Cavity engineering of Hubbard $U$ via phonon polaritons

Pump-probe experiments have suggested the possibility to control electronic correlations by driving infrared-active phonons with resonant midinfrared laser pulses. In this work we study two possible microscopic nonlinear electron-phonon interactions behind these observations, namely coupling of the squared lattice displacement either to the electronic density or to the double occupancy. We investigate whether photon-phonon coupling to quantized light in an optical cavity enables similar control over electronic correlations. We first show that inside a dark cavity electronic interactions increase, ruling out the possibility that $T_c$ in superconductors can be enhanced via effectively decreased electron-electron repulsion through nonlinear electron-phonon coupling in a cavity. We further find that upon driving the cavity, electronic interactions decrease. Two different regimes emerge: (i) a strong coupling regime where the phonons show a delayed response at a time proportional to the inverse coupling strength, and (ii) an ultra-strong coupling regime where the response is immediate when driving the phonon polaritons resonantly. We further identify a distinctive feature in the electronic spectral function when electrons couple to phonon polaritons involving an infrared-active phonon mode, namely the splitting of the shake-off band into three bands. This could potentially be observed by angle-resolved photoemission spectroscopy.

is their relatively short life time, typically in the picosecond range, with a recent extension to the nanosecond regime in K 3 C 60 . 52 As an alternative route to control over material properties, light-matter coupling in cavities has been suggested. [53][54][55][56][57] In these setups, instead of achieving strong modifications of material properties by strong driving, one focuses on realizing strong coupling between light and matter, supported by recent experimental advances. [58][59][60][61][62][63][64][65][66] This might enable the engineering of material properties already with few photons 67,68 or even inside a dark cavity, utilizing only the vacuum fluctuations of the light field. 69 Since in this case the material stays in its ground state or energetically close to it, effects from detrimental heating are expected to be reduced and life times to be longer.  112,113 and the modification of topological states of matter. [114][115][116][117] Taking a complementary approach, the photons in a cavity also couple to lattice vibrations forming hybrid light-matter excitations -namely phonon polaritons. Their potential for steering chemical reactions, 118,119 inducing superconductivity, 120 influencing the ferroelectric phase-transition, 77,121 , achieving the redistribution of energy between otherwise non-resonant phonon modes 122 , or influencing the electron-electron interaction mediated by phonons 123 has recently been investigated.
In this work we explore the possibility of replacing the laser drive for inducing transient superconducting-like states by coupling a material to an optical cavity. Among the proposals considered to explain the transient states is the suggestion that the laser effectively drives infrared-active (IR-active) phonons in the system that in turn lead to an effective attractive interaction between the electrons through one of two mechanisms put forward. 9,16,19,21 These mechanisms involve either a coupling of the phonon coordinate X to the electronic density of the form X 2 n, or to the double occupation of the form X 2 n ↑ n ↓ . Such couplings are distinct from the paradigmatic BCS mechanism since the phonons involved are IR-active. Therefore, a coupling to the electrons proportional to an odd power of the lattice displacement, including a linear coupling as in the BCS mechanism, is in general forbidden by symmetry 19,124,125 in inversion-symmetric crystals. In particular in Ref. 9 a superconducting response of the system was only observed when driving specific phonon modes in the chargetransfer salt κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br, abbreviated as κ-salt from here on out. Our modelling therefore focuses on this κ-salt but is kept sufficiently general to be applicable in a broader sense. Since the considered mechanisms stem from an electron-phonon interaction, we neglect a direct coupling of the electrons to the photons of the cavity. The remaining coupling between the cavity and the phonons will naturally lead to the formation of phonon polaritons.
After introducing our model we investigate the effect of coupling the phonons in the system to the vacuum fluctuations of a cavity. For both considered electron-phonon coupling mechanisms this yields an increase of electronic interactions. This rules out the possibility to enhance T c in superconductors by reducing the electron-electron repulsion via vacuum fluctuations through both proposed phonon mechanisms. Next we consider a weak drive of the cavity, populating the cavity with few, O(1) photons. Similar to the case of classical driving of the phonons, this decreases electronic interactions. We find that in the strong coupling (SC) but not ultra-strong coupling (USC) regime an increase of the light-matter coupling (LMC) does not necessarily lead to a more pronounced effect. Instead a LMC that exceeds cavity losses is needed since it determines the time scale on which the photons transfer their energy to the phonons. Complementary to this, when increasing the LMC to become comparable to the bare cavity frequency hence entering the USC regime, we find that driving the emerging phonon polaritons resonantly at their respective eigenfrequency induces an immediate response in the electronic system. Thus in this USC regime a LMC that outweighs cavity losses is not strictly required anymore. Finally, we consider the effects of polariton formation on the electronic spectral function that is in principle observable in an angle-resolved photoemission spectroscopy (ARPES) measurement. To this end we derive an effective model which we show to capture the dynamics of the system well. We show a distinctive feature of electrons coupling to polaritons that stem from an IR-active phonon.
The shake-off band 126 that is predicted to appear at a distance from the main spectral peak that equals twice the phonon frequency 127 splits into three bands. We discuss the feasibility ω + (green curve) and ω − (brown curve) for equal bare phonon and photon frequencies. c.) Double occupancy Eq. (10) for different values of the light-matter coupling parametrized by ω P inside a dark cavity. For both coupling mechanisms g 1 = 0, g 2 = −0.2J (X 2 n ↑ n ↓ , green curve) and g 1 = 0.5J, g 2 = 0 (X 2 n, brown curve) an increase of the light-matter coupling leads to a decrease in double occupancy that is associated with an increase in the effective electron-electron interaction. The black line (g = 0) marks the value for the uncoupled Hubbard dimer for these parameters. We used as a cutoff of the bosonic part of the Hilbert space N B = 16. Other parameters are the same as in the main text, Sec. II. of experimentally measuring this feature.

II. MODEL
We orient our modelling on the κ-salts discussed in Ref. 9 where they were described using a Hubbard model. Other molecular compounds, such as ET-F 2 TCNQ studied in Refs. 19 and 21, were also found to be described well by a Hubbard model. 128,129 We therefore also consider a Hubbard model for the matter degrees of freedom, focusing on the two-site version of this model -the Hubbard dimer. The Hamiltonian readŝ Here,ĉ j,σ annihilates -;ĉ † j,σ creates an electron at one of the two sites j ∈ {1, 2} with spin σ ∈ {↑, ↓}. J denotes the hopping integral, U the onsite repulsion of the electrons and we usedn el j,σ =ĉ † j,σĉ j,σ .
We couple each site to an optically active phonon for which the bare Hamiltonian is expressed asĤ In this expressionb j annihilates -;b † j creates a phonon with frequency ω phon at site j. Since the molecules forming the studied solids are centrosymmetric, a coupling between electrons and phonons that is linear in the phonon displacementX phon,j = 1 √ 2ω phon b j +b † j is forbidden. 19,124,125 The most general term for the electron-phonon interaction where the electrons couple to the quadratic displacement of the phonons readŝ Here g 1 parametrizes the coupling of the phonons to the linear electronic density and g 2 that of the phonons to the double occupancy. In previous works both a coupling that involves a term proportional to the double occupancy 19,21 as well as one that only incorporates a coupling to the linear electronic density 16 have been considered to understand the optical control of electronic correlations. In this work we will investigate both mechanisms separately, hence either setting g 1 = 0 and g 2 = 0 or vice versa.
We model the light degrees of freedom of the optical resonator by a single bosonic mode.
The photon of the cavity is coupled to the optically active phonon whereas its coupling to the electrons is neglected. Thus we write the total Hamiltonian of the system, including the photon-phonon interaction 130  H =Ĥ e − +Ĥ phon +Ĥ phon−e − +Ĥ phot ω photâ †â Here,â annihilates -;â † creates a photon in the effective single cavity mode. ω phot denotes the bare cavity frequency, ω P the polariton frequency that parametrizes the phonon-photon or light-matter coupling. The model is illustrated in Fig. 1 a.).
The coupling between phonons and photons will lead to the formation of hybrid lightmatter states, phonon polaritons. Their effective frequencies are calculated as 123 (see Appendix F) For identical phonon and photon frequency, the polariton frequencies are plotted as a function of the coupling ω P in Fig. 1 b.). We call the polariton with the effectively higher frequency ω + the upper polariton and that with the effective lower frequency ω − the lower polariton.
In what follows the hopping J defines the unit of energy. For the onsite repulsion U we take an intermediate value of U = 5J that was found in first principles calculations for the κ-salts. 9 The C − C breathing mode of the κ-salts has an effective frequency of ω eff phon ≈ 2J that is composed the bare phonon-frequency and contributions stemming from the coupling to the electrons. In Ref. 19 it was shown for the molecular compound ET-F 2 TCNQ that the contribution from the coupling to the electrons can be comparable to or even dominate that from the bare phonon frequency. We therefore choose parameters such that the two contributions are close to equal in the case of the coupling to the linear electronic density, where we set We determine the bare phonon frequency ω phon such that the effective phonon frequency is equal to the value previously determined for the κ-salts ω eff phon = 2J. We find to fulfill this condition. The exact procedure how to obtain the bare phonon frequency is outlined in Appendix A. In the case of coupling exclusively to the double occupancy, the coupling constant g 2 is expected to be negative 21 for the considered solids. We anticipate a somewhat smaller absolute value |g 2 | < |g 1 | compared to the coupling to the linear electronic density (see Eq. (6)) and thus choose We note that the detailed values of these couplings do not fundamentally alter our conclusions. Choosing the bare phonon frequency as creates a resonance of the phonons at frequency 2J (also see Appendix A). We couple the phonons resonantly to the cavity and therefore set ω phot = ω eff phon = 2J. The results for different values of ω P are shown in Fig. 1 c.). Without a cavity (ω P = 0) the coupling to the phonons leads to a slight increase of the double occupancy for both We also consider the effect of finite temperature on the cavity-induced increase in effective electron-electron interactions discussed above. For this we calculate the thermal expectation value of the double-occupancy in the canonical ensemble according to where Z = n e −βEn is the partition function, E n is the n th eigenenergy of the system, |ψ n the corresponding eigenstate, and β = 1 k B T the inverse temperature. To obtain concrete temperature values we take J = 80meV, which is the value found in ab initio simulations for the κ-salts performed in Ref. 9. Since these were performed for a triangular-lattice Hubbard model, this can only give a rough order-of-magnitude scale for the temperatures.
The result are presented in Fig. 2. Overall, higher temperatures result in a reduction of the double-occupancy but the effect from the coupling to the cavity remains present.

IV. WEAK DRIVING OF THE CAVITY
In this part we apply a weak coherent drive to the cavity and investigate the dynamical change of electronic interactions. Adding a coherent drive, the time-dependent Hamiltonian Here,Ĥ is the Hamiltonian of the undriven system Eq.(4),Â phot = 1 √ 2ω phot â † +â is the quantized cavity field and F (t) is a pump pulse for which we choose a Gaussian envelope We 2 as parameters for the driving envelope. At t = 0 the system is prepared in its GS and then evolved forward in time via a commutator-free scheme according to Ref. 131. Details about the numerical scheme including a convergence study in the finite time-step used as well as the cutoff of the bosonic part of the Hilbert space can be found in Appendix C.
The coupling strength between light and matter inside a cavity is typically classified by comparing it to two distinct quantities: once to the losses of the cavity, where strong coupling (SC) refers to a situation in which the coupling exceeds the losses; and once by comparing the coupling to the bare cavity resonance. When the coupling reaches one tenth of the resonance frequency one speaks of ultra-strong coupling (USC). 54 We do not consider any losses of the cavity, and since we modelled the solid within the cavity with the Hubbard dimer there are no true heating effects either. We are therefore automatically in the SC regime since all time scales are shorter than the (infinite) decay time of the cavity excitation. Effects from including a finite cavity life time are discussed later in this section. Comparing the strength of the LMC parametrized in our case by ω P to the bare cavity resonance we consider two different regimes: two values below USC of ω P = The time evolution of the GS of the full coupled system for a coupling of the phonons to the linear electronic density (g 1 = 0.5J and g 2 = 0 -also see Eq. (3)) is shown in Fig. 3 and that for the coupling of the phonons to the double occupancy (g 1 = 0 and g 2 = −0.2Jalso see Eq. (3)) of the electrons in Fig. 4. Both coupling mechanisms display qualitatively similar behaviour. In the case of strong, but not ultra-strong coupling, the pump drives the cavity into an excited state with an increased photon number N phot =â †â within the time duration of the pump. The strength of the drive is such that only few photons N phot = O(1) are created. The energy of the photon excitation is subsequently completely transferred to the phonons on a time scale that is approximately π ω P , as marked in the plot. When considering even longer times the excitation of the cavity mode and the phonons oscillates back and forth with a period τ beat ≈ 4π ω P .
In the USC case ω P = ω phot 2 , driving the cavity at its bare resonance frequency ω Drive = ω phot only yields a weak response. However, when driving at an increased frequency of ω Drive = 2.56J = ω + that coincides with the upper polariton frequency ω + or a decreased frequency of ω Drive = 1.56J = ω − coinciding with the lower polariton frequency ω − again a sizeable response is obtained. In contrast to the SC regime, the phonon system reacts immediately in the USC regime. No periodic oscillations between light and matter excitations are observed in this case. Instead both the phonon number N phon = jb † jb j and the photon number N phot reach a plateau after the drive, with some oscillations on top.
The dynamics of the cavity mode and the phonons can be understood as that of two coupled harmonic oscillators with coupling constant ω P . To see this we first note that the cavity only couples to the even superposition of the phonon modes on the two sites, where we have introduced the even combination of bosonic operatorŝ to which a complementary odd combination exists, In the strong, but not ultra-strong, coupling regime the two oscillators are weakly coupled when comparing with their bare frequency ω P ω phot and ω P ω phon . The drive of the cavity displaces one of the oscillators (the photons) such that in the subsequent coupled motion one observes beats -a phenomenon well known from classical physics. The period of these beats is classically expected to be τ beat = 4π ω P , since ω P equals the splitting of the two eigenmodes of the system, such that one expects the first maximum in the phonon occupation after a quarter period τ beat 4 = π ω P . This matches well with the observations in Fig. 3 and Fig. 4.
In the USC regime light and matter excitations are completely hybridized forming phonon polaritons: One upper polariton with an increased effective frequency of ω + ≈ 2.56J; and one lower polariton with a decreased effective frequency of ω − ≈ 1.56J according to Eq. (5). This explains why only a small response is observed when driving the system at its bare resonance frequency ω phot = 2J -one simply drives the effective oscillators off-resonantly. When the polaritons are driven at their true resonances instead, with ω Drive = ω + or ω Drive = ω − , both phonon and photon degrees of freedom show an immediate response which is a direct consequence of the hybridization of light and matter degrees of freedom.
In a more realistic setup the cavity-matter system experiences losses, either through imperfect mirrors or heating of the material, that might be parametrized by an energy constant γ loss . In the case of smaller LMC ω P = ω phot 20 and ω P = ω phot 40 the response of the system is only triggered with a certain time delay π ω P . In a realistic setup, in order to get a sizeable effect, one thus need a LMC of to the bare cavity frequency does not necessarily yield a larger effect as becomes apparent both from Fig. 3 and Fig. 4. The comparison of the LMC to the cavity losses is therefore the more relevant one in this regime.
In the USC regime the response is immediate. One therefore does not need ω P > γ loss but the challenge lies in reaching a LMC that is of comparable size to the bare cavity frequency ω P ≈ ω phot . This is, in turn, precisely the definition of USC 54,132 , which is in particular not a subset of SC.
Both electron-phonon coupling mechanisms display qualitatively similar dynamics. In Appendix D we compare again both mechanisms also for classically driven phonons, finding essentially similar behaviour. Eq. (4). Additionally, we restore the particle-hole symmetry of the system by writing the electron-phonon coupling aŝ the electronic part of the Hilbert space and is therefore expected to not change the dynamics.
We calculate the spectral function from the time-evolved states according to the general formalism of time-resolved photoemission spectroscopy, 133 with T the time ordering operator.
The choice of a particular site or spin orientation does not matter due to symmetry. As we strictly work at zero temperature the expectation value is calculated with the GS of the system |ψ GS , which is determined via ED. Here s t 1 ,t 2 ,τ (t 0 ) denotes a Gaussian probe pulse defined as In the following we take σ = 1.6/J and t 0 = 5/J. As parameters for the model we set J = 1 and U = 5J, take ω phot = 2J and an electron-phonon coupling ofg 2 = −0.2J, unless explicitly denoted otherwise.
The spectral functions for the Hubbard dimer coupled to the phonon mode without cavity, 127 ω P = 0, are shown in Fig. 5 a.) and b.). We first focus on the case without hopping, J = 0, Fig. 5 a.). By construction the spectrum is particle-hole symmetric which is why we only show the lower part. The spectrum of the uncoupled Hubbard dimer is shown in gray exhibiting the well-known lower Hubbard band. When coupling the electrons to the phonons two shake-off bands at distances 2ω phon and 4ω phon from the main peak emerge. No side peaks at uneven multiples of the frequency ω phon are observed, which is a result of the non-linear coupling proportional to the squared phonon displacement X 2 phonon . 127 This coupling essentially squeezes the phonon which leads to the response of the system at twice the bare phonon frequency -a phenomenon that has previously been predicted and measured. 134,135 Additionally the bare value of U is slightly modified -the effect is however quite small and can hardly be seen. The small wiggles in the spectrum are an artefact of the Gaussian probe pulse.
The spectrum in the intermediate coupling case U = 5J, Fig. 5 b.) similarly exhibits side bands at distances that are compatible with multiples of 2ω phon from the main bands.
In fact, one would expect the shake-off peaks to appear at a slightly different distance due to the effective frequency of the phonons changing upon coupling to the electrons. This change, however, lies within 1% of the bare frequency (also see Sec. II) and can therefore not be discerned in the plot. Now allowing for a LMC larger that zero ω P > 0 the spectral function for the case of vanishing hopping J = 0 is shown in Fig. 5 c.). For ω P = 0 the spectrum coincides with that shown in Fig. 5 a.) exhibiting the previously discussed 2ω phon replica band. Upon turning on the coupling ω P > 0 we observe a split of this shake-off band into three separate peaks.
We mark in the plot distances from the lower Hubbard band that equal combinations of the polariton frequencies ω + and ω − namely 2ω + , 2ω − and ω + + ω − . These match the positions of all observed peaks well for all considered coupling strengths.
We show the details of the transformation of the coupling between electrons and phonons to a coupling between electrons and polaritons in Appendix F. The term in the resulting electron-polariton coupling proportional toX 2 + generates a replica peak at distance 2ω + , while the term proportional toX 2 − the one at distance 2ω − . Additionally a mixed term proportional toX +X− appears that generates the peak at ω + + ω − distance. Together this explains the splitting of the electronic shake-off peaks as a unique feature of the electrons coupling to the quadratic displacement of an IR-active phonon.

VI. DISCUSSION AND OUTLOOK
In this work we have investigated the effect of phonon polaritons on electronic interactions. We have considered two distinct coupling mechanisms between electrons of a strongly correlated material and IR-active phonons, which are in turn coupled to an optical resonator.
Our first finding is that the vacuum fluctuations of the cavity increase the effective electronelectron repulsion. This might open the path to control electronic interactions in a way that is to date only possible in cold-atom systems. 136 One possible application would be the triggering of a metal-to-insulator transition by increased rather than decreased electronic correlations. To date there are several examples of inducing an insulator-to-metal transition by driving. 128,[137][138][139][140][141] In particular a photo-induced insulator-to-metal transition was observed in in the one-dimensional Mott-insulator ET-F 2 TCNQ 128,141 for which the possibility of controlling electronic interactions through driving of an IR-active phonon with a laser has previously been demonstrated. 19,21 Similarly, effectively reduced correlations by electronic screening through laser-induced electronic excitations have been proposed theoretically 142,143 and reported experimentally 144,145 .
By contrast, we predict that coupling an IR-active phonon to the vacuum fluctuations of an optical cavity will increase electronic correlations, with the possibility of inducing a metal-to-insulator transition. However, more sophisticated calculations are needed to put our prediction on firmer ground. The effect of taking the thermodynamic limit should be investigated, 68,74 and a more detailed description of both the material as well as the cavity is needed -possibly by building on first principles methods that have recently been extended to cavity QED settings. [146][147][148] The range of realistically achievable changes of effective interactions depends on whether one considers a dark or a driven cavity. In a dark cavity, the relevant quantity is the achievable light-matter coupling strength. Provided that light-matter couplings in the ultrastrongcoupling regime can be attained with quantum materials, modifications of effective interactions in the few-percent range appear realistic. The situation is different in driven cavities.
For classically driven systems, changes in effective U of up to 10 percent or even more have been estimated. 9,21 Similarly large changes are found in our model simulations of a driven cavity. Therefore, we expect that significant light-induced changes (e.g., potentially cavity-induced superconductivity) might be possible in a driven cavity, presumably at laser intensities below the ones required without a cavity.
One question that has motivated our work is whether a cavity and phonon polaritons can be used to decrease electronic interactions to enable light-induced superconductivity in a similar manner as discussed in Ref. 9. Despite having practically ruled out this possibility using a dark cavity, a decrease of interactions is being achieved when driving the cavity.
We have investigated the behaviour in two distinct regimes: Once in the strong-coupling case where we have found a delayed response of the matter part with a time delay given by π ω P , where ω P is the splitting of the two polaritons frequencies; and once in the ultrastrongcoupling regime where we have found a prompt response of the matter system, Sec. IV. For further investigation one might promote the model for the matter degrees of freedom to a more sophisticated one. In a first step possibly, one could investigate a one-dimensional chain that would give access to studying the thermodynamic limit. 68,74 In order to research such a model for a sufficiently large system, full diagonalization is not feasible in general anymore due to the exponential growth of the computational cost in the system size. Instead one might revert to dynamical mean-field theory for correlated electron-boson systems 149,150 , tensor-network based methods 151,152 , or the more recently developed methods based on neural network quantum states. 153,154 For the model where the IR-active phonon is coupled to the local electronic density introduced in Ref. 16 a calculation using a 1D chain to model the electronic system as well as a classical drive of the phonons was performed 152 using the infinite time-evolving block decimation (iTEBD) 155 method. The authors found quick decoherence of the phonon motion and phonon-induced disorder in the electronic system. No superconductivity was observed. It would be interesting to investigate whether similar effects can be found when coupling the phonons to an optical cavity. In a more sophisticated model it would also be interesting to study the effects of heating of the material or a finite cavity life time. In our work we have found that large light-matter coupling might not be strictly necessary to achieve sizeable effects, but a light-matter coupling that exceeds losses might be sufficient. Such a strong-coupling regime has already been reached several decades ago 156,157 and can nowadays be realized in different platforms including array defect cavities 59 and semiconductor heterostructure cavities. 62 Recently also another interesting route to enhance superconducting fluctuations through a parametric drive of IR-active phonons -possibly with the use of an optical cavity -has been explored. 158 Another possible direction is to make a prediction that helps determine which of the two electron-phonon coupling mechanisms investigated in this work is dominant. In Ref. 19,21 the observed drop in reflectivity upon laser driving was explained by a coupling that involved the double occupancy of the electrons. It was, however, later realized in Ref. 16 and also becomes apparent from the findings in this work, see Appendix D, that the observations might also be explained by a coupling to the linear electronic density. Within our model we have not found qualitative differences between the two coupling mechanisms, neither for ground-state properties nor for their dynamical behavior. It might be necessary to explore larger system sizes and study the effect of the cavity on longer-range correlation functions and instabilities towards ordered phases in order to identify potential clear distinctions between the two coupling mechanisms.
Finally, we have investigated the electronic spectral function. For this we have focused on the coupling of the phonons to the double occupancy of the electrons and have derived a simplified model displaying qualitatively and quantitatively similar dynamics upon driving the cavity. We have identified a distinctive feature of the coupling between electrons and phonon polaritons stemming from IR-active phonons, namely the split-up of the observed shake-off bands into three bands. Such replica bands due to the coupling between electrons and phonons are well-known in the literature. 126 We note that while we have focussed here on a local, on-site photoemission spectrum without momentum resolution, the corresponding shake-off peaks are expected to appear in a similar fashion in a momentum-resolved ARPES spectrum. This is due to the fact that the long-wavelength photons carry zero momentum transfer compared to the size of the electronic Brillouin zone, thus leading to shake-off peaks separately for each electronic momentum (also see Ref. 68). To observe the split-up of the replica band due to the coupling to an optical resonator proposed here, the linewidth needs to be smaller than the splitting. For broadening stemming from losses intrinsic to the cavity setup this should be well within reach since the necessary condition is simply the strongcoupling condition. The question is therefore whether it is possible to achieve a sufficiently strong light-matter coupling to induce spectral weight in the polaritonic shake-off bands that can be detected by an ARPES experiment.   For the cavity driven system we set the LMC to ω P = 0.2J. Otherwise the used parameters are the same as those in Sec. IV and the time-evolution is calculated in the same way as in that section. For the classical phonon driving we consider the system uncoupled from the cavity thus setting ω P = 0. As initial state we take the the GS of the system -in this case that without the cavity coupled. The coherent drive is realized by adding a time-dependent term to the Hamiltonian that readŝ where we have usedX j,phon = 1 √ 2ω phon b † j +b j . We drive the system resonantly at the effective phonon frequency ω D = 2J. F (t) is a Gaussian pulse defined in the main part in Eq. (13). The bare strength of the pump F 0 cannot be compared directly between the classical and cavity drive since very different matrix elements enter in it: Once the coupling of a drive to the cavity; and once the coupling of the phonons to an external laser. Since we are interested in a qualitative comparison we simply take the same value for both cases Otherwise the parameters for the classically driven system are identical to those of the cavity coupled system. The results for the time-evolution are shown in Fig. 8.
In case of coupling to the double occupancy (shown on the right) the classical drive simply promotes the phonons into a coherent state that oscillates without any damping. The double occupancy of the electrons also starts oscillating, however, around an average value that is increased from its GS value indicating that the drive effectively decreases the electronelectron repulsion. The here shown plot can be directly compared to that obtained in Ref. 9 and Ref. 21 displaying essentially the same phenomenology, albeit without any damping.
In the case of the coupling to the linear density (shown on the left), the coherent state that the phonons are driven into is not as clean as for the quadratic coupling which we attribute to the fact that we here use a larger electron-phonon coupling. This shows in oscillations in the phonon number, that are larger than in the case of the coupling to the quadratic density. There also seem to be some overlaying oscillations in the evolution of the double occupancy that are however not reflected in the phonon number. Whether this is an intrinsic property of this coupling type or some artefact from the model remains unclear at this point.
Qualitatively, the phenomenology between the two couplings is, nevertheless, the same -the driving induces an increased phonon population that in turn leads to an oscillating doubleoccupancy that is, on average, increased. A linear coupling of the phonons to the local electronic density might therefore not be ruled out to explain the observations in Ref. 21 and Ref. 19 as was previously noted in Ref. 16.
Comparing these results to the cavity driven system, the most prominent difference is an overlaying oscillation between excitations of the phonons and consequently oscillations in the double occupancy; and excitations of the cavity. This is simply a beating motion of two coupled oscillators after initial displacement of one of the two (the photons in this case).
Otherwise the phenomenology is qualitatively similar. π and thus approximate the electron-phonon coupling aŝ H e−phon →H e−phon = 1 2 This leaves us with a model hosting only a single phonon mode coupling to the cavity with the same strength as beforeω P = ω P since the additional factor of √ 2 cancels the previously present 1 √ 2 term in Eq. (4). This phonon couples to the entire double occupancy of the electronsD =D 1 +D 2 with a coupling constant ofg 2 = 1 2 g 2 . To show that the approximate model has similar dynamical properties as the original one we again forward propagate the GS of each system in time under a coherent driving of the cavity mode. As parameters we use g 2 = −0.2J, ω P = 0.2J and correspondinglyg 2 = −0.1J.
We set the frequency of the effective model again such that we expect an effective phonon frequency of ω eff phon = 2J by choosing ω phon ≈ 2.01J (E4) (also see Appendix A). Otherwise the parameters are as in Sec. IV. The result are shown in Fig. 9.
Here, one can see that the dynamics for both models is comparable with two differences.
The effective frequency of the model hosting only a single phonon seems to be slightly higher than in the model hosting two phonons. This can be observed in the oscillations of the double occupancy but also in the less complete energy transfer of the cavity to the phonons that indicates that photons and phonons are not quite resonant. Additionally the beating frequency for the one-phonon model is slightly higher since the frequencies of the effective oscillators lie further apart. No attempt to correct this slight frequency mismatch was made. The second difference is the lower double occupancy of the one-phonon model in the GS seen at times preceding the pump. We attribute this to the fact that we dropped some terms in the electron-phonon coupling.
Overall our findings justify the approximation of neglecting the odd modeb ( †) π when investigating dynamics induced through the coupling to the cavity with an effective onemode model.
To diagonalize the Hamiltonian taking into account the last equations (Eq. (F4), Eq. (F5)), a rotation of −θ is applied onX phot andX phon giving respectivelyX + andX − -the canonical coordinate operators of the upper and lower polariton respectively. The same transformation is applied toP phot andP phon to giveP + andP − . Performing these transformations, the Hamiltonian can be expressed as: The polariton frequencies ω + and ω − have already been reported in the main text Eq. (5).
Defining raising and lowering operators for the upper (λ = +) and lower (λ = −) in the usual way we can write the diagonalized polariton Hamiltonian aŝ Using the polariton transformation we can write the coupling term between electrons and phonons Eq. (3) with the new α † ± and α ± operators: with u + (u − ) the phononic contribution of the upper (lower) polariton: Due to the transformation made in Eq. (F4) the canonical momenta of the polaritons now couple to the electrons instead of their displacement which is at this point just a matter of definition. Nevertheless, the bosonic operators appearing here show that both polaritons effectively couple to the electrons explaining the immediate response of the whole system to a drive in the USC regime discussed in Sec. IV. The presence of three coupling terms with different combinations of bosonic operators also explain the split of the shake-off peak in the electronic spectra into three peaks discussed in Sec. V.