3.5 The H 2 molecule in an external field
3.5.2 Lochfraß in H 2 molecules
10-10 10-9 10-8 10-7
0 10 20 30 40 50 60 70
0 500 1000 1500 2000
Harmonic order
(a)
10-10 10-9 10-8 10-7
0 10 20 30 40 50 60 70
0 500 1000 1500 2000
Harmonic order
Time (as) (b)
10-10 10-9 10-8 10-7
0 10 20 30 40 50 60 70
0 500 1000 1500 2000
Harmonic order
(c)
10-10 10-9 10-8 10-7
0 10 20 30 40 50 60 70
0 500 1000 1500 2000
Harmonic order
Time (as) (d)
Figure 3.21: Gabor analysis obtained from the HHG spectra of the 1D H2 molecule in a laser pulse with intensity I = 3 ×1014 W/cm2 using (a) Hartree, (b) two-SPFs and (c) eight-SPFs MCTDH, and (d) exact calculations.
3.5.2 Lochfraß in H2 molecules 53
5 10 15 20 25 30 35 40 45 50 55 60
Harmonic order 10-8
10-7 10-6 10-5 10-4
Harmonic intensity (arb. units)
1 SPF 2 SPFs 4 SPFs 8 SPFs 16 SPFs 32 SPFs Exact
Figure 3.22: Harmonic intensity obtained from the Gabor analysis of 1D H2molecule in a laser pulse with intensity I = 3 × 1014 W/cm2 using Hartree, MCTDH, and exact calculations.
vibrational wave packet is created in the electronic ground state of the molecule that is mainly in a superposition ofν = 0 and ν = 1. We calculate the expectation value of the internuclear distancehRiin order to observe the R-dependent depletion process. In (3.4.1) the nuclear density was obtained by integrating the wave function over all values of the electronic coordinates, while in the present case the nuclear density is obtained by integrating the wave function over a limited range of the electronic coordinates to isolate the bound part and to observe the depletion. The R-expectation valuehRib of the bound part is given by
Γlochfrass =
Z xbox
−xbox
dx Z ybox
−ybox
|ψ(x, y, R)|2dy, (3.5.4) hRib =
Z Rmax
0
ΓlochfrassR dR, (3.5.5)
where|xbox| = |ybox| = 20 a.u. In Figs. 3.23(a)-(d) we plot hRib for the H2 and D2
molecules subjected to laser pulses with wavelengths 200 nm, 400 nm, 800 nm, and 1200 nm for the intensity 6 × 1014 W/cm2 as a function of time. A sin2 n-cycle function with n2 turn-on and turn-off times is used as the field envelope. The pulse length for all the wavelengths is 8 fs and the duration of the field-free propagation is 42 fs. As mentioned earlier in the case of H+2 and H2 calculations, complex absorbing potentials are employed to avoid reflections from the boundary. The field strength attains the maximum around 3.5 fs. Looking at Fig, 3.23(c), we can see that up to this timehRib increases from the equilibrium internuclear distance. From 3.5 fs onwards, due to the rapid ionization at large internuclear distances, the molecules deplete at large R. This causes hRib to decrease. When the field becomes smaller,
1.0 1.2 1.4 1.6 1.8
<R> b (a.u.)
H2 molecule D2 molecule
1.4 1.5 1.6 1.7 1.8 1.9
<R> b (a.u.)
0 5 10 15 20 25 30 35 40 45 50
Time (fs) 1.5
1.6 1.7 1.8 1.9
<R> b(a.u.)
0 5 10 15 20 25 30 35 40 45 50
Time (fs) 1.5
1.6 1.7 1.8 1.9
<R> b(a.u.)
(a) (c)
(b) (d)
200 nm 800 nm
400 nm 1200 nm
Figure 3.23: hRib as function of time for H2 and D2 molecules in a laser pulse with intensity 6 × 1014 W/cm2 and wavelengths (a) 200 nm, (b) 400 nm, (c) 800 nm, and (a) 1200 nm using eight-SPFs MCTDH calculation. The pulse length in all the plots is 8 fs.
the intensity is not strong enough to cause any further ionization and as a result the wave packet starts to move towards large R leading to increasinghRib. The decrease and increase of hRib continues repeatedly due to the oscillation of the electric field till the end of the pulse. The wave packet, created by this process, is formed from the vibrational states of the electronic ground state of the H2 or D2 molecule. The superposition of vibrational states remains in coherence for an extremely long time.
In Fig. 3.23 we can see that, from 10 fs onwards the oscillation in hRib are periodic over time for both H2 and D2 molecules. The period of oscillation for the H2 and D2 molecules is roughly 8 fs and 11 fs, respectively. The R-dependent depletion process is known as lochfraß.
It is interesting to see that in Figs. 3.23(a)-(d) the depletion of the wave packet is strongest for the 200 nm wavelength while weak for the 1200 nm wavelength. This gives a clear message that lochfraß effect is more pronounced for high-frequency fields and less pronounced for low frequency fields.
In Figs. 3.24 and 3.25 we plot the nuclear density obtained as a function of time for D2 and H2 molecules. It is evident that the pump pulse excites oscillations in the density. In the region of small R, it is noteworthy that the D2 nuclear density oscillates coherently with a vibrational period of 11.5 fs. The nuclear density plot for H2 molecule in Fig. 3.25 shows the same oscillating nature with a vibrational period of 8 fs. These vibrational periods are in agreement with the values obtained from the Fig. 3.23.
3.5.2 Lochfraß in H2 molecules 55
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
200 nm (a)
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
400 nm (b)
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
800 nm (c)
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
1200 nm (d)
Figure 3.24: Nuclear density as a function of time for the D2 molecule in a laser pulse with pump wavelengths (a) 200 nm, (b) 400 nm, (c) 800 nm, and (d) 1200 nm and intensity 6 × 1014 W/cm2 using eight-SPFs MCTDH calculation. The pump pulse is 8 fs long.
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
200 nm (a)
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
400 nm (b)
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
800 nm (c)
10-6 10-5 10-4 10-3 10-2 10-1 100
0 2 4 6 8 10
0 10 20 30 40 50
Internuclear distance (au)
Time (fs)
1200 nm (d)
Figure 3.25: Nuclear density as a function of time for the H2 molecule in a laser pulse with pump wavelength (a) 200 nm, (b) 400 nm, (c) 800 nm, and (d) 1200 nm and intensity 6 × 1014 W/cm2 using eight-SPFs MCTDH calculations. The pump pulse is 8 fs long.
-1.72 -1.71 -1.70 -1.69 -1.68 -1.67 -1.66 -1.65 -1.64
Energy (a.u.) 10-4
10-2 100 102 104
Spectral power
Autocorrelation spectra 200 nm
400 nm 800 nm 1200 nm
-1.72 -1.71 -1.70 -1.69 -1.68 -1.67 -1.66 -1.65 -1.64
Energy (a.u.) 10-4
10-2 100 102 104
Spectral power
(a) D2 molecule (b) H2 molecule
Figure 3.26: Spectral power for (a) D2 molecule and (b) H2 molecule obtained by taking a Fourier transform ofhRib in a laser pulse with pump wavelengths 200 nm, 400 nm, 800 nm, and 1200 nm and intensity 6 × 1014 W/cm2 using eight-SPFs MCTDH calculations. The pump pulse is 8 fs long. The dotted curves show spectra obtained by taking the Fourier transform of the autocorrelation function obtained using an arbitrary initial state.
By calculating the autocorrelation spectra for the molecules, we can easily identify the occupied vibrational levels. To obtain these spectra, we have propagated an arbitrary initial state in real-time using the model Hamiltonian. The autocorrelation function from the propagated wave function is obtained at each time step and later Fourier transformed to obtain the spectrum. In Figs. 3.26(a) and 3.26(b), the dotted lines show the autocorrelation spectra for D2 and H2 molecule calculated using exact calculation. Here we have shown seven and five vibrational levels for D2 and H2
molecules, respectively. To see the levels occupied by the D2 and H2 molecules in the case of laser excitation, we take the Fourier transform of hRib. The energy axis has been shifted to compare the results with the autocorrelation spectrum.
Figs. 3.26(a) and 3.26(b) show that the pump pulses of different wavelengths create vibrational wave packets in the ground and first excited vibrational states. The vibrational periodTvib =2π/∆Eis obtained from the energy difference ∆E between the ground and the first excited vibrational level. For our 1D model calculation, the vibrational time periods obtained from the exact calculation for the D2 and the H2
molecules are 10.93 fs and 8.33 fs respectively. The vibrational time periods obtained from the eight-SPFs MCTDH calculations for D2 and H2 in good agreement with these values.
Lochfraß needs to be distinguished from another strong-field process known as bond softening, which refers to the formation of an adiabatic potential energy curve with increased equilibrium distance and smaller dissociation threshold [168]. Bond-softening causes the molecule to stretch to large R as long as the pulse lasts. When
3.5.2 Lochfraß in H2 molecules 57
-5 0 5 10 15 20 25 30 35 40 45
Time delay (fs) 1.6
2.0 2.4
<R> at probe maximum (a.u.)
200nm---delay--800nm 400nm---delay--800nm 800nm---delay--800nm 1200nm--delay--800nm
-5 0 5 10 15 20 25 30 35 40 45
Time delay (fs) 1.6
2.0 2.4 2.8 3.2
<R> at probe maximum (a.u.)
D2 molecule H2 molecule
(a) (b)
Figure 3.27: hRib as a function of delay time for the (a) D2 molecule and (b) H2
molecule in a laser pulse with pump wavelengths 200 nm, 400 nm, 800 nm, and 1200 nm and intensity 6 × 1014 W/cm2 using eight-SPFs MCTDH calculations. The pump and probe pulses are 8 fs long.
the pulse is over, the molecule moves to small R. Bond softening is dominant when the field strength is not too strong and the pulse length is longer. When the laser pulse is short and intense, the lochfraß mechanism is dominant: due to the prefer-ential ionization at large R values, the molecule depletes at large R and thus moves inwards towards small R. The depleted molecule later moves outwards towards large R when the field is turned off. Knowing the phase of oscillation we can distinguish the lochfraß and the bond softening processes. When comparing the theoretical calculations with the experimental data [167, 169], contribution from both mecha-nisms have to be taken into account. The contribution from each mechanism varies depending on the pulse length and intensity.
We now probe the lochfraß process, created by the application of a pump pulse, by an additional pulse at various delay times. This probe pulse is of 800 nm wavelength and 8 fs long. The delay time between the pump and the probe pulse is varied from -5 fs to 50 fs in steps of 1 fs. The probe pulse hasnprobe = 3 cycles. Asin2 function with nprobe2 turn-on and turn-off times is used as the field envelope. The field intensity 6× 1014 W/cm2 is used.
Figs. 3.27(a) and 3.27(b) show hRib as a function of time delay for D2 and H2
molecules respectively. The hRib values are taken at the center of the probe pulse.
The calculation for the 200 nm pump pulse for D2 and H2 show that the molecule is stretched to distances much larger than the equilibrium internuclear distance.
As the wavelength increases the nuclear excursions decrease and the oscillation are limited to the vicinity of the equilibrium internuclear distance. Different structures appearing in the plot ofhRib indicate that the nuclear wave packet contains not only
0.00 0.25 0.50 0.75 1.00
PSU
200nm---delay--800nm 400nm---delay--800nm
0.00 0.25 0.50 0.75 1.00
PSU
10-8 10-6 10-4 10-2
PPD
10-8 10-6 10-4 10-2
PPD
-5 0 5 10 15 20 25 30 35 40 45 50
Time delay (fs)
0.00 0.25 0.50 0.75 1.00
PCE
800nm---delay--800nm 1200nm--delay--800nm
-5 0 5 10 15 20 25 30 35 40 45 50
Time delay (fs)
0.00 0.25 0.50 0.75 1.00
PCE
(a)
(b)
(c) (f)
(e) (d)
D2 molecule H2 molecule
Figure 3.28: PSU, PP D, and PCO for D2 molecule (panels (a), (b), and (c) ) and H2
molecule (panels (d), (e), and (f)) obtained in a laser pulse with pump wavelengths 200 nm, 400 nm, 800 nm, and 1200 nm and intensity 6 × 1014 W/cm2 using eight-SPFs MCTDH calculation. The pump and probe pulses are 8 fs long.
the ground and the first vibrational levels but also some higher vibrational levels.
Next we calculate various observables for the pump-probe calculation. All the ob-servables are obtained at the end of the probe pulse. Fig. 3.28 shows the survival, photodissociation, and Coulomb explosion probabilities plotted as functions of time delay. The box parameters used here are the same as used in Eqs. (3.4.2) - (3.4.4) for the H+2 calculation. In Figs. 3.28(a) and 3.28(d) we plot the survival probability for the D2 and the H2 molecules. The pump and the probe pulse overlap for the small delay times. For these delay times if the pulses interfere constructively, the survival probability is reduced. This can be seen in panel (a) and (d) of Fig. 3.28 in the range from -5 to 5 fs. From 5 fs onwards, the survival probability shows periodic oscillations. Figs. 3.28(b) and 3.28(e) show the photodissociation probability while Figs. 3.28(c) and 3.28(f) show the Coulomb explosion probability for the D2 and the H2 molecules. The photodissociation probability is negligible for the negative time delays. The maximum values are reached after around 5 fs of time delay and then the signal oscillates periodically. The Coulomb explosion probability is large for delay times close to zero due to the overlap between pump and probe pulses.
For later times, the Coulomb explosion probability oscillates periodically. The os-cillations are explained by wave-packet formation. The pump pulse creates a wave packet in the electronic ground state of D2/H2 molecule. In Fig. 3.28(a) we find that when the survival probability attains a deep minimum (at 5 fs, 17 fs, 29 fs, and 41 fs in the plots for the D2 molecule) the photodissociation and Coulomb explosion probabilities in Figs. 3.28(b) and 3.28(c) attain a maximum at the same values.
3.5.3 Discussion 59