Mott polaritons in cavity-coupled quantum materials

TL;DR

This paper demonstrates that strong electron-electron interactions in cavity-coupled quantum materials can lead to ultrastrong collective light-matter coupling, with potential for experimental observation through optical conductivity measurements.

Contribution

It introduces a model showing how electron correlations enhance collective light-matter interactions, surpassing single-electron coupling limits.

Findings

Collective coupling strength scales with the square root of the number of electronic sites.

The maximum effective coupling can exceed the width of the first Hubbard band.

Enhanced interactions can be observed via optical conductivity measurements.

Abstract

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.

Equations

$${\hat{H}}_0=\hat{P}+\hat{D},$$

$$\hat{P}=\Omega {\hat{a}}^{†}\hat{a}$$

$$\hat{D}=$$

$${\hat{\mathcal{P}}}_n^{(j)}={\hat{\mathcal{P}}}_{n-j}^D\otimes {\hat{\mathcal{P}}}_j^P,$$

$${\hat{\mathcal{P}}}_n=\sum _{j=0}^n{\hat{\mathcal{P}}}_n^{(j)}$$

$$\hat{T}=$$

$$\hat{V}=g(\hat{a}+{\hat{a}}^{†})\hat{\mathcal{J}}-\frac{1}{2}\frac{g^2}{t^2}(\hat{a}+{\hat{a}}^{†}{)}^2\hat{T},$$

$$\hat{\mathcal{J}}=$$

$$\eta =\frac{de}{\sqrt{2\hslash {\epsilon }_0{\omega }_{c}v}}$$

$$\hat{H}={\hat{H}}_0+{\hat{H}}_1,$$

$${\hat{H}}_1$$

$$=\left[1-\frac{1}{2}\frac{g^2}{t^2}(\hat{a}+{\hat{a}}^{†}{)}^2\right]\hat{T}+g(\hat{a}+{\hat{a}}^{†})\hat{\mathcal{J}}.$$

$${\hat{H}}_{\text{eff}}^{(0)}={\hat{H}}_{\text{S}}\otimes {\hat{\mathcal{P}}}_0^P,$$

$${\hat{H}}_{\text{S}}=$$

$${\hat{b}}_{kl}^{†}=\left({\hat{c}}_{k,\uparrow }^{†}{\hat{c}}_{l,\downarrow }^{†}-{\hat{c}}_{k,\downarrow }^{†}{\hat{c}}_{l,\uparrow }^{†}\right)/\sqrt{2}$$

$$J_c$$

$${\mathcal{R}}_c$$

$${\hat{H}}_{\text{eff}}^{(1)}=U{\hat{\mathcal{P}}}_1+{\mathcal{R}}_c{\hat{\mathcal{P}}}_1^{(0)}\hat{T}{\hat{\mathcal{P}}}_1^{(0)}+{\hat{H}}_{D-P}.$$

$${\hat{H}}_{D-P}=g\left({\hat{\mathcal{D}}}^{†}\otimes \hat{\mathcal{A}}+\hat{\mathcal{D}}\otimes {\hat{\mathcal{A}}}^{†}\right),$$

$$\hat{\mathcal{A}}=|0_P⟩⟨1_P|$$

$$\hat{\mathcal{D}}={\hat{\mathcal{P}}}_0^D\hat{\mathcal{J}}{\hat{\mathcal{P}}}_1^D$$

$$\hat{\mathcal{D}}=-\mathrm{i}\left({\hat{\mathcal{D}}}_R-{\hat{\mathcal{D}}}_L\right),$$

$${\hat{\mathcal{D}}}_R=$$

$${\hat{\mathcal{D}}}_L=$$

$${\hat{H}}_{\text{S}}|g_j⟩={\mathcal{E}}_j|g_j⟩$$

$$|{\psi }_{D-P}^j{⟩}_{\pm }=\frac{1}{\sqrt{2}}\left[|g_j⟩\otimes |1_P⟩\pm \frac{1}{2{ϵ}_j}\left({\hat{\mathcal{D}}}^{†}|g_j⟩\right)\otimes |0_P⟩\right],$$

$${ϵ}_j=\sqrt{-{\mathcal{E}}_j/J_c}$$

$${\hat{H}}_{D-P}|{\psi }_{D-P}^{(j)}{⟩}_{\pm }=\pm 2g{ϵ}_j|{\psi }_{D-P}^{(j)}{⟩}_{\pm }.$$

$${ϵ}_{\text{max}}=\sqrt{-{\mathcal{E}}_G/J_c}\approx \sqrt{L\log 2}$$

$$g_{\text{eff}}=4g\sqrt{L\log 2}\approx 3.33g\sqrt{L},$$

$$\sigma (\omega )=\frac{\pi e^2{t}^2}{d{\hslash }^3}\sum _{m>0}\frac{|⟨{\psi }_m|\hat{\mathcal{J}}|{\psi }_0⟩{|}^2}{{\omega }_m-{\omega }_0}\delta [\omega -({\omega }_m-{\omega }_0)],$$

Full text

I 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 Dicke (1954); Garraway (2011), which comprises an ensemble of independent two-level atoms interacting with a single mode of the radiation field.

Collective light-matter interactions are conveniently described within the framework of polaritons, which are combined excitations of light and matter. A prominent example is given by dark-state polaritons in laser-driven atomic gases Fleischhauer et al. (2005) and more recently, polaritons have been investigated in various solid state systems coupled to cavities Deng et al. (2010); Carusotto and Ciuti (2013); Orgiu et al. (2015); Schwartz et al. (2011); Kèna-Cohen et al. (2013); Zhang et al. (2014); Tabuchi et al. (2014); Yao et al. (2015); siv ; abd ; Mergenthaler et al. (2017); Hagenmüller et al. (2010); Scalari et al. (2012); Zhang et al. (2016); Li et al. (2018); Paravicini-Bagliani et al. (2019); Bartolo and Ciuti (2018). For example, Bose-Einstein condensation of exciton polaritons in semiconductor materials attracted considerable attention Deng et al. (2010); Carusotto and Ciuti (2013), and molecular systems Orgiu et al. (2015); Schwartz et al. (2011); Kèna-Cohen et al. (2013) coupled to cavities can exhibit giant Rabi splittings between polariton branches. The strong coupling of magnetic excitations to microwave cavities was investigated in  Zhang et al. (2014); Tabuchi et al. (2014); Yao et al. (2015); siv ; abd ; Mergenthaler et al. (2017), and two-dimensional electron gases coupled to THz cavities were studied in Hagenmüller et al. (2010); Scalari et al. (2012); Zhang et al. (2016); Li et al. (2018); Paravicini-Bagliani et al. (2019); Bartolo and Ciuti (2018). In all these systems Deng et al. (2010); Carusotto and Ciuti (2013); Orgiu et al. (2015); Schwartz et al. (2011); Kèna-Cohen et al. (2013); Zhang et al. (2014); Tabuchi et al. (2014); Yao et al. (2015); siv ; abd ; Mergenthaler et al. (2017); Hagenmüller et al. (2010); Scalari et al. (2012); Zhang et al. (2016); Li et al. (2018); Paravicini-Bagliani et al. (2019); Bartolo and Ciuti (2018), Coulomb interactions between electrons play a minor role and are not directly involved in the formation of polaritons.

A particularly intriguing yet challenging platform for investigating light-matter interactions are quantum materials Editorial (2016); Powell and McKenzie (2006, 2011); Kato (2004). 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 Mott (1949); Imada et al. (1998) which can be modelled within the Fermi-Hubbard model Essler et al. (2005).

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 Mentink et al. (2015); Coulthard et al. (2017); Görg et al. (2018); Stepanov et al. (2017), and superradiance of quantum materials coupled to a cavity field was predicted in Mazza and Georges (2019). The possibility of inducing superconductivity by coupling electron systems to terahertz and microwave cavities was explored in Laplace et al. (2016); Schlawin et al. (2019); Curtis et al. (2019); Sentef et al. (2018). Furthermore, it was shown in Kiffner et al. (2019a, b) 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 cavity-coupled quantum materials can enable collectively enhanced light-matter interactions that change the macroscopic properties of the quantum material. In particular, we consider a one-dimensional Hubbard model coupled to a single-mode cavity as shown in Fig. 1(a). We find that the optical conductivity of the quantum material features two peaks that are separated in energy by the collectively enhanced vacuum Rabi frequency $g_{\text{eff}}\propto\sqrt{2L}$, 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 $g_{\text{eff}}$ is attained if the material fills the entire cavity. In this case, the effective coupling constant becomes independent of $L$ and $g_{\text{eff}}\propto 1/\sqrt{v_{\text{uc}}}$, where $v_{\text{uc}}$ is the volume of the unit cell of the crystal. As an example, for the quantum material $\text{ET-F}_{2}\text{TCNQ}$, which is well described by a one-dimensional Hubbard model, $g_{\text{eff}}$ can exceed $250\text{meV}$. This is several orders of magnitude larger than collective energy shifts in atomic systems Raizen et al. (1989); Thompson et al. (1992); Baumann et al. (2010) and comparable to the extremely large energy splittings achieved in cavity-coupled molecular materials Orgiu et al. (2015); Schwartz et al. (2011); Kèna-Cohen et al. (2013).

The resonant light-matter interactions considered here are schematically shown in Fig. 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 $de$ ($d:$ lattice spacing, $e$ elementary charge), which is comparable to strong transitions in alkali metal atoms ste . A single-mode cavity is tuned in resonance with this transition between electronic states.

We find that this resonant electron-photon interaction leads to a quasi-continuum of polariton states. The two branches with the largest energy splitting $g_{\text{eff}}$ can be constructed from the electronic ground state. The collective energy splitting $g_{\text{eff}}\propto\sqrt{2L}$ of these branches gives rise to the two peaks in the optical conductivity. A comparison of our results with the Dicke model Dicke (1954); Tavis and Cummings (1968, 1969); Garraway (2011) reveals two important differences between the two systems. First, the quantum correlations in the ground state of the electronic system lead to an enhancement of $g_{\text{eff}}$ by $\approx 18\%$. Second, $g_{\text{eff}}$ for the electronic system is larger by a factor of $\sqrt{2}$ 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 organized as follows. The theoretical model describing the system shown in Fig. 1(a) is introduced in Sec. II. Our results are presented in Sec. III, and the derivation of the Mott polaritons in the manifold with one excitation is outlined in Sec. III.1. We then show in Sec. III.2 that a direct signature of the light-matter hybridisation appears in the optical conductivity. The discussion in Sec. IV 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 Sec. V, and a summary of our results is provided in Sec. VI.

II Model

In this section we present the theoretical model established in Kiffner et al. (2019a, b) for the quantum hybrid system shown in Fig. 1(a). The electronic system is described by the one-dimensional Fermi-Hubbard model Essler et al. (2005) with on-site energy $U$ and hopping amplitude $t$. The electrons are weakly coupled to a single-mode cavity with resonance frequency $\omega_{c}$, and $\Omega=\hbar\omega_{c}$ is the photon energy.

The gross energy structure of our system in the parameter regime of interest ($U,\,\Omega\gg t$) is determined by the Hamiltonian

[TABLE]

where

[TABLE]

describes the cavity photons and $\hat{a}^{\dagger}$ ($\hat{a}$) is the bosonic photon creation (annihilation) operator. The operator $\hat{D}$ in Eq. (1) accounts for the on-site Coulomb repulsion between electrons,

[TABLE]

where $U$ is the interaction energy and $\hat{\mathcal{P}}_{k}^{D}$ is the projector onto the manifold with $k$ doubly-occupied sites Kiffner et al. (2019a). In the following we refer to these excitations as doublons. The eigenstates of $\hat{H}_{0}$ are tensor products of photon number states $|j_{P}\rangle$ with $j$ photons and Wannier states Essler et al. (2005) with $k$ doublons. The associated eigenvalues $j\Omega+kU$ are generally highly degenerate and form manifolds as shown in Fig. 2. We denote the projector onto a manifold with $j$ photons and $k=n-j$ doublons by

[TABLE]

where $n$ is the total number of excitations, $\hat{\mathcal{P}}_{j}^{P}=|j_{P}\rangle\langle j_{P}|$ projects onto the subspace with $j$ photons and

[TABLE]

projects onto all sub-manifolds with $n$ excitations.

Modifications to the simple energy structure shown in Fig. 1 arise from the electron hopping and the electron-photon interaction. The hopping operator is

[TABLE]

where $\langle jk\rangle$ denotes neighbouring sites with $j<k$ and $\hat{c}_{j,\sigma}^{\dagger}$ ($\hat{c}_{j,\sigma}$) creates (annihilates) an electron at site $j$ in spin state $\sigma\in\{\uparrow,\downarrow\}$.

The electron-photon interaction was derived in Kiffner et al. (2019a, b) via the Peierls substitution Essler et al. (2005) and by expanding the resulting interaction Hamiltonian up to second order in the electron-cavity coupling,

[TABLE]

where

[TABLE]

is the dimensionless current operator. The parameter $g=t\eta$ in $\hat{V}$ determines the coupling strength between the electrons and photons, and

[TABLE]

is a dimensionless parameter that depends on the lattice constant $d$ and the cavity mode volume $v$ ($e$: elementary charge, $\varepsilon_{0}$: vacuum permittivity, $\hbar$: reduced Planck’s constant). The derivation of $\hat{V}$ assumes $\eta\ll 1$, and this condition is also required to grant the validity of the single-mode cavity approximation Muñoz et al. (2018).

With the preceding definitions we arrive at the total Hamiltonian for the quantum hybrid system in Fig. 1,

[TABLE]

where

[TABLE]

For the parameters of interest ($U,\Omega\gg t\gg g$), $H_{1}$ can be treated as a perturbation to the gross energy structure dictated by $H_{0}$.

In the following Sec. III we investigate the formation of Mott polaritons through resonant electron-photon interactions in $\hat{\mathcal{P}}_{1}$. To set the stage for this we recall how the cavity modifies the physics in $\hat{\mathcal{P}}_{0}$, which was investigated in Kiffner et al. (2019a, b) using second-order perturbation theory. For the special case of $\Omega=U$ and an electronic system at half filling, the effective Hamiltonian in $\hat{\mathcal{P}}_{0}$ is given by Kiffner et al. (2019a, b)

[TABLE]

where

[TABLE]

and

[TABLE]

creates a singlet pair at sites $k$ and $l$. $\hat{H}_{\text{S}}$ acts only on the electronic system and is an isotropic Heisenberg model [see Appendix A] with coupling

[TABLE]

where

[TABLE]

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 $g\ll t$ and thus the ground state $|G\rangle$ of $\hat{H}_{\text{S}}$ is an antiferromagnetic state Essler et al. (2005).

III Results

Throughout this section we consider an electronic system at half filling and $\Omega=U$. In Sec. III.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 Sec. III.2.

III.1 Mott polaritons

The first excited manifold $\hat{\mathcal{P}}_{1}$ contains all states with either one doublon or one photon. The effective Hamiltonian in $\hat{\mathcal{P}}_{1}$ and in first order in $H_{1}$ is [see Appendix B]

[TABLE]

Higher-order terms in $H_{1}$ are neglected in Eq. (17) and become negligible in the limit $U\gg t,g$. The first term in Eq. (17) is a constant energy offset of the states in $\hat{\mathcal{P}}_{1}$. The second term describes the dynamics of the doublon and holon in $\hat{\mathcal{P}}_{1}$ and gives rise to the first excited Hubbard band. At $g=0$, the width of this band is $8t$ Gallagher and Mazumdar (1997); Jeckelmann (2003), and the scaling factor $\mathcal{R}_{c}$ reduces this width slightly for $g>0$. The last term in Eq. (17) accounts for the resonant doublon-photon interaction and is given by [see Appendix C],

[TABLE]

where

[TABLE]

is a transition operator between the vacuum and the one-photon state and

[TABLE]

mediates a transition between one and zero doublon states. We note that the definition of $\hat{\mathcal{J}}$ allows us to write $\hat{\mathcal{D}}$ as

[TABLE]

where

[TABLE]

Since $j<k$ in $\langle jk\rangle$, this means that $\hat{\mathcal{D}}_{R}$ $(\hat{\mathcal{D}}_{R}^{\dagger})$ annihilates (creates) a doublon-holon pair where the doublon is to the right of the holon. Similarly, $\hat{\mathcal{D}}_{L}$ $(\hat{\mathcal{D}}_{L}^{\dagger})$ annihilates (creates) a doublon-holon pair where the doublon is to the left of the holon.

We emphasize that $\hat{H}_{D-P}$ 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 Kiffner et al. (2019a, b). We thus expect that the electron-cavity coupling has a much stronger effect in $\hat{\mathcal{P}}_{1}$ than in $\hat{\mathcal{P}}_{0}$ for a fixed value of $g$.

The resonant electron-photon coupling described by $\hat{H}_{D-P}$ in Eq. (18) results in the formation of doublon-photon polaritons. All eigenstates of $\hat{H}_{D-P}$ with non-zero eigenvalues can be constructed from the eigenstates of $\hat{H}_{\text{S}}$ with non-zero eigenvalues [see Appendix C]. For each eigenstate $|g_{j}\rangle$ with

[TABLE]

and $\mathcal{E}_{j}<0$ the corresponding pair of polariton states is

[TABLE]

where

[TABLE]

and

[TABLE]

Each polariton state in Eq. (24) 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 $\epsilon_{\text{max}}$ corresponds to the ground state $|G\rangle$ of $\hat{H}_{\text{S}}$ in Eq. (12) with energy $\mathcal{E}_{G}$, and thus

[TABLE]

for $L\gg 1$ mos . The energy difference between the corresponding polariton states $|\psi_{D-P}^{\text{max}}\rangle_{\pm}$ is

[TABLE]

and carries a direct signature of the collective doublon-photon coupling. If the material fills the mode volume $v$ of the cavity we have $v=Lv_{\text{uc}}$, where $v_{\text{uc}}$ is the volume per lattice site. Since $g=\eta t\propto 1/\sqrt{v}$ [see Eq. (9)], the value of $g_{\text{eff}}$ is independent of $L$ and just depends on $v_{\text{uc}}$. This shows that nanoplasmonic cavities are not required to achieve large values of $g_{\text{eff}}$.

The values of $\epsilon_{j}$ for a system with $L=12$ sites are shown in Fig. 4(a), and the corresponding density of states is shown in Fig. 4(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 $\epsilon_{j}$.

Note that the states $|\psi_{D-P}^{j}\rangle_{\pm}$ in Eq. (24) are not eigenstates of the full effective Hamiltonian in Eq. (17) due to the kinetic energy term $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}\hat{\mathcal{P}}_{1}^{(0)}$. However, we show in Appendix C that the states $|\psi_{D-P}^{j}\rangle_{\pm}$ are approximate eigenstates of $\hat{H}_{\text{eff}}^{(1)}$ if their energy splitting is much larger than $t$. In this case, $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}\hat{\mathcal{P}}_{1}^{(0)}$ only leads to a broadening of the polariton states by coupling them off-resonantly to the quasi-continuum of the first Hubbard band.

III.2 Optical conductivity

A direct signature of the collective doublon-photon coupling in the first excited manifold can be found in the optical conductivity Essler et al. (2005),

[TABLE]

where $|\psi_{m}\rangle$ are the eigenstates of the full Hamiltonian $\hat{H}$ with energies $E_{m}=\hbar\omega_{m}$ and $E_{0}=\hbar\omega_{0}$ is the energy of the ground state $|\psi_{0}\rangle$. We calculate the optical conductivity with the full system Hamiltonian using Krylov subspace methods Hochbruck and Lubich (1997) for a half-filled electronic system with $L=12$ sites. Figure 4(c) shows a density plot of the optical conductivity spectrum as a function of $g_{\text{eff}}$ and $\omega$. At $g_{\text{eff}}=0$ the optical conductivity maps out the first excited Hubbard band of width $8t$ that describes the kinematic excitations of a single doublon. At $g_{\text{eff}}/t\approx 3$ the optical conductivity splits into two branches that become narrower with increasing $g_{\text{eff}}$. The peaks of the optical conductivity signal approximately follow the energies of the polariton branches $|\psi_{D-P}^{\text{max}}\rangle_{\pm}$.

These results suggest that the optical conductivity signal is mostly dominated by the two outermost polariton branches $|\psi_{D-P}^{\text{max}}\rangle_{\pm}$ for $g_{\text{eff}}/t\geq 3$ 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 $|\psi_{m}\rangle=|\psi_{D-P}^{\text{max}}\rangle_{\pm}$ are the only polariton states contributing to the sum in Eq. (29) if we approximate the ground state by $|\psi_{0}\rangle\approx|G\rangle\otimes|0_{P}\rangle$. The finite width of the optical conductivity signal is caused by the coupling of polariton states to the first Hubbard band via $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}\hat{\mathcal{P}}_{1}^{(0)}$, and this coupling becomes less effective with increasing energy splitting $\propto g_{\text{eff}}$ [see Appendix C]. The slight asymmetry in the intensity and position of the two conductivity branches as well as the slight increase of the energy splitting compared with the analytical result is a consequence of the higher order terms that are neglected in Eq. (17) but taken into account in the numerical evaluation of $\sigma(\omega)$.

IV Discussion

In Sec. III we have shown that there is a one-to-one correspondence between the polariton branches in the manifold $\hat{\mathcal{P}}_{1}$ and the eigenstates of $\hat{H}_{\text{S}}$ in Eq. (13) with zero excitations. The two branches with the largest splitting $g_{\text{eff}}$ contribute significantly to the optical conductivity signal and correspond to the electronic ground state $|G\rangle$ of $\hat{H}_{\text{S}}$. A rigorous derivation of the results presented in Sec. III is provided in Appendix C. Here we give an alternative and approximate derivation of the two polariton branches with the largest energy splitting $g_{\text{eff}}$. 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 $g_{\text{eff}}$ starts by approximating the electronic ground state $|G\rangle$ of $\hat{H}_{\text{S}}$ by $|G\rangle\approx|G_{\text{N\'{e}el}}\rangle$, where

[TABLE]

is the antiferromagnetic Néel state. Note that we also employed this state for $L=5$ to illustrate the electronic ground state in Fig. 1(b). Applying the doublon-holon creation operator $\hat{\mathcal{D}}^{\dagger}$ [see Eq. (20)] to this state results in a state with one doublon excitation,

[TABLE]

where $\mathcal{C}$ is a normalisation constant. Assuming open boundary conditions, the operators $\hat{\mathcal{D}}_{L}^{\dagger}$ and $\hat{\mathcal{D}}_{R}^{\dagger}$ each create $L-1$ states with a holon-doublon pair. Since all these states are orthonormal, we have $\mathcal{C}=1/\sqrt{2(L-1)}\approx 1/\sqrt{2L}$ for $L\gg 1$. Ignoring boundary effects the matrix element of $\hat{H}_{D-P}$ between the states $|G_{\text{N\'{e}el}}\rangle\otimes|1_{P}\rangle$ and $|E_{\text{N\'{e}el}}\rangle\otimes|0_{P}\rangle$ is thus

[TABLE]

Diagonalisation of $\hat{H}_{D-P}$ in the two-dimensional subspace spanned by $|G_{\text{N\'{e}el}}\rangle\otimes|1_{P}\rangle$ and $|E_{\text{N\'{e}el}}\rangle\otimes|0_{P}\rangle$ results in two polariton states with energy splitting

[TABLE]

This value needs to be compared to $g_{\text{eff}}$ in Eq. (28). We find that $g_{\text{eff}}$ is larger than $g_{\text{eff}}[\text{N\'{e}el}]$ by a factor of $\sqrt{2\log 2}\approx 1.18$, i.e., $g_{\text{eff}}/g_{\text{N\'{e}el}}[\text{N\'{e}el}]=\sqrt{2\log 2}$. The reason for this is that $|G_{\text{N\'{e}el}}\rangle$ 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 Tavis and Cummings (1968, 1969) or Dicke Dicke (1954) 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]

[TABLE]

which is smaller than $g_{\text{eff}}[\text{N\'{e}el}]$ in Eq. (33) by a factor of $\sqrt{2}$. 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 $|G_{\text{N\'{e}el}}\rangle$ the operator $\hat{H}_{D-P}$ 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 $\sqrt{2L}$ and $\sqrt{L}$ in the case of electrons and atoms, respectively.

V Experimental realization

To discuss the experimental observation of the collectively enhanced light-matter coupling in our system we consider $\text{ET-F}_{2}\text{TCNQ}$ Hasegawa et al. (1997, 2000); Wall et al. (2011); Mitrano et al. (2014), which is a generic example of a one-dimensional Mott insulator where $U\gg t$. In order to observe the splitting of the optical conductivity spectrum shown in Fig. 4 we require $g_{\text{eff}}/t\gtrsim 3$. In addition, the photon-doublon coupling must be much faster than the cavity decay rate $\kappa$, i.e., $g_{\text{eff}}\gg\hbar\kappa$. In the case of $\text{ET-F}_{2}\text{TCNQ}$ Hasegawa et al. (1997) we find $g_{\text{eff}}/t\approx 6.4$. It follows that the two branches in the optical conductivity should be clearly visible, and their energy splitting can be as large as $g_{\text{eff}}\approx 250\text{meV}$. Even larger values of $g_{\text{eff}}/t$ are possible in materials with a smaller Mott gap or smaller unit cells. The condition $g_{\text{eff}}\gg\hbar\kappa$ is also fulfilled in $\text{ET-F}_{2}\text{TCNQ}$ where $t/\hbar\approx 2\pi\times 10\text{THz}$ Hasegawa et al. (1997, 2000), which is at least two orders of magnitude larger than cavity decay rates of lossy microcavities with frequencies in the low THz regime Keller et al. (2017).

Finally we address the finite lifetime $\tau_{D}$ of doublon excitations which increases exponentially with $U/t$ Strohmaier et al. (2010). The experimentally measured value for $\text{ET-F}_{2}\text{TCNQ}$ at ambient pressure is $\tau_{D}\approx 0.5\text{ps}$ Mitrano et al. (2014), which corresponds to a decay rate of $\kappa_{D}\approx 0.2t/\hbar$. This decay rate is smaller than the artificial broadening introduced in the numerical evaluation of Eq. (29), where each term of the sum was broadened with a Lorentzian of width $0.5t/\hbar$. We thus conclude that the finite lifetime of doublons does not hinder the observation of the two peaks in the optical conductivity.

VI 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 $\mathcal{E}_{j}<0$ gives rise to two polariton branches, and the magnitude of their energy splitting is proportional to $\sqrt{-\mathcal{E}_{j}}$. The two branches with the largest energy splitting are thus associated with the ground state of the isotropic Heisenberg chain, and their energy splitting $g_{\text{eff}}$ is proportional to $\sqrt{2L}$, where $L$ is the number of electronic sites.

An approximate derivation for $g_{\text{eff}}$ in Sec. IV illustrates that quantum correlations in the ground state result in an enhancement of the polariton splitting by $18\%$. Furthermore, $g_{\text{eff}}\propto\sqrt{2L}$ 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 $\sqrt{2}$ 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 $g_{\text{eff}}\propto\sqrt{2L}$. 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 $1/\sqrt{v_{\text{uc}}}$, where $v_{\text{uc}}$ is the volume of the unit cell of the crystal. As a generic example of a one-dimensional Mott insulator we consider $\text{ET-F}_{2}\text{TCNQ}$, 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 $g_{\text{eff}}\propto 1/\sqrt{v_{\text{uc}}}$ together with the small unit cells in solid state materials can result in macroscopically large polariton splittings $g_{\text{eff}}$. In the case of $\text{ET-F}_{2}\text{TCNQ}$, we find $g_{\text{eff}}\approx 250\text{meV}$, which is several orders of magnitude larger than what has been achieved in atomic systems Raizen et al. (1989); Thompson et al. (1992); Baumann et al. (2010). In addition, we note that the near-resonant electron-photon coupling described in this work is much larger than the effects described in Kiffner et al. (2019a, b), which are mediated by virtual, second-order electron-photon interactions.

In this paper we focussed on the resonant electron-photon coupling in the manifold with one excitation. An intriguing prospect for future studies is to investigate higher-excited manifolds where different sub-manifolds are resonantly coupled via the electron-photon interaction as indicated in Fig. 2. Since the electron-photon interaction increases with the number of photons $j$ as $\sqrt{j}$, the energy spectrum is anharmonic. Like in atomic systems Schuster et al. (2008) this feature results in giant photon nonlinearities and further amplifies the intrinsically large optical nonlinearity of Mott insulators Kishida et al. (2000). 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 polarization vector Kiffner et al. (2019a, b). In $k$-dimensional systems where the cavity couples to all spatial directions one expects $g_{\text{eff}}\propto\sqrt{2k}$ due to the additional excitation pathways to nearest-neighbour sites, and thus a further enhancement of the effective coupling strength.

Acknowledgements.

MK and DJ acknowledge financial support from the National Research Foundation, Prime Minister’s Office, Singapore, and the Ministry of Education, Singapore, under the Research Centres of Excellence program. DJ,FS and JC acknowledge funding from the European Research Council under the European Unionʼs Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement no. 319286, Q-MAC. DJ acknowledges funding from EPSRC grant no. EP/P009565/1. MK, AA and DJ thank Andrea Cavalleri for discussions.

Appendix A Isotropic Heisenberg model

The effective Hamiltonian $\hat{H}_{\text{S}}$ in Eq. (13) can be cast into the form Essler et al. (2005)

[TABLE]

where $\hat{n}_{j}=\hat{c}_{j,\uparrow}^{\dagger}\hat{c}_{j,\uparrow}+\hat{c}_{j,\downarrow}^{\dagger}\hat{c}_{j,\downarrow}$ is the number operator at site $j$ and the components of the local spin operator $\boldsymbol{\hat{S}_{j}}=(\hat{S}_{x}^{j},\hat{S}_{y}^{j},\hat{S}_{z}^{j})$ are defined as

[TABLE]

At half filling every site is occupied precisely by one electron, and thus

[TABLE]

is an isotropic spin-1/2 Heisenberg chain with exchange coupling $J_{c}$.

Appendix B Effective Hamiltonian in $\hat{\mathcal{P}}_{1}$

An approximate effective Hamiltonian in the manifold $\hat{\mathcal{P}}_{1}$ that only takes into account the perturbation $\hat{H}_{1}$ in first order and for $\Omega=U$ is Cohen-Tannoudji et al. (1998); Essler et al. (2005)

[TABLE]

The eigenvalues of $\hat{H}_{\text{eff}}^{(1)}$ will coincide with the eigenvalues of the full Hamiltonian $\hat{H}$ in $\hat{\mathcal{P}}_{1}$ in the limit $U\gg g,t$ where higher-order terms in $\hat{H}_{1}$ are negligible. The first term in Eq. (38) represents the unperturbed energy of the manifold with one excitation, which can be either a photon with energy $\Omega=U$ or a doublon. The second term in Eq. (38) can be written as

[TABLE]

where we used $\hat{\mathcal{P}}_{1}^{(1)}\hat{T}\hat{\mathcal{P}}_{1}^{(1)}=0$ at half filling. The second term in Eq. (39) is

[TABLE]

where $\hat{H}_{D-P}$ is defined in Eq. (18). Combining Eqs. (39), (40) and Eq. (38) shows that the expression for $\hat{H}_{\text{eff}}^{(1)}$ in Eq. (38) is the same as Eq. (17).

A comparison of the eigenvalues of the effective Hamiltonian $\hat{H}_{\text{eff}}^{(1)}$ and the system Hamiltonian $\hat{H}$ is shown in Fig. 4 for a system with $L=4$ sites at half filling and as a function of the cavity coupling $g$. The eigenvalues are in very good agreement for large values of $U=\Omega$, and their differences are of the order of higher-order corrections $\mathcal{O}(g^{2}/U,t^{2}/U,gt/U)$ that are neglected in Eq. (38).

Appendix C Spectrum of $\hat{H}_{D-P}$

Here we investigate the spectrum of the Hamiltonian $\hat{H}_{D-P}$ in Eq. (18). To this end we consider matrix elements of $\hat{H}_{D-P}$ between states in the $\hat{\mathcal{P}}_{1}$ manifold. Note that $\hat{H}_{D-P}$ can only couple states with one photon and no doublon to states with no photon and one doublon. In a first step, we construct electronic states $|e_{j}\rangle$ with exactly one doublon from the eigenstates $|g_{j}\rangle$ of $\hat{H}_{\text{S}}$ with non-zero eigenvalue $\mathcal{E}_{j}<0$,

[TABLE]

where $\hat{\mathcal{D}}$ and $\epsilon_{j}$ are defined in Eqs. (20) and (25), respectively. At half filling it is straightforward to prove the operator identity

[TABLE]

and hence the states $|e_{j}\rangle$ are orthonormal,

[TABLE]

Next we define the following states in $\hat{\mathcal{P}}_{1}$,

[TABLE]

The matrix elements of $\hat{H}_{D-P}$ with respect to these states can be found via Eq. (43) and are given by

[TABLE]

It follows that the matrix representation of $\hat{H}_{D-P}$ reduces to a simple $2\times 2$ block diagonal form in the states defined in Eq. (44), and diagonalizing these blocks leads to the polariton states in Eq. (24).

It remains to show that the matrix elements in Eq. (45) and their complex conjugates are the only non-zero matrix elements of $\hat{H}_{D-P}$ in $\hat{\mathcal{P}}_{1}$. This can be understood as follows. First, we consider states $|\psi_{D}^{(k)}\rangle_{\perp}$ that complement the states $|\psi_{D}^{(j)}\rangle$ to an orthonormal basis in $\hat{\mathcal{P}}_{1}^{(0)}$. Since

[TABLE]

according to Eqs. (18) and (41), we find

[TABLE]

for all values of $j$ and $k$. Second, we consider the eigenstates $|g_{l}^{0}\rangle$ of $\hat{H}_{\text{S}}$ with eigenvalue zero. The states

[TABLE]

complement the states $|\psi_{P}^{(l)}\rangle$ to a basis in $\hat{\mathcal{P}}_{1}^{(1)}$. According to Eq. (42) we have

[TABLE]

and thus $\hat{\mathcal{D}}^{\dagger}|g_{l}^{0}\rangle=0$. It follows that all matrix elements of $\hat{H}_{D-P}$ involving $|\psi_{P}^{(l)}\rangle_{\perp}$ vanish,

[TABLE]

which concludes our proof of the spectrum of $\hat{H}_{D-P}$.

Finally, we note that the polariton states $|\psi_{D-P}^{(j)}\rangle_{\pm}$ are not coupled by the kinetic energy term $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}\hat{\mathcal{P}}_{1}^{(0)}$,

[TABLE]

The second equality in Eq. (51) follows from the fact that in $|e_{i}\rangle$, the doubly occupied site has an adjacent empty site to its right or left. On the other hand, $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}|e_{j}\rangle$ is either zero or describes a state where the doublon and the holon are separated by a singly occupied site, and hence this state is orthogonal to $|e_{i}\rangle$.

Note that $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}\hat{\mathcal{P}}_{1}^{(0)}$ has non-zero matrix elements between the states $|\psi_{D}^{(k)}\rangle_{\perp}$ that form the quasi-continuum of the first Hubbard band. Furthermore, $\hat{\mathcal{P}}_{1}^{(0)}\hat{T}\hat{\mathcal{P}}_{1}^{(0)}$ couples $|\psi_{D}^{(k)}\rangle_{\perp}$ to the polariton states $|\psi_{D-P}^{(j)}\rangle_{\pm}$ via the states $|\psi_{D}^{(j)}\rangle$. This coupling leads to a broadening of the polariton states but becomes less effective if the polariton splitting exceeds the tunneling amplitude $t$. We thus expect the resonances in the optical conductivity to become sharper when the collective coupling increases.

Appendix D Dicke model

The Tavis-Cummings or Dicke Hamiltonian for a system of $L$ independent two-level atoms interacting with a single cavity mode and in rotating-wave approximation is given by Dicke (1954); Tavis and Cummings (1968, 1969); Garraway (2011)

[TABLE]

where $g$ is the light-matter coupling constant. The collective atomic operators are defined as

[TABLE]

where $|g_{k}\rangle$ ($|e_{k}\rangle$) denotes the ground (excited) state for the $k$th atom. There are two eigenstates of $H_{\text{Dicke}}$ in the subspace of one excitation, and their energy difference for $\Omega=U$ is $g_{\text{eff}}[\text{Dicke}]$ defined in Eq. (34).