7 Ultrafast Laser-Induced Processes Described by Ab Initio Molecular
163
APLIP requires that the frequency of the first laser, ω 1 , is shifted to the blue of
the Franck-Condon transition. This shift must be compensated by the second frequency ω 2 , so that the initial and target potentials, V 1 and V 3 , are in two-photon
resonance. In the simulations presented here we have chosen ω 1 = 0.981 eV and
ω 2 = 2.192 eV. The APLIP scheme can be understood in the dressed state picture
using only this pathway as follows (see Fig. 7.8a): the first laser pulse, with frequency ω 1 , induces an electronic repulsion between the potentials V 2 − ω 2 and
V 3 − (ω 1 + ω 2 ) modifying the target potential V 3 − (ω 1 + ω 2 ). Then, the second
pulse, with frequency ω 2 , induces a second Stark shift between the potentials V 1 and
V 2 − ω 2 , modifying the initial potential V 1 . During the time both pulses overlap
the barrier between the V 1 and V 3 − (ω 1 + ω 2 ) in the LIP is removed and then a
slow displacement from the well corresponding to V 1 to the well corresponding to
V 3 − (ω 1 + ω 2 ) is possible, accounting for a very robust and selective population
transfer. This is the standard APLIP mechanism proposed by Garraway and coworkers [102]. However, a second two-photon resonant process cannot be neglected in
this case. For this set of potentials the second pathway describes a “red-detuning”
intuitive pulse sequence that also leads to an effective APLIP process [104, 105].
Here, we consider the effect of both pathways in the dynamics, by constructing the
Hamiltonian matrix in the Floquet picture as:
H
el
=
⎛
⎜
⎜
⎜
⎜
⎝
V 1
0
−μ 12 E 1 /2 −μ 12 E 2 /2
0
V 3 − (ω 1 + ω 2 ) −μ 23 E 2 /2 −μ 23 E 1 /2
−μ 21 E 1 /2
−μ 32 E 2 /2
V 2 − ω 2
0
−μ 21 E 2 /2
−μ 32 E 1 /2
0
V 2 − ω 1
⎞
⎟
⎟
⎟
⎟
⎠
(7.32)
where μ ij is the transition dipole moment between electronic states i and j . The envelopes in time domain of the laser pulses are chosen of the form cosh
−2 ((t −t 0 )/τ ),
centered at t 0 and with width τ in all simulations. Note that the first three rows and
columns describe the blue-detunning pathway while the last column/row takes into
account the red-detunning alternative. The potentials included in this Hamiltonian
are shown in Fig. 7.8a.
As explained before, the success of the APLIP control scheme, and in general of
adiabatic control schemes, relies on the nuclear wave function remaining in a single
LIP and, for high selectivity, following in a quasi-static way the time-dependent
structural changes of the LIP. Thus, it is necessary that the semiclassical dynamics
follows the dynamics of LIPs, which is exactly the idea underlying the SHARC
methodology [74, 85, 86]. As an example, Fig. 7.8b shows the time evolution of
a particular trajectory in SHARC on top of the LIPs. This trajectory was created
with zero momentum in the minimum of the potential V 1 (the most likely situation
in the Wigner distribution) using the Hamiltonian of Eq. (7.32) and the following
laser parameters: τ = 5.5 ps, τ = 4.5 ps and E 1 = E 2 = 0.006 e/a 2
0 . Due to the
electric field interaction, the molecule adapts to the minimum of the LIP and evolves
following the light-induced gradients. In this special case, with almost no kinetic
energy, the internuclear distance and the energy of the trajectory gives directly the
position of the LIP equilibrium geometry and its energy.
163
APLIP requires that the frequency of the first laser, ω 1 , is shifted to the blue of
the Franck-Condon transition. This shift must be compensated by the second frequency ω 2 , so that the initial and target potentials, V 1 and V 3 , are in two-photon
resonance. In the simulations presented here we have chosen ω 1 = 0.981 eV and
ω 2 = 2.192 eV. The APLIP scheme can be understood in the dressed state picture
using only this pathway as follows (see Fig. 7.8a): the first laser pulse, with frequency ω 1 , induces an electronic repulsion between the potentials V 2 − ω 2 and
V 3 − (ω 1 + ω 2 ) modifying the target potential V 3 − (ω 1 + ω 2 ). Then, the second
pulse, with frequency ω 2 , induces a second Stark shift between the potentials V 1 and
V 2 − ω 2 , modifying the initial potential V 1 . During the time both pulses overlap
the barrier between the V 1 and V 3 − (ω 1 + ω 2 ) in the LIP is removed and then a
slow displacement from the well corresponding to V 1 to the well corresponding to
V 3 − (ω 1 + ω 2 ) is possible, accounting for a very robust and selective population
transfer. This is the standard APLIP mechanism proposed by Garraway and coworkers [102]. However, a second two-photon resonant process cannot be neglected in
this case. For this set of potentials the second pathway describes a “red-detuning”
intuitive pulse sequence that also leads to an effective APLIP process [104, 105].
Here, we consider the effect of both pathways in the dynamics, by constructing the
Hamiltonian matrix in the Floquet picture as:
H
el
=
⎛
⎜
⎜
⎜
⎜
⎝
V 1
0
−μ 12 E 1 /2 −μ 12 E 2 /2
0
V 3 − (ω 1 + ω 2 ) −μ 23 E 2 /2 −μ 23 E 1 /2
−μ 21 E 1 /2
−μ 32 E 2 /2
V 2 − ω 2
0
−μ 21 E 2 /2
−μ 32 E 1 /2
0
V 2 − ω 1
⎞
⎟
⎟
⎟
⎟
⎠
(7.32)
where μ ij is the transition dipole moment between electronic states i and j . The envelopes in time domain of the laser pulses are chosen of the form cosh
−2 ((t −t 0 )/τ ),
centered at t 0 and with width τ in all simulations. Note that the first three rows and
columns describe the blue-detunning pathway while the last column/row takes into
account the red-detunning alternative. The potentials included in this Hamiltonian
are shown in Fig. 7.8a.
As explained before, the success of the APLIP control scheme, and in general of
adiabatic control schemes, relies on the nuclear wave function remaining in a single
LIP and, for high selectivity, following in a quasi-static way the time-dependent
structural changes of the LIP. Thus, it is necessary that the semiclassical dynamics
follows the dynamics of LIPs, which is exactly the idea underlying the SHARC
methodology [74, 85, 86]. As an example, Fig. 7.8b shows the time evolution of
a particular trajectory in SHARC on top of the LIPs. This trajectory was created
with zero momentum in the minimum of the potential V 1 (the most likely situation
in the Wigner distribution) using the Hamiltonian of Eq. (7.32) and the following
laser parameters: τ = 5.5 ps, τ = 4.5 ps and E 1 = E 2 = 0.006 e/a 2
0 . Due to the
electric field interaction, the molecule adapts to the minimum of the LIP and evolves
following the light-induced gradients. In this special case, with almost no kinetic
energy, the internuclear distance and the energy of the trajectory gives directly the
position of the LIP equilibrium geometry and its energy.