4 Light-Dressed Spectroscopy of Molecules
87
4.3 Computational Details
To illustrate the numerical results one can derive via the theory introduced in Sect. 4.2,
we investigate the Na 2 molecule. In the simulations we consider the X
1
+
g ground
state and the first excited A
1
+
u electronic state of Na 2 , which we represent with
the potential energy curves (PEC) on [40]. The transition dipole function between
these two states is taken from [41]. The field-free rovibrational wave functions of
Na 2 on the V X (R) and V A (R) PECs are computed using 200 spherical-DVR basis
functions [42] with the related grid points placed in the internuclear coordinate range
(0, 10) bohr. All rovibrational eigenstates with J < 16 and whose energy does not
exceed the zero-point energy of the respective PEC by more than 2000 cm
−1 were
included in the basis representing the Floquet Hamiltonian of (4.19), which was then
diagonalized to obtain the light-dressed states.
In all computations the polarization vector of the pump pulse was assumed to be
parallel to the polarization vector of the probe pulse. Therefore, the projection of the
total angular momentum onto this axis is a conserved quantity in our simulations.
When investigating the effect of the turn-on time of the dressing field
(see Sect. 4.4.5), the TDSE was solved using the simple formula (t + dt) =
e
−(i/)H(t)dt
(t). Due to the small size of H(t) (few thousand by few thousand)
the exponential function could be constructed by diagonalizing H(t) at each time
step.
4.4 Results and Discussion
4.4.1 Interpretation of the Light-Dressed Spectrum
Before the light-dressed spectra are investigated in detail, it is worth considering
their expected structure qualitatively. Naturally, the light-dressed spectrum strongly
depends on the molecule investigated and the properties of the dressing field. For
Na 2 the rotational, vibrational, and electronic transition wavenumbers considered in
this work are of the orders of 1,100, and 15,000 cm
−1 , respectively. Figure 4.1 depicts
the landscape of light-dressed PECs for the Na 2 molecule dressed by light whose
wavelength is λ = 662 nm. As can be seen in Fig. 4.1, the manifolds of light-dressed
states labeled by n are well separated from each other. Based on (4.40), arrows are
drawn to indicate the physical origin of possible absorption and stimulated emission processes induced by the probe pulse. Absorption is described by the first term
in (4.40), in which the initial light-dressed state contributes through its X ground
electronic state component and the final state contributes through its A excited electronic state component. On the other hand, stimulated emission originates from the
second term in (4.40), in which the initial light-dressed state contributes through its
A excited electronic state component and the final state contributes through its X
ground electronic state component.
87
4.3 Computational Details
To illustrate the numerical results one can derive via the theory introduced in Sect. 4.2,
we investigate the Na 2 molecule. In the simulations we consider the X
1
+
g ground
state and the first excited A
1
+
u electronic state of Na 2 , which we represent with
the potential energy curves (PEC) on [40]. The transition dipole function between
these two states is taken from [41]. The field-free rovibrational wave functions of
Na 2 on the V X (R) and V A (R) PECs are computed using 200 spherical-DVR basis
functions [42] with the related grid points placed in the internuclear coordinate range
(0, 10) bohr. All rovibrational eigenstates with J < 16 and whose energy does not
exceed the zero-point energy of the respective PEC by more than 2000 cm
−1 were
included in the basis representing the Floquet Hamiltonian of (4.19), which was then
diagonalized to obtain the light-dressed states.
In all computations the polarization vector of the pump pulse was assumed to be
parallel to the polarization vector of the probe pulse. Therefore, the projection of the
total angular momentum onto this axis is a conserved quantity in our simulations.
When investigating the effect of the turn-on time of the dressing field
(see Sect. 4.4.5), the TDSE was solved using the simple formula (t + dt) =
e
−(i/)H(t)dt
(t). Due to the small size of H(t) (few thousand by few thousand)
the exponential function could be constructed by diagonalizing H(t) at each time
step.
4.4 Results and Discussion
4.4.1 Interpretation of the Light-Dressed Spectrum
Before the light-dressed spectra are investigated in detail, it is worth considering
their expected structure qualitatively. Naturally, the light-dressed spectrum strongly
depends on the molecule investigated and the properties of the dressing field. For
Na 2 the rotational, vibrational, and electronic transition wavenumbers considered in
this work are of the orders of 1,100, and 15,000 cm
−1 , respectively. Figure 4.1 depicts
the landscape of light-dressed PECs for the Na 2 molecule dressed by light whose
wavelength is λ = 662 nm. As can be seen in Fig. 4.1, the manifolds of light-dressed
states labeled by n are well separated from each other. Based on (4.40), arrows are
drawn to indicate the physical origin of possible absorption and stimulated emission processes induced by the probe pulse. Absorption is described by the first term
in (4.40), in which the initial light-dressed state contributes through its X ground
electronic state component and the final state contributes through its A excited electronic state component. On the other hand, stimulated emission originates from the
second term in (4.40), in which the initial light-dressed state contributes through its
A excited electronic state component and the final state contributes through its X
ground electronic state component.