Superfluid-Mott-insulator phase transition of light in a two-mode cavity array with ultrastrong coupling

TL;DR

This paper introduces a novel cavity array model with ultrastrong coupling that exhibits a superfluid-Mott-insulator phase transition of light, influenced by a unique symmetry and atom-photon interactions, with potential implementation in circuit QED.

Contribution

The work presents a new two-mode cavity array model with ultrastrong coupling, supporting stable Mott-lobe structures and a continuous symmetry, expanding understanding of strongly-correlated photonic systems.

Findings

Supports a global conserved excitation and continuous U(1) symmetry.

Displays stable Mott-lobe structures of photons.

Predicts a second-order superfluid-Mott-insulator phase transition.

Abstract

In this paper we construct a new type of cavity array, in each cavity of which multiple two-level atoms interact with two independent photon modes. This system can be totally governed by a two-mode Dicke-lattice model, which includes all of the counter-rotating terms and therefore works well in the ultrastrong coupling regime achieved in recent experiments. Attributed to its special atom-photon coupling scheme, this model supports a global conserved excitation and a continuous $U(1)$ symmetry, rather than the discrete $Z_{2}$ symmetry in the standard Dicke-lattice model. This distinct change of symmetry via adding an extra photon mode strongly impacts the nature of photon localization/delocalization behavior. Specifically, the atom-photon interaction features stable Mott-lobe structures of photons and a second-order superfluid-Mott-insulator phase transition, which share similarities with the Jaynes-Cummings-lattice and Bose-Hubbard models. More interestingly, the Mott-lobe structures predicted here depend crucially on the atom number of each site. We also show that our model can be mapped into a continuous $XX$ spin model. Finally, we propose a scheme to implement the introduced cavity array in circuit quantum electrodynamics. This work broadens our understanding of strongly-correlated photons.

Equations

$${\hat{H}}_{\text{T}}=\sum _j{\hat{H}}_j^{\text{TD}}-t\sum _{⟨j,k⟩}\sum _{m=1,2}{\hat{a}}_{m,j}^{†}{\hat{a}}_{m,k},$$

$${\hat{H}}_j^{\text{TD}}$$

$${\hat{H}}_j^{\text{D}}=\omega {\hat{a}}_{1,j}^{†}{\hat{a}}_{1,j}+{\omega }_0{\hat{J}}_{z,j}+g({\hat{a}}_{1,j}+{\hat{a}}_{1,j}^{†}){\hat{J}}_{x,j},$$

$${\hat{N}}_e=\sum _j{\hat{N}}_{e,j}=\sum _j({\hat{J}}_{z,j}+{\hat{a}}_{1,j}^{†}{\hat{a}}_{2,j}+{\hat{a}}_{2,j}^{†}{\hat{a}}_{1,j}),$$

$${\hat{H}}_{\mathcal{M}\mathcal{F}}=\sum _j{\hat{H}}_j^{\text{TD}}-zt\sum _{j,m}\left[{\psi }_m\left({\hat{a}}_{m,j}^{†}+{\hat{a}}_{m,j}\right)-{\left|{\psi }_m\right|}^2\right],$$

$$E_n({\psi }_1,{\psi }_2)=E_n^{(0)}+E_n^{(2)}+O(tz\psi {)}^4,$$

$$E_n^{(2)}=\sum _{m=1,2}(zt+z^2{t}^2{R}_{m,n}){\left|{\psi }_m\right|}^2+2z^2{t}^2{T}_n{\psi }_1{\psi }_2.$$

$$R_{m,n}=\sum _{k\ne n}\frac{{\left|⟨n|({\hat{a}}_{m,j}+{\hat{a}}_{m,j}^{†})|k⟩\right|}^2}{E_n^{(0)}-E_k^{(0)}},$$

$$T_n=\sum _{k\ne n}\frac{\left[⟨n|({\hat{a}}_{1,j}+{\hat{a}}_{1,j}^{†})|k⟩⟨k|({\hat{a}}_{2,j}+{\hat{a}}_{2,j}^{†})|n⟩+c.c\right]}{2(E_n^{(0)}-E_k^{(0)})}.$$

$$t_c=\min (t_1^N,t_2^N).$$

$${\hat{H}}_{\text{G}}$$

$${\hat{a}}_{1,j}\to \alpha {\hat{\Sigma }}_j^{-}+\beta {\hat{\Sigma }}_j^{+}\text{, }{\hat{a}}_{2,j}\to \alpha {\hat{\Sigma }}_j^{-}-\beta {\hat{\Sigma }}_j^{+},$$

$$\hat{H}=\frac{\Delta }{2}\sum _i{\hat{\Sigma }}_i^z-J\sum _{⟨i,j⟩}\left({\hat{\Sigma }}_i^x{\hat{\Sigma }}_j^x+{\hat{\Sigma }}_i^y{\hat{\Sigma }}_j^y\right),$$

$$\mathcal{L}$$

$$\left(\begin{array}[]{c}\dot{\phi}_{J}\\ \dot{\phi}_{b}^{s}\end{array}\right)=\frac{1}{C_{\Sigma}}\left(\begin{array}[]{cc}C_{b}+\tilde{C}_{g},&-\tilde{C}_{g}\\ -\tilde{C}_{g},&C_{J}+\tilde{C}_{g}\end{array}\right)\left(\begin{array}[]{c}Q_{J}\\ Q_{b}^{s}\end{array}\right)$$

$${\dot{\varphi }}_m^{i\ne s}=\frac{1}{C_m}{Q}_m^{i\ne s}\text{ (}m=1,2\text{)},$$

$$H_{\text{s}}=H_{\text{res}}+H_{\text{at}}+H_{\text{int}}.$$

$$H_{\text{res}}$$

$$H_{\text{at}}$$

$$H_{\text{int}}=\frac{{\tilde{L}}_c}{2L_{\Sigma }^2{L}_2{L}_a}{\varphi }_J({\varphi }_a^s-{\varphi }_a^{s-1})-\frac{{\tilde{C}}_g}{C_{\Sigma }}{Q}_J{Q}_b^s.$$

$${\hat{\varphi }}_m(x_i)$$

$${\hat{Q}}_m(x_i)$$

$${\hat{\varphi }}_J\iff ⟨\downarrow |{\hat{\varphi }}_J|\uparrow ⟩{\hat{\sigma }}_x$$

$${\hat{Q}}_J\iff \frac{{\omega }_0}{4e{E}_Q{\varphi }_0}⟨\downarrow |{\hat{\varphi }}_J|\uparrow ⟩{\hat{\sigma }}_y,$$

$${\hat{H}}_s$$

$$g_1=-\frac{{\tilde{L}}_c\sqrt{{\omega }_1/L_{a}D}\sin (\pi x_s/D)⟨\downarrow |{\hat{\varphi }}_J|\uparrow ⟩}{2L_{\Sigma }^2{L}_2},$$

$$g_2=\frac{{\tilde{C}}_g{\omega }_0\sqrt{{\omega }_2{C}_{b}D}\cos (\pi x_s/D)⟨\downarrow |{\hat{\varphi }}_J|\uparrow ⟩}{4\pi e{E}_Q{\varphi }_0{C}_{\Sigma }},$$

$${\omega }_1=\frac{\pi }{D\sqrt{L_a{C}_a}},$$

$${\omega }_2=\frac{\pi }{D\sqrt{L_b{C}_b}}.$$

Full text

I Introduction

Photons are excellent information carriers in nature, and generally pass through each other without consequence. The realization of coherent manipulation and controlling of photons allows us to achieve photon quantum information processing TEN14 as well as to explore exotic many-body phenomena of photons IC13 . Cavity array MJHF08 ; AAH12 ; SS13 ; IMG14 ; CN17 , in which each single-mode cavity interacts with a two-level atom, is a promising platform to accomplish the required target and has now been considered extensively ADG06 ; MFA06 ; MJH07 ; DGA07 ; MIM08 ; SS09 ; JK09 ; KT13 ; GK13 ; KKM13 ; SF14 ; BB14 ; MB15 ; ZY15 ; KS15 ; MS16 ; HMJ16 ; AM16 ; ALCH16 . On one hand, this platform has a novel interplay between strong local nonlinearities and photon hopping of the nearest-neighbor cavities, which has a phenomenological analogy to those of the Bose-Hubbard model MPA89 realized, for example, by ultracold atoms in optical lattices IB08 . More importantly, compared with the condensed-matter or atomic physics, cavity array has a unique property that the fundamental many-body phenomena depend crucially on the intrinsic atom-photon coupling strength MJHF08 ; AAH12 ; SS13 ; IMG14 ; CN17 .

For the weak and moderately-strong coupling regimes, the counter-rotating terms of the single-site Hamiltonian are usually neglected by employing the rotating-wave approximation. As a result, the property of cavity array is governed by a Jaynes-Cummings-lattice model MJHF08 ; AAH12 ; SS13 ; IMG14 ; CN17 . Since this Jaynes-Cummings-lattice model preserves a global excitation number, a series of Mott insulator (MI) phase of photons form a lobe structure and a second-order superfluid(SF)-MI phase transition take place across the edge of each lobe. This Mott-lobe structure makes it a photonic counterpart of the Bose-Hubbard model MPA89 , which simulates massive bosons in lattice and also supports a similar lobe structure. However, it should be noticed that a complete description of the light-matter interaction should always incorporate the counter-rotating terms, especially considering the fact that recent experiments of circuit quantum electrodynamics (QED) have accessed the ultrastrong coupling regime (i.e., the atom-photon coupling strength has the same order of the photon frequency) TN10 ; PF10 ; PJ16 ; FY16 , in which the rotating-wave approximation totally breaks down. In such a case, a proper description of the system dynamics should resort to a Rabi-lattice model. Since the counter-rotating terms in the Rabi-lattice model breaks the conservation of excitation number, there is, in principle, no similar MI as that of the Bose-Hubbard model and the transition between the SF and MI should be replaced by the coherent and incoherent type MSCH12 ; MSCH13 . These essential changes of equilibrium properties motivate us to ask a question: could the Mott-lobe structure still exist even though all of the counter-rotating terms of the atom-photon coupling are taken into consideration?

In the present paper, we try to answer this question by constructing a new type of cavity array, in each cavity of which multiple two-level atoms interact with two independent photon modes. This system can be totally governed by a two-mode Dicke-lattice (TMDL) model, which includes all of the counter-rotating terms and therefore works well in the ultrastrong coupling regime. Unlike the Rabi-lattice model, the TMDL model has a global conserved excitation and a continuous $U(1)$ symmetry. This distinct change of symmetry via adding an extra photon mode induces some interesting many-body physics of strongly-correlated photons. Specifically, the atom-photon interaction features stable Mott-lobe structures of photons and a second-order SF-MI phase transition, which share similarities with the Jaynes-Cummings-lattice MJHF08 ; AAH12 ; SS13 ; IMG14 ; CN17 and Bose-Hubbard MPA89 models. However, in contrast to these models, the Mott-lobe structures predicted here depend crucially on the atom number of each site, reflecting its particularity among lattice models. We also show that the TMDL model can be mapped into a continuous $XX$ spin model under proper parameter conditions. Finally, motivated by recent experimental achievements of cavity array JR14 ; CE14 ; MF16 and multimode cavity MM11 ; NMS15 ; DCM15 in circuit QED, we propose a scheme to realize the TMDL model in a two-mode superconducting stripline cavity array. This work broadens our understanding of strongly-correlated photons.

II Model and Hamiltonian

We study a photon lattice system composed by an array of identical coupled cavities, inside each of which multiple two-level atoms interact with two degenerate photon modes. Such a system is governed by the TMDL Hamiltonian

[TABLE]

where the single-site Hamiltonian

[TABLE]

In the Hamiltonians (1) and (II), $\hat{a}_{m,j}^{{\dagger}}$ and $\hat{a}_{m,j}$ are the creation and annihilation operators of the $m$th photon mode of site $j$, $\hat{J}_{i,j}(i=x,y,z)=$ $\sum_{l=1}^{N}\hat{\sigma}_{i,j}^{l}/2$, with $\hat{\sigma}_{i,j}^{l}$ being the Pauli spin operator, is the collective spin operator of site $j$, $\omega$ is the frequency of the degenerate photon modes, $\omega_{0}$ is the atom resonant frequency, $g$ is the atom-photon coupling strength, $t$ is the hopping rate, and $\left\langle j,k\right\rangle$ denotes the photon hopping between the nearest-neighbor sites $j$ and $k$.

An intriguing feature of the Hamiltonian (II) is that the spin operator couples to the two independent photon modes via its two orthogonal components $\hat{J}_{x}$ and $\hat{J}_{y}$, respectively. Without the coupling term $i(\hat{a}_{2,j}-\hat{a}_{2,j}^{{\dagger}})J_{y,j}$, the Hamiltonian (II) reduces to the standard Dicke model

[TABLE]

and the corresponding Hamiltonian (1) is thus called the Dicke-lattice model LJZ14 (Rabi-lattice model for $N=1$ HZ11 ; MSCH12 ; MSCH13 ; TF16 ; BS13 , with $N$ being the atom number of each site). Obviously, since the rotating-wave approximation is not employed, the TMDL model is able to completely describe potential effects arising from the counter-rotating terms and is therefore reasonable in the ultrastrong coupling regime, which has been achieved in current experiments of circuit QED TN10 ; PF10 ; PJ16 ; FY16 .

The emergence of the so-called counter-rotating terms in the Dicke Hamiltonian (3) reduces the conservation of its excitation number, $\hat{N}_{s,j}=$ $\hat{J}_{z,j}+\hat{a}_{1,j}^{{\dagger}}\hat{a}_{1,j}$, to a parity $\Pi=\exp(i\pi\hat{N}_{s,j})$. However, by introducing an extra degenerate photon mode $\hat{a}_{2,j}$, the Hamiltonian (II) exhibits a special conserved excitation JF14 , $\hat{N}_{e,j}=\hat{J}_{z,j}+\hat{a}_{1,j}^{{\dagger}}\hat{a}_{2,j}+\hat{a}_{2,j}^{{\dagger}}\hat{a}_{1,j}$, apart from the known conserved parity CT03 , even if the rotating-wave approximation is not applied. When the photon hopping is triggered on, this conserved local excitation $\hat{N}_{e,j}$ is replaced by a global one,

[TABLE]

which manifests the $U(1)$ symmetry of the Hamiltonian (1). The conserved global excitation $\hat{N}_{e}$ and its induced $U(1)$ symmetry distinguish the TMDL model from the standard Dicke-lattice model (with a discrete $Z_{2}$ symmetry and without conserved excitation). This complete change of symmetry are expected to deeply impact the behavior of strongly-correlated photons.

III Ground-state phase diagram

Since the knowledge of the single-site limit is crucial for a further understanding of many-body physics, before proceeding, we first catch some instructive insights into the Hamiltonian (II). In the absence of the photon hopping ($t=0$), the excitation density $\hat{N}_{e,j}$ commutes with the Hamiltonian (1) and each eigenstate is thus characterized by a certain excitation number. With an increasing of the system parameter, the level-crossings of the lowest eigenstates are expected to take place, switching a definite excitation density of the ground state. Armed with this argument, we plot the ground-state mean excitation density, $n=\left\langle\hat{N}_{e,j}\right\rangle$, of the single-site Hamiltonian (II) as a function of $g$ in Fig. 1. The evolution of $n$ reflects a conspicuous staircase, whose jump points are associated with the crossover points of the lowest energy levels. For $N=1$, $n$ remains a constant, whereas when increasing $N$, the staircase appears and becomes more and more crowded, showing that the level crossing occurs only for $N\geqslant 2$. This property is totally different from the standard Dicke model (3), where no staircase can be found for any $N$ [see the insert part of Fig. 1], due to the nonconservation of its excitation density $\hat{N}_{s,j}$.

We now pay attention to the TMDL Hamiltonian (1). By applying a mean-field decoupling approximation MPA89 , i.e., $\hat{a}_{m,j}^{{\dagger}}\hat{a}_{m,k}=\langle\hat{a}_{m,j}^{{\dagger}}\rangle\hat{a}_{m,k}+\langle\hat{a}_{m,k}\rangle\hat{a}_{m,j}^{{\dagger}}-\langle\hat{a}_{m,j}^{{\dagger}}\rangle\langle\hat{a}_{m,k}\rangle$, the many-body Hamiltonian (1) reduces to an effective mean-field Hamiltonian

[TABLE]

where $z$ denotes the number of nearest neighbors, and $\psi_{m}=\langle\hat{a}_{m,j}\rangle$ ($m=1,2$) is the variational SF order parameter, which is taken to be real for simplicity ADG06 ; SCR08 . $\psi_{m}$ can be determined self-consistently by minimizing the ground-state energy $E(\psi_{1},\psi_{2})$ of the mean-field Hamiltonian (5) ADG06 .

The effective mean-field Hamiltonian (5) reveals an intimate connections between the single-site Hamiltonian (II) and the many-body properties. In general, even though the global excitation $\hat{N}_{e}$ is a conserved quantity, the excitation number $\hat{N}_{e,j}$ of each site does not conserve, due to the photon hopping. However, as shown in the Hamiltonian (5), if both $\psi_{1}$ and $\psi_{2}$ vanish, the system dynamics is dominated by the single-site Hamiltonian (II), and the photons at each site are thus effectively frozen and characterized by a specific excitation number $n$. We accordingly denote this case as a MI phase, in which the $U(1)$ symmetry is preserved. Whereas a $U(1)$ symmetry-broken phase, associated with the breaking of the conservation of $\hat{N}_{e,j}$, is symbolized by a nonzero value of $\psi_{m}$ and can be anticipated across a critical hopping rate $t_{c}(g)$. In this condition, the photon mode $m$ governs a macroscopic coherence over the lattice and we have a SF phase of the mode $m$. It was generally believed that the complete inclusion of the counter-rotating terms would demolish the MI phase since they couple states with different numbers of the dressed photons and therefore inhibit the formation of photon blockade, which is crucially necessary for the MI phase HZ11 ; MSCH12 ; MSCH13 ; TF16 . In such a case, the notion “SF/MI” should be replaced by “coherent/incoherent”. Nevertheless, the TMDL model we introduced here offers a superb exception $--$ although still breaking the conventional conservation of $\hat{N}_{s,j}$, the counter-rotating terms in the TMDL model preserve the hybridized two-mode excitation $\hat{N}_{e,j}$, attributed to the special atom-photon coupling scheme in the Hamiltonian (II), and thus retain the possibility to form the SF-MI phase transition.

Based on above considerations, we plot the ground-state phase diagram in the $t-g$ plane for different $N$ in Fig. 2. These results show two typical phases: the $U(1)$ symmetry-preserved MI with $\psi_{1}=\psi_{2}=0$ and the symmetry-broken SF with nonzero $\psi_{1}$ and $\psi_{2}$. A further analysis of $\psi_{m}$ near the critical point demonstrates that the transition between these two phases is of second order. According to the Landau’s theory JPS06 ; KH87 , the phase boundary of such a continuous transition can be obtained by a perturbation method, in which the ground-state energy $E_{n}(\psi_{1},\psi_{2})$ is expanded up to second order in $\psi_{m}$ JK09 ; MSCH13 . We expand $E_{n}(\psi_{1},\psi_{2})$ of the $n$th MI phase around the critical value of the order parameter $\psi_{m}=0$. The expanded ground-state energy in powers of $zt\psi$ reads

[TABLE]

where the second-order energy correction

[TABLE]

The coefficients $R_{m,n}$ and $T_{n}$ in Eq. (7) are derived from the second-order perturbation theory by

[TABLE]

and

[TABLE]

where $E_{k}^{(0)}$ and $\left|k\right\rangle$ arise from the eigenequation $\hat{H}_{j}^{\text{TD}}\left|k\right\rangle=E_{k}^{(0)}\left|k\right\rangle$.

The critical hopping rate $t_{c}$ can be obtained by the following procedure. (i) We first write a $2\times 2$ Hessian matrix in terms of Eq. (7), i.e., $\mathcal{M}_{ij}=\partial^{2}E_{n}^{(2)}/\partial\psi_{i}\partial\psi_{j}$, and then derive its two eigenvalues $\varepsilon_{1}$ and $\varepsilon_{2}$. (ii) These two eigenvalues generate two equations, $\varepsilon_{1}=0$ and $\varepsilon_{2}=0$, with respect to $t$. Each of these equations, say $\varepsilon_{m}=0$, supports a trivial solution $t_{m}^{T}=0$ and a nontrivial solution $t_{m}^{N}\neq 0$. (iii) The critical transition point is finally given by

[TABLE]

The obtained boundaries are shown by the black solid curves in Fig. 2. The most important finding, as expected, is that the missing Mott lobes in the standard Dicke-lattice model HZ11 ; MSCH12 (see the red dashed curve in Fig. 2) reappear. More interestingly, our predicted Mott-lobe structure depends crucially on the atom number $N$, which has no counterpart in the Jaynes-Cummings-lattice MJHF08 ; AAH12 ; SS13 ; IMG14 ; CN17 and Bose-Hubbard MPA89 models (Note that the $N$-dependent phase diagram for the Tavis-Cummings-lattice, which is nothing but the Dicke-lattice after the rotating-wave approximation, has been investigated previously SCR08 ; DR07 ; MK10 . In that case, the atom number $N$ only slightly shifts the phase boundary of each lobe, rather than its total structure). Specifically, when $N=1$, the atom-photon coupling features only a single Mott lobe, as shown in Fig. 2(a). With the increasing of $N$, however, more and more Mott lobes emerge, as shown in Figs. 2(b)-2(d). This $N$-dependent behavior of the Mott lobes is a direct legacy of the $N$-dependent staircase of $n$ governed by the single-site Hamiltonian (II). In fact, since in the MI phase, the mean-field Hamiltonian (5) equals to the single-site Hamiltonian (II), there exists a one-to-one correspondence between Fig. 1 and Fig. 2. As a result, each Mott lobe is specified by a definite mean excitation density $n$.

We emphasize that in the TMDL model, on the one hand, no chemical potential is needed to engineer the Mott lobes, which is here stabilized by the atom-photon coupling instead MSCH12 ; MSCH13 . This is in sharp contrast to both the cases of the Jaynes-Cummings-lattice MJHF08 ; AAH12 ; SS13 ; IMG14 ; CN17 and Bose-Hubbard MPA89 models, which are often studied within the framework of grand canonical ensemble where a chemical potential is introduced to fix the (conserved) number of excitations on the lattice ADG06 ; JK09 . On the other hand, the standard Dicke- or Rabi-lattice model does not support any conserved excitations, due to the inclusion of the counter-rotating terms. This makes the description of grand canonical ensemble irrelevant to some extent and no well-defined chemical potential thus exists JPS06 ; KH87 . However, the conserved excitation in the TMDL model motivates us to introduce a chemical potential $\mu$ and access a theory of grand canonical ensemble. We now extend Eq. (1) to the following Hamiltonian in grand canonical ensemble:

[TABLE]

where the on-site two-mode Dicke Hamiltonian becomes $\hat{H}_{j}^{\text{GTD}}=\hat{H}_{j}^{\text{TD}}-\mu\hat{N}_{e,j}$. Following the same mean-field theory, we plot the phase diagram in the $t-\mu$ plane in Fig. 3. As shown in Fig. 3(a), the engineered chemical potential $\mu$ still features the Mott lobes, which is a direct analog of those of the Bose-Hubbard model MPA89 . Once again, a clear interpretation of this lobe structure is still based on the dynamics of the single-site limit, which is governed by the Hamiltonian $\hat{H}_{j}^{\text{GTD}}$. As the chemical potential couples to a conserved quantity $\hat{N}_{e,j}$ in the Hamiltonian $\hat{H}_{j}^{\text{GTD}}$, the eigenstates are independent of $\mu$, due to the simultaneous diagonalization of $\hat{H}_{j}^{\text{TD}}$ and $\hat{N}_{e,j}$. Thus, the ground-state competition leads to a staircase behavior of the excitation density $\hat{N}_{e,j}$ when varying $\mu$, as shown in Fig. 3(b). And accordingly, each Mott lobe in Fig. 3(a) is characterized by the corresponding plateaux.

IV Effective spin model: the continuous $XX$ model

It has been well established that the Jaynes-Cummings-lattice model, respecting a $U(1)$ symmetry, can be mapped to a continuous $XX$ spin model (the isotropic $XY$ spin model) DGA07 ; JK09 , whereas the Rabi-lattice model with the counter-rotating terms has been demonstrated to be in the Ising universality class, owing to its discrete $Z_{2}$ symmetry MSCH12 ; MSCH13 ; BS13 . As revealed in this paper, however, the inclusion of the counter-rotating terms does not always break the continuous symmetry. Especially, for our TMDL model, the $U(1)$ symmetry associated with the conserved excitation number is a signature of its intimate connection with the continuous spin model. To confirm this argument, we focus on the system dynamics in the $t-g$ plane, which is governed by the Hamiltonian (1). We first consider the case of $N\geqslant 2$, which supports a multi-lobe structure in the phase diagram.

When parameters are tuned close to the degenerate point in the MI phase with $t\ll g$, i.e., the boundary between two nearest Mott lobes, we can truncate the Hilbert space to two of the excitation number eigenstates $\left|n\right\rangle$ and $\left|n+1\right\rangle$, where $\left|n\right\rangle$ denotes the eigenstate of the excitation density $\hat{N}_{e,j}$ with eigenvalue $n$ (as verified numerically below, $n$ varies only by one across the degenerate point). Utilizing the commutation relations between the photon annihilation operator $\hat{a}_{m,j}$ and the excitation number $\hat{N}_{e,j}$, we can map $\hat{a}_{m,j}$ in the reduced Hilbert space $\left\{\left|n\right\rangle,\left|n+1\right\rangle\right\}$ into

[TABLE]

where $\hat{\Sigma}_{j}^{+}=\left|n\right\rangle\left\langle n+1\right|$ and $\hat{\Sigma}_{j}^{-}=\left|n+1\right\rangle\left\langle n\right|$ are the redefined Pauli spin ladder operators, and the coefficients $\alpha$ and $\beta$ can be determined numerically (see Appendix A for details). Therefore, the effective spin Hamiltonian of the TMDL model reads

[TABLE]

where $\Delta$ is the energy gap between the two states $\left|n\right\rangle$ and $\left|n+1\right\rangle$ and acts as a longitudinal field, and $J=2t(\left|\alpha\right|^{2}+\left|\beta\right|^{2})$ is the isotropic exchange interaction. As expected, we reproduce the continuous $XX$ model even taking the counter-rotating terms into account.

We now turn to the special case of $N=1$, where only a single Mott lobe exists. In this case, the mapping procedure of $N\geqslant 2$ can not be employed directly. However, similar to Ref. MSCH12 , the energy gap between the two lowest energy levels is of higher-order small, compared with the gap to the next energy level in the ultrastrong coupling regime, and the numerical calculation verifies that these two lowest levels are still characterized by two nearest excitation numbers $n$ and $n+1$ (see Appendix B). Based on these facts, in the ultrastrong coupling regime, we can still obtain the effective Hamiltonian (13) in the subspace spanned by the two lowest energy levels.

V Possible experimental implementation

Having revealed some striking features of the two-mode cavity array, we now turn to the experimental implementation of the Hamiltonian (1). Motivated by recent experimental achievements of cavity array JR14 ; CE14 ; MF16 and multimode cavity MM11 ; NMS15 ; DCM15 in circuit QED, we propose a scheme, depicted in Fig. 4, to implement the TMDL model. As shown in Fig. 4(a), the structure we consider is a series of identical circuit QED elements coupled through capacities. Each of these elements simulates the single-site two-mode Dicke model (II) and the capacitive coupling gives rise to the photon hopping of different elements. The effective circuit diagram of each element is shown in Fig. 4(b). A Josephson junction, acting as an artificial two-level atom, is coupled to two different superconducting stripline resonators.

We first focus on the circuit QED element labeled in Fig. 4(a) with $N=1$. According to the theory of circuit QED, we can regard the flux $\phi$ and the charge $Q$ as the canonical coordinate and momentum, respectively. In this sense, the Lagrangian of a circuit QED element in Fig. 4(b) is written as

[TABLE]

where $\tilde{C}_{g}=C_{g}+C_{a}$ and $\phi_{ext}$ is the external flux of the Josephson junction. Notice that in deriving Eq. (V), the relation $\dot{\phi}_{a}^{s}=\dot{\phi}_{b}^{s}+\dot{\phi}_{J}$ has been used. Moreover, in terms of the Kirchoff’s law at the point, there exists an extra constraint relation $\phi_{f}=\left(L_{1}L_{a}+L_{1}L_{2}\right)\phi_{J}/L_{\Sigma}+L_{1}L_{2}\left(\phi_{a}^{s-1}-\phi_{a}^{s}\right)/L_{\Sigma}$, where $L_{\Sigma}=L_{2}L_{a}+L_{1}L_{a}+L_{1}L_{2}$.

Using $Q_{k}^{j}=\partial\mathcal{L}/\partial\dot{\phi}_{k}^{j}$, we obtain the expression of $\dot{\phi}_{k}^{j}$ in terms of $Q_{k}^{j}$, i.e.,

[TABLE]

and

[TABLE]

where $C_{\Sigma}=\tilde{C}_{g}C_{b}+\tilde{C}_{g}C_{J}+C_{J}C_{b}$.

By means of Eqs. (V)-(16), together with the relation between the Lagrangian and the Hamiltonian, we expand the Hamiltonian of the circuit QED element as a sum of three contributions, i.e.,

[TABLE]

In Eq. (17), the Hamiltonian of the stripline resonator is given by

[TABLE]

Since the last three terms in the Hamiltonian (18) do not involve a sum over sites, their contributions can be neglected in the continuous limit, where the number of the sites becomes infinite. Based on this consideration, we obtain

[TABLE]

The Hamiltonian of the artificial atom reads

[TABLE]

The interaction between the artificial atom and the resonator is governed by the Hamiltonian

[TABLE]

In Eqs. (18)-(21), $\tilde{L}_{J}=L_{1}^{2}L_{2}^{3}+3L_{1}^{2}L_{2}^{2}L_{a}+3L_{1}^{2}L_{2}^{2}L_{a}+3L_{1}^{2}L_{2}L_{a}^{2}+L_{1}^{2}L_{a}^{3}+L_{1}L_{2}^{3}L_{a}+2L_{1}L_{2}^{2}L_{a}^{2}+L_{1}L_{2}L_{a}^{3}-2L_{\Sigma}L_{1}L_{2}^{2}-4L_{\Sigma}L_{1}L_{2}L_{a}-2L_{\Sigma}L_{1}L_{a}^{2}+L_{\Sigma}^{2}L_{2}+L_{\Sigma}^{2}L_{a}$, $\tilde{L}_{s}=L_{1}^{2}L_{2}^{3}+L_{1}^{2}L_{2}^{2}L_{a}+L_{1}L_{2}^{3}L_{a}-2L_{\Sigma}L_{1}L_{2}^{2}+L_{\Sigma}^{2}L_{2}$, and $\tilde{L}_{c}=4L_{\Sigma}L_{1}L_{2}^{2}+4L_{\Sigma}L_{1}L_{2}L_{a}-2L_{1}^{2}L_{2}^{3}-4L_{1}^{2}L_{2}^{2}L_{a}-2L_{1}^{2}L_{2}L_{a}^{2}-2L_{1}L_{2}^{3}L_{a}-2L_{1}L_{2}^{2}L_{a}^{2}-2L_{\Sigma}^{2}L_{2}$.

We thus take the continuous limit of the canonical parameters in the superconducting stripline resonators, i.e., $\phi_{m}^{i}\rightarrow\phi_{m}(x_{i})$ and $Q_{m}^{i}\rightarrow Q_{m}(x_{i})$, and then promote them to quantum operators obeying the canonical commutation relation $\left[\hat{\phi}_{m}(x),\hat{Q}_{n}(y)\right]=i\delta(x-y)\delta_{m,n}$. Following the standard quantization procedure in circuit QED AB04 , the quantized canonical parameters are expressed as

[TABLE]

[TABLE]

where $\omega_{m,n}=n\pi/(D\sqrt{L_{m}C_{m}})$ is the eigenfrequency, $D$ is the length of the resonator, and $n_{o}$ and $n_{e}$ are odd and even integers, respectively.

When the external flux is set to $\phi_{ext}/\phi_{0}=\pi$, the two-level approximation of the Josephson junction gives that PC10 ; AC14

[TABLE]

and

[TABLE]

with $E_{Q}=(\tilde{C}_{g}+C_{b})/(2C_{\Sigma})$, where $\omega_{0}$ is the resonant frequency of the two-level system, $\left|\downarrow\right\rangle$ and $\left|\uparrow\right\rangle$ are the two lowest macroscopic states of the Hamiltonian $H_{\text{at}}$, and $\hat{\sigma}_{i}$ ($i=x,y,z$) is the Pauli spin operator spanned by these two macroscopic states.

At low temperature, we only keep the mode resonate with the artificial atom (i.e., $n=1$) and neglect other non-resonate terms. Under this single-mode approximation of the resonator and the two-level approximation of the artificial atom, the Hamiltonian of the considered circuit QED element is finally expressed as

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The tunability of the inductance and the capacitance of the two superconducting stipline resonators allows us to set $\omega_{1}=\omega_{2}=\omega$ and $g_{1}=g_{2}=g_{0}$, under which the Hamiltonian (V) reduces to the single-site two-mode Rabi model. Using the same procedure, the Hamiltonian (V) can be extended straightforwardly to the case with several two-level artificial atoms, i.e., the single-site two-mode Dicke Hamiltonian (II). When a series of such circuit QED elements are coupled capacitively with the hopping rate $t$ [see Fig. 4(a)], the TMDL Hamiltonian (1) can be achieved.

We emphasize that the improvement of current experimental techniques in the ultrastrong-coupling circuit QED TN10 ; PF10 ; PJ16 ; FY16 makes our proposal a promising candidate to exhibit relevant physics of the TMDL model.

VI Discussions

Up to now, our discussions are restricted to the case of the degenerate photon modes ($\omega_{1}=\omega_{2}=\omega$) and the equal atom-photon coupling strengths ($g_{1}=g_{2}=g$). If these conditions are not fulfilled, there would not be a strict conservation law of $\hat{N}_{e}$, and an instructive question is whether the Mott-lobe structure still exists in such a case or not. To briefly show the influence of a slight deviation of these two equalities, $\omega_{1}=\omega_{2}=\omega$ and $g_{1}=g_{2}=g$, we plot the phase diagrams in the $t-g$ plane for different $\omega_{2}/\omega_{1}$ [Fig. 5(a)] or $g_{2}/g_{1}$ [Fig. 5(b)], when $N=3$. It can be seen clearly from these figures that a slight deviation of the ideal condition does not break the Mott-lobe structure but merely shift the phase boundary.

VII Conclusion

In summary, we have constructed a new type of cavity array system, which is governed by the TMDL model. This model incorporates all of the counter-rotating terms of the atom-photon coupling and therefore works well in the ultrastrong coupling regime achieved in recent experiments. Unlike the standard Dicke-lattice model, the TMDL has a global conserved excitation and a continuous $U(1)$ symmetry. This distinct change of symmetry strongly impacts the nature of photon localization/delocalization behavior. Specifically, the atom-photon interaction features Mott-lobe structures of photons and a second-order SF-MI phase transition, which share similarities with the Jaynes-Cummings-lattice and Bose-Hubbard models. However, the Mott-lobe structures predicted here depend crucially on the atom number of each site, reflecting its particularity among lattice models. We have also shown that the TMDL model can be mapped into a continuous $XX$ spin model under proper parameter conditions. Finally, we have proposed an experimentally-feasible scheme to realize the TMDL model in a two-mode superconducting stripline cavity array.

VIII Acknowledgements

This work is supported partly by the NSFC under Grants No. 11422433, No. 11674200, No. 11604392, No. 11434007, and No. 61378049; the FANEDD under Grant No. 201316; SFSSSP; OYTPSP; and SSCC.

Appendix A Mapping $\hat{a}_{1,j}$ and $\hat{a}_{2,j}$ to the spin operators

We first notice that the commutation relations between the photon annihilation operator and the excitation number operator satisfy $\left[\hat{a}_{1,j},\hat{N}_{e,j}\right]=\hat{a}_{2,j}$ and $\left[\hat{a}_{2,j},\hat{N}_{e,j}\right]=\hat{a}_{1,j}$. Taking these two equations into account, the matrix elements of $\hat{a}_{1,j}+\hat{a}_{2,j}$ and $\hat{a}_{1,j}-\hat{a}_{2,j}$ in the basis of the excitation eigenstates $\left|n\right\rangle$ and $\left|m\right\rangle$ are expressed respectively as

[TABLE]

To obtain a nonzero value of $\left\langle n\right|\hat{a}_{1,j}+\hat{a}_{2,j}\left|m\right\rangle$ ($\left\langle n\right|\hat{a}_{1,j}-\hat{a}_{2,j}\left|m\right\rangle$), we should have $m=n+1$ ($m=n-1$), and in the reduced Hilbert space $\left\{\left|n\right\rangle,\left|n+1\right\rangle\right\}$, the operators $\hat{a}_{1,j}+\hat{a}_{2,j}$ and $\hat{a}_{1,j}-\hat{a}_{2,j}$ thus read

[TABLE]

from which we can straightforwardly obtain $\hat{a}_{1,j}\rightarrow$ $\alpha\hat{\Sigma}_{j}^{-}+\beta\hat{\Sigma}_{j}^{+}$ and $a_{2,j}\rightarrow$ $\alpha\hat{\Sigma}_{j}^{-}-\beta\hat{\Sigma}_{j}^{+}$, i.e., Eq. (12) of the main text. The coefficients $\alpha$ and $\beta$ can be determined numerically.

Appendix B Numerical demonstration of the two state subspace $\left\{\left|n\right\rangle,\left|n+1\right\rangle\right\}$ in the

ultrastrong coupling regime for $N=1$

Figure LABEL:TOCD_Energylevel shows the low-lying spectrum of the Hamiltonian (II) with $N=1$, from which we can see clearly that the two lowest energy levels become quasi-degenerate in the ultrastrong coupling regime. Moreover, as shown in the inset of this figure, both of these two levels support the well-defined excitation numbers, whose difference remains one. This guarantees the validity of the truncation of the Hilbert space to an effective two state subspace $\left\{\left|n\right\rangle,\left|n+1\right\rangle\right\}$ for a large $g/\omega$.