Mott polaritons in cavity-coupled quantum materials

We show that strong electron-electron interactions in cavity-coupled quantum materials can enable collectively enhanced light-matter interactions with ultrastrong effective coupling strengths. As a paradigmatic example we consider a Fermi-Hubbard model coupled to a single-mode cavity and find that resonant electron-cavity interactions result in the formation of a quasi-continuum of polariton branches. The vacuum Rabi splitting of the two outermost branches is collectively enhanced and scales with $g_{\text{eff}}\propto\sqrt{2L}$, where $L$ is the number of electronic sites, and the maximal achievable value for $g_{\text{eff}}$ is determined by the volume of the unit cell of the crystal. We find that $g_{\text{eff}}$ for existing quantum materials can by far exceed the width of the first excited Hubbard band. This effect can be experimentally observed via measurements of the optical conductivity and does not require ultra-strong coupling on the single-electron level. Quantum correlations in the electronic ground state as well as the microscopic nature of the light-matter interaction enhance the collective light-matter interaction compared to an ensemble of independent two-level atoms interacting with a cavity mode.


Introduction
Collective phenomena in light-matter interactions are of tremendous interest in quantum physics. The characteristic feature of these phenomena is that observable quantities increase with the number of emitters, and thus intrinsically small quantum effects can be elevated to a macroscopic level. One of the first studied examples is superradiance within the Dicke model [1,2], which comprises an ensemble of independent two-level atoms interacting with a single mode of the radiation field.
A particularly intriguing yet challenging platform for investigating light-matter interactions are quantum materials [21][22][23][24]. In these systems strong electron-electron interactions give rise a plethora of physical effects that are difficult to describe due to their intrinsically quantum many-body nature. An example is given by the Mott metal-insulator transition [25,26] which can be modelled within the Fermi-Hubbard model [27].
First steps investigating how quantum materials couple to classical and quantum light have been undertaken recently. The interaction of quantum materials with strong, classical light fields was investigated in [28][29][30][31], and superradiance of quantum materials coupled to a cavity field was predicted in [32]. The possibility of inducing superconductivity by coupling electron systems to terahertz and microwave cavities was explored in [33][34][35][36].
Furthermore, it was shown in [37,38] that second-order electron-cavity interactions reduce the magnetic exchange energy in cavity-coupled quantum materials and lead to a collectively enhanced momentum-space pairing effect for electrons.
Here we show that strong electron-electron interactions in quantum materials can give rise to electronic transitions that couple strongly to cavity fields. Collective enhancement of these interactions results in ultrastrong effective coupling strengths that can change the macroscopic properties of the quantum material. As a canonical example we consider a one-dimensional Hubbard model coupled to a single-mode cavity as shown in figure 1(a). We find that the optical conductivity of this system features two peaks that are separated in energy by the collectively enhanced vacuum Rabi frequency µ g L 2 eff , where L is the number of electronic sites. Macroscopically large energy splittings are thus even possible for weakly coupled electron-photon systems. The largest possible value of g eff is attained if the material fills the entire cavity. In this case, the effective coupling constant becomes independent of L and µ g v 1 eff uc , where v uc is the volume of the unit cell of the crystal. As an example, for the quantum material ET-F 2 TCNQ, which is well described by a one-dimensional Hubbard model, g eff can exceed 250 meV. This is several orders of magnitude larger than collective energy shifts in atomic systems [39][40][41] and comparable to the extremely large energy splittings achieved in cavity-coupled molecular materials [6][7][8].
The resonant light-matter interactions considered here are schematically shown in figure 1(b). An electronic state at half filling and with no electronic excitation is resonantly coupled via a cavity photon to an electronic state with one doubly occupied state (doublon) and an empty site (holon) next to it. These states differ in energy by the on-site Coulomb interaction U, which corresponds to the Mott gap of the quantum material. The transition dipole moment between these two states is of the order of d e (d: lattice spacing, e: elementary charge), which is comparable to strong transitions in alkali metal atoms [42]. We find that the resonant coupling of this transition with a single-mode cavity gives rise to polariton states.
Note that the underlying principle of this effect is that strong electron-electron interactions split the electronic energy levels into different bands. Transitions between these bands can exhibit large dipole moments and couple strongly to cavity fields. In this sense our results are quite generic and do not only apply to the system shown in figure 1(a), but to quantum materials with strong on-site electron-electron interactions in general. In the special case studied here, the resonant electron-photon interactions lead to a quasi-continuum of polariton states, and the two branches with the largest energy splitting g eff can be constructed from the electronic ground state. The collective energy splitting µ g L 2 eff of these branches gives rise to the two peaks in the optical conductivity.
Our scheme is qualitatively different to conventional quantum optics systems like the Dicke model [1,2,43,44] where dipole transitions of non-interacting atoms couple collectively to the cavity field. Comparing our system with the Dicke model reveals two important features of our scheme. First, the quantum correlations in the ground state of the electronic system lead to an enhancement of g eff by ≈18%. Second, g eff for the electronic system is larger by a factor of 2 than in the Dicke model. We show that this difference is caused by the different microscopic nature of the light-matter coupling in these systems. This paper is organised as follows. The theoretical model describing the system shown in figure 1(a) is introduced in section 2. Our results are presented in section 3, and the derivation of the Mott polaritons in the manifold with one excitation is outlined in section 3.1. We then show in section 3.2 that a direct signature of the light-matter hybridisation appears in the optical conductivity. The discussion in section 4 gives an intuitive explanation for the collective enhancement of the polariton splitting and illustrates similarities and differences of our system with the Dicke model. The experimental observation of the predicted effects is discussed in section 5, and a summary of our results is provided in section 6.

Model
In this section we present the theoretical model established in [37,38] for the quantum hybrid system shown in figure 1(a). The electronic system is described by the one-dimensional Fermi-Hubbard model [27] with on-site energy U and hopping amplitude t. The electrons are weakly coupled to a single-mode cavity with resonance frequency ω c , and Ω=ÿω c is the photon energy.
The gross energy structure of our system in the parameter regime of interest (U, Ω?t) is determined by the Hamiltonian describes the cavity photons andˆ † a (â) is the bosonic photon creation (annihilation) operator. The operatorD in equation (1) accounts for the on-site Coulomb repulsion between electrons where U is the interaction energy and  k D is the projector onto the manifold with k doubly-occupied sites [37].
In the following we refer to these excitations as doublons. The eigenstates ofĤ 0 are tensor products of photon number states ñ | j P with j photons and Wannier states [27] with k doublons. The associated eigenvalues W + j kU are generally highly degenerate and form manifolds as shown in figure 2. We denote the projector onto a manifold with j photons and k=n − j doublons by  Modifications to the simple energy structure shown in figure 1 arise from the electron hopping and the electron-photon interaction. The hopping operator is where á ñ jk denotes neighbouring sites with j<k and ŝ † c j, ( ŝ c j, ) creates (annihilates) an electron at site j in spin state s Î   { } , . The electron-photon interaction was derived in [37,38] via the Peierls substitution [27] and by expanding the resulting interaction Hamiltonian up to second order in the electron-cavity coupling is the dimensionless current operator. The parameter g=tη inV determines the coupling strength between the electrons and photons, and is a dimensionless parameter that depends on the lattice constant d and the cavity mode volume v (e: elementary charge, ε 0 : vacuum permittivity, ÿ: reduced Planck's constant). The derivation ofV assumesη=1, and this condition is also required to grant the validity of the single-mode cavity approximation [45].
With the preceding definitions we arrive at the total Hamiltonian for the quantum hybrid system in ), H 1 can be treated as a perturbation to the gross energy structure dictated by H 0 . The general effective HamiltonianĤ eff corresponding toĤ can be written as  [46]. In the following section 3 we investigate the formation of Mott polaritons through resonant electronphoton interactions in  1 . To set the stage for this we recall how the cavity modifies the physics in  0 , which was investigated in [37,38] using second-order perturbation theory. For the special case of Ω=U and an electronic system at half filling, the effective Hamiltonian in  0 is given by [37,38] creates a singlet pair at sites k and l.Ĥ S acts only on the electronic system and is an isotropic Heisenberg model (see appendix A) with coupling is a dimensionless scaling factor. Note that R c is equal to unity for g=0 and R c <1 for g>0, and thus the cavity reduces the magnetic exchange interaction. In addition, we have J c >0 for all permitted values of g=t and thus the ground state ñ |G ofĤ S is an antiferromagnetic state [27].

Results
Throughout this section we consider an electronic system at half filling and Ω=U. In section 3.1 we show that resonant electron-photon interactions result in the formation of polaritons, and the energy splitting of the two outermost polariton branches is collectively enhanced. Evidence for this light-matter hybridisation can be found in the optical conductivity as shown in section 3.2.

Mott polaritons
The first excited manifold  1 contains all states with either one doublon or one photon. The effective Hamiltonian in  1 and in first order in H 1 is (see appendix B) Higher-order terms in H 1 are neglected in equation (18) and become negligible in the limit U?t, g. The first term in equation (18) is a constant energy offset of the states in  1 . The second term describes the dynamics of the doublon and holon in  1 and gives rise to the first excited Hubbard band. At g=0, the width of this band is 8t [47,48], and the scaling factor  c reduces this width slightly for g>0. The last term in equation (18) accounts for the resonant doublon-photon interaction and is given by (see appendix C) is a transition operator between the vacuum and the one-photon state and mediates a transition between one and zero doublon states. We note that the definition of  allows us to write  as Since j<k in á ñ jk , this means that  R  (ˆ) † R annihilates (creates) a doublon-holon pair where the doublon is to the right of the holon. Similarly,  L  (ˆ) † L annihilates (creates) a doublon-holon pair where the doublon is to the left of the holon.
We emphasise that -H D P is of first order in the electron-photon coupling since it is proportional to the coupling strength g. This is in contrast to the ground-state manifold where the leading term is of second order in the electron-photon coupling [37,38]. We thus expect that the electron-cavity coupling has a much stronger effect in  1 than in  0 for a fixed value of g.
The resonant electron-photon coupling described by -H D P in equation (19) results in the formation of doublon-photon polaritons. All eigenstates of -H D P with non-zero eigenvalues can be constructed from the eigenstates ofĤ S with non-zero eigenvalues (see appendix C). For each eigenstate ñ |g j with j j S and  < 0 j the corresponding pair of polariton states is is a rescaled eigenvalue ofĤ S with ò j >0 and Each polariton state in equation (25) is a maximally entangled superposition of a state with one doublon and no photon, and a state with no doublon and one photon. The largest value ò max corresponds to the ground state ñ |G ofĤ S in equation (13) with energy  G , and thus for L?1, see appendix A. The energy difference between the corresponding polariton states y ñ 3.33 , 29 eff and carries a direct signature of the collective doublon-photon coupling. Note that we also provide an approximate but intuitive derivation for the value of g eff in section 4. An upper bound for g eff can be obtained by assuming that the material fills the mode volume v of the cavity such that = v Lv uc , where v uc is the volume per lattice site. Furthermore, we assume  l v c 3 (λ c : cavity wavelength) such that the spatial dependence of the cavity mode function is small and . It follows that the value of g eff is independent of L and just depends on v uc .
The values of ò j for a system with L=12 sites are shown in figure 3(a), and the corresponding density of states is shown in figure 3(b). Even a relatively small system with L=12 sites exhibits a quasi-continuum of polariton states, with the largest density of states for intermediate values of

Optical conductivity
A direct signature of the collective doublon-photon coupling in the first excited manifold can be found in the optical conductivity [27]  where y ñ | m are the eigenstates of the full HamiltonianĤ with energies w = E m m and E 0 =ÿω 0 is the energy of the ground state y ñ | 0 . We calculate the optical conductivity with the full system Hamiltonian using Krylov subspace methods [49] for a half-filled electronic system with L=12 sites. Figure 3(c) shows a density plot of the optical conductivity spectrum as a function of g eff and ω. At g eff =0 the optical conductivity maps out the first excited Hubbard band of width 8t that describes the kinematic excitations of a single doublon. At g eff /t≈3 the optical conductivity splits into two branches that become narrower with increasing g eff . The peaks of the optical conductivity signal approximately follow the energies of the polariton branches y ñ - | D P max . These results suggest that the optical conductivity signal is mostly dominated by the two outermost polariton branches y ñ which can be understood as follows. States which are split strongly by the cavity's light field are also expected to couple strongly to an externally applied light field, and thus they show a strong signal in the optical conductivity. This can also be confirmed by noting that y y ñ = ñ

Discussion
In section 3 we have shown that there is a one-to-one correspondence between the polariton branches in the manifold  1 and the eigenstates ofĤ S in equation (14) with zero excitations. The two branches with the largest splitting g eff contribute significantly to the optical conductivity signal and correspond to the electronic ground state ñ |G ofĤ S . A rigorous derivation of the results presented in section 3 is provided in appendix C. Here we give an alternative and approximate derivation of the two polariton branches with the largest energy splitting g eff . This more intuitive picture allows us to gain further insights into our system and highlights similarities and differences with other polariton systems.
Our elementary derivation of g eff starts by approximating the electronic ground state ñ is the antiferromagnetic Néel state. Note that we also employed this state for L=5 to illustrate the electronic ground state in figure 1(b). Applying the doublon-holon creation operator  † (see equation (21)) to this state results in a state with one doublon excitation, This value needs to be compared to g eff in equation (29). We find that g eff is larger than . The reason for this is that ñ |Ǵ Neel is not the true ground state of the electronic system, which is an entangled superposition of Wannier states. It follows that the correlations in the true ground state of the electronic system enhance the polariton splitting by about 18%.
Next we compare our results to those obtained for L independent two-level atoms with transition energy U that interact resonantly with a single cavity mode. This model is a special case of the so-called Tavis-Cummings [43,44] or Dicke [1] model, and in the following we refer to it as the Dicke model. A brief description of the Dicke Hamiltonian is given in appendix D. The manifold with zero excitations has only one non-degenerate ground state where the cavity is in the vacuum state and all atoms are in the ground state. The manifold with one excitation contains two states that are split by (see appendix D) which is smaller than [́] g Neel eff in equation (34) by a factor of 2 . This difference can be attributed to the different nature of the light-matter interaction for atoms and electrons: The cavity field couples to the atomic density in the Dicke model, whereas the light-matter coupling in the electronic system is proportional to the current operator. Starting from ñ |Ǵ Neel the operator -H D P can create a doublon with the holon either to the left or two the right, giving rise to 2L possible states as discussed above. On the contrary, the corresponding Hamiltonian for the atoms can only locally excite one atom at site k, and there are only L different states. Taking into account the normalisation of the corresponding states gives rise to collective coupling strengths proportional to L 2 in the case of electrons and L in the case of atoms.

Experimental realisation
To discuss the experimental observation of the collectively enhanced light-matter coupling in our system we consider ET-F 2 TCNQ [50][51][52][53], which is a generic example of a one-dimensional Mott insulator where U?t.
The optical conductivity σ is the central materials quantity of interest since it determines the dielectric function of the material and thus its reflectivity and transmittance, for example. Typical experiments measure the optical conductivity indirectly, e.g. via reflectivity measurements and self-consistent calculations utilising Kramers-Kronig relations [53]. Optimal strategies for measuring σ depend on the specifics of the experimental setup and are beyond the scope of this work. In order to observe the splitting of the optical conductivity spectrum shown in figure 3 we require g eff /t3. In addition, the photon-doublon coupling must be much faster than the cavity decay rate κ, i.e. g eff ?ÿκ. In the case of ET-F 2 TCNQ [50] we find g eff /t≈6.4. It follows that the two branches in the optical conductivity should be clearly visible, and their energy splitting can be as large as g eff ≈250 meV. Even larger values of g eff /t are possible in materials with a smaller Mott gap or smaller unit cells. The condition g eff ?ÿκ is also fulfilled in ET-F 2 TCNQ where  p »t 2 10THz [50,51], which is at least two orders of magnitude larger than cavity decay rates of lossy microcavities with frequencies in the low THz regime [54].
Finally we address the finite lifetime τ D of doublon excitations which increases exponentially with U/t [55]. The experimentally measured value for ET-F 2 TCNQ at ambient pressure is t » 0.5 ps D [53], which corresponds to a decay rate of  k » t 0.2 D . This decay rate is smaller than the artificial broadening introduced in the numerical evaluation of equation (30), where each term of the sum was broadened with a Lorentzian of width 0.5t/ÿ. We thus conclude that the finite lifetime of doublons does not hinder the observation of the two peaks in the optical conductivity.

Summary
We have shown that the resonant coupling between strongly correlated electrons and a single-mode cavity results in the formation of Mott polaritons. The manifold with one excitation exhibits a dense spectrum of polariton branches which can be derived from the eigenstates in the zero excitation manifold. At half filling the effective Hamiltonian in the manifold with zero excitations is an isotropic Heisenberg chain. Each eigenstate with non-zero eigenvalue  < 0 j gives rise to two polariton branches, and the magnitude of their energy splitting is proportional to  j . The two branches with the largest energy splitting are thus associated with the ground state of the isotropic Heisenberg chain, and their energy splitting g eff is proportional to L 2 , where L is the number of electronic sites.
An approximate derivation for g eff in section 4 illustrates that quantum correlations in the ground state result in an enhancement of the polariton splitting by 18%. Furthermore, µ g L 2 eff is a direct consequence of the fact that the electron-photon interaction is mediated by the current operator. The absorption of a photon is associated with an electronic hopping process creating a holon-doublon pair where the doublon is either to the right or the left of the holon. This two-fold excitation pathway is in contrast to atomic systems where the atomic density couples to the cavity field, allowing only for one local excitation when absorbing a photon. The collective polariton splitting for L independent two-level atoms and in the manifold with one excitation is consequently smaller by a factor of 2 compared to our electronic system.
We find that the collectively enhanced polariton splitting is directly observable in the optical conductivity, which features two peaks separated by µ g L 2 eff . If the material fills the whole mode volume of the cavity, the magnitude of the splitting is independent of the mode volume and just depends on v 1 uc , where v uc is the volume of the unit cell of the crystal. As a generic example of a one-dimensional Mott insulator we consider ET-F 2 TCNQ, and find that its unit cell is small enough such that the splitting of the optical conductivity signal exceeds the width of the first Hubbard band. The optical conductivity thus carries a clear signature of the collective electron-photon coupling.
We emphasise that µ g v 1 eff uc together with the small unit cells in solid state materials can result in macroscopically large polariton splittings g eff . In the case of ET-F 2 TCNQ, we find g eff ≈250 meV, which is several orders of magnitude larger than what has been achieved in atomic systems [39][40][41]. In addition, we note that the near-resonant electron-photon coupling described in this work is much larger than the effects described in [37,38], which are mediated by virtual, second-order electron-photon interactions.
In this paper we focused on the resonant electron-photon coupling in the manifold with one excitation. In conventional quantum optics systems like the Dicke model one can approximately diagonalize the Hamiltonian in several excitation subspaces simultaneously via bosonization techniques. Since this approach is in general not applicable to our fermionic system [27], it is an intriguing yet challenging prospect for future studies to investigate near-resonant electron-photon interactions in higher-excited manifolds, see figure 2. Since the electron-photon interaction increases with the number of photons j as j , the energy spectrum is anharmonic. Like in atomic systems [56] this feature results in giant photon nonlinearities and further amplifies the intrinsically large optical nonlinearity of Mott insulators [57]. Furthermore, the physics in manifolds with a large number of excitations will be fundamentally different from the Dicke model. The reason is that the maximal number of atomic excitations within the Dicke model is L, but at most L/2 doublons can be created in the electronic system.
A further intriguing avenue for future studies is the investigation of higher-dimensional systems. For example, the electron-cavity interaction in higher-dimensional systems can be tuned via the relative orientation between the crystal and the cavity polarisation vector [37,38]. In k-dimensional systems where the cavity couples to all spatial directions one expects µ g k 2 eff due to the additional excitation pathways to nearest-neighbour sites, and thus a further enhancement of the effective coupling strength. is an isotropic spin-1/2 Heisenberg chain with exchange coupling J c . The ground state energy ofĤ S per electron and in the thermodynamic limit (  ¥ L ) is  = -L J log 2 G c [58], and thus equation (28) follows. where ñ |g k ( ñ |e k ) denotes the ground (excited) state for the kth atom. There are two eigenstates of H Dicke in the subspace of one excitation, and their energy difference for Ω=U is g eff [Dicke] defined in equation (35).