4 Light-Dressed Spectroscopy of Molecules
83
that the well-known adiabatic theorem could be used to predict the temporal changes
in (t). Therefore, in the limit of the dressing light intensity going to zero (g XA → 0),
the field-free eigenstates are eigenstates of H F as well, and the adiabatic turn-on of the
dressing field will convert an initial field-free eigenstate into a single light-dressed
state. This means that light-dressed states and field-free states can be correlated
in a one-to-one fashion. However, it is important to mention that the one-to-one
correlation is not possible if ω 1 is in exact resonance with an allowed transition,
because in that case the field-free eigenstates are not eigenstates of H F but are linear
combinations of H F eigenstates even for infinitesimal light-matter coupling strengths.
Further information on the temporal evolution of light-dressed states in the Floquet
formalism and its possible utilization can be found, for example, in Refs. [35–38].
4.2.3 Transitions Between Light-Dressed States
Once the light-dressed states are determined, one can compute the transition probabilities between the different light-dressed states induced by the weak probe pulse.
Following the standard approach of molecular spectroscopy [39], we use first-order
time-dependent perturbation theory (TDPT) to derive the relevant equations.
4.2.3.1 General Considerations
Let us assume that the molecule interacts with two periodic electric fields, oscillating
with frequencies ω 1 and ω 2 . The full Hamiltonian then reads as
ˆ
H (t) = ˆ
H mol + ˆ
W 1 (t) + ˆ
W 2 (t),
(4.22)
where ˆ
W 1 (t) and ˆ
W 2 (t) represent the interaction between the molecule and the two
fields. In the dipole approximation
ˆ
W 1 (t) = −E 1 ˆ
μcos(ω 1 t + φ)
ˆ
W 2 (t) = −E 2 ˆ
μcos(ω 2 t).
(4.23)
ˆ
W 1 (t) generates the light-dressed states, while ˆ
W 2 (t) represents the interaction with
a weak probe pulse used to record the spectrum of the light-dressed molecule. The
formation of light-dressed states by ˆ
W 1 (t) is described within the Floquet approach,
as overviewed in Sect. 4.2.1. For computing the ˆ
W 2 (t)-induced transition amplitudes
between individual light-dressed states or between the superpositions of light-dressed
states, first-order time-dependent perturbation theory is used. To derive our working
equations, we start with the TDSE containing the interaction with both the dressing
and the probe fields,
83
that the well-known adiabatic theorem could be used to predict the temporal changes
in (t). Therefore, in the limit of the dressing light intensity going to zero (g XA → 0),
the field-free eigenstates are eigenstates of H F as well, and the adiabatic turn-on of the
dressing field will convert an initial field-free eigenstate into a single light-dressed
state. This means that light-dressed states and field-free states can be correlated
in a one-to-one fashion. However, it is important to mention that the one-to-one
correlation is not possible if ω 1 is in exact resonance with an allowed transition,
because in that case the field-free eigenstates are not eigenstates of H F but are linear
combinations of H F eigenstates even for infinitesimal light-matter coupling strengths.
Further information on the temporal evolution of light-dressed states in the Floquet
formalism and its possible utilization can be found, for example, in Refs. [35–38].
4.2.3 Transitions Between Light-Dressed States
Once the light-dressed states are determined, one can compute the transition probabilities between the different light-dressed states induced by the weak probe pulse.
Following the standard approach of molecular spectroscopy [39], we use first-order
time-dependent perturbation theory (TDPT) to derive the relevant equations.
4.2.3.1 General Considerations
Let us assume that the molecule interacts with two periodic electric fields, oscillating
with frequencies ω 1 and ω 2 . The full Hamiltonian then reads as
ˆ
H (t) = ˆ
H mol + ˆ
W 1 (t) + ˆ
W 2 (t),
(4.22)
where ˆ
W 1 (t) and ˆ
W 2 (t) represent the interaction between the molecule and the two
fields. In the dipole approximation
ˆ
W 1 (t) = −E 1 ˆ
μcos(ω 1 t + φ)
ˆ
W 2 (t) = −E 2 ˆ
μcos(ω 2 t).
(4.23)
ˆ
W 1 (t) generates the light-dressed states, while ˆ
W 2 (t) represents the interaction with
a weak probe pulse used to record the spectrum of the light-dressed molecule. The
formation of light-dressed states by ˆ
W 1 (t) is described within the Floquet approach,
as overviewed in Sect. 4.2.1. For computing the ˆ
W 2 (t)-induced transition amplitudes
between individual light-dressed states or between the superpositions of light-dressed
states, first-order time-dependent perturbation theory is used. To derive our working
equations, we start with the TDSE containing the interaction with both the dressing
and the probe fields,