Superfluid–Mott-insulator transition of light in the Jaynes-Cummings lattice

Regular arrays of electromagnetic resonators, in turn coupled coherently to individual quantum two-level systems, exhibit a quantum phase transition of polaritons from a superfluid phase to a Mott-insulating phase. The critical behavior of such a Jaynes-Cummings lattice thus resembles the physics of the Bose-Hubbard model. We explore this analogy by elaborating on the mean-field theory of the phase transition and by presenting several useful mappings which pinpoint both similarities and differences of the two models. We show that a field-theory approach can be applied to prove the existence of multicritical curves analogous to the multicritical points of the Bose-Hubbard model, and we provide analytical expressions for the position of these curves.

Figure 1

(Color online) Schematic of the Jaynes-Cummings lattice system, consisting of an array of electromagnetic resonators [see (a)], with a coupling between nearest-neighbor lattice sites due to photon hopping. Each resonator is coherently coupled to a two-level system shown in (b).

Figure 2

(Color online) Properties of the JC lattice model in the resonant case $\Delta =0$. (a) Ground states of the JC lattice system in the atomic limit $\kappa =0$ as a function of detuning $\Delta$. The ground state of the system is given by a product state of Jaynes-Cummings eigenstates on each lattice site $j$. Depending on the chemical potential, the system assumes either the state ${|0⟩}^{{\otimes }_j}$ or one of the antisymmetric states ${|n-⟩}^{{\otimes }_j}$. Degeneracies between $|n-⟩$ and $|\left(n+1\right)-⟩$ mark the onset of superfluidity, occurring for finite photon hopping $\kappa >0$. The onset points, for zero detuning located at $\left(\mu -\omega \right)/g=\sqrt{n}-\sqrt{n+1}$, become dense as $\mu$ approaches $\omega$. For $\mu >\omega$, the system becomes unstable. Degeneracies occur at the same chemical potential for negative and positive detunings $\Delta$ (solid curves), except for the lowest degeneracy between ${|0⟩}^{{\otimes }_j}$ and ${|1-⟩}^{{\otimes }_j}$ where the $\Delta >0$ case is given by the dashed curve. (b) Mean-field phase diagram of the resonant JC lattice system as a function of the effective chemical potential $\mu -\omega$ and photon-hopping strength $\kappa$. The color/gray scale shows the magnitude of the order parameter $\psi =⟨a⟩$. The value of $\psi$ reveals the Mott-insulating phases (denoted “MI”) with $\psi =0$ and fixed number $n=0,1,\dots$ of polaritons per site, and the superfluid phase (“SF”) with $\psi >0$. The phase boundary can be obtained analytically [cf. Eq. (20)] and is marked by a black curve. For sufficiently large photon-hopping strength, the system becomes unstable with respect to the addition of polaritons. A crude estimate of the onset of instability is given by $\left(\mu -\omega \right)=z_c\kappa$ depicted by the white dashed curve. In the hatched region close to $\mu -\omega =0$, numerical results are unreliable when using a fixed cutoff for the maximum photon number. (c) Improved numerical results near $\mu -\omega =0$ [range of $\mu -\omega$ and $\kappa$ as marked by the rectangle in panel (b)] can be obtained by employing a “sliding” truncation (see text) centered at the photon occupation number obtained in the atomic limit.

Figure 3

(Color online) Mean-field phase boundary as a function of effective chemical potential $\left(\mu -\omega \right)$ and atom-resonator detuning $\Delta$. The cross section shows the Mott-insulating lobes at zero detuning (in orange/light gray). With an exception of the $|0⟩$ lobe, all lobes become narrower when increasing the detuning $\Delta$. Arrows and curves in red/dark gray mark the positions of multicritical lines where the system switches from the universality class of the generic superfluid–Mott-insulator transition to the universality class of the soft-spin $\left(d+1\right)$-dimensional $XY$ model.

Figure 4

(Color online) Change of variables shifts multicritical points away from lobe tips. While multicritical points appear at lobe tips when plotting the phase boundary as a function of $\left(\omega -\mu \right)$ and $\kappa$ for constant atomic energy $\epsilon$, their position is shifted when plotting the phase boundary for constant detuning $\Delta$ [see Eq. (65)].

Figure 5

(Color online) Possible realization of the JC lattice model, building upon circuit quantum electrodynamics: superconducting microwave resonators can be patterned onto a chip and coupled to superconducting qubits (shown in the zoom in).