¡yxDz9©¾ ~ ~Az ∆t 1¡&¬Ax§¡y Áx§J¡&x{ÄfËÈÛܬAx§¢A ÅÅx{¡`z9¥¾~d¡«Åx{ ©½C¡y¬AxÝAð`̶Ïx5¡&¬A©qz]wx¡y< ÁÄA ¦{¡Ø| ©iÀ¶Áx{Á©wJ|A&Åxm¾~Azß½ùJ¡§¦©ÏÄ99¡«¾¡y ©~Am¡&ÏxÝx¥x{~0ÀƬAx~ÖyÏ m¡y ÁxÝ¡&xÄAwxÏ9yx{zfË ÛܬAxÇAð`Ì Áx¡&¬9©/z,À$©wyª£Ü¥xw|,À$x{f~,ñyA ¾¡&©~ACÀÆ¡&¬}¾~ë©wyzA~VwJ|wxÚfÔ<Á ס&©~A ~ ÀƬA¦«¬ß Å~A©wyx`¡&¬AxÁ wwyx¥x{wy ¢9 xÝzAx{¦{Ú|©½Þ£V~d¡yA ¡«Ò¡&x{Ë,ÛܬAx,Að`ÌàÁx¡y¬A©/zó Nª£~A©iÀÆ~ß¡y© ¢Rx9~A¡&¢A x}ÀƬAx~~ûx5Ã/ÄR©~Ax{~d¡yjÜzAx¦ÚÚ|û Ý ~9¦{ 9zAx{z?Ë'ÛܬAxwyx5½Ó©wyx¾¡Ý¦{~)~A©¾¡Ï¢Rx}Ayxz)~ ÄAw©¢A xÏÀƬ9x{wyx`¡y¬AxNwyx{ ÒÃ9¾¡&©~ë Ü~A¦{AzAxzz9 wyx¦¡y|,~ë¡&¬AxÇÔ<¾Ï¡&©~Aj~fË ð ~AxÜ©½V¡&¬9xÜ~§ÄAw©¢Ax{ÁT ~¡y¬AxÏÄAx{Áx{~d¡«Ò¡& ©~È©½V¡y¬Ax1Að`ÌÁx¡&¬9©/zAT mwx{jÒ¡&x{zÈ¡&© ¡&¬9x ~9¡& A¦©~AzAס& ©~A{qy~A¦{x1¡&¬AxN9ðNÌ Ïx5¡&¬A©qzwx{Þ£A wx{¡y¬Ax<ª£~A©iÀÆx{zAÅx<©½;¡ØÀ©§ ~Aס&j¾V¦{©~Az9¡&©~A ~Az |ψ(r, ∆t)i ËØ~Åx{~Axw& ¡&¬9xÀÚ¥xÆÄV¾¦«ªx¡Êmª£~A©iÀÆ~¾¡T¡&¬Ax¡y Áx t = 0 d¬A©iÀÂx5¥x{w |ψ(r, 0)i À$xzA©ë~A©¡`ª£~A©iÀ+¡&¬AxÈñ ¾¡N¾~£|½Ó9¡yAwyx¡y ÁxË ÛܬAxÀÂÚ|!¡y©ëy©×¥x§¡y¬A Ä9wy©¢Ax{Ñ 1¡&©ëAxÝ Ûi|q©wÜx5Ã/ÄV~9y ©~ ÷ 9Ë ø ı∆t H(t) |ψ(r, t)i + O[(∆t)] |ψ(r, t + ∆t)i = 1 − l ~ * ®s(K° « ²¹¯°¡p²¸¹¯6¹«°®° ¬ \\3²¸¢«´ ÀƬA¦«¬~Axx{zA1©~A×|,©~9xÇ~A¡yj_¥¾ 9x |ψ(r, 0)i Ë ÛܬAx`9ðNÌý¡&x¦«¬A~A Þ£Ax÷ 9Ë×bøm¡&w¡y$½Ów© |ψ(r, 0)i ~9z |ψ(r, ∆t/2)i ¦©ÏÄ99¡&xzÏϪ£ ~AÅÇAx ¡&¬9xÛTÚ|q ©w<x5Ã/ÄV~A ©~;÷ 9Ë ø5ËNÛܬAx{~ |ψ(r, ∆t)i <¦Ú¦{A ¾¡&xzAy~AÅ}çÞ_ËC÷9Ë×bøD~¡&¬Ax~A¡&~d¡ ÀÆ¡y¬¡y¬AxÇ¡&xÄ ∆t/2 Ë$Ûܬ£A{ɾ½Y¡&xwÜ¡&¬Ax§Að`Ì ¦«¬Ax{Áxë;÷ 9ËiøÜÆAyxz!¡&©Á¦{©ÁÄA9¡yx`¡&¬Ax t = ∆t/2 ÀÂi¥x`ÄV¦«ª¾x¡Ü½Ó©wC¡&¬AxD½Ó9¡&Awx<¡&Ïx{ËÛܬAxNAð`Ì Å©wyס&¬Aí ¬A©iÀÆ~ëy¦«¬AxÒ¡& ¦{ ×|Ý ~vT ÅAË 9Ë×Ë ÛܬAxD|q¢R© v C¡&¬9xD¦©ÏÄ9 x5ÃÝ¥x{¦5¡&©w©¾½ÇÀÂÚ¥x ÄV¾¦«ªx¡Ü¾¡$¡y¬Ax<~A¡&~d¡ t ~Ïz9ª ©Éx{wyx~d¡ÄR© ~d¡& ©½m¡y¬Ax¦©/©wzA ~A¾t¡&xÇyÄV¾¦{xËDØ~!¡&¬9xx Dx{¦§ÄAwy©Åw& ¡y¬Axj¾~A¦«ª¦©~AJ¡«~d¡ ~ <9yx{z]~!A~9¡& ©½T9ËÎ?Ë b½ÓD¾~Azë¡&¬Ax`ñ |ψ(r, t)i ¬V¾Æ¡y¬AxNx{A¦ zAxÚ~ l2 Ø~A©wôxÞqA?¡&©}Ë § A¡1¢RxǪ£~A©iÀÆ~~Az? ∆t ÜÄAw©ÄÉxwy×| yx¡Æ¡y©ÏÚ¥© z~£AÏxwy¦Ú?~A¡&¢A ¡y xÂ~9zxwyw©wy vt=0 Ë /Ë Ox~Ày ;"§y*;{Jx vt=0.5∆t = vt=∆t = vt=0 − [ı∆t/(2~)] Hvt=0 l jmz {J5¦¥§y {Jx vt=0 − (ı∆t/~) Hvt=0.5∆t l ½Ó©wܪ / . . . 9Ë vt=(k+1)∆t = vt=(k−1)∆t − (2ı∆t/~) Hvt=k∆t vTJÝ˦9ËáØÏÄAx{Áx{~d¡«Ò¡& ©~,©½T¡y¬AxNyx¦{©~Az©wzAx{wÆzAÉ©fxwyx{~9¦{ ~9ÅÁÄ9wy©ÄV¾Åd¾¡&©~}Å©wyס&¬AË r l ÜÛ ¬Ax]9ðNÌ Áx¡&¬9©/z'À¾ÏAx{zû¡&©óy©×¥xçCÞ£Ë ÷ /ËÚdø~Az ÷¿/Ë×bø§½Ó©wz9ª©Éx{wyx~d¡Á© x¦{A w |qJ¡&x{ÁÇ÷ùx{xNvTÅAËD9Ë ø5ËmÛP©ÝÚ¥©zë~q9Ïxwy ¦{Rx{wyw©wy$ÀÂx`Ax{zë¡&¬9x ~d¡&xÅw&Ò¡& ©~¡yx{ÄA ∆t ©½;¡&¬9x ©wzAx{w©¾½Å~A¡yAzAx1©½Ci −4− i −5 ½ÓËÊ/A¦«¬}yÏ_¡y Áx<J¡&x{Ä9Æwyx<¢A x1¡&©ª¾x{xĦ©~A¡&~d¡¡&¬9x ~A©w4©½?¡&¬9xDñ'~Az,¦©~Ax{w¥x ¡&¬Ax<Ú¥x{wyÅxDx{~AxwyÅ|Ï©½?¡y¬Ax<J|/J¡&xö½Ó©wC¡&¬Ax1¡&ÏxÆ ~9zAx{ÄRx{~Az9x{~d¡ Ô<¾Ï¡&©~Aj~9{Ë v¤£ 839UQ 5«x&9U57'*³?@+(%(' Q ),%('*?}8 ÛܬAxN¨ÜzVÁJج9½Ó©w¡y¬x{¦©~Az/©wyzAxwܽөwy9j ÜÅ¥x~}¢£|Rá |ψ(r, t+∆t)i = |ψ(r, t)i− ÷ /Ëdø ı∆t 3H(t)|ψ(r, t)i−H(t−∆t)|ψ(r, t−∆t)i +O[(∆t)2 ]. 2~ <Ì AxÈ¡&¬AxÈyx{¦©~Az/©wyz9x{w<¡y¬Ax§¨ÜzV¾Ï¬9½Ó©wJ¡&¬!Áx¡y¬A©qzÖ÷ù¨<ÜæøD¦{~¢Rx§ÏÄAx{Áx{~d¡&xz~¡ØÀ$© zA¡&~A¦¡§ÀÚ|q{ ~A¦x}¡wx{Þ£A wx{¡&¬9xë¦{©ÁÄA9¡&¾¡&©~Ö©½ H(t − ∆t)|ψ(r, t − ∆t)i ËÖÛܬ9x}ìAwy¡ ÄR©y ¢AxÏÄ9 x{Áx{~d¡&¾¡&©~?m¡&¬A¾¡ÏÀÆ ¢Rx¦Ú x{zÖ¡y¬Ax½ù¾¡Ï¦{©ÁÄA9¡&¾¡&©~Vܾ Å©w¡y¬A ÷ùvPШ`ø5 ÄRx{w½Ó©wyÁÏ¡&¬9x¦Ú¦{A ¾¡&©~Aݪxx{ÄA~AÅ' ~û¡&¬Ax¦©ÁÄA9¡&xw,Áx{Á©w|Ö¡y¬Ax!¥¾ 9x{,©½`¡y¬AxϾ¡&w×ã ¥x¦¡y©wNA¡y ÄA ¦{¾¡&©~A H(t)|ψ(r, t) ©½C¡&¬9xÁ¦Awywx{~d¡Ç¡yx{Ä?;ÀƬA¦«¬ßwxAyxzÙ~¡&¬AxÝ~AxÃq¡ÈJ¡&xÄ?Ë ÑN«°3´3>Ðm°´A)qÊA\3²¸¢ / &>4$ os 85,$ís &>*G !us ;: /M3wR$!o "*%A$n;ß/ ß<! 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úúIAú PEGF + $Dþ4ÿBÿ LùúLþ ùrþ4"û *ý Iý 2:û9ú $ÿ %ý Rþ$? rþ % z5ú wùrþ ý >þVÿ úùrþ :' ∆r(t) 0' ý + "û@ú $@ý i43cþ Dÿ\ý $ú4J 3 $ýBÿ\ÿ²þ 9ýV3 hr(t)i þ bþ9û@ú>þ i4V3 9ùú$EGF Z3 9úIgû 9ùú 0 '& û9ú;ütý\üVþ4ÿ 9ý5ú T = 1210 .3 Gzù)ú Zùúû9ú Iúlý Cû9ú (%Zû@ú Dþ9ùúEGF ÿ Iþ4ÿBý\AIú Cþ41þ4ý W ¥ {û >þ4yÿ ' rev ' ÿ²þ 9ý]9 ù Tp9ùrþ c|zùfzùú¢EGFªþ Wzùú& 'Âû9Uþ H9ú 9û$#û@ú (%Zû@Yú 9ùYú 9ùrþVú zù$û 4? ù 9ùú¾û9\ú ' ütý\üVþ4ÿ®úIû9ý Dþ#zùrkþ b9ùúlý zù*403 9ùúl& ' ûzSþ #9ú 9û$ ∆Ω(∆t) = phΩ2 (∆t)i − hΩ(∆t)i2 ùrþ þ4ÿ *"m9ùf ú zþ 5H ú zýDH ú úIú úIúþ ∆r(t) ' Lù< ú @? úIúAIý h 4g3 9ùúPEGFWý jIú"û wþVý 9ý5ú [ý ? * ú $"9û$\zýMüZúý zúIûÕ3¼úIû@ú ú5lùý wù ÿ >þVÿBýBAú 9ùú EGFÆý Nj þ $Dþ4ÿBÿLû@úIZý }4F 3 9ùú| zúzý²þVÿúÿBÿ Lùú*EGO F @? úIúIAý Áý 5? ý zú ' ý 5| Z"û zþQ3¼"û ~9ùúL| Zý {V3ütý\úN43û9ú>þ zýzû$4ÿ Lù)ú (zû"9ý\üZúLý zúIûÕ3¼úIû@ú úþVÿBÿ T ' %Rý Iû@ú>þ @H ú zùú:û@ú>þ \zý2ûzþ 9ú[þ Azùú[ú #Iý\ú \A4J3 wùú 5ý Iþ4ÿKû@ú>þ \zý Lùú|EG} F 9? úúIAIý ' >þ * ú ýBû@ú 9ÿ$û ú #/zùú ! −& ' ûzSþ P 5 ' $û )ú 9ú \zû"$Z Dþ ·ý ý ¾þ:$û 5ý @ý 4 %ZÿKý izû"Z0 ÿ 43Qû9úIþ 9ý ' ]2^ éi2 ÅbeQl:2dÇ dh©ci2 eQdc[kiv ÇÆlH :b2eCdlÄ [ k;d Pklbf Lgb ú?f%û"ûþ$9úzýa%izùú#Dúwùrþý$V3ùþ9ú#9ú9ýzý\ü4ú*ÿBúIþ4ûf rþ5ý[lùýwù $û bý GI þ "9û$ !ì%rúÿ Lù ú Zû@ýBZý P4? 3 zùý 2ùrþ @ ú úIú úIú ý ;9ùj ú 4|U ' û9ú $rþ 4 ' ý9úIûzþ zý/4|3 zùúl DR%rúÿ Rlý9ù39ùú *4ÿBú ÿBú 4lùý w# ù ú 5ú ¾ý 5X 4"û wþClùú 3zùú!þ ý 3¼û9ú? ú \ G(R) +-, þ4"û 1þ wùú l9ùú3¼û@úI? ú 4G3 9ùúütý ûzþ 9ýþ40ÿ zû9þ 9ý zý0 ω 9ùý '§¹10 10 ')4 10 >þ 9ú zùúý zú9ý 2#4V 3 zùú 5 ÿ 9úý >"9û$ú Z< ù %ütý ZÿBþ zúzùú$!KE ·lùý wùýB 4û9ú & zùú:3-þ ""IýBÿ\ÿ²þ 9ý@ zúI"û * +-, 'MT.10 cos ((ωL + ω10 )t + 2ϕL ) ý ?zùb ú rþ 5ý Iþ4ÿ·úI? rþ 9ý + @úIúLFQþ4úûQ Lùý Cý J9û$ú»lùú 9ùúû@ú $rþ úÿBú \zû"þV ú 9ý 0;' %rúÿ ýûzkþ zùúûúIþ <zùý gIþ 9ú ÿ *@ÿ T "IýBÿ\ÿ²þ 9ý ¾lýzù3¼û@úI? ú ω − ω þ4û@F ú VúI 10 L ' 4 Y È ® r °2Ix1¶r4µ6t;btZ;±²|ªµ$´< 1 ±²bZ;±B¨±²6±B4; V³B lùý\ÿBúHzùú:3-þ""IýBÿ\ÿ²þ9ý ¿lý9ù23¼û9úI?ú Water dimer Od r H* β Excited state 400 0.4 0 0.4 0 Oa R 350 Time (fs) ω OH ωL (infrared transition) 200 150 ground state 100 ω OO ' Ground state 375.00fs R* 329.10fs 0.4 0 0.4 0 0.4 0 0.4 0 250 306.15fs 283.20fs 260.25fs 237.30fs 0.4 0 0.4 0 214.36fs 191.41fs 0.4 0 0.4 0 0.4 0 0.4 0 E(t) (arb. units) A 352.05fs 0.4 0 300 excited state þ4û@úfúIZÿ\ú\zú ω10 + ωL ¸ 168.46fs 145.51fs 122.56fs 100.00fs 5 6 7 8 5 R (O−O) (a.u.) 6 7 8 B úIû + ¦ F¢ùrþ9 úúúúIú,43zùú)@?rþ4û9ú439ùúþ>ü4úrþC4ú$ , J + LùúLþzúûý5 (4 LR' 'MM &|0 ' 0 4309ùú$4û$ þDú;&Iýzú;"zþzúLþbý\UúIû@úb9ý5ú Lùú$ÿ\ú3Z»rþúIÿ|9ùTg9ùú$úÿBú\zû@ýl%rúIÿ K ' 43m9ùú D ÿ@ú 74 ÿBý þ=rþ9ùú6ÿBýú?4û9û9ú9|p$ EL (t) = EL (t) cos(ωL t + ϕL ) ' þ ϕ = π/2 cû@ú@úzýMüZúÿ ûzþkzý!V3gzùúD D ÿ9ú τ = 50 3.-9ùúDú>þ ϕL = 0 L L '¡ |9ýzýAZû@û9ú@X)% t = 250 3.9ùú:úIþ ý9ú@ ý22ý 15 E¬« 2 IL = 10 L ' EÁúY9ýúIûcTzùú¢û@ú$rþCýzúIû9þ9ýWV39ùú !Á%rúIÿ[lýzùW9ùú»¥ ütýûzþ9ýþ4ÿ#ú 4 7 V0 3 zù ú þ 9úIm û ý 5úIû + FCþVúû¿ <lùýwù2ýþû$%2túF43cþû9þzùúIûg#* ωL ≈ ω10 ≡ ωOH 0 >þ9, ú 403 #ZÿBú ÿ\ú lýzù3Xzù#9ÿ TÍþD3-þ "¾ütý û9þ zýrþ4ÿ *9ý g¦ zù3zùúlAIú$û ' X 4ý »úúû9 ' þ a9ùúRú úû9 E4Y 3 zù/ ú ú;& ¥ ü<ý ûzkþ zýrþ4Jÿ "wþkzú/úú hzùú5¥:5 ¥ ý(wþú + KCý\ , 7 ' ' MM + &|090' EÁ,ú 9úrþ4ûzkþ zú)zùúl3-þ (¥ 7 þ #9ÿ TG¥:¥Oütý û9þ zýrþ4 ÿ *ú¾ý 39ùúg¦ Z$û 'Õ¥|ú ùúý 5úIû þ4$û >&<ý Dþ zý0lùý wùÁý |ü4þVÿBý åùúû93 ú ú >þ 9# ú zùúPû9ú ú pDþ $f4b3 zùúD¥ *úRý ?wù 7 $Dþ4ÿ\ÿBúIû zùrþ 34X3 zùú¥:[ ¥ #ú Lùl ú Zùúû9ú !Nûzþ ý²þ 9ýRý ú zû9þ @ý zý{ú ¿úIú <zùú ' 4 Âb5¥ ¿úIÿBÿ þ ;Iû@ú>þ 9úþ>üZú|rþ C4ú $Lý *X9ù0·lùý wù2úIûÕ.3 Z$û 5"Iý\ÿBÿ²kþ zý¾þ4ÿ @ 9ùú[¥:¥ 7 | ' ¥ H û @ý ÿBþ zýW@ù T5zùrþ ?zùú| Z ÿBþ zýf4m3 9ùúD4$û + ρ þ =9ùúú& Iý 9ú + ρ g0 e0 "zþzú + KCý\ , ú;&<úIû9ý\ú úLúIþ #ÿ²þ 9ýklý9ù/zù)ú ÿBú¿3¼û9úI? ú 3V|3 9ùúLÿ²þ 9úIû + ' ' MM &[0@0 %rúÿ 0flùý wùÁþ4úIþ4W û 3 ú %Azù# ú '"!KE zû9ý 9ý + T$!KE Lù# ú wùrþ4ûzþ zúf û 4b3 9ùú @ú & &[0' ¿ú>þ h$Iý\ÿBÿBþ zýlý H@ý 5ýBÿBþ4)û %;zù/ ú T$!KE # ÿ²þ 9ý ý Ezù3 ú T$_ ¥ #Zÿ\ú ÿBú + 9úúRý @ú [ý & KCý\ , zû9þ4"û p$åX Z ÿ²þ 9ý ρ þ ρ zù6 ú þ Dý Vl 3 9ùú2þ>üZúbrþ C4ú Dý ' ' w + &|090'¾ g e "9û$Zÿ }ý +ú Iú =azù* ú T!KE 9úI"û * + KQýB , + ¦ Lùú5û9úIþ $ .3 ZW û 9ùý :ý H9ùrþ fzùú & ' ' MM 0@0;' û@ú>þ Ta4{ 3 9ùúH!KE þVUú $Fzù ú 9þ zýBþ4cÿ ý(zû9ý 9ýi43 9ù ú ÿBú>þVûþ>üZúRrþ CVú Wwù & ¸< c1 w1Ix − xKb ItkHcxTItxt "9û$ZúIû)zùþi$wùþ*ýzúI4ûzþ4ÿ0wùþ4ûzþ\zúû9ý(zý:þ)zùú:|ZÿBþzý 7rþ4 9ù$lV3Qþ>üZúþC4ú\%FV3-9ùú:Zû$ þ*ú&Iý9ú6"zþzú ' + KCýB ' , 'MM + ¦ 090 9ù zùrþ zùúWý\UúIû@úúWú\Âúúbþ>üZú:rþC4ú$lý Iúi<9ùú[5%rúÿlý9ùbþHùrþ@ú ϕ = 0 þ L -þx&<ýþ4ÿ1lùú/zùúýzú 9ý2@43+9ùúL5%rúÿPþ]Zùú Iú9ùúL!þ ý·3¼û9úI?ú +-, ý π/2 '§¹10 û9úIþwùúHþ&ý + 9úúÿ\ú3Zrþúÿ-V3KCý\ , + ¦ E ú?9úIú@zùrþ 9ùúþ>üZúþ C4\ú %Wúú ' 'MM 090;' izùú? ÿBú*ùþ9ú + ú?$#9ùú@T$!KE zúû$# úIú 0zù? ú 9ý ÿ²kþ zým9 ù zùrþ 2ϕL & ' 4 > zùúþTü4úHrþCVúF<úFFwùrþZúý%,9ùrþVúlùú 2ϕ = 0 → 2π 9ûzþ4(û 6$*9ù ú wùrþ Zú L 43Jzùú:ùþ9ú ý π/2 Lùú:ùþ9úúUú\,zþ4ú lDþx&<ý üVþ4ÿ ú|lùú 2ϕ = 0 → π ϕL ' ' L ]2^? ©deQ i2dg;i2dbbiv l2I[kl ÅP ;Å2eCd2I ¿da2bgAv R d:eC#;db\uljfc[kl6dh#vg;dÄg d2dlÄ © dgw la2g; ú Aþ4ÿ`zùúE9ùú4û"}43|&'Âû9þU¤ D' û$ú y H 9ú\zû"$Zú;üZúÿZúWý¬û9úütý69ú\zýi%Íþ H O O O x O #Zû9 ú 5ÿBúa & "(zú E / ú "$ h9ù3 ú Zû@\ú ' ù ZÿBúHý ' ' C N C N ú ¨ý9ûzþ #ZÿBú ÿBþ4f û rþ 5ý 43$û %=9ûzþ @3¼úIû C H C H 43$Zÿ T&þ4ÿ *>&ý Dú + 9úh ú KCý\ , ÿ\ú>þ ý + L 0 ' 'MxwZ0 H H %izùú5.3 Z"û kþ zýa4g 3 9ù/ ú wþ $5úIû '2ý zû"$tú 9ùú Zÿ w GM NE + TS 0;' Lùú$4ÿ >&þ4ÿ * T&<ý D)ú #ZÿBú ÿ\)ú zþ4ý YÂb wùú Dý >þ4ÿ\ÿ [' úI? ýMü4þVÿBú a>& tZú þ $*}¥ + T& ' K 4(LR' , 'Mxw J L ÿ>&þ4ÿ* T&<ýDú + L 0 1 ý 5ý þ ¤¥ + 4ú $ 0 zùrkþ RX$9ú $#ý(zýRû"ZÿBú Rý þ w '2ýzû"$tú9ùú Zÿ + T$S 0;' 0 2 zùúù û$4Zú ;|J¢¥ ý ¿þ:ù û$Z4ú ;Zû¢lùýBÿ\ú¥ ý»þ:ùû"ZZú2þúI $Zû Lùú$ýýBAIþk' 1 2 ' zýA4c3 zùú 1s úÿBú zû"i4c3 zùW ú Z)û Zûþ úI $Zû T& tZú43 ~ÿBúIþm$39ùú|3.Zû"þ9ý*43 L Âb 2A0 Zû9ú 'Âý ýBAú ý²þ þ zý?(wþ9ú$lùý wa ù 9 ù < þ $û "9ý + þk q = q 43zùúýBûXzú9ý²þ4ÿ c0 û@üZú + þ 4 þ ÿ / zù ú ¥ (zû@ú %wùý * *ú 9 û ú ² ÿ þ 9úi%3zùúý zû9þ #ZÿBú ÿBþ4û q = qc ^?wt'MTÊ ' '/0 q 7 "û %izû9þ Õ3¼úIû wùrþ úIÿ + KCý\ , Lùúütý $û ý WÿBý + 4G3 9ùú @@ ú (wþ9ú + ψ (r, q) 1 ' ' MT.10' L0 þ 4J 3 zù ú f 9þ DW ú (*5ú 9û" ψ2 (r, q) 0 È 1 1 d dC12 (q) =− 2C12 (q) + , 2mH dq dq 2 1 2 d |ψ2 (r, q)i, dq +-, 'M , 0 rþ4û(zýBþ4ÿBÿ/úIÿIþ4ÿBý\AIú>9ùúWZû@ú|ùZÿ\úLþ2þVUú\%mzùúH rþ5ým43Vzùú|û$%;9ûzþÕ 3¼úIûý<zùú H12 = −H21 4û9ú\' ýýBAIúA"zþzú[lý9ùAzùú H= 7 C12 (q) = hψ1 (r, q)| þDý\ÿ%ý²þ H11 H12 H21 H22 ! , Hnn = − 1 d2 + En (q) − Ec . 2mH dq 2 +-, 'MXZ0 ® <ª³Bt±²X bV±BZV 6´±\I<±\ <³²±²X ±²fjtI ±B<±x wZxtY³\< ·³²b< ±²bPb³Bxx³B ¸<¸ Lùúbû"%p9ûzþ @3¼úIû3 rþDýIþNú2ý1ü4ú"9ýB1þkzú`@ý&' ûzþSI9ú\zû"$Zÿ\ýVúD&'ÂûzþS !þ þ 6">þ "zúû9ý %r$û ($ýDû@ú "9ý ý ,9ùrþ zù@ ú Z"û ýþ4"û Ó >F 76zúwù ýB? úý @úIÿBú $ùúIû@ú 'm& ú >þ 9úPý Hþ4 Fzù3 ú rþ Dý |ý åi þ þ4û9"û zKrûzþ C' û@úIZý aúIþ4H û zùúPú? ýBÿBý û9ý ' T¿úü4úIû ¾¿úlýBÿBY ÿ @úIA ú úIÿ T_zùþ 9ùý ý zû"ú EOùú 8ú2ú&<ÿ Zû@< ú zù; ú Z$û ýrþ4(û Ó >F 7 7 ' 9ú \zû9þbý }þ ýzýE$;zùú Ó > F 7a@ú 9ûzA þ 4Y3 zùú #ZÿBú ÿ\@ ú û@ý\ü4ú j}þ ! %rúIÿ Lùú L 4 ' 4ùúIû@ú !ø%úIÿ =wùrþ Zú zù< ú 9ý $rþ 9ý ? þ4ÿBý wþ9ý\üZúÿ úIú ]> < ú %h9ùú ! 5= zùú 4 ' 4 4 #ZÿBú ÿ\ú1þ4ý ¢ú ZùPú úIû9 %:"û *zúg9ùúl$û $Pý 9ùúúIþ ·ÿ /| D"û %zûzþ @3¼úû ¿úIÿBÿclùý wù}ý |þ | |æ ú Æùý\ZùúIûý ªú úIû@ E9ùrþ E9ùúHZÿ rþ4cÿ 5ý ý ú %;9ùý 3-þ ' '$¡ zùú !Gý ú ;ÿBúIþ4û¿þ>üZú[rþ CVú φ(t) úû@.3 Z"û *brþ CDþ 3.Z(û zù"û 9ú3¼"û Ozùú $ 4 L zù@ ú TSNúÿBÿ + KCýB , T,TQzùú:"û ú[& ' ûzSþ 2 ÿ 9? ú >þ ú;& ý zúW9ù@ ú #Zÿ\ú ÿBú|ý 6ý\UúIû9ú ' 'MT.10' û9úZý Y403 zùl ú Zû9ú 'Âý ýBAú 5X9úzýBþ4ÿ®þ #zùúû9ú *>þ <#ý $Z{û zùl ú rþ Dý >403 zùú"û % zû9þ Õ3¼úIûþ 5rþ ý\ú i;zùú[ütý $û ý WÿBý ' Lùú$û þ ýBÿ\ý 2#4-3 Zû@\ú ' ý ý\A>þ 9ý<@ý (%4J3 Âg#9û9ý zý P (Ω) = hϕc (Ω)|ϕc (Ω)i = hϕ1 (Ω)|ϕ1 (Ω)i + hϕ2 (Ω)|ϕ2 (Ω)i. û9úÿ²þ9úA%/9ùúWÂg5ù$ZýýBAIþzý;wùrþ úIÿ$¥ |ϕn (Ω)i = Z∞ 1 (1s −1 ) þ¥ dt e−ıΩt EX (t) |ϕn (t)i, 2 (1s −1 ) rû9ú9ú9ý\ü4úIÿ n = 1, 2. '»7 +-, ' MT¹Z0 úIû@ú +¼, 'M½Z0 −∞ φc (t) = φ1 (t) φ2 (t) ! = eıHt ζφ(t), ζ= ζ1 ζ2 ! , 1 ζn = (Dn0 ·eX ), 2 +-, 'MTÊ10 lùúû9ú Ω = BE − I ý9ùú2û9úÿ²þ9ý\ü4ú;ý ýúúû9 + ¦S ý 1s ^ ωX − ε 0 I1s = Ec − E0 zùú@"þ4ÿ\ÿBú"þýBþrþ9ýWZû9úýýBA>þkzý2X9úzýBþ4ÿ + ú¿úIúhzùú:¿úIÿBÿ þ þ ε ý)9ùú M wZ0 ú úIû9 4F 3 zùú*ù $<úÿBú \zû" LùE ú ÿBú>þVûDþTü4ú}rþCVú% +-, þ +¼, þ4û9úrþ%ûzþ4ÿ ' ' M½Z0 'MTÊ10 Zú úIû9þ4ÿBý\A>þ 9ý;V-3 9ùú[EGYF ý 9û$Iú bú>þ4û@ÿBýBúûý 7<ú ' w<' w<' X LùúP ZúIûzkþ %Zû +¼, , ú@úIûÕüZú 6 þ @ú IýBþ4{ÿ *5ú jzù3 ú @ý ÿBþ zý$¿ú39ú3zùú 'M 0 '34 $Iþ4ÿ²þVû.3 Z$û 43 9ùú[ ZúIû9þ %Zû +-, ' MUZ0 H12 = H21 = λq ú >þ 93 ú zùú £û9úû9ú 9ú zþzý ý [û@ú3¼úIû9þ ÿ\úH3¼$û 9ù3 ú Dúû9ý >þ4ÿ| Zý H43¿ütý\ú Ö"òÍLùú @ú 7 ' û9úû9ú 9ú zþzý6ý\UúIûD3¼$û úIþ w[ ù zùúûbý Pzùú úÿBú zû"ý ªþ>üZúå.3 \zý2lùý w[ ù úú rþ4û9þ 5ú zû@ý Iþ4ÿBÿ `Pzùa ú ÿBúIþ4A û Z$û ýrþ 9úý Pzùúª%$û "iIþ 9ú +-, , lùýBÿB ú zùú}úÿBú \zû"ý 'M 0 :Ö 1 g þTü4ú .3 9ý5þ4û@; ú wþVú ãþ R%&<ú NIÿ\ú>þ4ûRZú 5ú zû(8ý `9ùi ú 9ú N>þ @ú +-, ¦ 9ù ' MU10' þ4$û 1þ wùú 9ù ÿ bZýMüZF ú 9ùf ú zþ 5ú|û9ú $ÿ lý\c3 zùH ú %zþ4ÿ.3 zý2ý Lú&<rþ úTü4úIû þ 5ÿ\ú zú úIÿ\ú \zû$ý Wrþ 9ý )@ú ' ¸ c1 w1Ix − xKb ItkHcxTItxt 544 543 EO (1s-1)(q) EO (1s-1)(q) 1 542 2 541 Energy (eV) 540 NE2 GM1 539 qc 538 6 TS GM2 5 NE1 t=0 t = 300 fs t = 396 fs t = 430 fs t = 476 fs 4 3 E0(q) 2 1 0 1 1.5 2 2.5 q (a.u.) 3 3.5 4 JF-9úzýBþ4ÿ¾úúû9!û@üZú43Zû$ 8þ =4û9ú5ýý\AIú!(wþ9úþ4ÿ*þ6>þ4û(zú9ý²þ m5û@úIÿBþ zú`$!9ùúb$û %Izû9þ Õ3¼úI# û wùrþ úÿ þ @ùú ãÿBý ú /Dþ4$û !zùú*þ ýBþ rþ 9ý 'ã¡ |zú9ý²þ4ÿ E (q) *ý ý _43 E (q) ý m@ý $rþ zúbþ q ú ýBþ rþ 9ý$| 9úzýBþ4ÿ , þ ' 'M' LùW ' ± 0 0 ^MZ'/½ ù zùúûý i9ùúX Zý q þ Lùúý ý 9ý²þ4ÿ1þ "9ýBþ þ>üZúrþ C4\ú + "ZÿBýbÿBýú 0 $û "úIþ wE c ^fw<' MTÊ ' 'M' ú;&ýzúªýzùúùýBZùúIûú úû9 *û9úZý *úû@.3 Z"û *,rþ C*þ *3.Z"û 9ù$û zú3¼$û 9ùú $*zùú L T$S¤úÿBÿ ' , K (4 LR' 'MT. Zû"ýrþkzú ú 9ú \zû9` þ V3þ Zû9ý\úzú #ZÿBúÿ\ú + ~e , 'M , ý9ÿBSþ 4 ! −& 'Âû9Uþ N 5 'Â$û h L L ' & 3.Zû/ý\UúIû9ú5üVþ4ÿú343,zùúbütýû$ýAÿBýa"zþ Ný\UúIû9ú*úIÿ²þSp9ý5ú þ λ 0 ú ¿úIú Pzùú& 'Âû9Sþ ¤þ !( ÿ @ú ú D û r úIÿ#@ùT Ó F>7¨û$4%ÿBúR3.Zû þ ∆t = tX − tL 4 'ÍÐ ¡ æ lùýBÿB? ú zùúÿ Túû$rþ úIÿ H9ù T~zù ú 9ú 9ûz; þ 4Y3 #Zÿ\ú ÿBú lû9ýMüZú a6zùú ! %rúÿ Lùú 4 ' L^ rþ4û9þ 5ú zú$û 4J3 zùú ! ÿ @úþVû9@ ú ¡ & æ 14 E¬k« 2 æZæ.3 τ æH.3 ϕ æ ûzþ L ^fw<'/. L ^fw L ^M L^ 4 M '/X4. þ æ ú L ù ú ûzþ 9ýI4, 3 zù 2 ú & ' ûzSþ 6 ÿ 9úbý .3 Lù6 ú "zúNþ , ωL = ω10 ^ '§.Z. τX ^ ' ' rþ@ùú ÿBý úPþ4û9/ ú 9ùúDþ4"û 9ý²þ4ÿQù $Zý ýBA>kþ zýª$û rþ ý\ÿBý 9ýBú P (Ω) þ P (Ω) cû9ú @ú 9ý\üZúÿ ' 1 2 Lùúlütý û9þ zýrþ4ÿ\ÿ û9ú $ZÿMüZú Ó >F 7R$û 4%ÿBú úIý 9ú5ý3zùúlÿ\úZ3 YúIû»rþ úIÿþbIþ4ÿ ÿ²kþ zúR3.Zû þRÿ ZúIû& 'Âû9Sþ 2 ÿ 9ú τ = 15 .3 X ' þ4ÿ ÿ²kþ zý-@ù T`zùrþ Gzùú»ü<ý $û ý bÿBý û9úIþ zýH 4ùúIû@ú Y$úIû@X 9ý 9ý@43®ÿ >þVÿ ' KCýB ® <ª³Bt±²X bV±BZV 6´±\I<±\ <³²±²X ±²fjtI ±B<±x wZxtY³\< ·³²b< ±²bPb³Bxx³B 600 λqc = 0 GM2 = 0.08 eV = 0.16 eV = 0.33 eV Intensity (arb. units) = 0.65 eV GM1 500 400 ¸Ú IL=0 NE1 TS NE2 ∆t=100 fs 300 =196 fs 200 =230 fs 100 =276 fs 0 K -1 0 1 2 3 4(LR' , 'M , J 4 ! -1 0 1 2 3 − -1 0 1 2 3 Ω (eV) -1 0 1 2 3 -2 -1 0 1 2 3 4 &' ûzþS2 D' û$úW@úzû9þ#V3-9ùú L ¢*4ÿBúÿBú ' ýBAú/(wþ9ú·ý +úIú{zùú)@úzû9þ[úüZú5lýzù ¾þ !¨%rúIÿ + 9úúúIûrþúIÿ¢ý/KCý\ , , 4 ' 'M 0;' FCþV"û zýBþ4ÿ úIÿ Iþ4ÿBý\A>þ 9ýf4 3 zùmú Zû9ú¿ù Zÿ\ú Cÿ\ú>þ -%,zùúg$û9ú"9ý®«Vúùrþú5úY43+9ùúgú;lÿ 3.Z"û Dú Eû@ýBZù «krþ4$û AZû9ú 'Âý ýBAú "wþkzú + KCý\ , Lù@ ú Zû9ý\Zý 64{3 9ùý úUú |ýú;&<ÿ²þVý ú ' 'MXZ0;' ý@ÿ²Sþ ý =úzþ4ýBÿ¾ý 8FCþ4úIû Lùú Ó bF 7Áý zú 9ý 9ýBú (d /d)2 þ (d /d)2 + KCý\ , ' ' MT X &[0 4w' + − ? rþ4ÿ\ý wþkzý\ü4úIÿ ;ýMUúû9ú FúIú úú W6zù? ú ÿBý * (zû@ú 9ù λq @ ú %#zù? ú "9û$\zý\ü4ú c þ iú"9û$\zý\ü4úý zúIûÕ3¼úIû@ú ? ú V3Cÿ Iþ4ÿBý\AIú <Zû@\ú ' ù ZÿBW ú (wþ9ú>þ 9ú EOùú λ > 0 ú $' >þ /9úI) ú 9ùú? ú wùý f 43 9ù>ú o:þ4$û q ψ "zþzúlþ zùúLú ùrþ Iú 5ú Y43 9ùú Ó >F 7Hý 9ú@ý 2@43 − zùú o:û9ýB4ù (q)(wþ9ú ψ lý9ù*zùú$ý Iû9úIþ 9, ú 43 λ Lùl ú rþ4"û Dþ <û@ýBZù b"zþzúþ4û@úý 9úI"û wùrþ Zú ' + ý\c3 9ùf ú 9ýB 6V-3 9ùú[ ý Lý ·ü4úI"û 9ú ] λ < 0 ' EOùú zù? ú *4ÿBú ÿBú[ý lú;&<X 9ú a # þ "zû" !# ÿ @? ú Âb<ú£rþ |þ4û@ý @úý 69ùú Ó >F 7 4 "û 4%rÿBú Lùú 9f ú rþ )Zû9ý\Zý Dý <9ùW ú Zû@ú|ý ýBAIþ zý53¼"û _zùú[û@úIZý ;úIþ4gû zùú|úIþ ·ÿ *X ' ¿úIÿBÿ 43 zùYú wþ %5úIû 'eý 9û$"<\ú zùú Zÿ LùúÿBþ 9úûcý ú @ IÿBúIþ4ûkþ>üZúrþ C4\ú {#TüZú cý fzùú w ' Z"û "wþkzú:úÿBdÿ lùý wE ù Dþ Vú zùúý zú9ý zý\ú 4G3 9ù? ú úlÿ2.3 Z$û 5ú Ó >F 7rþ ,9ú 9ý 9ý\ü4ú %/9ùf ú úIÿBUþ <zý5H ú ú ¿úIú Ezùú$& ' ûzSþ bþ izùú !¬ ÿ @ú 4 ' !ú;3¼úIû@û9ý } $NKQýB , Hj ú >þ ¬û@ú>þ4ÿ\ýBAIh ú 9ù! ú X"zÿ\ú ú C4 3 i û "% JãE ú @úIj ú zùrþ ' ' MT¹ zùúDþTü4ú2rþ CVú Pý@ 9ý úIûzþ ÿ !úÿ >þ4ÿ\ýBAIú 7wù¨E þ úIÿ >þVÿBýBAIþ zý}û9ú "9û9ý \%9ù< ú @rþ 9ý²þ4ÿ ' û9ú $Zÿ zý=Vm3 zùý ?9úwù ýB? ú Yú Iþ 9úý @<ú Rþ4ÿ\ÿ Tz$ú;& ý zú*zù* ú IÿBúIþ4 û "((zú ú 9ùrþ4ý * 4{3 zù ú rþ4û@$û T¤ÿBú>þVûþTü4ú ú;&<ÿ\ý ý zÿý ª* þ ú9ý\û9ú ªX4ý q Lùý l5ú>þ lzùþ l9ù? ' rþ CVú #ú9úIûÕüZú #@ú IýBþ4ÿþ "zú9ý]¾þ$û ÿBú è¿úIÿ\ÿ¿bZ"û 9ù`$h9ý úIû LùA ú þ4ý Áû@ú>þ " ' 4b 3 úIÿ Iþ4ÿBý\A>þ 9ý4>3 zùúRþTü4úDþ C4\ú ýaf û >þ @úPý WT0Jrþ 5úIÿ *ý $[ûzþ 9ùúIû:| 4H û 9ú ' zû9þ4)ÿ D| 9ý zý úIú ]gzùúbütý û9þ zýrþ4ÿ¿ÿBú;üZúIÿ ν = 4 9û9ý zúþVÿ #" æ ¥% φ(t) ' 4 p ¹ ¸tä c1 w1Ix − xKb ItkHcxTItxt 2.0 (A) 2 (d±/d) λ<0 λqc = + 0.33 eV Intensity (arb. units) 1.5 1.0 (B) λqc = − 0.33 eV d+ λ>0 0.5 d− 0.0 -1.0 -0.5 0.0 ζ/2 0.5 1.0 0 -1 0 1 Ω (eV) 2 3 7 ýzú9 ýzý\ú üZúû$";zùú6ÿBý!"zû@úzù0 ζ/2 = λq/∆(q) , J Ó Fb¨ (d± /d)2 (4 LR' 'MTX &|0 ' ú ¦ >7a9ú9ûzþ3.Zû[|9ýzýMüZúRþ !úI1þ9ý\ü4úH F "9û9úzù + ! %rúÿ Ó ∆(q0 ) = 1.02 ' 0 4 0' ¥ zùúûlrþ4ûzþDú\zúIû"lþ4û9úW9ùúfzþ5ú:þlý6KCý\ , , F ' 'M ' K + FQþ4úIû¿ 4Õ0;' E ú| b"9û$ZúIû¿ÿ>þ4ÿ\ýBA>þkzý/43 φ(t) ý*zùúþzúûmý5úIû"¦29ú;&ýzý@ýÿwþú\' @ÿ <þütýûzþ9ýþ4ÿÿ\úüZúÿ ' jIÿ ú;9ùý @ú \zýIúA>þIzþSpzùþPþ "zû@úzù λq ÿBþ4û94úIû@9ùrþ¨æ ú Ïý 'M c þ Nþ4rþ4û@ú 2ú&þVZZúIû9þ zý¨.3 Z* û Zû9ú*ý ýBA>kþ zý} ú $zùúE$Dþ4ÿ\ÿ)ü4úIû@ÿ²þ4PV3 D3. 9ý M ÿ>þVÿBýBAú aý\UúIû@ú ?>& tZú 6þ $* Tú;üZúIû ú5ý Iÿ ú ý !ûþ rþVÿ 9ý þVÿ $bÿBþ4û9Zú λ '7 üVþ4ÿ ú W4úúIý ý jDý hzùú5$û $azûzþ @3¼úû:lùý w! ù >þ jûþVÿ $A úIûù %Zý ýBAIþ zý 43cüVþ4ÿBú If ú #ZÿBú ÿBþ4gû Z"û ý zþ4ÿ ¿lùúIû@f ú 9ùf ú ÿBý Pý )"9û$ZúIû + 9úIúFCþVúûl 4Õ0;' 300 × 1.76 250 × 1.76 × 1.76 2 150 |φ(t)| t - tL (fs) 200 × 3.3 100 50 0 -50 × 1.0 1 1.5 2 2.5 q (a.u.) 3 3.5 4 , JG<7 ?r þ4û@ú5þ>üZúlrþC4 ú |φ(t)|2 þPý$Gzû9þÉ@ú\%Zû( hφ(t)|q|φ(t)i + " ZÿBýRÿBýú 0 xüZúû$$ (4 LR' ' MT¹ zýDú ¥Fzùúûlrþ4ûzþDú\zúIû"lþ4û9úW9ùúfzþ5ú:þlý6KCý\ , , ' ' 'M ' K ® ;§ ·4b4:t¾;5Ixx<±²³ck¨cx ]2^î ©G ibbl ¸ê lPødhÍci2èeQldg¨éål¿ ú % @ý úûQ& 'Âû9Uþ ù %Zý ýBAIþ zý?43 È 4û9ú6ÿBú;üZúÿlýzùQzùú6ù$W3¼û9ú?ú\ ú(' 546 E (r) 9úzý²þVÿBÿ bÿBþ4û94úI û zùrþ EZû@úý ýBAIþ zýAzùû@ú9ù' 544 Zÿ EOùú Íþ 3-þ ( ù %túIÿ\ú 9û$ãý 5ú"É@ú zú ' zù/ ú *Zÿ\ú ÿBúú;&<úIû9ý\ú ú þû9ú Zý\ÿ Lù/ ú #' 542 ' 5ú %«4+ 3 zùúLù $<úÿBú \zû" p ý Gzû9þ Õ3¼úIû9ú 300 |zùQ$PzùI ú ú zú! û V3D4ûzþ>ütý 2þ ª¿úIÿBÿHþ E (r) 298 %#zùúý zú$û rþVÿ ÿBú>þVl û #zý + ütý û9þ zý 0;' R ¦ú >þ @H ú 403 9ùý 9ùúütý ûzþ 9ýrþVÿr$û 4%ÿBú "zþ4"û $ 296 %iúú !azùúR ù %åú úIû@ ¿úü4úIû L E (r) G 'P7 4 %lZ\ú QþLütý 9ý ÿB ú wùrþ ZYú 4 3 9ùú¢ü<ý ûzkþ zýrþ4ÿ4$û ' %rÿ\ú 9ùúLú úû9 4+3 zùúLù $<úÿBú \zû"(Yú 2 ûzkþ zùúûùý\Zù + > 2 Vú ¦$B úú 0G9ùúû@ú ZýBÿ t = 698.9 fs t = 700.4 fs 0;' 4 úUú \Gú 5ú cÿ²þ4û@Zúlùú ?zùú¾þTü4úÿ\ú 9ùf43 0 2 3 1.5 2.5 r (a.u.) zùúLù $<úÿBú \zû"/ú 5ú G5rþ4û9þ ÿBú»lý9ù K , Jª7wùúDú[43 4 ! −&'Âû9þUø D' 4(LR' 'M½ zù? ú @ýBAú a 4-3 zùú[ý ý 9ý²þ4ÿ®ütý û9þ zýrþ4Xÿ "wþkzú û$úH9ûzþ@ý9ý¿3.Zûmzùú « ¥ #Zÿ\úÿBú ' +¼, '¬w æ 0 ap ∼ 1 Lùý bDþ 4ú >9ù,ú DúIþ $û@ú 5ú %>4X3 zùúû9ú Zý\ÿrúUú »üZú"û #ý*ÿ ¦ÂÒ !lú ú zÿ·úû9ú Z ýBAIú ' zùþ @9ùúû@úI? ýBû@ú 5ú +¼, æ >þ ú5.3 ÿ\%rÿ\ÿBú .3 Zûû9úIþ $rþ ÿ\úù %túIÿBú 9û$ ú úû9Zý\ú ýM3¿ú 'w 0 ý Iû9úIþ 9i ú zùE ú 9ý\AI6 ú 4F 3 9ùú Z$û (wþ9ú ütý ûzþ 9ýrþVÿþ>ü4úªrþ CVú A@ý þ (zû" !( ÿ @ú 4 + @úIE ú KCýB , þ ÍFCþVúû ú *úIZý ¤lýzù`zùú ù 9ý Iþ4ÿý \%û@6 ú VF 3 zùú !(ý Iú ' 'MT½ 4 0;' È 4 ú ùrþ ú 5ú f4Y 3 zùúû9ú ZýBÿcúUú \ Lùú& 'Âû9Sþ ù %Zý ý\A>þ 9ý6"û WþVý ÿ Aú>þVF û zùúÿBúZ3 ' ú KCýB , ¿úü4úIû Xzùú + Èc0 þ izùúû9ý\Zù + ! 0 $$û ý 5X Zý %4{3 9ùú 4 !Wý ú EGJF 9úI ' ' M½<'L7 ù %túIÿBú 9û$3·ý CYzùú Iÿ\úIý®þ #Dþ Vú zù)ú 9ûzþ @ý 9ý/' ü4úI"û 9ýIþ4ÿ + KCý\ , Zý Dÿ tý ' ' MTÊZ0 zùþ mzùú|ÿBú;Z3 Lþ2û@ýBZù g9ý úrþ )4V3 zùú Ó >F 7;9ú \zû$ ú;&<úIû@ýBú IW ú 9ùý\Z3 %gF ú $zùú|û@ú ZýBÿ úUú\ (i) (mB /mA ) cos2 θ Erec +-, 'wM0 ≈ (ωX − I1s ) . ∆ω = 1 − F0 /Fc M (1 − F0 /Fc ) úû9ú m ý 3zù6 ú þ $34F 3 9ùE ú Zû9ú 'Âý ýBAú ¤þ$ ý `zù6 ú ý²kþ %5ý ;#Zÿ\ú ÿBú l ¦ M = 7 & & A 7wù}3 þ ú.3 Z"û kþ zý6VG3 zùú Ó bF 7i9ú \zû$ Àý 9úIú hIÿ\ú>þ4û@ÿ ý KCýB , lú mA + m B ' ' 'MS'g& >þ A9úIú3¼$û S¢? 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Ó σ(ω, ω1 ) = 7Aû$"l@ú\zý ýIÿú,">þ"zúû9ý3$39ýZÿBú\þAzû@ýBÿ\ú%+þ4ÿV"zþzú X " |Ff (S)|2 Φ(ω1 − ω + ωf (S),0 , γ) + X |Ffm (T )|2 Φ(ω1 − ω + ωf (T ),0 , γ) # + Xt'/.10 lùúû9ú ω ý,9ùú3¼û@úI?úE43{zû9þ@ýzýb3¼û$ÏZû"6%%+þ4ÿc9ýZÿBú\FZû f (q),0 = Ef (q) − E0 zû@ýBÿ\ú-"zþzú Lùúm@úzû9þ4ÿ<3. 9ýV3®ýýúCûzþý²þ9ýýþ$$ Dú%HúLþ þ"9ý²þlý9ù ' L zùú[ùrþ4ÿ\3klý zù þLùþ4ÿ\3JDþx&<ý + E ú?rþ4ÿ]$ γ Lùú?">þ"zúû9ý%/zùú|%rþ4]ÿ 9û9ýBÿBú 7 7 0 ' "zþzúRý|þ4ÿ\ÿT¿úaú$;zùú/7¥ý9úIû9þ9ýªýEzùú '2@ùúIÿ\ÿ úIú]c9ùúþ>üZúR.3 \zýa43 w 'W4 zùú?Zû9ú:ú& ý9ú6"wþkzú:ýlþ/$úû9|9ýzý6V3G@ýZÿ\úlþ69û9ý\ÿBú\)"zþzú f m=1,0,−1 + X<' , 0 ΦΛ = aΦ(S) + bΦ(T ) ý9úIû9þ9ý5ý3zùúlZû@úF@ùúIÿ\ÿ úl$@zùú|þ5ýM&$û9ú4309ùúzû9ý\ÿBú\>"wþkzú ú>þ9úHV3]zùúW7¥ '¡ ú5ý"9ý2ýlþ4ÿ\ÿT¿ú þ4ÿ"/%/zùúW9û9ýBÿBú%+rþ4ÿV(wþ9ú ' Lùúlú;&<úIû9ý Dú ¢þ¾úIûÕ.3 Z$û 5ú Rlý9ùR1þ ¢ùþ 9ú#ZÿBúÿ\ú¢þ Plýzù5þ$%&<úDþ4ÿBú χ = 0 ú ¿úIú þ 69ùú|þTü4úü4ú \%Z û 4J3 zùú[ú Dý $9úbù $ k Azùý )>þ @ú ¿úùþTü4? ú $/9ú:ý e ' 4 1Y S¢? + zùúþ4"û 9ý²þ40ÿ zû@ý 9ý Lþ>üZúIû9þ4Zú ETüZúl û #ZÿBú ÿBþ4gû 4û9ýBú wþ9ý ' X<'§.10 |Ff h (S)|2 12 12 1 2XX = 9 Λ=1 Λ =1 (ω1 − ωΛ,f (S) + ıΓ)(ω1 − ωΛ1 ,f (S) − ıΓ) 1 (Λ)∗ × {D(Λ) (S) · D(Λ1 )∗ (S)}{Df (Λ1 ) (S) · Df (S)} 3 (1 − 3 cos2 χ) 20 (Λ ) (Λ)∗ × {D(Λ) (S) · Df (S)}{D(Λ1 )∗ (S) · Df 1 (S)} + (Λ ) (Λ)∗ +{D(Λ) (S) · Df 1 (S)}{D(Λ1 )∗ (S) · Df (S)} i 2 (Λ) (Λ1 ) (Λ)∗ (Λ1 )∗ (S)}{Df (S) · Df (S)} − {D (S) · D 3 + X<'YX40 Ú ³{Ð II,III Iå1tI;±B |Ffm (T )|2 = h ¸ 12 12 1 2XX 9 Λ=1 Λ =1 (ω1 − ωΛ,f (T ) + ıΓ)(ω1 − ωΛ1 ,f (T ) − ıΓ) 1 (Λ)∗ (Λ ) × {D(Λ) (S) · D(Λ1 )∗ (S)}{Df m (T ) · Df m1 (T )} 3 (1 − 3 cos2 χ) 20 (Λ)∗ (Λ ) × {D(Λ) (S) · Df m (T )}{D(Λ1 )∗ (S) · Df m1 (T )} + (Λ ) 7 ·ü4úzýrþ4ÿJDÿ\ú;&6xÉ"1þkzý43-zùú@ûzþPüZú%4û #Dú%FV3»þ$4û9 9ý + X<'/¹Z0 (Λ)∗ +{D(Λ) (S) · Df m1 (T )}{D(Λ1 )∗ (S) · Df m (T )} i 2 (Λ)∗ (Λ ) − {D(Λ) (S) · D(Λ1 )∗ (S)}{Df m (T ) · Df m1 (T )} 3 úû9úú*ùrþ>üZúªý9û$ úN9ùúE"> þ4ÿBþ4ûPû$\;43Fzùú65ÿ\ú;&¨üZú$Zû$ J a {a · b} = P a þ a k bk ' lýzù LùúýB| ZÿBú b (Λ) zûzþ9ýzý$þVû9ú3' + D(Λ) (S)0 þ }ú5ý$@ý + D(Λ) f (S) Df m (T ) 0 9ú@9ýl9ùú¿úýBZú1üZú\%Z"û G43zùúY%wþ4ÿ þDý\ÿ$ý²þ + þ?zùú>ú'ÂúÿBú\zû"?9ûzþ 9ý9ý 7 X<'M0 ý\XZÿ\úH#Dú% + 3.4û)*Zû@W ú úzþ4ýBÿ m9úúFQþ4úIûL 4"4$4Õ0;' ËP^zî ©gIÑ II,III k=x,y,z Ä5eC Å fd:eC2k;d KCý\ @ ù Tgzùl ú zùú Zû@ú zý>þ4ÿ ÿ L− & ' ûzSþ Dþ $4û9 9ý*9ú9û$R>þVÿÿ²þ9ú<Dþ·ý@@ú t' X<'w?& zùú:3.4ÿBÿTlý3.Zû"ÿ²þ3.Zûmzùú:ù %1þ$Zû@ zýiIû"$9ú9ý 2 X X (Λ) + Xt'Y½Z0 σ(ω) = |D (S)|2 ∆(ω − ωΛ0 , Γ), ωΛ0 = EΛ − E0 , 3 ν Λ=1 3.Zûzùú@zûzþ9ýzý 2p → 6σ, 7σ, 8σ, 9σ þ 2p → 3π, 4π, 1δ Lùúû9úIÿBþzýMütý"9ýf>þVÿÿ²þ9ý ' rþ@úpzùú53.û?5X ú/7tlS Ó zúw ù ýB? úÙ29/9 ùT øû9þzùúû@@ý5ýBÿBþ4û|ý9ú@ýzý\ú[3.ZûWzùú 12 %rû""@ý'eÿBú + 6σ ZþRýQZýMüZú ∆ ≈ 1.6 ú + KQýB xlùýwùRýGÿ@úb%lzùúú&<úû(' SO 0 'VX<'w|L0 ý 5ú zþ4V ÿ 9ý '24$û ýwþ4ÿ0@ÿBý$9ý ∆ ≈ 1.75 ú ¦g9ùa 7 KNþû@úIÿBþzýMü<ý"9ýW>þVÿÿ²þ9ý SO ' 9 ù «zùúH% / ú "9û$\%û@# ú V3¿úIþwùj5|úf43Y9ùú/9ý'2 ÿ\úWIþ9újEzùú/#ZÿBúÿBþ4û Z"û ý zþ4) ÿ 9ÿ\ý "zý! VH 3 zù ú Zû@E ú @ùúIÿ\dÿ þ D¿úIÿBÿþ *}9ùú ÿ Ný 9úIûzþ zý}\ú Âúú Pzùú 9ùúÿBÿkþ 6zùú[üVþ Iþ Wª¥ E @ ú @úI? ú þVý ÿ *9û9ýBÿBú LþE ÿ\ú L%+? ú "9û$\%û@ ú 4J3 zùú LII,III ' þ rþ û@ú @ú zýMüZúÿ E úÿ²þ úI0 ÿ 9ù@ ú Zû9ú 'Âú & ý 9úE(wþ9ú |2p−1ν 1 i þ ν rlý9ù 6σ3/2 6σ1/2 ' j j L ù [ ú þ û9þ *4V 3 g û 9ý ÿBþ zý43 þ "((zú*lý ¾üZú"û *9ú 9ý zý\ü4F ú % j = 1/2, 3/2 ' σ π zùú 769rþ ú DZú |þ*ú "zúIû$þ4Zû@úIú 5ú lý9ùú;&<úIû@ý 5ú S®ú@9ùýMZ3 zúhýM.3 Z$û 5ÿ *zùú π & g ' 9ú \zû" + rþ úÿ¦ $ÿ T¿úIû|ú úIû@ ªû@úIZý Eªæ ú øþ H@ù Tåý aKCý\ ûzþ 9ùúIû 0 '/ÊZX 'X<'w*&'X& 9ý Dý\ÿ²þ4, û ý9ÿBþ Iú Dú H4b3 9ùú π ÿBú;üZúIÿ [û9úÿ²þ 9ý\üZ@ ú %Azùú σ ""(zú þf@úIû@ü4ú ú>þ4û@ÿBýBúû$ý >þVÿ ÿ²þ 9ýg43 Ó 7A4J 3 zùú¥ 7i#Zÿ\ú ÿBú $Ù B & ' c·Ú ÛV<·Î 0.6 0.5 0.4 0.3 0.2 0.1 0 A 0.5 B 0.4 3π3/2 1δ3/2 4π3/2 8σ 1/2 6σ3/2 ×10 6σ1/2 7σ 3/2 7σ1/2 8σ3/2 1δ1/2 4π1/2 3π1/2 9σ3/2 9σ1/2 1δ3/2 4π3/2 0.3 1δ1/2 4π1/2 0.2 3π1/2 0.1 0 200 3π3/2 202 204 206 ω (eV) XAS cross section (arb. units) XAS cross section (arb. units) t II,III 0.006 Û· · I}xZwx>±BX <bϳ 7σ3/2 C 0.005 0.004 6σ3/2 0.003 0.002 ∆SO = 1.62 eV 6σ1/2 0.001 0 208 210 212 202 203 ω (eV) 204 205 JCLùúZû9ú\zýIþ4ÿ ÿ L &' ûzþS5þ"Zû9zý<9ú\zû"V3 ÿ þ@ùú2ÿ\ýú¿ýDþúIÿ II,III 7| '¢¡ 7V) 3 zùú π þ δ $ ""9ú# + úý\zú6ýÁrþúIÿ¦ @ùý\3Z9úpÁæ ú Ó & & 0 '/ÊZX ' ú C F þ úIÿ @ùT#9ùúbû9ú$ÿ%343û9úIÿBþzýMütý"9ý*9ýÿ²þ9ý 43,zùú2%rû""3zùû@úIú Γ = 0.0465 ' ú>þlý Azùú ÿ L 7i@ú9û$®43 ÿ Ó II,III & 7| ' K 4(LR'<X<'¬w @ ùT _zùú & )zùúFúZýý¿úWDzú;9ùú$%rû""@ý'eÿBúû9úÿ²þ9ú;%Zû@ú[ú;&Iýwþ9ý#%zùú ¥ +D ª þ4û"4úãýNKCýB þ 6σ þ 6σ wþ·ý ý% þ;ÿp9ùúÿ\ý\3¼ú9ý5ú 6σ '¾X<'w 3/2 1/2 0 û"1 þúý Γ T¿úü4úIûlzùúh9ûzþ@ý9ý$pzùú 6σ ª¥ú;&<úû9ý\úúi"9û$Iû$1þúý '©7 ú> þ9 ú>zùú»%rû$(GZ û9úÂ' ú;& ýzú(wþ9úýcý$" ý²þ9ý\üZú + CK ýB E ú>9ýÿ²þ9úWzùúgý$$IýBþzýMüZú 'TX<'/.Z0;' ÙÂÒ Lùúû@ú$ÿ%Y43|zùú ' >þVÿÿ²þ9ý¿þ4û@úH9ùT ýAKQýB <9ùúW"zú'2rþ@ùúbÿ\ýú KCýB û@úIû" IúþVÿBÿ 'X<'w@& ' ' Xt'¬w& ú;&<úIû@ý5úwþ4ÿ3¼ú>þ$û9úý;zùú Ó 7;9ú\zû" + KCý\ , þV3Zzúûmzùú:þXTüZf ú 5ú zýú6@ùý\3Zm43 & 'X<' 0 zùú π þ δ û@ú$rþú T¿úü4úIûJzùúPý9ú@ý2}û9þzý2ý|3-þ4û|3¼û"úIý búû@3¼ú # ú $ 'D7 '5¡ zùýk¿úDû@ú$Iþ4ÿBú9ùú/zûzþ9ýzý!ýB|ZÿBú/#5ú%f}%$9ýif û 9ùú 4û9ú 9ýIþ4ÿ¢$û V%rÿB/ ú %izùú ú;&<úIû@ý5ú + KCýB , Lùúlù ZÿBú?zùúZû9ú\zýIþ4ÿ Ó 769ú\zû9þ4ÿC"û 4%rÿ\ú |KCýB ý ,9ùýMZ3 zú 'X<' 0;' & 'X<¬' w3& %5þRÿT¿úIûlúúIû9bû9úI4ýA ú MZ'wZX ' û"1þúýf43XzùúL%rû"">9ý'eÿ\úY@ýfzùúlþ>üZúrþCVúg9úwùýB?ú ËP^@_ Ò <el Ì k:fkeQiv ¦g9ù*ú&<úû9ý5úzþ4ÿkþ6zùúZû9ú\zýIþ4ÿk! 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MX MU1½40;' á , F 6Zû9ý *þ Tú úIû ®F¢ù !ú;ü \ú $ ¹ â ' 4w' ' '®È |' æ|ê MS MT. + MU4ÊZ¹10;' á L ´a²´z¶«· à Ǻd[r¿dg;deð ibfklbfk;~ÄD<el a 2eCdÅ} f}l¿feQdkdP MON MPN QSRUTWVYX6Z4[6\&];^ _`N aCTWVcb\dZFe'^ MPN Qf\dgihjVkR'lmXonUpdqr^XonUsutvNwxe=ZF\&n y'z${}|i~Fr >}~ > oo$3 − $x1; H±B 1 1Vi$DcP±²kx V³)¶®·V´d·pÏd?<4 Úê PHYSICAL REVIEW A 70, 062504 (2004) Two-color phase-sensitive x-ray pump-probe spectroscopy 1 F. F. Guimarães,1,2 V. Kimberg,1 F. Gel’mukhanov,1 and H. Ågren1 Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden 2 Departamento de Química, Universidade Federal de Minas Gerais, Avenida Antonio Carlos, 6627, CEP-31270-901, Belo Horizonte, Minais Gerais, Brazil (Received 9 September 2004; published 14 December 2004) X-ray pump-probe spectroscopy is studied theoretically. It is shown that two-color—optical +x-ray—excitation with constant phase of the pump radiation exhibits strong interference between the one- and two-photon excitation channels. This effect is found to be large for both long and short pump pulses, while the interference vanishes for x-ray pulses longer than one cycle of the pump field. It is predicted that the spectral shape of x-ray absorption is strongly influenced by the absolute phase of the pump light. A strong sensitivity of the x-ray absorption and/or photoionization profile to the phase and detuning of the pump field is predicted, as well as to the duration of the x-ray pulse. Our simulations display oscillations of x-ray absorption as a function of the delay time. This effect allows the synchronization of the x-ray pulse relative to the “comb” of the pump radiation. The interference pattern copies the temporal and space distribution of the pump field. We pay special attention to the role of molecular orientation for the interference effect. DOI: 10.1103/PhysRevA.70.062504 PACS number(s): 33.20.Rm, 33.80.Eh, 33.70.Ca, 34.50.Gb I. INTRODUCTION Ultrafast x-ray spectroscopy inherently exhibits a spectral evolution on the time scale of molecular motion and can therefore map the dynamics of a material system. Despite considerable progress in time-resolved pump-probe optical spectroscopy, except for a few real pump-probe x-ray measurements with the femtosecond resolution [1,2] actual x-ray measurements are performed with continuum wave light sources or with long pulses. The generation of femtosecond and subfemtosecond intense x-ray pulses constitutes one of the major problems of current experimental studies [1,3–6]. An important problem of the pump-probe measurements in the x-ray region is the synchronization and phase matching of the pump and probe pulses [7,8]. The experimental investigation of the relaxation dynamics of core excited krypton [1] serve here as a striking application of time-resolved pump-probe spectroscopy in the x-ray region. We refer to the recent extensive and profound review of ultrafast x-ray absorption spectroscopy in Ref. [2], where one can find new ideas, as well as an analysis of current and future experiments. Different pump-probe schemes in resonant x-ray Raman scattering have been also studied in Ref. [9]. It is shown that many-electron effects strongly influence the laser assisted x-ray absorption by atoms [10,11] and molecules [12]. The works alluded to above exemplify that quantum interference, one of the basic features of quantum mechanics, is the key for the many effects and applications of x-ray spectroscopy that have been reported recently. The main aim of this paper is to conduct a theoretical study of two-color pump-probe x-ray spectroscopy where the absolute phase of the pump field is important. It is necessary to note that our idea has nothing to do with the role of the phase in few-cycle experiments [13–15]. The significance of the phase in photoionization was recognized in studies of anisotropy in photoionization caused by interference of one- and two-photon channels [16,17]. This asymmetry was induced by the laser 1050-2947/2004/70(6)/062504(9)/$22.50 field consisting of two frequencies, one with an ultraviolet field capable of photoionizing the atom through the absorption of single photon, and the second a visible field for which the absorption of two photons is required for photoionization. The frequency of the first phase-shifted field was precisely twice that of the second. Due to this circumstance, any beating phenomenon was absent in the experiment [17] and the signal was sensitive to the relative phase between these two fields. In spite of the efforts made for different applications of ultrashort x-ray pulses, the investigation of the role of the phase in x-ray pump-probe spectroscopy is limited. We suggest here an experiment which allows measurement of the effect of the absolute phase of the pump laser on x-ray absorption. It is important to note that the effect discussed here does not depend on the phase of the x-ray pulse, except the case of few-cycle x-ray pulses. Due to this circumstance, we avoid the problem of phase matching of the pump laser and x-ray fields. We study a three-level system exposed to lowfrequency pump and x-ray probe radiations and focus on the role of the interference of one- and two-photon excitation channels which takes place when the phase of the pump field is constant. Contrary to optical experiments [16,17], the interference studied here is influenced strongly by the beating with the frequency of the pump field and the absolute phase of the pump field. We show that such an interference results in strong oscillations of the x-ray absorption and/or ionization as a function of the delay time. This effect opens a way to synchronize the x-ray pulse relative to the “comb” of optical pump radiation. We show that the interference effect discussed here is quenched when the duration of the x-ray pulse is longer than the inverse frequency of pump radiation, while the interference is large in a wide range of the pump durations. We shall begin by briefly reviewing the theory of twocolor pump-probe experiments in Sec. II. The results of numerical simulations are analyzed in Sec. III. In Sec. III A, the role of phase and detuning of the pump field on an x-ray 062504-1 ©2004 The American Physical Society Úñ °cµ <³B<K1; 4;±\;±\´< x}b Itc;kxTIt;t PHYSICAL REVIEW A 70, 062504 (2004) GUIMARÃES et al. where I is the ionization potential of a core electron. Due to this, we only here consider the case when the core-excited 2 is discrete [Fig. 1(A)]. First of all, we would like to stress the role of the phase of the pump field in the process studied. The upper level 2 is populated though the one-photon (OP) and two-photon (TP) absorption channels 0 → 2, FIG. 1. Two-color excitation. Pump laser + x-ray field. Interference of the 0-1-2 and 0-2 channels: vL , G. (A) X-ray transition to discrete state 2. (B) Core-ionization with the energy of photoelectron «. spectrum is studied. Three qualitatively different contributions into the total phase shift of one- and two-photon excitation channels are discussed in Sec. III B. We predict in this section that the interference results in an oscillatory pattern of the x-ray absorption. The role of molecular orientation on the discussed interference effect is studied in Sec. IV. An analysis of the experimental possibilities for observation and some applications are given in Sec. V. The main results are summarized in Sec. VI. II. TWO-COLOR EXCITATION OF THREE-LEVEL MOLECULES We consider three-level molecules which interact with the low-frequency pump field sLd and high-frequency probe x-ray radiation sXd (see Fig. 1): «astd = eaEastdcossvat − ka · R + wad, a = L,X. s1d Here, R is the radius vector between the light source and a molecule. The pump and probe fields are characterized by the polarization vectors ea, envelopes Eastd, frequencies va, and phases wa. These fields mix the molecular states 0, 1, and 2 (Fig. 1) Cstd = a0stdC0e−ıe0t + a1stdC1e−ıe1t + a2stdC2e−ıe2t , s2d where Cn and en are the wave function and energy of the nth state. Atomic units are used unless otherwise stated. The ground, 0, and the first excited, 1, states are coupled via the pump radiation. The probe x-ray field couples the coreexcited state, 2, with both lower states, 0 and 1, because the spacing sv10 = e1 − e0d between levels 0 and 1 is comparable with the lifetime broadening G2 of the core-excited state 2 and the inverse width of the x-ray pulse, 1 / t. Two qualitatively different situations depicted in Figs. 1(A) and 1(B) are possible. The first one [Fig. 1(A)] corresponds to the excitation of the core electron to the unoccupied molecular orbital (excitation to discrete level 2 lying below the core-ionization threshold). In the second case, the frequency of the x-ray photon exceeds the core-ionization threshold and the core electron is excited into a continuum with the kinetic energy « [see Fig. 1(B)]. The treatment of the corresponding photoelectron spectrum is the same as the core excitation in a discrete state. The only change is the replacement e2 → I + «, 0 → 1 → 2. s3d The two-photon path differs from the one-photon channel only by the phase of the pump field e−ıswL+vLtd = e−ıswL+vLDtde−ıvLt8 . s4d Here, we ignore for a while the phase kL · R [see Eq. (1)], the role of which will be discussed in Sec. IV C. We observe that the phase wX of the probe radiation does not influence the interference, which depends only on the absolute phase of the pump field wL. The reason for this is that the same x-ray photon participates in the channels 0 → 2 and 0 → 1 → 2 (see also Fig. 1). This situation differs qualitatively from the scheme studied in Ref. [16], where the interference of OP and TP channels depends on the relative of pump and probe radiations. When the phase wL is fixed two channels (3) are indistinguishable; and, they interfere with each other. However, the factor exps−ıvLt8d results in fast oscillations which can wash out such an interference due to integration over local time, t8 = t − Dt, in the frame of the x-ray pulse. One can detect such a beating if the temporal resolution of the x-ray measurements is better than 1 / v10. This, however, is not the case of common x-ray experiments [2]. Thus, a key point for an observation of the interference is to use a temporal width t of the x-ray pulse that is smaller or comparable with the inverse frequency of the pump field, vL tvL & 1 s5d The interference between one- and two-photon channels (3) depends, according to Eq. (4), on the effective phase, wL + vLDt, which will be discussed in more detail in Sec. III. From the Schrödinger equation in the interaction picture, we get—using the rotating-wave approximation—the equations of motion for the probability amplitudes a1std and a2std: S S D ] + G1 a1std = − ıV10std, ]t D ] + G2 a2std = − ıV21stda1std − ıV20std. ]t s6d Here V10std = exphıfsv10 − vLdt − wLgjseL · d10dELstd / 2 and V2nstd = exphıfsv2n − vXdt − wXgjseX · d2ndEXstd / 2; v2n = e2 − en. dnm is the transition dipole moment between states n and m, and Gn is the decay rate of the nth excited state. To make the analysis more transparent, we assume the pump field to be too weak to considerably depopulate the ground state: a0std < 1. We also neglect small depopulations of the levels 0 and 1 by the weak x-ray radiation. The role of the high intensity of the pump field is discussed qualitatively in Sec. IV A and in more detail in Ref. [18], where the nuclear degrees of freedom are described strictly by using a direct numerical 062504-2 H±B 1 1Vi$DcP±²kx V³)¶®·V´d·pÏd?<4 Úö PHYSICAL REVIEW A 70, 062504 (2004) TWO-COLOR PHASE-SENSITIVE X-RAY PUMP-PROBE… g3 = G2 − ıV, where erfcszd is the error function [19] and V = vX − v20 is the detuning of the x-ray frequency from the resonant frequency of the direct channel 0 → 2. We assume a rather poor temporal resolution of the x-ray detector. This means that the x-ray signal must be integrated over a long detection time. In other words, the cross section of x-ray absorption is given by the following integral of the population of the core-excited state ua2stdu2 FIG. 2. Temporal shape of pump and probe (x-ray) fields. solution of the Schrödinger equation for the ground electronic state of the NO molecule. We proceed by considering the experimental situation in which the continuous-wave (cw) pump field is switched on at the instant t = 0. Then, with the delay Dt, the molecules are exposed by the probe pulse with the width t. We assume rectangular and Gaussian shapes for the probe and pump fields, respectively (see Fig. 2), ELstd = ELustd, S EXstd = EX exp − D st − Dtd2 , 2t2 where ustd is the step function. The modeling of the pump field by a step function makes the analysis much more transparent while it reproduces all essential effects of the studied probe. The Gaussian shape of the pump field is investigated elsewhere [18]. The probability amplitude of the first excited state reads S D s8d Here, VL = vL − v10 is the detuning of the frequency of the pump field from the resonant one. The substitution of this expression in Eq. (6) results in the following equation for the probability amplitude of the coreexcited state 2: a2std = − a Î p fAOPstd + ATPstdg 2 s9d with amplitudes of one-photon AOPstd and two-photon ATPstd channels AOPstd = t L3stde−G2t , ATPstd = − ssV,VLd = E ` −` ssV,VL ;tddt, ssV,VL ;td = ua2stdu2 , s12d where the inessential prefactor is omitted. We would like to note that the integrand in Eq. (12) can also be measured if ultrafast x-ray detectors are available [2]. III. RESULTS AND DISCUSSION s7d e−G1t − e−ıVLt ı ustd. a1std = − seL · d10dELe−ıwL 2 G1 − ıVL s11d In our calculations of the cross section (12), the following parameters are used (except Fig. 9): G1 = 10−4 eV, G2 = 0.1 eV, a1 = v10 = 0.4 eV, 1 Î2 < 0.7. s13d We assume in the simulations that the molecules are oriented and the polarization vectors and transition dipole moments are parallel to each other: eL i eX i d10 i d20 i d21. This our assumption is discussed in detail in Sec. IV devoted to the role of the molecular orientation. A. Role of phase and detuning of pump field on the x-ray spectrum As was shown above, the phase dependence arises only due to interference of the one-photon and two-photon channels (except the case of a few-cycle pump pulse [14,15] which is not considered here). To give insight into the physics, it is instructive to start from the simplified expression * sintsV,VL,td = pa2 RefAOPstdATP stdg H ıtustda1exps− ıwLd hL1std − L1s0d − fL2std 1 − ıVL/G1 − L2s0dgje−G2t . , Re eısc−wLd ÎVL2 + G21 fe −sG1+ıv10dt − e−ıvLtg J s10d s14d The dimensionless parameter ua1u is the population of the level 1 for t = ` and VL = 0. Here for the interference of the OP and TP channels with the amplitudes (10). This equation is an approximation valid for long x-ray pulses and G2 @ G1, uVLu. One can see that the interference term depends on the absolute phase of the pump field wL and on the phase 2 2 S LkstdegkDt+sgktd /2erfc ı a = e−ıwXEXseX · d20d, 2 D t + g kt 2 − t Î2t a1 = g1 = G2 − G1 − ısV + v10d, D , k = 1,2,3, seX · d21d EL seL · d10d , 2G1 seX · d20d g2 = G2 − ısV + vLd, S D c = arctan VL G1 s15d which arises when the pump frequency is tuned from the 0 → 1 resonance. The phase c originates in the resonant de062504-3 är °cµ <³B<K1; 4;±\;±\´< x}b Itc;kxTIt;t PHYSICAL REVIEW A 70, 062504 (2004) GUIMARÃES et al. FIG. 5. The interference Ds / s = fssV , VLdwL=0 − ssV , VLdwL=pg / ssV , VLdwL=0 versus the durations of the x-ray pulse, t. The quenching of the interference (phase dependence) for large durations of the x-ray pulse, t. T = 2p / v10 < 10.3 fs. VL = 0. V = −0.2 eV. Dt = 110 ps. The other parameters are collected in Eq. (13). FIG. 3. Temporal shape of the integrand ssV , VL ; td (12). VL = 0, V = −0.2 eV. Dt = 110 ps. t = 3.29 fs and 23.29 fs for upper and lower panels, respectively. The other parameters are collected in Eq. (13). nominator, G1 − ıVL, in the TP amplitude ATPstd (10). Equation (14) shows that the interference contribution to the integrand ssV , VL ; td (12) oscillates with the frequency vL < v10 = 0.4 eV which corresponds to the period of oscillations T = 2p / v10 < 10.3 fs. The integrand ssV , VL ; td experiences fast oscillations only when the duration of the x-ray pulse is longer than the period of oscillations T (see lower panel in Fig. 3). To obtain the cross section of x-ray absorption (12), we have to integrate ssV , VL ; td over time. Due to this integration, the interference term is suppressed strongly for a long pulse t @ T because of the fast oscillations. When the pulse duration is comparable or shorter than T, the interference term has no time to perform oscillations and the shape of the integrand ssV , VL ; td is a single peak (see the upper panel in Fig. 3). In this case, the interference of the TP and OP channels is big which causes the spectral shape of the x-ray absorption to strongly depend on the absolute phase of the pump radiation, wL (see Fig. 4). When the pulse duration FIG. 4. The dependence of the x-ray photoabsorption profile (12) on the delay time, Dt, and the phase of the pump field, wL. t = 3.29 fs. t / T < 0.3. VL = 0. The other parameters are collected in Eq. (13). increases, compared with the period of oscillations, the interference between the TP and OP channels is suppressed and the shape of the x-ray absorption profile ceases to depend on the phase wL (Fig. 5). The interference of the TP and OP channels leads to a periodical dependence (14) of the x-ray absorption cross section on the phase wL (Fig. 6). The amplitude of modulations displayed in Fig. 6 takes maximum value when the frequency of the x-ray photon is tuned between the resonant frequencies of the transitions 1 → 2 and 0 → 2: V = −v10 / 2 = −0.2 eV as the interference is largest in that case. As was mentioned above, the effective phase of the interference term is influenced also by the phase c (15) which changes from −p to +p when the frequency of the pump radiation crosses the resonance VL = 0. The strong dependence of c on VL (Fig. 7) near the 0 → 1 resonance suVL / G1u & 1d yields a strong variation of the x-ray absorption profile when VL crosses zero (Fig. 8). Indeed, on the way from VL = 0 to VL = 2G1, the phase c increases from 0 to <p. B. Total phase shift, x-ray absorption versus pump frequency and delay time Equation (4) indicates that the delay of the probe and pump pulses Dt leads to an extra phase vLDt. One can also FIG. 6. Intensity of the x-ray absorption (12) versus the phase of the pump radiation for different detuning of the x-ray field, V. t = 3.29 fs. Dt = 110 ps. VL = 0. The other parameters are collected in Eq. (13). 062504-4 H±B 1 1Vi$DcP±²kx V³)¶®·V´d·pÏd?<4 PHYSICAL REVIEW A 70, 062504 (2004) TWO-COLOR PHASE-SENSITIVE X-RAY PUMP-PROBE… FIG. 7. Phases c (15) versus detuning of the pump field VL. The parameters are collected in Eq. (13). see this directly from the interference term (14) * std , exps−ıcdexps−ıvLt8d, which, when written in AOPstdATP terms of the local time in the frame of the x-ray pulse, shows directly that x-ray absorption is affected by the total phase: w = w L − c + v LD t . ä s16d Here we assumed for simplicity that G1Dt @ 1. The total phase w consists of three terms. We discussed already the role of two of them, wL and c. Now, we would like to point out the importance of the third contribution, vLDt, which is the product of the delay time and the frequency of the pump radiation. Due to the phase vLDt, the x-ray absorption cross section experiences nondamping oscillations with the period 2p / vL when Dt changes (Fig. 9). The character of these oscillations is sensitive to the detuning of the pump field VL, which leads to a beating of the oscillations with the period 2p / VL. The origin of these beatings can be traced to the interference of two terms at the right-hand side of Eq. (14) which oscillate with different frequencies v10 and vL = v10 + VL. These beatings damp for large delay times because the first term at the right-hand side of Eq. (14) decreases when G1Dt . 1. Let us note that we use a smaller value of v10 in Fig. 9 compared with the previous figures only to make both oscillation and beating visible on the same plot. Figure 9 shows the suppression of the amplitude of oscillations when Dt is smaller than the lifetime of the state 1. This is seen directly from Eq. (14) FIG. 8. X-ray spectra (12) for different detuning of the pump radiation, VL. t = 3.29 fs. Dt = 110 ps. wL = p / 2. The other parameters are collected in Eq. (13). FIG. 9. Intensity of the x-ray photoabsorption (12) versus delay time, Dt, for different detuning VL. V = −0.2 eV. The vertical arrow shows the lifetime, 1 / G1 = 6.6 ps, of state 1. Fast oscillations with the period 2p / vL < 1.03 ps. Beating with the period 2p / VL = 10.3 ps. sVL = 0.4 meVd and 2p / VL = 20.7 ps sVL = 0.2 meVd. The other parameters are collected in Eq. (13) except v10 = 4 meV. * AOPsDtdATP sDtd , 1 − e−sG1−ıVLdDt = H sG1 − ıVLdDt , 1, DtÎG21 + VL2 ! 1 DtÎG21 + VL2 @ 1. J s17d The physical reason for such a suppression is that the pump field has no time to populate the level 1 during Dt if the delay time is shorter than the lifetime of state 1. It is worthwhile to note that this statement is valid only when the Rabi frequency ELd10 / 2 is smaller or comparable with G1. IV. ROLE OF MOLECULAR ORIENTATION The results of our numerical simulations are valid only for oriented molecules. However, both oriented and disordered samples are investigated in experiments. This motivates us to explore the role of molecular orientation in more detail. A. Randomly oriented molecules Two different experimental schemes are possible. The first one is shown in Fig. 10(A) with one-photon excitation of a 1s electron to an unoccupied molecular orbital (MO). This one-photon channel interferes with two-photon channel, where the pump field excites an electron from an occupied FIG. 10. Two possible schemes of interfering one- and twophoton channels (see text). 062504-5 ä< °cµ <³B<K1; 4;±\;±\´< x}b Itc;kxTIt;t PHYSICAL REVIEW A 70, 062504 (2004) GUIMARÃES et al. MO to an unoccupied MO and where the hole in the occupied MO is filled due to core-electron excitation. We focus our attention only on the second scheme shown in Fig. 10(B). Let us start from the case of a rather weak pump field which is able to populate only the first-vibrational level 1 of the ground electronic state of the molecule u0d [Fig. 10(B)]. The transition dipole moment of the IR transition, 0 → 1, is s0d d10 , where the superscript designates the ground electronic state. The x-ray photon core excites the 1s electron to the particular unoccupied molecular orbital ck with the electronic transition dipole moment dk,1s. Let us denote this final electronic state as ufd ; u1s−1ckd. To be specific, we consider here the following interfering channels [compare with Eq. (3)] x-ray IR u0,0l → uf,0l, x-ray u0,0l→ u0,1l → uf,0l. s18d The bra vector ui , nl indicates the nth vibrational level of the electronic state i = f,0. Equation (18) says that we have to replace the transition matrix elements used in Eq. (11) acs0d cording to d10 → d10 , d20 → dk,1skf , 0 u 0 , 0l, d21 → dk,1skf , 0 u 0 , 1l, where kf , n8 u 0 , nl is the Franck-Condon amplitude. Now we are at the stage to extract the factors seX · dk,1sd s0d and seL · d10 d from the absorption amplitude (9) and which depend on the molecular orientation s0d a2std = seX · dk,1sdfA + B e−ıwLseL · d10 dg, s19d where all unessential parameters, such as Franck-Condon amplitudes, are collected in the coefficients A and B. The first and the second terms at the right-hand side of Eq. (19) describe the one- and two-photon channels, respectively. One can immediately see that the interference of OP and TP channels, s0d sintsV,VL,td = 2 ResA*B e−ıwLdueX · dk,1su2seL · d10 d, s20d is the odd function of the transition dipole moments. Quite often the x-ray measurements are performed in gas phase, where molecules are randomly oriented. According to Eq. (20), the averaging over molecular orientations results in the zero-interference term ksintsV,VL,tdl = 0. s21d It is necessary to note that the orientational quenching of the interference is absent in scheme studied in Refs. [16,17]. The main reason for such a distinction is that in our case the electronic dipole moments of core excitation are the same for both channels. Thus, the x-ray absorption is not influenced by the phase of the pump field when the sample is disordered and the pump radiation is rather weak. The situation changes drastically for higher intensities of the pump radiation. In this case, the IR field also populates higher vibrational levels: n = 2 , 3 , …. The amplitude of photoabsorption (19) now becomes s0d s0d a2std = seX · dk,1sdfA + B1e−ıwLseL · d10 d + B2e−2ıwLseL · d21 d s0d d + ¯ g. 3seL · d10 s22d Here, Bn is the probability amplitude of population of the nth vibrational level due to n-photon absorption. One can see that the averaging over orientations only quenches the terms with odd nsB1 , B3 , …d at the right-hand side of Eq. (22). Now, the interference term averaged over orientations is not equal to zero ksintsV,VL,tdl s0d = 2kResA*B2e−2ıwLdueX · dk,1su2seL · d21dseL · d10 d+ ¯l Þ0 s23d and it includes all even terms sB2 , B4 , …d caused by interference of the one-photon channels with sn + 1d-photon channels sn = 2 , 4 , 6 , …d. B. Fixed-in-space molecules We have shown above that the interference pattern can be observed for gas phase molecules if the pump radiation is quite strong. This interference effect is suppressed for disordered systems when the intensity of pump field is not sufficient to populate higher vibrational levels. Therefore, when the pump intensity is rather weak, the discussed interference effect can be observed only for oriented molecules. There are two ways to do this. The first one is based on the measurement of the photoabsorption or detection of photoelectrons from molecules adsorbed on surfaces. Quite often the adsorbed molecules have preferential orientation. It is the case, for example, for surface adsorbed NO and CO molecules [20–23]. Another method is based on the measurement of x-ray absorption in the ion-yield mode [24–27]. The core hole state has large energy and, due to this, such a state has many Auger decay channels into different final ionic states. Many of these states are dissociative. The detection of the dissociating ion selects a certain orientation of the molecules and nowadays this method constitutes a powerful technique to study x-ray spectra of fixed-in-space molecules [24–27]. A strong pump field can also dissociate a molecule and this can mask the ion yield caused by x-ray absorption. However, quite often the dissociation of the gas-phase molecule by an IR field from the ground electronic state yields only neutral fragments. Due to this fact, the dissociation of the molecule by pump field does not influence the ion current induced by the x-ray probe field. C. Dephasing caused by spatial dependence of the phase According to Eq. (1) we have to take into account the phase, kLz, which arises due to the propagation of the pump field w L → w L − k Lz s24d where z is the coordinate of a molecule along the direction of pump field propagation. To be specific, we assume perpendicular intersection of x-ray and pump fields. Because of the 062504-6 H±B 1 1Vi$DcP±²kx V³)¶®·V´d·pÏd?<4 ä<¸ PHYSICAL REVIEW A 70, 062504 (2004) TWO-COLOR PHASE-SENSITIVE X-RAY PUMP-PROBE… phase, kLz, the transverse distribution of the x-ray beam after the region of intersection experiences a modulation with the wavelength of the pump radiation, lL, due to the modulation of photoabsorption. Such a space “comb” can be detected if the spatial of the x-ray detector is better than lL. If the x-ray detector does not have such a resolution, the interference term, sint ~ expf−ikLsz0 + Dzdg, is suppressed by the factor lL , , s25d when , . lL. Here, , is the size of the crossing of the light beams. This is due to averaging over deviations, Dz = z − z0, from the center of the region of intersection, z0. To avoid this dephasing one can use an x-ray beam with a diameter, ,, smaller than the wavelength of the pump field, lL. Experiment [28] shows that the x-ray beam can be focused in a spot with the size , , 10−3 cm. The discussed interference is large when lL * ,. In principle, the x-ray pulse can be focused in much smaller spot size which is comparable with the x-ray wave length: , , lX cm. Eq. (24) indicates that the interference pattern depends periodically on the center of the illuminated region z0. FIG. 11. Mapping of the pump field making use the train of x-ray probe pulses. See text for details. can lead to dephasing and can suppress interference between one- and two-photon channels. To understand the role of such a dephasing, it is useful to write down the following estimate of interference term for the case of strict resonance vLc = v10: sint , e−ıv10Dt−ıwL 3 V. POSSIBILITY OF EXPERIMENTAL OBSERVATION A. Long pump pulses or continuous-wave pump laser H 2 G−1 1 expf− sDt/tLd /2g, G1tL @ 1 tLÎp/2, G1tL ! 1. J s26d Let us consider the possibility of experimental observation of the dependence of x-ray absorption on the absolute phase of pump radiation discussed here, wL, and applications of that effect for the purpose of synchronization of the optical and x-ray pulses. It is worth noting that one can control only a physical relevant component, 0 , dwL , 2p, of the total phase wL = dwL + 2pN, where N is an integer. First of all, we will discuss the case of long pump pulses or cw pump laser fields. Apparently, measurements with different delay times Dt have to be performed within the coherence time of the cw pump laser: Dt , tcoh. This time can be very long for cw lasers, for example, the He-Ne laser has a coherence time about tcoh , 10−3 s. The x-ray free-electron laser (FEL) can drive a train of short pulses with time separation around dt , 100 ns [7]. This allows one to get tcoh / dt , 104 measurements of x-ray absorption for different delay times during the time of the coherence, tcoh. Such a snapshot of the pump field gives the phase of pump laser, wL, relative to the peak of x-ray pulse. That means synchronization of x-ray pulses with respect to the comb of the pump field. A scheme of a possible experiment is shown in Fig. 11. This mapping of the pump field is possible when the ratio vLdt / 2p is not an integer. B. Short pump pulses The effects discussed above take place also for the short pump pulse. The case of short pulses is considered in more details elsewhere [18]. Here, we would like to discuss this problem only qualitatively. When the pump pulse has finite duration, tL, it experiences a spectral broadening, DvL , 1 / tL, around the carrier frequency, vLc. This broadening One can see, that when the pump pulse is longer than the lifetime of the first-excited state stL @ G−1 1 d, the interference term is large only for small delay times, Dt , tL. The dephasing leads to suppression of interference for large delay times, Dt . tL [see Eq. (26)]. It is surprising that the dephasing does not play any role when the lifetime G−1 1 is bigger than the pulse duration tL. In this case, interference is large [see Eq. (26)], which is related to the memory of the molecular system about the phase of pump radiation. Such a phase memory allows to get snapshots of the pump comb considerably later of the instant of the interaction between pump field and molecule. The molecule remembers about the pump pulse during large time, G−1 1 . For example, the lifetime of vibrational states G−1 1 in diatomic molecules are in the range of milliseconds to a few seconds. Due to this circumstance, one can get a snapshot of a short pump comb making use of sequences of short x-ray pulses with delay times much bigger than both the period of oscillations of the pump field, 2p / vL, and the duration of the pump pulse. This effect allows high precision synchronization (with time smaller than the period of the pump field) of the x-ray and pump pulses even if these pulses are strongly delayed relative to each other. Let us note that the interference effect discussed here can be observed also for electronic transitions in the visible region (instead of x-ray transitions) for sufficient short probe pulses s,1 – 10 fsd. The fact that the molecule can keep information about the phase of the pump field, Est , xd, for a long while makes the use of the interference of one- and two-photon channels very promising in design of optical memory elements. 062504-7 ä °cµ <³B<K1; 4;±\;±\´< x}b Itc;kxTIt;t PHYSICAL REVIEW A 70, 062504 (2004) GUIMARÃES et al. VI. SUMMARY In summary, we have shown that two-color pump-probe absorption and/or ionization is strongly modified by phase sensitive interference between one- and two-photon excitation channels. The x-ray absorption profile displays a maximum interference pattern when the duration of the probe x-ray pulse is shorter than the inverse frequency of the pump field. There is here an important distinction from standard few-cycle optical experiments: In our case, the duration of both pump and probe pulses are longer than the inverse frequencies of the corresponding fields. We have found that the spectral shape of x-ray absorption is strongly influenced by the absolute phase of the pump light and that it does not depend on the phase of the x-ray radiation. Due to the interference effect, the x-ray absorption also experiences oscillations as a function of the delay time. This allows synchronization of the x-ray pulses relative to the comb of the optical pump field. The interference pattern also allows us to map the comb of pump radiation. We showed the importance of the concept of the total phase shift between the interfering pathways. The total phase consists of three contributions: the absolute phase of the pump field, the phase lag between interfering pathways due to the delay time, and the phase shift which arises when the frequency of the pump field crosses the optical resonance. [1] M. Drescher, M. Hentschel, R. Klenberger, M. Ulberacker, V. Yakovlev, A. Scrinzl, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, Nature (London) 419, 803 (2002). [2] C. Bressler and M. Chergui, Chem. Rev. (Washington, D.C.) 104, 1781 (2004). [3] R. W. Schoenlein, W. P. Leemans, A. H. Chin, P. Volfbeyn, T. E. Glover, P. Balling, M. Zolotorev, K. J. Kim, S. Chattopadhyay, and C. V. Shank, Science 274, 236 (1996). [4] R. W. Schoenlein, S. Chattopadhyay, H. H. W. Chong, T. E. Glover, P. A. Heimann, W. P. Leemans, C. V. Shank, A. Zholents, and M. Zolotorev, Appl. Phys. B: Lasers Opt. 71, 1 (2000). [5] E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, Opt. Commun. 212, 377 (2002). [6] A. A. Zholents and W. M. Fawley, Phys. Rev. Lett. 92, 224801 (2004). [7] J. R. Schneider, Nucl. Instrum. Methods Phys. Res. A 398, 41 (1997). [8] T. Tschentscher, Chem. Phys. 299, 221 (2004). [9] F. Gel’mukhanov, P. Cronstrand, and H. Ågren, Phys. Rev. A 61, 022503 (2000). [10] D. Cubaynes, S. Diehl, L. Journel, B. Rouvellou, J.-M. Bizau, S. Al Moussalami, F. J. Wuilleumier, N. Berrah, L. Vo Ky, P. Faucher, A. Hibbert, C. Blancard, E. Kennedy, T. J. Morgan, J. Bozek, and A. S. Schlachter, Phys. Rev. Lett. 77, 2194 (1996). [11] H. L. Zhou, S. T. Manson, L. Vo Ky, P. Faucher, F. BelyDubau, A. Hibbert, S. Diehl, D. Cubaynes, J.-M. Bizau, L. Journel, and F. W. Wuilleumier, Phys. Rev. A 59, 462 (1999). [12] T. Privalov, F. Gel’mukhanov, and H. Ågren, J. Electron Spectrosc. Relat. Phenom. 129, 43 (2003). We have shown that when the pump field is rather weak, its phase does not influence x-ray absorption if the molecules are randomly oriented. In this case, the discussed interference effect can be observed only for oriented molecules. One can do this by making use of surface adsorbed molecules or detection of x-ray absorption in the ion yield mode. The interference pattern for randomly oriented molecules starts to grow when the intensity of pump radiation increases and the pump field is able to populate even vibrational levels. Thus, a third way to detect the phase sensitivity of x-ray absorption is to use rather high intensities of the pump field. We have mainly focused in this paper on IR pump fields which excite vibrational levels of the ground electronic states. Similar effects take place for microwave pump fields which change the populations of the rotational levels. We propose that the discussed interference effect can be useful in creation of optical memory elements. ACKNOWLEDGMENTS We would like to thank Professor A. Cesar for stimulating discussions. This work was supported by the Swedish Research Council (VR). F.F.G. acknowledges financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (Brazil). [13] T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000). [14] A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdorfer, T. W. Hansch, and F. Krausz, Phys. Rev. Lett. 92, 073902 (2004). [15] S. Chelkowski, A. D. Bandrauk, and A. Apolonski, Opt. Lett. 29, 1557 (2004). [16] N. B. Baranova and B. Y. Zel’dovich, J. Opt. Soc. Am. B 8, 27 (1991). [17] Y.-Y. Yin, C. Chen, D. S. Eliott, and A. V. Smith, Phys. Rev. Lett. 69, 2353 (1992). [18] F. F. Guimarães et al. (unpublished). [19] Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (National Bureau of Standards, Washington, D.C. 1996). [20] A. Nilsson, O. Björneholm, E. O. F. Zdansky, H. Tillborg, N. Mårtensson, J. N. Andersen, and R. Nyholm, Chem. Phys. Lett. 197, 12 (1992). [21] O. Björneholm, A. Nilsson, E. O. F. Zdansky, A. Sandell, B. Hernnäs, H. Tillborg, J. N. Andersen, and N. Mårtensson, Phys. Rev. B 46, 10353 (1992). [22] A. Föhlisch, J. Hasselström, P. Bennich, N. Wassdahl, O. Karis, A. Nilsson, L. Triguero, M. Nyberg, and L. G. M. Pettersson, Phys. Rev. B 61, 16229 (2000). [23] B. Gilbert, F. Huang, H. Zhang, G. A. Waychunas, and J. F. Banfield, Science 305, 651 (2004). [24] A. Yagishita, E. Shigemasa, and N. Kosugi, Phys. Rev. Lett. 72, 3961 (1994). [25] R. Guillemin, E. Shigemasa, K. Le Guen, D. Ceolin, C. Miron, N. Leqlercl, K. Ueda, P. Morin, and M. Simon, Rev. Sci. Instrum. 71, 4387 (2000). 062504-8 H±B 1 1Vi$DcP±²kx V³)¶®·V´d·pÏd?<4 äÚ PHYSICAL REVIEW A 70, 062504 (2004) TWO-COLOR PHASE-SENSITIVE X-RAY PUMP-PROBE… [26] K. Ueda, J. Phys. B 36, R1 (2003). [27] F. Gel’mukhanov and I. Minkov, Phys. Rev. A 70, 032507 (2004). [28] J. Wang, A. K. Sood, P. V. Satyam, Y. Feng, X.-Z. Wu, Z. Cai, W. Yun, and S. K. Sinha, Phys. Rev. Lett. 80, 1110 (1998). 062504-9 ´a²´«¶z· L}L 2he[eQl Ä5eC a 2eCdÅ} − feCdkd5 dhÍci2u¡Ù d;g la2g; k}l¾ MON/MON/QfRUT¢V£XoZ4[6\&];^_`NaCT¢Vkb\dZEe'^_`N/¤CNM\dgWT¦¥d§¢]E¨ ]4T¢V©pª^MON«Qf\dg¬hVkR'lmXnUp;qr^S®kN«¤¯\&]X6Z&^XonUs tvN wxe=ZF\&n y'z${}|i~FK E}~° >Y±6²³;}4´dE3 − }4´;>3µ¯1$ 6 R±² 1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d 5QV1HK ·Ï2d@ tIV ä<ö PHYSICAL REVIEW A 72, 012714 2005 Infrared–x-ray pump-probe spectroscopy of the NO molecule F. F. Guimarães,1,2 V. Kimberg,1 V. C. Felicíssimo,1,2 F. Gel’mukhanov,1 A. Cesar,2 and H. Ågren1 1 Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden Departamento de Química, Universidade Federal de Minas Gerais, Avenida Antonio Carlos, 6627, CEP-31270-901, Belo Horizonte, Minas Gerais, Brazil Received 24 November 2004; published 22 July 2005 2 Two color infrared–x-ray pump-probe spectroscopy of the NO molecule is studied theoretically and numerically in order to obtain a deeper insight of the underlying physics and of the potential of this suggested technology. From the theoretical investigation a number of conclusions could be drawn: It is found that the phase of the infrared field strongly influences the trajectory of the nuclear wave packet, and hence, the x-ray spectrum. The trajectory experiences fast oscillations with the vibrational frequency with a modulation due to the anharmonicity of the potential. The dependences of the x-ray spectra on the delay time, the duration, and the shape of the pulses are studied in detail. It is shown that the x-ray spectrum keep memory about the infrared phase after the pump field left the system. This memory effect is sensitive to the time of switching-off the pump field and the Rabi frequency. The phase effect takes maximum value when the duration of the x-ray pulse is one-fourth of the infrared field period, and can be enhanced by a proper control of the duration and intensity of the pump pulse. The manifestation of the phase is different for oriented and disordered molecules and depends strongly on the intensity of the pump radiation. DOI: 10.1103/PhysRevA.72.012714 PACS numbers: 33.80.Eh, 33.70.Ca I. INTRODUCTION Coherent superposition of states is a key concept in contemporary quantum physics. Various superpositions of molecular states or wave packets can be created in strong fields generated by infrared or optical lasers. One can expect that coherent properties of the light are transferred to the molecule, which means that the evolution of the wave packet must be sensitive to the phase of the pump radiation. This leads to the idea to probe the phase sensitive dynamics of the molecular wave packet by means of x-ray radiation. According to our knowledge the influence of the phase of the pump radiation on the wave packet trajectory and, hence, on x-ray probe signals has not been studied for a real system yet: In our companion paper we presented infrared–x-ray pump probe theory and applied this theory to study the proton transfer in the water dimer which constitutes an important prototype system containing a hydrogen bond 1. The viability to perform core excitation in regions of the potential energy surface that are unavailable by standard x-ray absorption was there demonstrated 1, something that indicates the power of this kind of new experimental tool. In the present paper we address infrared ir–x-ray pump-probe spectroscopy of the NO molecule, as this simple one-dimensional system allows great numerical detail with strict solutions of the quantum equations and scrutiny of the underlying physics. This in turn allows one to pinpoint the optimal experimental conditions for ir–x-ray pump-probe spectroscopy and for measurements. We point out the striking applications of time-resolved and phase sensitive ir–x-ray pump-probe experiments with a few-cycle 750-nm laser field and with a duration of the x-ray pulse of the order of femtoseconds 2. Studies of phase sensitive dynamics of the molecular wave packet WP require quite short x-ray pulses, with a time lapse of 1 – 100 fs. As 1050-2947/2005/721/01271412/$23.00 reviewed in 1 several kinds of such sources are available already today 3–6, in fact, ultrashort x-ray pulses as short as 250 as 2 have recently been reported. This indicates that x-ray pump-probe spectroscopy is able to explore the nuclear dynamics even by the use of current light sources. The main aim of our paper is to investigate time and phase resolved x-ray absorption of nitrogen monoxide driven by a strong ir pulse. The pump radiation affects the x-ray absorption in two qualitatively different ways. The first one arises from an incoherent population of higher vibrational levels “heating” effect which takes place when the pump radiation is incoherent. Our simulations show very different x-ray spectra of the NO molecule in different initial vibrational states. A coherent pump pulse with permanent phase changes the scenario drastically as a coherent superposition of vibrational states then is created. This second way makes the dynamics of the nuclear wave packet phase sensitive. Due to the phase dependence of the trajectory of the wave packet, the x-ray spectrum becomes different for different phases and delay times. Because of the long lifetime of the vibrational levels the WP keeps the memory about the ir phase, which results in a memory effect in x-ray absorption. We would like to note that the phase effect which we study has nothing to do with the role of the phase in few-cycle experiments 7–9. The paper is organized as follows. We start in Sec. II describing the physical picture of phase dependence of x-ray absorption. The wave packet formalism used in the simulations is described in Sec. III. The details of computations are elucidated in Sec. IV. We analyze the results of calculations of x-ray absorption of NO driven by a strong ir field in Sec. V. The experimental conditions for observation of the discussed phase effect are analyzed in Sec. VI. Our findings are summarized in Sec. VII. 012714-1 ©2005 The American Physical Society ê w-I1V − IªK <kHkVT<<ª<¾;¸·<øb³Bxx³B r PHYSICAL REVIEW A 72, 012714 2005 GUIMARÃES et al. larizations e and wave vectors k, envelopes, Et, frequencies, , and phases, = t, which in general are all time-dependent. The key idea behind x-ray pump-probe spectroscopy is straightforward. The strong ir field mixes coherently the vibrational levels in the ground electronic state. Because of this, the nuclear wave packet starts to move in the potential well. The proper choice of delay time for the x-ray pulse allows one to obtain snapshots of x-ray spectra at different site positions of the nuclear wave packet see Fig. 1. As one can see from Fig. 1 such a technique allows one to map the shape of the excited state potential. We show that the x-ray spectra are very sensitive to the phase of ir field. The interaction between the probe x-ray radiation and the molecule is influenced by the strong ir field which changes the populations of the vibrational levels of the ground electronic state. We ignore the ir mixing of vibrational levels in the final core-excited electronic state which is small due to the large lifetime broadening. Different channels of the x-ray absorption are possible, for example, direct one-photon absorption or + 1 absorption channel where j , is the electron-vibrational state. The second channel corresponds to x-ray absorption from the vibrational level of the ground electronic state excited due to absorption of ir photons: 0 → 1 → 2 → ¯ . The photoabsorption amplitude is the sum of one- and + 1-photon contributions, which can approximately be written as FIG. 1. Main spectral features of the O 1s x-ray absorption spectra 2 → 2− of the NO molecule in the strong ir field. L = 0. L = 10 = 0.241 eV. eL d. = t − 0 see Eq. 23. L = 100 fs. The wave packets and corresponding x-ray spectra are marked by labels A, B, and C. A: IL = 0, X = 15 fs. B: IL = 2.3 1012 W / cm2, X = 3 fs, t = 1035 fs. C: IL = 2.3 1012 W / cm2, X = 3 fs, t = 1025 fs. D: IL = 2.3 1012 W / cm2, X = 15 fs, t 2L + X. II. PHYSICAL PICTURE OF THE PHASE SENSITIVITY OF X-RAY ABSORPTION SPECTRUM To give insight into the physics of the studied effect it is instructive to start from a simplified picture. The phase effect 10 can be explained in two qualitatively different ways. We present both interpretations because they shed light on different aspects of the phase problem. A. Interference of one- and many-photon absorption channels We consider molecules which interact with the ir pump field L and high-frequency probe x-ray radiation field X see Fig. 1: Et = Etcost − k · R + , = L,X. Dc0 · EX + d10 · EL ¯ d,−1 · EL X − ci,00 − i =1 e−iLt+L+kL·R L − i0 Dc0 · EX . X − ci,0 − i 2 Here L = L − 10 is the detuning of the ir field relative to the frequency of the resonance transition between first and lowest vibrational levels; and 0 are lifetime broadenings of the core excited state and vibrational levels of the ground state, respectively; ci,0 is the resonant energy of the electron-vibrational transition 0 , → c , i; d,−1 is the transition dipole moment between adjusted vibrational levels, while Dc0 is the dipole moment of the electronic transition between ground and core-excited electronic states. The expression for the absorption amplitude 2 represents a rather rough approximation. However, the simplicity of this expression makes it easy to understand the role of the phase of the x-ray absorption. The x-ray absorption probability, which is the square of the amplitude 2, contains the interference term Pintt 1 We use atomic units everywhere except in Sec. IV. The pump and probe fields, Et = eEt, are characterized by the po- e−iLt+L+kL·R d10 · EL Dc0 · eX2EX* EX L − i0 EXte−iLt+L, = 1,2, . . . . 3 We assume in this estimate that the transition matrix ele- 012714-2 6 R±² 1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d 5QV1HK ·Ï2d@ tIV ê PHYSICAL REVIEW A 72, 012714 2005 IR–X-RAY PUMP-PROBE SPECTROSCOPY OF THE… t = ate−it , 0 = 0, tt = 1. 4 Here and are the vibrational energy and eigenvector of the ground electronic state. The nuclear wave packet of the ground state 4 obeys the Schrödinger equation i t = Htt, t Ht = H0 − d · ELtcosLt + L, FIG. 2. Qualitative illustration of the dependence of the x-ray absorption probability on the ir phase L and on the delay time t. The ir field for different phases is depicted by solid and broken lines. The labels A, B, and C mark short x-ray pulses X TL for different delay times. The curve marked by stars shows long x-ray pulse X TL. TL = 2 / L is the period of oscillations of the ir field. ments of the ir transitions, → + 1, are collinear; which is the case for diatomic molecules. Equation 3 shows directly the strong sensitivity of the x-ray absorption on the phase of the pump field, L. Figure 2 shows the interference 3 of one- and two-photon channels = 1 which results in that the signal beats with the ir frequency, L. Such beats can be directly observed if the time resolution of the x-ray detector is better than the period of the oscillations of the ir field, TL = 2 / L. According to our knowledge the current x-ray instrumentation allows one to measure only the signal integrated over time. In this case the phase sensitive interference term is quenched when the duration of the x-ray pulse, X, is much longer than TL Fig. 2. The situation changes drastically for a short x-ray pulse, X TL, as such a pulse may select positive or negative parts of the oscillating ir field Fig. 2. As one can see from Fig. 2, the sign and magnitude of the area of the ir field selected by a short x-ray pulse, SL = 5 with H0 as the nuclear Hamiltonian of the ground electronic state. The broadening of the vibrational levels of the ground state, 0, is ignored here. This approximation is justified for diatomic molecules in gas phase which have very large vibrational state lifetimes 1 ms, much longer than the delay time and duration of x-ray pulse considered in this paper. Let us note that the spatial phases kL · R and kX · R are ignored in the remaining part of the paper. The role of these phases can be important and this approximation was already discussed in Ref. 10. For instance, these phases can be neglected for perpendicular intersection of x-ray and ir pulses. We solve the Schrödinger equation 5 using the rotating wave approximation, assuming the pump field to be weak and L = 10: a1t = ie−iL t dt1d10 · ELt1 = e−iLc1t, a0t 1. − 6 One can see that contrary to a0t 1 the amplitude of the first vibrational states depends on the phase, L. Now we are in a position to guess how wave packets 4 of the harmonic oscillators depend on the phase when L = 10 t cte−iL+t , 7 =0 EXtcosLt + Ldt 0, are very sensitive to the phase of the ir field as well as to the delay time of the x-ray pulse relative to the pump pulse. One can expect that the x-ray spectrum of the molecule in the strong ir field also is sensitive to the phase of the ir field. More precisely, it is sensitive to the peak position of the x-ray pulse relative to the “comb” of the ir field. This means that the phase of the ir field and the delay time play quite similar roles, namely, one can change the peak position of the probe pulse relative to the ir comb with help of the ir phase or delay time see Fig. 2. Let us explore this effect in more detail making use of a strict formalism. where c0t 1, c1t is defined by Eq. 6, and c2t c21t. The coefficients ct do not depend on L in the rotating wave approximation. Numerical simulations Fig. 3 which will be discussed in detail in Sec. V show that the trajectory of the center of gravity of the WP r̄t = trt is very sensitive to the phase of the ir light. Taking into account Fig. 1 it easy to understand that the spectral shape of the x-ray absorption driven by a strong ir radiation depends strongly on the phase. III. X-RAY PHOTOABSORPTION OF MOLECULES DRIVEN BY A STRONG IR FIELD B. Role of phase of ir field on the wave packet dynamics The pump ir pulse ELt induces transitions between vibrational levels of the ground electronic state or dissociates and creates the nuclear WP 8 We consider the case when a resonant x-ray field excites a molecule from the ground to core-excited electron- 012714-3 w-I1V − IªK <kHkVT<<ª<¾;¸·<øb³Bxx³B ê PHYSICAL REVIEW A 72, 012714 2005 GUIMARÃES et al. ait = ie−t t dt1EXt1e−i−i+t1−iXt1it1, − 1 = eX · Dc0 2 13 in Eq. 10 results in the following expression for the instantaneous transition probability: Pt, = 2EXt Re t dt1EXt1ei−t−t1t − t1 − ctct1, 14 where FIG. 3. Trajectory of the WP r̄t Eq. 8 vs phase of pump field. The case of oriented molecule: d eL. IL = 2.3 1012 W / cm2. L = 01 = 0.241 eV. tL = 700 fs. L = 100 fs. kL = 1. Solid lines show the trajectories of the WP. The shaded areas on the right-hand side panel represent wave packets. Ta 1210 fs. ct = eiHctt. Here we introduced the correlation function describing the phase fluctuations of the x-ray field we ignore the amplitude fluctuations t − t1 = eiXt−Xt1, el vibrational states el The population, 0 t → c it. iit , , of the ith vibrational level of the core-excited state evolves according to the balance equation + 2 iit, = Pit,, t 9 where 2 is the decay rate of the core-excited state. The total probability of the transitions from ground state to coreexcited state Pt, = Pit, = − 2 Im it,Vit 10 i i depends on the density matrix of the molecule, ijt , = aita*j t, and on the interaction of the molecule with the x-ray pulse, Vit. Due to near resonant conditions it is possible to use the rotating-wave approximation Vit = − e · D0c EXtiei−it+iXt . 2 15 = 1 2 d ei . 16 − The angular brackets implicate averaging over the phase fluctuations of the x-ray field. The Fourier transform of the real correlation function, , is the spectral function of the x-ray radiation. We assume that the pump field is coherent and has a long correlation time. As it was pointed out above the duration of standard x-ray measurements is longer than the pulse duration. This motivates us to focus attention only on the integral probability P = dtPt, = − d P0 − K. 17 − Here we introduced the absorption probability for K = P0 = c− c− 11 18 and the WP in the frequency domain Here = X − c0 is the detuning of the x-ray field from the resonant frequency of the pure electronic transition 0 → c, c0 = Ecrce − E0r0e is the adiabatic excitation energy which constitutes the difference between minima of core excited and ground state potentials, i = i − is the difference between vibrational energies of the core excited and ground states. The x-ray field is assumed to be weak, which means that it does not affect the ground state wave packet 4. In other words, the density matrix element it , = aita*t in Eq. 10 has the same coefficient at as the ground state wave packet 4. The amplitude of the ith vibrational level of the core excited state obeys the equation + ait = − i Vitat. t The substitution of the solution of this equation 12 c− = dt e−itEXtct. 19 − The convolution of the Lorentzian , = / 2 + 2, the spectral function of the x-ray field results in the total spectral function K = = 1 Re d e−−i 0 d111 − ,. 20 − This function becomes a Voigt profile when the spectral function of the x-ray field is a Gaussian. It is worthwhile to note that the finite duration of the x-ray pulse produces an extra broadening of the spectral lines of 012714-4 6 R±² 1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d 5QV1HK ·Ï2d@ tIV ê¸ PHYSICAL REVIEW A 72, 012714 2005 IR–X-RAY PUMP-PROBE SPECTROSCOPY OF THE… TABLE I. Spectroscopic constants 13 of NO used in the simulations: vibrational frequencies, anharmonicity constants, internuclear distances, probabilities of O 1s → 2 transitions, f. Excited States NO* Spectral constant e / cm−1 exe / cm−1 re / Å / eV f / f 2+ c0 / eV NO 2 2 − 1943.79 13.71 1.146 1121.11 9.68 1.339 0.0870 3.52 531.30 the x-ray absorption. This effect is included in P0 via the amplitude of the x-ray field, EXt. IV. COMPUTATIONAL DETAILS Our simulations are divided in four blocks: 1 calculation of the nuclear WP t in the ground electronic state, solving numerically the Schrödinger equation 5 without any assumption about the intensity of the ir field, 2 evaluation of the nuclear wave packet ct in the potential of the core excited state 15, 3 Fourier transform, c− 19, of the wave packet ct, 4 calculation of the norm c− c− which is nothing else than the probability of x-ray absorption P0 18. To control the contribution of individual vibrational levels in the WP āt; L = t, = āt; L2 2 21 we calculated also the vibrational frequencies and stationary wave functions in the ground electronic state solving the stationary Schrödinger equation: H0 = . Here is the population of the th vibrational level in the ground electronic state. Both wave packets, t and ct, are calculated employing time dependent techniques 11 using the ESPEC program 12. The second order differential scheme is applied in the propagation of the WP with a time step of 5 10−5 fs. These parameters preserve the norm of the WP, t, being equal to one during the propagation. In the simulations we neglect the lifetime broadening 0.08 eV of the core excited state O 1s → 2 of the NO molecule except Sec. V B, as well as the broadening due to the spectral function of the x-ray field, . This is a reasonable approximation considering that the rather short x-ray pulses using X 3 – 20 fs give quite large spectral broadening 1 / L 0.5– 0.07 eV. The propagation of the WPs is calculated using computed Morse potentials 13 for the core excited 2− 1 12 23 24 21 45 22 2 and ground 2 12223242145221 states of the NO molecule. The detuning of the x-ray field = − c0 is defined relative to the adiabatic excitation energy 13 c0 = 531.3 eV. The other parameters of the Morse potential are listed in Table I. The potential energy curve is mapped from 0.5 until 2.8 Å 1282.42 8.87 1.295 0.0875 2.13 532.20 2 + 1306.61 9.68 1.290 0.0865 1.00 533.64 with 256 points. It is noteworthy that the strong pump field can mix different electronic states and, hence it can change the molecular potential 14,15. This effect is neglected here because it is small for the NO molecule, with large spacing between first excited and ground electronic states, 5 eV 16. The dynamics of the ground state wave packet t 5 is simulated using an r-dependent dipole moment d = dr Fig. 4, which was computed by the CAS-MCSCF method with DALTON 17. The active space is formed by 11 electrons in 10 orbitals comprising the nitrogen and oxygen second shell. The N K and O K electrons are kept inactive. Two different basis sets are used in the calculations, aug-cc-pVDZ and augcc-pVTZ, giving quite similar results see Fig. 4. As one can see from Table II the r-dependence of this dipole moment results in a rather weak breakdown of the dipole selection rules. The r-dependence of the transition dipole moment of the core excitation Dc0 is neglected. This is a good approximation due to the strong localization of the O 1s orbital. We have used the following relation between parameters in atomic units and in SI units: dEa.u. = dD IW / cm2 2.1132 10−9 a.u., ta.u. = a.u. tfs 41.3417. Here I = c0Et2 / 2 is the intensity of radiation. The ir radiation is assumed to be in resonance with the first vibrational transition, L = 10, everywhere except for the situation depicted in Fig. 9 where L = 20. FIG. 4. Dependence of permanent dipole moment, dr, of nitrogen monoxide molecule in ground state on the internuclear distance, r. Both dipole moment and ground state energy E0r are in a.u. 012714-5 w-I1V − IªK <kHkVT<<ª<¾;¸·<øb³Bxx³B ê4 PHYSICAL REVIEW A 72, 012714 2005 GUIMARÃES et al. TABLE II. Dipole moments d of ir transitions in the ground state of the NO molecule in D. → d → d → d 0→1 0→2 0→3 0→4 0→5 0.07284 0.00615 0.00076 0.00008 0.00001 1→2 1→3 1→4 1→5 2→3 0.10271 0.01056 0.00156 0.00018 0.12552 2→4 2→5 3→4 3→5 4→5 0.01476 0.00264 0.14488 0.01863 0.16224 The temporal shape of the ir and x-ray pulses is modeled in the calculations as follows: I exp − t − t ¯ 2k , ¯ = , ln 21/2k 22 where is the half width at half maximum HWHM, k = 1 , 2 , . . .. This expression is convenient because it describes a smooth transition from a Gaussian k = 1 to a rectangular function k 1. In the simulations we used the Gaussian shape k = 1 of the pump and probe pulses and assumed the following values for the peak position tL = 700 fs and the duration L = 100 fs of the pump pulse except Fig. 7b. Through all calculations it is assumed that the x-ray pulse probes the system after the ir pulse leaves it: t 2L + X. Everywhere except for the cases reproduced in Figs. 8 and 9 our simulations were performed for oriented NO molecules: d e L. V. RESULTS A. Gross spectral features It is instructive to start analyzing the gross spectral features of the x-ray pump-probe spectra. Throughout the paper we focus our attention only on the x-ray transition 2 → 2− in NO, except in Sec. V B. The reference spectrum is the ordinary x-ray absorption profile of NO without ir field spectrum A in Fig. 1. The spectrum changes qualitatively if the molecule is exposed to a strong ir field. Due to the r-dependence of the permanent dipole moment dr Fig. 1, the ir field populates higher vibrational levels of the ground state and creates a coherent superposition, t. The WP performs back and forth oscillations in the ground state potential. The time dependence of the peak position, r, of t affects the probe spectrum measured at a certain instant. When the x-ray pulse is short X = 3 fs the proper choice of the delay time, t = tX − tL or ir phase, allows one to get a snapshot of the x-ray spectra for WPs localized near the left B or right C turning points. The B and C spectra Fig. 1 differ qualitatively because their vertical transitions have different energies. The x-ray spectrum is approximately the sum of spectra B and C if the duration of the x-ray pulse is longer or comparable with the period of oscillations of the wave packet: X 2 / 10 17 fs see spectrum D in Fig. 1. This is because the WP has time to move from left to right turning points in this case. When the probe and pump pulses overlap the x-ray spectrum depends on the delay time both for short and long pulses due to the sensitivity of the populations of vibrational levels to t. When the ir pulse leaves the system the x-ray spectrum continues to depend on t if the x-ray pulse is short; when the x-ray pulse is long the spectrum ceases to depend on the delay time see Sec. V F. It is worthwhile to note that the ir field changes the mean vibrational energy of the ground state = t − 0, FIG. 5. The partial O K x-ray absorption profiles P Eq. 24 of NO excited in ground state vibrational levels = 0 , 1 , 2 for different core-excited states: 2−, 2, and 2+. The total spectral profiles are shown by the thin solid lines. Narrow resonances display the spectral distribution of the Franck-Condon factors for the 2 − core-excited state. t = tH0t. 23 This results in a shift of the center of gravity of the x-ray spectrum. In the general case this shift depends on the delay time and the intensity of the ir pulse. Such a shift, 0.23 eV, is shown in Fig. 1 for t 2L + X when ceases to depend on the delay time. The ir intensity affects the x-ray spectrum because a larger corresponds to a larger distance between the left and right turning points of the wave packet in the potential see Fig. 1. The influence of the x-ray pulse duration on the spectral resolution deserves a short remark. When the x-ray pulse is rather long X 2 / 10 17 fs, the spectra display vibrational structure see the spectra A and D in Fig. 1, while the vibrational resolution is washed out if the x-ray pulse is short Fig. 1 spectra B and C because of the uncertainty relation between time and energy. 012714-6 6 R±² 1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d 5QV1HK ·Ï2d@ tIV ê·Ú PHYSICAL REVIEW A 72, 012714 2005 IR–X-RAY PUMP-PROBE SPECTROSCOPY OF THE… B. X-ray absorption of NO molecules in the field of incoherent pump radiation The molecule is excited from lowest to higher vibrational levels by the strong ir field. Such an excitation influences the x-ray absorption due to the change of the populations of vibrational states as well as due to the coherence between these states created by the ir field see Sec. II B. We consider in this section the incoherent ir pulse with randomly fluctuating phase, Lt. Due to this randomness the coherence between different vibrational states of the WP, t Eq. 7, is destroyed and the x-ray absorption probability becomes a simple sum of partial contributions P = P. 24 Here P is the probability of x-ray transition from vibrational level with the population Eq. 21. Figure 5 shows the partial x-ray spectra, P, of O 1s → 2 absorption for three different initial vibrational states, = 0 , 1 , 2. In this figure we show the total and partial absorption spectra which correspond to the core-excitation in different close-lying final states 13, 2−, 2, and 2+. The spectroscopic parameters used in the simulations are collected in Table I. One can see that the different initial vibrational states result in very different x-ray spectra, something that can be referred to the different distributions of the Franck-Condon factors, 0 i2, 1 i2, and 2 i2 and the quite large displacement of the potential well of the coreexcited state relative to the ground state potential. Apparently, the x-ray absorption of NO driven by an incoherent pump field 24 does not depend on the phase L due to phase independence of the populations Fig. 5. However, the x-ray spectrum is sensitive to the delay time because of the time dependence of populations. Such a time dependence takes place in the studied case only when the x-ray pulse overlaps with the ir incoherent pulse. Let us now investigate ir–x-ray pump probe spectroscopy in the field of coherent ir radiation see Eq. 18. In order to focus our attention on the physics we will only study the x-ray absorption band related to the lowest final state 2−. C. Dynamics of the nuclear wave packet versus phase and Rabi oscillations. Phase memory versus the shape of the pulse The coherent pump field prepares the wave packet t which is probed by the x-ray pulse. We have shown in Sec. II B that the WP and, hence, the x-ray absorption is sensitive to the phase of the strong ir field, L. The simulations indicate a strong dependence of the trajectory of the wave packet on the phase, Fig. 3. As one can see from this figure the WP performs fast back and forth oscillations with the vibrational frequency 10, and is modulated with a lower frequency: a = 10 − 21 = 2exe = 2 13.71 cm−1 see Table I. This modulation caused by the anharmonicity of the ground state potential has the period Ta = 2 / a 1210 fs see inset of Fig. 3. One can show that the trajectories of the center of gravity of the x-ray spectrum and of the wave packet are quite similar 18. The measurement of the time interval be- FIG. 6. The phase dependence of the contributions ã 21 of different vibrational states = 0 , 1 , 2 , 3 in the wave packet, t: a = ãL = 0 − ãL = / 2. The left and right panels show the real and imaginary part of a. IL = 2.3 1012 W / cm2. L = 01 = 0.241 eV. tL = 700 fs. L = 100 fs. kL = 1. The vertical arrows shows the instant where the ir intensity is decreased in two times. tween the adjacent nodes of the trajectory gives directly the revival time, Ta. This allows one to measure the anharmonicity constant. The phase effect can be explained also with help of classical mechanics. According to Ehrenfest’s theorem the mean value of the force which affects the center of gravity of the wave packet F = t d ELt · drcosLt + Lt dr depends on the ir phase, L. For example, this force changes the sign when L → L + . This means that the evolution of the WP in the potential well depends on the phase. The phase sensitivity is clearly seen from the strong phase dependence in the projections of the WP on the stationary vibrational state, ã 21 which was calculated for L = 0 and L = / 2 see Fig. 6. The simulations are in nice agreement with the simple equation 7 ã e−iL+t, = 0,1.2, . . . , 25 which is valid for L = 10. According to this equation the amplitude of the lowest vibrational level, ã0, does not depend on the phase. The simulations display very weak phase dependence of ã0 which, probably, is due to the slight break down of the rotating wave approximation which is used in Eqs. 7 and 25, while the amplitudes of higher vibrational levels of ã1, ã2, and ã3 strongly depend on the phase Fig. 6. Equation 25 indicates that ã oscillates with the period T = 2 / , for example, T1 = 11.56 fs, T2 = 6.98 fs, and T3 = 5.03 fs. These values agree perfectly with the simulations based on the strict solution of the Schrödinger equation 5 see Fig. 6. It can be also seen in Fig. 6 that the amplitudes ã and, hence the wave packet t depend on the phase L during the interaction with the pump pulse, as well as later when the pulse leaves the system. The main reason for such a long memory about the phase is the long lifetime of the vibrational levels of the ground state this time is assumed to 012714-7 w-I1V − IªK <kHkVT<<ª<¾;¸·<øb³Bxx³B êä PHYSICAL REVIEW A 72, 012714 2005 GUIMARÃES et al. be infinite in our simulations. Clearly, the x-ray spectrum also keeps the phase memory when the pump pulse leaves the system. Contrary to the real and imaginary parts of ã, the populations of vibrational levels = ã2 21 almost do not depend on the phase. Figure 7a shows only a weak modulation of GR cos2Lt + 2L L 26 with twice the frequency of the ir field. These oscillations origin in the off-resonant interaction with the field and they depend on the phase L. The inset in Fig. 7a displays different time dependences for L = 0 and L = / 2. Simulations show that the phase L = gives the same results as L = 0 in agreement with Eq. 26. The off-resonant interaction can be important when the Rabi frequency, GR = EL · d10, TR = 2 , GR 27 approaches the ir frequency, L when the rotating wave approximation breaks down. Evidence that the weak modulation of the populations with the frequency 2L are related to the Rabi frequency is given by the absence of these oscillations in the region where the ir intensity is very low Fig. 7a. The ir intensity used in the simulations shown in Fig. 7 corresponds to the Rabi period 27 around 750 fs which is longer than the duration of the ir pulse, L = 100 fs. This corresponds to the limiting case of a sudden switching of the pump field: The ir field is shut off faster than the Rabi period and the molecule remains in the vibrationally excited state after the pulse leaves the system Fig. 7a which then keeps the phase memory. The scenario changes drastically when the time for the switching-off of the ir pulse, T 715 fs, is long and becomes comparable with the Rabi period, TR 750 fs see Fig. 7b. In this case the field is shut off slowly and the system follows adiabatically the slow decrease of the light intensity up to zero where only the lowest vibrational level is populated Fig. 7b. In this case the phase will influence the x-ray spectrum only when the x-ray and pump pulses overlap in the time domain. The phase sensitivity is absent when the x-ray pulse exposes the molecules after the ir pulse leaves the system; moreover, the x-ray spectrum coincides, in this case, with the x-ray spectrum of the molecules without the ir field. So in the adiabatic limit, T TR, any memory about the ir pulse is absent. D. Influence of the phase of the ir field on x-ray absorption by oriented molecules The phase of the ir field strongly influences the shape of x-ray absorption. This is seen clearly from Figs. 8 and 9 where the x-ray absorption probabilities are shown for four different phases, L = 0, / 2, , and 3 / 2. The physical mechanism of such a phase sensitivity, described in Sec. II is directly related to the phase dependence of the dynamics of the WP. It is important to note here that the x-ray pulse FIG. 7. Populations, Eq. 21 of the vibrational levels of the ground electronic state vs time for different durations of switching off the pump field which is L = 100 fs for short pulse a, and T 715 fs for long rectangular pulse b. Lt = ELt / ELmaxt = Lt − tLcosLt is the time distribution of the pump field with the peak value normalized to one see Eq. 22. L = 01 = 0.241 eV. IL = 2.3 1012 W / cm2. L = 0. a tL = 700 fs, L = 100 fs. kL = 1. The inset shows the case L = / 2. b tL = 2 ps, L = 1 ps. kL = 3. probes the system after the pump pulse leaves the system. Thus both Figs. 8 and 9 evidence the above discussed effect of the phase memory. Figures 8a and 8b show x-ray spectra when L = 10 = 0.241 eV for two different ir intensities, 1.5 1012 and 2.3 1012 W / cm2, respectively. The weaker pump field populates mainly the two first vibrational states and creates the following distribution of vibrational populations; 0 = 0.374, 1 = 0.527, 2 = 0.097, 3 = 0.002. The population distribution in the stronger ir field, 0 = 0.208, 1 = 0.605, 2 = 0.182, 3 = 0.005, shows the population of the level, = 2, also. The phase influence is strong in both cases. However, we see below that the population of level = 2 becomes very important for randomly oriented molecules. Let us note that the transition energy is influenced also by the mean energy of the wave packet t Eq. 23. This energy depends on the time when the ir pulse interacts with the molecule and it is time-independent when the pulse leaves the system. After the pump pulse leaves, t = 0.29 and 0.35 eV for the param- 012714-8 6 R±² 1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d 5QV1HK ·Ï2d@ tIV ê<ê PHYSICAL REVIEW A 72, 012714 2005 IR–X-RAY PUMP-PROBE SPECTROSCOPY OF THE… FIG. 9. Phase dependence of the probability of O K x-ray absorption of NO Eq. 18. The frequency of the pump is tuned in resonance with the second vibrational level: L = 02 = 0.4717 eV. IL = 3.0966 1013 W / cm2. The other parameters are the same as in Fig. 8. 1 Pint t d10 · eLDc0 · eX2 = 0 FIG. 8. Phase dependence of the probability of O K x-ray absorption of NO Eq. 18. The frequency of the ir field is tuned in resonance with the first vibrational level: L = 01 = 0.241 eV. The spectra averaged over molecular orientations 29 are marked by L with L = 0 and / 2. X = 3 fs. The delay time: t = 610 fs. a IL = 1.5 1012 W / cm2. b IL = 2.3 = − 531.3 eV. 1012 W / cm2. eters used in Figs. 8a and 8b, respectively. When the pump field is tuned in resonance with the second vibrational level, L = 20 = 0.4717 eV, the efficiency of the population decreases strongly because the corresponding transition dipole moment, being proportional to 0dr2 d0r22, is small. In order to populate the higher vibrational levels an ir intensity of IL = 3.0966 1013 W / cm2 was used. In this case it only populates efficiently the ground state and the second vibrational level = 2: 0 = 0.817, 1 = 0.000, 2 = 0.182, and 3 = 0.001. The mean energy of the wave packet is in this case t = 0.21 eV. Here and above the populations as well as t correspond to times after that the ir pulse left the system. E. Probe signal from randomly oriented molecules and nonlinearity Usually molecules are randomly oriented and the probe signal 18 has to be averaged over molecular orientations. The orientational averaging strongly affects the phase dependence which origins in the interference term 3. When the pump intensity is weak, the ir field populates only the first vibrational levels, = 1. In this case the interference term is equal to zero 28 and the phase dependence is absent. Such a quenching of the interference is due to that the two opposite molecular orientations, d10 and −d10, cancel each other. As one can see from Eq. 1 the change of the sign d10 · eL is equivalent to the change of the phase L → L + . We performed orientational averaging taking only into account the most important molecular orientations, namely the two opposite orientations: 1 P̄0, L P0, L + P0, L + . 2 29 The x-ray spectra averaged over molecular orientations are marked in Figs. 8 and 9 as L. Figure 8 compare spectra 0 and / 2 displays a suppression of the phase effect for the ir intensity 1.5 1012 W / cm2 and a strong phase dependence for higher intensity, IL = 2.3 1012 W / cm2. Mainly levels = 0 and 1 are populated in the first case: 0 = 0.374, 1 = 0.527, and 2 = 0.097. Equation 28 explains the small phase effect for randomly oriented molecules in such a field. For higher intensity, 2.3 1012 W / cm2, the level = 2 is higher populated: 0 = 0.208, 1 = 0.605, 2 = 0.182, and 3 = 0.005. This results in a nonzero interference term 3: 2 Pint t d10 · eL2Dc0 · eX2 0. Figure 9 indicates a strong suppression of the phase dependence when the ir frequency is tuned in resonance with the transition 0 → 2. The reason for this is the small value of the transition dipole moment d20 which suppresses the population of higher vibrational levels, 3. In this case mainly the one-photon channel 0 → 2 influences x-ray absorption and the interference term similar to Eq. 28 is quenched: d20 · eLDc0 · eX2 = 0. F. Phase effect versus x-ray frequency, duration of x-ray pulse, and delay time Any phase effect is related to interference. In our case the interference operates between one- and many-photon absorp- 012714-9 êñ w-I1V − IªK <kHkVT<<ª<¾;¸·<øb³Bxx³B PHYSICAL REVIEW A 72, 012714 2005 GUIMARÃES et al. FIG. 10. Difference between x-ray absorption spectra 18 for L = 0 and L = / 2. 2.3 1012 W / cm2. Delay time is t = 610 fs. tion channels. The contributions of these different channels strongly depend on the frequency of the x-ray field, X. This means that the interference, as well as the phase effect, are different for different X, something that is clearly illustrated in Fig. 10. We already recognized Fig. 2 that the phase of the pump field influences the x-ray spectrum when X / 10 = 8.58 fs. This simple estimate is in accord with results of the strict simulations, Figs. 10 and 11, which show that the phase dependence decreases for X 5 fs. It is interesting that the phase effect also vanishes for small durations of the x-ray pulse. The short x-ray pulse has many harmonics with different frequencies, X. Thus to understand what is going on with the probe signal for shorter pulses we have to convolute the probe spectrum Fig. 8 with the spectral distribution with FIG. 11. Probability of x-ray absorption O 1s → 2 of NO vs duration of x-ray pulse, X for different phases lower panel. Upper panel displays the phase dependence for different . IL = 2.3 1012 W / cm2, delay time t = 610 fs. = 0.4 eV in the lower panel. FIG. 12. Probability of x-ray absorption O 1s → 2 of NO Eq. 18 vs delay time, t = tX − tL, and the phase of ir field, L. Solid line: L = 0. Broken line: L = / 2. X = 4 fs. = 0.4 eV. L = 10 = 0.241 eV. IL = 2.3 1012 W / cm2. tL = 700 fs. L = 100 fs. the width 1 / X. As one can see from Fig. 8 such a convolution diminishes the difference between x-ray spectra for different phases if X is small. The phase effect takes maximum value when X TL / 4. Figure 12 shows the dependence of x-ray absorption on the delay time for two different phases: L = 0 , / 2. The results of simulations presented here confirm perfectly the qualitative picture, Fig. 2. Figure 12 says that the phase and the delay time play a similar role. VI. POSSIBILITY OF EXPERIMENTAL OBSERVATIONS A few words about the possibility of experimental observation of the phase sensitivity of x-ray absorption are in place. First of all, our simulations are performed for a real molecular system and from this point of view our predictions operate with realistic quantities. An intensity of the ir pulse around 1012 W / cm2 is sufficient to populate the first vibrational levels and to see the phase effect. This means that the intensity of the x-ray pulse is the same as is used in standard x-ray absorption measurements. From this point of view one can use conventional synchrotron radiation light sources. The main problem which we face now is the duration of the x-ray pulse. To observe the discussed phase effect this duration time must be comparable with the period of the vibrational mode. For the NO molecule the duration must be around X 5 – 10 fs. This requirement can be weaker for polyatomic or heavy molecules which have much smaller vibrational frequencies. This makes the duration essentially longer, X 100 fs. Already now ultrashort x-ray pulses generated by high harmonics with X 1 fs are available 2. The current state of affairs for creation of short x-ray pulses was overviewed briefly in Ref. 1. As shown above, a rather strong ir pulse, 1012 W / cm2, is needed to create a nuclear wave packet. Tunable wavelengths in the range from 1 up to 20 m can be produced via optical parametric amplification or via the second harmonic generation of a 10.6 m CO2 laser 19. The desired intensity, 1012 W / cm2, can be obtained by tight focusing of the light beam in the spot with appropriate diameter. It is worth 012714-10 6 R±² 1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d 5QV1HK ·Ï2d@ tIV êö PHYSICAL REVIEW A 72, 012714 2005 IR–X-RAY PUMP-PROBE SPECTROSCOPY OF THE… mentioning that many other molecular systems have different vibrational frequencies, which match properly with available ir or mid ir powerful lasers. Some of these molecules can be resonantly excited by ir pulses with the wavelength 10.6 m CO2 laser or by 1064 nm Nd:YAG laser 20 or by lasers produced using the technique of chirped pulse amplification CPA, which is an extremely promising tool to generate powerful lasers. An advantage of the optical parametric amplification over the CPA technique is to allow the study of several different molecular systems, due to its intrinsic tunability via phase matching in the nonlinear crystal. Commercial lasers 21 exist which generate about 10 J, 30 fs pulses at 1 – 10 m. Another advantage is that such pulses are automatically locked in time to a powerful titanium-sapphire pump, which in principle is able to produce x-ray radiation via high order harmonic generation. We have shown that the phase effect can be observed both for oriented and disordered molecules. In the first case the effect is stronger. Fortunately, there exist powerful techniques to measure x-ray absorption from fixed-in-space molecules in coincidence experiments applied on randomly oriented molecules 22–24. Let us mention also the possibility of alignment and orientation of the molecules by a strong ir field 25,26. VII. SUMMARY stant through the revival time of the center of gravity of the x-ray spectrum. The phase sensitivity of the trajectory results in a dependence of the x-ray absorption profile on the phase and delay time. The x-ray absorption profile displays a maximum interference pattern when the duration of the probe x-ray pulse is one-fourth of the infrared field period. There is here an important distinction from standard few-cycle optical experiments, namely that in our case the duration of both pump and probe pulses are longer than the inverse frequencies of the corresponding fields. The phase effect is found to be sensitive to the duration of the x-ray pulse. We have found a phase memory effect, namely that the x-ray spectrum keeps the memory about the phase after the ir pulse leaves the system. Such a phase memory strongly depends on the relation between the time of switching-off the ir pulse and the Rabi frequency. We have shown that when the pump field is weak its phase does not influence the x-ray absorption if the molecules are randomly oriented. In this case the discussed interference effect can be observed only for oriented molecules. One can accomplish orientation by making use of surface adsorbed molecules or by detection of x-ray absorption in the ion yield mode. The interference pattern for randomly oriented molecules starts to grow when the intensity of the pump radiation increases and the pump field is able to populate even vibrational levels. Thus a third way to detect the phase sensitivity of the x-ray absorption of disordered molecules is to use rather high intensities of the pump field. In that case the interference between the different photon excitation channels can be enhanced controlling the intensity and time duration of the infrared pulse. In this work we have theoretically predicted different x-ray absorption spectra of the NO molecule driven by a strong ir field. The x-ray absorption excited incoherently in different vibrational levels of the ground state was found to demonstrate a strong dependence of the absorption profile on the initial vibrational state and on the final electronic state. Special attention was paid on the coherent superposition of ground state vibrational levels created by an ir laser. In this case the simulations displayed a strong dependence of the trajectory of the vibrational wave packet on the phase of the ir field. The trajectory of the wave packet experiences oscillations in the potential well with two qualitatively different frequencies. The wave packet performs fast back and forth oscillations with the vibrational frequency but which are modulated by the anharmonicity of the potential. This fact allows in principle one to measure the anharmonicity con- This work was supported by the Swedish Research Council VR and by the STINT foundation. V.C.F. and F.F.G. acknowledge financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior CAPES Brazil. F.G. acknowledge also financial support from the Russian Foundation for Basic Research Project No. 04-02-81020-Bel 2004. We acknowledge Professor Michael Meyer for fruitful discussions of current and future possibilities of experimental measurements of the phase effect. 1 V. C. Felicíssimo, F. F. Guimarães, F. Gel’mukhanov, A. Cesar, and H. Ågren, J. Chem. Phys. 122, 094319 2005. 2 R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, Nature London 427, 817 2004. 3 B. W. Adams, Rev. Sci. Instrum. 75, 1982 2004. 4 R. W. Schoenlein, W. P. Leemans, A. H. Chin, P. Volfbeyn, T. E. Glover, P. Balling, M. Zolotorev, K. J. Kim, S. Chattopadhyay, and C. V. Shank, Science 274, 236 1996. 5 R. W. Schoenlein, S. Chattopadhyay, H. H. W. Chong, T. E. Glover, P. A. Heimann, W. P. Leemans, C. V. Shank, A. A. Zholents, and M. Zolotorev, Appl. Phys. B: Lasers Opt. 71, 1 2000. A. A. Zholents and W. M. Fawley, Phys. Rev. Lett. 92, 224801 2004. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 2000. A. 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Steinfeld, Molecules and Radiation: An Introduction to Modern Molecular Spectroscopy Harper & Row, New York, 1974. 17 T. Helgaker et al., DALTON a molecular electronic structure program—release 1.2, 2001, see http://www.kjemi.uio.no/ software/dalton/dalton.html. 18 F. F. Guimarães, F. Gel’mukhanov, A. Cesar, and H. Ågren, Chem. Phys. Lett. 405, 398 2005. 19 B. Zhao and S. Zhu, Opt. Eng. 38, 2129 1999. 20 J. Ohkubo, T. Kato, H. Kono, and Y. Fujimura, J. Chem. Phys. 120, 9123 2004. 21 TOPAS Model 4 / 800 Femtosecond version wavelength range from 1150 to 2600 nm, see http://www.lightcon.com/lc/ scientific/scientific.htm. 22 A. Yagishita, E. Shigemasa, and N. Kosugi, Phys. Rev. Lett. 72, 3961 1994. 23 R. Guillemin, E. Shigemasa, K. Le Guen, D. Ceolin, C. Miron, N. Leclercq, K. Ueda, P. Morin, and M. Simon, Rev. Sci. Instrum. 71, 4387 2000. 24 K. Ueda, J. Phys. B 36, R1 2003. 25 S. Chelkowski, P. B. Corkum, and A. D. Bandrauk, Phys. Rev. Lett. 82, 3416 1999. 26 D. Sugny, A. Keller, O. Atabek, D. Daems, C. M. Dion, S. Guerin, and H. R. Jauslin, Phys. Rev. A 69, 033402 2004. 012714-12 ´a²´«¶z· `a Rca L;L}L ÇÆ WeQ[:g Ì ¹ ÄD<eC ab 2eQdÅ}jf}l¿feQdkd5 MONºMPN6QfRUT¢V£XoZ4[6\&];^,MONºQf\dg¬hVkR'lmXnUp;qr^6®kN6¤»\&]XoZ XonUsutvNªwCeKZF\&n ¼ z} F½¾~F³y'z${}|i~E|À¿K >Á¬Ái F¬|¸Ã:=Ä,ÆÅÇ − &ÅC1$ H±B 1 1Vi }HV³)¶®·4´ddPQx1H·Ï;@tIV ñ<¸ Chemical Physics Letters 405 (2005) 398–403 www.elsevier.com/locate/cplett Quantum wave packet revivals in IR + X-ray pump–probe spectroscopy F.F. Guimarães b a,b,* , F. Gelmukhanov a, A. Cesar b, H. Ågren a a Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden Departamento de Quı́mica, Universidade Federal de Minas Gerais, Av. Antonio Carlos 6627, CEP-31270-901, Belo Horizonte, Minas Gerais, Brazil Received 10 January 2005; in final form 14 February 2005 Available online 16 March 2005 Abstract The wave packet revivals constitute a central concept of X-ray spectroscopy with ultra-high spectral resolution. The revival phenomenon allows to resolve the anharmonical shift or rotational structure by means of time dependent measurements and makes X-ray pump–probe spectroscopy a powerful technique to study long-term dynamics of molecules in different phases. We study the revivals referring to the X-ray absorption spectrum of the NO molecule driven by strong infrared pulse. It is shown that the phase sensitive trajectories of the center of gravity of the wave packets and the X-ray spectra copy each other. 2005 Elsevier B.V. All rights reserved. 1. Introduction Advances in the physics and chemistry of laser interactions with atoms and molecules have brought the concept of wave-packets and their dynamics into the limelight. One of the major reasons for studying wave packet dynamics in the context of molecules is related to laser catalysis [1,2] and the control of chemical reactions [3–8] by careful application of pulses of light with optimal frequency, intensity, duration, and timing. Among these studies a special attention has been paid to the long-term dynamics related to quantum wave packet revivals. Classical as well as quantum dynamics of systems of coupled oscillators with slightly different frequencies experience fast oscillations with vibrational frequency and slow modulations. These modulations, defined by the differences between the frequencies of individual oscillators, are named revivals. Revival phenomena are * Corresponding author. Fax: +46 8 5537 8590. E-mail addresses: [email protected], [email protected] (F.F. Guimarães). 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.061 observed in many branches of physics. The first studies related to the revival time are traced to the Poincare recurrences for a rotation map [9], where the revival is analyzed in classical systems. Quantum revivals are usually associated with the dynamics of wave packets (WP) invented in physics by Schrödinger [10]. Quantum mechanical dephasing caused by anharmonicity (nonequidistant spectrum) leads to a delocalization of the initial wave packet which regains its initial shape after the revival period. The wave packet revival, predicted by Parker and Stroud [11], was experimentally confirmed a few years later [12]. Wave packet revivals are recognized to be important, for instance, in the motion of Rydberg wave packets [11,13], for rotational and vibrational degrees of freedom [14–17], in core-excited states [18,19], for pulse shaping [20,21], and for isotope separation [22]. The current state of theoretical and experimental studies of the wave packet revivals in atomic, molecular, condensed matter and optical systems has been reviewed by Robinett [23] a short time ago. The quantum revival is essentially a coherent phenomenon related to the dynamics of coherent superposition of quantum states. The wave packets can be ñ È ·; µ$´D1¶<I´±\´1·³B±²åzÛÊÉ I};Kkx T It ;Ittk 399 2. Physical picture of quantum revivals in an X-ray absorption spectrum We consider molecules that interact with IR pump field (L) and probe X-ray radiation (X) (see Fig. 1): Ea ðtÞ ¼ Ea ðtÞ cosðxa t ka R þ ua Þ; a ¼ L; X : ð1Þ The pump and probe fields Ea(t) = eaEa(t) are characterized by the polarization ea, the wave vectors envelopes ka, the frequencies xa and the phases ua. Atomic units are assumed everywhere in this Letter. The IR radiation influences the X-ray absorption in two different ways: due to the population of vibrational levels (m) and through the IR induced coherence between these levels. The coherent IR light mixes different vibrational states and creates their coherent superposition or the wave packet. Since the wave packet is not an eigenfunction of the ground state Hamiltonian, it starts to move. The coherently created wave packet performs the back and forth propagation in the potential well of the ground electronic state. X-ray snapshots of the nuclear 534.0 Energy (eV) generated in the potential energy surface due to optical [14] or X-ray transitions [24] as well as by means of vibrational–rotational transitions in the field of an infrared (IR) laser. The coherent properties of wave packets induced by an IR field are related directly to their phases and gives the opportunity to construct different superpositions of quantum states which are sensitive to the phase of the pump radiation. This constitutes the background of phase sensitive IR–X-ray pump–probe spectroscopy [25–28]. The main aim of our article is to study the role of quantum revivals in IR–X-ray pump–probe spectroscopy. Contrary to optical and IR spectroscopies, conventional X-ray spectroscopy has rather poor spectral resolution, mainly due to the rather large lifetime broadening and because of the quite large instrumental broadening. We show that the center of gravity of X-ray probe spectrum experiences modulation in the time domain with the revival period inversely proportional to the anharmonicity constant which is smaller than the lifetime broadening of X-ray resonance. Thus, the revival phenomenon makes this pump–probe setup very promising in X-ray spectroscopy with ultra-high resolution, far beyond the limitations due to the lifetime broadening. Measurements of the revival period give direct information about hyperfine structure of the molecular spectrum. We will also show how the X-ray pump–probe spectroscopy can be used in studies of inter- and intramolecular interactions. We deal with revivals related to the anharmonicity of the interatomic potential in the ground electronic state. Ω (eV) F.F. Guimarães et al. / Chemical Physics Letters 405 (2005) 398–403 533.0 A 532.0 B 531.0 Absorption probability 5.0 4.0 3.0 2.0 1.0 0.0 1.5 A 2.0 B 2.5 3.0 3.5 r (a.u.) Fig. 1. Scheme of OK X-ray absorption of NO driven by a strong IR field. At a certain instant a short X-ray pulse promotes the wave packet (A or B) to a certain point of the core excited potential. The X-ray absorption profiles A and B show the dependence of the X-ray spectra on the delay between X-ray and IR pulses. wave packet at different site positions yield the X-ray spectra (see Fig. 1). This technique maps the trajectory of the WP and is useful in studies of intramolecular interaction in excited states [28,27]. The ground-states nuclear wave packet obeys the Schrödinger equation i o ^ þ C /ðtÞ ¼ ½H 0 ðd EL ðtÞÞ cosðxL t þ uL Þ/ðtÞ; ot ð2Þ ^ and with the Hamiltonian H0, the relaxation matrix C initial condition j/(0)æ = j0æ. We neglect the spatial phases kL Æ R and kX Æ R the role of which was already discussed [25]. Before embarking on the details of the calculations, we employ simple arguments based on the rotating wave approximation (xL x10): when the IR pulse leaves the system the dynamics of the wave packet (2) is straightforwardly described by X jmiam eðim þCm Þðtt0 Þ ; j/ðtÞi ¼ m am ¼ i e iuL Z 1 dt1 ðdm;m1 EL ðt1 ÞÞ 1 e½iðxm;m1 xL ÞþCm Cm1 ðt1 t0 Þ am1 ðt1 Þ ¼ jam jeiðmuL þvm Þ ; where vm is the intrinsic phase of the vibrational state m, which depends on the frequency and the shape of the IR pulse; t0 is the time when the IR pulse leaves the system, xm,m1 = m m1; m and jmæ are the vibrational energy and eigenvector of the ground electronic state, and; Cm is the decay rate of the vibrational level m (C0 = 0). It can be seen that even after that the IR pulse H±B 1 1Vi }HV³)¶®·4´ddPQx1H·Ï;@tIV 400 F.F. Guimarães et al. / Chemical Physics Letters 405 (2005) 398–403 left the system the wave packet continues to be coherent and keeps the memory about the laser phase uL during the lifetime of the vibrational levels, C1 m . The physical understanding of the revival phenomenon can be achieved by considering the IR field of moderate intensity which mixes only the first three vibrational levels, m = 0,1,2. It is instructive to analyze the trajectory of the wave packet which is characterized by its center of gravity hrðtÞi ¼ h/ðtÞjrj/ðtÞi ¼ re þ 2Re q10 eðix10 þC10 Þðtt0 Þ r01 þq21 eðix21 þC21 Þðtt0 Þ r12 : ð3Þ Here, qmm1 ¼ am am1 is the density matrix, rmm1 ¼ hmjrjm1 i, re is the equilibrium interatomic separation, and Cmm1 ¼ Cm þ Cm1 . When the molecule is embedded in a bath (gas or liquid), it experiences collisions which quench the coherence qm,m1 with the rate cm,m1 c. Such a dephasing increases the decay rate of the coherence Cm;m1 ¼ Cm þ Cm1 þ c: ð4Þ To be specific, we assume that the density of the sample is quite high: Cm,m1 c. The resulting expression for the center of gravity (3) can be written as follows pffiffiffiffiffiffiffiffiffiffiffiffi hrðtÞi ¼ re þ 2r01 q00 q11 ð1 fÞ cosðx10 t0 þ u10 Þ t0 0 þ 2f cosðx10 t0 þ uþ Þ cos 2p þ u ect ; TR 0 ð5Þ where t = t t0 P 0, u± = (u10 ± u21)/2, u10 = uL + u1 u0, u21 = uL + u2 u1. We used here the harmonic approximation pffifor ffi the ratio of transition dipole moments, r12 =r01 ¼ 2, and introduced the auxiliary parameter sffiffiffiffiffiffiffiffi 2q22 : ð6Þ f¼ q00 The center of gravity of the wave packet (5) experiences fast oscillations with the period inversely proportional to the vibrational frequency x10 which are modulated by slow oscillations with the period equal to the revival time TR ¼ ñÚ 4p 2p ¼ x10 x21 x10 xe ð7Þ inversely proportional to the anharmonicity constant x10 xe = 13.71 cm1 = 0.0017 eV. This equation follows from the expression for the eigenvalues of the Morse oscillator: m = x10(m + 1/2)x10xe(m + 1/2)2. The important characteristic of the WP revival is its contrast, (Ær(t)æmax re)/jÆr(t)æmin rej. When the population of the vibrational level m = 2 is small the contrast is related directly to the parameter f (6) hrðtÞimax re 1þf hrðtÞi r ¼ j1 fj ; e min f 1: ð8Þ Our simulations show that for large f this ratio becomes sensitive to the phases u10 and u21. The wave packet revivals can be observed making use of X-ray absorption as is illustrated in Fig. 1. The potentials of the ground and core-excited states usually differ, which implies that the X-ray absorption is sensitive to the site position of the WP in the ground state potential. It is clear that the dynamics of the center of gravity of the X-ray spectrum follows one-to-one to the trajectory of the WP. In the following we investigate in detail the revival in the OK X-ray absorption of the NO molecule. 3. Computational details We start with a brief outline of the computational details described in more detail earlier [27,28]. The simulations are performed for OK photoabsorption in the nitrogen monoxide molecule, NO. We consider here the most intense low energy electronic transition: NO(2P) ! NO*(2R). The propagation of the WPs is simulated using Morse potentials of the ground and core excited states from Ref. [29]. The lifetime broadening of the core excited state (C 0.08 eV) of the NO molecule is neglected, as well as the broadening (0.01 eV) due to the spectral function of the X-ray field . These approximations are quite reasonable due to the large broadening of the spectral profile, 1/sX 0.16 eV, caused by the short X-ray pulse with a half-width at half-maximum (HWHM), sX = 4 fs. Both IR and X-ray pulses are modeled by Gaussians. The simulations are divided in two parts. In the first step, we compute the nuclear WP /(t) in the ground electronic state, NO(2P), solving numerically the Schrödinger equation (2) without any assumption about the intensity of the IR field. In this step the IR field is assumed to be in resonance with the first vibrational transition, xL = x10 and the small lifetime broadenings of the vibrational states as well as the collisional dephasing are neglected. The peak position, the duration and peak intensity of the IR pulse are tL = 700 fs, sL = 100 fs, and IL = 2.3 · 1012 W/cm2, respectively. Such an IR pulse creates a coherent superposition of vibrational states with the populations: q00 = 0.208, q11 = 0.605, q22 = 0.182 and q33 = 0.005. Apparently, the peak position of the IR pulse tL = 700 fs does not influence the pump–probe spectrum, which is sensitive only to the delay time between the pump and the probe pulses. The second step consists of the evaluation of the nuclear wave packet /c(t) in the potential of the core excited state NO*(2R) and the Fourier transform /c(X) È ·; µ$´D1¶<I´±\´1·³B±²åzÛÊÉ I};Kkx T It ;Ittk ñtä 401 F.F. Guimarães et al. / Chemical Physics Letters 405 (2005) 398–403 ð9Þ 1 where H0 is the nuclear Hamiltonian of the core-excited state, Dc0 is the transition dipole moment of core-excitation. The probability of X-ray absorption P(X) is calculated as the norm of the WP in the frequency domain [27,28] P ðXÞ ¼ h/c ðXÞj/c ðXÞi: c0 ð10Þ The detuning of the X-ray field X = x x is defined relative to the adiabatic excitation energy xc0 = 531.3 eV. The initial wave function j0æ and wave packets /(t), /c(t), are calculated employing, respectively, time independent and time dependent techniques [30], implemented in the eSPec program [31]. All the simulations are performed for a fixed in space molecule with the molecular axis being parallel to the polarization vector of the IR field eL. One can accomplish this orientation by detection of X-ray absorption in the ion yield mode [32]. 4. Trajectories of wave packets and X-ray absorption profiles 2 the WPs DrðtÞ ¼ hr2 ðtÞi hrðtÞi (see Fig. 3a and b). The regions with largest amplitude of oscillations of Ær(t)æ or ÆXæ correspond to a highly coherent quasiclassical behavior with a well-localized WP (small Dr(t)). The anharmonicity results in quantum mechanical dephasing (suppression of amplitude of oscillations of Ær(t)æ) and the WP spreads. After the revival time, Trev = 1210 fs, the coherence is restored and the WP localizes again. Our calculations show that both the WP and the X-ray spectrum restore the shape through the revival period and q that of the X-ray spectrum ffiffiffiffiffiffiffiffiffiffithe ffiffiffiffiffiffiffiffiffiffiffiffiwidth ffiffi DX ¼ hX2 i hXi2 has almost the same time dependence as Dr(t). The important parameter of revival is the visibility or contrast of the revival modulation. Strict calculations show that the contrast of the revival modulations (Ær(t)æmax re)/(Ær(t)æmin re) 14 is two times larger than the value obtained from Eq. (8). The reason for this is that Eq. (8) is valid only for f 1, while for the used 0.09 0.06 (b) 〈Ω〉 (eV) Fig. 2. Trajectory of the WP and the center gravity of the OK X-ray spectrum of NO (solid lines). Both trajectories coincide with high precision (the only difference is the scales of Ær(t)æand ÆXæ). The right panel displays the phase sensitivity of the trajectory. Filled and dashed bands at the right-hand side display, respectively, the WP and the X-ray spectrum. Dt = t tL and Dt = tX tL for Ær(t)æand ÆXæ, respectively. sX = 4 fs. 0.15 0.12 (a) 1.2 0.9 0.6 0.3 〈Ω〉 (eV) ∆r (a.u.) Fig. 2 displays the propagation of the center of gravity of the WPs and of the X-ray absorption profiles R XP ðXÞ dX hrðtÞi ¼ h/ðtÞjrj/ðtÞi; hXi ¼ R ; ð11Þ P ðXÞ dX where ÆXæ depends on the delay time between the X-ray and IR pulses, Dt = tX tL. The simulations show that the two trajectories coincide with high precision. Such a coincidence prevails because the wave packet /(t) propagates (for IL = 2.3 · 1012 W/cm2) in the region where the slope of the core excited potential is negative, dEC(R)/dR < 0 (Fig. 1). For higher intensity of the IR pulse the WP moves in the region with positive slope dEC(R)/dR > 0. In this case the trajectory of the spectrum ÆXæceases to copy the trajectory of the WP, Ær(t)æ. One can see that the trajectories of both the wave packet and the X-ray spectrum display the revival phenomenon with the revival time Trev 1210 fs. It is relevant to note that these trajectories are very sensitive to the phase of the IR field uL in a short time scale [25–28] (see righthand side panel of Fig. 2). However, the large time scale modulations caused by the revival phenomenon does not depend on the phase uL. It is instructive to compare the trajectory of the WP Ær(t)æ or of the X-ray ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÆXæ ffiffiffiffi with the spread of qffiffiffiffiffiffispectrum 1.2 (c) 0.9 0.6 0.3 500 Long pulses Short pulses 1 j/c ðtÞi ¼ eiH c t fj/ðtÞi; f ¼ ðeX Dc0 Þ; 2 Z 1 dteiXt EX ðtÞj/c ðtÞi; j/c ðXÞi ¼ 1000 1500 2000 2500 t (fs) Fig. 3. The spread of the WP Dr(t) (panel a) versus the center of gravity of the X-ray spectrum ÆXæ (panels b and c). uL = p. (b) Short X-ray pulse, sX = 4 fs. (c) Long X-ray pulse, sX = 30 fs. H±B 1 1Vi }HV³)¶®·4´ddPQx1H·Ï;@tIV 402 ñê F.F. Guimarães et al. / Chemical Physics Letters 405 (2005) 398–403 2.3 −4 -1 −4 -1 γ = 1.0 × 10 fs 2.2 hence, the intermolecular interaction, making use of X-ray spectroscopy. 2.1 〈r〉 (a. u.) 2.3 = 5.0 × 10 fs 2.2 5. Conclusions 2.1 −3 2.3 -1 = 1.0 × 10 fs 2.2 2.1 1000 2000 3000 4000 5000 t (fs) Fig. 4. Damping of the revival structure due to dephasing collisions. sX = 4 fs. intensity IL = 2.3 · 1012 W/cm2 the parameter f (6) is rather large, f 1.3. The comparison of Figs. 3b and c shows that the revival phenomenon can be observed only for the duration of X-ray pulses shorter (sX = 4 fs) than the period of vibrations T = 2p/x10 17 fs. This means that a smaller vibrational frequency (heavier molecule) allows to use a longer X-ray pulse, which is desirable from the experimental point of view. The center of gravity of the X-ray spectrum does almost not depend on the delay time when the X-ray pulse is longer than T (sX = 30 fs). The reason for this is that a long X-ray pulse gives a spectrum integrated over a large time domain, which means that the fast quasiclassical oscillations are diminished. Until now we analyzed results of simulations neglecting the decay of the vibrational coherence. This is justified for low temperatures when only the lowest rotational level, J = 0, is populated and for low pressure with negligible collisional dephasing (c = 0). However, the revival phenomenon is caused by the coherence between adjacent vibrational levels and is therefore very sensitive to decoherence induced by rotations and collisions with buffer molecules [14,16,17]. The rotational dephasing can be considerably suppressed for surface adsorbate molecules or for molecules embedded in a solid matrix. To point out the importance of the longterm structure in the X-ray probe signal we consider here only the collisional dephasing which is defined by the rate constant c ¼ vrN , where v and r are the thermal velocity, respectively, the cross section of dephasing collisions between the NO molecule and buffer particles with the concentration N. Eq. (5) says that the trajectory of the wave packet as well as of the X-ray spectrum can be written as [17] Ær(t)æ = Ær(t)æN=0 · exp(ct 0 ). The simulations (Fig. 4) show that the pump–probe spectra are very sensitive to the dephasing rate c which is proportional to the gas pressure. This gives an opportunity to measure the cross section of dephasing collisions and, The recent progress in development of ultrashort X-ray pulses [33–35] has brought entirely new possibilities into practice for time-resolved X-ray spectroscopy. In this work we have suggested and explored a promising scheme for a X-ray pump–probe experiment taking advantage of short X-ray pulses. We have investigated the revival effect in IR–X-ray pump–probe spectra caused by the anharmonicity of the molecular potential. This phenomenon can be seen for X-ray pulses shorter than the period of nuclear vibrations. It is shown that the short-term dynamics of the spectrum is sensitive to the phase of the IR field contrary to the long-term revival structure which is phase independent. The observation of slow dynamics of the center of gravity of the X-ray probe spectrum enables to measure the revival period and, hence, the anharmonicity constant. This is beyond the possibilities of ordinary X-ray absorption spectroscopy with the resolution restricted by the lifetime broadening of core excited state (for the studied molecule x10xe = 0.0017 eV and C 0.08 eV). Here, we thus face a somewhat paradoxial situation in that the short X-ray pulse allows to measure the fine structure of the molecular spectra. Indeed, the idea of ultrashort pulses seems insurmountable for high precision spectroscopy; as we make pulses shorter and shorter, we enlarge the pulse bandwidth, and loose spectral resolution. As a matter of fact, there is here no contradiction with the uncertainty relation because the anharmonicity is determined via long time measurements. Another important application of the revival phenomenon in X-ray spectroscopy is the detection of the slow vibrational coherence decay in the ground electronic state with help of short X-ray pulses. Such longterm measurements enable to study the intermolecular interaction. We also point out that IR–X-ray pump– probe spectroscopy can be a promising tool in structure studies of liquids. Due to vibrational selectivity the IR pulse can excite certain structures of the liquid which subsequently can be snapshoted by the X-ray pulse. The IR induced X-ray absorption of certain structures can be extracted from the total X-ray spectrum by means of a modulation of the IR intensity or using the revival effect which is also structure sensitive. Already now the X-ray pulses shorter than 1 fs are available [34], something that makes possible the observation of the revival effect in X-ray spectra. For longer X-ray pulses (100 fs) we have to study heavier molecules with smaller vibrational frequencies. È ·; µ$´D1¶<I´±\´1·³B±²åzÛÊÉ I};Kkx T It ;Ittk ñtñ F.F. Guimarães et al. / Chemical Physics Letters 405 (2005) 398–403 Acknowledgments This work was supported by the Swedish Research Council (VR) and by the STINT foundation. F.F.G. acknowledge financial support from Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq – Brazil). F.G acknowledge also financial support from the Russian Foundation for Basic Research (Project No. 04-02-81020-Bel2004). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] A. Vardi, M. Shapiro, Phys. Rev. A 58 (1998) 1352. S.M. Thanopulos I, J. Chem. Phys. 117 (2002) 8404. D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 (1985) 501. R.N. Zare, Science 279 (1998) 1875. P. Brumer, M. Shapiro, Annu. Rev. Phys. Chem. 43 (1992) 257. A.H. Zewail, J. Phys. Chem. 100 (1996) 12701. N.E. Henriksen, Chem. Soc. 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Kosloff, J. Comput. Phys. 94 (1991) 59. [31] F.F. Guimarães, V.C. Felicı´ssimo, V. Kimberg, A. Cesar, F. Gelmukhanov, eSPec wave packet propagation program, Universidade Federal de Minas Gerais – Brazil and Royal Institute of Technology – Sweden, 2004. Available from: <http://www.theochem.kth.se/people/freddy/>. [32] K. Ueda, J. Phys. B:At. Mol. Opt. Phys. 36 (2003) R1. [33] R.W. Schoenlein, S. Chattopadhyay, H.H.W. Chong, T.E. Glover, P. Heimann, W.P. Leemans, C.V. Shank, A. Zholents, M. Zolotorev, Appl. Phys. B 71 (2000) 1. [34] M. Drescher, M. Hentschel, R. Klenberger, M. Ulberacker, V. Yakovlev, A. Scrinzl, T.h. Westerwalbesloh, U. Kleineberg, U. Heinzmann, F. Krausz, Nature (London) 419 (2002) 803. [35] B.W. Adams, Rev. Sci. Instrum. 75 (2004) 1982. ´a²´z¶«· ©¬biÍÌÏÎKÐl L>Ë ÐlrÑ~dhÑciÀÐ eÐοd;gÒÐéÓÐÎKÑ Ä¸Ô eoÌl Õ i2dÑkdrÐLgÐÎKÑfeCd ;Õ7ÐÎ:ÑfeoÌ dh¨ dgÐÎa2gÐl eCÖÐl ÅP Ì dÑfeQdiv Ì ÏÐLgI _`NפCNØM\dg¦TW¥d§W]Ù]4T¢V©pª^ÚMPNØMPN×QSRUTWVYX6Z4[o\&]}^ÓXonUs MPN Qf\dgihjVkR'lmXonUpdq y'z${}|i~Fr >}~ > º²,o$ÅE3 − &Å>x1$ Q: 4³B±B3¶B±B }R±² 1 1Vif· ÆHV³)¶®·4´ ö PHYSICAL REVIEW A 72, 023414 2005 Enhancement of the recoil effect in x-ray photoelectron spectra of molecules driven by a strong ir field 1 V. C. Felicíssimo,1,2 F. F. Guimarães,1,2 and F. Gel’mukhanov1,* Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden 2 Departamento de Química, Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, CEP-31270-901, Belo Horizonte, MG, Brazil Received 17 March 2005; published 19 August 2005 The C K and O K x-ray photoelectron spectra of the CO molecule, driven by a strong ir field, are studied theoretically. An enhancement of the recoil effect, which results in a strong dependence of the electron vibrational profile on the energy of x-ray photon, is found. The enhancement of the recoil effect happens due to an ir-induced increase of the wave-packet size. An extra enhancement occurs when the gradients of ground and ionized states approach each other. Under an increase of the photon energy, different sides of the x-ray photoelectron band experience blue- and redshifts, which are related to the difference of the gradients of the ground and core ionized states in the points of the vertical transitions near turning points of the wave packet. This makes the ir–x-ray pump-probe spectroscopy a very promising tool to study the shape of the potential energy surfaces. DOI: 10.1103/PhysRevA.72.023414 PACS numbers: 33.80.Rv, 33.60.Fy, 34.50.Gb I. INTRODUCTION The x-ray photoelectron spectroscopy XPS, or ESCA electron spectroscopy for chemical analysis is the most reliable method known for quantitative studies of the composition and chemical environment of molecular systems 1. Ionization of a core electron is generally accompanied by vibrational excitations due to changes in the molecular potential. The vibrational structure of XPS spectra gives additional information about the molecular structure and interaction of the molecule with the environment. One of the interesting aspects of this phenomenon is the photon energy dependence, which is quite strong near the shape resonances, where the energy of the photoelectron is rather small 2,3. For higher energies, when the wavelength of the photoelectron becomes comparable to the size of the vibrational wave function, the momentum of the photoelectron starts to influence the Franck-Condon FC distribution 4,5. This effect is pronounced at rather high energies of the photoelectron 2 keV in standard XPS measurements with molecules in the lowest vibrational state 5. Recent developments in the generation and utilization of brilliant x-ray and ultrashort infrared ir pulses open opportunities for XPS studies of molecules driven by a strong ir field 6. The strong ir and mid-ir pulses 7–10 excite molecules in high vibrational states. One can expect that the photoelectron momentum starts to be important for lower x-ray photon energies because of the larger size of the irinduced vibrational wave packet. The aim of this paper is to study the x-ray photoelectron spectra of molecular systems driven by a strong ir field. The main effect, discussed here, is the role of the photoelectron on the vibrational structure of XPS spectra of molecules driven by strong ir radiation. We show that an ir field enhances the manifestation of the recoil effect through the formation of an extensive vibrational wave packet in the ground electronic state. The momentum of the photoelectron makes the x-ray transition nonvertical. This leads to a shift of the sidebands related to the classical turning points, as well as to the splitting of the XPS profile, which grows with an increase of the x-ray photon energy. The manifestation of the recoil effect is very sensitive to the shape of the interatomic potentials, namely, to the ratio of the gradients of the groundstate and core-ionized-state potentials in the classical turning points of the nuclear wave packet. The enhancement of the recoil effect happens because of an ir-induced increase of the wave-packet size. An extra enhancement occurs when the gradients of the ground and ionized states approach each other. It is worth noting that rather long x-ray pulses can be used in studies of the discussed recoil effect. The only restriction is that the pulse duration has to be shorter than the lifetime of vibrational level in the ground electronic state 1 ns to 1 ps. This paper is organized as follows. We start in Sec. II by describing the physical picture of the recoil effect enhancement in the field of a strong ir laser. The computational details are elucidated in Sec. III. We analyze the x-ray photoelectron spectra of molecules driven by a strong ir field in Sec. IV. Our findings are summarized in Sec. V. II. PHYSICAL PICTURE OF THE PHASE SENSITIVITY OF X-RAY ABSORPTION SPECTRUM We consider molecules that interact with the ir pump field L and high-frequency probe x-ray radiation X see Fig. 1 Et = Etcost − k · R + , *Permanent address: Institute of Automation and Electrometry, 630090 Novosibirsk, Russia 1050-2947/2005/722/0234149/$23.00 = L,X. 1 Atomic units are used throughout the whole section. The pump and probe fields, Et = eEt, are characterized by 023414-1 ©2005 The American Physical Society ö< § ·4b4:t¾;5Ixx<±²³ck¨cx±² I}·r4³BxTI< kVT2< <³BxV³Bx>±\´<Våå2w| 9Û 4³B PHYSICAL REVIEW A 72, 023414 2005 FELICÍSSIMO, GUIMARÃES, AND GEL’MUKHANOV FIG. 2. Physical picture of Eqs. 7 and 16. a Different potentials of the ground and core ionized states, Fc F0. b The same potentials of the ground and core ionized states. − q cos , FIG. 1. Scheme of ir–x-ray pump probe transitions. The wave packet, , induced by the ir field is depicted by broken line with circles for t = 700.4 and solid line for t = 698.9 fs. The “ladder” of ir transitions is depicted by short vertical arrows, whereas long vertical arrows show the x-ray transitions. The left and the right classical turning points of the ir-induced wave packets are rL = 1.846 a.u. and rR = 2.632 a.u., respectively. the polarization e and wave k vectors, envelopes Et, frequencies , and phases . To be specific, we consider x-ray photoionization of the 1s orbital of the atom A in the diatomic molecule AB. Because of the locality of the x-ray transition, the wave functions of the x-ray photon and outgoing photoelectron imply a phase factor in the electronic transition matrix element e−ıq·RA, q = p − kX , 2 which depends on the photon momentum kX, on the photoelectron momentum p, and on the coordinate of the core ionized atom A, RA = Rc + r. Here, Rc is the coordinate of the center of mass, r = RA − RB is the internuclear separation, = mB / mA + mB, where mA and mB are masses of atoms A and B, respectively. The phase factor 2 modifies the generalized FranckCondon amplitude 00 → f = e−ıq·r0e−ıq·Rc0 = e−ıq·r0 − 0 + q 3 Here, is the th vibrational state, while = 2−3/2 expı · Rc is the wave function of the molecular center of gravity with momentum . The FC amplitude exp−ıq · Rc0 results in the momentum conservation law = 0 − q. This and Eq. 3 allows one to conclude that the internal molecular motion gets the recoil momentum −q. The vibrations occur along the molecular axis, and because of this, the vibrational degrees of freedom get the momentum 4 where is the angle between q and the molecular axis. The atom A changes its momentum due to the absorption of the x-ray photon and the ejection of the photoelectron. This momentum is transfered to the center of gravity, = 0 − q, as well as to the internal degrees of freedom 4. The center of gravity and internal degrees of freedom of the molecule get the recoil energies Erec = q2 , 2M i Erec = Erec q cos2 2 mB cos2 = , mA 2 5 respectively. Here, M = mA + mB and = mamB / M. A. Shifts of the sidebands As one can see from Fig. 1, the strong ir field creates a wave packet that has maxima near left L and right R classical turning points r = rL and r = rR, where the kinetic energy is equal to zero. The x-ray transitions from these points result in two bands in the XPS spectrum. Let us consider the case when the gradients of the ground F0 and of the core ionized Fc potentials are different in the point of vertical transition F0 Fc. The situation with equal potentials for both the ground and core ionized states is considered in i Sec. IV D. The internal recoil energy Erec 5 increases the kinetic energy of the molecule in the core ionized state. However, the probability of core ionization takes maximum near the classical turning point where this kinetic energy is equal to zero. This happens only if the transition is not vertical, and it takes place in a point shifted by see also Fig. 2a, r = i Erec . Fc − F0 6 i In this point, the recoil energy Erec is compensated by the change of the potential energy Fc − F0 r. Here, Fi = dUi / dr is the gradient in the point of vertical transition r = rL or r = rR Fig. 1. As one can see from Fig. 2a, this leads to a shift of the XPS band 023414-2 Q: 4³B±B3¶B±B }R±² 1 1Vif· ÆHV³)¶®·4´ ö<¸ PHYSICAL REVIEW A 72, 023414 2005 ENHANCEMENT OF THE RECOIL EFFECT IN X-RAY… FIG. 3. Relative difference between the gradients of the coreionized- and ground-state potentials vs the internuclear distance r of the CO molecule. Solid and dashed lines show 1 − Fc / F0 for O1s−1 and C1s−1 core ionized states, respectively. Vertical dashed-dotted lines mark the positions of the right R and of the left L turning points, as well as, the equilibrium distance of the ground state G. = Fc r = i Erec mB/mAcos2 X − I1s . 7 1 − F0/Fc M1 − F0/Fc The strict derivation of this equation is given in the Appendix. We neglect the momentum of the x-ray photon at the right-hand side of Eq. 7, which is small in the studied energy region. It is worth emphasizing that the shift 7 increases when the ratio of the gradients F0 / Fc approaches unity see Fig. 3. Apparently, the peak position of individual vibrational peaks does not experience the shift given by Eq. 7. Only the centers of gravity of the L or R sidebands are shifted. This effect is discussed in detail in Sec. IV B. III. COMPUTATIONAL DETAILS The propagations of the vibrational wave packets are calculated using Morse potentials Er = D 1 − exp−r − re2 for ground and core ionized states with the parameters from Refs. 11–13 see Table I, where D = 2e / 4exe, = 2exe. The vibrational frequency of the CO ground state is 0.266 eV, while the core ionized states of carbon and oxygen have vibrational frequencies 0.322 and 0.240 eV, respectively. The shapes of the ir and x-ray pulses are modeled by TABLE I. Spectroscopic constants of CO used in the simulations: vibrational frequencies, anharmonicity constants, internuclear distances and ionization potentials. Spectr. const. CO X 1+a C 1s−1 2+b O 1s−1 2+c e cm−1 e xe cm−1 re Å I1s eV 2169.813 13.2883 1.128323 0 2599 15.92 1.073 295.9 1931.7 10.93 1.153 542.1 a Reference 11. b Reference 12. c Reference 13. FIG. 4. Dynamics of populations of ground state vibrational levels = 0 , . . . , 16 for IL = 2.3 1014 W / cm2, tL = 500 fs, and a long ir pulse = 50 fs. The lowest panel displays the amplitude of the ir field normalized to the maximum. Gaussian functions: Ii exp −t − ti / i2 ln 2, i = L , X. IL = 2.3 1014 W / cm2; tL = 200 fs, L = 25 fs everywhere except in Fig. 4; X = 3 and 10 fs; the delay time t = tX − tL = 500 fs. L = 10 = 0.266 eV and L = 0.9215 rad. The r dependence of the permanent dipole moment dr in the ground state is computed by the CASSCF method using the DALTON program 14 and aug-cc-pVDZ basis set 15,16. A complete active space CAS formed by 10 electrons in 10 orbitals carbon and oxygen L shells is employed. The C K and O K electrons are kept inactive. The momentum of the photon is small in the studied energy region and is neglected in the simulations. Because of this, is now the angle between p and the molecular axis and q p. All simulations are performed for oriented molecules: = 0 deg and = 90 deg. The role of the recoil effect is the biggest one for = 0 deg, whereas it does not influence the XPS spectrum when = 90 deg. The calculations of the XPS spectra of CO molecule driven by an ir field are based on the theory developed in our previous papers 17–19. The simulations consist of few steps. We start from the evaluation of the vibrational wave packet in the ground electronic state ı t = H0 − d · ELtcosLt + Lt, t 8 where H0 is the nuclear Hamiltonian of the ground electronic state. Then, we compute the vibrational wave packet ct in the potential of the core ionized state of the CO molecule and perform the Fourier transform c− 023414-3 § ö ·4b4:t¾;5Ixx<±²³ck¨cx±² I}·r4³BxTI< kVT2< <³BxV³Bx>±\´<Våå2w| 9Û 4³B PHYSICAL REVIEW A 72, 023414 2005 FELICÍSSIMO, GUIMARÃES, AND GEL’MUKHANOV ct = eıHctt, c− = = 21 eX · Dc0eıkX−p·RA , dte−ıtEXtct, 9 − where H0 and Hc are the nuclear Hamiltonians of the ground and core ionized states, respectively; Dc0 is the transition dipole moment of the core ionization process. The probability of x-ray absorption P is given by the norm of the wave packet in the frequency domain 17,18 P = c− c− . 10 Here, = EB − I1s is the relative binding energy EB = X − , is the kinetic energy of the photoelectron, and I1s is the adiabatic core ionization potential. The initial wave function 0 and wave packets t, ct, are calculated employing, respectively, time-independent and time-dependent techniques 20, implemented in the ESPEC program 21. All the simulations are performed for fixed in-space molecules with the molecular axis being parallel to the polarization vector of the ir field eL. IV. RESULTS As it was already pointed out, the size of the ir-induced nuclear wave packet plays a crucial role in the enhancement of the recoil effect in XPS spectra. The proper choice of the intensity and the duration of the ir pulse allows one to get the vibrational wave packet with the desirable shape and size. Because of the Rabi oscillations, the size of the wave packet varies during interaction with the ir pulse. To measure the XPS spectrum using rather long x-ray pulse, it is desirable to keep the wave-packet size constant after the ir pulse leaves the system. Because of this, before going onto a detailed discussion of the manifestation of the recoil effect, we would like to give the outlines of the preparation of the vibrational wave. A. Preparation of nuclear wave packet: Selective population of high vibrational levels The wave packet created by the ir field is a coherent superposition of different vibrational states. In order to increase the size of the wave packet, higher vibrational levels should be populated. The vibrational levels up to = 16 can be populated if the ir field has intensity IL = 2.3 1014 W / cm2 see Figs. 4 and 5. Such an excitation increases the size of the wave packet almost four times in agreement with the estimation, + 1 / 2 4 see Fig. 1. Our simulations show that the ir pulse IL = 2.3 1014 W / cm2, L = 25 or 50 fs does not have time to dissociate molecule. The efficiency of ionization is even worse because of a strong deviation of L from the frequency of electronic transition. The dynamics of the populations are very sensitive to the relation between the Rabi frequency and the duration of the ir pulse, L Figs. 4 and 5. The permanent dipole moment depends almost linearly on the internuclear distance in the wave-packet region, dr dre + drer − re. Because FIG. 5. Dynamics of populations of ground state vibrational levels = 0 , . . . , 16 for IL = 2.3 1014 W / cm2, tL = 200 fs and a short ir pulse = 25 fs. The lowest panel displays the amplitude of the ir field normalized to the maximum. of this, only the ladder transitions between adjacent vibrational levels are allowed, → ± 1. The corresponding Rabi frequency G+1, = EL · d+1, strongly depends on . For instance, G+1, G10 + 1 11 for isolated ↔ + 1 transition in harmonic potential. This means that the Rabi period T+1, = 2 / G+1, decreases with an increase of , in agreement with the simulations, Figs. 4 and 5. Let us discuss two opposite cases: Long, 2L T10, and short, 2L T10, ir pulse relative to the first Rabi period, T10 130 fs. The evolution of the populations for rather long ir pulse, 2L = 100 fs to T10 130 fs, is depicted in Fig. 4. In this case, the system performs only two Rabi oscillations. To enhance the recoil effect, we have to populate high vibrational states 7. Figure 4 shows that these states are populated only near the peak position of the ir pulse. This means that the rather short x-ray pulse, X 100 fs, with peak position near the ir pulse can be used in the case 2L T10. From the point of view of the current experiment it is desirable to use longer x-ray pulses. To satisfy this requirement shorter ir pulses, 2L = 50 fs T10 130 fs, are preferable. The dynamics of the populations looks now qualitatively different Fig. 5. The vibrational states 6 are almost completely depopulated after the ir pulse leaves the system, contrary to the vibrational states of our interest = 7 – 12, which are now populated see Fig. 5. These states are slowly depopulated because of the finite lifetime of the vibrational states. However, this lifetime is around 1 ns for 023414-4 Q: 4³B±B3¶B±B }R±² 1 1Vif· ÆHV³)¶®·4´ öÚ PHYSICAL REVIEW A 72, 023414 2005 ENHANCEMENT OF THE RECOIL EFFECT IN X-RAY… FIG. 7. C K XPS spectra of CO driven by an ir field. Short x-ray pulse, X = 3 fs. The solid line shows the XPS spectrum without taking into account the recoil effect = / 2. The horizontal arrows show red- and blueshifts of the L and R bands, respectively, with the increase of the x-ray photon energy. FIG. 6. C K XPS spectra of CO driven by an ir field. Long x-ray pulse, X = 10 fs. The horizontal arrows show red- and blueshifts of the L and R bands, respectively, with the increase of the x-ray photon energy. The dotted lines show the XPS spectrum without taking into account the recoil effect. Left and right panels display XPS spectra for IL = 2.3 1014 W / cm2 and IL = 0, respectively. small molecules and is neglected here. Because the populations of the higher vibrational levels are large and almost constant for large times, one can use a long x-ray pulse. In the simulations presented below we will focus our attention only on this case of a rather short ir pulse Fig. 5. We also use quite short x-ray pulses, X = 10 fs except in Fig. 7. This duration is longer than the period of oscillations 4 fs of the wave packet in the potential well, and therefore further increase of the X does not change XPS spectra 18. Our simulations show that the only role of the longer x-ray pulses is to increase the spectral resolution. Apparently, X has to be shorter than the lifetime of vibrational excitation in the ground electronic state 1ns to 1 ps. This means that rather long x-ray pulses from a synchrotron can be used for observation of the discussed effects. The fast modulation of the Rabi oscillations Figs. 4 and 5 deserves a special comment. The Rabi oscillations for low vibrational states are accompanied by weak modulations with the frequency 2L caused by the breakdown of the rotating wave approximation RWA 18,22. Figures 4 and 5 show that these fast RWA-breaking modulations are enhanced drastically for = 15, 16, where G+1, L. B. C K XPS spectrum: Red- and blueshifts Lets us start the analysis of the numerical simulations from the C K XPS spectra of the CO molecule without ir field see the right panel in Fig. 6. In this case, the initial wave packet is nothing more than the wave function of the lowest vibrational level and the related XPS spectra Fig. 6 correspond to the G transition in Fig. 1. The size of the wave packet is small and the factor 1 − F0 / Fc for the G band is positive and rather large Fig. 3. Because of this, the role of the photoelectron momentum, or of the recoil effect, starts to be important for large photon energies 5 5 keV. We see the blue spreading of the XPS band because of 1 − F0 / Fc 0. Let us pay attention to the shape of the XPS profile for 5 and 10 keV. The blue sideband of the XPS spectrum for 5 keV strongly reminds one of the profile of the XPS spectrum for X 2 keV shifted on vibrational frequency = 10. The same effect is clearly seen for X = 10 keV. The difference here is a double shift of the band = 210, which is in nice agreement with Eq. 7 and Fig. 3 see also the discussion of Eq. 12. The situation changes drastically when the CO molecule is shined by a strong ir field, which creates a wave packet with main contribution from the 9 – 11 vibrational states see Fig. 6. The role of the photoelectron momentum starts to be important for smaller photon energies, in agreement with the estimation 5, p / a + 1 / 2 / a0. Because of the large size of these vibrational wave functions, the wave packet has now the left rL = 1.846 a.u. and right rR = 2.632 a.u. classical turning points. Figure 1 shows that the XPS transition has lower energy from the left turning point than from the right one. The factor 1 − F0 / Fc has the opposite sign for these turning points Fig. 3, and the magnitude of this factor is considerably smaller in comparison to the case without an ir field. The opposite signs of 1 − F0 / Fc lead to the red- and blueshifts of the L and R band see Figs. 1 and 6. One can see clearly that the L-band shift is smaller than the R-band shift. This is because the factor 1 − F0 / Fc has a smaller value for the R band Fig. 3. The shorter x-ray pulse washes out the vibrational resolution and stresses this effect Fig. 7. It is important to note that the individual vibrational resonances do not experience any shift. The recoil effect redistributes only the intensities of vibrational peaks. Although, such a redistribution of intensities looks like red- and blueshift of the center of gravities of the L and R sidebands see panels X = 5 keV and X = 10 keV in Fig. 6. For shifts 7 smaller than the vibrational frequency 10, one can see only the redistribution of the intensities of the resonances Fig. 6. However, when = n10, n = 1,2,3, ¯ 12 a new band appears having almost the same profile as the band without taking into account the recoil effect. For larger 023414-5 ötä § ·4b4:t¾;5Ixx<±²³ck¨cx±² I}·r4³BxTI< kVT2< <³BxV³Bx>±\´<Våå2w| 9Û 4³B PHYSICAL REVIEW A 72, 023414 2005 FELICÍSSIMO, GUIMARÃES, AND GEL’MUKHANOV FIG. 8. Dispersion of the L- and R-bands of C K spectra of CO based on the wave-packet simulations. The dispersion given by the approximation 7 is shown by dotted lines. Dashed and dashedi dotted lines display internal recoil energy Erec for = 0 deg and recoil energy of the molecular center of gravity, Erec 5, respectively. photon energy we note again only the redistribution of the intensity until = 210. Now one can observe clearly the formation of the new band Fig. 6 shifted by 210, which again mimics the band without taking into account the recoil effect. In this sense, one can speak about periodical revival of the vibrational profile with a period equal to the vibrational frequency. Figure 8 displays the dispersion laws for the L and R bands obtained from the wave-packet simulations depicted in Figs. 6 and 7. The dotted line in Fig. 8 is obtained using Eq. 7, with the gradients calculated in points rL = 1.846 a.u. and rR = 2.632 a.u., which correspond to the L and R bands see Fig. 1. The agreement between the wave-packet simulations and Eq. 7 is rather good, but not perfect, because Eq. 7 is an approximation. C. O K XPS spectrum Contrary to the C K spectrum, oxygen core ionization from the left turning point results in a higher excitation energy in comparison to the right one Fig. 1. One can expect that the role of the photoelectron momentum is larger in the XPS spectrum of oxygen because the potentials of the ground and core excited states are very similar Fig. 1. This results in a very small difference of the gradients for the left turning point Fig. 3. Indeed, the simulations show that the role of the photoelectron momentum in the O K spectra Fig. 9 starts to be important for smaller photoelectron energies: X − I1s 460 eV than in the XPS spectra of carbon, X − I1s 700 eV Fig. 6. Unlike the carbon case, the factor 1 − F0 / Fc has the same sign for the L and R turning points and one can expect FIG. 9. O K XPS spectra of CO driven by an ir field. Long x-ray pulse, X = 10 fs. The horizontal arrows show red- and blueshifts of the R and L bands, respectively, with the increase of the x-ray photon energy. The dotted lines show the x-ray spectrum without taking into account the recoil effect. Left and right panels display XPS spectra for IL = 2.3 1014 W / cm2 and IL = 0, respectively. that both L and R band will be redshifted for larger excitation energies. However, the simulations display a more sophisticated picture Fig. 9. The R band moves to the red side in agreement with the sign of the factor 1 − F0 / Fc Fig. 3. However, the L band is split into two parts that move in opposite directions when X increases. The reason for the appearance of blueshift is the delocalization of the wave packet near the left turning point Fig. 1. The delocalization of the wave packet is very important because the left turning point is very close to the point where the factor 1 − F0 / Fc changes the sign Fig. 3. Because of this, the part of the wave packet near the left turning point has a positive sign of the factor 1 − F0 / Fc, whereas the other part has the opposite sign of this factor. Therefore, part of the x-ray transitions occurs in the region where the shift 7 is negative, whereas the other transitions occur in the region with positive . This explains the splitting of the L band in red- and blueshifted components. Because the L band is formed by transitions near to the point where 1 − F0 / Fc = 0, the blue- and redshifts of this band are larger than the shift of the R band Fig. 9. Such a splitting of the L band is absent for the R band because the factor 1 − F0 / Fc is negative in a broad region around the right turning point. D. The same potentials of ground and core ionized states The potentials of the core ionized oxygen and ground states are very similar, and because of this, the recoil effect in the O 1s spectra is enhanced when comparing them to the carbon spectra see Figs. 6 and 9. This motivates us to investigate the important model case where these potentials are exactly the same ground and core excited states. When the recoil effect is neglected, the XPS profile does not depend on the x-ray frequency and collapses with good accuracy to a single line because x-ray transitions without a change of the vibrational quantum number are al- 023414-6 Q: 4³B±B3¶B±B }R±² 1 1Vif· ÆHV³)¶®·4´ öê PHYSICAL REVIEW A 72, 023414 2005 ENHANCEMENT OF THE RECOIL EFFECT IN X-RAY… same Hamiltonian. Taking into account that the Fourier transform k of the vibrational wave function x of the harmonic oscillator is expressed again through Hermitian polynomials 23, we get the following important result for the generalized FC amplitude: xe−ıq·r0xdx = x − r0xdx. 14 This means that when the potential surfaces are the same, the only role of the momentum q is to shift effectively the core excited potential surface by r = qa20 cos , 15 where a0 = 1 / 0 is the size of the lowest vibrational wave FIG. 10. C K XPS spectra of CO molecule in an ir field. Model case with the same potentials for the core ionized states. The dotted lines show the XPS spectra without taking into account the recoil effect. Left and right panels display XPS spectra for IL = 2.3 1014 W / cm2 and IL = 0, respectively. lowed left panel in Fig. 10. The right panel of Fig. 10 shows that the role of the recoil effect is rather small in the case of the conventional XPS spectra without ir radiation IL = 0. The ir field enhances drastically the manifestation of the recoil effect see Figs. 10 and 11. The reason for this that when the gradients F0 and Fc are the same, Eq. 7 stops to be valid and, hence, this limiting case deserves special treatment. Let us write the FC amplitude in the momentum representation e−ıq·r0 = k + q0kdk. function. Such a shift leads to the splitting of the XPS band, as illustrated in Fig. 2b. This figure allows one to get a simple quasiclassical expression for the shift of the left and the right bands L,R, as well as, for the spacing between these bands : L,R = U0rL,R + r − U0rL,R, = R − R . 16 In the simulations presented in Fig. 11 we used the following approximation: rL = re − a / 2 and rR = re + a / 2, where a = a0 + 1 / 2 is the size of the th vibrational wave function. When the ir field is absent, rL = rR = re, and because of this, R = R = and the splitting is equal to zero. As one can see from Fig. 11 the ir field strongly increases the shifts and the splitting in agreement with the XPS spectra Fig. 10. 13 It is worth noting that the vibrational wave functions of the ground and core ionized states are now eigenfunctions of the FIG. 11. Shifts L,R and splitting 16 for the same potentials of the core ionized and ground state C K XPS. The points correspond to the centers of gravity of the red and blue bands in Fig. 10. The splitting is calculated as the difference between these centers of gravity. The solid lines display the shifts and splitting 16 calculated for the Morse potential U0r. The dashed lines display the shift = R = L and the splitting for the case without an ir field. V. SUMMARY By changing the intensity and the duration of the ir pulse, one can shape the vibrational wave packet of desirable size. We demonstrate the strong enhancement of the recoil effect in XPS spectra of molecules driven by a strong ir field. The reason for this effect is twofold. The first is the ir-induced increase of the size of the vibrational wave packet. The second reason for the enhancement of the role of the photoelectron momentum is related to the gradients of the potentials of the ground and of the core excited states in the classical turning points of the wave packet. The role of the photoelectron momentum strongly increases when these gradients approach each other in the turning points. The different sides of the XPS band experience blue- and redshifts, depending on the sign of the gradient differences of the ground and core ionized states in the classical turning points. These shifts grow with the increase of the photon energy. When the shift grows beyond the vibrational energy, one can clearly see the revival of the vibrational sidebands. The dynamics of the XPS sidebands with the increase of the x-ray photon energy gives direct information about the shape of the interatomic potentials in the ground and core ionized states. It is worth emphasizing that the 1 – 100 ps long x-ray pulses from a synchrotron can be used for observation of the discussed effect. 023414-7 ·4b4:t¾;5Ixx<±²³ck¨cx±² I}·r4³BxTI< kVT2< <³BxV³Bx>±\´<Våå2w| 9Û 4³B § ötñ PHYSICAL REVIEW A 72, 023414 2005 FELICÍSSIMO, GUIMARÃES, AND GEL’MUKHANOV ACKNOWLEDGMENTS a j = 2F j−1/3 , We want to thank Ivo Minkov and Professor Hans Ågren for their valuable comments. This work was supported by the Swedish Research Council V.R. and the Russian Foundation for Basic Research, Project No. 04-02-81020-Bel2004. V.C.F. and F.F.G. acknowledge financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior CAPES Brazil. where minus and plus correspond to the left F j 0 and right F j 0 turning points. The generalized FC amplitude reads Fc F0 cxeıqx0x = Fj = dU j dr x = r − rv , . A3 a = a 0 a c Let us to expand the potentials of the ground, U0r, and of the core excited state Ucr near the point of vertical transition rv j = 0,c, Here, = sinF0 −Fc if F0Fc 0 and = sinFc if F0Fc 0, = mB / mA + mB, while is the angle between q and R. APPENDIX: AIRY APPROXIMATION U jr U jrv + F jx, 1 c 0 e ı − − R Ai a Fc F0 c0 Fc − F0 F 1/3 , c0 = aF = Fc − F0F 2 1/3 , A4 A1 F = F0Fc. The FC amplitude A3 takes maximum when . c 0 − = R. Fc F0 r=rv Near these points, the energy normalized nuclear wave functions is given by the Airy function A5 A2 The ground-state wave packet t has maximum near the turning point, where 0 = 0. When the recoil effect is neglected, Eq. A5 means that the transition is vertical and c = 0. The recoil effect makes the x-ray transition nonvertical, R 0, and this results in the shift of the XPS sideband by = c = Fc R see Eq. 7. 1 S. Svanberg, Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications Springer, New York, 2001. 2 K. J. Randall, A. L. D. Kilcoyne, H. M. Köppe, J. Feldhaus, A. M. Bradshaw, J.-E. Rubensson, W. Eberhardt, Z. Xu, P. D. Johnson, and Y. Ma, Phys. Rev. Lett. 71, 1156 1993. 3 A. Föhlisch, J. Hasselström, O. Karis, P. Väterlein, N. Mårtensson, A. Nilsson, C. Heske, M. Stichler, C. Keller, W. Wurth, and D. Menzel, Chem. Phys. Lett. 315, 194 1999. 4 W. Domcke and L. S. Cederbaum, Chem. Phys. Lett. 13, 161 1978. 5 F. Gel’mukhanov, P. Sałek, and H. Ågren, Phys. Rev. A 64, 012504 2001. 6 M. Drescher, Z. phys. Chem. 218, 1147 2004. 7 G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priory, and S. De Silvestri, Nature London 414, 182 2001. 8 T. Beddar, W. Sibbett, D. T. Reid, J. Garduno-Mejia, N. Jamasbi, and M. Mohebi, Opt. Lett. 24, 163 1999. 9 J. A. Gruetzmacher and N. F. Scherer, Rev. Sci. Instrum. 73, 2227 2002. 10 H.-S. Tan and W. S. Warren, Opt. Express 11, 1021 2003. 11 K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV, Van Nostrand Reinhold, New York, 1979. 12 M. Tronc, G. C. King, and F. Read, J. Phys. B 12, 137 1979. 13 U. Gelius, S. Svensson, H. Siegbahn, E. Basilier, Å. Faxälv, and K. Siegbahn, Chem. Phys. Lett. 28, 1 1974. 14 T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Ågren, A. A. Auer, K. L. Bak, V. Bakken, O. Christiansen, S. Coriani, P. Dahle, E. K. Dalskov, T. Enevoldsen, B. Fernandez, C. Hättig, K. Hald, A. Halkier, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. B. Pedersen, T. A. Ruden, A. Sanchez, T. Saue, S. P. A. Sauer, B. Schimmelpfenning, K. O. Sylvester-Hvid, P. R. Taylor, O. Vahtras, Dalton, A Molecular Electronic Structure Program, Release 1.2, 2001. See http://www.kjemi.uio.no/software/dalton/dalton.html 15 T. H. DunningJr., J. Chem. Phys. 90, 1007 1989. 16 R. A. Kendall, T. H. Dunning Jr., and R. J. Harrison, J. Chem. Phys. 96, 6796 1992. 17 V. C. Felicíssimo, F. F. Guimarães, F. Gel’mukhanov, A. Cesar, and H. Ågren, J. Chem. Phys. 122, 094319 2005. 18 F. F. Guimarães, V. Kimberg, V. C. Felicíssimo, F. Gel’mukhanov, A. Cesar, and H. Ågren, Phys. Rev. A 72, 012714 2005. 19 F. F. Guimarães, F. Gel’mukhanov, A. Cesar, and H. Ågren, Chem. Phys. Lett. 405, 398 2005. 20 C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, V. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 1991. 21 F. F. Guimarães, V. C. Felicíssimo, V. Kimberg, A. Cesar, and jx = 2a jAi x − j/F j , aj j = E j − U jRt, 023414-8 Q: 4³B±B3¶B±B }R±² 1 1Vif· ÆHV³)¶®·4´ ö<ö PHYSICAL REVIEW A 72, 023414 2005 ENHANCEMENT OF THE RECOIL EFFECT IN X-RAY… F. Gel’mukhanov, eSPec wave packet propagation program, Universidade Federal de Minas Gerais, Brazil and Royal Institute of Technology, Sweden 2004, See http:// www.theochem.kth.se/people/freddy/ 22 M. S. Shahriar, P. Pradhan, and J. Morzinski, Phys. Rev. A 69, 032308 2004. 23 L. D. Landau and E. M. Lifshitz, Quantum Mechanics Pergamon Press, London, 1962. 023414-9 ´a²´«¶z· Ë iÏÌdÐ=ÔrdÐlbfÑkÖÐ ÇkÌ5ÖÐ=Ô4ÕÏÌÎoÛÐ,ÑuÜ2ÍÌ Î Î,Ìa dÐÜ ÅPuÌÏÅ2eÐÌUÛ¸ÜbdÇ( dhÏÑciÀкeQdÑ}Ì'Ñkiv'Ô ÇÝÌÖÐÞÌÕÍÕ2eQdÄI Ì'Ñkd _`NaCT¢Vkb\dZEeU^MON/MON/QfRUT¢V£XoZ4[6\&];^_`N/¤CNM\dgWT¦¥d§¢]E¨ ]4T¢V©pª^XnUsuMONQf\dgihjVkR'lmXonUpdq y'z${}|i~Fr >}~ > oßo$Å$µ3 − &Å$4x1& ®5±BkV àH±B 1 1Vi ~: V³²±B3¶B;±²b · 4³¯ ¶®·4´ r<¸ PHYSICAL REVIEW A 73, 023409 2006 Phase-sensitive wave-packet dynamics caused by a breakdown of the rotating-wave approximation 1 V. Kimberg,1 F. F. Guimarães,1,2 V. C. Felicíssimo,1,2 and F. Gel’mukhanov1,* Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden 2 Departamento de Química, Universidade Federal de Minas Gerais, Avenue Antonio Carlos, 6627, CEP-31270-901, Belo Horizonte, MG, Brazil Received 6 October 2005; published 10 February 2006 The water dimer driven by strong infrared field is studied in the two-vibrational mode approximation. A pump pulse excites the OH vibrational modes and creates a coherent superposition of vibrational states of the low-frequency OO mode. The solution of the Schrödinger equation in the adiabatic approximation shows a strong sensitivity of the OO vibrational wave-packet dynamics to the absolute phase of the pump field. This effect appears due to a break down of the rotating-wave approximation when the Rabi frequency of the OH vibrational transition approaches the frequency of the OH mode. The violation of the rotating wave approximation modifies considerably the interaction of the probe radiation with the laser-driven molecule. DOI: 10.1103/PhysRevA.73.023409 PACS numbers: 42.50.Md, 33.80.b, 33.70.Ca, 34.50.Gb I. INTRODUCTION The recent development of femtosecond control of coherent molecular dynamics have excited theoretical 1–3 and experimental 4 studies aiming at realization of logical gates and quantum computing algorithm in atoms and molecules, especially using the feedback control methods 5. The possibility of molecular quantum computing using two vibrational modes of the NH3 molecule as qubits was shown in Ref. 6. In Ref. 7, the connection between quantum computation and quantum control of the rotational and vibrational wave packets is established. The analogs of several quantum bits within the shape of a single wave packet were found there. These bits are based on wave-packet symmetries. In quantum bit applications, one is interested in performing the transition as fast as possible. This demands strong Rabi frequencies 8, which are enough for invalidating of the rotating-wave approximation RWA. In Refs. 9,10 it was shown that the Rabi oscillation is accompanied by another oscillation at twice the transition frequency. The amplitude of this modulation is rather small 10% . This oscillation carries information about the absolute phase of the driving field. One can detect this phase by simply measuring the population of the excited state, by coupling this state to the other state with a short laser pulse and monitoring the resulting fluorescence 12. The effect of phase sensitivity appears when the Rabi frequency of the driving field is strong enough for RWA to break down. In the present paper, we show that, contrary to population, the dynamics of the wave packet is strongly influenced by the breakdown of the rotating-wave approximation BRWA effect. The main reason for this is that the BRWA affects the spatial distribution of the nuclear or electronic wave packet much stronger than such an integral characteristic as population. Here we explore the dynamics of the vibrational wave *On leave from Institute of Automation and Electrometry, Novosibirsk, Russia. 1050-2947/2006/732/0234096/$23.00 packet in the water dimer driven by strong coherent infrared IR field which is resonant to the frequency of the OH mode. The IR radiation creates a coherent superposition of OO vibrational states or a wave packet WP. We have chosen the water dimer as prototype for polyatomic molecules with slow and fast vibrational modes. Typical examples of such systems are molecules with intramolecular hydrogen bonds. Fast and slow nuclear modes can be separated and can be treated similar to the Born-Oppenheimer approximation in electron-nuclear dynamics. The laser-induced nuclear dynamics in such systems is rather unique and interesting and is one of the main objectives of our study. Simulations show that contrary to populations, the change of the IR phase leads to a strong variation of the WP up to 60%. This effect can result in significant errors and must be taken into account in low-energy fast qubit operations based on WP quantum control 7. We also investigate the influence of the phase of the pump radiation on the interaction of the probe radiation with molecules. The BRWA effect can be detected by using short probe pulse with duration shorter than the Rabi period. We shall begin by outlining the two-mode model of the water dimer molecule in Sec. II. The theory of interaction of the molecule with strong pump and weak probe fields is described in Secs. II B and II C. The numerical simulations are analyzed in Sec. III. The main results are summarized in Sec. IV. II. THEORY The water dimer is formed essentially by two hydrogenbonded water molecules, and possess two nonequivalent oxygen atoms, the so-called donor Od and acceptor Oa oxygens Fig. 1. The hydrogen atom which takes part in the hydrogen bonding is shown in Fig. 1 as H*. The notations r and R will be used to represent the Od-H* and Od-Oa interatomic distances, respectively. To focus our attention on the physics, we take into account only nuclear motion along Od-H* and Od-Oa bonds. 023409-1 ©2006 The American Physical Society r 01 V;±\;±\´<Dµ´ 1¶<· ±B|·x8}2V1¶r4µå< ;5I·Z;±²X µ x´<1I ±² Z;±\ PHYSICAL REVIEW A 73, 023409 2006 KIMBERG et al. potentials EnR cross each other such a situation does not occur for the water dimer. B. IR pump-probe spectroscopy FIG. 1. The water dimer. A. Born-Oppenheimer or adiabatic approximation Generally, to explore this model a solution of a twodimensional 2D Schrödingier equation is required. However, the reduced mass of the OH subsystem mr is much smaller then the reduced mass of OO subsystem mR. This allows us to split our model system in two subsystems: fast hydrogen-oxygen and slow oxygen-oxygen motions. This adiabatic approximation, being a direct analog of the BornOppenheimer BO approximation, allows one to reduce the 2D-problem to a one-dimensional one. The total wave function of the system reads r,R,t nr,RR,t. 2 3 evolves in the potential of the nth OH vibrational mode EnR. The Hamiltonian Hn of this mode has the following eigenfunctions and eigenvalues: Hnn = Enn, 1 2 Hn = − + EnR, 2mR R2 5 The wave packet t obeys the following time-dependent Schrödinger equation: ı t = Htt, Ht = H + Vt. t 6 The substitution of Eq. 5 in Eq. 6 results in ı = H, t = g , e H= Hg Vge Veg He , 4 = 0 , 1 , 2 , . . .. Apparently, the BO approximation 1 breaks down near the points where two different OH vibrational 7 where VgeR,t = Veg*R,t = gVe = − dge · Etcost + − k · r − dge · E ptcos pt + p − k p · r where the OO distance R is a parameter. Now the energy EnR of the nth vibrational mode depends on the oxygenoxygen distance. The wave packet of the slow OO stretch mode ı R,t = HnR,t t t = gr,RgR,t + er,ReR,t. 1 The fast hydrogen-oxygen stretch motion is described by the equation 1 2 − + Ur,R nr,R = EnRnr,R, 2mr r2 We assume that the molecule interacts with strong pump Et and weak probe E pt fields whose frequencies and p are close to the vibrational frequency eg = EeR0 − EgR0 of the OH mode R0 is the equilibrium OO separation. This allows one to neglect the radiative transitions in EgR and EeR wells. The pump and the probe fields mix the lowest g and the first excited e vibrational level of the OH mode Fig. 2a: 8 and dge = dgeR = *gr,Rd*e r,Rdr. 9 In our simulations, we assume that E and E p are parallel to the transition dipole moment dge. We need to solve the time-dependent coupled Schrödinger equations 7 with the initial conditions t = 0 = g0 . 0 10 Because both g and e states are bound, the matrix elements are real Vge = Veg. In the water dimer, the potential energy surfaces of the ground and the excited OH vibrational states are almost the FIG. 2. a Energy levels scheme of the water dimer. Laser field with wave length = 2780 nm is resonant to g , 0 → e , 0 transition. b Transiy tion dipole moment dge = dxge , dge , dzge. 023409-2 ®5±BkV àH±B 1 1Vi ~: V³²±B3¶B;±²b · 4³¯ ¶®·4´ ˙ e = Hg + ege + Vegg, ı Ú PHYSICAL REVIEW A 73, 023409 2006 PHASE-SENSITIVE WAVE-PACKET DYNAMICS CAUSED… same see Fig. 2a: EeR EgR + eg. This rather general property is equivalent to He Hg + eg. With this approximation Eq. 7 becomes r − k p · r in Eq. 15 and define P p as a variation P pt = − ˙ g = Hgg + Vgee . ı 2 ge Im gVge p e + gV p e, 16 11 To see the terms which break the RWA, we introduce the ¯ e ± g which obey the equations auxiliary wave packets ± = where n = npump+ probe − npump, n = npump, n = g , e. ˙ ± = Hg 1 G1 + cos + G pcos 1 ı 2 III. NUMERICAL SIMULATIONS ı + cos 2 ± ± G sin + G psin 1 + sin 2 , 2 12 ıegt+ ¯ e = ee where , G = dge · Et, and G p = dge · E pt are the Rabi frequencies of the pump and probe fields, respectively, and = 2t + , 1 = + pt + + p , 13 2 = − pt + − p . Here and in the numerical simulations, we assume that the pump radiation is tuned in resonance with the transition to the first OH vibrational level = eg. This allows us to neglect excitation of higher vibrational states of the OH mode. In the rotating-wave approximation, the fast oscillating terms the terms with and 1 are neglected. However, when the Rabi frequency approaches the light frequency, this terms influence strongly the dynamics of the system which start to depend on the double phase 2. The terms with 2 − p show a strong sensitivity of the dynamics of the WPs on the relative phase = − p. These terms play an important role even for small intensities, where the RWA approximation is valid see Sec. III. C. Probe signal Our computational approach is based on numerical solution of the coupled Schrödinger equations in the framework of the adiabatic approximation 7. This approach allows us to reduce the time of calculation drastically: instead of solving 2D problem we are working in one dimension. That means, that time of computation increases linearly with increasing the number of mesh points. In our simulation, the potential energy curves are mapped from R = 4.0 to 15.0 a.u. with 512 points. These potential curves can be approximated by Morse potentials EnR = EnR0 n = g , e, with parameters + Di1 − exp−iR − R02, EgR0 = 0.2309 eV, EeR0 = 0.6768 eV, R0 = 5.9298 a.u., Dg = 0.1290 eV, De = 0.1355 eV, g = 0.7855, e = 0.7756. The WP propagation is calculated employing time-dependent techniques 14. The second order differential scheme is applied in the propagation of the wave packet with a time step of 5 10−5 fs. The potentials of the ground and excited states Fig. 2a, as well as the R-dependent transition dipole moment Fig. 2b were found 15 from ab initio calculations using the complete active space multiconfigurational selfconsistent method CAS MCSCF implemented in the DALTON 16 program. The frequency of the ground-excited transition eg is equal to 0.4459 eV. In all calculations the frequency of the pump and the probe fields are resonant to this transition. We use the Gaussian shape for both the pump and the probe pulses with half width at half maximum = 50 fs and p = 300 fs, respectively. The population of the excited state e,e evolves according to the density matrix equation + e,e = Pe , t 14 where the small decay rate of the excited state is ignored in our simulations. The rate of transitions from the ground to the excited state Pe forms the total “absorption” probability 13 Pt = Pe = − =− 2 Im eeVee 2 Im gVgee, 15 where = is the density matrix of the molecule. To find the “photoabsorption” of a weak probe field P p we make the replacement Vge → Vge p = −dge · E ptcos pt + p A. Wave packet dynamics beyond the rotating wave approximation The results of the numerical simulations of WP propagation are shown in Fig. 3. The left panel shows the time dependence of the electric field of the pump pulse. The right panel shows snapshots of WPs of the ground and excited states for different times. One can see that the change of the phase of the pump pulse results in a change of the wave packets. This change takes maximum near the maximum of the pump field. The phase dependence arises due to the break down of the RWA through the double phase 2 see Eqs. 12 and 13. Indeed, the simulations show that the wave packet does not change the shape when 2 = 0 → 2, contrary to the change of the phase in / 2. The phase effect takes maximum when 2 = 0 → . The relative change of the wave packet caused by the change of the absolute phase onto / 2: 023409-3 01 V;±\;±\´<Dµ´ 1¶<· ±B|·x8}2V1¶r4µå< r<ä ;5I·Z;±²X µ x´<1I ±² Z;±\ PHYSICAL REVIEW A 73, 023409 2006 KIMBERG et al. FIG. 5. Dynamics of the ground and excited states wave packets at the point R* = 5.42 a.u. see Fig. 3. Solid and dashed lines correspond to = 0 and = / 2, respectively. Wave packets oscillate with the doubled frequency of the pump radiation 2. All parameters are the same as in Fig. 3. FIG. 3. Phase dependence of square of the wave packets of ground g2 and excited e2 state at different times. The left panel shows the electric field of the pump pulse Et = Etcost + . Solid and dashed lines correspond to = 0 and = / 2, respectively. Duration of the pump pulse = 50 fs; the peak position corresponds to T = 250 fs; the peak intensity is I = 1015 W / cm2. n = maxn = 02 − n = /22 , maxg = 0,t = 267fs2 quency eg = 0.4459 eV of the resonant transition frequency of the OH mode and with the frequency of the OO mode g10 = 0.0147 eV. Both Figs. 3 and 4 show that the molecule loses the phase memory after the pump pulse has left the system. Simulations show very strong change of the local value of the wave packets Figs. 3 and 5 when the phase changes to / 2 contrary to the area or population n = g,e n = 17 is depicted in Fig. 4. One can see that n approaches 60% when the Rabi frequencies are comparable with the fre- n2dr, 18 whose dependence on the phase is essentially weaker 9,10 see also below. Figure 5 displays the oscillations of the wave packets with the double frequency 2. Such a fast modulation caused by the break down of the RWA is related to the nonresonant terms cos2t + in Eqs. 12 and 13. B. Work of the probe field The absorption or enhancement of the weak probe radiation is the way to observe the effects related to the break down of the rotating wave approximation. We calculated the work of the probe field using Eq. 16. The results of our simulations are summarized in Figs. 6 and 8. The intensity of the probe radiation used in the simulations was I p = 107 W / cm2. This intensity of the probe pulse with duration p = 300 fs makes it rather weak. To be confirm that the probe pulse is weak, we calculated P pt for smaller intensities and did not observe any changes. 1. Role of the R dependence of the transition dipole moment FIG. 4. Intensity dependence of the phase sensitivity of the wave packet n 17. a Time dependence of the Rabi frequencies for different transitions G = E · dge. b e. c g. The parameters are the same as in Fig. 3. One can see in Fig. 6a that the modulation depths of the populations e and g decrease strongly near the maximum of the pulse where the intensity is large. Such an effect is caused by the R dependence of the transition dipole moment dgeR. Simulations show that both e and g display ordinary Rabi oscillations with amplitude 1 when dgeR = const. In this case, we have two-level system where 023409-4 ®5±BkV àH±B 1 1Vi ~: V³²±B3¶B;±²b · 4³¯ ¶®·4´ r ê PHYSICAL REVIEW A 73, 023409 2006 PHASE-SENSITIVE WAVE-PACKET DYNAMICS CAUSED… FIG. 8. The effect of absolute phase. a Population of the ground state. b The work of the probe field 16. Solid and dashed lines correspond to the case c = p = 0 and d = p = / 2 of Fig. 6, respectively. TR = 2 / G00 17 fs is the Rabi period. The population oscillates with frequency 2. FIG. 6. Time dependence of the populations of the ground solid line end excited dashed line states for = 0 a. Electric field of the pump pulse Et = Etcost + in arb. units b. Phase dependence of the absorption probability in arb. units of the probe field: relative phase effect = 0, p = 0 c; = / 2, p = / 2 d; = / 2, p = 0 e. I = 1015 W / cm2, = 50 fs, T = 250 fs, = 0.4459 eV, I p 107 W / cm2, p = 300 fs, T p = 250 fs, p = 0.4459 eV. only g , 0 → e , 0 transition is allowed see Fig. 7a. However, dgeR depends on the OO bond distance. Due to this, the transitions with a change of the vibrational quantum number are allowed. In linear approximation, R dgeR dgeR0 + R − R0dge 19 these transitions are n → n and n → n ± 1 see Fig. 7b. FIG. 7. Illustrating the role of the R dependence of the transition dipole moment. Due to this, the dynamics of the populations of vibrational sublevels of the ground and excited states becomes more complex 11. When the pulse is very long, the populations e and g take periodically minimal 0 and maximal 1 values through the so-called revival time. However, when the pulse is short the only role of the R dependence of dgeR is to reduce the modulation depth of populations near the peak of pulse as it is displayed in Fig. 6a. 2. Role of the phase Figure 6a illustrates the time evolution of the population of the ground and excited states. One can see a weak modulation of the Rabi oscillations of the populations related to break down of RWA in the regions t 220 fs and t 310 fs. The nonresonant terms see Eq. 12 have strongest impact on the populations near the peak intensity of the pump pulse, 220 fs t 310 fs, where the Rabi frequency becomes comparable with the frequency of the pump radiation compare Figs. 6a and 6b. Simulations show fine structure inside the Rabi period TR caused by the break down of RWA see Fig. 8a. These fast non-Rabi oscillations of frequency 2 depend strongly on the absolute phase . The relative phase of the pump and the probe fields = − p strongly affects the work of the probe field P pt: compare panels c,d, where = 0 with the panel e where = / 2 Fig. 6. The effect of the relative phase appears in the rotating wave approximation and the origin of this effect is the coherent interaction of the pump and probe fields 17,18. To see the effect of the absolute phase, we need to compare panels c and d of Fig. 6 with different absolute phases = p = 0 and = p = / 2. At first glance, the work of the probe field is almost the same in both panels. However, the fine time structure of the populations and of P pt see Fig. 8 depends strongly on the absolute phase. Due to the break down of the RWA, the populations of the ground and excited states, as well as the work of the probe field P pt, oscillate with double frequency 2. One can see that the change of the absolute phases = p = 0 → / 2 results in a shift of the fine structure of the populations and 023409-5 r<ñ 01 V;±\;±\´<Dµ´ 1¶<· ±B|·x8}2V1¶r4µå< ;5I·Z;±²X µ x´<1I ±² Z;±\ PHYSICAL REVIEW A 73, 023409 2006 KIMBERG et al. P pt on t = 2 / 2 = 2 / 4 in agreement with Eqs. 12 and 13 which demonstrate that the probe signal depends on 2 and + p. It is interesting to note that in the region of strong pump field the probe signal Fig. 8b experiences Rabi oscillation with period TR = 2 / G00 17, fs which is strongly suppressed for the populations Fig. 6a. It is worth pointing out that the role of the absolute phase is stronger for the WP Figs. 3 and 4 than for the populations Fig. 8a. IV. CONCLUSIONS It is found that the manifestation of the rotating wave approximation break down is strongly enhanced in the wavepacket dynamics comparing to the time evolution of such an integral characteristics as the populations. With the help of the infrared pump-probe spectroscopy it is shown that the absorption profile of the probe field also experiences dependence on the absolute phase of the pump field due to violation of the rotating-wave approximation. The fast temporal oscillation of the probe field profile has a slower envelope which corresponds to the Rabi oscillation of the system. It is necessary to note the strong suppression of the Rabi oscillations near the peak position of the pulse due to the R dependence of the transition dipole moment. The presented theory and obtained results can be applied to any kind of two-mode system in both infrared and optical range. In this paper we studied theoretically the infrared excitation of the vibrational wave packet in the water dimer. Initially a two-dimensional problem was reduced to a onedimensional one by splitting the system into two subsystems of fast and slow vibrational motions. 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"$#&%('*)*+-,/.10-)*23)5462,87 +-9:<;$=:'*9?>@:A7CB*9 .CD)*23.1E89?.F=G)7)*,/>@+-9HI7)5;9?0-)*280-J9HIDK.C:.C9D ÷ ôù5õ ,ú ô,õBöF÷ ú_úû;þ5ÿBö/òDúôù%úVð@ý;÷dó Sõ jöFø9ôù%ú÷ñ +ÿBúóò ó úVþdó ú+ôjúIû +õ(ô,úeð ý úeó rû;ü\òDóeý a òDó/ð@õHóFôõBö/÷ öF÷îôù%ú eö/ò,úhõBö/÷5õ ú+ð ôIóôDú ù%úhõBö/÷5õ óFô,õBöF÷ öFøö/÷%úZú+ÿBú +ôòDöF÷½õ(÷ (1s) ö/ò (1s) ö/ò 5õ(ôIóÿ 1 2 öFø >ÿ(ýöeû%óÿ Gö/÷%öeû@õ Gú õ ú÷%ú+òDóFô,ú bò,ú þú +ôõ úÿ(ý ôù%ú ð@õHó dóFôõ Lú+ÿBú +ôòDö/÷5õ ôIóôDú |ψ i óF÷%ð |ψ i 1 2 ù5õ ,ù ,ù%ö ]ó òDö õ(÷ LöFøôù%ú+õ(òþö/ôDú÷ôõHóFÿbú÷%ú+ò >ý +ñ5ò ú E ó % ÷ ð EO (1s−1) ≡ E2 (q) O (1s−1) ≡ E1 (q) óFôNôù%ú?÷ñ ÿBúóFò eö;ö/ò,ð@õ(÷dóFô,ú q = q Dú+ú õ ù%ú?ô,ö/ôIóÿ ó ú?øñ5÷ ô,õBöF÷_öøôù%ú eö/ò,ú?õBö/÷5õ eúeð ,ôDóFô,ú 1õ c òõ(ôôDú÷Úó L M S _ ` K T ZY\[ U N T I G ] ^ O P O A^ X 3 Q R WV ?V a ^@ b[ X XO 1 P f dce 2 O b T 3 Yhg Ψc = (ψ1 φ1 (t) + ψ2 φ2 (t))e−ıEc t , ] ù%úòDú E õ ð5ú %÷%ú+ðjó ôù%úÿBö bú ô1ú÷%ú+ò >ý ú+ô ú+ú+÷jôù%úô bö eöFòDúõBöF÷5õ ú+ðLþö/ô,ú+÷ô,õHóFÿ E (q) óF÷%ð E (q) c 1 2 ó % ÷ ð F ó D ò ú ô % ù j ú + ú B ÿ ú , ô , ò / ö 5 ÷ õ j ô õ G ú \ ü ( õ % ÷ 5 ð + ú þ ú % ÷ 5 ð + ú ÷ ô ó L ú ø 5 ñ ÷ + ô B õ / ö ÷ hö F N ø , ô % ù ú e F ö D ò L ú B õ / ö 5 ÷ õ e e ú ð ô D F ó D ô ú Gõ ( ÷ ôù%ú ψ1 ψ2 öFò,÷@ü þ5þú÷5ù%ú+õ Gúò1óFþ5þ5ò,öeû@õ 7óô,õBö/÷ ù%ú1÷ñ +ÿBúóò ó úNþdó úô φ óF÷%ð φ öøôù%úNúÿBú +ôò,ö/÷5õ ôIóôDú ψ 1 2 1 óF÷%ð ψ 5ò,ú þú +ôõ úÿ(ý @øJö/ò ôù%úô ö +ö Sþö/÷%ú÷ô÷ñ ÿBúóFò ó >ú^þdó ú+ôöFøô,ù%ú +ö/ò,úü\õBö/÷5õ eúeð ,ôDóFô,ú Ci FV n ^ X j A^ 6 k S O ! ! l O T O l KP G K O m T W P a T 2 |φc (t)i = ù5õ ,ùmö ú+ý ô,ù%ú eö/ñ5þ5ÿBú+ðLú ñdóFôõBö/÷ e a p ı |φ1 (t)i |φ2 (t)i Yho ] ∂ φ1 (t) = H11 φ1 (t) + H12 φ2 (t) + V10 (t)φ(t), ∂t Y!q ı L ô õ ñ DúIøJñ5ÿô,ö r ] ∂ φ2 (t) = H21 φ1 (t) + H22 φ2 (t) + V20 (t)φ(t). ∂t òõ(ô,ú^ð5ö ÷Lôù%ú 7óFôòõ¨ûLò,ú+þ@òDú Dú÷ôIóô,õBö/÷möFøôù%ú ,ú^ú ñdóô,õBö/÷ a ∂ ı φc (t) = Hφc (t) + V(t)φ(t), ∂t V(t) = p V10 (t) V20 (t) , φc (t) = φ1 (t) φ2 (t) Yhs ] %ù úòDú V (t) = −(D ·E (t)) cos(ω t) exp(ı(ε+E −E )t) õ ôù%ú 7óô,òõ¨ûGú+ÿBú Gú+÷ô«öFøôù%úõ(÷ô,ú+òDó ô,õBö/÷ n0 n0 X X c 0 ú+ô búeú+÷jôù%ú Gö/ÿBú +ñ5ÿBú?ó÷%ðLôù%ú1û;ü\òDóeý dú+ÿBð ε õ ô,ù%úNú+÷%úò >ý öFøôù%úNþ5ù%ö/ô,ö;ú+ÿBú +ôòDöF÷ ω e dó÷%ð E (t) ! t ji m C m ^ X ^ X X & ¶ VUWXU BYDZ%[\ ]] [%^`_ ÏlU;vu dYDi i U ba [%^ced U Y ] Y ] ced fU6W YDZ ^ U a3gBh c$i3_Rj c$iFk wfwfx yz{<|5z{}X~@{XA{CA|*X {\z{5{CA|*~@-y zyjA{CKzylX {{{XzX/{lX {G*hzKyb/{Cd eX EX (t) Dn0 } {aXzKy}Xe~@{a??{5 {az{ay3 fX {e5A{ A {}zX@{}X {$e*/yr}X {yA{~/yA{ φc (t) EX (t) = n = 1, 2 NX {!(`~{NzK{a{3}XKyK{} X {t5{yzyrK-y H= H11 H12 H21 H22 . h< {CzK{ {e-y A/y {zX?} Hnn = − 1 d2 + En (q) − Ec , 2mH dq 2 n = 1, 2. {}zX@{` {`{C~@{{AjA{y zl?XWv {e-y/yX~@{5X-y} zK{1!{N{C{Cz$}X1y K{} £ {3¤8!e-y</y{zX?}| }X~d{-y r?{C5 £ F¡¢A En (q) = hψn |Hel |ψn i tX { ψn Hnm |G K ¥yzK{Nz{}X~@}{zSekye/y IzKy}XX}|t{}K{Cz<{}y (?e¦¤d{z{5y~~zAy {}y zK{t}{jz{zK{ydX }K y$~z{|/y?{CA| X {e¦¤d{z{5X-yzX§ fX {a¨©ª~@{zKyz H12 = −H21 = − d dC12 (q) 1 2C12 (q) + , 2mH dq dq y`X {a}Ky-yzGz{zX@{zX £ {{Cz £ C12 (q) = hψ1 (r, q)| d |ψ2 (r, q)i dq zK{C~zK{}K{C5Ky H12 = H21 = λq £ F¡<¡ X¡«A {}K{bz{~ezK{}K{C51y }e¦¤{Czt\z¬{y _X {CzaX {r{{XzryA{b\X}a K _{~@{CI~/yzKy ?{XzXy__ {r5C{y zr*ze/y{}$_X {rhz?{Czy}{<| {lX {l{{XzKly<{?X}byz{ KyA{®®e{®A{y z?A{r{Xz_vX {S}K{®y}{3¯ ¡°<±!I² X Uy~~zK<yK {}l} ®<<{SX {S}1y ?{ z{}X¦X {K1y8\C`}³{Ce~/y {S<{zy$r~{CK{a{C{Cz/y}}³}K{C ( {b¨©µ~@{zKyz *{CQ¯ ¡¢±! £ F¡«A d{y }K{bX {r {{1{C~r}6·*{ ¨ £ ²3¡ ´ {}K{G {}y -yz³zX z{~z{}{5KyX_}t~zK{1{zKy{b\z¶X {b5?{Czy~d5$ {CzK{³³{³?*{X {³*z~ H12 5ry5y }K}X-y ¸¹»º³¼½5¾@¿¾dÀ-Á-ÀÂFÃN½*ÄÅeÆk¼¿ ÃNÇdÈ8½5Â1½*À-½*É8À-Ê ¿<Â1À-½*É3½*Ä˽*Á-ÌÍ Î8Á-ÌÏÐ/¼ÀÑ*ÌÉ3¾ÃI¿jÏKÂ1¼½*É8ÒSÓKÔÖÕ@ÌÁ-Ð × {y3}KA{ez{X {~{I{5/yX} hØ< ?3X {$~ yX_z<}K}}K{Xv¯ ¡Ù ±! ³{CA{z|e }65?{zXy }K {?{}6z1y {Cz{1e~@{}XA{<|5@{y}{Ky`y$}~@{XzÖ({{{l}K¦A{ 0 dE_Ý_>iÞÝdEc$i ]ß YDdÈ^Y§k[cSÝY§káà¬âÏj3[à6dE_ãiF[\b\p_ a3ä Z%[%i [_ãiF[kÛpYDk wfw Ú ] ÝcSÝY ] _ ßlu Z+â>_RåBc$Z%^`_ãi3_Rå3[%^`Y ä6a ^ ÜÝAÞ/ßàáâãäåâæGÜßçèéßêÞÜaâåàXèÜtëèâàâ*ÜêÜçàXæKâã3ÜCãÜæXì<í ò ßäÜñâãàXèÜtåhßçààè/ß àGàXèÜaëèâàâáâãáóßàáâãIëæâ ò u [%iÞ_Rå5â u YDiæ\p_dY ] Ý a k[Y§k£à¬â£[%i ß dcedEYDk ä ç ä d_àGY ]ä å¬ç;dcpâ − Y§\®Ýd_ ] \p_ ä â ÜåâêêâðMáãäXàÜßñßãßêàÜæXã/ßàXáéAÜàÜçèãáÝ5ÞÜ ε îï ß ò áêáà!í`áäàXèÜaãâæôõâåàèÜaçâæÜaáâãáóÜñðßéAÜë/ßçöAÜCà áã`àèÜå\æKÜÝ5ÞÜãçCí`ñâôlßáã P (Ω) = hϕc (Ω)|ϕc (Ω)i = hϕ1 (Ω)|ϕ1 (Ω)i + hϕ2 (Ω)|ϕ2 (Ω)i. û èáäßãñ_àXèÜlÜÝ5Þ/ß àáâãä ò Üêâðüì<ÜãÜæKßêáóÜjâÞæ$ëæÜé*áâÞä$æÜäXÞêàäâ ò àKßáãÜñåâæã5ÞçCêÜßæbñeí5ã/ßôráçä$áã®ß ø ÿ äXáãì<êÜ$ÜCýçCáàÜñÜêÜçàXæKâãáçäàKßàÜlþ ÿ ð³ßéAÜë/ßçöAÜCà ÿ ÷ ÷FøùAú âÞæXáÜæàæKßãäFåâæôõâåÝ û î èÜ î ÷Fø <ú ì<áéAÜäßrà!ð³âjçâôrë@âãÜãAà âåàXèÜçâæKÜaèâêÜäàKßàÜtáãàXèÜå\æKÜÝ5ÞÜãçCí`ñâôlßáã ú |ϕn (Ω)i = Z∞ dt e−ıΩt EX (t) |ϕn (t)i, −∞ n = 1, 2. ÷Xø 5ú 1 ζn = (Dn0 ·eX ). 2 ÷Xø <ú âÞæXáÜæGàæKßãäFåâæô§âåàXèÜtãÜðQðßé<Üë/ßçöAÜà ðèáçèáäàXèÜ ϕc (t) = ÜCæKÜ Ω = BE − I1s ϕ1 (t) =e ϕ2 (t) áäbàXèÜSæÜê-ß àáé<Ü ıHt Γc = 0 φc (t) ÷ ε î äXáô$Þê-ßàXáâãä û î èÜSçßêçÞê-ß àáâãvâå t φ(t) ÷hùAú àXèÜtêá¦åÜCàáô?Ü àXèÜð³ßéAÜë/ßçKöAÜCà ϕc (t) ϕc (t) aÞëSàKâ t î ô?âô?ÜãAà t î÷FøùAú %bðáàXèàèÜáãáàá-ß àáêê ϕc (t) ≡ ϕc (0, t) 2 (1s ê áãäXàKÜßñ3âå û èÜGý !æKßí ÷ áì φc (t) áã3àèÜ$ã5Þô?ÜæXáçßê ðèÜæÜjàèÜlìæKâÞãñ îfùAú /ÜCêñjëæKâô?âàKÜä φ(t) '&)(*! 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"# hϕc (Ω)|ϕc (Ω)i 4 dm (q) ý`öeúöUï*ôAñ"öL÷6ïZñóQü;ïúRøóQô?ôAøUòhû;ñ6úRöLïQë P (Ω) u 4 1 > [%iÞ_Rå5â dx 0 (7 u ] Ý a k[Y§k£à¬â£[%i ß dcedEYDk ôõöL÷6ï*êZþÿùôü;ïó6ü;ï'û;ñöL÷6ïøUô6ñò $%&' ()*+&!,- !."/0,0(((21-43 52"/61 8:9;<= ?A@B5C3 5D"E+1F.#*.*."0CG&.C6"/"G+#*.6H* &() '$G+IJ%.,/KJL6MN +&L(#*.#O+N6H (-PQRS6T.")"E#VU(9XWY2=Z[&6+ H-"H+"\H]_^`"/N4&/ +#OP& &/+Ha(- P &\)<" H Hb6"/*(766Hb$G+IJ*.,KJ+&c$%(de(#OO&/.dX%fX3C9g −5 h&c) iN +6"F+ "H](,/ q j"# g]= k<fiO;]= k<W*]= 4= &()ei&.,XH]&+"+(7XXl;]9gm.`(N/&6= > '(_h+(#OG0"JH()J&Ln ')<" H*&EI (0"E,X+IJ&'&o$'+(n&p, %+"/E5D(7+HO&/ " +)6/+H(q6+ &!H] P/*"/ +"\& "\?:@rX Hs3N52"/61e.&6&6= tmuwv xny<z{*yJ|n}bzCy<~6yJ{E{~+P Sxn{O x]TzCnE{ > qh(6$%()"0C+2$++[.,X"EX#O++"&q(/#*b(& H([ ?O(/&q"l &/+H( X+ &6 dE[a.u.] = d[Debye] × 41.3417 ='!+" I = cε0 |E(t)|2 /2 H-3 5D"/61b8 p I[W/cm2 ] × 2.1132 · 10−9 a.u. Ωt[a.u.] = Ω(a.u.) × t(f s) × &' !(N+ &(21-4"EH]4= α = X '.&/+&!"#O H6Hs(s .&6&6= τα \/#*.C"/`&,.C\ !?:@8 α = L &!0N1saX &/& h t − t 2 i α Iα exp − , τ̄α $% "/ > τα τ̄α = √ ln 2 &p %,_`$%H]e,_C#aX3](#c#8h!TbpX H tα 8:9XW &p %.CKm.C<&(aC \?A@"3 5D"/61 ¶ VUWXU BYDZ%[\ ]] [%^`_ ; U vu dYDi λqc = 0 600 GM2 Y ] %fU6W = 0.08 eV Y ] ced YDZ ^ U\ = 0.16 eV a3gBh c$i3_Rj = 0.33 eV c$iFk %å³w w = 0.65 eV GM1 500 IL=0 Intensity (arb. units) i UTba [%^ced U 400 NE1 TS NE2 ∆t=100 fs 300 =196 fs 200 =230 fs 100 =276 fs 0 -1 0 1 2 3 -1 0 1 2 3 -1 0 1 2 3 -1 0 1 2 3 -2 -1 0 1 2 3 4 Ω (eV) n2TC<TDj :¡+¢O£J¤<¥P£<j£J A¦X§ ¨m©2£ ¨Eª«A :¡i¦¬S«:<¨c¦X A®¯¨E°X«:¨E±sG²³¥P¦X´ ¨ª¤J´¯¨Oµ ¹®¯§< :¦°J®_ªª¦X¤J£<´¯®¯°JºPª¦X°N©D«:¡°« λ ¡°J±m±<® ¸]¨/ :¨/°«±<¨E´¯¡+¢P«A®¯¥P¨E© ¡J ¤¯4Á£J£ ¨/ p£J¡°J¨/´_©S©2J¦+»ÃÂLÄoÅc£J A¦ÆN´¯¨E©¬¦ pÇ ~eL ¶¯¶ N·¬¦ T±<® ¸]¨/ A¨E°«\¹6¡´ ¤J¨©G¦¬S«:<¨ § ¨«D»S¨/¨/°cj :¡+¢¡6°N±cD¼£J¤<´_©A¨E©/¯ ∆t = tX − tL qc ½G¾ ¯¿EÀ »pJ®¯´¯¨L«:<¨´¯¦+»S¨/ £N¡°J¨/´_©o©AJ¦+»b«AJ¨©2£ ¨ª«: :¡\¦6¬,¥P¦X´ ¨ª¤J´¯¨©o±< A®¯¹X¨/° L ½'ÈJÉ §¢m«AJ¨\2ÊÆN¨/´_±'ËSJ¨\£J¡ :¡¥P¨«:¨/ :©¦6¬`«AJ¨\DÌ£J¤J´_©2¨!¡6 :¨!Ç ¬© ¿ ¬© 6Í ÎÊÏ ª/¥ 2 « É L ½G¾ÈÈ É τL ½ È É ϕL ½È :¡X±e¡°N± ÍÍm¨hÐ!,ËSJ¨±<¤J :¡6«A®¯¦°-¦6¬«:J¨Tj :¡+¢i£J¤J´_©2¨®_© ¬©/cËSJ¨±<¦6«2«:¨E±-¡°N±e±J¡©AJ¨E±a´¯®¯°J¨c¡6 A¨ ωL = ω10 ½È τX ½'Ñ «AJ¨*£N¡ 2«A®_¡´p£JJ¦6«:¦®¯¦X°J®¯ÒE¡6«A®¯¦X°l£J A¦X§J¡§J®¯´¯® «A®¯¨E© ¡°J± A¨©2£ ¨Eª«A®¯¹X¨/´¯¢ ¡X©T¨/<£<´_¡®¯°J¨E±l®¯°dŨEª622NÓ'ÃËSJ¨ P1 (Ω) P2 (Ω) É É ¹®¯§< ¡+«:®¯¦°N¡´¯´¯¢a A¨E©A¦´¯¹X¨E±-ÂLÄoÅ-£J :¦6ÆN´¯¨m±<¨/£J®_ª«:¨E±s®¯°s«:<¨m´¯¨/¬(«!¤J£J£ ¨/ T£N¡6°J¨E´»S¡X©!ª/¡´_ª¤J´_¡6«A¨E±e¬¦X !´¯¦°JºX¨/ %j :¡+¢O£<¤J´_©2¨ τX = 15 ÍP-¿ L ½G¾ È 14 ÔGÕ Öm×,ØÙEØXÚOÖ+ÛÖ+ÙÜcÝÞ'ÛÕ ÖißAà×áâÜÖFá Ü/Ö+ãbÛÝeäJÖå Ö+Ù/ØÛ/ÖiÛÕ ÖFä<ÙÝáå ãbÜÛ/ØÛÖiæGØ+çJÖi×,ØèéJÖ+ÛmØÙ/ÖmÛ τL êmîì ÞDÜ6íoï L êPë ð ñ-òBîì 14 óô è+Ú ÛÕ Ö%ö ÙÜÛLç ÷(øÙEØXÛ÷Ýå,ØânÛÙ/Øå Ü÷(Û÷Ýå ÚaØ ò øCÖÖ 2í ϕL êPì]ð õ<ñ Ù/Øã ð ÖDù ωL ê ω10 êPì ð ñ<ñ L êPëì<ì ÞhÜí ÔGÕ ÖOßAàö,Ö+âãb÷ÜmØÜÜáÚOÖ6ãbÛ/Ý-øCÖ*÷(åbÙ/Ö+Ü/Ýå,ØXå è6ÖOæ%÷(ÛÕ ÔGÕ % Ö ×Õ,ØÜÖFú ϕL ð ÷(ÚcáÚV÷(åNÛÖ+å Ü÷(Û2ýbÝXÞ'ÛÕ ÖmÞhÖ+æþAèý]è+âÖißAàÿ×áâÜÖ*ØÛTÛÕ ÖF(÷ å ÜÛEØXåNÛÛ = ì ð õ<ñ Ù/Øã,ûæØÜØãXüá ÜÛÖ6ãaÛÝcäJÖÛ'ÛÕ Ö ó ÖFá ÜÖiØeÜÕ ÝÙÛPßAà×áâÜÖ<í ð ÞhÜínØâ(âÝ6æ\Ü!ÛÝeÝçJÖÙ/è+ÝÚOÖmÛÕ ÖiØåÕ,ØÙÚOÝå÷è÷(ÛAýlÝÞpÛÕ ÖmÕ÷(ä<Õ Ö+Ù!ç ÷(øÙ/ØÛ÷Ýå,ØâSÜÛ/ØÛÖ6Ü\æGÕ÷è/ÕbÜÕ ÝáâãÛ/Ý L τL = 10 É ¬©E τX ê ãÖ+âØ+ýeÛ÷(ÚOÖ+Ü%øCÖ+Û2æ'Ö6Öå âÖØ+çJÖiÛÕ ÖiÜý]ÜÛÖ+Ú ÞhÜ è+÷(ÛÖ6ãsÛÝiÙ/Ö6Øè/ÕqÛÕ Ö×ÙÝÛ/ÝåqÛÙ/Øå Ü ÞhÖÙ%ÙÖ+ä<÷ÝåBújÜÖ6Ö o÷(ä û ò ð,ë ð Ô ÝOÚaØéJÖmÜå,Ø× ÜÕ ÝÛÜ!ÝÞÛÕ Öä<Ù/Ýáå l ã ÜÛ/ØÛ/ÖæGØ6ç<Öm×,ØèéJÖ+Û æ%÷(ÛÕ-ÛÕ ÖPã]áÙEØXÛ÷Ýå = ë<ì<ì ÞhÜPúhÖ ò è6Ö×Û o÷(ä ð õ þDÙ/Ø6ý-Øå ãsßAà×áâÜÖ6Ü6í ò ÞhÝÙÜá è/ÕbãÖ+âØ+ý]Ü ÛÕ ÖcÜEØXÚOÖTÞhÝÙ!ØXâ(âÛÕ Ö6ÜÖãÖâØ6ý]Ü ð ð o÷(ä Øå ã φ(t) ð û ð æ'Öcá ÜÖÙEØXÛÕ ÖÙPÜÕ ÝÙÛ þDÙEØ+ý-×Ù/ÝøCÖc×á]âÜ/Ö ò aÜ×CÖ6èÛÙEØiØÙÖPè6Øâè+áâØXÛ/Ö+ã-ÞhÝÙ%ã]÷ `Ö+ÙÖ+åJÛ ÔÕ Ö í í Øå ã ÞhÜ ÔGÕ ÖPßAà×áâÜÖTÕ,ØÜÛ÷(ÚOÖTÛ/Ý îì<ì î ë<ñ<ì ë ð á ÖFÛ/ÝeÛÕ÷ÜPÛÕ ÖiÜ×CÖ+è+ÛÙEØâpè+ÝÚ*×CÝ<Ü÷(Û÷ÝåÃÝÞ'ÛÕ ÖFæØ6çJÖ*× Øè/éJÖÛ ÷Ü ∆t = φ(t) ç·dE_Ý_>iÞÝdEc$i [_ãiF[kÛpYDk %åå w ]ß YDdÈ^Y§k[cSÝY§káà¬âÏj3[à6dE_ãiF[\b\p_ a3ä Z%[%i ] ÝcSÝY ] _ ßlu Z+â>_RåBc$Z%^`_ãi3_Rå3[%^`Y ä6a ^ Linear GM2 Intensity (arb. units) GM2 = 0.07 eV = 0.16 eV = 0.15 eV = 0.33 eV = 0.30 eV = 0.65 eV = 0.60 eV 1 Ω (eV) u ä ç ä d_àGY ]ä YDiæ\p_dY − å¬ç;dcpâ Y§\®Ýd_ ] \p_ ä â GM1 λqc = 0 = 0.08 eV 0 [%iÞ_Rå5â Gaussian GM1 λqc = 0 u ] Ý a k[Y§k£à¬â£[%i ß dcedEYDk 2 0 1 Ω (eV) 2 !#"%$'&(*),+.-0/213)4%576/98;:)<-0/2=?>-0/@BA 576)>C5DFEG+./?-H'4I)<(3-0A0J/2K(MLFNPOQ3KJH3K>.171AKJRLSO<TQ9UVW:)HC/2-1? 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BXCEDn 9;: S4UVGTQEH$"S4I$HPQI$FN4|!Ig!L"BVHH4OEU$7C)|!C)ª|!T[[³´OT|!pINPOEIQTS4UVN4UVC)|pH7JIg|!CT~NPS8BCEHPYZIS8IgDYZS4UV}NPOEI N8U~?A@ = = HPN4SPGEFgNPGTS8IgH(0NPOEIH4|!CTVpInBCED¦DTUGTJTpI~F8O"BS8BVFN8IS8HUXYyNPOEI? BCED% JUVCEDTH$S4I$HPQI$FN4|!Ig!L«BS4IK|!CMIS4N4I$D [ v OT|pHD{SWBVHPNP|pF~F8O"BCTInUVC¦NPOEI}~UVpIgFgGTjBSH4IpIgN4UVCD{GTS4|!CTNPOEIK|!CMNPSWB}~UpI$FgG{jBSQTS8UN8UC%NPSWBXCEHPYZIgS;S8I$BVFgNP|pUVC 9½ UWXU  U vu ¾ ]] YDZ%[\ dYDi ¿ ½(U9½(UTba [%^`_ iÀ [%^ced eY ] ¿ ¿ ½(U\ 2Á Y ] ced fU6W YDZ .^ ¿5c$iFk a3gBh c$i3_Rj %å w -0.3 -0.4 19 18 17 16 15 13 -0.5 14 12 11 -0.6 10 -0.7 Orbital energy (a.u.) ¼ 9 -0.8 -0.9 8 -1.0 7 -1.1 6 -1.2 5 -10.3 4 -14.4 3 2 1 -19.2 1 2 3 q(a.u.) µ¶&·¸{¹º »0¼½8¾3¿(ÀÁ½ÂÃ(ÄÃÅ8Å4ÆÇÀÁ½WÈÉ;ÃXÊÁ½WÅ8ÆÊË(¾aÃX¾sÌÀ Í3Ë(Ê i (q) Î 4 ½W¾3ÂsÆ«ÍsϽy¼VÆMÅ4ÊÁ½Ë$¾.Å4ÃVÃX¾3ÈÀÁ¼Ë$Ís½ µ¶&·¸Mк«ÑÅWÅ4ÆÇÀÁ½WÈÉ;ÃXÊÁ½WÅ4ÆÊË(¾aÃX¾sÌÀ Í3Ë(Ê«Ã(ÄÍsϽA·RÒÓÉ;ÃXÊÁ½WÅ4ÆÊÁ½X¸ q ¸ ç·dE_Ý_>iÞÝdEc$i %å w ï [_ãiF[kÛpYDk ]ß YDdÈ^Y§k[cSÝY§káà¬âÏj3[à6dE_ãiF[\b\p_ a3ä Z%[%i ] ÝcSÝY ] _ ßlu Z+â>_RåBc$Z%^`_ãi3_Rå3[%^`Y ä6a ^ ÔpÕyÖT×4ØVÙ"ÚXÙTÛ!ÜÝPÞEß;ànÚXÔ!á×4ß(ÚÕ8ØVáØâ«ÛjÚ×PãßãV×WÚä{ÔpßgáMÝ4Õ ÔpÕ7×4ß(ÚÕ8ØVá"ÚXÙTÛpßÝ4Ø~ÚVÕ8ÕPèTà~ßxÚKÕPÔ!àÔ!ÛjÚ×ÕPÝP×8ØáTã q− ∂6 (q)/∂q ÚáEä u [%iÞ_Rå5â u ] Ý a k[Y§k£à¬â£[%i ß dcedEYDk ∂7 (q)/∂q ä ç ä d_àGY ]ä YDiæ\p_dY − å¬ç;dcpâ Y§\®Ýd_ ] \p_ ä â áEß(Ú×yÝ4ÞEß;åg×4ØÕ8ÕPÔ!áTãÖØVÔ!áMÝ$æç&Ý äTßÖ ßáEäTßgáEågßxØâ«ÝPÞEßxå$ØV×P×4ß$ÕPÖØVáEä{Ô!áTãné©êÕ(æaë.×4ØVÙ"ÚÙTÛ!ÜÝPÞEß$Õ4ß à~ØVÛpßgågèTÛjÚ×AØV×PÙTÔ!ÝWÚÛpÕ7ãÔ!ìßànÚ(í4Ø×å$ØáMÝ4×PÔ!ÙTèTÝPÔpØVáEÕÔ!áÝPÞEßîïðågØVáEÕPÝWÚáÝ C12 ñ3òòXó ÚáEä ôTÞEßgáEågßô λ ñPò(õó æ ö÷ùø;ú3ûgüXýþÿXü Wÿ ÷RöAûÿ;üüû"!#Wü$%! & ßgÝ«èEÕ«ÙßgãÔ!á('yÔ!Ý4ÞÝPÞEßRå(ÚVÕ4ß)'yÞEßgáç+*×8ÚVä{ÔjÚÝPÔpØVáÔpÕÚÙEÕ4ßgáMÝ$ô IL = 0 æ-,RÞEßRØ×8ä{Ô!á"Ú×PÜ/.;ë)0ÕPÖß$ågÝP×8ÚÔ!áÝPÞTÔpÕ å(ÚÕ8ßxÚX×8ßxäTßÖTÔpågÝ4ß$äÔ!áÝPÞEßèTÖTÖßg×yÖ"ÚáEßgÛpÕØâ21Ô!ãEæ3âZØV×7ä{Ô54 ßg×4ßgáMÝ7îïðågØVáEÕPÝWÚáÝ8Õ$ô Ý4ØÝPÞEßìß×4ÝPÔpå(ÚXÛÝP×WÚXáEÕ4Ô!ÝPÔpØVáEÕ;â×4ØVà E1 (q) = EO1 (1s−1 ) (q) ÚáEä Ý4ÞEßÙØÝ4Ý4ØVà 2 1 2 E2 (q) = EO2 (1s−1 ) (q) Ùß$å$ÚèEÕ4ßÝ4ÞEß/8D×8Ú$Ü)ÖTèTÛpÕ4ßÔpÕÕ4ÞEØ×4Ý$ô =;é æ76ÕÚÛ!×8ß$ÚVä{Üß8ÖTÛjÚÔ!áEß$ä Ù"ÚXáEäTÕ 1 Ô!ãTæ ô'yÞTÔpå8ÞågØV×4×4ß$ÕPÖØVáEä ñ õó ØâRÝPÞEßKã×4ØVèTáEä¦Õ4Ý8ÚÝ4ßÖØVÝ4ßgáMÝPÔjÚÛ Ý4ØÝPÞEß('ßgÛ!ÛpÕ ñ q0 = 1.74 ó ô×4ß$ÕPÖß$åÝ4Ô!ìßÛ!ÜæBA¦ßäTØáEØVÝ~Õ8ßgßÚáMÜC;EáEßÕ4ÝP×4èEåÝ4èT×4ß Ô!á90ß$åVæMç8ç«ÝPÞEß:.;ë)0KÖT×8Ø<;EÛpßå$ØáEÕ4ÔpÕPÝ8ÕØâ0á"Ú×P×8Ø'>=;é ÚXáEä?'7ÔpäTßg×@=;é λ τX = 4 âZÕ$æEØF'ßgìß×(ô0ÝPÞEßìÔ!ÙT×WÚXÝ4ÔpØVá"ÚXÛÕPÝP×4èEåÝ4èT×4ßØâÙØVÝ4ÞC=;é 1 ÚáEä Ù"ÚáEäTÕ7Ùß$å$Øà~ß$Õ×4ßgÕ8ØVÛ!ìß$ä9'7ÞEßáÝPÞEßxä{èT×8ÚÝPÔpØVáØâÝ4ÞEßÖTèTÛpÕ4ßÔpÕyÛpØVáTãßg×Ý4Þ"ÚXá)ÝPÞEßìÔ!ÙT×8ÚÝPÔpØVá"ÚÛÖßg×PÔpØä Õ8ßgßyÛpßâÝaèTÖTÖßg×.Ö"ÚáEßÛTÔ!áG1Ô!ãEæ$3 æ2HØF'°Ý4ÞEß:=;é Ù"ÚXáEä~å$ØVáEÕPÔpÕ4Ý4Õ.Øâ0Ú;Õ4ÝP×4ØVáTã/I<D+I;Û!Ô!áEß7ÚáEänÚìß×PÜJ'Rß$Ú%K?I<D ñ ó ò 2 ì Ô!ÙT×8ÚÝPÔpØVá"ÚÛ.Ý4×8ÚáEÕPÔ!Ý4ÔpØá0æL,RÞTÔpÕÔpÕ;Ùßgå(ÚèEÕ4ßKÝ4ÞEßàÔ!áTÔ!àèTà ÖØÕPÔ!ÝPÔpØVá æ M NÚæ è0æ ØâÝPÞEß 'ßgÛ!Û EO2 (1s−1 ) (q) ñPò ó ÔpÕ×8ÚÝPÞEßg×xåÛpØÕ4ßÝ8ØÝPÞEßßOMèTÔ!Û!Ô!ÙT×PÔ!èTà Ú{æ è0æ!æL,yÞEßG=;é Ù"ÚáEä¦Þ"ÚÕ×4Ôpå4ÞEßg×ìÔ!ÙT×WÚXÝ4ÔpØVá"ÚXÛÕPÝ4×PèEåÝ4èT×4ß ä{èEßÝ4ØÛjÚ×Pãßä{ÔpÕPÖTÛjÚVågßgà~ßgáÝAØâÝPÞEß ØâaÝ4ÞTß(=;é 1 q0 = 1.74 EO1 (1s−1 ) (q) 1 'RßÛ!Ûâ×8ØVà ÝPÞEßßPOMèTÔ!Û!Ô!ÙT×4Ô!èTàæ7,yÞEßAìÔ!ÙT×WÚXÝ4ÔpØVá"ÚXÛ â×4ßOMèEßáEågÜ Ù"ÚáEäÔpÕAßgÕ8Õ4ßgáMÝPÔjÚÛ!Û!ÜÕPànÚÛ!Ûpßg×Ý4Þ"Úá)Ô!á)ÝPÞEß(=;é OMèTÔ!Ý4ßxÕPÞ"ÚÛ!ÛpØF' Ô!áÝPÞEßQ1T×8ÚáEåKD3ïØVáEäTØVá×8ßãÔpØVá0æ 2 'RßÛ!Û«Ùßgå(ÚèEÕ4ßxÝPÞEß ,yÞEß;èTÖTÖßg×AÖ"ÚXáEßgÛpÕØXâR1«Ô!ãEæ3Õ4ÞEØ' ÝPÞEßÕ4èTÖTÖT×4ß$Õ4ÕPÔpØVáØâ.ÝPÞEß(=;é 2 EO1 (1s−1 ) (q) ÖØVÝ4ßgáMÝPÔjÚÛ«ÔpÕ Ù"ÚXáEä'yÞEßgá)ÝPÞEßå$ØVèTÖTÛ!Ô!áTãågØVáEÕPÝWÚáÝ ã×4ØF'7Õ(æ7,yÞTÔpÕ7ß#4 ß$åÝyÔpÕ7Õ4ß$ßáågÛpß$Ú×4Û!ÜÔ!áÝPÞEßS.;ë)0Õ4ÖßgågÝP×WÚ/'yÔ!ÝPÞ)ÚXáEä9'yÔ!ÝPÞEØVèTÝç+*T;"ßÛpä æR,ØKèTáEäTßg×4Õ4Ý8ÚáEäÝPÞEß ØV×PÔ!ãÔ!áØâÝ4ÞTÔpÕ7ÖTÞEßgáEØVà~ßáEØVáU'RßxÚá"ÚÛ!ÜV$ßQ.;ë)0ÕPÖß$åÝ4×8ÚâZØV×ÚJ;W8TßgäáMèEågÛpß$Ú×ågØVá;EãèT×8ÚÝPÔpØVá0æX,RÞEßxìÔ!ÙT×8ØáTÔpå å$ØèTÖTÛ!Ô!áTã H12 = λq àÔ58TßgÕ7ÝY'RØnä{ÔjÚXÙ"ÚÝ4ÔpåxßgÛpß$åÝ4×4ØVáTÔpåÕ4Ý8ÚÝ4ß$Õ ψ+ = ψ1 sin β + ψ2 cos β, ψ1 ÚáEä ψ2 ÚáEä×4ß$ÕPèTÛ!Ý8ÕyÔ!áÚä{ÔjÚÙ"ÚÝPÔpåÕ4Ý8ÚÝ4ß$Õ ñZõ{òXó ψ− = ψ1 cos β − ψ2 sin β, '7Ô!ÝPÞÝ4ÞEßßgáEß×4ãÔpßgÕ E± (q) = q 1 2 . E1 (q) + E2 (q) ± (E2 (q) − E1 (q))2 + 4H12 2 ñõõó 0MèEå4Þ Ú%àÔ58{Ô!áTãäTßgÛpØå$ÚÛ!ÔZV$ß$ÕÝ4ÞEßågØV×8ßÞEØVÛpßgÕÚáEäð×4ß$ÕPèTÛ!Ý8ÕÔ!áðÝPÞEß)âZØVÛ!ÛpØF'yÔ!áTã*ß8ÖT×8ßgÕ8ÕPÔpØVá´âZØV×ÝPÞEßä{Ô!Ö ØÛpß à~ØVà~ßáMÝ ØâÝP×WÚXáEÕ4Ô!ÝPÔpØVáÝ8ØÝPÞEßgÕ8ß;áEß' ÕPÝWÚÝ4ß$Õ hψ± |d|ψ0 i = d(cos β ± sin β) r r d 1 1 λ d± = √ 1∓ 1 ± , ± 1 + ζ 2 |λ| 1 + ζ2 2 ζ = tan 2β = 2λq , ∆(q) ñõ$[ó 9½  vu ¿ ½(U9½(UTba [%^`_ dYDi iÀ [%^ced eY ] ¿ fU6W ¿ ½(U\ Y ] ced 2Á ¿5c$iFk YDZ .^ a3gBh c$i3_Rj %å(x w 2.0 (A) λqc = − 0.33 eV d+ 1.5 2 U ¾ ]] YDZ%[\ (d±/d) UWXU 1.0 λ<0 λ>0 0.5 (B) λqc = + 0.33 eV Intensity (arb. units) ¼ d− 0.0 -1.0 -0.5 \]Y^S_`acbcdceRfhgGikj%l+m#jnYi l+ikm#n 0.0 ζ/2 0.5 1.0 0 -1 0 1 Ω (eV) 2 3 qo pr d)sFm#tunYvn)l+wmxzy<v{$|kikj}Ln~lut+mj}Fl+w ζ/2 = λq/∆(q) _ ∆(q0 ) = 1.02 m_ -d2eRfhgLn+{m#xzl+tu yFtR{y<n+i l+iks<m@ FjJj$m#}< l+iks<mX)nYl+t+m#j}l+wn o jy]~m#|kdz_-Rl+wmtR{ Ftu Fmzl+m#tun- tum)l+wmn+ Fm@ <n2ikj (d± /d)2 \ik}_ _ @ ∆(q) = |E (q) − E (q)| J z%PZW" cP % z< E (q) < E (q) ¡C¢ ¤£ ¥F% 2 1 2 1 %¥Z¥ Qz¦ ψ+ % ψ+ z% § z¨ ¤©Sª 1 < ©ª 2 % W« z P¥ £ Z¬$¡G ® zP¨ Z %cZ Z ¯+ P $¬ z $%9 %zZ¥ 9 Z @¨ ¨? z@<§° z¯ WZ±F%¥ ²%§³ P´¬ c z ? @ ¬?%§ z G < R Z 7 ¨? « d1 = d2 = d « @ ¡ ¢ %? ) ¤ %%¥ Pz E (q ) = E (q ) ¶)$ £ @ ·S¸ ¨? P ¥%S<: $z zZWL Z qc µ X 2 c 1 c 2|λ|qc = E+ (qc ) − E− (qc ) µ~¹%º ¶ ¢ Z zZ¥ (d+ /d)2 % (d− /d)2 µ¼» Z ¡½¦¾¶: ¥W¬"¿®%ZZ%z £$ Z¬U 5À Pz P Q %®WWZZW(z z W$ λqc ® zQ ¥ ® z £ % ¤ ¥ ®z £ Z § <§³F%ZZ± ¤ #¯~ ¥#% µ~¹Á ¶:F<® U¬ ·¸ ¡Â λ > 0 F%à P ¿® ZW9%§R Ä ¥ÅÆ ψ− ¥#% % < ¨? L%§X LÇSÈcÉ Z ¥Z+¬%§X JÄ Z Æ z% ψ Z Z z % %§ λ ¡¤¾@P ZW"z + Ê ¿¡ µ¹$Ë ¶@ zÅ % Z Q¥#< z Z P %W Ã5§) zZ Ã<§ ·¸ S®W £ λ < 0 ¡Â :P Z¬¤§ ¨ ® X %®W ¥z ¨ % µ¹$Ë ¶ µ» Z¡½(¾¶c ) Z2 c§ ¨ ¥ P:z ¨ ®W%z% µ¼» ZW¡½(Ì@¶¡ ç·dE_Ý_>iÞÝdEc$i [_ãiF[kÛpYDk %åfÚ w ]ß YDdÈ^Y§k[cSÝY§káà¬âÏj3[à6dE_ãiF[\b\p_ a3ä Z%[%i ] ÝcSÝY ] _ ßlu Z+â>_RåBc$Z%^`_ãi3_Rå3[%^`Y u [%iÞ_Rå5â YDiæ\p_dY u ] Ý a k[Y§k£à¬â£[%i ß dcedEYDk ä6a ^ ä ç ä d_àGY ]ä − å¬ç;dcpâ Y§\®Ýd_ ] \p_ ä â Í<ÎÐÏRÑPÒ ÓuÓzÔÑPÕGÖ×ØFÙÚ×~ÛXÜ#ÔÝÞ+×ÕÔßߥ×PÒàÚ ÔZÕ á âYã¤ä åWæXç¥èZé/åWêë%ì¥èä ãçcícî@ï~ä%êZêäFíTì¥ðîSñPä ãòîãìzèä%ãë%ê³óSôé?äõWîPê2öZ÷Fø<ùíXèZìzð¤êZèZãîë%æXñPä åWúWêZèZãWû ë çzçzåWéLèZãWû λ ìzä/ÿîSñPä ãç¥ì#ë%ãìSäFí)îPòîæWì¥ðîòèZÿWæzä ãWèññPä åWúWêZèZãWû?ñë%ã¦ðëòîç¥ì¥æä ãWû$îPæXõWîPúîPãõWîãñîä%ã éLèZéLèñQç¥åñðUëJúä$çzçzèZÿèZêZèZì¦í)îé?äõWîPê³ì¥ðîóôTèZãìîæë ñPì¥èä ãUÿ9ëJûë%åçzçzèë<ãö ý÷Pù 2 2 H12 = λqc e−(q − qc ) /a . hèZû²õèç¥úWêëç/ì¥ðîÃúWæzäêîGä%ï@ìzðWî ç¥ìzæ¥åñPì¥åWæîícîQåçzîJë(æë%ì¥ðîPæêä%ãWû!"Yæë¦úWåWêçzî ä%ï7ìzðî& 2 ÷ýþ H12 = λq ü q °ä ü ý $þ é?ä 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»²LÁ°XÀÍÉ;ËÍÎvÏÑаX±! ²ÀÍÉu¶JÊ6¿ ¹HÀ1 ( ° Ó3³m² ÊO°m ´ ÊO¹7ÊO° ´ Ò »¶*¸À' BÄ "!#"$&% Ô)ÕlÖp×7ØÙ|ØÚ#Û+Ü+ÝÞvßÞà[Ú#Ù|Øá!Ü1âJáãØBä[áÚåØ#Û7Ýæ8Øçv×1ÛÕlØ.èvÚÕlÜ+éêvÖ|Þlë!ØÜ Øìà[áÛ+Ö|ÝÞí ı ñ>ØÜ+Øò ∂ + Γ̂ Ψ(t) = H(t)Ψ, ∂t Γ̂ ' î@ïð H(t) = H − V (t) Öp×1Û+ÕlØ>ó2ÝÙ|ØÚàvÙpá!Ü7ñ;áó2Ö|Ù Û+ÝÞlÖpáÞCô H ÔJÕvØÖ|Þl×Ûá!ÞÛ"áÞlØÝ!à[×uälÜ+Ýæ[á!ævÖ|Ù|Ö Ûç¡Ý!öáæ[×ÝÜ+ävÛ+Ö|ÝÞ älÜ+Ýæ8Ø [ØÙpê Öp×1Û+ÕlØ>ó2áÛ+ÜÖ õ,Ý!ö P (t, Ω) = 2 =m hφc |Vxc0 |φ0 i n o = EX (t)=m hφ0 |d0c (R)|φc i eı(Ωt+ϕX ) î(ð Ü+ØÙpá-õláÛ+Ö|ÝÞ ×ØØeæ8ØÙ|Ýâ ô ò8á!Þ[ê (c) î ð E0 = U0 (R0 ) Ec = Uc (R0 ) á!Ü+Ø;Û+ÕlØ.ó2Ö|ÞlÖ|ó3àlóØÞlØÜ+ëÖ Ø×7 Ý!ö=Û+ÕlØeë!ÜÝ!àlÞ[êáÞ[ê ØõvÚÖ Û+Øêä8Ý!ß Û+ØÞÛ+ÖpáÙp× áÞ[ê òCÜ+Ø×ä8ØÚ"Û+Ö|ãØÙ|ç÷ Öp×>Û+ÕlØAêvÖp×+×Ý!ß U (R) U (R) E ÚÖ|áÛ+Ö|ãØ.Ù|Ö|ó20Ö Û òlÖ cöGÛ+ÕlØeØõlÚ#Ö Û+Øê9×ÛáÛ+ØeÖ|×BcêvÖp×+×ÝvÚ#ÖpáÛ+Ö|ãØ!÷ Uc (∞) ÔJÕlØ9×+àlæl×Û+Ö Û+àv Û+Ö|ÝÞøÝ!ö Ö|Þøú7ì8÷ Ü+Ø×àlÙ Û×2Ö|ÞnÛÕvØ9öÝÙ ß îù!ð î@ïð Ù|ÝâJÖ|Þlë$ó2áÛ+ÜÖ õ'Øìà[á-ÛÖ ÝÞ'öÝÜeÛÕlØAÞà[Ú#Ù|ØáÜ3â)á-ãØ,ä[áÚåØ#Û"×.Ö|Þ ëÜ+ÝàlÞlê á!Þ[êØ#õlÚ#Ö ÛØê ×ÛáÛ+Ø×í î φ0 ð î φc ð ı ∂ + Γ̂ φ = Hφ, ∂t Γ̂ = 0 0 0 Γ îûð φ= φ0 φc , H= h0 + VL00 VXc0 VX0c hc + VLcc ÝöÛ+ÕlØ Öp×4ÞlÝ!Û+ÕlÖ|Þvë,ØÙp×ØBÛÕlá!Þ<Û+ÕlØ.Ü"á-ÛØ>Ý!ö=ä8Ý!älàlÙpáÛ+Ö|Ý!ÞÝ!öCÛ+ÕlØ.Ø#õlÚ#Ö Û+Øê ØÙ|ØÚ"Û+ÜÝ!ÞlÖpÚe×ÛáÛ+Ø )+*ð ∂ + 2Γ ρcc (t) = P (t, Ω). ∂t î Ô)ÕlÖp×uÜáÛ+ØøØìà[á-ÛÖ|Ý!Þ öÝÜ'Û+ÕlØÛ+Ý!Ûá!ÙAä8Ý!älàlÙpáÛ+Ö|Ý!Þ ρ (t) = Ýö<ØõvÚÖ Û+Øê¡×Û"á-ÛØöÝ!Ù|Ù|ÝâB×Ö|ó2ó2ØêvÖpáÛ+ØÙ|çöÜ+cc Ýó Û+ÕlØ hφc |φc i èvÚÕlÜ+éêvÖ|ÞvëØÜeØìà[áÛ+Ö|ÝÞ ÷ÔJÕvØêvàlÜáÛ+Ö|ÝÞ¢Ý!öB×Ûá!Þ[êláÜ"ê'õß îûð Üá-çáÞ[ê ó,Øá!×+àlÜ+Øó2ØÞÛ"×;Öp×Ù|Ý!ÞlëØÜ.Û+Õ[áÞuÛ+ÕlØ2älàlÙp×ØAêvàvß ÜáÛ+Ö|ÝÞC÷(Ô)ÕlÖp×4ó2Ý!Û+Ö|ã-áÛ+Ø×4à[×4ÛÝ*à[×Ø;Û+ÕlØ;Ö|ÞÛ+ØëÜáÙ8älÜÝ!æ[á!ævÖ|Ù|Ö Ûçò âJÕlÖpÚÕëÖ|ãØ×4ÛÕvØ3×ä8ØÚ"Û+Üàló Ý!öGÛ+ÕlØälÜ+Ýæ[Ø×Ö|ëÞ[áÙ -,/. âJÖ ÛÕÛ+ÕlØeÖ|ÞlÖ Û+Öpá!Ù=Ú#ÝÞ[êÖ ÛÖ|Ý!Þ φ(t = 0) = |0i 0 , îüð âJÕlØÜ+Ø Öp×=Û+ÕlØ4ýØÜÝßä8ÝÖ|ÞÛGãÖ|ælÜáÛ+Ö|ÝÞ[áÙ×Û"á-ÛØ4Ý!övÛÕvØ)ë!ÜÝ!àlÞ[ê |0i ØÙ ØÚ"ÛÜ+ÝÞvÖpÚ3Ù|ØãØÙ÷*ÔJÕvØ2êvØÚáç9óAáÛ+Ü+Ö õ Û"áåØ×>Ö|ÞÛ+Ý9áÚÚ#ÝàlÞÛ Γ̂ ÝÞlÙ ç3Û+ÕlØ;êØÚá-ç3ÜáÛ+Ø)Ý!öÛ+ÕlØJØõvÚÖ Û+Øê2×ÛáÛ+Ø á!Þ[ê*Ö|ëÞlÝ!ÜØ×GÛÕvØ Γ 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Ý!öÛ+ÕlØ.Úá!Ü+Ü+Ö|ØÜ Ω = ωX − ωc0 öÜ+ØìàvØÞ[Ú#ç ÝöÛ+ÕlØälÜ+Ýæ8Ø lØÙpê Ü Ø ÙpáÛ+Ö|ã!Ø9ÛÝ¢Û+ÕlØ ωX ωc0 = ÷ Ec − E 0 Z ∞ )0)ð î ρcc (t) dt. −∞ −∞ 2( 54 76 ×;Ö Ûeâ)á×>×ÕlÝâJÞØáÜÙ Ö|ØÜ ò8Û+ÕlØ,ävÕ[á×Ø,Ý!ö6Û+ÕlØ2Ú#ÝÕlØÜ+ØÞÛ ï á 8ØÚ"Û"×=×Û+ÜÝ!ÞlëÙ|çJÛ+ÕlØ1älÜ+Ýæ8Ø7×+ä[ØÚ"ÛÜ+àló âBÕvØÞ ϕ Û+ÕlØJêvàlÜáÛ+Ö|ÝÞ*Ý!L ö[Û+ÕlØJälÜÝ!æ8Ø4älàlÙp×ØBÖp×6×ÕlÝÜÛ+ØÜ=Û+Õ[áÞ,Û+ÕlØJä8ØÜ+Ö|Ývê Ý!ö)ãÖ ælÜ"á-ÛÖ ÝÞ[×÷<ÔJÕvØAÜØá×ÝÞöÝ!ÜeÛ+ÕlÖp×3Ö|×eÛ+Õ[áÛeÛÕlØ<êvçÞ[áó2ÖpÚ× ÷6Ô)ÕlØ Ý!öCÛ+ÕlØ.Þà[Ú#Ù|ØáÜ)â)áãØ>äláÚåØÛ)êvØä8ØÞ[êl×J×Û+Ü+ÝÞlë!Ù|ç2ÝÞ ϕL á!æl×+Ý!ÜäÛÖ|Ý!ÞÝö3Û+ÕlØuälÜ+Ýæ8Ø'älàlÙp×ØuêvÝØ×ÞlÝÛêvØä[ØÞ[ê ÝÞÖ Û× älÕ[á!×Ø Ø#õlÚ#ØävÛ9Ö|Þ Û+ÕlØuÚá!×+ØÝö3Û+Ö|ó2Ø#ß/êvØä8ØÞ[êvØÞÛ9älÕlá×Ø ϕX ÝÜ) öØâ)ß/ÚçÚÙ|Ø.älÜ+Ýæ8ØeävàlÙp×Ø÷ uÜ"á!êvÖpáÛ+Ö|Ý!Þ 7 ϕX (t) 8:9;=<5>#?A@-BCDEBFHG:;A<I;H>BEJ@ P (t, Ω) dt = 2Γ hφc3 |φc i 'Ø7×+ØÙ|ØÚ"ÛCöÝÜCÝ!àlÜ=×Û+à[êvç>Ûâ4ÝBó2Ý!Ù|ØÚ#àlÙ|Ø×íLK :MIN á!Þ[ê 3÷Ô)ÕlØ õ ßÜá-çeälàló2äß/ävÜÝ!æ8Ø>×ä8ØÚ#Û+Üáá!Ü+ØJ×+Ö ó*àlÙpáÛ+Ø2ê*öÝÜ − 1s → π ∗ õßÜá-çuáæ[×+Ý!Ü+ävÛÖ ÝÞ¢ÛÜá!Þl×+Ö Û+Ö|Ý!ÞÖ|ÞnÞlÖ Û+Ü+ÝëØÞ >ú è ÞvØáÜ î ØêvëØ;õßÜ"áç<á!æl×+Ý!ÜäÛÖ|Ý!Þ [ÞlØ×Û+Ü+à[Ú#Û+àlÜ+Ø ×ä8ØÚ#Û+Ü+àló9÷1ÔJÕlØeä[áß ð Üá!ó2Ø#Û+ØÜ×=Ý!ö[Û+ÕlØ $ÝÜ×Ø)ä8ÝÛØÞÛ+Öpá!ÙvêvØ×+Ú#ÜÖ|ælÖ Þlë;Û+ÕlØ)Ö|ÞlÖ Û+Öpá!Ùvá!Þ[ê [Þ[áÙ×ÛáÛ+Ø7áÜØ1Ú#ÝÙ|Ù|ØÚ#Û+ØêeÖ|ÞÔGá!ælÙ|Ø "÷ uØ7êvÖp×+Ú#à[×+×õßÜ"áç>ävÜÝ!æ8Ø ×ä8ØÚ"Û+Ü"á3öÝÜJæ8Ý!Û+Õ9Ù|ÝvÚá!Ù|Ö|ýØêáÞ[êêvØÙ|ÝvÚá!Ù|Ö|ýØêÚ#ÝÜ+Ø.ÕvÝÙ|Ø!÷ ψi∗ (r, R)dψj∗ (r, R)dr ∞ 1)+* î !ð Öp×<êvÖ 8ØÜ+ØÞÛAÖ|ÞøëÜ+ÝàlÞ[êá!ÞlêøÖ|ÞøØ#õlÚÖ Û+Øêø×ÛáÛ+Ø×AêvàlØÛ+ÝuÛÕvØ êvÖ ØÜ+ØÞ[Ú#Ø Ýö9ÛÕvØêvÖ|ä[ÝÙ|Øó2Ýó2ØÞÛ× áÞ[ê ÷ ÔJÕvØ d d êvØä8ØÞlêvØÞ[Ú#ØJÝ!öÛ+ÕlØ×Ø>êvÖ|ä8ÝÙ|ØJó2Ý!ó2ØÞ00 Û"×1Öp×1Ú#ÜàlÚÖpcc á!ÙöÝÜ6Û+ÕlØ R− Ö|ÞÛ+ØÜ"á!Ú#Û+Ö|ÝÞâJÖ Û+ÕÛ+ÕlØ [ØÙpêò7æ8ØÚá!à[×ØãÖ ælÜ"á-Û+Ö|ÝÞ[á!Ù1Û+Üá!Þvß ×Ö ÛÖ ÝÞ[×AáÜØ9áæ[×ØÞÛ2Ö ö ÷ 'Ø9Û+ÜØáÛ2Û+ÕlØ$Ö|Þß d (R) = const Û+ØÜáÚ"Û+Ö|ÝÞ$âJÖ ÛÕâ4Øá!åälÜ+ii Ýæ8Ø [ØÙpêÖ Þ$ÛÕlØ3öÜ"áó2Øâ)Ý!ÜåÝö6ÛÕvØ Ü+Ý!ÛáÛ+Ö|Þlë,â)á-ã!ØeáälälÜ+Ý-õvÖ|ó2áÛ+Ö|ÝÞCí Z Ô)Õà[×òeá!ÚÚ#ÝÜêvÖ|ÞlëÛ+Ý ú7ì8÷ ò;Û+ÕlØuälÜ+Ýæ8Ø¢×ä8ØÚ"Û+Üàvó Úá!Þ î ð æ ØAÚ#Ýó,älàvÛ+Øêà[×Ö|ÞlëÛ+ÕlØ,Ö|Þ[×Û"áÞÛá!ÞlØÝà[×>älÜ+Ýæ[á!ævÖ|Ù|Ö Ûç 8 Ý!Ü î ð Û+ÕlØä8Ý!älàlÙpáÛ+Ö|Ý!ÞnÝö>Û+ÕlØØ#õlÚ#Ö ÛØêØÙ|ØÚ#Û+Ü+ÝÞlÖpÚA×ÛáÛ+Ø!ò ρcc (t) = ÷ îþð á ×â4ØÙ|Ù3á×æçÛ+ÕlØuä[áÜ"áó2ØÛ+ØÜ"×<Ý!ö*Û+ÕlØuälàló,äÿá!Þlê älÜ+Ýæ8Ø älàlÙp×Ø×÷ ñBØÜ+Ø!ò Öp×Û+ÕlØuåÖ|ÞlØ#Û+ÖpÚØÞlØÜ+ëçøÝä8ØÜ"á-ÛÝ!ÜÝ!öÛÕvØ T Þà[Ú#Ù|ØÖ÷4Ô)ÕlØÖ ÞÛ+ØÜáÚ"ÛÖ|Ý!Þ Ý!öGÛ+ÕlØ3ó,ÝÙ|ØÚàlÙ Ø VLii = (ψi |VL |ψi ) âJÖ ÛÕÛ+ÕlØ [ØÙpê VLii = −(dii · EL (t)) cos(ωL t + ϕL ), P (Ω) = HS OK QP 3'R A T /Þ Û+ÕlØÚá!×+Ø'Ýö2Úá!Ü+æ8ÝÞó2ÝÞlÝõvÖpêvØ!ò.â4ØuÖ|ÞãØ×ÛÖ|ëáÛ+ØÛ+ÕlØ ä Ü+Ýæ8Ø ×ä8ØÚ"Û+Üàvó l Ý!ö9ã-á!Ù|ØÞ[ÚØ älÕlÝÛ+ÝÖ|ÝÞlÖ ýáÛ+Ö|Ý!Þ Ý!öÛ+ÕlØ π ó2Ý!Ù|ØÚ#àlÙpá!ÜøÙ|ØãØÙò'Þlá!ó2ØÙ|çòÛ+ÕlØÛÜá!Þ[×Ö Û+Ö|ÝÞ î 1 Σ+ ð → ÷ ) Ý Û ÿ Õ + Û l Õ n Ø 8 ä ! 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10ÞTÛjÕYÎ t/yû x/á "/íAóÖ1àgÎ"äÛjâ#ÙØðÖÛË7â1ÛÖêÛjÜëäÕYË ÝYî0Î ÕYÎÝÛjÕYÎå×ÏÝwÛjÎÝuÖj× Ûù Ü Þ ÛjÕYÎ Ð â#Ë0é×ÚË]Û#×ÏÜ Ý ÎYäÎ+Î+éYÖ~Ð ≈ 5 þ × 1013 * ]äØ 2 ÓÊIÕY×ÏÖ ÕuË7àYàgÎ+ÝYÖ þ ÕÎ+ÝãÛjÕY Î /Ë7ÌY×Þâ#/ÎLwÙYÎÝuäù Ü7Þ'ÛjÕYÎ Ð Ûjâ#Ë ÝuÖ1× Ûj×ÏÜ0Ý Ë ààYâ#Ü0Ë0ä#ÕYÎ"ÖAÛjÕYÎ'ÍÏ× ÞÎÛj×ÏØ~ÎEÌYâjÜwË0é|Î+ÝY×ÏÝî:Ü Þ Gνν = EL hν|d|ν 0 i ÛjÕYÎEäÜ0âjÎHÎ Yä× ÛjÎ"éÖ1Û#Ë7ÛjÎ Γ = 0.0575 Î 5ÓÊIÕ&Î tHyû x&"/íQóßàYâ#Ü7á uÍÏÎHË ÍÚÖ1ÜéÎ+àgÎÝuéYÖÖÛ#âjÜ0Ýî0ÍÏù5Ü ÝÇÛjÕYÎIÞâ#/ Î wÙYÎÝuäù:Ü ÞgÛjÕYÎI Ð IuÎÍÚé 0 q ¸1µÇÆeÈÆ»B¶BÉu° ´ u Æ °X³=ÊO»B¶ ´ ³¶BÆ;Ë Å + Intensity (arb. units) 5 4 (a) Ï/Õ\ÕÏÞfÓ Æ ÝÙaÙaÖAÔ ÆaÇ ÔÏÑ ∗ 15 N2 1s → π 2 IL = 5×10 W/cm ÆÞ Õ Ó\ÝÑÓÕÓÍA×1Ícà Æ ØÖRÓÔÏ<ç2èé-ê Åyëòä Ñ\Í ï8ÙaÏ&ØÍÑ2Ñ\Ý ÓÔÏÑ2Ô ÆaÇ Ô Æ Þcð Ê ÏÚÝÛ8ÕÏÍØ&Ó\ÔcÏvÕÔÍÑÓ 13 óô Ú× 2 Ò L > 5 × 10 Ù Æ ØÏ3Ó Æ ×1ÏRÍØÓÔÏRÚ3Í0ÑÏ1Ï3ßÚ Æ Ó\Ï/àdÕÓ\Ý Ó Ï È % æ ÔcÏ Ç ÑÍ0ÛÞ8àvÕ Ó\Ý ÓÏ Æ Þcð 14 2 13 2 12 2 ÓÏ/Ñ\Ý0ÚÓ Æ Í0Þ1Ý(õ?ÏÚÓ\ÕÓÔÏ ä ÑÍ Ê Ï2Õ ä ÏÚÓÑ\Û×ØÍ0ÑÙaÍ ÎÏÑ Æ ÞfÓÏ/Þ8Õ Æ Ó Æ ÏÕÒ í IL = 5×10 W/cm 401 Photon energy (eV) 402 (b) ' ∗ N2 1s → π 4 2 1 ÍÞfÓÑÝ ÑÖ¬ÓͬÓÔÏ&Þ Æ Ó\ÑÍ Ç Ï/ÞR×FÍ0ÙaÏ/ÚÛÙaÏÒ0ÓÔÏ&à Æaä Í0ÙaÏ+×1Í0×FÏ/Þ0Ó à Æ õ?ÏÑ%ØÑ\Í× ωL = 0.300 á ÏÑÍ Èyã Æ ÚÕ eV ωL = 0.236 eV ωL = 0.202 eV ωL = 0.150 eV ä Û× ä ÆÞ Ñ\Ý0à Æ Ý Ó Æ ÍÞ í îOÑÝà Æ Ý Ó Æ Í0Þ 401 ÆÞ Ç ÑÍ0ÛÞàqÝÞ8à Æ Õ Æ ÞAÑ\Ï/ÕÍ0Þ8ÝÞÚÏ2Î ωL Þ8Ý Ù Æ Õ&Ý(õ?ÏÚÓ\Ï/à×AÝ Æ ÞÙaÖ Ð ÆaÊ ÑÝ(Ó Æ ÍÞ8ÝÙ8Ù Ï/Ð0ÏÙÕ Æ ÞRÓÔÏ =0 ÄÅPÆaÇ8È_É<Ê_ËÈ Æ Õ Ç Ï/Ñ\ÝàcÏFÍ0Ñ Æ Ó\Ô#Ú3Í0ÑÏAÔÍ0ÙaÏ ÝÞ8àqÑ ÆaÇ Ô0Ó Ý ÓÍ0×AÕâ Ψ(N N ∗ ) ÛÏLÓÍ1ÓÔ Æ ÕÒÓÔϬà Æ ä Í0ÙaÏL×FÍ0×1ÏÞfÓ2ÍØ ∗ Ψ(N N) Èã ÓÑÍ Ç ÏÞq×1ÍÙaÏÚ3ÛÙaÏ Æ Õ%Ï/ÜfÛ8Ý Ù?Ó\Í á ÏÑ\Í Ä å(É0Ë d(N ∗ N ) + d(N N ∗ ) = 0. æKÔ Æ ÕK×FÏÝ Þ8ÕyÓ\Ô8Ý(Ó%ÓÔÏ&ç+èyé-ê Åë Õ ä Ï/Ú3ÓÑÛ×ìÍتÔÍ×1Í0ÞfÛ8ÚÙaÏ/ÝÑ Æ ÚF×FÍ0ÙaÏÚ3ÛÙaÏ1àcÍ4Ï/ÕLÞÍÓLàcÏ ä Ï/Þ8àcÕLÍ0ÞXÓÔÏ ÓÔÏLí îï8Ï/Ùà Æ ØÓÔÏLÚÍÑ\Ï-ÔÍ0ÙaÏ Æ Þ0Ó\ÏÞ8Õ Æ Ó ÖÍØ æ%ÔfÛ8Õ/Ò4ÓÔÏ&×1Ï/Ý0ÕÛÑÏ/×1ÏÞfÓ\ÕKÍ ØªÓÔÏí î Æ Ú − ä Û× ß4ðÂÑ\Ý(Ö Ç0Æ ÐÏLà Æ ÑÏ/Ú3Ó ä ð ä ÑÍ Ê Ï ä ÑÍ4ÍØÍ Øyà4Ïð ÙaÍcÚÝÙ Æaá Ý Ó Æ Í0ÞAÍØ|ÓÔÏLÚ3Í0ÑÏ-ÔcÍ0ÙaÏ È íÞ8àcÏÏà?ÒàcÛÏ&ÓÍ<ÓÔ Æ ÕKàcÏÙaÍcÚÝÙ ð Æaá Ý Ó Æ Í0ÞLÍ Ø8Ú3Í0ÑÏyÔcÍ0ÙaÏÒÓÔÏí îdï8ÏÙà¬àcÍfÏÕ|ÞÍ Ó Æ ÞfÓÏ/Ñ\Ý0ÚӪΠ×1ÍÙaÏÚ3ÛÙaÏ Æ Þvà Æ ä Í0ÙaÏAÝ ää ÑÍ(ß Æ ×AÝ Ó Æ Í0Þ Èë Û8Ú\Ô Æ Ó\ÔÓ\ÔcÏ Æ ÞfÓÏ/Ñ\Ý0ÚÓ Æ Í0ÞXÚÝÞ ÍcÚÚÛÑLÍ0ÞÙaÖvà4ÛÏAÓÍXÜfÛ8Ý0à4Ñ\Û ä 0 Í ÙaÏA×1Í×1ÏÞfÓ<Í Ø%ÓÔÏR×FÍ0ÙaÏ/ÚÛÙaÏ È ñÛcÓ/ÒPÓÔÏRÑÏ/ÜfÛ Æ ÑÏàØÍÑLÓÔÏ Æ ÞfÓÏÞ8Õ Æ ÓÖXÍ ØKÓÔÏ ä Û× ä ÙaÏÐ0Ï/Ù Æ Õ ä Û× ä ÛÙÕÏ Æ Õ-Í ää ÍfÕ Æ ÓÏ äÆ Þ Ç Ç ÑÍ0ÛÞ8àAÕ Ó\Ý ÓÏ ÅPÆaÇ8È*0ËÈ-, Í ä ÛÙÝ Ó Æ Í0Þ8ÕÍ ØÙ Ï/Ð0ÏÙÕ ÆÞ ÍØy×AÝ Æ ÞÙaÖqØÍ0ÛcÑ Ä ÕÏ/Ï-Û ää ÏÑ ä ÝÞÏÙ Æ Þ ä ÏÑØÍÑ\×ìÝ ää ÑÍ(ßfð ν = 0, 1, 2ä Æ ×AÝ(ÓÏ/ÙaÖ¬Í0ÞÏ2î+Ý ÊÆ ÍfÕÚ Æ ÙaÙÝ Ó Æ Í0ÞFàcÛÑ Æ Þ Ç ÛÙÕÏ Æ ÞfÓÏ/Ñ\Ý0ÚÓ Æ Í0Þ È ÓÔÏ æKÔ Æ ÕK×1ÏÝ Þ8ÕKÓ\Ô8Ý(Ó2ÎÏÛ8ÕÏ-ÓÔϬî2Ý ÊÆ|ä ÛÙÕÏ È æ%ÔÏÐ ÆaÊ Ñ\Ý Ó Æ ÍÞ8ÝÙ ä Í ä ÛÙÝ Ó Æ Í0ÞÕ1ÍØ&Ó\ÔcÏ Æ Í0Þ Æaá ÏàIÕ Ó\Ý ÓÏÔ8Ý(ÐÏòÞÍÓAÓ Æ ×1ÏÓÍ î2Ý ÊÆ ÍfÕÚ Æ ÙaÙÝ Ó Æ Í0Þ8Õ ä ÏÑð ä ä ÏÑ ä ÝÞcÏ/ÙyÍØ Õ Ï/ÏAÛ (0) Ä ω = ω10 ÅPÆaÇ8È/.fË ÒªÎ%Ô Æ ÙaÏAÓÔÏÖX×AÝ&0ÏAÍ0L ÞÏ1î+Ý ÊcÆ Í0Õ\Ú Æ ÙaÙÝ(Ó Æ ÍÞvÎ2ÔÏÞdÓÔÏ ä Æ Õ ï8ÏÙà ÆØ Æ ÞqÑ\Ï/ÕÍÞ8ÝÞ8Ú3ÏÎ Æ ÓÔqÓÔÏLí îÓ\Ñ\ÝÞÕ Æ Ó Æ Í0Þ ÆÞ Æ Í0Þcð í í<ÚÍÙaÙaÏÚÓ\ÕFÓÔÏ (c) ÄLÅPÆaÇ8È-*fËÈ æÝ Ê ÙaÏ ωL = ω10 î2Ý ÊÆ ØÑÏÜ0ÛÏ/ÞÚ Æ Ï/Õ Ø Í Ñ+ÓÔÏ Ç ÑÍ0ÛÞ8à (α) G 0 = EL hνÈ 0 |dαα (q)|νi ÝÞà Æ ÍÞ Æaá ÏàOÕÓ\Ý ÓÏ/Õ/Ò ν ν æKÔÏî+Ý ÊÆ Í0\ Õ Ú Æ ÙaÙÝ(Ó Æ ÍÞ8Õ1ÍØ Æaá ÏàµÕ ÓÝ(Ó\ÏÒ α = 0, c Ç ÑÍ0ÛÞ8à<ÕÓ\Ý ÓÏ ÓÔÏ ÓÍÓÔÏ ä Í ä ÛcÙÝ Ó Æ Í0Þ8ÕyÝÑÏ%ÕÙ ÆaÇ ÔfÓÙaÖ<×FÍcàcÛÙÝ(ÓÏànÒfà4ÛÏ Ê ÑÏÝ10wà4Í Î%ÞµÍ Ø2ÓÔÏqÑÍÓ\Ý Ó Æ Þ Ç ÎKÝÐ0ÏkÝ äcä Ñ\Íß Æ ×AÝ(Ó Æ Í0Þ Ä î ó ÓÔÏ Æ Í0Þ Æaá ÏàÕ Ó\Ý ÓϬÝÑÏÏÕÕÏ/ÞfÓ Æ ÝÙ ÙaÖqÕÓÑÍ0Þ Ç Ï/Ñ ÏÙÕ ÆÞ ê ËÈ æ%ÔÏ×FÍcàcÛÙÝ(Ó Æ Í0Þ8Õ1Í Ø&Ð Æ Ê ÑÝ(Ó Æ Í0Þ8ÝÙ ä Í ä ÛÙÝ Ó Æ Í0ÞÕ1ÍØ ÅPÆaÇ8È2* ÝÞ8à ÅPÆaÇ8È+.4ËÈ ×1Í4àcÛÙÝ Ó Æ ÍÞ8Õ Æ ÕªÓ\ÔcÏ Ê Ï/Ý Ó Æ Þ Ç Ú/Ý Û8ÕÏ/à ä ÝÞcð Ê ÖÓÔÏKà Æ õ?ÏÑ\ÏÞ8Ú3ÏÍ ØÓÔÏ Ç ÑÍ0ÛcÞ8àkÝÞ8à Ê ÖqÓÔÏFî+Ý ÊÆ Õ ä Ù Æ ÓÓ Æ Þ ÇÈ Ó ÎÍFÙaÍ ÎÏ/ÕÓ ä Ý ÞÏ/ÙÕÍØ ÅPÆaÇÈ3* Ä ÕÏÏLÙaÍ(ÎÏ/Ñ æKÔÏòÑÏÝÕÍÞ~ØÍÑ1ÓÔÏÕÏÕ ÓÑ\ÍÞ Ç Ð ÆaÊ ÑÝ(Ó Æ ÍÞ8ÝÙ_ØÑÏ/ÜfÛÏÞ8Ú Æ Ï/Õ Æ ÞqÓ\ÔcÏ Ý0Õ2ÎÏ/ÙaÙÝÕ Æ Õ2àcÏ/ÙaÍ4Ú/Ý Ù Æaá Ï/à È Õ ä ÏÚÓÑÝFÍ ØÔÍ0×FÍ0Þ4Û8Ú3ÙaÏÝ Ñ2à Æ Ý ÓÍ0× äÆ ÚÓ\ÛcÑ\Ï æ%ÔcÏ á Ï/ÑÍð ä Í Æ ÞfÓ2Ð ÆaÊ Ñ\Ý Ó Æ Í0Þ8Ý ÙªÙaÏ/ÐÏ/ÙÝÞà ä Û× Ì+Í(ÎKÏÐ0ÏÑÒ8ÓÔÏAÕÖ4×F×1Ï3Ó\ÑÖÍØyÓÔÏ1×FÍ0ÙaÏÚ3ÛÙaÏ<Ñ\Ï/ÜfÛ Æ ÑÏ/Õ+ÓÔ8Ý Ó ÓÔϬÚ3Í0ÑÏ&Ï3ßÚ Ó\Ï/àqÞ ä ÑÍ Ê ÏRÕ ÆaÇ ð Æ ÍÞ Æaá ÏàÕ Ó\Ý ÓÏÕ%ØÍ0Ñ Ý Þ8à (0) (c) Ò ωL = ω ä ä ωL = ω ÑÏÕ ä Ï/Ú3Ó Æ Ð0ÏÙaÖ È íÞqÓÔÏïÑÕ Ó+Ú/Ý0ÕÏ&ÎKÏLÕÏÏLà410 Ï Í ÛÙÝ Ó Æ Í0ÞqÍØPÓ10 ÔÏ ØÍ0Ñ× Ψ(N ∗ N ) ± Ψ(N NÆ ∗ ) Æ Ï)(<ÒªÓÔÏ ÞfÓÏ/Ñ\Ý0ÚÓ Í0ÞÍØÓÔÏ<í î È ÅPÆaÇ8È+* ÕÔÍ Î2Õ¬ÓÔÏ )Ï ( (c) ω = ω10 = 0.193ÆaÊ Æ ÚÕÍ Ø ä Í ä LÛÙ Ý(Ó Æ Í0Þ8 ÕÍ ØKÓÔÏ1Ð Ñ Ý Ó Æ Í0ÞÝÙÙaÏ/ÐÏ/ÙÕ-ÍØÓÔÏ \ ¹¹&¥\· ¹ /º/» ¥a 3¶(²f\f¢\ \·4 \¡c\¢\£ ± ® (c) τX= τ ¸ \¢\£ ²f¿ω = ω10¤¸ Á(§R«(® ´¼+_©¨0L½0¡c L\ £¤n%|K | AL¡c ¾ /L¹Àª +L ¢\ ¢ \\ ©¨fKªK| -¡¯4 ¥_ § \0/£§¡c\4§&£¨0y³v¯f § ¥ 2 £ ¨f+¢ ¨f/ & 2§ (¢¤/ ¦\§q ¤/f§R¢/ f¢\ §0\Ãy £ ¨R£ ¨f-¡c\¢£ ²0 K£ ¨f£½£ ¥ ÙaÍcÚÝÙ Æaá ÏàRÍ0ÞqÓÔϬÙaÏ3ØÓ (0) ÖkÓÔÏ Æ ÓÔwÓÔÏRÐ ÆaÊ Ñ\Ý(Ó Æ ÍÞ8ÝÙØÑÏÜfÛcÏ/Þ8Ú3ÖvÍØKÓÔÏ Ç ÑÍ0ÛcÞ8àÝÞ8à ÊÆ Þ8Ý(Ó Æ ÍÞdÍØ2Î%ÝÐ0ÏFØÛcÞ8Ú3Ó Æ Í0ÞÕ<Î Æ ÍÞ Æaá Ï/àqÕ Ó\Ý ÓÏÕ È ó ÔÏÞòÓÔÏ Ç ÑÍ0ÛÞ8à1ÕÓ\Ý ÓÏ2Ð ÆaÊ Ñ\Ý ð =Ê ω10 = 0.269 Æ Æ ?-4 P %/f f K|%| L¡c\¢£¤&/¥n ¥/ → π∗ (¢3¤ ¦\3§<¢ %¨0 /ª©¨0£ /¡k¤(P¤4§¬«4/££ 5«cª¡f £ y§ 2¡f ¤\® £ ¨fK%|K| ¬¡0 ¯4 K¥/§0 °\ \ £y±/¤/ ²0y¥n´³µ (£ 0 £ ¤/f§ ÛÞ Ç Ï/Ñ\Ý0àcÏAÚÍ0× ÍØKÓ\ÔcÏRí î Æ ÓÔAÓÔÏ Ç ÑÍ0ÛÞ8àÕ Ó\Ý ÓÏ<×1ÍÙaÏÚ3ÛÙaÏ/Õ È àcÖfÞ8Ý× ÓÔÏAÎKÝ(ÐÏ<ØÛÞ8ÚÓ Æ Í0ÞdÍØÓÔÏRÚÍÑÏFÏßcÚ Æ ÓÏ/àwÕ Ó\Ý ÓÏ æ%ÔcÏkàcÏÓÛÞ Æ Þ Ç Æ ÍÞ Æaá Ï/à#ÕÓ\Ý ÓÏÒ 402 Photon energy (eV) Æ ÍÞ Æaá Ïà¬Õ Ó\Ý ÓÏ/Õ Æ Þ4Ð0Ï/ÕÓ ÆaÇ Ý ÓÏ&ÕÏ ä ÝÑ\Ý ÓÏ/ÙaÖFÓ\ÔÏ&ÑÍ0ÙaÏ-ÍتÓÔÏ ÚÝ0ÕÏAÍØ%ÑÏÕÍ0Þ8Ý Þ8ÚÏ1Î IL = 0 Ç Ñ\ÍÛÞ8àLÝ Þ8à Ê Í Ó\Ô#Õ Ó\Ý ÓÏÕ È Ó Æ Í0Þ8Ý ÙyØÑ\Ï/ÜfÛÏÞ8ÚÖÒ Æ ÓÔ Ê ÍÓÔ ÛÏÓ\Í<Ó\Ô Æ Õ/ÒÓÔÏíîï8ÏÙàòÝ õ?Ï/Ú3Ó\ÕKÓ\ÔÏLÞ4Ûcð ØÑÏÜ0ÛÏ/ÞÚÖRÝ ÙaÙaÍ Î2ÕÓÍ Î 0 400 ë Ï/Ú È_ËÈ ÍØ4Ó\ÔcÏKÚÝÑ Ê Í0Þ×1Í0ÞcÍ(ß Æ àcÏ Æ Þ Ú3ÙaÏÝ Ñ<àcÖ4ÞÝ× 3 Ä ÕÏÏ 2 ÷2øfùaúfû_üúAýnþ8ÿ\ÿaÿûøaÿû~ÿ cú0ûþú ý _ù \ ú ! û 3 ÿ ? û vøc û "aÿ# û ($ ú %ø$ú& IL = 0 400 óöô Ú3× ∼ 1012 L IL = 5×10 W/cm 5 à Æ Ý(Ó\Í× Æ Þcð àcÏ×1Í0ÞÕÓÑ\Ý ÓÏÓ\Ô8Ý(ÓyÓ\ÔÏ Æ Ó\ÔdÓÔÏRÚ3Í0ÑÏ1Ï3ßÚ Æ Ó\Ï/àd×1Í0ÙaÏ/ÚÛcÙaÏ ÓÏ/ÞÕ Æ Ó Æ Ï/Õ/Ò%í 2 L ÅPÆaÇÈ4É ÓÏ/Ñ\Ý0ÚÓ Æ Í0ÞvÍØÓÔÏRí î5ï8ÏÙàdÎ IL = 5×10 W/cm 1 Æ ÞqÓÔ Æ Õ2ÚÝ0ÕÏ È æ%ÔcÏ2ÑÏÕÛÙ Ó\ÕyàcÏ äÆ Ú3ÓÏ/à 3 0 Intensity (arb. units) ¶J̵º¶*±!°m³m¸±!° ´ »²LÁ°XÀÍÉ;ËÍÎvÏÑаX±! ²ÀÍÉu¶JÊ6¿ ¹HÀ1 ( ° Ó3³m² ÊO°m ´ ÊO¹7ÊO° ´ Ò »¶*¸À' æKÔÏFÚ3Í0× Æ Í0Þ Æaá ÏàòÕ Ó\Ý ÓÏÕÒ ä Ý Ñ Æ ÕÍÞÍØÓÔÏ Ç Ñ\ÍÛÞ8àRÕ Ó\Ý ÓÏ ÕÔÍ(Î+ÕÓÔ8Ý ÓKÓÔÏ Æ ÞfÓÏÑÝÚ3Ó Æ Í0ÞLÝÙa×FÍfÕ ÓPàcÍ4Ï/ÕPÞcÍÓ Æ Þ48ÛcÏ/Þ8Ú3ÏyÓ\ÔÏÐ ÆaÊ ÑÝ(Ó Æ Í0Þ8ÝÙ ä Í ä ð ÛÙÝ(Ó Æ Í0Þ8Õ+ÍØÓ\ÔcÏ Æ Í0Þ Æaá Ï/àXÕÓ\Ý ÓÏÒ Ê ÏÚÝÛ8ÕϬÓÔÏFí î Æ Þ~ÑÏÕÍ0Þ8Ý Þ8ÚÏ1Î Æ ÓÔ~Ð ÆaÊ Ñ\Ý(Ó Æ ÍÞ8ÝÙÓÑÝ Þ8Õ Æ Ó Æ Í0Þ8Õ Æ Õ+ÞÍ Ó ï8Ï/Ùà Æ ÞwÓÔÏ Ç Ñ\ÍÛÞ8à Õ Ó\Ý ÓÏ È ÅPÆaÇ8È65 à Æ Õ ä ÙaÝ(ÖcÕRÓÔÏdÕ Æ ×<ÛÙaÝ ÓÏà à Æ õ?ÏÑÏ/Þ0Ó~ØÑÏ/ÜfÛÏÞ8Ú Æ Ï/ÕdÍØkÓÔÏOí î ä ÑÍ Ê ÏdÕ ä ÏÚÓ\Ñ\ÝwØÍ0ÑqÓ ÎÍ ï8ÏÙànÒ (0) ÝÞ8à ωL = ω10 787 _ ¸²!µº¹U»:9U° ´ ¹HÀ' 7 _ q 2q °X±;µ½¸'¾3¿1¹HÀ1¶XÁ <:=?>A@B3C)CED<#FEGH$IKJ LEJ MNHIPO$J QMNR SUTVMNTSH1LKI M W+LKXSVYFKMNZ$HOGHO\[NG]R SH^SUJ MHJ _S`OIaLEG`LESIbMWc+deTVMR S^Z$R SgfhJ HiSj1k1S`lmGHOnG]jJ dνν 0 ^ M]F)FESI)QMHO$I+LKMPLKXSQSGxU[NG]R ZSMWLKX$SmCtnyJ H1LKSHI)J Ltk WFKS`opZSH^qk 0 fhJ HrStsmlEuv?SFKSw 12 zy{ ^T 2 u Gνν 0 = EL hν|d|ν i ν→ν | → } | → | → | → } → } → } → → → → IL = 5 × 10 EL fhimSj1kpS`l u |p~]N } } u ||N | u ||N|] ~ u ||N||N } u pN } u| | }N} p} u ||N| N } u ] } } u | ~ N ] } u p ] p}N} (0) dνν 0 0 | | | | | | | | | | (0) Gν 0 ν | u N } | u |N | u ~`p | u | u N } | u } | u }} | u N | u |N | u fhStsml ν→ν | → } | → | → | → } → } → } → → → → ×10−1 ×10−3 ×10−4 ×10−5 ×10−1 ×10−2 ×10−3 ×10−1 ×10−2 ×10−1 hf iSjpk1S`l u p } } u | |N } u |N||N } u |N||N| u | u | N|N } u |N| N } u ~`Np~ u | N } u N (C) fhStsml G 0 | u ν| ν } ×10−1 | u N } ×10−2 | u ×10−4 | u }` ×10−5 | u ×10−1 | u p~ ×10−2 | u | } ×10−3 | u ×10−1 | u p~] ×10−2 | u p −1 (C) dνν 0 0 | | | | | | | | | | ×10 -7 8×10 0 1×10 ν=0 ν=1 ν=2 ν=3 ν=4 -1 8×10 -1 -7 6×10 -7 4×10 ν=0 ν=3 ν=4 -7 2×10 0 ρν ρν 6×10 -1 4×10 -8 9×10 -8 6×10 -1 2×10 -8 3×10 0 0 200 400 t (fs) 0 300 800 600 IL = 0 -5 2×10 0 -6 3×10 ρν 500 t (fs) 600 700 :C)bu D6fcAMR M]F6MHR J H$S]l-i?k$HG]TVJ ^I2MW/LKXSbQMQZR G`LKJ MNH8MW[&J j$FEG]µ LKJ MHGR$R S[pSR IMWLKX$S+HGR$J MNH$J ^2I)LEG`LKS?= 2 MW:c+d + u (0) ωL = ω10 = Sasbu Stsu=lC Π~ 12 zy{ ^T 2 uy<AX$S Ω ×10 0.269 X = −0.036 L¶ QMQZR G]LKJ MHIMW#R S[NSR I G]HO wN·2X$J ^EXG]FKS2I)J TVJ R G]FLKM w =G1R R SqF2QM QZR G`LKJ MNH$I?G`FKS6H$ML?O$SQJ ^qνLKS`= G]I-·ASR R:G]I2R S[NSR I2·-J LKXνI)TV OUJ4H LKXJ I?QR MLu6>+l-C ~ 13 zy{ ^T 2 umdH$R krLKXSb[&J j&FqG`LKJ MNHGRR S[pSR w$·-XJ ^EXXLG`[N¶ SLK×10 X$SmR G`FKYNSIaLQMNQZ$R G]LKJ MH#wJ I-I)XM·-H#u -5 4×10 400 -6 2×10 -6 1×10 ν=3 0 -6 2×10 -6 d00 = 0 0 ¦ E q#pVqg`$qEª$¢#®£¢¸\EªNq¦ 1r&¬$EPq ¦q&¬&Am£?¬rN&¢2q#pqb£¢q$q£$¢ £ q¦ ¨K©¹$®£1q£$¢ £ qº¸$q&«¢#®3E1q!£Kq&¢¸$3E«ªªqNqEN® ¢³q!¨K© ¡§qN»$«¢]g£q«¢®¡&m¡§&¬¼V¸$q$«¢#®\Kp½£ ² 1q£&¢#& ¡§qN»$«¢]&gm81¬8`$¢#`«#¥ E£$¢£½1&£®¡§&q8¸&&«¢#® (0) K1q£¢q]&`q£$¢ ± ωL = ω10 Eª#&¢m 0 &¢#® 0 ¯ ´?&¬ª#1£E$¢"1¡¾£¸+¿¤ 0 &¢#®¾£¸#A¿¤® 0 qNVq#1Uq ¬81£¢ qNE« 6&¡AqVqNE$¢&¢m£¢q]¦&`q£$¢ 1¡A¨E©ÁÀ#]®\£¢ q ¸$q$«¢#®ºK1q£q"& ² &¢#® £Á&q£q¦ \®«qyq ½£ ² p£&¢#&:`Â`£ 1q£$¢\£¢\q¸$q$«¢#®g¦ E1q ]t m& ² &¢®V1¦ ¬KU®£1ªªN1 ]¢"qr¨E©À#®£ q«¢®£¢qNE$¢&¢#` £ q¥ ½£ ² p£&¢#&q&¢#E£ q£$¢1¡/q £&¢£Ã®¤q$¢£rKp E]¾£¸#¿ ac −¦ cc ¯m£P£ ² ]° ]&«#E q!£¢q$q£$¢3&¡P!¨K©Äª«q £ q¦ ¹q¤¸&&«¢#® K1q½£ ² 1q£$¢&½$]£&Å°aqNE$¢#1¢¢ º¨)¢#K$®y&¡ 1×10 400 t (fs) 600 800 :C)u D?fcAMR M]FMHR J HS]l-<MQ#D:MNQ$ZR G`LEJ MNHI2M]W/[&J j$FEG`LKJ MNHGRR S[pSR I MW2LKXSUYFKMNZ$HOIaLEG]LKS 1 + MWmc+du Sas Σ Ω = −0.036 ω = Stsbu1>ML)LKMNTrD:sJ j$FEG]LKJ MHG]RQXMNQ$ZR G`LKJ MNH$IM]WLKXSALHGR (0) ω10 = 0.269 J MHJ ^gIaLEG]LKS\= 2 MWPc+d + u Stsb (c) Π ΩX = 0.0336 ωL = ω10 = Stsbu ^T 2 u IL = 5 × 1012 zy{ 0.193 (c) ω ¥§¦ L = ω10 ¦ E]"`&q!q$ &¡U]Nq&¦ ¢£¤Kp £!£¢q]&` £ q¤¦ qr¨K©ª«¬ª!$®£p£&¢¯ ¥ r]&`«° 1q®8&EVqEªN £ q d00 = 0 $ dcc = 0 E]ª#1¢ c ±b² c &¢#®³ 0 ±b² 0 ¯`´?&¬ª#&q£E$¢&¡¤ª#&¢] ² c 1¢#® c ¶J̵º¶*±!°m³m¸±!° ´ »²LÁ°XÀÍÉ;ËÍÎvÏÑаX±! ²ÀÍÉu¶JÊ6¿ ¹HÀ1 ( ° Ó3³m² ÊO°m ´ ÊO¹7ÊO° ´ Ò »¶*¸À' 0.5 0 0 1.0 0 0 0→ ΩX (eV) 5 0→ 0 0→ 9 0→ 10 0→ 11 0→ 6 0→ 7 0→ 8 0 IL = 0 0→ 3 0→ 4 0→ 1 0→ 2 (d0) 2 2.0 1.5 0 1→2 →0 0→ 1 1→3 0→ 2 1→4 0→ 3 0; 4 →3 1→ 0 1→ 1 2→ 1 0→ 0→ 5 0→ 3 0→ 7 0.0 (c0) 4 0→ 8 0→ 9 0→ 10 0→ 11 0 2 6 IL = 0 0→ 6 2 (dc) 0→ 4 4 0 0 0→ 2 6 TPA 0→ 1 0 2 3→2 2→1; 4→4 1 d00 = 0 4 2→2; 4→5 2 (cc) 0→ 0 0 TPA (b0) 4→2 3→1 1 4→ 0 d00 = 0 6 0 0→ 0 Intensity (arb. units) 0 (bc) 3→ TPA 0→ 0 Intensity (arb. units) 2 dcc = 0 0→ 1 2 0 (a0) 0 dcc = 0 0→ 0 4 2 2→ (ac) 4→ 0 4→1 6 ¸1µÇÆeÈÆ»B¶BÉu° ´ u Æ °X³=ÊO»B¶ ´ ³¶BÆ;Ë 1→ Å 3→ q Æ -1.0 -0.5 0.0 0.5 1.0 ΩX (eV) 1.5 2.0 Þäå Ë Ç:È)ÉÊË1ÌmÍÎAÏÐ ÏÑmÏÒÐ Ó Ò$Ô]Õ-Ö/×Ï]ØKÏ1ÔÐ ÔÙqØ)ÑKÏNÒÚÛÒÜrÝ 1 + Ï]ÞÙ`Û`ÑKÚÏNÒ8ßVÏÒÏ`à&Ó Ü$ÔÊ Ùß 2 Ê2èA×ÔbÈaé τX á τL ábâãNã IL × 12 æyç Σ → X2Π ê Ô Ð Ü\Ó ä6ØKëÒÔÜÓ ÒäaØ)ÑKÓ ÙØ6ÑKÔäKÏÒÛ]ÒÙÔUì-Ó ØK×\ØK×ÔVÔqà$ÙÓ ØKÔÜ\äaØqÛ`ØKÔUí&Ó Ú$ÑEÛ]ØKÓ ÏÒÛÐÞÑKÔî1ë$ÔÒ$Ùïpå Ê Ôtóá ÍhÐ ÔqÞØPâ`ãôÛÒÔÐÕ6ÛÒÜì2Ó ØE×\ØK×$Ô (c) ω ω í&Ó Ú&ÑEÛ]ØKÓ ÏÒÛÐÞ ÑEÔî1ë$ÔÒ$ÙïÏÞØK×Ô6õÑKÏëÒÜäaØEÛ`ØEÔå Ê ö÷ ÔtóøÍÑEÓ õN×pØôÛÒÔÐÕEÊ-è+×$ÔbLÈtá é ê 01ÔÐ áð ÜÓ ã ÒpØKâÔñNÑEò Û]ÙØKä-ì-Ó ØK×ØK×Ô6ßVÏNÐ ÔÙëÐ Ô6Ó ÒÚÏØK× (0) ω ñ L á ω01 áºã õÑKÏëÒÜÛÒÜÔà&ÙÓ ØEÔÜäaØEÛ]ØKÔä6ÍhôÛÒ$ÔÐ ä-Ù ÛÒÜUÙ ÕEÊ c 0 ù]úûù]ü#ýùýþÿ&û1ü#ý ùE ùùþùq ùÿ$ü ù ù ΩX − 0 = ωL ù!ÿ 0ûqþù# "$%" û&&1ü' û'ÿ&ü()Eùù ûþù ù +,* '$ùý+-/.10324ü5 6 ac − cc +.þùÿ & ü"ÿ 7 ûqþ8 ù #86ûqþù/`ÿ$þù!qùüûmû ÿ $ þÿ1ûÿ&ü Eÿ- û'ÿ$ü9).1032 ωX + ωL ù`û ù]ùüûqþù;: ù!q ÿ $ ÿ- ü$û=<>&1 û' ÿ$üù!<$ù8-ÿ7:ûþù -q ÿ-? üý@& ü#ýù]úA! ûqùNý@E û&1 ûqù |0, 0i → |c, 0i Bm. þù;.1032CDqÿ>' E ù û& ÿ$ü,8 q ÿ ÿ q û' ÿ$ü ûqÿEF&G-? q ù6ÿ:7 ûqþùH.1032C* û'#? ýù JIK σ(ωX + ωL ) = |F |2 δ(ωX + ωL − ωc0,00 ). . þù;!-E ù ) >L ` ÿ-'q ù ÿ$ü#ýA ûqÿF'*M? û&& üù]ÿ-? N;O û'1 üPd û&00 ÿ$ü,=7Qq 0ÿ-* ûqþùL -q cÿ- ? ü#ý,E û&1 ûqù |0, 0i ûqÿûqþùR ÿ&ü]: ùNý ù ù û&q ÿ$üA8D$S<TAp û& ÿ&ü =K û&1 ûqù |c, νi & ü#ý"ûqþùUJV û'&& üP û' ÿ$ü üûqþùE ÿ&ü]: ùNýK ûp ûqù ν → 0 ;m . þùE &!1 ûEûqù' ü , * û'#? ýùÿ 7 ûqþ8W þ1 üüù!=E ù]ùF X/2 ù$ ý F ∝ X hc, 0|eL · dcc (R)|c, νi(eX · dc0 )hc, ν|0, 0i ωX − ωcν,00 ν ∝ (eL · dcc (R0 ))(eX · dc0 ) hc, 0|0, 0i. ωL JIY Z þùü )6 @[ ûqþù(.032 qÿA]ù'\`ÿ-'qù!$ ÿ$ü#ýA6dûqccÿ# =qÿ 0 û'ÿ&ü\ÿ 7=üA7Q&c q ùNý þÿ&ûqÿ$üü ÿ ?ü#ý]Kû&1ûqù |0, 0i → |0, νi -`ÿ-* &üùý^T_ qÿ û'ÿ&ü!ÿ 72ûþù qÿ-ù þÿ1ûqÿ$ü |0, νi → |c, 0i ) AX-` A.þùR]ÿ-'qù ÿ$üýAü E.1032 * û'#? ýùþ-? ûþùR§7 ÿ-'*]a (eL · dcc (R0 ))(eX · dc0 ) JI F ∝ hc, 0|0, 0i. (0) ωL − ω10 b ÿ-* '8qÿ&üyÿ7ûqþùýùüÿ *@ü1ûqÿ-&E üûqþù'!1ûEûù!'ü + */$ ûþùf.1032 ù û'#? ýùcPI!Y & ü#ýdJIe ù`ú 8 ü þT¼ ü ;\ c 8g'? A ù'qùýC`ÿ-* 'ü ûþºûqþù"ÿ$üùü E c =aU û8 ÿ q ûqþüÿ1û& ü ûqþ1 ûW@E ùG-? ùü$û&8Bh û ÿ $%K ûqù û ÿ $ þÿ1ûÿ&ü û&&&ü' û'ÿ&ü ûÿûqþù |c, 0i KûpûùM8H8qÿ ÿ>''Aù- h ÿ ù&< ù ûþùgû ÿ $%K ûqù qÿ û& ÿ&ü\§7 ÿ ûqþù]!-E ù d00 = 0 þ<$ù+üÿ&ûiqùqÿ&ü&ü`ù+üùûþù1.1032 ùREþÿ üHüE )U ü\ûqþùE!-E ù d = 0 ûqþùEE ùG> ? ù]üû'8 q ÿA` ù' $< ùW1 üÿ&ûqþù!c cc qùEÿ$ü&ü`ù ÿ>P û'ÿ$üja ωX = ωc0,00 − ω10(0) h ÿ ù<&ù ûþ û ÿ $%K ûù û ÿ $ þÿ&ûqÿ&ü q ÿA] ù'= ù $ ù!+ ûþ1 ü8ûqþùH` ÿ$þù!q ùüû .1 032cJIe ù! ?Eùûþù;Dÿ-&BEù`û' ÿ$üE8 ÿ ÿ-Eû& ÿ$ü $ûÿPûqþù 'G>? ùýyýù'<e1 û'$< ù ÿ b7 ûqþù -q ÿ-? üý5K û&1 ûqù¤ýA ÿ ù]* ÿ-r* ùüûa 787 _ ¸²!µº¹U»:9U° ´ ¹HÀ' 7 _ °X±;µ½¸'¾3¿1¹HÀ1¶XÁ d 00= 0 d cc= 0 0 ωL 0 ωX ωX ωL 0 0 B A kml%nLo3pq/r)stu tvHtw>u x w>yDz;{j|} ~%x D!u>x 'J-vJyMt!BS 6t!)>|>t!JtwgvJy&~%t wT!w>&y~r~%yyB{\Ay~=x w/kmx oAzPoi1z6kmx wTuA~%PDJyx weJy&vP!&Jx tw oBz=nvJt>wTM~%PDJyWx weJy&vP'Jx tw d = 0 o d =0 00 cc >'&-'-d 5'4J¡ !¢8£d¤W '¥'4 ¦£ § ¢D¨¢R'©'¨A¢ ©1g'R; ªAi©'«¢ ''¬A¤WA8] F! ]©P ! § «&A¬]®¬>© 6¯ ª ££ #° ©P!/'M'@!±mDF 'EW ªi©P«¢ P'¬,c!' ©'©cQ-;A¬-!W²>ª&'-¢3¢² ¢8©! ³ ¬-'! § !>/¤W '³ ª¢´' ]¬-'¤1&µ L'^;ª Q'¶>¨AD·,¤ 'g!'-©P; ν 8©W£T¨L'E'HD&-©PLj&A '©P & ]£A«¸ ¢ § § !> dν,ν+1 ∝ √ν + 1 ° 1£A;©'!1W ªA>©P«¢ P& ¬L¤!F'¹P+ !¢8£F >'&-© ¤ &º¬-'-¨A£³©J&' § -¢ D¨¢µ»)®6 ¬ ¯ 0 ¼ ¾½ '-ªª¢·\&A © § ¢¢©P«¢ P'¬_ 1'#'©P-!©R¤ '\¢8 '¬- 8©M£A¨' '#W ª©P«¢ P&A¬ _¿ @ 1&A#«¸>©'©Pª¢,' ©'ν©R;©P¨& ]¶>¨¢ &'²-,£T ±m'!!/ª¸!J¤!+¢!M £'¬->H« A¢8©H ®¬ ¹¯ Àª¸g'e,'JÁ«¨¢8©P] ©,£A¨&'-µ¤8&³8© ! § «ª¢F¤ '&A/W ªi«¸!&T£ H¿ E! ^©PR';&A ²Tª&' ¢«¸-«¨¢8e& -©3 >&A¹ ¦!£R©P&'¹£A £A8©P«¢8· W ª6>©'! ¢¢8'-©WQ- 12 °C § 2 jD->'& '·g' 'g!-©P, ¬-AE >'IL=©P J· 5 ×IL10= 5 × 1013 °Â § 2 »©P ®¬ à LÄ ¨F&,&A8©m /D-¨¢8£DÅA«¸D;&A/W ª6©P«¢ P& ¬ Q-¼ ¬ >'©' '©1' IL = 5 × 1012 °C § 2 q à «'T¢D''-_©P«¸D''¨ § j¤WA¢E'@-&!&-D' ^ =&A@J !¢8£F¤ 'M'!ÅA! 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II,III & Ä '(' Å ( A Ãjà ÿ Z Ç*)f - +-,/.0,/.2143658795;:=<?>[email protected]:7FB1G,H:JI-79,/.LKBM8MNBOP58<AQR</B<SUT".219B63R:74195WVX.YMN.23Z.2<[7FI ı 1, c = y, c1 = x hΨc (S)|VSO |Ψc1 ,0 (T )i = ζ 2 −1, c = x, c = y, 1 hΨc (S)|VSO |Ψc1 ,1 (T )i = hΨc (S)|VSO |Ψc1 ,−1 (T )i∗ = −hΨc (T )|VSO |Ψc1 ,1 (T )i −ı, c = y, c1 = z 1 ı, c = z, c1 = y = hΨc (T )|VSO |Ψc1 ,−1 (T )i∗ = √ ζ 2 2 −1, c = x, c1 = z 1, c = z, c = x, 1 1, c = y, c1 = x ı hΨc,m (T )|VSO |Ψc1 ,m (T )i = mζ −1, c = x, c = y. 2 \U]J^ 1 +-,/._I9BJM8`A745NBJ<BKG74,/._a"BJ`ADAMN.2bc.2de`H:745NBJ</I \f=^ :</b \gh^ DA1FBeb`/a".jiXb5Wkl.219.2<e76.Y58Q[.2<emJ:M8`/."IO-5879,nobABJ`ApAM8q bA.2Qh.2</.Y1:79."bXMN.Ym[.2MNI"r/b`/.s7FBt74,/.I4qu363Z.27414qjBK74,/.s3ZBMN."a2`AMN.Jv wZxzy9{(|J};~[FY; "~~[X~[{=~hFY|= ";{( +-,/.-:36DAM85874`/bA.GB=KE74,/.1F.2IFBJ<H:=<e758</.YM;:JI4795NaVuSU1F:"q0:3R:<ZI9a:=7979.21458<AQ 0.Y5NIF.Y<epE.214Q6KBJ143`AM;: \ U> ^ 5NIQJ58m[.2<6peq79,/.L1F:3Z.219I4S \ L ^ Ff (q) = 12 X h0|(e · D)|ΦΛ ihΦΛ |(e1 · D)|Ψf (q)i , ω1 − ωΛ,f (q) + ıΓ \i ^ q = S, T, Λ=1 OP,/.Y1F. Γ 5NIs74,/.6M85WK.274583Z.6pA1FBh:JbA.2<A58<AQoBK79,/.Za2BJ19.YSU.YVAaY587F.2bI47F:7F.J r ω e r:=</b r ω 1 e1 :19.679,/.tK19."de`/.Y</a25N.2I :</bDEBJM;:1458T=:745NBJ<cm[."aY7FB1FI6BKP79,/.R58</aY5NbA.2<e7R:=</b74,/._IFa":747F.Y1F.2bVuS1F:"q?DA,/BJ7FB</Ir1F.2I9DE.2a27458m[.2M8qE EΛ − Ef (q) I458<AQhMN.27 \ ωΛ,f (q) = 5NI_79,/.o1F.2IFB<H:<e7jK1F.2de`/.2</aYqBK.2365NI9I95NBJ<791F:</I458795NB<K1FB379,/.a2BJ1F.SU.YVAaY587F.2b7FB79,/./<H:M Ψf (q) = Ψf (S) ^ B1-741958DAMN.Y7 \ `A</Bua2a2`ADA5N."b3ZBJMN."aY`AM;:1BJ19pA587F:M ψν Ψf (q) = Ψm f (T ) \ 58Q/v f=^ v+,/.58</bA.V ^ I47F:7F.2Iv+-,/.0a"BJ19.YSU.YVa25879."bI97F:79.sbA.2DE.Y</bAIPBJ<j74,/. ν BK/74,A5NI ?@¡5NII4¢u58DADE."bLKB1pA19.2me587qBK/</BJ7F:795NB</Iv U7PO-58M8MpE.L19."I479BJ1F.2bX58<>u.2ahv/9FA£¤7FBZI4,/BO¥74,/.s5836DEBJ147F:</a".B=K74,/.58<e79.21K.219.2</a2.BK79,/..YM;:JI4795NaI9a:=7979.21458<AQ a9,H:<A</.2MNIj74,A1FB`AQh,b5Wkl.219.2<[7`A</Bea"a2`ADA5N.2b @0I"v¥+-,/.791F:</I458795NB<b58DEBJMN.3ZBJ3Z.Y<e7FIBKa"BJ19.o.YVAa2587F:745NBJ< h0|D|Ψc (S)i ≈ 2pc √ :19.6.YVDA19.2IFI9."bo79,A19BJ`AQh,?74,/.6BJ</.YSU.YMN."a27419BJ<741F:</I95W795NBJ<b58DEBJMN.t3ZBJ3Z.Y<e79IpE.27UO.2.2< 2dcν :=</bc19.2M;:=V."bc`A</Bea"a2`ADA5N.2b¤ @ 741:</I458745NBJ</I ψf → 2pc Bua2a2`ADA5N."bo @ ψf :</bo741958DAMN.Y7I97F:79."I ψν v?+-,/.j3R:=79145WV¡.YMN.23Z.Y<e7FI hΨc (q)|D|Ψf (q)i ≈ dcf BK74,/.jbA."a":"q :1F.:DADA19B"V583R:79."b_peq79,/.B</.YSU.2MN.2a2741FB<j791F:</I458795NBJ<_b58DEBMN.G3ZBJ3Z.Y<e7FI"rupE.27UO.2.2<_74,/. :=</bo74,/. g p a2BJ1F.,/BJMN.hv0+,/.6>A@C58<e79.21F:Ja2745NBJ<58<o79,/.a2BJ19.YSU.YVAaY587F.2bI47F:7F.s365WVA."ILI458<AQhMN.Y7 \¦[^ v§0`/.79B_79,A5NI"rE58<:JbAbu58795NBJ<79Bj74,/. S →S→S IFa":747F.Y1958<AQXa9,H:<A</.2MNIL79,/.741458DAMN.27 ú ) =à þaÿ ú û¶ü:ý û¨ !" 9(ÿ ûÄ ûTÅ Æ ÿ û ZÇT ÿ û Ç8 $#% à & TÈ û '( 7 û¨ ÿ) ; * #^þ !,+ .- û0/ Ç8 ÿ, 7 û 7 û¨ û ν >%$> Cl Z 5σ 4σ 3/2 2p H 1/2 ©EªU«¬/=®%¯h°4±h²³L²´µ(¶±[²G·¶¸e¹hº»²¹¶¼4½"¾e·Uº ¶º»´¾e· ¿9ÀHÁÂAÂ/Ã2ÄÅ S → T → T ÅAÆEÃ2¿"ÇJÈZÃÁJ¿"¿2Ã"É9É9Ê8ÆAÄNÃJËÌ-À/ÃLÍGÎÏjÁÈ6ÐAÄ8Ê8Ñ4Ò/ÓAÃÇ=ÔÑ4À/Ã"É9Ãs¿9ÀHÁÂAÂ/Ã2ÄNÉGÕFÃ"ÁJÓAÉ Ff (S) = (Λ)) ∗ 12 √ X (S)) (e · D(Λ) (S))(e1 · Df 2 , ω1 − ωΛ,f (S) + ıΓ Λ=1 Ffm (T ) = Ö×hØ (Λ)) ∗ 12 √ X (e · D(Λ) (S))(e1 · Df m (T )) . 2 ω1 − ωΛ,f (T ) + ıΓ Λ=1 Ù ÃYÕFÃJÅ D(Λ) (S) = X Cc(Λ) (S)dcν X Cc(Λ) (S)dcf , c=x,y,z (Λ) Df (S) = ÖÚÛhØ (Λ) Df m (T ) = X (Λ) Ccm (T )dcf . c=x,y,z c=x,y,z Ì-À/ÃLÍGÎÏ_¿YÕFÇhÉ9ÉGÉ9Ã"¿YÑ9ÊNÇÂÊ8Â/¿YÄ8Ò/ÓAÃ"ÉPÑ9À/ÃsÉF¿"ÁÑ9Ñ9Ã2Õ4Ê8ÂAÜZÑ9Ç6É9Ê8ÂAÜhÄNÃYÑ0Á=Â/ÓXÑ4Õ4Ê8ÐAÄNÃ2ÑPÝ/ÂHÁ=ÄÉ4ÑFÁÑFÃ2É σ(ω, ω1 ) = X f Þ À/ÃYÕFà |Ff (S)|2 Φ(ω1 − ω + ωf (S),0 , γ) + X m=1,0,−1 |Ffm (T )|2 Φ(ω1 − ω + ωf (T ),0 , γ) ÖÚhÚØ ωf (q),0 = Ef (q) − E0 ÊNÉÑ4À/ÃRÔÕFÃ2ßeÒ/Ã2Â/¿YàÇ=ÔGÑ4ÕÁÂ/É4Ê8Ñ4ÊNÇJÂcÔÕ9ÇJÈáÜhÕFÇÒAÂ/ÓcÑ9ÇÝ/Â/ÁÄPÉ4Ê8ÂAÜhÄNÃ2ÑÇJÕÑ9Õ4Ê8ÐAÄNÃYÑ É4ÑÁÑ9ÃhËÌ-À/ÃsÉ4ÐEÃ"¿YÑ9ÕFÁÄlÔÒAÂ/¿2Ñ4ÊNÇJÂÇÔÊ8Â/¿2ÊNÓAÃYÂeÑÕFÁJÓÊ;ÁÑ4ÊNÇJÂÊNÉÁÉFÉ4ÒAÈZÃ"ÓjÑ9Ç6ÆEÃÁZâsÁÒ/É9É9Ê;Á= 1 Φ(ω1 − ω + ωf (q),0 , γ) = γ Þ Ê8Ñ4À r ln 2 exp − π ω1 − ω + ωf (q),0 γ 2 ln 2 ! ÖÚãhØ ÙGäåÙ0æ Ë γ ÁJÉ ç Ã2Ñ-Ò/ÉGÑ4ÒAÂ/ÃLÑ4À/ÃÃYèA¿YÊ8ÑÁ=Ñ9ÊNÇJÂÃYÂ/Ã2Õ4Ühà p ω ÔÁÕ-ÆEÃ2ÄNÇ Þ Ñ9À/ÃÁ=Æ/ÉFÇJÕ4ÐAÑ4ÊNÇJÂoÃ"ÓÜ[à Ω2 + Γ2 ∆SO , ∆ST , Ω = ω − ωedge ÖÚéhØ >%$ Å ÿ à ? Æ ê0ëYìFëJí ë ì ë ë-ù ï4î8òAóhô ð4õð î8öAô ð ï4öAô8î8ð4ð9î8òAó÷=ø(ð9ñ ì ù ûAë ù ë ë ï4ñ/÷ø÷ ï ñ ðAòAî8òAó ÷8ô ô;üö/ï ð9ñ ÿ Aô8ð9î8öAô ∆ST îNïð4ñ ë Ff (S) ≈ √ Ä '(' Å ( A Ãjà ÿ Z Ç*)f - II,III & ì9ë ë úAù ë"û ëJýþ ë ì ù ë2ì ûAë õ î8ð 4ï ðüð ñ öHü ð9î;ü=ô/ï üð9ð î8òAóüÿ6öô8î8ð ï ë ì ù ìFë ë ë ð0ï4ð ð 8î òoütï4î8òAóhô S → S → S ô8î8ò ÷ X 2 (e · dνc )(dcf · e1 ), Ω + ıΓ c=x,y,z Ffm (T ) ≈ 0 þ ò ëù ñAîNï üòï ëlë2ù ë2ëù ë ë ë ù ë ù ë û ë ë " ù ë ë ý S → T → T ñHü=òAò ÷Jÿ6öAô ð ò ñAî8òAó÷øð4ñ ô üò 9ð ñ ü /ï ò ÷øð9ñ ï9öAô8î8ð4ð9î8òAó ì ëYì ì ëù ë ì ì ù ì9ë û " ë ûXë ì ëYì í eí ! í "# ý ðGîNï ü=ð9ñ 4ï î8ÿ6î8ô;ü 9ð ÷tð4ñ ÷Jô8ô;ü=ö/ï = ÷ øuî üð4îN÷JòHüôï4ð ð 9ï ð î ü 8ô î $ %'&)(!*,+.-0/213/2*,45-6178195+:(1;5*,-6; þ ë ë ë"ù ë û <AúAë2û = ë ë ë2ë û ë ü=ôNïPî8ð4ñóeüïöAñHüJï ÿZ÷Jô Aô ï6üò üòAóJô χ ð> ò e = ü ò ð4ñ ë ë2ù ì ë ë2û ýþ ë ?A@ Rù2ì ë"ù ë2ì ë2û ë2ì ë"ù ì ì ë ü! ð9÷ ÷ø ÿ6î8ð4ð öAñ/÷ðF÷Jò k1 ñ ÷hï9ïGï ð9îN÷Jòoü üó ÷! üô8ô Z ÿ ÷Jô Aô;ü ÷ î òeFð üð9îN÷ò/ï ë ë ë DE(ý 3 ë ì ù ì ì9ë ù ë"û " ë ë2ì ë"û ë îNïPóJî òBCjð4ñ ïü=ÿ Pî8ð4ñXð9ñ öHü ð4î;üô ÷òeð î FAð4îN÷Jò/ï öAô;ü Cjð4ñ ü ! üó ÷Jò ï ñ ö ì9ë ï ë ò[ð ëYú ëYì ö ë î8ÿ ûAë òeð |Ff (S)|2 = h × {D (Λ) 12 12 2X X 1 9 (ω1 − ωΛ,f (S) + ıΓ)(ω1 − ωΛ1 ,f (S) − ıΓ) Λ=1 Λ1 =1 (Λ)∗ (S) · D(Λ1 )∗ (S)}{Df (Λ1 ) (S) · Df (S)} 3 (1 − 3 cos2 χ) 20 (Λ)∗ (Λ ) × {D(Λ) (S) · Df (S)}{D(Λ1 )∗ (S) · Df 1 (S)} + (Λ ) !G (Λ)∗ +{D(Λ) (S) · Df 1 (S)}{D(Λ1 )∗ (S) · Df (S)} i 2 (Λ)∗ (Λ ) − {D(Λ) (S) · D(Λ1 )∗ (S)}{Df (S) · Df 1 (S)} 3 12 12 2X X 1 9 (ω1 − ωΛ,f (T ) + ıΓ)(ω1 − ωΛ1 ,f (T ) − ıΓ) |Ffm (T )|2 = h × {D (Λ) Λ=1 Λ1 =1 (Λ)∗ (Λ ) (S) · D(Λ1 )∗ (S)}{Df m (T ) · Df m1 (T )} 3 (1 − 3 cos2 χ) 20 (Λ)∗ (Λ ) × {D(Λ) (S) · Df m (T )}{D(Λ1 )∗ (S) · Df m1 (T )} + (Λ ) ù ê0ëYìFëJí ë ì û î8òeð ÷ ÷ò:IAóeüð4îN÷Jòo÷øð9ñ ù2ë"û ë ì ð4ñ ë ü ï ù ë2ù ü=ô;ü ð9÷ ì ì ö ì ÷ û ù ð÷=ø/ð9ñ a J {a · b} = !H (Λ)∗ +{D(Λ) (S) · Df m1 (T )}{D(Λ1 )∗ (S) · Df m (T )} i 2 (Λ)∗ (Λ ) − {D(Λ) (S) · D(Λ1 )∗ (S)}{Df m (T ) · Df m1 (T )} 3 ëù ÷Jÿ6öAô P k=x,y,z ëú ak bk ý ë2ù ð9÷ ì ï a üò û b Pî8ð4ñ/÷.Að ù ÷Jò ë òeð4îN÷JòHüô ù ÷Jÿ6öAô ëYú ú ) =à þaÿ ú û¶ü:ý û¨ !" 9(ÿ ûÄ ûTÅ Æ ÿ û ZÇT ÿ û Ç8 $#% à & TÈ û '( 7 û¨ ÿ) ; * #^þ !,+ .- û0/ Ç8 ÿ, 7 û 7 û¨ û >%$K L MONQP,R3S2T2U,VW0XYRZ=[R\BR]^SVR3T2T2_:YU,]` acbed^df,g.hijlkmhnk!g:iindojpFqsrtvuEwyxFz|{0j~}dohg:fji2gijdf o.ibedmjpedf,g.hijlktAw xdkg:ehdmibedBepg:f hi2g:idk.jpekjl{Fdh)Ajibibedq3o.Fpe{=hi2g:id:ibed|.fld!kFfld3ed inibFjlh.pFf BibFd S→S→S kbg:pFpedf jlhg:ffl!Qd{pe:ibedoj.oing:pi:jpijlhibgiibedFpe5kkFFjld{=h!icbFjlknbibedk.od dfld!kio.pjlh Fo.|.id!{Fg:odvpe!¡ibedAjpido|d!{0j,g:idy.pedhjp¢ibedhkg:iidojpFqFo5k!d!hh 2p → ψν → 2p ibed hnk!g:iindojpFqgFfjie{Fdr£z.ovdf,g.hijlkyhkg:iidojpFq¢jpekfe{Fdhvibed hF¤!do F = ν ¥¦ ¦§ acbFjlh|dg:pehibg:i ¦¨:© 12 √ XX (e · D(Λν) (S))(e1 · D(Λν)) ∗ (S)) . 2 ω1 − ωΛν,0 + ıΓ ν r z ¦ª Λ=1 « :piong:o iBibed¢jpedf,g.hijlkhkg:iidojpFqeibed|dg:¬=3hjijl:p:Aibedkon.hnhhd!kijl.p:cibeddf,g.hijlkknbgpFpedf k!:jpekjl{FdhvAjibBibedyd®Fkjing:ijl.pmdpedoq. r σ(ω, ω1 ) = |F |2 Φ(ω1 − ω, γ) °dond±d {0jlhFf,g ibedjpe{Fd®:Fpe5kkFFjld{=²jp D(Λ) (S) → D(Λν) (S) g:pe{ ωΛ0 → ωΛν,0 z ¦¯ {0edi ibed^j:oing:pek!d^:yjpindodondpek!d=.p³ibed^df,g.hijkhkg:iidojpFqkbg:pFpedflhibFo.Fq3b´{0j~}dondpijpindod!{0j,g:id Fpe5kkFFjld!{¢=h jlhAq.jdpB uE6r ψν ¦¯ aQbedkon3hh±hd!kijl:p¢:df,g.hijlkvhkg:iindojpFqg!do2g:q3d!{3docgff|.fld!kFf,g:o±.ojldping:ijl:peh zQAjib |F |2 = h odFf,g:k!d!{B |F |2 × {D 12 12 2 XXX X 1 9 ν (ω − ω + ıΓ)(ω 1 Λν,0 1 − ωΛ1 ν1 ,0 − ıΓ) ν (Λν) Λ=1 (S) · D 1 Λ1 =1 (Λ1 ν1 )∗ (S)}{D(Λν)∗ (S) · D(Λ1 ν1 ) (S)} 3 (1 − 3 cos2 χ) 20 × {D(Λν) (S) · D(Λν)∗ (S)}{D(Λ1 ν1 )∗ (S) · D(Λ1 ν1 ) (S)} + r +{D(Λν) (S) · D(Λ1 ν1 ) (S)}{D(Λ1 ν1 )∗ (S) · D(Λν)∗ (S)} i 2 − {D(Λν) (S) · D(Λ1 ν1 )∗ (S)}{D(Λν)∗ (S) · D(Λ1 ν1 ) (S)} 3 µ¶ M'L)·y¸¹cº»8¼c» µ ·y½¼¾¿NQ»8¼ ¦ £z µ ¾ÁÀ acbed vÂ5avà |:fld!kFf,go)dfldkion:pFjlkhioekiFondg:kn¬3gqdQg.hehd!{ in¢k.FFidibed dpedoq3jldh): ¥ ¦Ä© ibed±hjpFq3fldig:pe{yiojFfldiÁk!:ond2ÅÆd2®Fkjid!{gpe{yepg:f0hi2gindh8° « fÇÉÈÊdEbg!3dcg:FFfjld{ibFd±mFfijlk.p0eq3Fong:ijl.pg:f x5df~ « .pehjlhidpiË8jldfl{r r « yxFz hdi ¥~¦ª© (6, 5, 5, 2) « x « Ëzfld3df0:k!g:flkFf,gijl.pk!.pehjl{FdojpFq dfld!kio.peh8jpgvk.Ffldidcg.kijdQhg.kd ¯ caQbed!hd)kgflkFf,g:ijl.pehAQdod)do.o|d!{cjibibedyk!:ondk!.oondf,g:id!{mg:Fq.Å>kkÅÇ «±Ì aÍ^g.hjlh EË8jqe0Î hbeAhQibed.idpij,g:fhFoÇg.kd!hQ:ibedvq3on:Fpe{ g:pe{¢ibedk!.odd2®Fkjid!{ 2π −1 6σ 1 hing:id!h ¥¦¯© Å ÿ à ? >%$ÁÏ II,III & Ä '(' Å ( A Ãjà ÿ Z Ç*)f - 216 208 Energy (eV) −1 1 −1 1 2p 6σ 200 40 32 4σ 6σ 24 8 GS 4 0 1 2 3 R (Å) Ð6Ñ>ÒÓ0Ô3ÕÖ×ØÆÙ2ÚØÛÝÜÞß>àáÆâ,ÜãnÙQ×âØÆäÙvåá×àÚæçèÒAéêQëÝìíîïðáß>Øã×áÆÙnñÙòãnÛ ØÆÙ2æ¢ëÝìíî óÁäÙæ3×!ØÆØÆÙ2æ|ÞÝÛÝÚÙ)æ3ÛÝßÆôÞÝÜõ3ßØä3ÙðÚÜ!Þ 4σ −1 6σ 1 2p−1 6σ 1 Ü!ÚæðÚÜ!Þ 4σ −1 6σ 1 ßÇØÜ!ØÆÙßnÓ ßÇØÜØÆÙô5×ØÆÙnÚ:ØÆÛ~ÜÞß>äÛ âØÆÙæ|ÛÝÚ¢ÙnÚÙ2áÆåõØÆ×ß>ä×öØÆäÙôÜ!áÜ!ÞÝÞÝÙnÞÝÛ~ßÆ÷ø×âØä3Ù ô5×ØÆÙnÚ:ØÆÛ~ÜÞ~ßnÓ ù.ú)û±üýý±ù.ú!þÉÿnüúFýÿ eüù:ýEúnùü 4σ −1 6σ 1 ÿ2ùüÿ^ùÿ eü nü!2ù"lÿ #$eü !ÿnü&%Æü&'"ü(ú2ùüû±üúÿ.ý)ü"(ü"*ùlÿ eú,+.-0) / û1 ∆ ü32 54 Êÿ 6(Fü!7ÿ983ü"eü ú:Fý8 SO = 1.75 )3ùýeüvûQü; : ü"< ÿ | = ü"(n üý,ù)>l ú?@! ù:ý?"Fý,ùl ÿ e úB A ù.úü¢ ( ÿ e ü; ÿ B! ÿ=C : ÿ e ü"; ;DFE ü?*eüû1eü !ÿnù !nü"(G*0ù8 %3&%3:2 @HJI 8K@:J(Aù.úlúvúü"LMONP QRTSU!VXW@YVZVU&[\][^`_Ba7b;_Bc7Ude_B\Cfhg7i#g7\CSjVU6dkYi5g1f;_lSmc7_;VY7infV goRpaqsr"tu$tKvZw!xr!yXz!{|x>} @ú=Fý,ùlÿeúcÿ L− '>%~nù|ùAeúÿ :lÿùnü;Aù.úü(|ÿ eü1ÿ.ýýlÿ!û;8ü&'n:nüúnúlÿ oÿ eüvùAeúÿ :nlÿ "ÿ3úúvúü!l ÿ ^ ù" ) ü"n ù 8 ü"^ ( ÿ ) ü"ù:ýý=|ÿ.ýlü!Fý,ùÿlü!2ùlÿeú 12 σ(ω) = 2 X X (Λ) |D (S)|2 ∆(ω − ωΛ0 , Γ), 3 ν Λ=1 ωΛ0 = EΛ − E0 , +~KK/ 4 eü;$ ; lú=7Fý,ùlÿ eú±ÿX'>%~2ùùAeúnÿ:lÿ ¢ú:ü"nùo+ E /ûQü12ù3ü;ÿù "!ÿ eü1ÿ.ýýlÿû18 2ùeúlÿ eú 2p → 6σ, 7σ, 8σ, 9σ ù( 2p → 3π, 4π, 1δ +.8 0 þ / K eü1üý,ù)>lú?;ù:ý?!Fý,ùlÿeúAù.úü( ÿ Ce ül ÿ ! ÿ=C:ÿ eü"1 ;DFE ü?*eüLMNPúeÿ!ûenùeü"ú=Cý,ùFü"eúlü!úZeü;ÿ =C:ÿ eü!núÿZeü úú:n%3(Fÿ AFýlü!þù(@8 )ü!ú ∆ ü~2j+.8 X 1/þeû;?lúýlÿ.únüÿoeü ü!'n:ü"=|ü"02ù:ýÁú:n% SO = 1.62 ÿ A2 ùýúF : ý8 ∆ ÿ $ ; Bù(üý,ù)>lú?ùý?"Fý,ùlÿ eúúeÿûeüeüú"nü ≈ 1.75 ü~2 Q SO ÿvüù ÿ =C:ÿ eü!ÿú:n%3(Fÿ AFýlü!ùeúü(A$eüo=|ÿ.ýlü""Fý,ùÿ Anù:ýQú:Fý8ÿeü!ÿ üúeüýýÇþEù:ú û±üýý±ù.úAeüo Qÿ Fýlÿ =Aü"nù !lÿ Aü">û±ü!ü"eü LII,III ú eüýý±ù($eü)3ù !ù7$ X üúnüü=¢ùFý ú ) û¨ !" XAS cross section (arb. units) ÿ ûÄ ûTÅ Æ ÿ û ZÇT ÿ û Ç8 $#% à & TÈ û '( û¨ ÿ&) * #^þ !,+ .- û0/ Ç8 ÿ, û û 0.6 0.5 0.4 0.3 0.2 0.1 0 A 0.5 B û¨ 3π3/2 1δ3/2 4π3/2 8σ 1/2 6σ3/2 6σ1/2 7σ 3/2 0.4 ×10 7σ1/2 8σ3/2 1δ1/2 4π1/2 3π1/2 XAS cross section (arb. units) =à þaÿ ú û¶ü:ý 9σ3/2 9σ1/2 1δ3/2 4π3/2 0.3 1δ1/2 4π1/2 0.2 3π1/2 0.1 3π3/2 0 200 202 204 206 ω (eV) 208 210 0.006 >%$$ 7σ3/2 C 0.005 0.004 6σ3/2 0.003 0.002 6σ1/2 ∆SO = 1.62 eV 0.001 0 212 202 203 ω (eV) 204 205 X3X0h 0¡6¢£s¡6¤s¥¦6§O¨©¨ £<§"¬o§®s¢O£s¯K¤<¥ ¢°®s¯¡!¦¤<£s±0²³¢O´Fµl©¨¶·B§®3 0¡&¸,¨¹¥¹°0¡!®1¥¹°¤< K¡¯§O°K¡&¨5º»®3 0¢¼k¤< K¡ LII,III ª « ½Bºl¾¢O´ §O°¸ ®3±0®3¬K®~¤s¡&²®¿|¸K¡&¯K¥¦¤s¡!¸¥¹°¯§°0¡6¨5ÀFÁ1®3 0¥ ´¤s¡&¸, ¬Â0 ÃÄ7¡~Å ¡.Å@ÆJ§O°0¡6¨©e®3 0¢¼®B¤s 0¡ π δ Γ = 0.0465 £s¡!®3±0¨ ¤<®h¢O´Z£<¡6¨§¤s¥¹Ç¥®3¤s¥¦B®s¥¹²±0¨§¤s¥¹¢O°®h´?¢O£F¤s 0¡lÈ£®~¤h¤s 0£<¡6¡;¯¡!§OÉ ®F¥¹°7¤s 0¡@©¨ ½Bºl¾7®s¯>¡&¦¤s£s±0²Ê¢´Jµl©5¨¶ LII,III ÑÒÓ ÌÐØÎZÐÙ!ËÍÝ0Ð!ÏÞ0ßFàÐÏ|ÑÖZÐ"ÏJËáÐÙÔ ÌÐ!â3Ð&ãÙ"ÍËÐÓ 6σ3/2 6σ1/2 Ü ØË&ÑËÐ ÍËá ßå1áÐCÑ ÙÙ!ÕÌ&Ñ Ù!ÞGÔÚ1ÔÕÌØÍæÕÏ|ÑËÍ?ÔÒØÔÚ ÑÒÓ ØÕÖØÞnØËÐ!æoØ 1 ÑØ ν |2p−1 j = 1/2, 3/2 σ π j ÜJä j ν i Í?ØÝ0Ð!ÌÞ#Ø6Ð!ÒØÍËÍÝ0Ð,ËÔ$ËáÐçlèéêØÎÛÑ ÙÐ ß#å5Ô$ëKÐ"ËoÑ9ÖZÐ"ËË6Ð!ÌÑëKÌÐÐ!æoÐ"ÒË ÍËáËáÐÐ&ãnÎZÐ"ÌÍæoÐ"ÒË ÐØáÍÚìËÐÓ ä Üä ØÎZÐÙ!ËÌÕæîíìÎÛÑÒÐ!ÏFïð;ËÔÏ?Ô Ð"ÌÐ!ÒÐ"ÌëKÞ9Ì6Ð!ëKÍ?Ô ÒÖÞñnß òKó,Ð~ô³Ñ Ø@ØáÔ ÒGÍÒ$õÍëßZöèßX÷ÑËáÐ"ÌØÍæCÍÏ|ÑÌ ËáÐ π ä ä ÓnÍ?ØÎÏ|Ñ ÙÐ!æoÐ"ÒË7ÔÚlËáÐ Ï?Ð!Ý0Ð!Ï?ØÌ6Ð!Ï|ÑËÍÝ0ÐËÔËáÐ ØÕÖØÞnØËÐ!æ Ñ ØÔ ÖØÐ"ÌÝ0Ð"ÓÐÑÌÏÍ?Ð"ÌÍÒGøèéÔÚËáÐùç;é π σ ä æoÔ Ï?Ð"Ù"ÕÏ?Ð7ú û ñü3ßFåláÐ;×ÌØËØÎÍÒnâsÓÔÕÖÏ?Ð"ËÌ6Ð!Ï|ÑË6Ð"ÓoË6ÔÙÔ ÌÐÐ&ãÙ"ÍË6ÑËÍ?Ô ÒË6Ô ùþíìæÑÌÿ0Ð"ÓÍÒõhÍëßöÑ Ø 6σ ý 6σ3/2 ÑÒÓ ðÍ?Ø7ÙÔæCÎÕË6Ð"Ó#Ë6Ñÿ>ÍÒë9ÍÒËÔGÑÙÙÔÕÒËCÔ ÒÏÞGÏÍÚ.Ð"ËÍæoÐoÖÌ6ÔKÑ ÓÐ"ÒÍÒë ß @Ô Ð"Ý0Ð!Ì ËáÐoËÌ6ÑÒØÍËÍ?Ô Ò 6σ1/2 Γ ä Ü ùjÍ?ØCÐ"Ø6ØÐ"Ò0ËÍ|ÑÏÏÞÖÌÔ0Ñ ÓÐ"ÌCÖZÐ"ÙOÑÕØÐ,ËáÐ,×ÌØËCÙÔ ÌÐ!â3Ð&ãÙ"ÍËÐÓkØË6ÑËÐÍ?ØÓ>Í?Ø6ØÔ>Ù"Í|ÑËÍÝ0Ð$íìõhÍëßû0ð&ßàÐ ËÔ 6σ ý ØÍæ7ÕÏ|ÑËÐÓoËáÐÓnÍ?ØØ6Ô>Ù!Í|ÑËÍÝ0Ð;ÖÌÔ0Ñ ÓÐ!ÒÍÒëÔÚZËáÐ1×ÌØËØÎÍÒnâ3ÓÔ ÕÖÏ?Ð!ËBÕØÍÒëËáÐ ÑÝ0Ð;ÎÛÑ Ù6ÿKÐ"ËË6Ð"Ù6áÒÍ ÕÐ7ú Oòü3ß ä å1áÐ;Ì6Ð"ØÕÏËØÔÚJËáÐØÍæ7ÕÏ|ÑËÍ?Ô ÒØ1ÑÌÐ@ØáÔ ÒÍÒõÍëß>öè`ÖÞCËáÐ@ÓÔ ËË6Ð"Ó>â3ÓÑ ØáÐ"ÓÏÍÒÐKßFõhÍëßöè ÌÐ"ÎÌÔ>ÓnÕÙÐ"Ø ä ÑÏÏÛÐ&ãnÎZÐ"ÌÍæoÐ"ÒË6ÑÏÚ.ÐOÑËÕÌÐØÍÒoËáÐ1øèéØÎZÐÙ!ËÌÕæ íìõhÍëß ðÑÚìËÐ"ÌËáÐ@ÑÖZÔOÝKÐ;æoÐ!ÒËÍ?Ô ÒÐÓ,ØáÍÚìËÔÚZËáÐ ÑÒÓ π ÌÐØÔ ÒÛÑÒÙÐ"ØOß @Ô Ð"Ý0Ð!Ì ËáÐÍÒË6Ð!ÒØÍË3ÞGÌ&ÑËÍ?Ô,Í?ØÚ~ÑÌÚìÌ6Ô æ]ÖZÐ"ÍÒëÎZÐ!ÌÚ.Ð"Ù"Ëß ÕÐËÔËáÍ?Ø ÐÌ6Ð"Ø6ÙÑÏ?Ð"Ó$ËáÐ δ ä Ü ÜJä ËÌ&ÑÒØÍËÍ?Ô Ò#Ó>ÍÎJÔÏ?ÐæoÔ æoÐ"Ò0Ë6ØÖÞ9×ËËÍÒë9ÔÚlÔ ÕÌËáÐÔÌ6Ð!ËÍ?ÙÑÏÎÌ6Ô×Ï?ÐCËÔ,ËáÐÐ&ãnÎZÐ"ÌÍæoÐ"Ò0Ë,í.õÍëß >ð&ß sÒGËáÍ?Ø ËÌÍÎÏ?Ð!Ë@ÑÒÓÓÔ ÕÖÏ?Ð!Ë;×ÒÐØËÌÕÙ!ËÕÌÐÔÚÖÛÑÒÓØ ×ËËÍÒë õÍëßÛöè XßBí0ðÑÒÓØáÍÚìË6Ð"ÓËáÐ ÐCÝKÑÌÍ?ÐÓËáÐCËÌ6ÑÒØÍËÍ?Ô Ò#ÓnÍÎZÔ Ï?ÐCæoÔ æoÐ!ÒË6Ø7ÍÒ ä Ü Ë6ÔoÑÏ?Ô ä Ð!ÌÐ!ÒÐ"ÌëKÞÌ6Ð!ëKÍ?Ô ÒÖÞ Kß ûKóCÐ~ôß ä áÔÏ?ÐoØÎZÐÙ!ËÌ6ÑÏÎÌÔ×Ï?Ð Ü "!#"$%'&)(*(*+,.-/$10 2 #3 σ ý Ð"ËÕØØË6ÑÌËÖÞËáÐ7ÍÏÏÕØËÌ6ÑËÍÒëÔÚËáÐÚ.Ô ÌæÑËÍ?Ô ÒGÔÚËáÐ÷;øé,ÎÌ6Ô×Ï?ÐÕÎZÔ ÒÐ!ãÙ"ÍË&ÑËÍ?Ô ÒËÔ 4Xßí5Oð Ü è;Ù"ÙÔ ÌÓnÍÒë9ËÔ ÙÔÒËÌÍÖÕËÍ?Ô ÒØ |Ff (S)|2 ùß ËáÐo÷;øé9ÎÌ6Ô×Ï?ÐCÍ?ØËáÐCÎÌÔ>ÓnÕÙ!Ë7ÔÚlËáÐoØÎZÐÙ"ËÌ6ÑÏhÚ.ÕÒÙ!ËÍ?ÔÒÑÒÓ$ÔÚlËáÐCÎÛÑÌËÍ|ÑÏ 5 í OóKðÑÒÓ |Ffm (T )|2 67 í Kð!ßõhÍëßFó9ÓnÍ?ØÎÏ|Ñ"ÞnØËáÐØÎZÐÙ!ËÌ6ÑÏlØáÛÑÎZÐÔÚ@ËáÐ,ÎÛÑÌËÍ|ÑÏ Å ÿ à ? >%$98 II,III & Ä '(' Å ( A Ãjà ÿ Z Ç*)f - 200 202 3/2 + 1δ 3/2 0 198 4π 1/2 + 9σ 6σ3/2 8σ 7σ3/2 1/2 Intensity (arb. units) 7σ1/2 + 4π3/2 8σ3/2 6σ1/2 9σ1/2 204 206 ω (eV) 208 1δ1/2 210 212 1: ;=<?>@BACEDFHGBI.JKMLD7N#D.OQPR"STSVUGWSXJKYFHS*NP5I.Z L []\_^ L=GWS'`5PJ6a"FbD.O]c_CdZ9L=GWS'`5PJ6a"Ffe/gD.PPS*ghZYKYNSikj9K PR#P6RS PR"S*D7J6SVPKM`*I.Z"L=KYFla"ZMIP6KYD7NBLe/LD7ZYKMg?ZYKYN"S'i1mBIL=S'glDN?PR"S9JS'L`*I.ZYKYN"noD.OWPJ5I.NBL=K P6KYD7NgKYGWD7ZYSF?DFHSXN,PQe/LSXSpP6R"S9PSXU,PVi5>rq_I7LR"S*g ZYKYN"S*LsL=R"DjtPR"SGBI.JPKMI.Zu`VDN,PJKYm"aPKYD7NBL*> Γ = 0.14 SvwODJsI7ZYZuLxP5IP6S*L_SVU"`VS*GPsgKMLLD)`XKMIP6K yS 6σ I7NBg 6σ LxP6I.PS'L j9K P6RTRBI7Z Oj9KMg)PRzI.P9RBI7Z OFIUKYFka"F{S'|aBI7Z}PD~> 7@?Svk> II,III 3/2 1/2 .,BV5 5 ))_u5X,Xs)BX, z)o 5 4σ , 5σ W .?.9_.V?,_ 5 ,V'/,XVVX,5 V ,V5T,V }T5V '5*X7ozV7VV5 #X5T, 5V 'k5*Xl s,7].}lXT5{ BVX)'V )# 5H.,X*=' V#5X,X) Q)V4d5.VH5 l5 l}HX¡5T¢ BVX)5 )u£5¤¡).'/,k'4V 5V ¥5)XV'x£1,k_'p)k..,V ,kV¦]))¤ § BX'*,5 )¨k5)V¥)©s 5§5#ª σ « )7B « 'x_¥V.'5*,]¬5 )x, } MrV"H'' *V )h'5 .o®6'?,9¯} Mr'X'B?,5V?,9V,V5 /,d.,BV5 5 )z°¬±p M²³k).)V} Vz Q´1E°6µµ³H°V.±d ¶"³*]k7,h,VzV.H5/,V5 7· ¸ )V¨'}'V ¤¡'B*,¨5.,X'V .,H¹sºt5.'VX,lV,©x,¥} MrVB'' X,V ,»'V) .,V 57s¼ §±d 9ªW_¥, 4σ _)z5 M¬X½V# 'z'V)¾½V ,t.,¤z,5 V4V¦s¦]± 7/,5 )4B¾#¿} ¿ÀxÁ o¡.,·} 6σ3/2 6σ1/2 .V « h5.' /,].)¤z¤¡'B7o¥)5 ¼,k5.V©V.V),.T 5 V7,5X'V #5V)Vz} VXB /,5 « z.,X*=' V.5*,V 2p−16σ °V.z±d u¿"³*©" ¤ /,5 ) V7ÂV 5z BX'V =¾,s5z,½² σ ¤ X¼5¤©,5,Vz "X'V =¾,s5zà σ r.,·¥Ä?zVV .£d' W')T,5VB5 )½)¾)¼,¾}V , 4σ → 2p Å ,XX¥ozV)VX,VzB.,T_ « )V·"p dX,¤¡)XVlVQ)VB5 /,,5o)V'='' X5*,VoT¤¡ « p¬V)¤Æ5], BV « 55 7, 5*,5 5 )TVk5QV ,, 'p,5_°±d ,¿³'dÄ?5 ?5]} VXW' /,5 ) T)X*='' XTVX,V£ ú ) =à þaÿ !" û û¨ 0.06 A 0.04 Intensity (arb. units) 4σ-band ∼∆SO 0.02 ∼∆ST ∼∆SO ∼∆ST 0 177 0.006 0.004 178 179 Singlet Triplet m = +1 Triplet m = 0 Triplet m = −1 180 181 5σ-band B ∼∆ST 0.002 ∼∆SO ∼∆SO 0 188 190 ∼∆ST 192 ω1 (eV) 194 1È É=Ê?ËÌ)ÍÎrÏ.ÐÑÒMÏ7Ó)ÔXÕ7Ö,ÑÐ6ÒY×ØÑ6Ò ÕÖBÙdÚ |F (S)| ÚxÛÌÜÏ.ÖBÝ |F (T )| ÚxÛ'Þ,ÜßÕ7ÐÑà"ápÔVÕÐápáVâ"ÔVÒ Ñ5ÏÑÒYÕÖlÕ.ß"Ñà"á]ã σ ÓYáXäá*Ó}ÚåÈ1ÒYæ"Ë7ç,Ü5Ë èpà"ákÒYÖBÙ=áVÑ_Ù=à"ÕéQÙdÑà"álÙ=ê"ÒYÖzÕ7ßÑà"áoëBÖBÏ7Ó1ÙxÑ6Ï.ÑásØ"ÖBÝ)á*Ð]Ù6ÔXÏ.Ñ=Ñá*ÐÒYÖ"æ"Ë ∆ ÒYÙ9Ñà"álÙ=ÒYÖ"æ7ÓYáXÑ=ìåÑÐ6ÒYêÓYáXÑ]Ù=ê"ÓYÒ Ñ=Ñ6ÒYÖæÕ7ßÑà"ásëBÖ"Ï7Ó ÙxÑ5ÏÑ6á7Ë 2 f m f 2 ST 6 ω = 206.9 eV −1 elasti 3 c 4σ 7σ1/2 1/2 − −1 7σ 5σ 7σ1/2 0 −1 4σ 7σ3/2 c ω = 205.2 eV 3/2 − elasti 6 7σ û¨ ÿ ûÄ ûTÅ Æ ÿ û ZÇT ÿ û Ç8 $#% à & TÈ û '( û¨ ÿ&) * #^þ !,+ .- û0/ Ç8 ÿ, û Intensity (arb. units) ú û¶ü:ý 3 −1 5σ 7σ3/2 0 180 190 ω1 (eV) 200 1È É=Ê?ËÞ"Í]íî_ï¡ÙêBá'Ô5Ñ6Ð6ÏÕ7ßdð_ñdÓßÕ7ÐoÝÒ ò}áXÐá*ÖÑoáVâ"ÔXÒ Ñ6Ï.ÑÒYÕ7Ö©á*Ö"áXÐæÒYá*Ù*Ë ω = 206.9 áó»Ï.ÖBÝ ω = 205.2 áówÔXÕÐÐá'Ù=êWÕ7ÖBÝ ÑÕzáXâÔXÒ Ñ6Ï.ÑÒYÕÖÕ.ßpÑà"á 7σ Ï.ÖBÝ 7σ ÙxÑ5ÏÑ6á*ÙXô1Ðá*ÙêWá*ÔVÑÒYä7á*ÓYõ4Ú/È1ÒYæBËçzö]ÜVËÉÖBÔVÒMÝá*ÖÑsâ)ì¬Ð6Ïõzêà"Õ7ÑÕÖBÙoéÒ ÑàÝÒ ò}áXÐá*Ö,Ñ áVâ"ÔXÒ Ñ6Ï.ÑÒYÕ7ÖTá*Ö"áXÐæÒYá*Ùr÷ÔVØÑÕ7ò1økÝÒ ò}á*Ðá*ÖÑêBÏ.Ð=Ñ5ÙpÕ7ßÑà"ásê"Ï7Ð=Ñ6ÒYÏ7ÓÔXÕ7Ö,ÑÐ6Ò ×"ØÑÒYÕÖBÙoÚ/ÈÒ æBËBÌÜpÏÔXÔXÕ7Ð5ÝÒ Ö"ælÑÕHùEúWËÚxÛÛÜ5Ë 3/2 1/2 >%$]Ç Å ÿ à ? >%$9û 4σ-band 4π3/2 ×3 9σ1/2 1δ3/2 4π1/2 9σ3/2 8σ1/2 7σ1/2 7σ1/2 8σ3/2 7σ3/2 6σ1/2 6σ1/2 ×0.2 ω = 207.3 eV [1δ3/2 + 4π1/2] 9σ3/2 8σ1/2 ω = 206.5 eV [9σ3/2 + 8σ1/2] 4π3/2 7σ1/2 8σ3/2 7σ3/2 6σ1/2 ×0.1 170 elastic ω = 208.2 eV [9σ1/2] 1δ3/2 4π1/2 9σ3/24π3/2 7σ1/2 8σ1/2 9σ3/24π3/2 8σ1/2 7σ1/2 4π3/2 Ä '(' Å ( A Ãjà ÿ Z Ç*)f - 5σ-band 9σ1/2 9σ3/2 II,III & ω = 206.1 eV 9σ1/2 4π1/2×0.2 1δ3/2 9σ3/2 4π3/2 ×0.2 9σ3/2 ω = 205.6 eV [4π3/2 + 7σ1/2] ω = 204.8 eV [8σ3/2] ω = 203.9 eV [7σ3/2] 4π3/2×0.1 7σ1/2 8σ3/2 7σ3/2 ×0.2 ω = 202.6 eV [6σ1/2] 6σ3/2 ×0.1 ω = 200.7 eV [6σ3/2] 6σ3/2 ×0.1 ω = 200.2 eV 180 1δ3/2 190 ω1 (eV) 200 210 ü1ý=þ?ÿ ! " #$&%('*)+ ,# -!.0/(12.3!45 67 2 "#" !* 8 ,)Xÿ:9; <=> > !? ; #" !@ 8A)+!?&)B C#"! ,)+!8< )! ,DE@ 45FG HAI>)+!!0J7: ü 8"ÿ KBÿ 9; 4 ! =# )+ ,#L?2 M#"!@N ,ON)+8L PH P.3 ! ü 8BBÿ KBLÿ 9; ! 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