EPJ manuscript No.(will be inserted by the editor)

Light-Matter Interactions via the Exact Factorization Approach

Norah M. Hoffmann1,2, Heiko Appel1, Angel Rubio1, and Neepa T. Maitra2,3

1 Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science and Department ofPhysics, Luruper Chaussee 149, 22761 Hamburg, Germany

2 Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NewYork 10065, USA

3 The Physics Program and the Chemistry Program of the Graduate Center of the City University of New York, New York 10065,USA

March 7, 2018

Abstract. The exact factorization approach, originally developed for electron-nuclear dynamics, is ex-tended to light-matter interactions within the dipole approximation. This allows for a Schrödinger equationfor the photonic wavefunction, in which the potential contains exactly the effects on the photon field of itscoupling to matter. We illustrate the formalism and potential for a two-level system representing the matter,coupled to an infinite number of photon modes in the Wigner-Weisskopf approximation, as well as a singlemode with various coupling strengths. Significant differences are found with the potential used in con-ventional approaches, especially for strong-couplings. We discuss how our exact factorization approach forlight-matter interactions can be used as a guideline to develop semiclassical trajectory methods for efficientsimulations of light-matter dynamics.

PACS. XX.XX.XX No PACS code given

1 Introduction

The interaction of light with matter involves the corre-lated dynamics of photons, electrons, and nuclei. Evenat a non-relativistic level the solution of Schrödinger’sequation for the coupled subsystems is a daunting com-putation. In a given situation however, one is often mea-suring properties of only one of these subsystems. Forexample, one might be wanting to know how the electri-cal conductivity of a molecule is affected by the photons,as in the recent experiment showing the increased con-ductivity of organic semiconductors due to hybridizationwith the vacuum field [1]. On the other hand, one mightwant to understand how molecular dissociation after elec-tronic excitation is affected in the presence of light, asin the recent study of light-induced versus intrinsic non-adiabatic dynamics in diatomics [2]. Or, one might wantto measure the superradiance from a collection of atoms[3]. In each of these three cases, the observable of inter-est involves one of the subsystems alone, electronic, nu-clear, and photonic, respectively, yet to capture the dy-namics of the relevant subsystem, clearly the effects of allsubsystems are needed. The question then arises: can wewrite a Schrödinger equation for one of the subsystemsalone, such that the solution yields the wavefunction ofthat subsystem? The potential appearing in the equationwould have to incorporate the couplings to the other sub-systems as well as to any externally applied fields.

Hardy Gross, with co-workers, in fact alreadyanswered exactly these questions [4,5,6] when the pho-tonic system is treated as a classical light field neglect-ing the magnetic field contribution. That is, for systemsof electrons and nuclei, interacting with each other via ascalar potential (usually taken as Coulomb), and in thepresence of an externally applied scalar potential, suchas the electric field of light, it was shown that one canexactly factorize the complete molecular wavefunctioninto a wavefunction describing the nuclear system, anda wavefunction describing the electronic system that isconditionally dependent on the nuclear subsystem [4,5,6,7]: Ψ(r,R, t) = χ(r, t)ΦR(r, t), where r = r1, ...rNe andR = R1...RNn represent all electronic and nuclear coor-dinates respectively. The equation for the nuclear subsys-tem has a Schrö- dinger form, with scalar and vector po-tentials that completely account for the coupling to theelectronic system. One can reverse the roles of the elec-tronic and nuclear subsystems, to instead get a Schrödingerequation for the electronic system, which is particularlyuseful when one is most interested in the electronic prop-erties [8], e.g. in field-induced molecular ionization.

Recently rapid experimental and theoretical advanceshave however drawn attention to fascinating phenom-ena that depend on the quantization of the light field inits interaction with matter. This includes few-photon co-herent nonlinear optics with single molecules [9], directexperimental sampling of electric-field vacuum fluctua-

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tions [10,11], multiple Rabi splittings under ultrastrongvibrational coupling [12], exciton-polariton condensates[13,14], or frustrated polaritons [15] among others. Opti-cal cavities can be used to tune the effective strength ofthe light-matter interaction, and, in the strong-couplingregime in particular, one finds for example non-radiativeenergy transfer well beyond the Förster limit betweenspatially separated donors and acceptors [16], strong cou-pling between chlorosomes of photosynthetic bacteria andconfined optical cavity modes [17], photochemical reac-tions can be suppressed with cavity modes [18], the posi-tion of conical intersections can be shifted or they can beremoved [19,2], or state-selective chemistry at room tem-perature can be achieved by strong vacuum-matter cou-pling [20]. Strong vacuum-coupling can change chemi-cal reactions, such as photoisomerization or a prototypi-cal deprotection reaction of alkynylsilane [20,21] This hasgiven rise to the burgeoning field now sometimes called”polaritonic chemistry” [22,23,24,25]. In addition, novelspectroscopies have been proposed which explicitly ex-ploit correlated states of the photon field. For example theuse of entangled photon pairs enables one to go beyondthe classical Fourier limit [26], or correlated photons canbe used to imprint correlation onto matter[19,27,28]

In this paper, we extend the exact factorization ap-proach to non-relativistic coupled photon-matter systemswithin the dipole approximation. We focus particularlyon finding the potential driving the system in the presentstudy. One motivation is towards developing mixedquantum-classical methods for the light-matter system.The observation that in a matter-free system, the pho-tonic Hamiltonian is a sum over harmonic Hamiltoniansfor each mode of the radiation field suggests that a classi-cal treatment of the photonic system would be accurate:if the system begins in a Gaussian wavepacket, classicalWigner dynamics exactly describes the motion [29]. Cou-pling to matter within the dipole approximation wherethe coupling operator is linear in the photonic variablepreserves the quadratic nature of the Hamiltonian, andone might then think that again a classical Wigner treat-ment would be exact. However, recently it was foundthat, although accurate, it was not exact [30]. This impliesthat the true potential driving the photonic motion is infact not quadratic. The exact factorization approach de-fines exactly what this potential should be. In this paperwe explain the formalism and give some examples of thispotential, that clearly show deviations from harmonic be-haviour throughout the dynamics.

The theory is described in Sec. 2, presenting the Hamil-tonian that we will consider, and the formalism of thefactorization approach. Section 3 demonstrates the ap-proach on two examples, that we choose as the simplestcases for this initial study. The matter system is describedby a two-level system while the photonic system is cho-sen to either be an infinite number of modes treated withinthe Wigner-Weisskopf approximation, or a single cavitymode chosen to be resonant with the spacing of the twolevels, explored over a range of coupling strengths. Wefind and interpret the potential driving the photonic sys-

tem, which depends significantly on whether the initialstate of the system is chosen correlated or fully factor-ized. Finally in Sec. 4 we summarize and discuss the rel-evance of this approach for future investigations of light-matter dynamics.

2 Theory

2.1 QED-Hamiltonian

In this work, we consider the non-relativistic limit of asystem ofNe electrons,Nn nuclei, andNp quantized pho-ton modes, treated within the dipole approximation inCoulomb gauge [31,32,33]. For now, we do not considerany classical external fields, and neglect spin-coupling.The Hamiltonian of this coupled system is then definedby [34,19,35,36,37]

Ĥ(q, r,R) = Ĥp+Ĥe+Ĥn+Ĥep+Ĥnp+Ĥen+Ĥpen, (1)

which operates in the space of: r = {r1..ri..rNe} repre-senting all electronic spatial coordinates, R = {R1..RI ..RNn}representing all nuclear coordinates, and q = {q1..qα..qNp}representing all photonic displacement coordinates. Thefirst term characterizes the cavity-photon Hamiltonian

Ĥp(q) =1

2

2Np∑α=1

p̂2α + ω2αq̂

2α

= T̂p(q) + V̂p(q). (2)Here q̂α =

∑α

√~

2ωα(â+α + âα) defines the photonic dis-

placement coordinate for theαth mode, with creation(a+)and annihilation(a) operators [34,35], and the commuta-tion relation [q̂α, p̂α′ ] = ı~δα,α′ . The photonic displace-ment coordinate is directly proportional to the mode- pro-jected electric displacement operator, D̂α = �0ωαλαq̂α,while p̂α is proportional to the magnetic field [35,36]. Theαth mode has frequency ωα = kαc = απc/V , with kα thewavevector and V the quantization volume. The electron-photon coupling strength is given by

λα =√

4πSα(kα · X)eα, (3)where Sα denotes the mode function, e.g. a sine-functionfor the case of a cubic cavity [38,35], kα the wave vector,and X the total dipole of the system. In particular, we em-phasize at this point that the mode functions introducea dependence of the coupling constants on the quanti-zation volume of the electromagnetic field. By confiningthis volume, for example with an optical cavity, one cantune the interaction strength. Finally, we note that thesum in Eq. (2) goes from 1 to 2Np, to take the two po-larization possibilities of the electro-magnetic field intoaccount. The second term of Eq. (1) denotes the electronicHamiltonian

Ĥe(r) =

Ne∑i=1

p̂2i2me

+e2

4π�0

Ne∑i>j

1

|ri − rj |

= T̂e(r) + V̂ee(r) , (4)

2

where me defines the electronic mass, p̂i the electronicmomentum operator conjugate to r̂i. The third term inEq. (1) denotes the nuclear Hamiltonian

Ĥn(R) =

Nn∑I=1

P̂I2

2MI+

e2

4π�0

Nn∑i>j

ZIZJ|RI −RJ |

(5)

= T̂n(R) + V̂nn(R); , (6)

with analogous identifications to the electronic Hamilto-nian and eZI here being the nuclear charge.

The remaining terms in Eq. (1) denote the couplingsbetween the subsystems. The electron-nuclear couplingappears as the usual Coulombic interaction:

Ĥen = −Ne∑i=1

Nn∑J=1

e2Z

|ri −RJ |(7)

The electron-photon coupling, in dipole approximation,

Ĥep = −2Np∑α=1

ωαq̂αλα ·Ne∑i=1

eri (8)

(where e is the magnitude of the electronic charge) bi-linearly couples the total electric dipole moment with theelectric field operator for each mode of the photonic field.Similarly, the nuclear-photon coupling is

Ĥnp =

2Np∑α=1

ωαq̂αλα ·Nn∑I=1

eZIRI (9)

Finally,Hpen represents the dipole self-energy of the mat-ter in the radiation field:

Ĥpen =1

2

2Np∑α=1

λα ·

(Nn∑I

ZIRI −Ne∑i

ri

)2(10)

This self-energy term is essential for a mathematicallywell defined light-matter interaction. Without this termthe Hamiltonian is not bound from below, and loses inaddition translational invariance (in case of a vanishingexternal potential) [39].

The dynamics of such a coupled system is given bythe solution of the time-dependent Schrödinger equation(TDSE)

ĤΨ(r,R,q, t) = i∂tΨ(r,R,q, t), (11)

where Ψ(r,R,q, t) is the full matter-photon wavefunc-tion, that contains the complete information of the cou-pled system. However it is difficult to obtain an intuitiveunderstanding and interpretation of such a coupled sys-tem from the high-dimensional Ψ(r,R,q, t), and more-over, we may not be interested in all the information aswe might be interested in one of the subsystems. If oneof these subsystems varies on a much slower time-scalethan the others (in particular the nuclei), what is often

done in coupled electron-nuclear systems is a Born- Op-penheimer adiabatic approximation where the faster-time-scale subsystem (in particular the electrons) are assumedto instantaneously adjust to the positions of the nuclei,and hence if they begin in an eigenstate, they remain inan eigenstate parameterized by the nuclear coordinate.The eigenenergy maps out a Born-Oppenheimer poten-tial energy surface (PES) which provides the potential forthe nuclear dynamics. These potential- energy surfaces(PES) are clearly an approximation within the adiabaticansatz, but in fact an exact PES can be defined quite gen-erally without the need for any adiabatic approximation,which brings us to the main point of this paper. For theelectron-nuclear problem, these arise from the exact fac-torization approach mentioned earlier in the introduc-tion. In the next section we will extend the idea of theexact factorization for electron-nuclei systems to coupledphoton-matter systems.

Before moving to this, we note that Eq. (1) is the mostgeneral form of Hamiltonian that we will consider in thepresent work. In later sections, in particular in the ex-plicit examples, we will simplify to just a two-level elec-tronic system interacting with the photonic field in a cav-ity. In that case, many of the terms in Eq. (1) are zero, andwe simplify the remaining terms even further to a modelHamiltonian

Ĥ = −ω02σ̂z+

∑α

(−1

2

∂2

∂q2α+

1

2ω2αq

2α

)+∑α

ωαλαq̂α(degσ̂x),

(12)Here σi are the Pauli matrices. The first term is the two-level system that replaces the electronic Hamiltonian (in-cluding the dipole self-energy, which simplifies to a con-stant energy shift for a two-level system), where the energy-level difference is ω0, and deg , appearing in the third term,is the dipole moment of the transition. The second termdescribes the free photon field, as in Eq. (2), while Eq. (8)reduces to the third term with λα as the coupling strengthevaluated at the position of the atom in the cavity. TheTDSE also simplifies, to

i~∂

∂t

−→Ψ (q, t) =

(−ω02 + Ĥp(q)

∑α ωαλαq̂αdeg∑

α ωαλαq̂αdegω02 + Ĥp(q)

)−→Ψ (q, t)

(13)where we use the notation

−→Ψ (q, t) being a 2-vector de-

fined at every q and t. A cartoon of the problem is givenin Fig. 1.

3

ωα

λα

Fig. 1. Cavity-setup: Particle (green) trapped in a cavity andcoupled by coupling strength λα to the α photon mode withthe photonic frequency ωα, where α = {1, 2, ..., 2np}.

2.2 Exact Factorization Approach

The exact factorization (EF) may be viewed as a refor-mulation of the quantum mechanics of interacting cou-pled systems where the wavefunction is factored into amarginal amplitude and a conditional amplitude [7,4,5,6]. With non-relativistic electron-nuclear systems in mind,the equations for these amplitudes were derived for Hamil-tonians of the form

Ĥ = T̂e + T̂n + V̂ (14)

where V̂ is a scalar potential that includes coupling be-tween the electrons and nuclei (usually Coulombic) andany externally applied fields. Here T̂e,n are kinetic energyoperators of the electronic and nuclear equations, just asin Eq. 4 and 6, that have the form of −

∑i(I)∇2i(I)/2mi(I)

(that is, no vector potential). The EF then proves that theexact full molecular wavefunction can be factored as

Ψ(r,R, t) = χ(R, t)ΦR(r, t) . (15)

The equation for the nuclear amplitude χ has a TDSEform [5,6,40,41], equipped with a time-dependent scalarpotential �(R, t) and a time-dependent vector potentialAI(R, t) that include entirely the effects of coupling tothe electronic system as well as external fields. The equa-tion for the conditional electronic amplitude ΦR has amore complicated form, involving a coupling operatorÛen, that acts on the parametric dependence of ΦR. Thefactorization is unique, provided ΦR satisfies the ”partialnormalization condition” (PNC),

∫dr|ΦR(r, t)|2 = 1, up

to a gauge-like transformation; under such a transforma-tion, � and A transform as scalar and vector potentialsdo in electrodynamics. The nuclear Nn-body probabilitydensity and current-density can be obtained in the usualway from the nuclear amplitude χ(R, t), so in this sense,can be identified as the nuclear wavefunction of the sys-tem.

The form of the EF Eq.(15) is similar to the Born- Op-penheimer (BO) approximation, however with the im-portant difference that Eq.(15) is an exact representation

of the wavefunction, not an approximation, and furtherthat it is valid for time-dependent systems, with time-dependent external fields, as well. The BO approxima-tion assumes that the electronic system remains alwaysin the instantaneous ground (or eigen)-state associatedwith the nuclear configuration R, and therefore missesall the physics associated with non-adiabatic effects, in-cluding wavepacket branching and decoherence. Theseeffects are contained exactly in the coupling terms in theEF equations: the scalar and vector potentials and thecoupling operator of the electronic equation. It is impor-tant to note that there is no assumption of differenttimescales in the EF approach, in contrast to the BO ap-proximation.

As the scalar potential plays a role analogous to theBO PES, but now for the exact system, it is denoted thetime-dependent potential energy surface (TDPES), whilethe vector potential (TDVP) is an exact time-dependentBerry connection. The gauge-freedom is a crucial part ofthe EF approach: in particular, whether a gauge exists inwhich the vector potential can be transformed into partof the TDPES has been explored in some works [42,43],especially since the common understanding is that Berryphases appear only out of an adiabatic separation of time-scales, while the EF is exact and does not assume anysuch separation. In fact, equally valid is the reverse fac-torization [8], Ψ(r,R, t) = χ(r, t)Φr(R, t), which is par-ticularly useful when one is interested in the electronicsystem, since in this factorization, the electronic systemfollows a TDSE in which the potentials can be analysedand interpreted.

2.3 Exact Factorization Approach for QED

Here, we extend the exact factorization to systems of cou-pled photons, electrons, and nuclei. Since all the kineticoperators in the Hamiltonian within the dipole approx-imation, Eq. (1), are of similar form to those that wereconsidered in the original EFA, Eq. (14), the mathemati-cal structure of the equations and coupling terms will besimilar when we make a factorization into two parts.

There are three possibilities for such a factorization,and we expect each to be useful in different contexts. Onepossibility, which is perhaps the most natural extensionof the factorization of Ref. [5,6], is to take the nuclear sys-tem as the marginal one,

Ψ(q, r,R; t) = χ(R; t)ΦR(q, r; t) . (16)

with the PNC

〈ΦR(t)|ΦR(t)〉q,r ≡∫dqdr|ΦR(q, r; t)|2 = 1 (17)

for every nuclear configuration R at each time t. Thiswould yield a TDSE for the nuclear system, much likein the original EFA, but now the TDPES and TDVP in-cludes not only the effects on the nuclei of coupling to theelectrons, but also to the photons. This would be a par-ticularly useful factorization for studying light-induced

4

non-adiabatic chemical dynamics phenomena, when thequantum nature of light is expected to play a role. In fact,an approximation based on the normal Born-Oppenheimerapproximation for the electron-ion dynamics has been usedto study the cavity-induced changes in the potential en-ergy surfaces in the strong coupling regime [44]. Thiswould be a particularly useful factorization for studyinglight-induced non-adiabatic phenomena, when the quan-tum nature of light is expected to play a role.

A second possibility is the natural extension of the re-verse factorization [8], where the electronic system is themarginal amplitude

Ψ(q, r,R; t) = χ(r; t)Φr(q,R; t) , (18)

with the PNC 〈Φr(t)|Φr(t)〉q,R ≡∫dqdR|Φr(q,R; t)|2 =

1, for all t and every electronic configuration r, whichwould yield a TDSE for electrons, with the e-TDPES ande-TDVP now incorporating the full effects on the elec-trons of coupling to the nuclei as well as the photons.This could be particularly useful for studying, for exam-ple, the impact of vacuum field on electrical conductivityin a molecule or semiconductor.

This leaves the third possibility, where the photonicsystem is chosen as the marginal:

Ψ(q, r,R; t) = χ(q; t)Φq(r,R; t) , (19)

with the PNC

〈Φq(t)|Φq(t)〉r,R ≡∫drdR|Φq(r,R; t)|2 = 1 , (20)

for each field-coordinate q and all times t. This is the fac-torization we will focus on in the present paper: it givesa TDSE for the photonic system, within which the scalarpotential, which we call the q-TDPES, and vector poten-tial, the q-TDVP, contain the feedback of the matter-systemon the radiation field. In free space, the potential actingon the photons is quadratic as is evident from Eq. (2),however, in the presence of matter, the potential deter-mining the photonic state deviates from its harmonic formdue to interactions with matter. The cavity-BO approachintroduced in Ref. [32] has demonstrated these deviationswithin the BO approximation. The EF approach now ren-ders this concept exact, beyond any adiabatic assump-tions.

The equations for each of these three factorizationsfollows from a straightforward generalization of the orig-inal EF equations, as the non-multiplicative operators (thekinetic operators) have the same form; hence the deriva-tion proceeds quite analogously to that given in Ref. [5,6,40]. In particular, for the factorization Eq. (19), we obtain(Ĥm(r,R,q; t)− �(q; t)

)Φq(r,R; t) = i∂tΦq(r,R; t),

(21)(2np∑α

1

2

(i∂

∂qα+Aα(q; t)

)2+ �(q; t)

)χ(q; t) = i∂tχ(q; t),

(22)

where the matter Hamiltonian Ĥm is given by

Ĥm(r,R,q; t) = ĤqBO + Ûep. (23)

with

ĤqBO = Ĥe+ Ĥn+ Ĥen+ Ĥpen+ Ĥep+ Ĥnp+1

2

2Np∑α=1

ωαq̂2α

(24)defined in an analogous way to the BO Hamiltonian, butnow for the photonic system. The electron-photon cou-pling potential Ûep is given by

Ûep[Φq, χ] =

2np∑α

[(−i∂qα −Aα(q; t))2

2+ (25)(

−i∂qαχ(q; t)χ(q; t)

+Aα(q; t)

)(−i∂qα −Aα(q; t)

)],

the q-TDPES by

�(q; t) =〈Φq(t)

∣∣∣ Ĥm(r,R,q; t)− i∂t ∣∣∣Φq(t)〉r,R

, (26)

and the q-TDVP by

Aα(q; t) = 〈Φq(t)| − i∂qαΦq(t)〉r,R

. (27)

with the notation 〈...|..〉r,R meaning an integration overelectronic and nuclear coordinates only in the expectationvalues (as in Eq. (20)). The factorization (19) is uniqueup to a gauge-like transformation, provided the PNC,Eq. (20) is satisfied. The gauge-like transformation hasthe structure of the usual one in electromagnetism, ex-cept here the scalar and vector potentials arise due tocoupling, rather than due to external fields, and they arepotentials on the photonic system, not on the matter sys-tem. The equations are form-invariant under the follow-ing transformation:

Φq(r,R, t)→ Φq(r,R, t) exp(iθ(q, t))

χ(q, t)→ χ(q, t) exp(−iθ(q, t))

Aα(q; t)→ Aα(q; t) + ∂αθ(q, t)

�(q; t)→ �(q; t) + ∂tθ(q, t) (28)

Further, one can show that the displacement-field den-sity represented by χ reproduces that of the full wave-function, i.e.

|χ(q; t)|2 =∫drdR|Ψ(q, r,R; t)|2 , (29)

and that the phase of χ together with the q-TDVP providethe displacement-field probability current in the naturalway:

Im〈Ψ |∂αΨ〉 = |χ(q; t)|2Aα(q; t) + ∂αS(q; t) , (30)

5

where χ(q; t) = |χ(q; t)| exp(iS(q, t)). This means, thatobservables associated with multiplication by q can beobtained directly from χ(q, t), for example, the electricfield

E(r; t) =∑α

ωαλα(r, t)

∫dq qα|χ(q; t)|2 , (31)

while the magnetic field is

B(r; t) =∑α

c

ωα∇×λα(r, t)

∫dq|χ(q; t)|2Aα(q; t)+∂αS(q; t),

(32)

2.4 Exact Factorization for Simplified ModelHamiltonian: Two-Level System in Radiation Field

For our exploration of the QED factorization in this pa-per, we will turn to the simplified model Hamiltonian ofEq. (12), where the matter system’s Hamiltonian is a 2×2matrix. First, it is useful to write Eq.( 12) as

Ĥ =∑α

1

2∂2qα12 + ĤqBO ,where (33)

ĤqBO = −ω02σ̂z +

∑α

1

2ω2αq

2α12 +

∑α

ωαλαq̂α(degσ̂x),(34)

Here ĤqBO is analogous to the BO Hamiltonian in theusual electron-nuclear case. We can define qBO-states asnormalized eigenstates:

ĤqBO−→Φ (1,2)q = �

(1,2)qBO(q)

−→Φ (1,2)q (35)

with−→Φ i,†q ·

−→Φ jq = δij . and these can be used as a basis to

expand the fully coupled wavefunction, i.e.−→Ψ (q, t) = χ1(q, t)

−→Φ (1)q + χ2(q, t)

−→Φ (2)q (36)

which would be analogous to the Born-Huang expansionbut now for the cavity-matter system.

Now in the EF approach, the fully coupled wavefunc-tion is instead factorized as a single product:

−→Ψ (q, t) = χ(q, t)

−→Φ q(t) (37)

where the PNC becomes−→Φ †q(t) ·

−→Φ q(t) = 1 , (38)

and holds for every q and each time t.We note that there are two useful bases for this prob-

lem. One is obtained from diagonalizing the field-freetwo-level system, i.e. that defined by eigenvectors of thePauli-σz matrix. The other basis is the qBO basis, definedby the eigenvectors of ĤqBO, as in Eq.( 35).

The EF equations follow directly from Eqs (21–27) butwith the much simplified ĤqBO above, and all 〈...〉mean-ing simply a 2×2 vector-multiply.

2.5 Photonic Time-Dependent Potential EnergySurface

Unlike the original electron-nuclear factorization, the q-TDVP can always be gauged away due to the one- di-mensional nature of each photon-displacement mode. Thismeans that one can always transform to a gauge in whichthe q-TDPES contains the entire effect of the coupling ofthe matter system on the radiation field, i.e. it is the onlypotential that is driving the photonic dynamics. For thematter system, both the q-TDPES and the photon-mattercoupling operator incorporate the effect of the photonicsystem on the matter. In the original electron-nuclear fac-torization, the TDPES proved to be a powerful tool to an-alyze and interpret the dynamics of the system in casesranging from dynamics of molecules in strong fields [5,6,8,45,46] to non-adiabatic proton-coupled electron- trans-fer [47], especially as it provides an exact generalizationof the adiabatic BO-PES.

In the present work, we will study the q-TDPES �(q, t)of Eq.(26) for the case of the radiation field coupled toa two level-system, using the model Hamiltonian (12).Given a solution

−→Ψ (q, t) for the coupled system, found

from Eq. (13), we will extract the exact q-TDPES by in-version.

To do this, we first ensure that we work in the gaugewhere Aα = 0. Similarly to previous work [6,48], thisgauge can be fixed by choosing the phase S(q, t) of thephotonic wavefunction, χ(q, t) = |χ(q, t)| exp(iS(q, t)),to satisfy

∂qαS(q, t) =Im[〈−→Ψ (t)|∂qα

−→Ψ (t)〉]

|χ(q, t)|2. (39)

Then, from the given solution−→Ψ (q, t), we compute

−→Φ q(r) =

−→Ψ (q, t)

|χ(q, t)|eiS(q,t)with|χ| =

√−→Ψ †(q, t) ·

−→Ψ (q, t)

(40)and insert into the q-TDPES

�(q, t) =−→Φ †q(t) · ĤqBO ·

−→Φ q(t)

+∑α

|∂α−→Φ q(t)|2 +

−→Φ †q(t) · (−i∂t

−→Φ q(t))

= �wBO(q, t) + �kin(q, t) + �GD(q, t) . (41)

where we have noted that in this gauge the electron-photoncoupling operator reduces to Uep =

∑2npα ∂

2α/2. We have

identified here the first term in Eq. (41) as �wBO, a ”weightedqBO” surface, weighted by the probabilities of being inthe qBO eigenstates: using the expansion Eq. (36), �wBO =(|χ1(q, t)|2�(1)qBO(q, t) + |χ2(q, t)|2�

(2)qBO(q, t)

)/|χ(qt)|2. The

second term arises from kinetic effects from the paramet-ric dependence of the conditional matter wavefunction,

6

hence we denote it as �kin. Both those terms are invari-ant under different gauge choices, while the last term isgauge-dependent, hence its name �GD.

3 Results and Discussion

We will consider two extremes within the simplified modelHamiltonian Eq. (12). The first is the Wigner-Weisskopflimit where the two-level system is coupled to an infinitenumber of cavity modes. The second is the two-level sys-tem coupled to a single resonant mode. As initial condi-tion, we consider the photon modes in the vacuum state,and the two level system in the excited state. In the lat-ter case, we will compare the effect of starting in a purelyfactorized matter-photonic state with that of starting in aqBO state.

We notice that the dipole matrix element and cou-pling parameter appear only together as a product in thismodel, degλ. Physically, these are fixed by the problem athand, specifically the volume of the cavity and the dipolecoupling between the two levels in the atom, apart fromfundamental constants. But here, in this model we choosethem arbitrarily, and compare dynamics for different degλthat range from relatively weak coupling to strong cou-pling.

3.1 Wigner-Weisskopf limit

We first consider the Wigner-Weisskopf limit, in whichour two-level system is coupled to an infinite number ofmodes. In this limit, the accepted well-known approxi-mate solution for the coupled system is known analyti-cally, which makes the q-TDPES particularly straightfor-ward to find.

The solution for−→Ψ of the coupled problem can be

found in the standard literature [49]. The initial state istaken to be a purely factorized state of the electron in theexcited state and all photon modes in their ground states,i.e.

−→Ψ (q, 0) = χ0(q)

(10

), (42)

where

χ0(q) =∏α

(ωαπ~

) 14

e−ωαq2α/2~ , (43)

which follows from the harmonic nature of the free pho-ton field. The coupling in the off-diagonal elements ofEq. (12) then cause Ψ to evolve in time, as

−→Ψ (q, t) = a(t)χ0(q)

(10

)+∑α

bα(t)χα(q)

(01

)(44)

under the reasonable assumption that the coefficients ofthe two-photon and higher states are negligible. Here theone-photon states of the photonic system are

χα(q) =

√2ωα~qα∏β

(ωβπ~

) 14

e−ωβq2β/2~ . (45)

The coefficients a(t) and bα(t) can be found by substi-tuting Eq. (44) into the TDSE Eq. (13). After making theWigner-Weisskopf approximations (taking the continuumlimit so V → ∞, taking a(t) to change with a rate muchslower than the resonant frequency ω0 and performing aMarkov rotating-wave approximation, and neglecting adivergent Lamb shift), we arrive at

a(t) = e−iω0t~ e−

Γt2 , (46)

bα(t) = eiωα

igα(ei(ωα−ω0)t−Γt/2 − 1)

i(ωα − ω0)− Γ/2. (47)

where gα =√

πωα2~ λαdeg and the decay (spontaneous emis-

sion rate), Γ = (degλ)2ω20V~c3 . The Wigner-Weisskopf so-

lution is accurate for weak coupling, so that in this limitthe solution also generates accurate q-TDPES.

With this Wigner-Weisskopf solution, we can then findthe corresponding ”exact” q-TDPES, Eq. (41), using Eq. (40)and (39). However, this yields an infinite dimensional sur-face, since q = (q1..qα...q∞), which is challenging to visu-alize. Instead, we plot some one-dimensional cross-sectionsof the q-TDPES, along the ith mode, setting qα6=i = 0.In the following, we use q̄ to denote all modes not equalto qi. We will abbreviate quantities such as �(qi, q̄ = 0, t)by �(qi, t), understood to be looking at the cross-sectionwhere the displacement-coordinate of all other modes iszero. We will choose two different modes to look along:one resonant with ω0, and the other slightly off-resonant.With this choice of cross-sections through the origin ofall modes but one, it can be shown that the phase of thenuclear wavefunction that satisfies the zero-q-TDVP con-dition, Eq. (27), S(qi, t) ≡ 0. This leads to some simplifi-cation in the components of �(qi, t).

In Fig.2 we plot the autocorrelation function

AΦ(t) =∣∣∣∫ dqi(−→Φ †(qi; t = 0) · −→Φ (qi; t))∣∣∣2. (48)

as this gives an indication of what to expect for time-scales for the behavior of the q-TDPES �(qi, t) for differ-ent coupling strengths degλ = {0.01, 0.1, 0.4}. In the up-per panel, we have chosen to plot the q-TDPES along themode of the radiation field that is resonant with the two-level system. In this case, the decay of the autocorrelationdepends primarily on (degλ)2, through Γ , i.e. AΦ(t) ∝e−Γt, although there are some small polynomial correc-tions.

In the slightly off-resonant case, we have chosen ωi =0.411 while ω0 = 0.4. In fact, we observe partial revivalsin in the autocorrelation function for very long times inthe case of the weakest coupling shown (degλ = 0.01), asshown in the inset, with the amplitude decreasing witheach revival. However the initial decay follows a sim-ilar degλ-scaling pattern to that of the on-resonant sec-tion through the autocorrelation function. In either case,the dynamics of the decay is essentially the same for allcoupling strengths, provided the time is scaled appropri-ately, and their q-TDPES’s also map on to each other atthe corresponding times. In the following we show the

7

graphs for degλ = 0.01 for the cross-section taken alongthe on-resonant mode in Fig. 3, and that along the off-resonant mode in Fig. 4.

0 0.2 0.4 0.6 0.8

1

0 50 100 150 200 250 300t

AΦ,On(t)dl = 0.01dl = 0.1dl = 0.4

0 0.2 0.4 0.6 0.8

1

0 50 100 150 200t

AΦ,Off(t)

0 0.2 0.4 0.6 0.8

1

0 1000 2000 3000 4000t

Zoom-Out

Fig. 2. The autocorrelation function AΦ(t) where the differentcolors describe the different coupling strengths of the system.The upper panel shows the decay of this function when wechoose to look on-resonance ωi = ω0 = 0.4. The lower panelillustrates the decay when looking along a slightly-off resonantmode ωi = ω0 + 0.011. The zoom-out shows the same off-resonance decay for a longer time.

In Fig. 3 and Fig.4 we show the different componentsof the q-TDPES �(qi, t) for different time snapshots whichare illustrated by the colored dots in the decay-plot. Thedisplacement-field density, |χ(qi, t)|2 = |χ(qi, q̄ = 0, t)|2at these time-snapshots is shown in the top middle panel,and we observe the gradual evolution from the vacuumstate towards the state with one photon during the de-cay. This is also seen in the conditional probability am-plitudes shown in the top right panel, which we obtainfrom

|C1(2)(qi, t)|2 =−→Φ

(1(2))qi,q̄=0

·−→Φ qi,q̄=0(t) (49)

These are the coefficients of expansion of Φqi(t) in theBO basis and are equal to the coefficients in Eq. (36) viaCj(qi, t) = χj(qi, t)/χ(qi, t). C(1)(qi, t) and C(2)(qi, t) be-gin close to 0 and 1, respectively, as expected, and as thecoupling kicks in and the atom decays, one might expectthem to evolve to 1 and 0, respectively. This is in fact cor-rect for almost all qi, however non-uniformly in qi. Asexpected from the nature of the bilinear coupling Hamil-tonian Eq. (12), the conditional electronic amplitude asso-ciated with larger photonic displacements qi couple morestrongly than those associated with smaller ones, so theconditional amplitude on the upper surface falls away

from 1 starting on the outer edges and then moving in.In fact, the conditional amplitude at q = 0 remains for-ever stubbornly at the upper surface, unaffected by thecoupling to the field.

This non-uniformity is reflected in the q-TDPES, andleads to a strong deviation from the harmonic form it hasin the absence of matter. The potential, driving the pho-tonic motion, loses its harmonic form in the initial timesteps as the decay begins, peeling away starting from theouter qi. The potential nearer qi = 0 remains harmonicfor the initial stages, but as time goes on, more of thesurface peels away from the upper surface, while a peakstructure develops near qi = 0 that gets increasingly lo-calized and increasingly sharp as the atom decay pro-cess completes and the photon is fully emitted. It is thispeak structure in the potential driving the photonic sys-tem that excites the system from the zero-photon statetowards the one-photon state.

We turn now to the components of this exact surface.In the weighted BO surface, �wBO(qi, t) that is plotted inthe middle right panel, we see the same peeling awayfrom the outer edges, but sticking resolutely to the origi-nal upper surface at qi = 0. As the decay occurs, �wBO(qi, t)gradually melts to the lower surface everywhere exceptfor a shrinking region near the origin that sticks to theupper surface. The peak seen in the full q-TDPES on theother hand comes from �kin(qi, t), plotted in the lower leftpanel, which gets sharper and sharper as the photon isemitted. Mathematically, this structure follows from thechange in the conditional-dependence of Φq near qi = 0,as the electronic state associated with qi = 0 remainson the upper qBO surface while away from q = 0, in ashrinking region, the electronic state is associated withthe lower surface. This gets sharper as χ(qi = 0, t) getssmaller and smaller there. One can show from the ana-lytic solution, that, in the long-time limit, the surface atqi = 0 grows exponentially with t at a rate determinedby Γ , while for q 6= 0, �kin(qi 6= 0, t→∞)→ 0.

These features of �wBO and �kin are very similar forboth the cross-section that cuts along the resonant mode(Fig. 3) and that cutting along the slightly off-resonant(Fig. 4). The remaining component of the q-TDPES, �GDis much smaller than the other components, and has adifferent structure in the two cases. In fact, it is straight-forward to show from the analytic solution that �GD(qi =0, t) is independent of t, and that uniformly shifting�GD(qi, t) so that �GD(qi = 0, t) ≡ 0 yields �GD(qi 6= 0, t�Γ ) → ω0 − ωi for qi large. That is, there is a symmetricstep-like feature in �GD, of the size of the difference in themode frequency of interest and the resonant mode, andas t gets larger, this feature sharpens.

Thus, we can see that in the Wigner-Weisskopf limit,the potential driving the photonic modes deviates signif-icantly from its initial harmonic form during the decay,although once again becoming harmonic almost every-where (except at q = 0) in the long-time limit. The atom-photon correlation is required to capture these effects,and if one wanted to model this exact q-TDPES, the con-

8

0

0.5

1

0 50 100 150 200t

AΨ(t)

0 0.2 0.4 0.6 0.8

1

-6 -4 -2 0 2 4 6q

|Χ(q,t)|2

0

0.5

1

-6 -4 -2 0 2 4 6q

|Ci(q,t)|2

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

ε(q,t)

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

εwBO(q,t)

0

0.5

1

1.5

2

-6 -4 -2 0 2 4 6q

εkin(q,t)

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

-6 -4 -2 0 2 4 6q

εGD(q,t)

Fig. 3. Wigner-Weisskopf model, looking on-resonance ω0 =ωk. The top left panel shows the decay AΦ(t), where the differ-ent colored dots depict the times of the different time snapshotsof the dynamics shown within this plot. The middle and rightpanels along the top show the photonic distribution |χ(q, t)|2and each coefficient of the conditional electronic distribution|C(1)(qi; t)|2 (dashed), |C(2)(qi; t)|2 (solid) at the correspondingtime snapshots. The middle and lower panels show the differ-ent components of the �(qi; t) as well as the full scalar potentialat the given time snapshots. In the middle panels, the qBO sur-faces are shown in blue for reference.

ditional dependence of the electronic amplitude is crucialto include.

3.2 Two-level system coupled to a single resonantmode

We now turn to the other limit, and tune the cavity sothat there is just one mode that couples appreciably to thetwo-level atom, with a mode frequency that is resonantwith the atomic energy difference.

The q-BO surfaces can be easily found by diagonaliz-ing HqBO of Eq. (34), keeping only one mode with ωα =ω0 in the field:

�qBO(q) =1

2ω20q

2 ∓√ω20/4 + (degλω0)

2q2 (50)

0

0.5

1

0 50 100 150 200t

AΨ(t)

0 0.2 0.4 0.6 0.8

1

-6 -4 -2 0 2 4 6q

|Χ(q,t)|2

0

0.5

1

-6 -4 -2 0 2 4 6q

|Ci(q,t)|2

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

ε(q,t)

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

εwBO(q,t)

0

0.5

1

1.5

2

-6 -4 -2 0 2 4 6q

εkin(q,t)

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

-6 -4 -2 0 2 4 6q

εGD(q,t)

Fig. 4. As for Figure 3 but looking along the slightly-off resonantmode in the Wigner-Weisskopf model

For couplings λdeg � 1/2, the q-BO surfaces are approx-imately parallel and harmonic except at large q (see alsoRef. [32]). So in this case if the initial photonic state is avacuum, then the ensuing dynamics is driven by a largelyharmonic potential, without much perturbation from theatom, except at larger q. Deviations from parallel har-monic surfaces, and hence non-qBO behavior, occurs atlarger q and as the coupling increases. We will investi-gate the exact q-TDPES driving the photonic dynamicsfor three different coupling strengths, (dl = {0.01, 0.1, 0.4})and will include a plot of the two qBO surfaces with ourresults for comparison with the exact q-TDPES.

In Figure 5 we plot the exact q-TDPES for couplingstrength degλ = 0.01 beginning with the atom in the ex-cited BO level, multiplied by the photonic ground-state.On the upper panel (left) we plot the autocorrelation func-tion

AΨ (t) =

∣∣∣∣∫ dq−→Ψ †(q, 0) · −→Ψ (q, t)∣∣∣∣2 (51)to indicate the approximate periodicity of the system dy-namics. Comparing with Figures 6 and 7, we find a de-crease of the approximate period with the increase of cou-pling strength until the periodicity breaks down for thestrong coupling dl = 0.4. Further, we observe from Fig 8

9

0

0.5

1

0 200 400 600t

AΦ(t)

0

0.1

0.2

0.3

0.4

-4 -3 -2 -1 0 1 2 3 4q

|Χ(q,t)|2

0

0.5

1

-3 -2 -1 0 1 2 3q

|Ci(q,t)|2

0

0.5

1

1.5

2

2.5

-4 -3 -2 -1 0 1 2 3 4q

ε(q,t)

0

0.5

1

1.5

2

2.5

-4 -3 -2 -1 0 1 2 3 4q

εwBO(q,t)

0

0.5

1

1.5

2

-4 -3 -2 -1 0 1 2 3 4q

εkin(q,t)

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-3 -2 -1 0 1 2 3q

εGD(q,t)

Fig. 5. The �(q, t) for the excited BO initial state and couplingstrength dl = 0.01. The top left panel shows AΦ(t), where thedifferent colored dots depict the times of the dynamics withinthis plot. The top middle and right plots show the photonicdistribution |χ(q, t)|2 and the electronic coefficients in the BObasis, |C(1)(qi; t)|2 (dashed), |C(2)(qi; t)|2 (solid), for the timesnapshots shown. The middle and lower panel show the q-TDPES �(q, t) and its decomposition into components for thegiven time snapshots. The q-BO surfaces are shown in the mid-dle panel in blue for reference.

that the periodicity depends on the choice of initial state,as the time for one period decreases if we choose theinitial state to be fully factorized. The weakest couplingstrength we have chosen is on the borderline of being inthe Rabi regime [50], while the strongest is far from it.

The photonic distribution (middle) and conditionalelectronic coefficients (right) are shown in the topmostpanel of Figs. (5) - (10). The photonic system begins in thevacuum state, while the electronic distribution starts, asdefined in the initial condition, with an electron in the ex-cited state and no electron in the ground-state, either BOor field-free. The initial coefficients are C(1)(q, 0) = 1 andC(2)(q, 0) = 1 when beginning in the BO states. Whenbeginning in the fully factorized state, these coefficientsdeviate from these uniform values, especially for largerq, with deviation increasing with the coupling strength.

0

0.5

1

0 20 40 60 80 100t

AΦ(t)

0

0.1

0.2

0.3

0.4

-4 -3 -2 -1 0 1 2 3 4q

|Χ(q,t)|2

0

0.5

1

-3 -2 -1 0 1 2 3q

|Ci(q,t)|2

0

0.5

1

1.5

2

-4 -2 0 2 4q

ε(q,t)

0

0.5

1

1.5

2

-4 -2 0 2 4q

εwBO(q,t)

0

0.5

1

1.5

2

-4 -2 0 2 4q

εkin(q,t)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 4q

εGD(q,t)

Fig. 6. As in Figure 5 but with coupling strength degλ = 0.1.

The dynamics depends significantly on whether theinitial state is the correlated qBO state (Figs. 5–7) or afully factorized one (Figs. 8–10). The fully factorized ini-tial state would be the physical one when an excited atomis instantaneously brought into a closed cavity and justthen its dynamics is studied, while the excited qBO stateresults when there is initially an external dissipative cou-pling together with an applied resonant field to maintainthe atom in that excited state before the dynamics is ex-amined. We turn now to the dynamics and to the struc-ture of the exact q-TDPES for this latter case first.

After some time we see a transition of the electronfrom the excited state to the ground-state as indicated bythese coefficients. We observe that the transfer begins ear-lier for higher values of q and then is followed by lowerq-values, but again the conditional amplitude at q = 0sticks to the upper surface at all times in all cases as thereis no coupling for q = 0. The q-dependence of these co-efficients has a significant role in shaping the structureof the q-TDPES that we will shortly discuss. At the sametime, the probability of photon emission increases, as in-dicated by the morphing of the initial gaussian in χ(q, t)towards its first-excited profile. For the weakest couplingstrength after a half period, the system begins to moveback approximately to its initial state, as the photon is

10

0.5

0.8

1

0 20 40 60 80t

AΦ(t)

0

0.1

0.2

0.3

0.4

-6 -4 -2 0 2 4 6q

|Χ(q,t)|2

0

0.5

1

-6 -4 -2 0 2 4 6q

|Ci(q,t)|2

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

ε(q,t)

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

εwBO(q,t)

0

0.5

1

1.5

2

-6 -4 -2 0 2 4 6q

εkin(q,t)

-4

-3

-2

-1

0

1

2

3

4

-6 -4 -2 0 2 4 6q

εGD(q,t)

Fig. 7. As in Figure 5 but with coupling strength degλ = 0.4.

reabsorbed and atom becomes excited again. For strongcoupling dl = 0.4 the periodic character is lost and wefind more wells and structure appearing in the displacement-field density profile. With such strong coupling the qBOsurfaces are quite distorted from a pure harmonic, as ev-ident in the plot (blue lines in the middle panel), and theanharmonicity brings more frequencies into play. A one-photon state that is associated with the lower q-BO sur-face has a wider profile with density maxima further outthan a one-photon state associated with the upper surfacewould have, for example. In fact the character of the cou-pled cavity-matter system becomes quite mixed, as is ev-ident from the conditional electronic coefficients shownon the right, and as one goes along the photonic coordi-nate q one associates with different superpositions of theelectronic states. This leads to interesting structure in theexact q-TDPES, that, when decomposed in terms of theq-BO surfaces, has components that vary a lot with q (i.e.not just a piecewise combination).

The q-TDPES for initial state prepared in the upper q-BO state begins with the weighted BO component, �wBOon top of the upper q-BO surface as expected. For theweakest coupling, degλ = 0.01, �wBO(q, t) then melts downto the lower surface over half a cycle, peeling away fromthe outer higher q-values first, in a similar way to what

0

0.5

1

0 200 400 600t

AΦ(t)

0

0.1

0.2

0.3

0.4

-4 -3 -2 -1 0 1 2 3 4q

|Χ(q,t)|2

0

0.5

1

-3 -2 -1 0 1 2 3q

|Ci(q,t)|2

0

0.5

1

1.5

2

2.5

-4 -3 -2 -1 0 1 2 3 4q

ε(q,t)

0

0.5

1

1.5

2

2.5

-4 -3 -2 -1 0 1 2 3 4q

εwBO(q,t)

0

0.5

1

1.5

2

-4 -3 -2 -1 0 1 2 3 4q

εkin(q,t)

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-3 -2 -1 0 1 2 3q

εGD(q,t)

Fig. 8. As in Figure 5 with coupling strength degλ = 0.01 butwith the initial purely factorized state.

was seen in the Wigner-Weisskopf limit. This potentialapproaches the lower surface before returning back tothe upper BO-surface, but the region near q = 0 remainsbound to the upper surface. The time-dependent double-well structure in the potential is again important in driv-ing the photon emission. A similar trend is seen for thestronger coupling 0.1 in Fig. 6, but for the strongest cou-pling degλ = 0.4, �wBO(q, t) shows a more complicatedcorrelation in q, with structures mirroring those in thedisplacement-field density discussed above. As for thekinetic component, for the weaker couplings, a peak struc-ture in �kin(q, t) develops that grows and narrows dur-ing the photon emission stage, similar to what was seenin Wigner-Weisskopf, but this then reverses during thereabsorption here. Again for the stronger coupling, thestructure is more complicated, mirroring the more com-plicated dynamics. The gauge-dependent part, �GD is gen-erally a smaller contribution to the total q-TDPES com-pared to the other components, but again we see step-likefeatures for the weaker couplings, and more complicateddynamics for the strongest coupling.

For the fully factorized initial states, although the pho-tonic field still begins in the vacuum state, the electronicstate is not purely in the upper BO surface; the electronic

11

0

0.5

1

0 20 40 60 80 100t

AΦ(t)

0

0.1

0.2

0.3

0.4

-4 -3 -2 -1 0 1 2 3 4q

|Χ(q,t)|2

0

0.5

1

-3 -2 -1 0 1 2 3q

|Ci(q,t)|2

0

0.5

1

1.5

2

-4 -2 0 2 4q

ε(q,t)

0

0.5

1

1.5

2

-4 -2 0 2 4q

εwBO(q,t)

0

0.5

1

1.5

2

-4 -2 0 2 4q

εkin(q,t)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 4q

εGD(q,t)

Fig. 9. As in Figure 6 with coupling strength degλ = 0.1 butwith the initial purely factorized state.

state associated with larger q has already some compo-nent in the ground-state. So at these larger values of q,the initial �wBO(q, 0) surface dominates the q-TDPES andis anharmonic from the very start, lying intermediate be-tween the upper and lower q-BO surface. In the weakcoupling case, the differences are only large at values of qmuch larger than shown in the plot, and these are physi-cally unimportant given there is very little photonic fieldprobability there; hence Fig. 5 and 8 are almost identical.For strong couplings, comparing Fig 7 and 10 shows thatthe q-TDPES has a tamer structure for the fully-factorizedinitial state than for the correlated q-BO initial state, espe-cially at larger q; this is likely because less energy is avail-able at these larger q for the system to exchange betweenthe atomic and photonic systems because the atomic statecorrelated with large q is not completely in its excitedstate initially.

To summarize: At time zero the exact q-TDPES startson the upper q-BO-surface, which, depending on cou-pling strength and choice of initial state, ranges from ly-ing directly on top of the upper q-BO surface (weakercoupling and with q-BO initial state), to in between thetwo q-BO surfaces with deviations from the upper be-ing larger for larger q (stronger coupling, or fully fac-

0.5

0.8

1

0 20 40 60 80t

AΦ(t)

0

0.1

0.2

0.3

0.4

-6 -4 -2 0 2 4 6q

|Χ(q,t)|2

0

0.5

1

-6 -4 -2 0 2 4 6q

|Ci(q,t)|2

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

ε(q,t)

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

-6 -4 -2 0 2 4 6q

εwBO(q,t)

0

0.5

1

1.5

2

-6 -4 -2 0 2 4 6q

εkin(q,t)

-4

-3

-2

-1

0

1

2

3

4

-6 -4 -2 0 2 4 6q

εGD(q,t)

Fig. 10. As in Figure 7 with coupling strength degλ = 0.4 butwith the initial purely factorized state.

torized initial state). After some time the potential startsto melt down onto the lower BO-surface, first startingat higher q-values and then followed by lower q-values,with peak structures developing in the interior region.Around q = 0 the kinetic-component dominates, whichleads to an increasing and after half a period decreasingpeak. For stronger coupling we observe several peak fea-tures in the potential and significant deviations from thecurvature of the q-BO surfaces throughout q small con-tribution below the lower BO-surface; the deviations atlarger q arise from the gauge-dependent component.

4 Summary and Outlook

We have introduced here an extension of the exact- fac-torization approach, that was originally derived for cou-pled electron-nuclear systems, to light-matter systems inthe non- relativistic limit within the dipole approxima-tion. We have presented different possible choices for thefactorization but in this work have focussed on the onewhere the marginal is chosen as the photonic system andthe matter system is then conditionally-dependent on this.This choice is particularly relevant when one is primarilyinterested in the the state of the radiation field since the

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exact factorization yields a time-dependent Schrödingerequation for the marginal, while the conditional is de-scribed by an equation with an unusual matter-photoncoupling operator. The equation for the marginal is, in asense, simpler than that in the electron-nuclear case, sincethe vector potential, q-TDVP, appearing in the equationcan always be gauged away into a scalar potential, the q-TDPES. We have studied the potential appearing in thisequation in a gauge where the q-TDVP is zero, for a two-level system coupled to an infinite number of modes inthe Wigner-Weisskopf approximation, and for a two-levelsystem coupled to a single photonic field mode with arange of coupling strengths. In all cases we find a veryinteresting structure of the potential that drives the pho-tonic dynamics, and in particular, large deviations fromthe harmonic form of the free-photon field. These devia-tions completely incorporate the effect of the matter sys-tem on the photonic dynamics. We also studied the effectof beginning in an initially purely factorized light-matterstate, compared to a q-BO initial state, finding significantdifferences for larger coupling strengths in the ensuingdynamics, implying that in modelling these problems acareful consideration of the initial state is needed.

To use the exact factorization for realistic light-mattersystems, approximations will be needed, since solvingthe exact factorization equations is at least as computa-tionally expensive as solving the Schrödinger equationfor the fully coupled system. The success of such an ap-proximation depends on how well the q-TDPES is mod-elled. The components of the exact q-TDPES beyond theweighted BO depend significantly on the q-dependenceof the conditional probability amplitude; approximationsthat neglect this dependence (Ehrenfest-like) will likelylead to errors in the dynamics. It has been shown recentlythat mixed quantum-classical trajectory methods that arederived from the exact factorization approach can cor-rectly capture decoherence effects [51]. Since photons areintrinsically non-interacting and therefore even simplerto treat than nuclei, we expect in analogy to the electron-nuclear case that semiclassical trajectory methods derivedfrom systematic and controlled approximations to the fullexact factorization of the light-matter wavefunction willbe able to capture decoherence effects beyond the Ehren-fest limit for light-matter coupling. This will be subject offuture investigations.

We acknowledge financial support from the European ResearchCouncil(ERC-2015-AdG-694097), Grupos Consolidados (IT578-13), and European Union’s H2020 programme under GA no.676580(NOMAD) (NMH, HA, and AR). Financial support from the USNational Science Foundation CHE-1566197 is also gratefully ac-knowledged (NTM).

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1 Introduction2 Theory3 Results and Discussion4 Summary and Outlook