Conical Intersections Induced by Quantum Light: Field-Dressed Spectra from the Weak to the Ultrastrong Coupling Regimes
Tamás Szidarovszky, Attila. G. Császár, Gábor J. Halász, Lorenz S. Cederbaum, Ágnes Vibók
CConical Intersections Induced by Quantum Light:Field-Dressed Spectra from the Weak to the UltrastrongCoupling Regimes
Tam´as Szidarovszky ∗ and Attila G. Cs´asz´ar Laboratory of Molecular Structure and Dynamics, Institute of Chemistry,E¨otv¨os Lor´and University and MTA-ELTE Complex Chemical Systems Research Group,H-1117 Budapest, P´azm´any P´eter s´et´any 1/A, Hungary
G´abor J. Hal´asz
Department of Information Technology, University of Debrecen,P.O. Box 400, H-4002 Debrecen, Hungary
Lorenz S. Cederbaum
Theoretische Chemie, Physikalisch-Chemisches Institut,Universit¨at Heidelberg, D-69120, Heidelberg, Germany ´Agnes Vib´ok † Department of Theoretical Physics, University of Debrecen,P.O. Box 400, H-4002 Debrecen, Hungary and ELI-ALPS,ELI-HU Non-Profit Ltd., Dugonics t´er 13, H-6720 Szeged, Hungary (Dated: August 15, 2018) a r X i v : . [ phy s i c s . a t m - c l u s ] A ug bstract A fundamental theoretical framework is formulated for the investigation of rovibronic spectraresulting from the coupling of molecules to one mode of the radiation field in an optical cavity. Theapproach involves the computation of (1) cavity-field-dressed rovibronic states, which are hybridlight-matter eigenstates of the “molecule + cavity radiation field” system, and (2) the transitionamplitudes between these field-dressed states with respect to a weak probe pulse. The predictionsof the theory are shown for the homonuclear Na molecule. The field-dressed rovibronic spectrumdemonstrates undoubtedly that the Born–Oppenheimer approximation breaks down in the presenceof the cavity radiation field. A clear fingerprint of the strong nonadiabaticity is found, which canonly emerge in the close vicinity of conical intersections. In this work, the conical intersection isinduced by the quantized radiation field, and it is thus called a ”light-induced conical intersection”(LICI). Dependence of the cavity-field-dressed spectrum on the cavity-mode wavelength as well ason the light-matter coupling strength is investigated. Essential changes are identified in the spectrafrom the weak to the ultrastrong coupling regimes. ∗ [email protected] † [email protected] . INTRODUCTION Understanding the interaction of matter with strong and ultrastrong laser pulses is afundamental and rapidly developing field of research. With the remarkable advances in lasertechnology in the past few decades, experimental investigation of the interaction becamefeasible [1–3] and provided insight into the strongly nonlinear domain of optical processes,associated with various unique phenomena, such as high harmonic generation [4, 5], abovethreshold dissociation and ionization [6], bond softening and hardening effects [7–13], andlight-induced conical intersections (LICIs) [14–22][23].LICIs may form even in diatomic molecules, when the laser light not only rotates themolecule but can also couple the vibrational with the emerging rotational degree of free-dom. Theoretical and experimental studies have demonstrated that the light-induced nona-diabatic effects have significant impact on different observable dynamical properties, such asmolecular alignment, dissociation probability, or angular distribution of photofragments [14–20]. Recently, signatures of light-induced nonadiabatic phenomena have been successfullyidentified in the classical field-dressed static rovibronic spectrum of diatomics [24].As an alternative to interactions of atoms or molecules with intense laser fields, stronglight-matter coupling can also be achieved, both for atoms and molecules, by their con-finement in microscale or nanoscale optical cavities [25–27]. Such systems are usually de-scribed in terms of field-dressed or polariton states, which are the eigenstates of the full“atom/molecule + radiation field” system [28–31]. With decreasing cavity size the quan-tized nature of the radiation field eventually becomes important and strong photon-mattercoupling as well as a significant modification of the atomic and molecular properties mayoccur even if the photon number is (close to) zero [28–30, 32–35]. For example, quantummodeling efforts have shown that strong resonant coupling of a cavity radiation field withan electronic transition can decouple the electronic and nuclear degrees of freedom in molec-ular ensembles [32], strong coupling of molecules to a confined light mode could suppressphotoisomerization [33], and that collective motion of molecules could be triggered by asingle photon of a cavity [34–36]. In addition to exploring strong light-matter coupling,processes in nanoscale cavities could also be used to study how atoms/molecules interactwith non-classical states of the quantized light field [37, 38].Previous theoretical models mostly treated atoms or molecules in reduced dimensions3r via some simplified model [32, 33, 37–44], and most often utilized the concept of po-lariton states formed by coupling the electronic and photonic degrees of freedom. Nuclearmotion is then thought to proceed on the polariton surfaces. Alternatively, decoupling theelectronic motion from the nuclear and photonic degrees of freedom, known as the cavityBorn–Oppenheimer approximation, has sometimes been pursued [29, 45].On the experimental side, the coupling of molecules to cavity radiation field was shownto modify chemical landscapes and reaction dynamics [46], as well as the absorption spectra[47, 48] of molecules. Furthermore, intermolecular non-radiative energy transfer was foundto be enhanced by the formation of polariton states [49], and the hybridization of molec-ular vibrational states through strong light-matter coupling in a microcavity [50] was alsoobserved.For atoms or molecules interacting with a cavity mode (near) resonant to an electronictransition, one usually distinguishes three regimes of field-matter coupling strengths [29, 39],as depicted in Fig. 1: weak, strong, and ultrastrong. In the weak-coupling regime, see theleft panel of Fig. 1, the diabatic picture of photon-dressed potential energy curves (PECs)holds and the cavity mode only couples the excited electronic state with the ground electronicstate dressed by a photon. In the strong-coupling regime, shown in the middle panel of Fig.1, polariton states are formed and the adiabatic picture becomes appropriate for describingthe excited state manifold, while the ground state remains essentially unchanged. Finally, inthe ultrastrong-coupling regime, see the right panel of Fig. 1, nonresonant couplings becomestrong enough to significantly modify the electronic ground state, as well.In reality, the dashed polariton surfaces of Fig. 1 are strictly valid only if the molecularaxis is parallel to the preferred polarization direction of the cavity field. In contrast, whenthe molecular axis is perpendicular to the polarization direction of the cavity, the light-matter coupling vanishes and the diabatic picture (continuous potentials in Fig. 1) is therelevant one. In fact, the orientation of a rotating molecule can change continuously betweenthese two extreme positions, and the diabatic and cavity-induced polariton surfaces arecontinuously transformed into each other. Therefore, due to the rotation of the molecule,the upper and lower adiabatic surfaces are not completely separated but a conical intersection(CI) emerges between them, see Fig. 2, at which point the nonadiabatic couplings becomeinfinitely strong.At the vicinity of this CI, which is created by the quantum light and never present4 eak strong interatomicdistance E n e r gy ultrastrong + V V V ¯ hω c W W W FIG. 1. The three regimes of coupling strength , and the related field-dressed PECs, of a moleculeinteracting with a resonant cavity mode. The diabatic surfaces V and V are indicated withcontinuous line, while the polariton surfaces W , W and W are indicated with dashed lines. (cid:125) ω c is the cavity photon energy. in field-free diatomics, the Born − Oppenheimer picture [51, 52] breaks down. The nucleardynamics proceed on the coupled polariton surfaces and motions along the vibrational androtational coordinates become intricately coupled. It must be stressed that even in the caseof diatomics, considering rotations completely changes the paradigm and physical picturewith respect to the description when only the vibrational, electronic, and photonic degreesof freedom are taken into account. Moreover, in contrast to field-free polyatomic molecules,where conical intersections are dictated by nature, they are either present or not, light-induced conical intersections in the cavity are always present between the polariton surfaces.Even for a diatomic molecule, the appropriate description needs to account for rotations,which are coupled nonadiabatically to the vibrational, electronic, and photonic modes of thesystem.The purpose of the present study is to investigate the field-dressed rovibronic spectrumof diatomics in the framework of cavity quantum electrodynamics (QED). We complementprevious theoretical approaches by accounting for all molecular degrees of freedom, i.e. , wetreat rotational, vibrational, electronic, and photonic degrees of freedom on an equal footing.Furthermore, we incorporate for the first time the concept of LICIs with the quantizedradiation field.Our goals with are two-fold. First, we investigate the field-dressed rovibronic spectrum5
IG. 2. Two dimensional polariton surfaces ( W and W ) of the Na dimer in the cavity. The onephoton coupling of the cavity ( ε c ) corresponds to a classical field intensity of 64 GWcm − . Thecavity mode wavelength is λ c = 653 nm. The red arrow denotes the position of the light-inducedconical intersection. of our test system, the homonuclear Na molecule, in order to understand the effects ofthe cavity on the spectrum and to identify the direct signatures of a LICI created by thequantized cavity radiation field. Second, the coupling strength and cavity-mode wavelengthdependence of the spectrum from weak to ultrastrong coupling regimes is investigated. Weidentify the formation of polariton states in the strong coupling regime as well as the impactof nonresonant couplings on the spectrum in the ultrastrong coupling regime. Surprisingly,the effects of nonresonant couplings can be seen at coupling strengths much smaller thanthose necessary for significantly distorting the ground-state PEC, as long as molecular ro-tations are properly accounted for. An important conclusion of this study is that for thesimulation of freely rotating molecules confined in a cavity the appropriate treatment ofrotations as well as vibrations is mandatory. II. THEORETICAL APPROACH
For simulating the weak-field absorption spectrum of molecules confined in small opticalcavities, we first determine the field-dressed states, i.e. , the eigenstates of the full “molecule6 radiation field” system, and then we compute the dipole transition amplitudes between thefield-dressed states with respect to a probe pulse. We assume the probe pulse to be weak;therefore, transitions induced by it should be dominated by one-photon processes. Thisimplies that the standard approach [53] of using first-order time-dependent perturbationtheory to compute the transition amplitudes should be adequate.
A. The field-dressed states
Within the framework of QED and the electric dipole representation, the Hamiltonian ofa molecule interacting with a single cavity mode can be written as [54]ˆ H tot = ˆ H mol + ˆ H rad + ˆ H int , (1)where ˆ H mol is the field-free molecular Hamiltonian, ˆ H rad is the radiation-field Hamiltonian,and ˆ H int is the interaction term between the molecular dipole moment and the electric field.For a single radiation mode and an appropriate choice of the origin [54],ˆ H rad = (cid:125) ω c ˆ a † ˆ a (2)and ˆ H int = − (cid:114) (cid:125) ω c (cid:15) V ˆdˆe (cid:0) ˆ a † + ˆ a (cid:1) = − ε c √ ˆdˆe (cid:0) ˆ a † + ˆ a (cid:1) , (3)where ε c = (cid:112) (cid:125) ω c / ( (cid:15) V ) is the cavity one-photon field, ˆ a † and ˆ a are photon creation andannihilation operators, respectively, ω c is the frequency of the cavity mode, (cid:125) is Planck’sconstant divided by 2 π , (cid:15) is the electric constant, V is the volume of the electromagneticmode, ˆd is the molecular dipole moment, and ˆe is the polarization vector of the cavity mode.In the case of diatomic molecules, representing the Hamiltonian of Eq. (1) in a directproduct basis composed of two field-free molecular electronic states and the Fock states ofthe radiation field gives ˆ H = ˆ H m ˆ A (2) 0 . . . ˆ A † (2) ˆ H m + (cid:125) ω c ˆ A (3) . . . A † (3) ˆ H m + 2 (cid:125) ω c . . . ... ... ... . . . , (4)7here ˆ H m = ˆ T
00 ˆ T + V ( R ) 00 V ( R ) , (5)and ˆ A ( N ) = g ( R, θ ) √ N g ( R, θ ) √ Ng ( R, θ ) √ N g ( R, θ ) √ N , (6)with g ij ( R, θ ) = − (cid:114) (cid:125) ω (cid:15) V d ij ( R )cos( θ ) , (7)where R is the internuclear distance, V i ( R ) is the i th PEC, ˆ T is the nuclear kinetic energyoperator, d ij ( R ) is the transition dipole moment matrix element between the i th and j thelectronic states, and θ is the angle between the electric field polarization vector and thetransition dipole vector, assumed to be parallel to the molecular axis.In Eq. (4), the first, second, third, etc. columns(rows) correspond to zero, one, two,etc. photon number in the bra(ket) vectors of the cavity mode, respectively. ExpandingEq. (4) using Eqs. (5) and (6), and assuming a homonuclear diatomic molecule having nopermanent dipole, givesˆ H = ˆ T + V ( R ) 0 0 g ( R, θ ) √ · · · T + V ( R ) g ( R, θ ) √ · · · g ( R, θ ) √ T + V ( R ) + (cid:125) ω c · · · g ( R, θ ) √ T + V ( R ) + (cid:125) ω c g ( R, θ ) √ · · · g ( R, θ ) √ T + V ( R ) + 2 (cid:125) ω c · · · ... ... ... ... ... . . . , (8)which is the working Hamiltonian used in this study.The | Ψ FD i (cid:105) field-dressed states, i.e. , the eigenstates of the Hamiltonian of Eq. (1),ˆ H tot | Ψ FD i (cid:105) = E FD i | Ψ FD i (cid:105) (9)are obtained by diagonalizing the Hamiltonian of Eq. (8) in the basis of field-free rovibra-tional states. Then, the field-dressed states can be expressed as the linear combination ofproducts of field-free molecular rovibronic states and Fock states of the dressing field, i.e. , | Ψ FD i (cid:105) = (cid:88) J,v,α,N C i,αvJN | αvJ (cid:105)| N (cid:105) = (cid:88) J,v,N C i, vJN | vJ (cid:105)| N (cid:105) + (cid:88) J,v,N C i, vJN | vJ (cid:105)| N (cid:105) , (10)8here | jvJ (cid:105) is a field-free rovibronic state, in which the molecule is in the j th electronic, v thvibrational, and J th rotational state, | N (cid:105) is a Fock state of the dressing field with photonnumber N , and C i,jvJN are expansion coefficients obtained by diagonalizing the Hamiltonianof Eq. (8) in the basis of the field-free rovibrational states. B. Transitions between field-dressed states
Let us now compute the absorption spectrum with respect to a weak probe pulse, whosephoton number is represented by the letter M . Using first-order time-dependent perturba-tion theory, the transition amplitude between two field-dressed states, induced by the weakprobe pulse, can be expressed as [54, 55] (cid:104) Ψ FD i |(cid:104) M | ˆd ˆE | M (cid:48) (cid:105)| Ψ FD j (cid:105) = (cid:104) Ψ FD i | ˆ d cos( θ ) | Ψ FD j (cid:105)(cid:104) M | ˆ E | M (cid:48) (cid:105) . (11)In Eq. (11), the electric field operator ˆ E stands for the weak probe pulse, and we assumethat the probe pulse has a polarization axis identical to that of the cavity mode. Since ˆ E isproportional to the sum of a creation and an annihilation operator acting on | M (cid:48) (cid:105) , Eq. (11)leads to the well-known result that the transition amplitude is non-zero only if M = M (cid:48) ± i.e. , Eq. (11) accounts for single-photon absorption or stimulated emission.The matrix element of the operator ˆ d cos( θ ) between two field-dressed states of Eq. (10)gives (cid:104) Ψ FD i | ˆ d cos( θ ) | Ψ FD j (cid:105) == (cid:32) (cid:88) J,v,α,N C ∗ i,αvJN (cid:104) αvJ |(cid:104) N | (cid:33) ˆ d cos( θ ) (cid:32) (cid:88) J (cid:48) ,v (cid:48) ,α (cid:48) ,N (cid:48) C j,α (cid:48) v (cid:48) J (cid:48) N (cid:48) | α (cid:48) v (cid:48) J (cid:48) (cid:105)| N (cid:48) (cid:105) (cid:33) == (cid:88) J,v,α,N,J (cid:48) ,v (cid:48) ,α (cid:48) ,N (cid:48) C ∗ i,αvJN C j,α (cid:48) v (cid:48) J (cid:48) N (cid:48) (cid:104) αvJ | ˆ d cos( θ ) | α (cid:48) v (cid:48) J (cid:48) (cid:105) δ N,N (cid:48) == (cid:88) J,v,J (cid:48) ,v (cid:48) ,N C ∗ i, vJN C j, v (cid:48) J (cid:48) N (cid:104) vJ | ˆ d cos( θ ) | v (cid:48) J (cid:48) (cid:105) + (cid:88) J,v,J (cid:48) ,v (cid:48) ,N C ∗ i, vJN C j, v (cid:48) J (cid:48) N (cid:104) vJ | ˆ d cos( θ ) | v (cid:48) J (cid:48) (cid:105) . (12)In the last line of Eq. (12), the first(second) term represents transitions, in which thefirst(second) electronic state contributes from the i th field-dressed state and the second(first)electronic state contributes from the j th field-dressed state. Assuming that the i th stateis the initial state, the first term in the last line of Eq. (12) leads to the usual field-freeabsorption spectrum in the limit of the light-matter coupling going to zero. In all spectra9hown below, we plot the absolute square of the transition amplitudes, as computed by Eq.(12), or their convolution with a Gaussian function. III. COMPUTATIONAL DETAILS
We test the theoretical framework developed in Eqs. (1-12) on the Na molecule, forwhich the V ( R ) and V ( R ) PECs correspond to the X Σ +g and the A Σ +u electronic states,respectively. The PECs and the transition dipole are taken from Refs. 56 and 57, respec-tively. The field-free rovibrational eigenstates of Na on the V ( R ) and V ( R ) PECs arecomputed using 200 spherical-DVR basis function [58] with the related grid points placed inthe internuclear coordinate range (0 ,
10) bohr. Unless indicated otherwise, the set of field-free rovibrational eigenstates used to represent the Hamiltonian of Eq. (8) was composed ofall states with
J <
30 and an energy not exceeding the zero point energy of the respectivePEC by more than 5000 cm − . The maximum photon number in the cavity mode was setto two. IV. RESULTS AND DISCUSSION
The PECs of Na employed are given in Fig. 3, along with some important physicalprocesses. The main results of this study are conveniently depicted in Figs. 4–6. Eachfigure will be discussed separately. A. The field-dressed states
The left panel in Fig. 3 shows PECs of Na dressed with different number of photons inthe cavity radiation field, as well as vibrational probability densities for direct-product statesof the | jvJ (cid:105)| N (cid:105) form. As apparent from Eq. (8) and illustrated by the double-headed arrowsin Fig.3, light-matter interaction can give rise to resonant | v J (cid:105)| N (cid:105) ↔ | v (cid:48) J ± (cid:105)| N − (cid:105) and non-resonant | v J (cid:105)| N (cid:105) ↔ | v (cid:48) J ± (cid:105)| N + 1 (cid:105) type couplings, which lead to theformation of field-dressed states, see Eq. (10). The terms “resonant” and “non-resonant”indicate whether the direct-product states that are coupled are close in energy or not, seeFig. 3. Naturally, resonant couplings are much more efficient in mixing the direct-product10tates than non-resonant couplings. For comparison, the right panel of Fig. 3 shows thelight-dressed PECs of Na in a laser field [24]. Because nonresonant couplings are omittedin the usual Floquet description [59] of laser light-dressed molecules, these couplings are notshown in the right panel of Fig. 3. It is clear from Fig. 3 that the absorption spectrumof field-dressed molecules should be considerably different for the cavity-dressed and laser-dressed cases. The most significant difference is that while in the cavity the ground stateis primarily a field-free eigenstate in vacuum, which is only deformed at relatively largecoupling strengths through nonresonant couplings, the laser light-dressed state correlatingto the field-free ground state contains a mixture of field-free eigenstates due to the strongresonant coupling in this case. / c m - / bohr +++2 E R V V ¯ hω c V V ¯ hω c V ¯ hω c / c m - / bohr +(N-1)+N+(N+1)+N+(N+1)+(N+2) E R V ¯ hω L V ¯ hω L V ¯ hω L V ¯ hω L V ¯ hω L V ¯ hω L FIG. 3. Left: PECs of Na , dressed with different number of photons of the cavity field, obtainedwith a dressing-light wavelength of λ = 653 nm. Vibrational probability densities are drawn forstates of | (cid:105)| m (cid:105) type (dashed black lines on the V ( R )+ m (cid:125) ω c PECs), and for states of | (cid:105)| m (cid:105) type (dotted red lines on the V ( R ) + m (cid:125) ω c PECs). Couplings induced by the cavity radiationfield are indicated by the two double-headed arrows. The continuous green double-headed arrowrepresents | v J (cid:105)| m (cid:105) ↔ | v (cid:48) J ± (cid:105)| m − (cid:105) type resonant couplings, while the dashed purpledouble-headed arrow represents | v J (cid:105)| m (cid:105) ↔ | v (cid:48) J ± (cid:105)| m + 1 (cid:105) type non-resonant couplings.Finally, the vertical brown wavy arrow indicates transitions between the two manifolds of field-dressed states, resulting form the absorption of a photon of the weak probe pulse. Right: same asleft panel, but for Na dressed by laser light. .00.10.20.30.40.50.60.7 a b s o r p ti on / a . u . absorption / cm -1 c -2c -2c -2c -2c -2c -2c -2c -2c -2c -2c -2c -2 I E a b s o r p ti on / a . u . / cm -10.00.20.40.615758 15760 15762 15764 15766absorption / cm -10.00.10.215778 15780 15782 15784 15786 I EE FIG. 4. Field-dressed spectra obtained with different values of the light-matter coupling strengthsfor a cavity mode wavelength of λ = 653 nm. Coupling strength values are indicated by theintensity of a classical light field giving a coupling strength equal to the one-photon coupling ofthe cavity. The envelope lines depict the spectra convolved with a Gaussian function having astandard deviation of σ = 50 cm − . B. Spectra in the weak coupling regime
Figure 4 shows field-dressed spectra obtained with different values of the ε c = (cid:112) (cid:125) ω c / ( (cid:15) V )cavity one-photon field strength in the weak-coupling regime for a cavity-mode wavelength of λ = 653 nm. Although the light-matter coupling strength and the cavity-mode wavelengthare not completely independent in a cavity (see Eq. (7)), we treat them as independentparameters. This can be rationalized partially by Eq. (7), which shows that the couplingstrength could be changed independently by changing the cavity volume, while keeping thecavity length responsible for the considered cavity radiation mode fixed. The spectra in Fig.4 were computed assuming that the initial state is the ground state of the full system. Theleft panel of Fig. 4 reflects features similar to those observed in the spectrum of Na dressedby laser fields [24]. With increasing light-matter coupling, the overall intensity of the spec-trum slightly increases at almost all wavenumbers, with some shoulder features becomingmore pronounced in the spectrum envelope. In terms of spectroscopic nomenclature, such aphenomenon can be understood as an intensity-borrowing effect [60–62], which arises fromthe field-induced couplings between field-free states.On the other hand, the coupling strength dependence of the spectrum envelope is com-pletely absent if the spectra in Fig. 4 are generated from computations in which the ro-12 .00.10.20.30.40.50.60.7 a b s o r p ti on / a . u . absorption / cm -1 without rotation (1D) c -2 I E a b s o r p ti on / a . u . absorption / cm -1 with rotation (2D) c -2 I E / c m - / bohr -1001020 5.7 5.75 5.8 5.85 5.9 5.95 6.0 / bohr c -2 E RR a b s o r p ti on / a . u . absorption / cm -1 without rotation (1D) c -2 I E a b s o r p ti on / a . u . absorption / cm -1 with rotation (2D) c -2 I E / c m - / bohr -1001020 5.7 5.75 5.8 5.85 5.9 5.95 6.0 / bohr c -2 E RR a b s o r p ti on / a . u . absorption / cm -1 without rotation (1D) c
16 GWcm -2 I E a b s o r p ti on / a . u . absorption / cm -1 with rotation (2D) c
16 GWcm -2 I E / c m - / bohr -1001020 5.7 5.75 5.8 5.85 5.9 5.95 6.0 / bohr c
16 GWcm -2 E RR a b s o r p ti on / a . u . absorption / cm -1 without rotation (1D) c
64 GWcm -2 I E a b s o r p ti on / a . u . absorption / cm -1 with rotation (2D) c
64 GWcm -2 I E / c m - / bohr -1001020 5.7 5.75 5.8 5.85 5.9 5.95 6.0 / bohr c
64 GWcm -2 E RR FIG. 5. First two columns: field-dressed spectra obtained with different values of the light-mattercoupling strengths for a cavity mode wavelength of λ = 653 nm. Coupling strength values areindicated by the intensity of a classical light field giving a coupling strength equal to the one-photon coupling of the cavity. The envelope lines depict the spectra convoluted with a Gaussianfunction having a standard deviation of σ = 50 cm − . The labels “1D” and “2D” stand for vibrationonly and rovibrational calculations, defined by using J max = 1 and J max = 30, respectively. Solidand dashed lines correspond to calculations including or excluding the off resonant couplings in theHamiltonian, respectively. Third column: diabatic and adiabatic PECs at different light-mattercoupling strength values. tational motion is restricted by setting J max = 1. Such a rotationally-restricted model13 d r e ss i ng / c m - X absorption / cm -1 c -2 with rotation (2D) E E d r e ss i ng / c m - X absorption / cm -1 c -2 with rotation (2D) E E d r e ss i ng / c m - X absorption / cm -1 c -2 with rotation (2D) E E d r e ss i ng / c m - X absorption / cm -1 c -2 with rotation (2D) E E d r e ss i ng / c m - X absorption / cm -1 c
16 GWcm -2 with rotation (2D) E E d r e ss i ng / c m - X absorption / cm -1 c
16 GWcm -2 with rotation (2D) E E FIG. 6. Cavity mode wavenumber-dependence of the field-dressed spectrum obtained at threedifferent coupling strengths, computed using Eq. (12). Coupling strength values are indicated bythe intensity of a classical light field giving a coupling strength equal to the one-photon couplingof the cavity. The spectrum is convoluted at each fixed cavity mode wavelength with a Gaussianfunction having σ = 30 cm − . inherently lacks any signatures of a LICI, whose formation requires at least two nucleardegrees of freedom. Therefore, in terms of the adiabatic representation, the intensity bor-rowing effect visible in Fig. 4 can be attributed to the nonadiabatic couplings of a LICIcreated by the quantized cavity radiation field.Inspecting the individual transition lines in the spectra reveals that increasing the light-14atter coupling strength can result in the splitting of existing peaks and the appearance ofadditional peaks, as shown in the two panels on the right side of Fig. 4. The upper rightpanel of Fig. 4 shows the progression of three peaks, corresponding to transitions from theinitial state (essentially the | (cid:105)| (cid:105) state) to field-dressed states composed primarily of the | (cid:105)| (cid:105) , | (cid:105)| (cid:105) , and | (cid:105)| (cid:105) states, with | v J (cid:105)| (cid:105) -type states ( J even) contributingas well. With increasing light-matter coupling strength these transitions are red shifted, andthey can be interpreted as originating from the field-free transition | (cid:105) → | (cid:105) , whichis split due to the mixing of | (cid:105) with other states through the light-matter coupling withthe cavity mode. The lower right panel of Fig. 4 shows the progression of three peaks, whichdo not arise from the splitting of an existing field-free peak, but appear as new peaks. Thesetransitions are blue shifted with increasing light-matter coupling strength, and they occurbetween the initial state (essentially the | (cid:105)| (cid:105) state) and field-dressed states composedprimarily of the | (cid:105)| (cid:105) , | (cid:105)| (cid:105) , and | (cid:105)| (cid:105) states. Such transitions are forbiddenin the zero light-matter coupling limit; however, they become visible as the light-mattercoupling with the cavity mode contaminates the | J (cid:105)| (cid:105) states with | v (cid:105)| (cid:105) -type states,to which the initial state has allowed transitions. C. Spectra in the strong and ultrastrong coupling regimes
In Fig. 5 field-dressed spectra obtained with light-matter coupling strengths rangingfrom the weak to the ultrastrong coupling regimes are shown for a cavity-mode wavelengthof λ = 653 nm. The spectra labeled “1D” in Fig. 5 were obtained with a model havingrestricted rotational motion ( J max = 1). The results labeled “2D” fully account for rotationsas well as vibrations; therefore, incorporate the effects of a LICI on the spectrum. Bycomparing the “1D” and “2D” spectrum envelopes in Fig. 5, it is apparent that there is asignificant increase in absorption for the “2D” case. As discussed in the next subsection, thisis primarily due to nonresonant light-matter couplings between | J (cid:105)| (cid:105) and | v (cid:48) J ± (cid:105)| (cid:105) -type states and partially due to the intensity borrowing effect induced by the nonadiabaticcouplings of the LICI.Figure 5 also shows field-dressed PECs in the diabatic and adiabatic representations.As the coupling strength increases, two separate polariton surfaces are formed, and theabsorption spectrum splits into two distinct groups of peaks, corresponding to transitions15nto the two polariton states. At the largest coupling strength, a slight modification of theground-state PEC can also be seen, indicating that the ultrastrong coupling regime has beenreached. D. Impact of nonresonant coupling
Interestingly, nonresonant couplings seem to have an impact on the spectrum at muchsmaller coupling strengths than those required for a significant modification of the ground-state PEC. The spectrum envelopes depicted with dotted and continuous lines in Fig. 5indicate whether spectra were computed by using theˆ H = ˆ T + V ( R ) 0 00 ˆ T + V ( R ) g ( R, θ ) √ g ( R, θ ) √ T + V ( R ) + (cid:125) ω c , (13)upper left three-by-three block of the Hamiltonian in Eq. (8) or a six-by-six block, respec-tively. The deviation between these two types of spectra represents the effects of nonresonantcouplings, because the ˆ T + V ( R ) + (cid:125) ω c term and its couplings with ˆ T + V ( R ) are presentin Eq. 8, but absent in Eq. 13.The plots in Fig. 5 clearly demonstrate that in the vibration-only “1D” case nonresonantcouplings have no visible impact on the spectrum; however, for the “2D” case, in whichrotations are accounted for, nonresonant couplings lead to a visible increase in the absorptionsignal even at the lowest coupling strengths shown. The physical origin of the increase inabsorption is the contamination of the | (cid:105)| (cid:105) ground state with the | (cid:105)| (cid:105) , | (cid:105)| (cid:105) , etc. states, which allows for transitions onto the J = 3 , , ... components of therovibronic states in the excited polariton manifold. These results indicate that the effectsof nonresonant couplings can not be described in a vibration only model, and if one wishesto obtain meaningful simulation results for coupling strengths reaching or exceeding thoseshown in Fig. 5, it is necessary to properly account for molecular rotations. E. Cavity-mode wavelength dependence of the spectrum
Figure 6 shows the cavity-mode wavenumber dependence of the field-dressed spectrumobtained at the ε c = (cid:112) (cid:125) ω c / ( (cid:15) V ) cavity one-photon field strengths of 0.844 · − , 1.688 · − ,16nd 3.376 · − atomic units, corresponding to classical field intensities of 1, 4, and 16GWcm − , respectively. It can be concluded from Fig. 6 that, as expected, the field-dressed spectrum changes with the cavity-mode wavelength. Furthermore, the cavity-modewavelength dependent spectrum shows qualitative features considerably different from thedressing-field wavelength dependence of the spectrum when Na is dressed by medium in-tensity laser fields [24], as one might expect from Fig. 3.For all coupling strengths shown in Fig. 6, at large dressing-field photon energies, i.e. ,those exceeding 17000 cm − or so, the spectra resemble the field-free spectrum, depictingaround twenty lines corresponding to transitions to | v (cid:105)| (cid:105) -type states. This is expected,because for such large photon energies, the V ( R ) + (cid:125) ω c PEC crosses the V ( R ) PEC at shortinternuclear distances, and the V ( R ) PEC remains unperturbed in the Frank–Condon re-gion. As the photon energy of the dressing field is lowered and the crossing of the V ( R )+ (cid:125) ω c and V ( R ) PECs approaches the Frank–Condon region, the spectrum becomes perturbed.In the top row of Fig. 6, a decrease can be seen in the spectrum line intensities alongdiagonal lines in the plots, forming island-type features. Focusing on a specific vibrationalstate on V ( R ), corresponding to a vertical line in the plots, a decrease in the spectrumintensity occurs when this vibrational state becomes resonant with one of the | v J (cid:105)| (cid:105) states. Due to the resonance, a strong mixing occurs between the | v J (cid:105)| (cid:105) - and | v (cid:48) J (cid:48) (cid:105)| (cid:105) -type states, which leads to a decrease of the transition amplitude from the groundstate. Nonetheless, when the mixing of the states is not as efficient as in the resonantcase, i.e. , at the island-type features on the plots, an increase can be seen in the spectrumintensities with respect to the field-free case.As depicted in the middle and bottom rows of Fig. 6, when the coupling strength isincreased, the picture of a “perturbed spectrum” gradually changes into the picture of twodistinct spectra corresponding to the two polariton surfaces, in accordance with Fig. 5.The dressing-field wavenumber dependence of the spectrum in the bottom row of Fig. 6can easily be understood in terms of the wavenumber dependence of the polariton surfacesdepicted in the rightmost column of Fig. 5.17 . SUMMARY AND CONCLUSIONS We investigated the rovibronic spectrum of homonuclear diatomic molecules dressed bythe quantized radiation field of an optical cavity. Formation of light-induced conical inter-sections induced by the quantized radiation field is shown for the first time by identifyingthe robust light-induced nonadiabatic effects in the spectrum. The coupling strength andthe cavity mode wavelength dependence of the field-dressed spectrum was also investigatedfrom the weak to the ultrastrong coupling regimes. Formation of polariton states in thestrong coupling regime was demonstrated, and its was shown how nonresonant couplingslead to an increased absorption in the field-dressed spectrum even before the ultrastrongcoupling regime is reached. The numerical results demonstrate that the additional degreeof freedom (which is the rotation in the present diatomic situation) plays a crucial role inthe appropriate description of the light-induced nonadiabatic processes as well as in the effi-ciency of nonresonant couplings. Therefore, for physical scenarios when diatomic molecularrotations can proceed in the cavity, properly accounting for the rotational degrees of freedomis mandatory for obtaining reliable simulation results.We hope that our findings will stimulate photochemical cavity experiments, and also theextension of the theory for the proper description of polyatomic molecules. It did not escapeour attention that there is much potential in studying light-induced conical intersectionsin polyatomic molecules in cavity without rotations as there are many nuclear degrees offreedom to form such intersections which can also be used to selectively manipulate certainchemical and physical properties.
VI. ACKNOWLEDGEMENT
This research was supported by the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00005 and by the Deutsche Forschungsgemeinschaft (Project ID CE10/50-3). The authorsare grateful to NKFIH for support (Grant No. PD124623, K119658 and K128396). Theauthors thank P´eter Domokos for the fruitful discussions. [1] T. Brabec and F. Krausz, Rev. Mod. Phys. , 545 (2000).
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