Cavity engineering of Hubbard U via phonon polaritons

The ultrafast optical control of nonthermal phases of matter in quantum materials is a blossoming research field [1, 2]. Among the most intriguing experimental results are reports that suggest the possibility to induce transient superconducting-like states through laser driving [3–15]. This effect was observed in several classes of materials that share the common feature of a superconducting ground state (GS), implying the interpretation that the laser driving effectively raises the material's critical temperature. In order to explain the observed behaviour a number of different microscopic mechanisms were subsequently proposed [16–51]. However, to date no final and unifying conclusion could be drawn, neither on the nature of the transient states nor on the mechanism behind them.

From a practical point of view, a drawback of the transient superconducting-like states is their relatively short life time, typically in the picosecond range, with a recent extension to the nanosecond regime in K₃C₆₀ [52]. As an alternative route to control over material properties, light-matter coupling (LMC) in cavities has been suggested [53–57]. In these setups, instead of achieving strong modifications of material properties by strong driving, one focuses on realizing strong coupling (SC) between light and matter, supported by recent experimental advances [58–66]. 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 GS or energetically close to it, effects from detrimental heating are expected to be reduced and life times to be longer.

In general, photons couple to charged particles or excitations. Hence there are two main pathways to manipulate electronic properties of a material: either by employing the direct coupling of the light to the electrons; or by utilizing the coupling to other degrees of freedom of the system—for example lattice vibrations—that in turn couple to the electrons. Along the former path several studies have investigated cavity-induced phenomena both from a theoretical and experimental point of view, including superconductivity in which the photons of a cavity provide the pairing glue for the electrons similar to a Bardeen–Cooper–Schrieffer (BCS) description [70–73], suppression of the Drude peak [68, 74], superradiance [75–84] for which it remains an open question whether it can be realized in equilibrium, [85–90] coupling to magnetism [91–98] and in particular magnons, [99–104] excitons [105–111] forming exciton polaritons [112, 113] and the modification of topological states of matter [114–117]. 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 $X^{\,2}~n_\uparrow n_\downarrow$. 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 [9] a superconducting response of the system was only observed when driving specific phonon modes in the charge-transfer salt κ-(BEDT-TTF)₂Cu[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, $\mathcal{O}(1)$ photons. Similar to the case of classical driving of the phonons, this decreases electronic interactions. We find that in the SC but not ultra-strong coupling (USC) regime an increase of the 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 of experimentally measuring this feature.

We orient our modelling on the κ-salts discussed in [9] where they were described using a Hubbard model. Other molecular compounds, such as ET-F₂TCNQ studied in [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 reads

$$\begin{eqnarray} &&\hat{H}_{\textrm{e}^{-}} = - J \sum_{\sigma \in \{\uparrow, \downarrow\}} (\hat{c}_{1, \sigma}^{\dagger} \hat{c}_{2,\sigma} + h.c.) + U \sum_{j \in \{1,2\}} \left(\hat{n}^{\textrm{el}}_{j,\uparrow}-\frac{1}{2}\right) \left(\hat{n}^{\textrm{el}}_{j,\downarrow}-\frac{1}{2}\right). \end{eqnarray} \tag{ 1 }$$

Here, $\hat{c}_{j, \sigma}$ annihilates -; $\hat{c}_{j, \sigma}^{\dagger}$ creates an electron at one of the two sites $j \in \{1, 2\}$ with spin $\sigma \in \{\uparrow, \downarrow\}$. J denotes the hopping integral, U the onsite repulsion of the electrons and we used $\hat{n}^{\textrm{el}}_{j, \sigma} = \hat{c}_{j, \sigma}^{\dagger} \hat{c}_{j, \sigma}$.

We couple each site to an optically active phonon for which the bare Hamiltonian is expressed as

$$\begin{eqnarray} &&\hat{H}_{\textrm{phon}} = \sum_j \omega_{\textrm{phon}} \, \hat{b}_j^{\dagger} \hat{b}_j. \end{eqnarray} \tag{ 2 }$$

In this expression $\hat{b}_j$ annihilates -; $\hat{b}_j^{\dagger}$ 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 displacement $\hat{X}_{\textrm{phon}, j} = \frac{1}{\sqrt{2~\omega_{\textrm{phon}}}} \left(\hat{b}_j + \hat{b}_j^{\dagger} \right)$ 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 reads

$$\begin{equation} \hat{H}_{\textrm{phon}-\textrm{e}^{-}} = \sum_j g_1~\left( \hat{b}_j + \hat{b}_j^{\dagger} \right)^{\,2}~\left( \hat{n}^{\textrm{el}}_{j, \uparrow} + \hat{n}^{\textrm{el}}_{j, \downarrow} \right) + g_2~\left( \hat{b}_j + \hat{b}_j^{\dagger} \right)^{\,2}~\hat{n}^{\textrm{el}}_{j,\uparrow} \hat{n}^{\textrm{el}}_{j,\downarrow}. \end{equation} \tag{ 3 }$$

Here g₁ parametrizes the coupling of the phonons to the linear electronic density and g₂ 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 $g_1~\neq 0$ 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] and the bare photon energy and collecting the previously defined terms in equations (1)–(3)

$$\begin{equation} \begin{aligned} \hat{H} \,= \,& \hat{H}_{\textrm{e}^{-}} + \hat{H}_{\textrm{phon}} + \hat{H}_{\textrm{phon}-\textrm{e}^{-}} + \overbrace{\omega_{\textrm{phot}} \, \hat{a}^{\dagger} \hat{a}}^{\hat{H}_{\textrm{phot}}}\\ & +\underbrace{ \sum_j \left[ i\left(\frac{\omega_{\textrm{P}} \sqrt{\omega_{\textrm{phon}}}}{2~\sqrt{2} \sqrt{\omega_{\textrm{phot}}}}\right) \left( \hat{a} + \hat{a}^{\dagger} \right) \left( \hat{b}_j-\hat{b}_j^{\dagger} \right)\right] + \left( \frac{\omega_{\textrm{P}}^{\,2}}{4 \, \omega_{\textrm{phot}}} \right) \left(\hat{a} + \hat{a}^{\dagger}\right)^{\,2} }_{\hat{H}_{\textrm{phon}-\textrm{phot}}}. \end{aligned} \end{equation} \tag{ 4 }$$

Here, $\hat{a}$ annihilates -; $\hat{a}^{\dagger}$ 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 LMC. The model is illustrated in figure 1(a).

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The coupling between phonons and photons will lead to the formation of hybrid light-matter states, phonon polaritons. Their effective frequencies are calculated as [123] (see appendix F)

$$\begin{eqnarray} &&\omega_{\pm}^{\,2} = \frac{1}{2}\left(\omega_{\textrm{phot}}^{\,2}+ \omega_{\textrm{P}}^{\,2}+ \omega_{\textrm{phon}}^{\,2}~\pm \sqrt{\left(\omega_{\textrm{phot}}^{\,2}+ \omega_{\textrm{P}}^{\,2}+ \omega_{\textrm{phon}}^{\,2}\right)^{\,2} - 4\omega_{\textrm{phot}}^{\,2}\omega_{\textrm{phon}}^{\,2}} \right). \end{eqnarray} \tag{ 5 }$$

For identical phonon and photon frequency, the polariton frequencies are plotted as a function of the coupling ω_P in figure 1(b). We call the polariton with the effectively higher frequency $\omega_+$ the upper polariton and that with the effective lower frequency $\omega_-$ 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 $\omega_{\textrm{phon}}^{\textrm{eff}} \approx 2J$ that is composed the bare phonon-frequency and contributions stemming from the coupling to the electrons. In [19] it was shown for the molecular compound ET-F₂TCNQ 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

$$\begin{equation} g_1 = 0.5J ;\quad g_2 = 0. \end{equation} \tag{ 6 }$$

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 $\omega_{\textrm{phon}}^{\textrm{eff}} = 2J$. We find

$$\begin{equation} \omega_{\textrm{phon}} = \left(\sqrt{5} - 1~\right)J \approx 1.24J \end{equation} \tag{ 7 }$$

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 g₂ is expected to be negative [21] for the considered solids. We anticipate a somewhat smaller absolute value $|g_2| \lt |g_1|$ compared to the coupling to the linear electronic density (see equation (6)) and thus choose

$$\begin{equation} g_1 = 0 ;\quad g_2 = -0.2J. \end{equation} \tag{ 8 }$$

We note that the detailed values of these couplings do not fundamentally alter our conclusions. Choosing the bare phonon frequency as

$$\begin{equation} \omega_{\textrm{phon}} \approx 2.02J \end{equation} \tag{ 9 }$$

creates a resonance of the phonons at frequency 2 J (also see appendix A). We couple the phonons resonantly to the cavity and therefore set $\omega_{\textrm{phot}} = \omega_{\textrm{phon}}^{\textrm{eff}} = 2J$.

A classical drive of the phonons effectively decreases electron-electron interactions for both electron-phonon coupling mechanisms [16, 19, 21]. Here, we investigate the effect that the coupling of vacuum fluctuations of an optical cavity to an optical phonon have on the effective electron-electron repulsion. As a measure for the repulsion we compute the electronic double occupancy

$$\begin{equation} D = \langle \hat{D} \rangle = \langle \hat{D}_1 + \hat{D}_2~\rangle = \langle \hat{n}_{1, \uparrow}^{\textrm{el}} \hat{n}_{1, \downarrow}^{\textrm{el}} + \hat{n}_{2, \uparrow}^{\textrm{el}} \hat{n}_{2, \downarrow}^{\textrm{el}}\rangle. \end{equation} \tag{ 10 }$$

An increase in the double occupancy corresponds to a decrease in electronic interactions according to the notion that electrons repel each other less and vice versa. We set the temperature to T = 0 such that the expectation value in equation (10) is evaluated with respect to the GS. We obtain the GS via exact diagonalization (ED) introducing a cutoff N_B in the bosonic part of the Hilbert space. This is chosen as $N_{\textrm{B}} = 16$, and we have checked that all results are converged with respect to this cutoff. A more detailed analysis of the convergence in this parameter can be found in appendix B.

The results for different values of ω_P are shown in figure 1(c). Without a cavity ($\omega_{\textrm{P}} = 0$) the coupling to the phonons leads to a slight increase of the double occupancy for both coupling types—even without a coherent driving. The coupling of the cavity, however, reverses this effect and leads to a decrease of the double occupancy. From this observation one can deduce that the presence of the vacuum fluctuations of the cavity increases electronic interactions for both considered electron-phonon coupling mechanisms.

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

$$\begin{equation} \langle \hat{D} \rangle_{\textrm{therm}} = \frac{1}{Z}\sum_n \langle \psi_n | \hat{D} | \psi_n \rangle e^{-\beta E_n}, \end{equation} \tag{ 11 }$$

where $Z = \sum_n e^{-\beta E_n}$ is the partition function, E_n is the nth eigenenergy of the system, $| \psi_n \rangle$ the corresponding eigenstate, and $\beta = \frac{1}{k_B T}$ the inverse temperature. To obtain concrete temperature values we take $J = 80~\textrm{meV}$, which is the value found in ab initio simulations for the κ-salts performed in [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 figure 2. Overall, higher temperatures result in a reduction of the double-occupancy but the effect from the coupling to the cavity remains present.

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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 reads

$$\begin{equation} \hat{H}(t) = \hat{H} + F(t)\hat{A}_{\textrm{phot}}. \end{equation} \tag{ 12 }$$

Here, $\hat{H}$ is the Hamiltonian of the undriven system equation (4), $\hat{A}_{\textrm{phot}} = \frac{1}{\sqrt{2~\omega_{\textrm{phot}}}} \left( \hat{a}^{\dagger} + \hat{a} \right)$ is the quantized cavity field and F(t) is a pump pulse for which we choose a Gaussian envelope

$$\begin{equation} F(t) = F_0~\frac{1}{\sqrt{2~\pi} \sigma} \exp \left(- \frac{(t - t_0)^{\,2}}{2~\sigma^{\,2}} \right) \sin(\omega_{\textrm{Drive}} t). \end{equation} \tag{ 13 }$$

We use $t_0 = \frac{16~\pi}{\omega_{\textrm{phot}}}$, $\sigma = \frac{4~\pi}{\omega_{\textrm{phot}}}$ and $\frac{F_0}{\sqrt{2~\omega_{\textrm{phot}}}} = \frac{3J}{2}$ 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 [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 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 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 $\omega_{\textrm{P}} = \frac{\omega_{\textrm{phot}}}{40}$ and $\omega_{\textrm{P}} = \frac{\omega_{\textrm{phot}}}{20}$; and one value within the USC regime of $\omega_{\textrm{P}} = \frac{\omega_{\textrm{phot}}}{2}$.

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 equation (3)) is shown in figure 3 and that for the coupling of the phonons to the double occupancy ($g_1 = 0$ and $g_2 = -0.2J$—also see equation (3)) of the electrons in figure 4. Both coupling mechanisms display qualitatively similar behaviour. In the case of strong, but not USC, the pump drives the cavity into an excited state with an increased photon number $N_{\textrm{phot}} = \hat{a}^{\dagger} \hat{a}$ within the time duration of the pump. The strength of the drive is such that only few photons $N_{\textrm{phot}} = \mathcal{O}(1)$ are created. The energy of the photon excitation is subsequently completely transferred to the phonons on a time scale that is approximately $\frac{\pi}{\omega_{\textrm{P}}}$, 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 $\tau_{\textrm{beat}} \approx \frac{4~\pi}{\omega_P}$.

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In the USC case $\omega_{\textrm{P}} = \frac{\omega_{\textrm{phot}}}{2}$, driving the cavity at its bare resonance frequency $\omega_{\textrm{Drive}} = \omega_{\textrm{phot}}$ only yields a weak response. However, when driving at an increased frequency of $\omega_{\textrm{Drive}} = 2.56J = \omega_+$ that coincides with the upper polariton frequency $\omega_+$ or a decreased frequency of $\omega_{\textrm{Drive}} = 1.56J = \omega_-$ coinciding with the lower polariton frequency $\omega_-$ 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 $N_{\textrm{phon}} = \sum_j \hat{b}_j^{\dagger} \hat{b}_j$ 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,

$$\begin{equation} \begin{aligned} \hat{H}_{\textrm{phon}-\textrm{phot}} & = i \omega_{\textrm{P}} \frac{\sqrt{\omega_{\textrm{phon}}}}{2~\sqrt{2}\sqrt{\omega_{\textrm{phot}}}} \left( \hat{a}^{\dagger} + \hat{a} \right) \sum_j \left( \hat{b}_j - \hat{b}_j^{\dagger} \right) + \frac{{\omega_{\textrm{P}}^{\,2}}}{4~\omega_{\textrm{phot}}} \left( \hat{a}^{\dagger} + \hat{a} \right)^{\,2} \\ & = i \omega_{\textrm{P}} \frac{\sqrt{\omega_{\textrm{phon}}}}{2~\sqrt{2}\sqrt{\omega_{\textrm{phot}}}} \left( \hat{a}^{\dagger} + \hat{a} \right) \sqrt{2} \left( \hat{b}_0 - \hat{b}_0^{\dagger} \right) + \frac{{\omega_{\textrm{P}}^{\,2}}}{4~\omega_{\textrm{phot}}} \left( \hat{a}^{\dagger} + \hat{a} \right)^{\,2}, \end{aligned} \end{equation} \tag{ 14 }$$

where we have introduced the even combination of bosonic operators

$$\begin{equation} \hat{b}_0^{(\dagger)} = \frac{1}{\sqrt{2}} \left( \hat{b}^{(\dagger)}_1 + \hat{b}^{(\dagger)}_2~\right), \end{equation} \tag{ 15 }$$

to which a complementary odd combination exists,

$$\begin{equation} \hat{b}_{\pi}^{(\dagger)} = \frac{1}{\sqrt{2}} \left( \hat{b}^{(\dagger)}_1 - \hat{b}^{(\dagger)}_2~\right). \end{equation} \tag{ 16 }$$

In the strong, but not ultra-strong, coupling regime the two oscillators are weakly coupled when comparing with their bare frequency $\omega_{\textrm{P}} \ll \omega_{\textrm{phot}}$ and $\omega_{\textrm{P}} \ll \omega_{\textrm{phon}}$. 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 $\tau_{\textrm{beat}} = \frac{4~\pi}{\omega_{\textrm{P}}}$, 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 $\frac{\tau_{\textrm{beat}}}{4} = \frac{\pi}{\omega_{\textrm{P}}}$. This matches well with the observations in figures 3 and 4.

In the USC regime light and matter excitations are completely hybridized forming phonon polaritons: One upper polariton with an increased effective frequency of $\omega_+ \approx 2.56J$; and one lower polariton with a decreased effective frequency of $\omega_- \approx 1.56J$ according to equation (5). This explains why only a small response is observed when driving the system at its bare resonance frequency $\omega_{\textrm{phot}} = 2J$—one simply drives the effective oscillators off-resonantly. When the polaritons are driven at their true resonances instead, with $\omega_{\textrm{Drive}} = \omega_+$ or $\omega_{\textrm{Drive}} = \omega_-$, 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 $\omega_{\textrm{P}} = \frac{\omega_{\textrm{phot}}}{20}$ and $\omega_{\textrm{P}} = \frac{\omega_{\textrm{phot}}}{40}$ the response of the system is only triggered with a certain time delay $\frac{\pi}{\omega_{\textrm{P}}}$. In a realistic setup, in order to get a sizeable effect, one thus need a LMC of

$$\begin{equation} \omega_{\textrm{P}} \gt \gamma_{\textrm{loss}}. \end{equation} \tag{ 17 }$$

This is precisely the definition of the SC regime [54, 132]. Only increasing the LMC compared to the bare cavity frequency does not necessarily yield a larger effect as becomes apparent both from figures 3 and 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 $\omega_{\textrm{P}} \gt \gamma_{\textrm{loss}}$ but the challenge lies in reaching a LMC that is of comparable size to the bare cavity frequency $\omega_{\textrm{P}} \approx \omega_{\textrm{phot}}$. 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.

To further understand the effects of the cavity on electronic properties, we investigate the spectral function for single-particle excitations of the system. To this end, we focus on the coupling to the quadratic density ($g_1 = 0$, $g_2~\neq 0$) since both coupling mechanisms yield similar GS as well as dynamical properties (see sections 3 and 4). The model including only the coupling to the quadratic density allows for a further simplification when investigating changes induced through the presence of the cavity. The cavity only couples to the even superposition of the two phonon modes, as already noted in equation (14). We thus neglect the complementary bosonic mode that is given by the odd combination of phonon excitations. In appendix E we show that this leads to an effective model in which a single boson couples to the total double occupancy of the electrons $\hat{D}$ equation (10) with decreased strength $\tilde{g}_2 = 1 / 2 \, g_2$ and the same coupling to the cavity mode $\tilde{\omega}_{\textrm{P}} = \omega_{\textrm{P}}$. The dynamics induced through the cavity in this model is qualitatively the same as that in the full model equation (4). Additionally, we restore the particle-hole symmetry of the system by writing the electron-phonon coupling as

$$\begin{equation} \hat{H}_{\textrm{phon}-\textrm{e}^{-}} = \tilde{g}_2~\left(\hat{b}_+^{\dagger} + \hat{b}_+ \right)^{\,2}~\sum_j \left(\hat{n}^{\textrm{el}}_{j \uparrow} - \frac{1}{2} \right)\left(\hat{n}^{\textrm{el}}_{j \downarrow} - \frac{1}{2} \right). \end{equation} \tag{ 18 }$$

The operator added in the coupling to the electrons in this way only acts like the identity on 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],

$$\begin{align} A(\omega,t_0) = \textrm{Re} \int dt_1 dt_2 \, e^{i\omega (t_1-t_2)} \, s_{t_1,t_2,\tau}(t_0) & \Big[\langle{\Psi(t_2)}|\hat{c}^{\dagger}_{1,\uparrow}\mathcal{T} e^{-i \hat{H} (t_2 - t_1)} \, \hat{c}_{1,\uparrow}|{\Psi(t_1)}\rangle \nonumber \\ & +\langle{\Psi(t_1)}|\hat{c}_{1,\uparrow}\mathcal{T} e^{-i\hat{H} (t_1 - t_2)} \, \hat{c}^{\dagger}_{1,\uparrow}|{\Psi(t_2)}\rangle\Big], \end{align} \tag{ 19 }$$

with $\mathcal{T}$ 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 $| \psi_{\textrm{GS}} \rangle$, which is determined via ED. Here $s_{t_1,t_2,\tau}(t_0)$ denotes a Gaussian probe pulse defined as

$$\begin{equation} s_{t_1,t_2,\sigma}(t_0) = (2\pi^{3/2}\sigma)^{-1} \, \textrm{exp}(-(t_1-t_0)^{\,2}/(2\sigma^{\,2})) \, \textrm{exp}(-(t_2-t_0)^{\,2}/2\sigma^{\,2}). \end{equation} \tag{ 20 }$$

In the following we take $\sigma = 1.6 / J$ and $t_0 = 5 / J$. As parameters for the model we set J = 1 and $U = 5J$, take $\omega_{\textrm{phot}} = 2J$ and an electron-phonon coupling of $\tilde{g}_2 = -0.2J$, unless explicitly denoted otherwise.

The spectral functions for the Hubbard dimer coupled to the phonon mode without cavity [127], $\omega_{\textrm{P}} = 0$, are shown in figures 5(a) and (b). We first focus on the case without hopping, J = 0, figure 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 $2\omega_{\textrm{phon}}$ and $4~\omega_{\textrm{phon}}$ 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 $X_{\textrm{phon}on}^{\,2}$ [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.

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The spectrum in the intermediate coupling case $U = 5J$, figure 5(b) similarly exhibits side bands at distances that are compatible with multiples of $2~\omega_{\textrm{phon}}$ 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 section 2) and can therefore not be discerned in the plot.

Now allowing for a LMC larger that zero $\omega_{\textrm{P}} \gt 0$ the spectral function for the case of vanishing hopping J = 0 is shown in figure 5(c). For $\omega_{\textrm{P}} = 0$ the spectrum coincides with that shown in figure 5(a) exhibiting the previously discussed $2\omega_{\textrm{phon}}$ replica band. Upon turning on the coupling $\omega_{\textrm{P}} \gt 0$ 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 $\omega_+$ and $\omega_-$ namely $2\omega_+$, $2\omega_-$ and $\omega_+ + \omega_-$. These match the positions of all observed peaks well for all considered coupling strengths.

Essentially the same phenomenology is observed in the intermediate coupling $U = 5J$ case, figure 5(d). Here the situations complicated by the natural appearance of two peaks in the lower part of the spectrum of the uncoupled Hubbard dimer. Still one can observe the $2\omega_{\textrm{phon}}$ replica from both peaks and also track their subsequent split-up into three separate peaks. We again mark distances to the two main peaks consistent with the same combinations of polariton frequencies $2\omega_+$, $2\omega_-$ and $\omega_+ + \omega_-$ that match the appearing peaks well.

Now we explain the split-up with the formation of polaritons. The displacement of the phonon $\hat{X}_{\textrm{phon}}$ can be expressed in terms of a linear combination of the displacement of the upper and lower polariton mode, $\hat{X}_{\textrm{phon}} = c_+ \hat{X}_+ + c_- \hat{X}_-$ where $c_+$ and $c_-$ are two real numbers. Accordingly the quadratic displacement of the phonons that couples to the electrons equation (3) transforms under a polariton transformation according to

$$\begin{equation} \hat{X}_{\textrm{phon}}^{\,2} = c_+^{\,2}~\hat{X}_+^{\,2} + c_-^{\,2}~\hat{X}_-^{\,2} + 2~c_+ c_- \hat{X}_+ \hat{X}_-. \end{equation} \tag{ 21 }$$

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 to $\hat{X}_+^{\,2}$ generates a replica peak at distance $2\omega_+$, while the term proportional to $\hat{X}_-^{\,2}$ the one at distance $2\omega_-$. Additionally a mixed term proportional to $\hat{X}_+\hat{X}_-$ appears that generates the peak at $\omega_+ + \omega_-$ 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.

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 electron-electron 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, 137–141]. In particular a photo-induced insulator-to-metal transition was observed in in the one-dimensional Mott-insulator ET-F₂TCNQ [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 [146–148].

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 LMC strength. Provided that LMCs in the ultrastrong-coupling 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% 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 [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 $\frac{\pi}{\omega_{\textrm{P}}}$, where ω_P is the splitting of the two polaritons frequencies; and once in the ultrastrong-coupling regime where we have found a prompt response of the matter system, section 4. 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 [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 [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 LMC might not be strictly necessary to achieve sizeable effects, but a LMC 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 [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 [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 behaviour. 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 [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 strong-coupling condition. The question is therefore whether it is possible to achieve a sufficiently strong LMC to induce spectral weight in the polaritonic shake-off bands that can be detected by an ARPES experiment.

We thank Nicolas Tancogne-Dejean and Rashmi Singla for fruitful discussions. We thank Damian Hofmann for help with optimizing the computer code. Financial support by the DFG through the Emmy Noether program (SE 2558/2) is gratefully acknowledged. D M K acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through RTG 1995 and under Germany's Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769. D M K and M A S acknowledge support from the Max Planck-New York City Center for Non-Equilibrium Quantum Phenomena.

Code and data are publicly available at https://github.com/ce335805/PolaritonSqueezing/tree/submissionReference.

In this part we explain how we obtain the bare phonon-frequency that is a parameter in the Hamiltonian in order to obtain an effective phonon-frequency that is resonant with the cavity photon at $\omega_{\textrm{phot}} = 2J$. We start by considering the coupling of the phonons to the linear electronic density and write the corresponding electron-phonon Hamiltonian as well as the bare phonon Hamiltonian (compare to equations (2) and (3) of the main part)

$$\begin{equation} \hat{H}_{\textrm{phon}} + \hat{H}_{\textrm{phon}-\textrm{e}^{-}} = \sum_j \omega_{\textrm{phon}} \hat{b}_j^{\dagger} \hat{b}_j + \sum_j g_1~\left( \hat{b}_j + \hat{b}_j^{\dagger} \right)^{\,2}~\left( \hat{n}^{\textrm{el}}_{j, \uparrow} + \hat{n}^{\textrm{el}}_{j, \downarrow} \right). \end{equation} \tag{ A1 }$$

Next we introduce canonical coordinates and momenta

$$\begin{equation} \begin{aligned} \hat{X}_j & = \frac{1}{\sqrt{2~\omega_{\textrm{phon}}}} \left( \hat{b}_j + \hat{b}_j^{\dagger} \right)\\ \hat{P}_j & = i\frac{\sqrt{\omega_{\textrm{phon}}}}{\sqrt{2}}\left( \hat{b}_j^{\dagger} - \hat{b}_j \right) \end{aligned} \end{equation} \tag{ A2 }$$

and by inserting get

$$\begin{equation} \hat{H}_{\textrm{phon}} + \hat{H}_{\textrm{phon}-\textrm{e}^{-}} = \sum_j \frac{1}{2} \hat{P}_j^{\,2} + \frac{1}{2} \left( \omega_{\textrm{phon}}^{\,2} + 4~g_1~\omega_{\textrm{phon}} \left( \hat{n}^{\textrm{el}}_{j, \uparrow} + \hat{n}^{\textrm{el}}_{j, \downarrow} \right) \right) \hat{X}_j^{\,2}. \end{equation} \tag{ A3 }$$

We now define the effective phonon frequency $\omega_{\textrm{phon}}^{\textrm{eff}}$ as the term multiplying the canonical coordinate of the phonons $\hat{X}_j$ without the one-half. This contains an operator acting on the purely electronic part of the Hilbertspace of which we take the average to obtain a meaningful frequency. We set this equal to the anticipated value of $\omega_{\textrm{phon}}^{\textrm{eff}} = 2J$ and solve for the bare parameter ω_phon in the Hamiltonian

$$\begin{equation} \begin{aligned} \left(\omega_{\textrm{phon}}^{\textrm{eff}}\right)^{\,2} &: = \langle \omega_{\textrm{phon}}^{\,2} + 4~g_1~\omega_{\textrm{phon}} \left( \hat{n}^{\textrm{el}}_{j, \uparrow} + \hat{n}^{\textrm{el}}_{j, \downarrow} \right) \rangle_{\textrm{electronic}} = \omega_{\textrm{phon}}^{\,2} + 4~g_1~\omega_{\textrm{phon}} \overset{!}{ = } 4J^{\,2}\\ &\overset{g_1 = 0.5J}{\Rightarrow} \omega_{\textrm{phon}} = (\sqrt{5} - 1)J \approx 1.24J. \end{aligned} \end{equation} \tag{ A4 }$$

In the same way we determine the bare phonon frequency when coupling to the double occupancy of the electrons where we write for the part of the Hamiltonian only containing phonon and electron degrees of freedom

$$\begin{equation} \hat{H}_{\textrm{phon}} + \hat{H}_{\textrm{phon}-\textrm{e}^{-}} = \sum_j \frac{1}{2} \hat{P}_j^{\,2} + \frac{1}{2} \left( \omega_{\textrm{phon}}^{\,2} + 4~g_2~\omega_{\textrm{phon}} \hat{D}_j \right) \hat{X}_j^{\,2}. \end{equation} \tag{ A5 }$$

Hence we can obtain the effective phonon frequency as

$$\begin{equation} \begin{aligned} \left(\omega_{\textrm{phon}}^{\textrm{eff}}\right)^{\,2} &: = \langle \omega_{\textrm{phon}}^{\,2} + 4~g_2~\omega_{\textrm{phon}} \hat{D}_j \rangle_{\textrm{electronic}} = \omega_{\textrm{phon}}^{\,2} + 4~g_2~\omega_{\textrm{phon}} \langle \hat{D}_j \rangle_{\textrm{electronic}} \overset{!}{ = } 4J^{\,2}\\ &\overset{g_2 = -0.2J}{\Rightarrow} \omega_{\textrm{phon}} = 0.4J \langle \hat{D}_j \rangle_{\textrm{electronic}} + 2J\sqrt{0.04~\langle \hat{D}_j \rangle_{\textrm{electronic}}^{\,2} + 1}. \end{aligned} \end{equation} \tag{ A6 }$$

We calculate the expectation value of the double occupancy $\langle \hat{D}_j \rangle$ for the uncoupled Hubbard dimer with which we obtain for the bare phonon frequency

$$\begin{equation} \omega_{\textrm{phon}} \approx 2.02J. \end{equation} \tag{ A7 }$$

In this part we check the convergence of the ground-state (GS) properties, in particular the double occupancy, in the chosen cutoff of the bosonic Hilbert space N_B. We take the same model parameters as in section 3 and consider both coupling mechanisms between phonons and electrons discussed in the main part. The LMC is, however, considered up to $\omega_{\textrm{P}} = 6 \, \omega_{\textrm{phot}}$, to show how convergence depends on the coupling strength. The results are shown in figure 6. The double occupancy of the electrons is well converged with a bosonic cutoff of $N_{\textrm{B}} = 16$ as used in the main part for LMC of $\omega_{\textrm{P}} \leqslant 3 \, \omega_{\textrm{phot}}$. Further beyond that point, around $\omega_{\textrm{P}} \approx 4~\omega_{\textrm{phot}}$ one starts to see deviations between a cutoff of $N_{\textrm{B}} = 16$ and smaller values—indicating the incomplete convergence at this point. This result is also further underlined by the number of bosons in the system that remains below 1 for the values of the LMC used in the main part of the paper.

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In section 4 of the main part we propagate the GS of the full coupled system forward in time with the time-dependent Hamiltonian containing an additional coherent drive equation (12) according to

$$\begin{equation} |\psi(t) \rangle = \mathcal{T} e^{-i\int_0^{t} \hat{H}(t^{^{\prime}}) dt^{^{\prime}}} |\psi_{\textrm{GS}} \rangle \end{equation} \tag{ C1 }$$

where $| \psi_{\textrm{GS}} \rangle$ is the GS of the system. We approximate the exact time evolution equation (C1) using finite time-steps $\delta t$ choosing N_t steps within one driving period leading to $\delta t = \frac{2~\pi}{\omega_{\textrm{Drive}} N_t}$. The time evolution is then computed via the commutator-free scheme introduced in [131] for a single time-step according to

$$\begin{equation} |\psi(t + \delta t) \rangle = \mathcal{T} e^{-i\int_t^{t + \delta t} \hat{H}(t^{^{\prime}}) dt^{^{\prime}}} |\psi_{\textrm{GS}} \rangle \approx e^{-i \left( c_1~\hat{H}_1 + c_2~\hat{H}_2~\right)\delta t} e^{-i \left( c_2~\hat{H}_1 + c_1~\hat{H}_2~\right)\delta t}|\psi_{\textrm{GS}} \rangle \end{equation} \tag{ C2 }$$

where

$$\begin{equation} c_{1 / 2} = \frac{3~\mp 2~\sqrt{3}}{12}; \quad \hat{H}_{1 / 2} = \hat{H}\left(t + \left( \frac{1}{2} \mp \frac{\sqrt{3}}{6} \right) \delta t\right). \end{equation}$$

We investigate the convergence of this scheme both in the cutoff used for the bosonic part of the Hilbert space N_B and the length of the time-step $\delta t$—or the equivalently the number of time-points within one driving period N_t . We consider the double occupancy equation (10) since this is the quantity we are mainly interested in. Additionally, we found that bosonic quantities like N_phot or N_phon also considered in the main part usually converge much better with respect to the chosen bosonic cutoff or finite time-step. We focus on later times at the end of the pump pulse, since an occurring error might build up over time and choose the same parameters as considered in section 4 in the main text. As for the LMC we take $\omega_{P} = 0.2J = 0.1 \, \omega_{\textrm{phot}}$ which is on the verge of the USC regime. Overall, we checked that the convergence is similar for different values of the LMC.

The results of our analysis are shown in figure 7. For the convergence in the bosonic cutoff N_B, results obtained with the coupling to the double occupancy ($g_1 = 0$, $g_2 = -0.2$) are completely converged taking $N_{\textrm{B}} = 6$. For the coupling to the linear density convergence is significantly slower, however beyond $N_{\textrm{B}} = 8$ changes are relatively small and only quantitative; the qualitative behaviour remains unchanged. We conclude that a bosonic cutoff of $N_{\textrm{B}} = 10$ is sufficient for our means.

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Both coupling mechanisms converge similarly fast in the length of the finite time-step $\delta t$ where we find that $N_t = 80$ steps during a single driving cycle suffice to obtain converged results.

In this part we compare the time evolution of the system for the two different coupling mechanisms between electrons and phonons: once under a classical drive of the phonons and no coupling to the cavity ($\omega_{\textrm{P}} = 0$) and once when coupling to a driven cavity. This serves the purpose of comparing our results for the driven Cavity to earlier works where a classical drive of the phonons has been considered [16, 21] but also illustrates that both coupling mechanism in fact behave similarly under a classical drive of the phonons.

For the cavity driven system we set the LMC to $\omega_{\textrm{P}} = 0.2J$. Otherwise the used parameters are the same as those in section 4 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 $\omega_{\textrm{P}} = 0$. 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 reads

$$\begin{equation} \hat{H}_{\textrm{Drive}} = F(t) \frac{1}{\sqrt{2}} \left( \hat{X}_{1, \textrm{phon}} + \hat{X}_{2, \textrm{phon}} \right) \end{equation} \tag{ D1 }$$

where we have used $\hat{X}_{j, \textrm{phon}} = \frac{1}{\sqrt{2~\omega_{\textrm{phon}}}} \left( \hat{b}_j^{\dagger} + \hat{b}_j \right)$. We drive the system resonantly at the effective phonon frequency $\omega_{\textrm{D}} = 2J$. F(t) is a Gaussian pulse defined in the main part in equation (13). The bare strength of the pump F₀ 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 namely $\frac{1}{\sqrt{2~\omega_{\textrm{phot}}}} F_0 = \frac{3J}{2}$. 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 figure 8.

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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 electron-electron repulsion. The here shown plot can be directly compared to that obtained in [9] and [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 double-occupancy 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 [21] and [19] as was previously noted in [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.

In this part we discuss a simplification of the model in the case of the phonons coupling to the double occupancies of the electrons, i.e. the case of $g_1 = 0$ and $g_2~\neq 0$. We start by noting that the cavity only couples to the sum of the phonon displacements as already shown in section 4 where we already introduced the even combination of phonons that are annihilated (created) by $\hat{b}_0^{(\dagger)}$ equation (15) and odd combinations correspondingly being annihilated (created) by $\hat{b}_{\pi}^{(\dagger)}$ equation (16). We here reconsider the coupling between electrons and phonons equation (3) and also write it in terms of the even and odd phonon modes, yielding

$$\begin{eqnarray} &&\hat{H}_{\textrm{e}-\textrm{phon}} = \dfrac{1}{2} g_2~\left( \hat{b}_0^{\dagger} + \hat{b}_0\right)^{\,2}~\left( \hat{D}_1 + \hat{D}_2~\right) + \frac{1}{2} g_2~\left( \hat{b}_{\pi}^{\dagger} + \hat{b}_{\pi}\right)^{\,2}~\left( \hat{D}_1 + \hat{D}_2~\right) \end{eqnarray} \tag{ E1 }$$

$$\begin{eqnarray} &&+ g_2~\left( \hat{b}_0^{\dagger} + \hat{b}_0\right)\left( \hat{b}_{\pi}^{\dagger} + \hat{b}_{\pi}\right) \left( \hat{D}_1 - \hat{D}_2~\right). \end{eqnarray} \tag{ E2 }$$

Since equation (14) shows that only the even mode $\hat{b}_0^{(\dagger)}$ couples to the cavity and we are mainly interested in dynamics induced through the cavity we neglect the coupling to the odd mode $\hat{b}_{\pi}^{(\dagger)}$ and thus approximate the electron-phonon coupling as

$$\begin{equation} \hat{H}_{\textrm{e}-\textrm{phon}} \rightarrow \tilde{\hat{H}}_{\textrm{e}-\textrm{phon}} = \frac{1}{2} {g_2} \left( \hat{b}_0^{\dagger} + \hat{b}_0\right)^{\,2}~\hat{D}. \end{equation} \tag{ E3 }$$

This leaves us with a model hosting only a single phonon mode coupling to the cavity with the same strength as before $\tilde{\omega}_{\textrm{P}} = \omega_{\textrm{P}}$ since the additional factor of $\sqrt{2}$ cancels the previously present $\frac{1}{\sqrt{2}}$ term in equation (4). This phonon couples to the entire double occupancy of the electrons $\hat{D} = \hat{D}_1 + \hat{D}_2$ with a coupling constant of $\tilde{g}_2 = \frac{1}{2} g_2$.

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 $g_2 = -0.2J$, $\omega_{\textrm{P}} = 0.2J$ and correspondingly $\tilde{g}_2 = -0.1J$. We set the frequency of the effective model again such that we expect an effective phonon frequency of $\omega_{\textrm{phon}}^{\textrm{eff}} = 2J$ by choosing

$$\begin{equation} \omega_{\textrm{phon}} \approx 2.01J \end{equation} \tag{ E4 }$$

(also see appendix A). Otherwise the parameters are as in section 4. The result are shown in figure 9.

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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 mode $\hat{b}_{{\pi}}^{(\dagger)}$ when investigating dynamics induced through the coupling to the cavity with an effective one-mode model.

We show in this section a basic polaritonic transformation for two coupled oscillators modelling phonon and photon degrees of freedom of a system. We define canonical position and momentum operators for the photons and phonons

$$\begin{align} \hat{X}_{\textrm{phot}} &\equiv \sqrt{\frac{1}{2\omega_{\textrm{phot}}}}\left(\hat{a}+\hat{a}^{\dagger}\right) \, , \, \, \hat{P}_{\textrm{phot}} \equiv -i\sqrt{\frac{\omega_{\textrm{phot}}}{2}}\left(\hat{a}-\hat{a}^{\dagger}\right) \end{align} \tag{ F1 }$$

$$\begin{align} \hat{X}_{\textrm{phon}} &\equiv \sqrt{\frac{1}{2\omega_{\textrm{phon}}}}\left(\hat{b}+\hat{b}^{\dagger}\right) \, , \, \, \hat{P}_{\textrm{phon}} \equiv -i\sqrt{\frac{\omega_{\textrm{phon}}}{2}}\left(\hat{b}-\hat{b}^{\dagger}\right). \end{align} \tag{ F2 }$$

The total Hamiltonian of the coupled system can then be rewritten as (also compare with equation (4)):

$$\begin{align} {\hat{H}} = \frac{1}{2} {\Big(\hat{P}_{{\textrm{phot}}}^{\,2} + \hat{P}_{{\textrm{phon}}}^{\,2} + (\omega_{\textrm{phot}}^{\,2} + \omega_{\textrm{P}}^{\,2})\hat{X}_{{\textrm{phot}}}^{\,2} + \omega_{\textrm{phon}}^{\,2}~\hat{X}_{{\textrm{phon}}}^{\,2}-2\omega_{\textrm{P}}\hat{X}_{{\textrm{phot}}}\hat{P}_{{\textrm{phon}}}\Big)}. \end{align} \tag{ F3 }$$

It is now possible to diagonalize this Hamiltonian defining [123]:

$$\begin{align} \hat{\tilde{P}}_{{\textrm{phon}}} \equiv {\omega_{\textrm{phon}}} \, \hat{X}_{{\textrm{phon}}} \, , \, \, \hat{\tilde{X}}_{{\textrm{phon}}} \equiv -{\omega_{\textrm{phon}}}^{-1} \, \hat{P}_{{\textrm{phon}}}. \end{align} \tag{ F4 }$$

Next we define the angle θ as

$$\begin{align} &\cos(\theta) = \frac{\Sigma}{\sqrt{1+\Sigma^{2}}} \end{align} \tag{ F5 }$$

$$\begin{align} \Sigma & = \frac{\omega_{\textrm{phot}}^{\,2}+\omega_{\textrm{P}}^{\,2}-\omega_{\textrm{phon}}^{\,2}+\sqrt{(\omega_{\textrm{phot}}^{\,2}+\omega_{\textrm{P}}^{\,2}+\omega_{\textrm{phon}}^{\,2})^{\,2}-4\omega_{\textrm{phot}}^{\,2}\omega_{\textrm{phon}}^{\,2}}}{2\omega_{\textrm{phon}}\omega_{\textrm{P}}}. \end{align} \tag{ F6 }$$

To diagonalize the Hamiltonian taking into account the last equation (F4) and (F5), a rotation of $-\theta$ is applied on $\hat{X}_{{\textrm{phot}}}$ and $\hat{\tilde{X}}_{{\textrm{phon}}}$ giving respectively $\hat{X}_{+}$ and ${\hat{X}}_-$—the canonical coordinate operators of the upper and lower polariton respectively. The same transformation is applied to $\hat{P}_{{\textrm{phot}}}$ and $\hat{\tilde{P}}_{\textrm{phon}}$ to give $\hat{P}_{+}$ and ${\hat{P}}_{-}$. Performing these transformations, the Hamiltonian can be expressed as:

$$\begin{align} \hat{H} & = \frac{1}{2}{{\begin{pmatrix} \hat{P}_{+} & {\hat{P}}_{-} \end{pmatrix}} {{\begin{pmatrix} 1 & 0 \\ 0 & 1~\end{pmatrix}}} {\begin{pmatrix} \hat{P}_{+} \\ {\hat{P}}_{-} \end{pmatrix}}} + \frac{1}{2}{{\begin{pmatrix} \hat{X}_{+} & {\hat{X}}_{-} \end{pmatrix}} {{\begin{pmatrix} \omega_{+}^{\,2} & 0 \\ 0 & \omega_{-}^{\,2}~\end{pmatrix}}} {\begin{pmatrix} \hat{X}_{+} \\ {\hat{X}}_{-} \end{pmatrix}}}. \end{align} \tag{ F7 }$$

The polariton frequencies $\omega_+$ and $\omega_-$ have already been reported in the main text equation (5). Defining raising and lowering operators for the upper ($\lambda = +$) and lower ($\lambda = -$) in the usual way we can write the diagonalized polariton Hamiltonian as

$$\begin{align} \hat{H} = \sum_{\lambda = \pm}{\omega_{\lambda}\hat{\alpha}^{\dagger}_{\lambda}\hat{\alpha}_{\lambda}}. \end{align} \tag{ F8 }$$

Using the polariton transformation we can write the coupling term between electrons and phonons equation (3) with the new $\alpha_{\pm}^{\dagger}$ and $\alpha_{\pm}$ operators:

$$\begin{align} g_2 \, \left(\hat{b}+\hat{b}^{\dagger}\right)^{\,2} \, \sum_{i = 1,2} \left(\hat{n}^{\textrm{el}}_{i,\uparrow}-\frac{1}{2}\right) \left(\hat{n}^{\textrm{el}}_{i,\downarrow}-\frac{1}{2}\right) = &-g_2\left( u_+ \left(\hat{\alpha}_+-\hat{\alpha}_{+}^{\dagger}\right)^{\,2} + u_- \left(\hat{\alpha}_- -\hat{\alpha}_-^{\dagger}\right)^{\,2}\right.\nonumber\\ &\left. + 2\sqrt{u_+ \, u_-}\left(\hat{\alpha}_+-\hat{\alpha}_+^{\dagger}\right)\left(\hat{\alpha}_- -\hat{\alpha}_-^{\dagger}\right)\right)\nonumber\\ &\times\sum_{i = 1,2} \left(\hat{n}^{\textrm{el}}_{i,\uparrow}-\frac{1}{2}\right) \left(\hat{n}^{\textrm{el}}_{i,\downarrow}-\frac{1}{2}\right) \end{align} \tag{ F9 }$$

with $u_+$ ($u_-$) the phononic contribution of the upper (lower) polariton:

$$\begin{align} u_+ & = \sin^{\,2}(\theta) \, \frac{\omega_{+}}{\omega_{\textrm{phon}}} \end{align} \tag{ F10 }$$

$$\begin{align} u_- & = \cos^{\,2}(\theta)\, \frac{\omega_{-}}{\omega_{\textrm{phon}}}. \end{align} \tag{ F11 }$$

Due to the transformation made in equation (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 section 4. 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 section 5.