X-ray Spectroscopy of Molecules Driven by Strong Infrared Fields

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÷ CçÊAø½Ó©wÝ ©~AÅÄRx{w ©qzAÏ©¾½<¡&Ïx¾$z9Ax¡y©ó¡y¬Ax!©~AÅ ×½Óx,¡& Áx}©½<¥q¢Aw&¾¡y ©~9Á~¦©ÁÄVwyy©~
ÀÆ¡y¬¡&¬9xx5Ã/¦{×¡&xzëx{ x¦¡ywy©~A¦<J¡«¾¡yx{{ËÛÜ¬9xñû©¦{ jÒ¡& ©~ACwyxDÄAwy©¢Éxzë¢£|Ïy¬9©w¡ÂÃ£w&Ú|Ä9A x{{
zAyÄA i|q~AÅ¡&¬9xyx~Ay×¡&¥q×¡Ø|!©½Ã£ØwyÚ|¢9y©wÄ9¡&©~¡&©¡y¬Ax¡&¬9xñ©/¦{ üÚ¾¡y ©~~Azf?¬Ax{~9¦{x;¡y©
®s(úÌ\|A ¢¡Z ¬ \
E
Σ [NO*(1s)]
546
533.0
b
532.0
c
544
5.0
4.0
3.0
B
2.0
1.0 Pump
0.0
2.0
1.5
A
Absorption probability
2
C
Π [ΝΟ]
2.5
3.0
PT
540
Probe
538
8
3.5
Probe
B
PT
6
4
∆ε
B
542
d
531.0
Probe
Od core ionized state
Ω (eV)
2 −
Energy (eV)
Energy (eV)
534.0
Ω (eV)
¡&¬9x,¡& Áxz9x{jÚ|0¦Ú~'¢Rxëyxx{~'½Ówy© vT ÅAËÊË(9ËóÛÜ¬Axwyw©bÀÆ~0¡&¬AìVÅAwxë¾wyª¡&¬Ax,ñ
¦{x~d¡&x{w©¾½Æ~9z0¡&¬Ax,¦x{~d¡&xw©½ÆÅwyi¥q×¡Ø|Ù©¾½Ü¡&¬Ax,ÄÉx¦¡ywyAËó/ ~9¦{x¡y¬Ax,yÄRx{¦5¡&wyÂÄAw©ìVx
yx~Ay×¡&×¥xD¡y©È¡y¬AxDñ'ÄÉ©y¡y ©~fd©~Ax<¦{~,xÃ9ÄRx{¦5¡Ü§wx{ ¾¡&©~Ay¬9 ÄÝ¢Éx5¡ØÀÂxx{~ë¡&¬9xD¡&wyZ Yx{¦5¡&©wJ|Á©½?¡&¬9x
ÀÂi¥xÈÄA¦«ªx5¡`~Azë¡y¬AxN¦{x{~d¡yx{w1©½TÅw&Ú¥q¡Ø|ë©½P¡&¬9x`Ã£w&Ú|}¢A©wyÄ/¡& ©~?Ë
Ionization probability
Ground
state
A
2
0
1
2
Pump
3
4
vTJËÂË(9áf`A ×¡«¾¡y¥xÁ AJ¡&wy¾¡&©~Ù©¾½1¡y¬Axëê −Ã£w&Ú|0ÄA9ÏÄ/ÄAwy©¢Éx,ÄÉx¦¡ywy©¦{©Äd|ËßÛÜ¬Axë x5½Y¡
~Azßwy Å¬£¡§ÄV~Ax §wyx¾mwyxyÄRx{¦¡y¥x{|m¡&¬AxëðÁ÷Ú«øÃ£Øw&Ú|ß¢A©wyÄ9¡y ©~Ù©½Ü¡&¬9xëð>Á© x¦{Ax~9z
ðÝ÷JÚ&øÊÃ£Øw&Ú|ëÄ9¬A©¡&©qx{x{¦5¡&wy©~ÄÉx¦¡ywyA ©½P¡&¬9xNÀÂ¾¡yx{wÆzA Áx{wË
r (a.u.)
r(O-H) (a.u.)
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ÏZY©wÆÄVwJ¡©½m¡y¬Ax§Ï¾¡yx{wy ?ÄAwx{x{~d¡&xzó~¡y¬AxÈ½Ó© ©iÀÆ ~9Å¦«¬VÄ9¡yx{wy;ÆÀÂx {R ~¡&¬AxÈÄVÄRx{w
wyxÄAwy~d¡&x{z§ ~È¡y¬AxÄAwx{x{~d¡m¡&¬9x{y{ËmÛÜ¬9x~Ax5Ãq¡T¦«¬VÄ9¡yx{wyC¾wyxÂ©wyÅd~9 ü{xzP½Ó© ©iÀNËPÛÜ¬9xÂ¡&¬Ax©wyx5¡&¦Ú
¢V¦«ª£Åw©A~Az]D©9¡& ~Axz! ~ÐÂ¬VÄ9¡yx{w/Ë<Ø~ÐÂ¬AÄ9¡&xnw ÏÀ$x¬AÅ¬A Å¬d¡Ü¡&¬9x§~£AÁx{wy¦ÚPÁx¡&¬9©/zA
Ax{z~Ç¡y¬AxÂÄAw©ÄVÅ¾¡&©~N©½Añ{ËÛÜ¬AxÂÄA¬d|qy¦{m©¾½9¡&¬9xÂÄA¬V¾yxÂx{~A¡y¥xÊê −Ã£w&Ú|ÇÄAAÁÄ/ØÄAw©¢Rx
yÄRx{¦5¡&w©y¦©Äd|Êz9x{y¦wy¢Éxz ~ÁzAx5¡«V ~ÐÂ¬VÄ9¡yx{Hw AËCÐÂ¬VÄ/¡&x{w ` ÊzAx¥©¡&xz¡&©N¡&¬Ax1z9|q~VÁ ¦C©½
¡&¬9x§½Ó©wyÏ¾¡y ©~©½Cwyxy©~V¾~£¡Ã£w&Ú|ê1Ï~!ÄÉx¦¡yw&ë©½¡&¬9xÔÐÂmÁ©x{¦{9 xÇ~AxÚwD¡&¬AxÝÐÂ L
II,III
x{z9ÅxËCÛÜ¬AxÇ¾ ~,wx{y9¡&1wxNyAÁ¾wy üx{zë~!ÐÂ¬AÄ9¡&xw19Ë
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lSUTRVzkWJYX
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k
;ï x¡TAm¢Rx{Å~ÈÀÆ¡y¬È¡y¬AxÂÅx~Ax{wy9¦{yx$©½R¢A©wyÄ/¡& ©~§©½_DÄAwy©¢RxÊÃ£w&Ú|ÇÄAA x E (t) ¢d|§Á©x{¦{9 x{
zAw¥x~Ö¢d|0!J¡&w©~AÅ]ê¶ìVx{z E (t) ~0¢R©¡&¬ßÅwy©9~Az'~Az0xÃ/¦{¡yx{zÖx x{¦5¡&w©~AX ¦Ï¡&¾¡&x{Ë]ÛÜ¬Axyx
ìVx zAÆ ~d¡yx{w&¾¦¡ÜÀÆ¡y¬ë¡&¬AxNÁ© x¦{AL wÂzAÄÉ© xÁ©Áx{~d¡ d
÷ /Ëiø
V (t) = VX (t) + VL (t), Vn (t) = −(d · En (t)) cos(ωn t + ϕn ), n = L, X.
¨¡&©Ï¦]A~A¡y}wyxAx{z x¥xw|£ÀÆ¬Axwyx0A~9 x{}©¡&¬AxwÀÆyx¡&¾¡&xz?ËôÔ1x{wx ω ~9z ϕ wyx]¡&¬9x
½ÓwyxÞ£Ax{~A¦5|û~AzÖ¡&¬9xëÄA¬Vx©¾½1¡y¬Ax n¡y¬'ìAx{ zfÊwx{ÄÉx¦¡&×¥x|ËÙñßx} Å~A©wyxÏ¡&¬Anx,ÀÂÚ¥x}¥nx{¦5¡&©wÝ©½
¢R©¡&¬}ìVx{zA k R×¡&¬A©AÅ¬ë¡&¬AxÇyÏ?Ï©Ïx~£¡yA>©½mê ÄA¬A©¡y©~ k 9~AzAx{wD¦{xw¡«¾ ~¦{©~Az9¡&©~A
¦Ú¾~\~ [VAx~A¦{x`n¡&¬9x`Ã£w&Ú|ëÄAw©¢ÉxNÄÉx¦¡&wA>¡&¬9wy©AÅ¬}¡y¬AxNÄA¬Vx õ» L
÷¿qË ø
ϕ L → ϕ L − k L · Rα
ÀÆ¬Axwyx R <¡&¬Ax§¦{©q©wyzA~V¾¡yx§©½¡y¬AxÏ© x{¦Aj¾wD¦x{~d¡&xw`©½C¾y{ËNÛÜ¬Ax§Ã£w&Ú|!wyzA ¾¡&©~!Á×Ã/x{
Åw©A~Az ψα ~Az¦{©wx5x5Ã/¦{×¡&x{z ψ x{x{¦5¡&wy©~A ¦`J¡«¾¡yx{
0
c
÷¿qË dø
Ψ(t) = ψ0 Φ0 (R, t)e−ıE t + ψc Φc (R, t)e−ıE t .
ÛÜ¬A1x 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¬AxÏ¾¡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

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0
c
½Ó© ©iÀÆ ~AÅ§¦{©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
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Ï¾¡&w×Ã 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 ËÛÜ¬AxDz/|/~AÏ¦{Ê©½
¡&¬9xNÀÂÚ¥xÇÄV¦«ª¾x¡&D ÜzAx5ìV~Axz!¢d|,¡&¬9xÇ~£A¦ x{w1Ô<Á ×¡&©~A0 ~AÊ©½Åwy©9~A0 zÙ÷ h ø~Azx5Ã/¦{×¡&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¿ËÛÜ¬9x ~d¡&xw&¦5¡&©~ V ii = (ψ |V |ψ )
i L i
L
©½P¡&¬AxN~£A¦ x{?ÀÆ×¡&¬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/¦{×¡&xz¡&¾¡&x$z9Ax1¡y©Ç¡y¬AxÆzAª©Éx{wx{~A¦xD©¾½f¡&¬9x1Á©x{¦{9jwmz9 ÄR© x
Á©Ïx~d¡& d ~Az d ËÛÜ¬9x R−zAx{ÄRx{~9zAx{~A¦x]©½¡y¬Ax{x!zAÄÉ©x}Ï©Áx{~d¡yÁÏ¦wyA¦jÜ½Ó©w¡y¬Ax
~d¡yx{w&¾¦¡&©~00ÀÆ¡y¬Ï¡&¬9xDccê)ìVx z?£¢Rx{¦{Ayx1¥q ¢Awy¾¡&©~V9¡yw&~9y¡y ©~9CwyxD¢Ax{~d¡×½ d (R) = const Ë
¡`DÀ$©w¡y¬£ÀÆ¬9 xÈ¡&©ë~A©¡yx§¡&¬A¾¡<ÀÆ¬Ax~ 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©¢RxÈÃ£Øw&Ú|
ìVx z} ~ë¡&¬9xN½ÓwyÁxÀ$©wyª,©¾½¡y¬AxNwy©¡&¾¡&~AÅ§ÀÂi¥xÄ9ÄAwy©ÚÃ/ Ï¾¡&©~
1
VXc0 = VX0c ∗ = − (EX (t) · dc0 )Φ0 e−ı(Ωt+ϕX )
2
~Az ~d¡ywy©qzAA¦{x¡y¬AxNzAx¡yA~A~AÅ
Ω = ωX − ωc0
÷¿qË dø
÷¿/Ë×idø
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P (t, Ω) = 2 =m hΦc |Vxc0 |Φ0 i = EX (t)=m
hΦ0 |d0c (R)|Φc i eı(Ωt+ϕX )
÷¿/Ë×bø
Ü~9©¡&¬A~AÅÝx{yx¡&¬A~¡y¬AxNw&¾¡yx`©½ÄÉ©Ä9Aj¾¡y ©~ë©¾½¡y¬AxN¦{©wx5ØxÃ/¦{¡yx{z¡&¾¡&xN¢d|,¡&¬9x`Ã£w&Ú|ëÄAAyx
(
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∂
+ 2Γ)ρcc (t) = P (t, Ω).
∂t
ÛÜ¬Ax,z9Aw&¾¡y ©~Ù©½ÜJ¡«~AzAwyzÙÃ£w&Ú|ßÁxÚ¾yAwx{Áx{~d¡&È §©~AÅxwÇ¡y¬V~ß¡y¬AxÄ9A xzAAwy¾¡&©~?Ë}ÛÜ¬A
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P (Ω) =
Z
∞
P (t, Ω) dt = 2Γ
−∞
Z
∞
ρcc (t) dt.
÷¿qË*ø
−∞
ÛÜ¬£A{D¦¦{©wzA ~AÅÙ¡&©0çÞ_Ë`÷ /Ë×bø5¡y¬AxÃ£ØwyÚ|Ä9wy©¢RxyÄRx{¦5¡&wA ¦{~¢Rx!¦{©ÏÄA/¡&x{z)Ay~AÅß¡&¬9x
~9¡«¾~£¡&~Ax©A,ÄAw©¢V¢9 ×¡Ø| ÷¿/Ë×bøÝ©w¡y¬AxÄR©ÄAA ¾¡&©~)©½`¡y¬Ax¦©wyx]xÃ9¦¡yx{z¡&¾¡&x¾ ρ (t) =
cc
Ë
hΦc |Φc i
ñßx1ÀÆ Ayxx<¢Rx{ ©iÀ+÷Óyxx/x¦Ë AAË £ø¡y¬V¾¡C¡&¬9xDÄA¬AyxD©½f¡y¬Ax<¦©¬Ax{wx{~d¡êwyzAjÒ¡& ©~ ϕ ©fx¦¡y
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y¬9©w¡yx{wN¡&¬V~ó¡&¬AxÁÄRx{wy©/zó©½$¥/¢Aw&Ò¡& ©~AÁ÷ ÄÉxw§JøË,ÛÜ¬A Ç¬VÄ9ÄÉx~AÈ¢Rx{¦{AxÏ¡&¬9xÏê¶ ~9zAA¦{xz
ñz9|q~VÁ ¦{zAx{ÄRx{~9zAC¡&w©~AÅ×|Ç©~ ϕ ËTÛÜ¬AxÆ¾¢Ay©wÄ9¡&©~©½V¡y¬AxÄAw©¢RxÂìVx{zzA©qx{m~9©¡zAxÄÉx~Az
©~ë¡&¬9xÇÄ9¬VyxÇ©½P¡&¬AxÃ£ØwyÚ|ëÄAAyx ϕ xLÃ9¦x{Ä9¡< ~,¡y¬AxN¦ÚxÈ©½T¡&ÏxØzAxÄÉx~AzAx{~d¡1ÄA¬V¾yx ϕ (t) Ë
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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
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|φc (t)i = eıhc t ζ|φ(t)i,
|φc (t)i
Á¡y¬Ax
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ø
1
ζ = (dc0 ·eX ),
2
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Time (fs)
ω OH
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150
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375.00fs
R*
329.10fs
0.4
0
0.4
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0
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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

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0.4
0
300
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145.51fs
122.56fs
100.00fs
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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
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
Úö
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
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

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
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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
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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
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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.
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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).
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062504-8
H±B
äÚ
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W. Yun, and S. K. Sinha, Phys. Rev. Lett. 80, 1110 (1998).
062504-9
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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
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
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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

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
ê
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
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5QV1HK ·Ï2d@ tIV
ê¸
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
ê4
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
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1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d
5QV1HK ·Ï2d@ tIV
ê·Ú
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
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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
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1 1Vi;-P±²kxd¢: 4³²±\H¶\;;±²b 64³)¶®·4´d
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ê<ê
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
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
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,
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ACKNOWLEDGMENTS
6
7
8
9
10
012714-11
ñr
w-I1V − IªK
GUIMARÃES et al.
11 C. Leforestier et al., J. Comput. Phys. 94, 59 1991.
12 F. F. Guimarães, V. C. Felicíssimo, V. Kimberg, A. Cesar, and
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/.
13 R. Fink, J. Chem. Phys. 106, 4038 1997.
14 A. D. Bandrauk, E. Aubanel, and J.-M. Gauthier, in Molecules
in Laser Fields, edited by A. D. Bandrauk Marcel Dekker,
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15 C. Wunderlich, E. Kobler, H. Figger and Th. W. Hänsch, Phys.
Rev. Lett. 78, 2333 1997.
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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.
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20 J. Ohkubo, T. Kato, H. Kono, and Y. Fujimura, J. Chem. Phys.
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Guerin, and H. R. Jauslin, Phys. Rev. A 69, 033402 2004.
012714-12
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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
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
400
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)
T It ;Ittk
ñtä
401
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
402
ñê
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.
T It ;Ittk
ñtñ
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).
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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,*
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
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<³BxV³Bx>±\´<Våå2w|
9Û 4³B
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
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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

9Û 4³B
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
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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
§

9Û 4³B
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
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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

9Û 4³B
§
ötñ
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.
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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.
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Phys. 96, 6796 1992.
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Cesar, and H. Ågren, J. Chem. Phys. 122, 094319 2005.
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Gel’mukhanov, A. Cesar, and H. Ågren, Phys. Rev. A 72,
012714 2005.
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Chem. Phys. Lett. 405, 398 2005.
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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
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ö<ö
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
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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,*
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
r
01 V;±\;±\´<Dµ´ 1¶<·
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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
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;±\
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
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;±\
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. The results of our simulations show strong sensitivity of the dynamics of the vibrational wave packet on the absolute phase of the pump field.
This dependence arises due to the breakdown of the rotating
wave approximation when the Rabi frequency approaches
the frequency of the transmission. The observation of phase
sensitivity in this case does not require forming a special
pulse envelope, as was made in Refs. 19,20.
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. F.G. acknowledges financial support from
the Russian Foundation for Basic Research, Project No. 0402-81020-Bel2004.
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ACKNOWLEDGMENTS
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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$Ú%K?I<D
ñ
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ò
2
ì Ô!ÙT×8ÚÝPÔpØVá"ÚÛ.Ý4×8ÚáEÕPÔ!Ý4ÔpØá0æL,RÞTÔpÕÔpÕ;Ùßgå(ÚèEÕ4ßKÝ4ÞEßàÔ!áTÔ!àèTà ÖØÕPÔ!ÝPÔpØVá
æ M
NÚæ è0æ ØâÝPÞEß
'ßgÛ!Û
EO2 (1s−1 ) (q)
ñPò
ó
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Ú{æ è0æ!æL,yÞEßG=;é Ù"ÚáEä¦Þ"ÚÕ×4Ôpå4ÞEßg×ìÔ!ÙT×WÚXÝ4ÔpØVá"ÚXÛÕPÝ4×PèEåÝ4èT×4ß
ä{èEßÝ4ØÛjÚ×Pãßä{ÔpÕPÖTÛjÚVågßgà~ßgáÝAØâÝPÞEß
ØâaÝ4ÞTß(=;é
1
q0 = 1.74
EO1 (1s−1 ) (q)
1
'RßÛ!Ûâ×8ØVà ÝPÞEßßPOMèTÔ!Û!Ô!ÙT×4Ô!èTàæ7,yÞEßAìÔ!ÙT×WÚXÝ4ÔpØVá"ÚXÛâ×4ßOMèEßáEågÜ
Ù"ÚáEäÔpÕAßgÕ8Õ4ßgáMÝPÔjÚÛ!Û!ÜÕPànÚÛ!Ûpßg×Ý4Þ"Úá)Ô!á)ÝPÞEß(=;é
OMèTÔ!Ý4ßxÕPÞ"ÚÛ!ÛpØF' Ô!áÝPÞEßQ1T×8ÚáEåKD3ïØVáEäTØVá×8ßãÔpØVá0æ
2
'RßÛ!Û«Ùßgå(ÚèEÕ4ßxÝPÞEß
,yÞEß;èTÖTÖßg×AÖ"ÚXáEßgÛpÕØXâR1«Ô!ãEæ3Õ4ÞEØ' ÝPÞEßÕ4èTÖTÖT×4ß$Õ4ÕPÔpØVáØâ.ÝPÞEß(=;é
2
EO1 (1s−1 ) (q)
ÖØVÝ4ßgáMÝPÔjÚÛ«ÔpÕ
Ù"ÚXáEä'yÞEßgá)ÝPÞEßå$ØVèTÖTÛ!Ô!áTãågØVáEÕPÝWÚáÝ
ã×4ØF'7Õ(æ7,yÞTÔpÕ7ß#4ß$åÝyÔpÕ7Õ4ß$ßáågÛpß$Ú×4Û!ÜÔ!áÝPÞEßS.;ë)0Õ4ÖßgågÝP×WÚ/'yÔ!ÝPÞ)ÚXáEä9'yÔ!ÝPÞEØVèTÝç+*T;"ßÛpäæR,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!ÖØÛpß
à~ØVà~ßáMÝ
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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)
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9½
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vu
¿ ½(U9½(UTba
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λqc = − 0.33 eV
d+
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2
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1.0
λ<0
λ>0
0.5
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λqc = + 0.33 eV
¼
d−
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-0.5
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0.0
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Ω (eV)
2
3
qo pr d)sFm#tunYvn)l+wmxzy<v{$|kikj}Ln~lut+mj}Fl+w ζ/2 = λq/∆(q) _ ∆(q0 ) = 1.02 m_
-d2eRfhgLn+{m#xzl+tuyFtR{y<n+i l+iks<m@FjJj$m#}<l+iks<mX)nYl+t+m#j}l+wn o jy]~m#|kdz_-Rl+wmtR{FtuFmzl+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
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Pz E (q ) = E (q ) ¶)$ £ @
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Z qc µ X
2 c
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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

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¥#% µ~¹Á ¶:F<® U¬ ·¸ ¡Â λ > 0
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u
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ä ç ä d_àGY
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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
$þ
é?ä
êîPñPåWêîGí@èZì¥ðä
åWì(âîPêõ°äæîPçä
êZò$î?ìzðî?òèZÿWæë%ì¥èä
ãë%ê
τX = 15
ïç#hèZæç¥ìä%ïRë%êZêhä
ãîñFë%ãUçzîîì¥ðë%ìì¥ðî%$åîãñðWèZãWû
ÿë%ãõ²èZãUì¥ðîQûë<åçç¥èë%ãUé?äõWîPêhç¥ì#ë<æzìzç@ïä
æ:æ#ë%ì¥ðîPæðWèZû$ðUñä
åWúWêZèZãWû¦úë<æ#ë%é?îìîæç
λqc
ñPä
éLúë%æ¥èZãWû
í@èZì¥ð¦ì¥ðîSêZèZãîFë%æcé?äõWîê'Xðîæîë
çä%ã¤ï~ä%æcì¥ðWèçcèç)ìzðîSçzì¥æä%ãWû($åîPãñzðWèZãWû?ä%ï°ì¥ðîSûë%åçzç¥èë%ã9ñä
åWúWêZèZãWû
ì¥ðîQíXë)9ï¼æzä
é
ì¥ðî/ñPæzä$çzçzèZãWû?úä
èZãì
qc
ý þRä
ã
ü
+*cîñFë<åçîQì¥ðîéGë%ãWè5ïîPçzìë%ì¥èä
ã"ä%ï-ìzðîQóô èçç¥ìzæzä
ãWû$îPæèZãì¥ðîêZèZãîFë<æ
õ ÿë<ãõWçcçzîPúë%æë%ìzî:íXðîPãGì¥ðî:êZèZãîFë<æXñPä
åWúWêZèZãWû/û$æzäFí@ç
ñä%åWúWêZèZãWûJé?äõWîPê F÷ ýþ2ícî:çîPî@ì¥ðë%ìRìzðî ë%ã,
ü
1
2
XðWèç/èçJèZã ë%û
æîPîPé?îPãìLí@èZì¥ð-î úWæîPçç¥èä
ã
$ý ý$þïä
æJì¥ðî¦ç¥úë
ñèZãWûÃÿîì+í)îîãì¥ðîçzî¤ÿë%ãõWç
ü
E+ (q) − E− (q)
. ãä%ìzðîæ(èZãìîæîPçzì¥èZãWû"æîPçzåWêZì/èçJìzðë%ì/ìzð
î òèZÿWæë%ì¥èä
ãë%êXúWæzä êî¦ñPä
êZêë%úçzîçJìzäUì¥ðî¦ç¥èZãWû$êî¦ø/"+øêZèZãî?ïä%æ
êë%æ¥ûî
λqc
XðWèçcèçcÿîñë%åçzîSìzðîSéLèZãWèZé/åWé
ä%ï2ì¥ðîû$æzä
åWãõ²çzìë%ìzîícîêZê
ü q0 = 1.74
2
ä%ïhìzðîë
õèë%ÿë%ì¥èñúä
ìzîPãì¥èë%ê
E+ (q)
ë%úWúWæzäë
ñzðîç:ì¥ðîSéLèZãWèZé/åWé
ë0 å1 þ-Wí@ðîãì¥ðîQñPä
åWúWêZèZãWû?èç@êë<æzûî2
354768:9);/<>=@?BACDAFEGAFHBEGI/;2CDEGAFHJEGHJC-?BKML@K2NGOJAFPRQSC-9)AFHBT,USVW=@XBNGQDKQ
çzðä%æzì(âúWåWêçzî
.
ícëòî/úë
ñcbîì
φ(t) _
÷Fø9ïçþíXèZìzðìzðîGèZãìîãçzèZìZ\[ 0ý ]^ ÷Fø 14 _a` ñé 2 ñPæzîFë%ìzîç?ë9û$æzä
åWãõBçzìë%ìzî
ü τL Y
LY
ðîPã²ì¥ðî/â: úWåWêçîêîëòîPçì¥ðîQçdçzìzîPé ¥ì ðîúä
úWåWêë%ì¥èä
ãçSä<ï7òèZÿWæë%ì¥èä
ãë%ê°êîPò$îPêçñPîFë%çî
ìzäQõWîPúîãõGä
ã?ìzðîXìzèZé?îë%ãõLì¥ðî:éGë%èZãGñPä
ãì¥æzèZÿWåWì¥èä
ã
fe þhèZã
û$èZòîPç)ì¥ðî@ì¥ðî@òèZÿWæë%ì¥èä
ãë%êêîòîPê
ü %
φ(t)
ν=4
2èZûg@÷Føþ-hcðî¦îPãîPæ¥ûBçzìzä
æzîõBèZã ì¥ðWèçLíXëPòî9úë
ñSb$îPì
÷Z÷]Ãî~óQþ(ë<êZêäFí:çLìzäUæîë
ñðìzðî
ü
ü hφ(t)|H0 |φ(t)i =
æzîPû$èä
ãä%ïRúWæzä
ìä%ãì¥æ#ë<ãç¥ïîæèZã"ìzðîJû$æzä
åWãõ"ç¥ì#ë%ìzîJúä
ìzîPãì¥èë%ê çzî&
î 2èZû ý-þ 5Xðåçìzðî é?ä
êîPñPåWêî(ûë<èZãç
ü
ç¥å0iGñPèîPãìXîãîPæ¥ûLìäæîë
ñð¤ì¥ðî:í)îFë bjê Lÿä
åWãõ?úWæä%ìä
ãGì¥æë%ãç¥ïîæ)í)îPêZ'ê íXðîPæzî:ì¥ðl
î kmCì#ë<åWìä
é?îæcèç-ïä
æ¥é?î
õ
XðîXõ0ãë%éLèñPçRä<ï
φ(t) ü
hèZûW÷$÷Fþ2ç¥ðäFí@ç7ì¥ðë%ì2ïä
æ-ì¥ðîXì¥èZé?îPç
t & tL = 200
ïç2ì¥ðîXícëòîXúë
ñSb$îPì7ÿîPñä
éLîç
õWîPêäñFë%êZèjnîõGë%ãõLúîæ¥ïä
æ¥é ÿë
ñcbLë%ãõ(ïä
æ¥ìzðGä
çñèZêZêë%ìzèä%ãçRèZãLì¥ðî@û$æzä
åWãõ?ç¥ìë%ìîcícîêZê'o³îPì7åç-æîéLèZãõLìzðë<ìRì¥ðî
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ì¥ðîMkm í)îPêZê
"+ø0 rs(î~ó
ü
km
tL = 200
ïçqpSåWæ¥èZãWûì¥ðîçzîXä$çzñPèZêZêë<ìzèä
ãç
φ(t)
úîæzèäõèñë%êZêj(ë%úWúWæzäë
ñzðîç
ë 1
0
å þ-Xðî(òîæzì¥èñFë<êRì¥æ#ë<ãçzèZì¥èä
ãçï¼æzä
é ìzðWèçí)îPêZê-ïä
æ¥é ìYí)ä²ãîíUÿë<ãõWç/ë%ì
ü q = 3.05
þ7ë%ãõ¦ý0 ]/î~ó km þ-í@ðWèñzð²ë%æzîë%ì¥ìzæ¥èZÿWåWìzîõ¤ìäQìzðîúæä
ìzä
ã¤ì¥æ#ë<ãç¥ïîPæ)èZã¤ìzðî é?ä
êîñåWêî
ü 2
1
¼
9½
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U
vu
¾ ]]
¿ ½(U9½(UTba
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w
60
50
Population (%)

UWXU
40
30
20
10
0
0
1
2
3
4
5
7
6
Vibrational levels
8
9
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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
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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 ÓÏ¬ÝÑÏÏÕÕÏ/ÞfÓ Æ ÝÙ ÙaÖqÕÓÑÍ0Þ Ç Ï/Ñ
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ê
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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
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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 ∗ )
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î
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)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
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Ç ÑÍ0ÛcÞ8àÝÞ8à
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ó ÔÏÞòÓÔÏ
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=Ê ω10 =
0.269 Æ
Æ
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→ π∗
(¢3¤ ¦\3§<¢
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/¡k¤(P¤4§¬«4/££
5«cª¡f £
y§ 2¡f ¤\®
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àcÖfÞ8Ý×
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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-ÍØªÓÔÏ
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
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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
¹HÀ1Â (
Ò »¶*¸À'Â
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 ÏÚÝÛ8ÕÏ¬ÓÔÏFí
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'Â
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 Î2MW/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$J ÊXG]FKS2I)J TVJ R G]FLKM
w
=G1R R SqF2QM QZR G`LKJ MNH$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 ÊXXLG`[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£$¢ 1¡AË©ÁÀ#]®\£¢ q
¸$q$«¢#®ºK1q£q"& ² &¢#®
£Á&q£q¦ \®«qyq
½£ ² p£&¢#&:`Â`£ 1q£$¢\£¢\q¸$q$«¢#®g¦ E1q ]t
m& ² &¢®V1¦ ¬KU®£1ªªN1 ]¢"qrË©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
¹HÀ1Â (
Ò »¶*¸À'Â
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
0
(bc)
3→
TPA
0→
0
2
dcc = 0
0→
1
2
0
(a0)
0
dcc = 0
0→
0
4
2
2→
(ac)
4→
0
4→1
6
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)
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c1 =x,y,z
hΨc (S)|VSO |Ψc1 m (T )iCc(Λ)
(T ) = 0,
1m
[hΨcm (T )|VSO |Ψc1 (S)iCc(Λ)
(S)
1
(T )] = 0
hΨcm (T )|VSO |Ψc1 m1 (T )iCc(Λ)
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ı  1, c = y, c1 = x
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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
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Ff (q) =
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EΛ − Ef (q)
I458<AQhMN.27 \
ωΛ,f (q) =
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Ψf (q) = Ψf (S)
^ B1-741958DAMN.Y7 \
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12
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Cc(Λ) (S)dcν
X
Cc(Λ) (S)dcf ,
c=x,y,z
(Λ)
Df (S) =
ÖÚÛhØ
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(Λ)
Ccm
(T )dcf .
c=x,y,z
c=x,y,z
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X
f
Þ À/ÃYÕFÃ
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|Ff (S)|2 Φ(ω1 − ω + ωf (S),0 , γ) +
X
m=1,0,−1
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ÖÚhÚØ
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|Ff (S)|2 =
h
× {D
(Λ)
12
12
2X X
1
9
(ω1 − ωΛ,f (S) + ıΓ)(ω1 − ωΛ1 ,f (S) − ıΓ)
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(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
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|Ffm (T )|2 =
h
× {D
(Λ)
Λ=1 Λ1 =1
(Λ)∗
(Λ )
(S) · D(Λ1 )∗ (S)}{Df m (T ) · Df m1 (T )}
3
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20
(Λ)∗
(Λ )
× {D(Λ) (S) · Df m (T )}{D(Λ1 )∗ (S) · Df m1 (T )}
+
(Λ )
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ü
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ù
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ð9÷
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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)∗
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[9σ1/2]
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180
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190
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200
210
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X-ray Spectroscopy of Molecules Driven by Strong Infrared Fields

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