1. Fragmentation Dynamics of H2+ / D2+ Kansas State University
in Intense Ultrashort Laser Pulses
Kansas State
AMO PHYSICS
U. Thumm
B. Feuerstein
T. Niederhausen
Kansas State University
Introduction
Method of Calculation
Results:
initial vibrational state dependence
intensity dependence
pump-probe study of coherent vibrational motion

2. Laser pulse (Ti:sapphire)
INTRODUCTION
Laser pulse (Ti:sapphire)
H2+ (D2+)
Time scales
Tcycle = 2.7 fs
Tpulse = fs
Tv=0 = 14 (20) fs
Telectr = 0.01 fs
Energies
 = 1.5 eV
Ip = 30 eV
De = 2.8 eV
Length scales
l = a.u. (800 nm)
R0 = 2 a.u.

3. enhanced ionization (CREI)
H0 + H+
dissociation 2
dissociation 1
single ionization H2
H2+ 3
enhanced ionization (CREI) 4
fast Coulomb explosion
H+ + H+
Coulomb explosion

4. Dissociation and Ionization pathsg u
Charge resonance
enhanced ionization 1w
2(3)w CE
p + p
H2+
R [a.u.]
E [a.u.]
Dressed potential curves
(schematic) 3w 1w 2w 0w
weak field
strong field
Adiabatic dressed states (Floquet theory, CW laser)
-> distinguish 1w, 2w [effective (3-1)w – dipole selection], etc. dissociation paths via avoided crossings (tunneling and over barrier)
E.g.: 1w has threshold at v=5 (Saendig/Haensch et al did vibrationally resolved fragment momentum spectroscopy,
level shift due to potential widening observed as function of intensity.
trapping in 2w well = “bond hardening”)
Avoided crossings: mixing of g and u states -> localization ->CREI

5. 2D Crank-Nicholson split-operator propagation
METHOD OF CALCULATION
2x1D model R z
Laser field p e-
2D Crank-Nicholson split-operator propagation

6. Improved soft-core Coulomb potential
R-dep. softening function a(R)
+ fixed shape parameter b = 5
(Kulander et al PRA 53 (1996) 2562)
Fixed softening parameter a = 1
a(R) adjusted to
(exact) 3D pot. curve
present result

7. Dipole oscillator strength for sg – su transitions
} Kulander et al
PRA 53 (1996) 2562
This work (1D)

8. “virtual detector” method
Array for 2x1D collinear non-BO wave packet propagation
“virtual detector” method
z: electron coordinate
R: internuclear distance
Grid: z = 0.2 a.u.; R = 0.05 a.u.

9. “virtual detector”: data analysis
Integration over z and binning  fragment momentum distribution
Dissociation
Coulomb explosion
Integration over R and binning  fragment momentum distribution

10. RESULTS Time evolution of wave function and norm (on numerical grid)
Evolution of nuclear probability density r(R,t ) dissociation probability
ionization rate jz(R,t) CE probability
Kinetic energy spectra of the fragments
Single pulse (I = 0.05 – 0.5 PW/cm2, 25 fs):
vibrational state and intensity dependence
B) Pump-probe pulses (I = 0.3 PW/cm2, 25 fs):
CE-imaging of dissociating wave packets
C) Ultrashort pump-probe pulses (I = 1 PW/cm2, 5 fs):
CE-imaging of bound and dissociating wave packets

11. total fragment energy [eV]
0.2 PW/cm2
25 fs
PCE(t)
Dissociation
Coulomb explosion
(Coulomb energy)
PD (t)
Laser a
Norm(t) b c d
1 
2(3)  V
V 5 19
total fragment energy [eV]
log scale
Contours: jz(R,t)

12. - - - - - (Coulomb energy)
Norm(t)
v = 0
0.2 PW/cm2
25 fs
Dissociation
Coulomb explosion
1 
2(3)  V
V 5 19
(Coulomb energy)
Laser
PD (t)
PCE(t)
log scale

13. - - - - - (Coulomb energy)
v = 2
0.2 PW/cm2
25 fs
PCE(t)
PD (t)
Norm(t)
Dissociation
Coulomb explosion
1 
2(3)  V
V 5 19
(Coulomb energy)
Laser
log scale
Contours: jz(R,t)

14. - - - - - (Coulomb energy)
v = 4
0.2 PW/cm2
25 fs
PCE(t)
Dissociation
Coulomb explosion
(Coulomb energy)
PD (t)
Laser a
Norm(t) b c d
1 
2(3)  V
V 5 19
log scale
Contours: jz(R,t)

15. - - - - - (Coulomb energy)
PCE(t)
v = 6
0.2 PW/cm2
25 fs
Dissociation
Coulomb explosion
1 
2(3)  V
V 5 19
(Coulomb energy)
PD (t)
Laser
Norm(t)
log scale
Contours: jz(R,t)

16. - - - - - (Coulomb energy)
PCE(t)
v = 8
0.2 PW/cm2
25 fs
Dissociation
Coulomb explosion
1 
2(3)  V
V 5 19
(Coulomb energy)
Laser
Norm(t)
PD (t)
log scale
Contours: jz(R,t)

17. Branching ratio : Dissociation vs. Coulomb explosion

18. RESULTS II Single pulse (I = 0.05 – 0.5 PW/cm2, 25 fs):
vibrational state and intensity dependence
B) Pump-probe pulses (I = 0.3 PW/cm2, 25 fs):
CE-imaging of dissociating wave packets
C) Ultrashort pump-probe pulses (I = 1 PW/cm2, 5 fs):
CE-imaging of bound and dissociating wave packets

19. Pump-probe experiment
2(3)  CE
D2 target
0.1 PW/cm2
2 x 80 fs
variable delay fs
1 
Trump, Rottke and Sandner
PRA 59 (1999) 2858

20. Pump-probe (D2+) v = 0 0.3 PW/cm2 2 x 25 fs delay 30 fs Norm(t) PCE(t)
Dissociation
Coulomb explosion
Laser
PD (t)
(Coulomb only) b c a
log scale
Contours: jz(R,t)

21. Pump-probe (D2+) v = 0 0.3 PW/cm2 2 x 25 fs delay 50 fs Norm(t) PCE(t)
Dissociation
Coulomb explosion
PD (t)
Laser
(Coulomb only) b c a
log scale
Contours: jz(R,t) c b a

22. Pump-probe (D2+) v = 0 0.3 PW/cm2 2 x 25 fs delay 70 fs Norm(t) PCE(t)
Dissociation
Coulomb explosion
Laser
PD (t)
(Coulomb only) b c a
log scale
Contours: jz(R,t) c b a

23. RESULTS III Single pulse (I = 0.05 – 0.5 PW/cm2, 25 fs):
vibrational state and intensity dependence
B) Pump-probe pulses (I = 0.3 PW/cm2, 25 fs):
CE-imaging of dissociating wave packets
C) Ultrashort pump-probe pulses (I = 1 PW/cm2, 5 fs):
CE-imaging of bound and dissociating wave packets

24. Time evolution of a coherent superposition of states
Time dependent density matrix:
Time average:
Incoherent
mixture
Preparation of coherent states of molecular ions. How ?
Ion source: T  ms  incoherent ensemble
Ultrashort laser pulse: T  5 fs  coherence effects expected
H2+ (wkm-1 = 3 … 30 fs): produced by:

25. t autocorrelation pump 1 PW/cm2 5 fs probe 2 PW/cm2 5 fs D2+ D0 + D+
Model: vertical FC transition of D2 (v=0) -> D2+ potential curve.
Calculation shows complete ioinization within < 1fs (cf. extra plot). Survivals in remaining half pulse, we calculated:
H2+ ( period(v=0) = 14fs ) : 11% Diss., 4% CREI D
Explanation D2+ vibr. wave packet doesn’t reach avoided croissings within pulse duration.
WF collapse & partial revival in <R> and autocorrelation. t
probe 2 PW/cm2 5 fs
D0 + D+
D+ + D+
D2+
pump 1 PW/cm2 5 fs D2

26. Coulomb explosion imaging of nuclear wave packets
Fragment yield Y at Ekin :
Y(Ekin) dEkin = |(R)|2 dR  Y(Ekin) = R 2 |(R)|2
Kinetic energy
Ekin (R)
1/R
d + d
Probe
|(R,t)|2 R
D2+
Pump D2
initial |(R)|2

27. |(R)|2 reconstruction from CE fragment kin. energy spectra
 = 10 fs
|(R)|2
reconstructed |(R)|2
original |(R)|2
incoherent FC distr.
moving wave packet
relatively fast outward motion => noticable kinetic energy shift in reconstructed wp

28. |(R)|2 reconstruction from CE fragment kin. energy spectra
 = 20 fs
|(R)|2
reconstructed |(R)|2
original |(R)|2
incoherent FC distr.
turning point
Turning point => NO kin. energy shift and good agreement

29. |(R)|2 reconstruction from CE fragment kin. energy spectra
 = 40 fs
reconstructed |(R)|2
original |(R)|2
incoherent FC distr.
|(R)|2
wp moving back out again, but increasing broadening due to beginning collapse

30. |(R)|2 reconstruction from CE fragment kin. energy spectra
 = 580 fs
|(R)|2
reconstructed |(R)|2
original |(R)|2
incoherent FC distr.
‘revival’
Revival after about 20 oscillation periods => good reconstruction of initial wp