Rovibronic spectra of molecules dressed by light fields

TL;DR

This paper develops a quantum mechanical framework for understanding rovibronic spectra of molecules influenced by classical light fields, with detailed analysis on Na₂ and implications for spectroscopic measurements.

Contribution

It introduces a comprehensive quantum theory for light-dressed rovibronic spectra and simplifies the analysis for Na₂, elucidating physical processes and spectral dependencies.

Findings

Physical origin of spectral peaks explained

Spectral features depend on field intensity and wavelength

Implications for extracting field-free spectra discussed

Abstract

The theory of rovibronic spectroscopy of light-dressed molecules is presented within the framework of quantum mechanically treated molecules interacting with classical light fields. Numerical applications are demonstrated for the homonuclear diatomic molecule Na$_2$, for which the general formulae can be simplified considerably and the physical processes leading to the light-dressed spectra can be understood straightforwardly. The physical origin of different peaks in the light-dressed spectrum of Na$_2$ is given and the light-dressed spectrum is investigated in terms of its dependence on the dressing field's intensity and wavelength, the turn-on time of the dressing field, and the temperature. The important implications of light-dressed spectroscopy on deriving field-free spectroscopic quantities are also discussed.

Equations

$$|{\Phi }_k⟩=\sum _{n,\alpha ,v,J}{C}_{n,\alpha vJ}^{(k)}|\alpha vJ⟩|n⟩,$$

$$|{\Phi }_k(n)⟩=\sum _{v,J}{C}_{\mathrm{X}vJ}^{(k)}|\mathrm{X}vJ⟩|n⟩+\sum _{v,J}{C}_{\mathrm{A}vJ}^{(k)}|\mathrm{A}vJ⟩|n-1⟩,$$

$$\begin{array}{r}{T}_{l\leftarrow k}\propto \\ \sum _{v,J}\sum _{v^{\prime },J^{\prime }}{C}_{\mathrm{A}{v}^{\prime }{J}^{\prime }}^{(l)\ast }{C}_{\mathrm{X}vJ}^{(k)}⟨\mathrm{A}{v}^{\prime }{J}^{\prime }|{\mathbf{E}}_2\hat{\mathbf{\mu }}|\mathrm{X}vJ⟩\delta ({\epsilon }_k-{\epsilon }_l-\hslash {\omega }_1\pm \hslash {\omega }_2)+\\ \sum _{v,J}\sum _{v^{\prime },J^{\prime }}{C}_{\mathrm{X}{v}^{\prime }{J}^{\prime }}^{(l)\ast }{C}_{\mathrm{A}vJ}^{(k)}⟨\mathrm{X}{v}^{\prime }{J}^{\prime }|{\mathbf{E}}_2\hat{\mathbf{\mu }}|\mathrm{A}vJ⟩\delta ({\epsilon }_k-{\epsilon }_l+\hslash {\omega }_1\pm \hslash {\omega }_2),\end{array}$$

$$\frac{e^{-E_i^{\mathrm{FF}}/kT}}{Q(T)},$$

$${\hat{H}}_{\mathrm{d}}(t)={\hat{H}}_{\mathrm{mol}}+{\hat{W}}_1(t),$$

$${\hat{H}}_{\mathrm{d}}(t+T)={\hat{H}}_{\mathrm{d}}(t),$$

$$i\hslash {\partial }_t|\psi (t)⟩={\hat{H}}_{\mathrm{d}}(t)|\psi (t)⟩$$

$$|\psi (t)⟩=\sum _k{c}_k{e}^{-\frac{i}{\hslash }{\epsilon }_{k}t}|{\Phi }_k(t)⟩,$$

$$|{\Phi }_k(t+T)⟩=|{\Phi }_k(t)⟩,$$

$$({\hat{H}}_{\mathrm{d}}(t)-i\hslash {\partial }_t)|{\Phi }_k(t)⟩={\hat{H}}_{\mathrm{F}}(t)|{\Phi }_k(t)⟩={\epsilon }_k|{\Phi }_k(t)⟩.$$

$$|{\Phi }_k(t)⟩=\sum _n|{\phi }_{kn}⟩e^{in{\omega }_{1}t}.$$

$$({\hat{H}}_{\mathrm{mol}}+\hslash n{\omega }_1)\sum _n|{\phi }_{kn}⟩e^{in{\omega }_{1}t}-\frac{1}{2}{\mathbf{E}}_1\hat{\mathbf{\mu }}\sum _n|{\phi }_{kn}⟩(e^{i(n+1){\omega }_{1}t}+e^{i(n-1){\omega }_{1}t})={\epsilon }_k\sum _n|{\phi }_{kn}⟩e^{in{\omega }_{1}t}.$$

$$(\hat{H}_{\rm mol}+\hbar m\omega_{1})|\varphi_{km}\rangle-\frac{1}{2}\mathbf{E}_{1}\hat{\bm{\mu}}\big{(}|\varphi_{k,m-1}\rangle+|\varphi_{k,m+1}\rangle\big{)}=\varepsilon_{k}|\varphi_{km}\rangle.$$

$$|{\phi }_{km}⟩=\sum _{\alpha ,v,J}{C}_{m,\alpha vJ}^{(k)}|\alpha vJ⟩,$$

$$\sum _{n,\alpha ,v,J}({\mathbf{H}}_{\mathrm{F}}{)}_{m{\alpha }^{\prime }{v}^{\prime }{J}^{\prime },n\alpha vJ}{C}_{n,\alpha vJ}^{(k)}={\epsilon }_k{C}_{m,{\alpha }^{\prime }{v}^{\prime }{J}^{\prime }}^{(k)},$$

$$\begin{split}(\mathbf{H}_{\rm F})_{m\alpha^{\prime}v^{\prime}J^{\prime},n\alpha vJ}=\\ \big{(}\langle\alpha^{\prime}v^{\prime}J^{\prime}|\hat{H}_{\rm mol}|\alpha vJ\rangle+\hbar m\omega_{1}\delta_{\alpha\alpha^{\prime}}\delta_{vv^{\prime}}\delta_{JJ^{\prime}}\big{)}\delta_{nm}-\frac{1}{2}\langle\alpha^{\prime}v^{\prime}J^{\prime}|\mathbf{E}_{1}\hat{\bm{\mu}}|\alpha vJ\rangle\big{(}\delta_{n,m-1}+\delta_{n,m+1}\big{)}.\end{split}$$

$$\mathbf{H}_{\rm F}=\begin{bmatrix}\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\reflectbox{$\ddots$}\\ \cdots&\mathbf{H}_{\rm A}+\hbar\omega_{1}\mathbf{I}&\bm{\lambda}&\mathbf{g}_{\rm AA}&\mathbf{g}_{\rm AX}&0&0&\cdots\\ \cdots&\bm{\lambda}^{\dagger}&\mathbf{H}_{\rm X}+\hbar\omega_{1}\mathbf{I}&\mathbf{g}_{\rm XA}&\mathbf{g}_{\rm XX}&0&0&\cdots\\ \cdots&\mathbf{g}^{\dagger}_{\rm AA}&\mathbf{g}^{\dagger}_{\rm XA}&\mathbf{H}_{\rm A}&\bm{\lambda}&\mathbf{g}_{\rm AA}&\mathbf{g}_{\rm AX}&\cdots\\ \cdots&\mathbf{g}^{\dagger}_{\rm AX}&\mathbf{g}^{\dagger}_{\rm XX}&\bm{\lambda}^{\dagger}&\mathbf{H}_{\rm X}&\mathbf{g}_{\rm XA}&\mathbf{g}_{\rm XX}&\cdots\\ \cdots&0&0&\mathbf{g}^{\dagger}_{\rm AA}&\mathbf{g}^{\dagger}_{\rm XA}&\mathbf{H}_{\rm A}-\hbar\omega_{1}\mathbf{I}&\bm{\lambda}&\cdots\\ \cdots&0&0&\mathbf{g}^{\dagger}_{\rm AX}&\mathbf{g}^{\dagger}_{\rm XX}&\bm{\lambda}^{\dagger}&\mathbf{H}_{\rm X}-\hbar\omega_{1}\mathbf{I}&\cdots\\ \reflectbox{$\ddots$}&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix},$$

$$({\mathbf{H}}_{\mathrm{A}}{)}_{v^{\prime }{J}^{\prime },vJ}=⟨\mathrm{A}{v}^{\prime }{J}^{\prime }|{\hat{H}}_{\mathrm{mol}}|\mathrm{A}vJ⟩$$

$$({\mathbf{H}}_{\mathrm{X}}{)}_{v^{\prime }{J}^{\prime },vJ}=⟨\mathrm{X}{v}^{\prime }{J}^{\prime }|{\hat{H}}_{\mathrm{mol}}|\mathrm{X}vJ⟩$$

$$(\mathbf{\lambda }{)}_{v^{\prime }{J}^{\prime },vJ}=⟨\mathrm{A}{v}^{\prime }{J}^{\prime }|{\hat{H}}_{\mathrm{mol}}|\mathrm{X}vJ⟩$$

$$({\mathbf{g}}_{\alpha \beta }{)}_{v^{\prime }{J}^{\prime },vJ}=-\frac{1}{2}⟨\alpha v^{\prime }{J}^{\prime }|{\mathbf{E}}_1\hat{\mathbf{\mu }}|\beta vJ⟩$$

$${\mathbf{H}}_{\mathrm{F}}^{2\times 2}(n)=\left[\begin{array}{cc}{\mathbf{H}}_{\mathrm{X}}+n\hslash {\omega }_1& {\mathbf{g}}_{\mathrm{XA}}\\ {\mathbf{g}}_{\mathrm{XA}}^{†}& {\mathbf{H}}_{\mathrm{A}}+(n-1)\hslash {\omega }_1\mathbf{I}\end{array}\right].$$

$$\mathbf{\Psi }(t)=\sum _k{c}_k{e}^{-\frac{i}{\hslash }{\mathbf{H}}_{\mathrm{F}}t}{\mathbf{\Phi }}_k=e^{-\frac{i}{\hslash }{\mathbf{H}}_{\mathrm{F}}t}\sum _k{c}_k{\mathbf{\Phi }}_k=e^{-\frac{i}{\hslash }{\mathbf{H}}_{\mathrm{F}}t}\mathbf{\Psi }(t=0),$$

$$\hat{H}(t)={\hat{H}}_{\mathrm{mol}}+{\hat{W}}_1(t)+{\hat{W}}_2(t),$$

$$\begin{array}{r}{\hat{W}}_1(t)=-{\mathbf{E}}_1\hat{\mathbf{\mu }}\cos ({\omega }_{1}t+\varphi )\\ {\hat{W}}_2(t)=-{\mathbf{E}}_2\hat{\mathbf{\mu }}\cos ({\omega }_{2}t).\end{array}$$

$$i\hslash {\partial }_t|\Psi (t)⟩=({\hat{H}}_{\mathrm{mol}}+{\hat{W}}_1(t)+{\hat{W}}_2(t))|\Psi (t)⟩=({\hat{H}}_{\mathrm{d}}(t)+{\hat{W}}_2(t))|\Psi (t)⟩.$$

$$|{\Psi }_{\mathrm{I}}(t)⟩=e^{\frac{i}{\hslash }{\int }_{t_0}^t{\hat{H}}_{\mathrm{d}}(t^{\prime })d{t}^{\prime }}|\Psi (t)⟩,$$

$$i\hslash {\partial }_t|{\Psi }_{\mathrm{I}}(t)⟩={\hat{W}}_{2\mathrm{I}}(t)|{\Psi }_{\mathrm{I}}(t)⟩,$$

$$|{\Psi }_{\mathrm{I}}(t)⟩=\hat{U}(t,t_0)|{\Psi }_{\mathrm{I}}(t_0)⟩$$

$$\hat{U}(t,t_0)=\hat{I}+\frac{1}{i\hslash }{\int }_{t_0}^t{\hat{W}}_{2\mathrm{I}}(t^{\prime })d{t}^{\prime }+\frac{1}{(i\hslash {)}^2}{\int }_{t_0}^t{\hat{W}}_{2\mathrm{I}}(t^{\prime }){\int }_{t_0}^{t^{\prime }}{\hat{W}}_{2\mathrm{I}}(t^{\prime \prime })d{t}^{\prime \prime }d{t}^{\prime }+\cdots .$$

Full text

I Introduction

Atomic and molecular spectroscopy are among the most successful tools of science both in fundamental research and in practical applications. Despite the 200-year-long history of spectroscopy, new approaches and methods are being developed to this day Merkt and Quack (2011). This is strongly related to the remarkable advances in the available experimental techniques and light sources. Some notable developments include frequency-comb techniques Udem et al. (2002); Diddams (2010); Hall (2006); Hänsch (2006), facilitating extremely precise and accurate measurements in the frequency domain, and the ability to generate ultrashort and intense laser pulses Brabec and Krausz (2000); Krausz and Ivanov (2009), allowing for time-resolved spectroscopies on femtosecond or even attosecond Krausz and Ivanov (2009); Palacios and Martín (2019) timescales.

A common approach in the application of spectroscopic techniques is the use of two (or more) light pulses, with some pulses acting as so-called pump pulses, which induce specific changes in the system, while subsequent, so-called probe pulses are used (directly or indirectly) to measure the changes induced by the pump pulse(s). If the duration of both the pump and the probe pulses is short, then repeating the experiment with varying time delays between the pulses can lead to time-resolved dynamical information Zewail (2000); Dantus (2001) or by means of a Fourier transformation to multidimensional and/or high-resolution spectra Mukamel (2000); Ando et al. (2018). If the pump and probe pulses overlap in time, the signal recorded by the probe pulse represents the so-called field-dressed or light-dressed properties of the system Cohen-Tannoudji et al. (2004); Shirley (1965); George et al. (1977), where the pump pulse acts as the dressing field. If both the pump and probe pulses are long with respect to the timescales of the processes investigated, then one obtains static spectral properties of the light-dressed system. We investigate the last scenario in this work and henceforth call it light-dressed spectroscopy. The theoretical and experimental methods for investigating spectral transitions between atomic light-dressed states is well developed Cohen-Tannoudji et al. (2004). As to molecules, the concept of light-dressed electronic states, also called light-dressed potentials, has been utilized with success to understand nuclear dynamics both in experimental and theoretical studies Wunderlich et al. (1997); Garraway and Suominen (1998); Sato et al. (2003); Corrales et al. (2014); Kelkensberg et al. (2009); Atabek et al. (2010); Bucksbaum et al. (1990). To some extent light-dressed rovibrational spectroscopy has been adopted for molecular systems, as well; for example, inducing Autler–Townes-type splittings Autler and Townes (1955) of rotational transitions with microwave radiation have been used to deduce molecular parameters Sanli et al. (2017); Ahmed et al. (2012) or to promote the assignment of rovibrational spectra Lee and Pate (1997). Related to light-dressed spectroscopy, the optical absorption of electronic materials driven away from equilibrium by laser light was also investigated Gu and Franco (2018).

In a previous theoretical study of the present authors, in which all molecular degrees of freedom were incorporated into the concept of light-dressed states, the rovibronic spectrum of light-dressed Na2 was investigated in the context of how the presence of a Light-Induced Conical Intersection (LICI) Halász et al. (2011, 2012), generated by the dressing field, can be identified in the spectrum Szidarovszky et al. (2018a, b). The modeling work was carried out for both the classical dressing field of laser radiation Szidarovszky et al. (2018a) and the quantized dressing field of a microscopic cavity mode Szidarovszky et al. (2018b).

It is our firm belief that the spectroscopy on light-dressed molecules is of fundamental interest and could provide valuable scientific results on both light-dressed and field-free molecular systems. For example, because the response of light-dressed systems to external fields is different from that of the corresponding field-free systems, understanding and predicting the absorption and emission properties of light-dressed molecules can be useful for tuning or controlling optical processes, such as those in coherent control schemes or laser cooling Warren et al. (1993); Koch and Shapiro (2012); Vitanov et al. (2017); Atabek et al. (2011); Viteau et al. (2008). In this work we provide an introduction to the theory of computing light-dressed spectra induced by classical light fields and investigate certain aspects and spectroscopic implications of light-dressed spectroscopy, such as the effects of dressing-light intensity and frequency and those of finite non-zero temperature and the turn-on time of the dressing field. A procedure how to derive field-free transition frequencies from light-dressed spectra is also proposed.

II The theoretical approach

To make reading of this paper easier, the presentation of Floquet theory Chu and Telnov (2004); Shirley (1965); Chu (1985) and some of the detailed derivations have been moved to the Appendix, Section VI (VI.1, VI.2, and VI.3) of this paper. The Appendix contains a large number of equations and only a few equations are given in this section. Those not familiar with Floquet theory or those interested in modeling details may want to read the Appendix before reading this section.

We advocate the use of a three-step theoretical approach to compute the light-dressed spectra of molecules. First, compute the field-free rovibronic states of the investigated molecule, second, using the field-free eigenstates as basis functions compute the light-dressed states, and finally, compute the transitions between the light-dressed states.

As to the physical scenario to be simulated, we make a couple of important assumptions: (1) Initially the molecule is in a field-free eigenstate in the gas phase. (2) The molecule is exposed to a medium intensity dressing light ($I=10^{7}-5\cdot 10^{9}$ Wcm*-2* in the numerical examples of this work), which is turned on adiabatically, i.e., its envelope varies much slower than the rovibronic timescales characterizing the molecule (such a dressing pulse converts the molecular wave function into a light-dressed state or a superposition of light-dressed states, see Sections VI.1 and VI.2). In the numerical examples considered below, the slowest molecular timescale is around 100 ps; thus, dressing pulses of at least nanosecond length could be considered as turning on adiabatically. (3) The probe pulse, introduced to record the static rovibronic spectrum of the light-dressed molecule, is weak.

II.1 Light-dressed states

We determine the light-dressed states generated by the dressing light within the framework of Floquet theory Chu and Telnov (2004); Shirley (1965); Chu (1985). As detailed in Section VI.1, in the presence of a dressing field periodic in time, the solution of the time-dependent Schrödinger equation (TDSE) can be written as a superposition of light-dressed states (Floquet states) $|\Phi_{k}(t)\rangle$. With the initial wave function being a field-free eigenstate and the dressing-field being turned on adiabatically, the generated light-dressed wave function is composed of a single light-dressed state (see Sec. VI.2) Guérin and Jauslin (2003); Lefebvre et al. (2013). Because the $|\Phi_{k}(t)\rangle$ states are periodic in time, they can be expanded as a Fourier series,

[TABLE]

where we have used the notation called “Floquet-state nomenclature” Chu and Telnov (2004), in which $\langle t|n\rangle=e^{in\omega_{1}t}$ and $\omega_{1}=2\pi/T$ with $|\Phi_{k}(t+T)\rangle=|\Phi_{k}(t)\rangle$. In Eq. (1), $\alpha$, $v$, and $J$ represent electronic, vibrational, and rotational quantum numbers, respectively. The expansion coefficients $C^{(k)}_{n,\alpha vJ}$ are obtained by diagonalizing the Floquet Hamiltonian of Eq. (A13).

In practical applications, when the Floquet Hamiltonian can be simplified to the form of Eq. (A18), i.e., when molecules have no permanent dipole and no intrinsic nonadiabatic couplings (or when the transitions originating from the permanent dipole and intrinsic nonadiabatic couplings can be neglected), and the dressing-field intensity is moderate enough to allow for the two-by-two Floquet Hamiltonian approach Halász et al. (2012), light-dressed states have a simplified form of

[TABLE]

where X and A represent the two electronic states considered (as usual, X denotes the ground electronic state).

II.2 Transitions between light-dressed states

Once the light-dressed states have been determined, one can compute the transition probabilities between the different light-dressed states as induced by the weak probe pulse. Following the standard approach of molecular spectroscopy Bunker and Jensen (1998), we use first-order time-dependent perturbation theory (TDPT1), as detailed in Section VI.3.

For physical scenarios in which the light-dressed states have the form shown in Eq. (2), such as the light-dressed states of the Na2 molecule investigated below, the $T_{l\leftarrow k}$ TDPT1 transition amplitude induced between the kth and lth light-dressed states by an interaction of the form $\hat{W}_{2}(t)=-\mathbf{E}_{2}\hat{\bm{\mu}}\,{\rm cos}(\omega_{2}t)$ gives (see Eq. (A37))

[TABLE]

where $\mathbf{E}_{2}$ is the electric field vector of the probe pulse, $\mathbf{\mu}$ is the molecular dipole operator, $\hbar\omega_{1}$ and $\hbar\omega_{2}$ are the photon energies of the dressing and probe fields, respectively, and $\varepsilon_{i}$ is the quasienergy of the ith light-dressed state.

Based on the arguments of the delta functions in Eq. (3), the first term can be interpreted as a transition between light-dressed states having quasienergies $\varepsilon_{k}$ and $\varepsilon_{l}+\hbar\omega_{1}$, that is, a transition between $|\Phi_{k}(n)\rangle$ and $|\Phi_{l}(n^{\prime})\rangle$ with $n=n^{\prime}-1$. Similarly, the second term in Eq. (3) can be interpreted as a transition between $|\Phi_{k}(n)\rangle$ and $|\Phi_{l}(n^{\prime})\rangle$ with $n=n^{\prime}+1$.

II.3 Computational details

We choose to demonstrate the numerical application of the theory introduced above and in the Appendix (Section VI) on the Na2 molecule. In our simulations we consider the ${\rm X}\,^{1}\Sigma{\rm{}_{g}^{+}}$ ground and the first excited ${\rm A}\,^{1}\Sigma{\rm{}_{u}^{+}}$ electronic states of Na2, for which we use the potential energy curves (PEC) and the transition dipole function of Refs. Magnier et al. (1993) and Zemke et al. (1981), respectively. The field-free rovibrational eigenstates of Na2 on the $V_{\rm X}(R)$ and $V_{\rm A}(R)$ PECs are computed using 200 spherical-DVR basis function Szidarovszky et al. (2010) with the related grid points placed in the internuclear coordinate range $(0,10)$ bohr. All rovibrational eigenstates with $J<16$ and an energy not exceeding the zero-point energy of the respective PEC by more than 2000 cm*-1* were included into the basis representing the Floquet Hamiltonian of Eq. (A18), which was then diagonalized to obtain the light-dressed states.

In all computations the polarization vector of the probe pulse was assumed to be parallel to the polarization vector of the pump pulse, meaning that the projection of the total angular momentum onto this axis is a conserved quantity during our simulations.

When investigating the effect of the turn-on time of the dressing field (see Sec. III.5), the TDSE was solved using the simple formula $\mathbf{\Psi}(t+dt)=e^{-(i/\hbar)\mathbf{H}(t)dt}\mathbf{\Psi}(t)$. Due to the small size of $\mathbf{H}(t)$ (few thousand by few thousand) the exponential function could be constructed by diagonalizing $\mathbf{H}(t)$ at each time step.

III Results and discussion

III.1 Interpretation of the light-dressed spectrum

Before investigating the light-dressed spectra in detail, it is worth considering their expected structure qualitatively. Naturally, the light-dressed spectrum strongly depends on the specific molecule investigated and the properties of the dressing field. For Na2 the rotational, vibrational, and electronic transition frequencies considered in this work are around the order of 1, 100, and 15 000 cm*-1*, respectively. Figure 1 demonstrates the landscape of light-dressed PECs for the Na2 molecule dressed by a $\lambda=657$ nm wavelength light, which is near resonant with the transition between the $|{\rm X}$ [math] $0\rangle$ and $|{\rm A}$ $2$ $1\rangle$ states. As can be seen in Fig. 1, the manifolds of light-dressed states, labeled by n, are well separated. Based on Eq. (3), arrows are drawn to indicate the physical origin of possible absorption and stimulated emission processes induced by the probe pulse. As seen in Fig. 1, absorption originates from the first term in Eq. (3), in which the initial light-dressed state contributes with its X ground electronic state component and the final state contributes with its A excited electronic state component. On the other hand, stimulated emission originates from the second term in Eq. (3), in which the initial light-dressed state contributes with its A excited electronic state component and the final state contributes with its X ground electronic state component.

III.2 Intensity dependence of the light-dressed spectrum

Figure 2 shows the 0 K absorption and stimulated emission spectra of Na2, when the molecule is dressed by 657 nm wavelength light fields of different intensity. Note that there is no special reason for choosing 657 nm to demonstrate the intensity dependence of the light-dressed spectrum, it is merely a convenient choice for which the light-dressed PECs and some relevant states are already depicted in Fig. 1.

As seen in Fig. 2, with increasing dressing-field intensity the envelopes of both the absorption and the stimulated emission spectra increase. At the limit of zero dressing-field intensity, the stimulated emission peaks disappear, as expected. We point out here that the light-dressed states and the corresponding light-dressed spectra change if the dressing field wavelength is changed. Therefore, if dressing light wavelengths different from 657 nm are used, the envelope of the absorption spectrum might decrease or even show no monotonic behaviour with increasing dressing light intensity.

Inspecting the individual transition lines reveals that introducing the dressing field leads to the splitting of existing field-free absorption peaks as well as the appearance of new peaks, as shown in the upper and lower panels of Fig. 3, respectively. It is possible to understand and assign the transition peaks of the light-dressed spectrum based on the selection rules of transitions between field-free states and the fact that the light-dressed states can be described as a superposition of field-free states.

For example, the upper panel of Fig. 3 shows the progression of three peaks, corresponding to transitions from the initial state (light-dressed state correlating to the field-free ground state) composed primarily of the $|{\rm X}$ [math] $0\rangle$ state with smaller contributions from the $|{\rm X}$ [math] $2\rangle$ and $|{\rm A}$ $2$ $1\rangle$ states to light-dressed states composed primarily of the $|{\rm A}$ $7$ $1\rangle$, $|{\rm A}$ $7$ $3\rangle$, and $|{\rm A}$ $7$ $5\rangle$ states, with $|{\rm X}$ $v$ $J\rangle$-type states ($J$ even) contributing as well. These transitions can be interpreted as originating from the field-free transition $|{\rm A}$ $7$ $1\rangle\leftarrow|{\rm X}$ [math] $0\rangle$, which is split due to the mixing of field-free states through the light-matter coupling with the dressing field. Such type of peak splittings are similar in spirit to the well-known Autler–Townes effect Autler and Townes (1955) utilized in spectroscopy, see, for example, Refs. 26; 25; 24. Furthermore, the upper panel of Fig. 3 not only demonstrates splitting of the peak, during which the sum of individual peak intensities remain unchanged, but exhibits an overall increase in the peak intensities when the strength of the dressing field is increased. In a spectroscopic context the changes in transition peak intensities resulting from couplings between eigenstates of a zeroth-order Hamiltonian is usually called intensity borrowing Bunker and Jensen (1998).

The lower panel of Fig. 3 shows the progression of three peaks, which do not arise from the splitting of an existing field-free peak but appear as new peaks. These transitions occur between the initial state (light-dressed state correlating to the field-free ground state) composed primarily of $|{\rm X}$ [math] $0\rangle$ with smaller contributions from $|{\rm X}$ [math] $2\rangle$ and $|{\rm A}$ $2$ $1\rangle$ and light-dressed states composed primarily of the $|{\rm X}$ $3$ $0\rangle$, $|{\rm X}$ $3$ $2\rangle$, and $|{\rm X}$ $3$ $4\rangle$ states. Such transitions are forbidden in the limit of zero dressing-light intensity; however, they become visible as the light-matter coupling with the dressing field contaminates the $|{\rm X}$ $3$ $J\rangle$ states ($J$ even) with $|{\rm A}$ $v$ $1\rangle$-type states, to which $|{\rm X}$ [math] $0\rangle$ has allowed transitions. The appearance of transition peaks as a result of such a mixing phenomenon can be understood as an intensity borrowing effect.

As to the stimulated emission peaks shown in the lower panel of Fig. 2, they represent transitions from the initial state to light-dressed states composed primarily of vibrationally highly excited $|{\rm X}$ $v$ $0\rangle$- and $|{\rm X}$ $v$ $2\rangle$-type states, with $|{\rm A}$ $v^{\prime}$ $J\rangle$-type states ($J$ odd) contributing as well.

III.2.1 Predicting field-free properties via extrapolation

Although light-dressed spectroscopy might provide transition peaks forbidden in the field-free case, the transition frequencies between light-dressed states are in general different from the transition frequencies between field-free states. If one is interested in obtaining field-free transition frequencies, one might record the light-dressed spectrum at several dressing-field intensities and extrapolate to the zero intensity limit. Such a procedure is of course most valuable if the transition is forbidden in the field-free case.

As an example, we examine the stimulated emission peak at around 13 612.5 cm*-1*, see Fig. 4. The emission peak around 13 612.5 cm*-1* represents a transition in which the initial state is composed primarily of the $|{\rm X}$ [math] $0\rangle$ ground state with the $|{\rm X}$ [math] $2\rangle$ and $|{\rm A}$ $2$ $1\rangle$ states contributing as well, and in which the final state is composed primarily of $|{\rm X}$ $11$ $0\rangle$, with some contribution also from $|{\rm X}$ $11$ $2\rangle$ and $|{\rm A}$ $17$ $1\rangle$. In the limit of zero dressing-light intensity, the initial and final states correlate to $|{\rm X}$ [math] $0\rangle$ and $|{\rm X}$ $11$ $0\rangle$, respectively. Transition between these two field-free states is forbidden; nonetheless, their accurate transition frequency can be obtained by extrapolating the light-dressed transition frequency to the limit of zero dressing-light intensity. As expected and seen in Fig. 4, linear extrapolation might be pursued if data points at low dressing-light intensities are used. On the other hand, by increasing the dressing-light intensity above a certain point, the relation between intensity and transition frequency becomes nonlinear. As seen in Fig. 4, the value of the transition frequency extrapolated to zero intensity is 13 612.37 cm*-1*. Considering that in this transition the $|{\rm X}$ [math] $0\rangle$ and $|{\rm X}$ $11$ $0\rangle$ states belong to the $n$ and $n-1$ Fourier manifolds, see Fig. 1, the transition frequency between $|{\rm X}$ [math] $0\rangle$ and $|{\rm X}$ $11$ $0\rangle$ can be obtained as $(-13\,612.37+15\,220.70)$ $=1608.33$ cm*-1*, where $15\,220.70$ cm*-1* is the photon energy of the dressing light. The numerical value for the transition wavenumber, obtained as the difference between the computed field-free eigenenergies of $|{\rm X}$ [math] $0\rangle$ and $|{\rm X}$ $11$ $0\rangle$, is also $1608.33$ cm*-1*; thus, the extrapolation technique works perfectly.

Of course there are spectroscopic methods already available, such as Laser Induced Disperse Fluoroesence (LIDF) and Stimulated Emission Pumping (SEP) Dai and Field (1995), capable of producing data similar to that retrieved from the normally-forbidden transitions measured by our extrapolation scheme. Nonetheless, light-dressed spectroscopy could complement these existing emission-only spectroscopic methods and it is worth noting that the extrapolation scheme can be utilized in both absorption and emission measurements, and varying the dressing-light wavelength could provide control and selectivity over the transitions to be measured, see below.

III.3 Frequency dependence of the light-dressed spectrum

Figure 5 shows the light-dressed spectrum of Na2 when dressed by $I=10^{8}$ W cm*-2* intensity light fields of different wavelength. As seen in Fig. 5, both the absorption and the stimulated emission spectra vary strongly with the dressing-light wavelength. This is expected because with varying dressing-light wavelength the contribution of different field-free states in the light-dressed states also vary, leading to varying transition probabilities. Therefore, by changing the dressing-light wavelength, one has certain control over which type of transitions appear in the light-dressed spectrum. In the vicinity of dressing light wavelengths which are resonant with a $|{\rm A}$ $v$ $1\rangle\leftarrow|{\rm X}$ [math] $0\rangle$-type transition, the absorption spectrum signal decreases (vertical white lines in the left panel of Fig. 5) while stimulated emission increases simultaneously. This can be understood as resulting from the increased mixing of field-free states near resonance, i.e., the weight of $|{\rm X}$ [math] $0\rangle$ decreases (leading to a decrease in the absorption), while the weight of $|{\rm A}$ $v$ $1\rangle$ increases (leading to an increase in the emission) in the light-dressed state correlating to $|{\rm X}$ [math] $0\rangle$.

Interestingly, due to the values of the Frank–Condon overlaps between the vibrational states of the X and A electronic states of Na2, the number of vertical nodes in the stimulated emission spectrum at different dressing-light wavelengths can in some cases reveal which $v$ value of the $|{\rm A}$ $v$ $J\rangle$-type states contributes the most to the initial light-dressed state. For example, using 662 nm dressing light leads to emission lines whose transition amplitudes primarily originate from $|{\rm X}$ $v$ $J\pm 1\rangle\leftarrow|{\rm A}$ $1$ $J\rangle$-type transitions, while using 657 nm dressing light leads to emission lines whose transition amplitudes primarily originate from $|{\rm X}$ $v$ $J\pm 1\rangle\leftarrow|{\rm A}$ $2$ $J\rangle$-type transitions.

III.4 Light-dressed spectra at finite temperatures

Up to this point the light-dressed spectra shown correspond to $T=0$ K; that is, it was assumed that the initial light-dressed state correlates to the field-free rovibronic ground state of Na2. The physical picture behind this assumption is as follows: initially the field-free molecules are all in their ground state and these are changed into dressed states with the adiabatic turn-on of the dressing field. In a realistic experiment at a finite temperature, however, the molecules are not all necessarily in their ground state. Thus, thermal averaging of the computed spectrum needs to be carried out. Since it is assumed that thermal averaging occurs only prior but not after the light-dressing process, the thermal averaging can be done by weighting transitions with the Boltzmann weights of the field-free states correlating to the respective initial light-dressed states. That is, transitions from each $|\Phi_{i}\rangle$ light-dressed state are considered in the computed spectrum, but with all transitions from a given $|\Phi_{i}\rangle$ light-dressed state weighted by

[TABLE]

where $Q(T)=\sum_{i}e^{-E_{i}^{\rm FF}/kT}$ is the rovibronic partition function of the field-free molecule and $E_{i}^{\rm FF}$ is the energy of the field-free rovibronic state to which $|\Phi_{i}\rangle$ correlates in the limit of the dressing light intensity going to zero. An additional complication at finite temperatures is that one needs to take into account that the field-free rotational states of a closed-shell diatomic molecule are characterized not only by $J$, but also by the $m$ quantum number, which stands for the projection of the rotational angular momentum onto the chosen space-fixed quantization axis. Since a linearly polarized dressing field (and the weak probe pulse with identical polarization) can not mix states with different $m$ quantum numbers, one can simply work in the $m=0$ manifold when $T=0$ K. For finite-temperature calculations, however, one needs to determine the light-dressed states and corresponding transitions for the different $m$ manifolds, and include them into the spectrum with the appropriate weights shown in Eq. (4).

Figure 6 shows the light-dressed spectra of Na2 at different temperatures,, when Na2 is dressed by a 657 nm wavelength light field of 0.05 GW cm*-2* intensity. The upper panel of Fig. 6 demonstrates that although absorption peaks split and the peak height of individual lines decreases significantly with increasing temperature (due to low-lying rotational states being populated at finite temperature), the envelope of the spectrum changes to a much smaller extent. As for stimulated emission, increasing the temperature seems to have a more pronounced effect on the spectrum envelope than in the case of absorption, see the lower panel of Fig. 6.

Figure 7 shows the same dressing-light intensity dependence of the light-dressed spectra as in Fig. 2, but at $T=0.5$ K. The same conclusions apply as for Fig. 6; the number of individual lines and the peak amplitudes are much more affected by temperature than the spectrum envelopes.

III.5 Effects of the dressing-field turn-on time on the light-dressed states

Up to this point it was assumed that the dressing field is turned on adiabatically. This resulted in the fact that starting from an initial field-free state the generated light-dressed wave function is composed of a single light-dressed state, correlating to the initial field-free state. If the dressing field is not turned on adiabatically, then the generated light-dressed wave function becomes a superposition of light-dressed states, with the coefficients depending on the turn-on time. Interested readers can find further information and in-depth investigations on the temporal evolution of light-dressed states in the Floquet formalism, for example, in Refs. Guérin and Jauslin (2003); Lefebvre et al. (2013); Leclerc et al. (2016).

Figure 8 demonstrates the population of the different field-free eigenstates in the wave function for dressing-fields of different turn-on time ($\mathit{\Gamma}$) and intensity. The functional form of the dressing light was assumed to be $\bm{E}_{1}(t)=\bm{0}$ for $t<0$, $\bm{E}_{1}(t)=\bm{E}_{\rm max}{\rm sin}(\hbar\omega_{1}t){\rm sin}^{2}(\pi t/\mathit{\Gamma})$ for $0<t<\mathit{\Gamma}/2$, and $\bm{E}_{1}(t)=\bm{E}_{\rm max}$ for $\mathit{\Gamma}/2<t$. The populations shown in Fig. 8 were computed for $t=\mathit{\Gamma}/2$ by solving the TDSE directly.

Panels (b-d) of Fig. 8 demonstrate that when the turn-on time ($\mathit{\Gamma}$) is shorter than the characteristic timescale of molecular rotations (based on the $|{\rm X}$ [math] $0\rangle\leftrightarrow|{\rm X}$ [math] $1\rangle$ transition of Na2 this is around 100 ps), then the degree of rotational excitation is reduced and in fact in the excited electronic state it is limited to that required by the optical selection rules ($\Delta J=\pm 1$). Panels (c) and (d) of Fig. 8 also demonstrate that as the turn-on time of the dressing light becomes shorter in the time domain, it expands in the frequency domain, giving rise to significant populations in $|{\rm X}~{}v~{}0\rangle$- and $|{\rm A}~{}v~{}1\rangle$-type field-free eigenstates in a wider energy range.

After the dressing field reaches its peak intensity at $t=\mathit{\Gamma}/2$ (and is kept constant thereafter) the wave function can be expanded as a superposition of the light-dressed states, and the light-dressed spectrum can be calculated using Eq. (A33). Figure 9 depicts the population of the different light-dressed states in the wave function for dressing-fields of different turn-on time ($\mathit{\Gamma}$) and intensity. Fig. 9 demonstrates that by decreasing the turn-on time of the dressing field the initial wave function becomes a more and more pronounced mixture of light-dressed states. Nonetheless, if the probe pulse is long enough to average out interferences in the transition probability, then the light-dressed spectrum can be generated as a simple weighted sum of the spectra of individual light-dressed states.

To summarize, by changing the turn-on time of the dressing field, the initial wave function, and thus the transition peaks observed in the light-dressed spectrum can be influenced or controlled, although this might complicate the interpretation of the spectrum significantly.

IV Summary and conclusions

In this work we presented a theoretical method to compute the light-dressed spectra of molecules dressed by medium-intensity classical light fields. Such light-dressed spectroscopy is not only of fundamental interest, but can also be a useful practical tool for providing valuable spectroscopic data and related insight for both light-dressed and field-free molecular systems. The approach is based on Floquet theory for deriving the light-dressed states, which are expanded in the basis of field-free molecular eigenstates. Once the light-dressed states are determined, transition amplitudes between them, as recorded by a weak probe pulse, are computed with perturbation theory.

Numerical applications are demonstrated for the homonuclear diatomic molecule Na2, for which the general formulae can be simplified and the physical processes leading to its rovibronic light-dressed spectra can be understood.

It was found that with respect to the field-free spectrum, the light dressing process leads to the splitting of existing peaks as well as the appearance of new peaks, forbidden in the field-free case. Processes similar in spirit to the well-known Autler–Townes and intensity borrowing effects of molecular spectroscopy could be identified.

The dependence of the light-dressed rovibronic spectrum of Na2 on the dressing light intensity and the dressing light wavelength was also investigated. It was found that by manupulating the intensity and the wavelength of the dressing light one can influence and to some extent control the peaks appearing in the light-dressed spectra.

Implication to determine the frequencies of forbidden field-free transitions by extrapolating light-dressed transition frequencies to the limit of zero dressing light intensity was also made and the extrapolation scheme seemed to produce excellent results.

Finite temperature calculations, assuming different initial temperatures of the molecular ensemble prior to the dressing process revealed that the individual field-dressed spectral peaks are much more sensitive to the initial temperature than their envelope, similar to the field-free case.

Finally, it was shown that by changing the turn-on time of the dressing field from the adiabatic limit to shorter times, the initial wave function of the light-dressed system can be modified significantly, which is likely to result in changes in the light-dressed spectrum as well.

V Acknowledgement

This research was supported by the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00005. The authors are grateful to NKFIH for support (Grant No. PD124623 and K119658).

VI Appendix

VI.1 Determination of light-dressed states

VI.1.1 General considerations

In this section we summarize some aspects of the Floquet approach Chu and Telnov (2004); Shirley (1965); Chu (1985) used to compute the light-dressed states generated by a $\hat{W}_{1}(t)$ interaction between the molecule and the dressing field. For a Hamiltonian periodic in time, such as

[TABLE]

[TABLE]

where $T=2\pi/\omega_{1}$, the time-dependent Schrödinger equation (TDSE)

[TABLE]

has the general solution of the form

[TABLE]

where $\varepsilon_{k}$ are the so-called quasienergies and the $|\Phi_{k}(t)\rangle$ Floquet states (also termed light-dressed states in our work) satisfy

[TABLE]

and

[TABLE]

As can be verified using Eq. (A6), if $\varepsilon_{k}$ is a quasienergy, then $\varepsilon_{k}+\hbar m\omega_{1}$ is also a quasienergy with a corresponding Floquet state $e^{im\omega_{1}t}|\Phi_{k}(t)\rangle$. However, it is clear from Eq. (A4) that such a shifted quasienergy does not represent a new physical state, because $\varepsilon_{k}+\hbar m\omega_{1}$ with $e^{im\omega_{1}t}|\Phi_{k}(t)\rangle$ give the same contribution to the wave function as $\varepsilon_{k}$ with $|\Phi_{k}(t)\rangle$.

Because $|\Phi_{k}(t)\rangle$ are periodic in time, they can be expanded as a Fourier series

[TABLE]

Combining Eqs. (A6) and (A7) and assuming $\hat{W}_{1}(t)=-\mathbf{E}_{1}\hat{\bm{\mu}}{\rm cos}(\omega_{1}t)=-\frac{1}{2}\mathbf{E}_{1}\hat{\bm{\mu}}(e^{i\omega_{1}t}+e^{-i\omega_{1}t})$ gives

[TABLE]

Multiplying Eq. (A8) with $\frac{1}{T}e^{-im\omega_{1}t}$ and integrating on the time period $T$ leads to

[TABLE]

The Fourier components $|\varphi_{km}\rangle$ can further be expressed as a linear combination of rovibronic molecular states

[TABLE]

where $\alpha,v$ and $J$ represent electronic, vibrational and rotational quantum numbers, respectively. Using such an expansion, Eq. (A9) can be turned into the matrix eigenvalue problem

[TABLE]

where

[TABLE]

The pictorial representation of $\mathbf{H}_{\rm F}$ reads

[TABLE]

where different matrix elements represent different values of the $\alpha$ electronic and $n$ Fourier indices of Eq. (A12) (for the sake of simplicity, we assumed only two electronic states, labeled X and A), and each matrix element in Eq. (A13) is itself a matrix representation of different operators in the space of rovibrational states, i.e.,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and $\mathbf{I}$ is the identity matrix. $\mathbf{H}_{\rm A}$ and $\mathbf{H}_{\rm X}$ can be thought of as the rovibrational Hamiltonians in the adiabatic electronic states A and X, respectively, while $\bm{\lambda}$ accounts for intrinsic nonadiabatic couplings between the X and A electronic states. $\mathbf{g}_{\alpha\beta}$ naturally represents the coupling induced by the dressing field.

VI.1.2 Simplifying assumptions

In practical applications Eq. (A13) can often be simplified. (1) If intrinsic nonadiabatic couplings can be neglected in the system under investigation, then $\bm{\lambda}=\bm{0}$. (2) If the molecule has no permanent dipole, then $\mathbf{g}_{\rm XX}=\mathbf{g}_{\rm AA}=\bm{0}$. (3) Finally, if $\hbar\omega_{1}$ is resonant with the electronic excitation between the states X and A, then nonresonant coupling terms can be neglected up to moderate field strengths Halász et al. (2012). This leaves only those $\mathbf{g}_{\alpha\beta}$ matrices nonzero which connect $\mathbf{H}_{\rm X}+n\hbar\omega_{1}\mathbf{I}$ and $\mathbf{H}_{\rm A}+(n-1)\hbar\omega_{1}\mathbf{I}$ type elements. With the above three simplifications $\mathbf{H}_{\rm F}$ becomes block diagonal with each two-by-two block being identical up to a constant shift. The block labeled with the Fourier index n reads

[TABLE]

Therefore, instead of solving the general case of Eq. (A9), it becomes sufficient to solve the eigenvalue problem for $\mathbf{H}_{\rm F}^{2\times 2}(n)$ in order to obtain the light-dressed states and corresponding quasienergies.

VI.2 Temporal evolution of a light-dressed system

In the representation utilized in Eq. (A12) the $\mathbf{H}_{\rm F}$ matrix of the Floquet Hamiltonian and the $C^{(k)}_{n,\alpha vJ}$ expansion coefficients of the Floquet states are time independent. Based on this representation and Eqs. (A4) and (A6), the temporal evolution of a light-dressed system can be expressed as Shirley (1965)

[TABLE]

which is formally equivalent to the temporal evolution of a system with a time-independent Hamiltonian.

In a physical scenario when the dressing field amplitude is not constant but slowly changes in time, those matrix elements of $\mathbf{H}_{\rm F}$ which represent light-matter couplings also slowly change in time. For a dressing-field which is turned on much slower than the characteristic timescales of the field-free system (adiabatically), Eq. (A19) suggests that the well-known adiabatic theorem could be used to predict the temporal changes in $\bm{\Psi}(t)$. Therefore, in the limit of the dressing light intensity going to zero ($\mathbf{g}_{\rm XA}\rightarrow 0$) the field-free eigenstates are eigenstates of $\mathbf{H}_{\rm F}$ as well, therefore, an 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 previous two sentences are not true if $\omega_{1}$ is in exact resonance with an allowed transition, because this results in the field-free eigenstates not being eigenstates of $\mathbf{H}_{\rm F}$ (but being a linear combination of $\mathbf{H}_{\rm F}$ eigenstates) even for infinitezimal light-matter coupling strengths. Further information on the temporal evolution of light-dressed states in the Floquet formalism can be found for example in Ref. Guérin and Jauslin (2003), while Refs. Lefebvre et al. (2013); Leclerc et al. (2016); Fábri et al. (2019); Zhang et al. (2017) provide examples for the utilization of the adiabatic Floquet dynamics.

VI.3 Computing transitions between light-dressed states

VI.3.1 General considerations

We assume a molecule interacting with two periodic electric fields. The full Hamiltonian reads

[TABLE]

where $\hat{H}_{\rm mol}$ is the field-free molecular Hamiltonian, and $\hat{W}_{1}(t)$ and $\hat{W}_{2}(t)$ account for the interaction between the molecule and the two fields, i.e., in the dipole approximation

[TABLE]

$\hat{W}_{1}(t)$ is considered to be generating the light-dressed states, while $\hat{W}_{2}(t)$ originates from a weak probe pulse used to record the spectrum of the light-dressed molecule. The formation of light-dressed states by $\hat{W}_{1}(t)$ is accounted for within the Floquet approach, as described in Section VI.1.

For computing the $\hat{W}_{2}(t)$-induced transition amplitudes between the (superposition of) light-dressed states, first-order time-dependent perturbation theory (TDPT1) is used. To derive our working equations, we start with the TDSE containing the interaction with both the dressing and the probe fields,

[TABLE]

Eq. (A22) is transformed to the interaction picture using the transformation

[TABLE]

which leads to

[TABLE]

where $\hat{W}_{2{\rm I}}(t)=e^{\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{\rm d}(t^{\prime})dt^{\prime}}\hat{W}_{2}(t)e^{-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{\rm d}(t^{\prime})dt^{\prime}}$. Following the usual TDPT1 procedure of integrating Eq. (A24) from $t_{0}$ to $t$ and applying a successive approximation to express $|\Psi_{\rm I}(t)\rangle$ gives

[TABLE]

with

[TABLE]

Considering the first two terms of the propagator in Eq. (A26) leads to

[TABLE]

The transition amplitude to a final state $|\Psi^{\rm(F)}_{\rm I}\rangle$ at time $t$ is thus

[TABLE]

By expanding $|\Psi^{\rm(F)}_{\rm I}\rangle$ and $|\Psi_{\rm I}(t_{0})\rangle$ as a superposition of the $|\Phi_{k}(t_{0})\rangle$ Floquet states, i.e.,

[TABLE]

and

[TABLE]

Eq. (A28) gives

[TABLE]

Because $e^{-\frac{i}{\hbar}\varepsilon_{k}t}|\Phi_{k}(t)\rangle\Large$ is a solution of the TDSE of Eq. (A3), the effect of the $\hat{U}_{\rm d}(t^{\prime},t_{0})=e^{-\frac{i}{\hbar}\int_{t_{0}}^{t^{\prime}}\hat{H}_{\rm d}(t^{\prime\prime})dt^{\prime\prime}}$ operator, describing $\hat{H}_{\rm d}(t)$ governed time-evolution from $t_{0}$ to $t^{\prime}$, can be evaluated as $\hat{U}_{\rm d}(t^{\prime},t_{0})\Large(e^{-\frac{i}{\hbar}\varepsilon_{k}t_{0}}|\Phi_{k}(t_{0})\rangle\Large)=e^{-\frac{i}{\hbar}\varepsilon_{k}t^{\prime}}|\Phi_{k}(t^{\prime})\rangle$. By exploiting this fact, one arrives at

[TABLE]

Finally, using the explicit form of $\hat{W}_{2}(t)$ given in Eq. (A21) and expressing the periodic $|\Phi_{k}(t)\rangle$ functions under the integral with their Fourier series (see Eq. (A7)), leads to

[TABLE]

VI.3.2 Molecules with no permanent dipole

For molecules with no permanent dipole and neglectable intrinsic nonadiabatic couplings the light-dressed states determined within a Floquet approach, in which nonresonant coupling terms with the dressing field are neglected, see Eq. (A18), Eq. (A7) can be written as

[TABLE]

where $|\alpha_{kn}\rangle$ and $|\beta_{k(n-1)}\rangle$ represent the two manifolds of rovibronic states with Fourier indices $n$ and $n-1$, respectively. Because neglecting the nonresonant coupling terms in the Floquet approach means that the Floquet Hamiltonian becomes block diagonal, see Eq. (A18), and that $|\alpha_{kn}\rangle$ and $|\beta_{k(n-1)}\rangle$ become the same for all $n$, Eq. (A34) simplifies to

[TABLE]

where $|\Phi_{k}(t)\rangle$ is a Floquet state obtained from the nth two-by-two block of the Floquet Hamiltonian, and quasienergy corresponding to $|\Phi_{k}(t)\rangle$ may be written as $\varepsilon_{k}+\hbar n\omega_{1}$. As explained under Eq. (A6), if the quasienergy is shifted by $-\hbar n\omega_{1}$ and the Floquet state is multiplied with $e^{-in\omega_{1}t}$ one arrives to an equivalent physical state. Therefore, Eq. (A35) can be rewritten as

[TABLE]

with the corresponding quasienergy of $\varepsilon_{k}$. Using Eq. (A36) instead of Eq. (A7) leads to a simplified version of Eq. (A33), i.e.,

[TABLE]

where we exploited that $\langle\alpha_{l}|\mathbf{E}_{2}\hat{\bm{\mu}}|\alpha_{k}\rangle=\langle\beta_{l}|\mathbf{E}_{2}\hat{\bm{\mu}}|\beta_{k}\rangle=0$ for a molecule with no permanent dipole. Continuing with the standard TDPT1 procedure, the second and third terms in Eq. (A37) lead to the two terms of Eq. (3).