Photon Solid Phases in Driven Arrays of Nonlinearly Coupled Cavities

We introduce and study the properties of an array of QED cavities coupled by nonlinear elements, in the presence of photon leakage and driven by a coherent source. The nonlinear couplings lead to photon hopping and to nearest-neighbor Kerr terms. By tuning the system parameters, the steady state of the array can exhibit a photon crystal associated with a periodic modulation of the photon blockade. In some cases, the crystalline ordering may coexist with phase synchronization. The class of cavity arrays we consider can be built with superconducting circuits of existing technology.

Figure 1

(a) An array of QED cavities described by oscillator modes (red circles) that are coupled via nonlinear elements (crossed boxes). (b), (c) Implementation of its building blocks in circuit QED for one- and two-dimensional lattices. The circuit cavities are represented by a $LC$ circuit with capacitance $C$ and inductance $L$ and mutually coupled through a Josephson nanocircuit, with capacitance $C_J$ and Josephson energy $E_J$, that generates the on-site and cross-Kerr terms in Eq. (1). Details of this implementation can be found in Ref. 24. An alternative approach to cross-Kerr interactions is discussed in Ref. 26.

Figure 2

Order parameter $\Delta n$ for the photon crystal in the $U\mathrm{\text{−}}V$ plane at zero hopping. If the cross-Kerr term exceeds a critical threshold $V_c$, the steady state is characterized by a staggered order in which $\Delta n\ne 0$. Here we fixed $\Omega =0.75$ and $\delta =0$, for which $z{V}_c\approx 0.44$ at $U=0$, while $z{V}_c\approx 5.73$ in the hard-core limit ($U\to \infty$). In the inset, we show $\Delta n$ as a function of $\Omega$ and $V$ at a fixed value of $U=1$. Here and in the next figure, the color code signals the intensity of the order parameter, while dashed green lines are guides to the eye to locate the phase boundaries.

Figure 3

Order parameter for the photon crystal at finite hopping. $\Delta n$ is plotted as a function of $J$ and $V$, for $\delta =0$ and $U=1$ (a), and as a function of $J$ and $\delta$, for $zV=0.6$ and $U=0$ (b). Here we fixed $\Omega =0.75$. At finite values of the detuning, in addition to the normal (white) and crystalline (colored) phases, an intermediate region (shaded green), characterized by an oscillatory behavior in the asymptotic state, appears. As discussed in the main text, we suggest this last regime may be seen as a nonequilibrium analog of a supersolid.

Figure 4

(a) Time-dependent traces of the real and imaginary parts of $⟨a⟩$ for the two sublattices as a function of time in the steady-state regime of the intermediate oscillating phase of Fig. 3b. (b) The Wigner transform of the reduced single-site density matrix for sublattice $A$ is plotted at a given time in the intermediate oscillating phase of Fig. 3b. Here we used the same parameters as in Fig. 3b and fixed $zJ=0.2$ and $\delta =0.9$.

Figure 5

MPO results for linear chains of cavities. (a) $g^{(2)}(10,j)$ for $\delta =0$, $J=0$, $U=0.5$, $\Omega =0.4$, and $V$ as in the legend. (b) $g^{(2)}(11,j)$ for $\delta =0$, $U=1$, $V=1$, $\Omega =0.4$, and $J$ as in the legend.