Incompressible Polaritons in a Flat Band

We study the interplay of geometric frustration and interactions in a nonequilibrium photonic lattice system exhibiting a polariton flat band as described by a variant of the Jaynes-Cummings-Hubbard model. We show how to engineer strong photonic correlations in such a driven, dissipative system by quenching the kinetic energy through frustration. This produces an incompressible state of photons characterized by short-ranged crystalline order with period doubling. The latter manifests itself in strong spatial correlations, i.e., on-site and nearest-neighbor antibunching combined with extended density-wave oscillations at larger distances. We propose a state-of-the-art circuit QED realization of our system, which is tunable in situ.

Figure 1

Top: Sketch of a transmission line resonator chain including superconducting qubits in every other resonator. Bottom: Simplified lattice representation. Two resonators of type $A$ and $B$ (rectangles) are coupled by the photon hopping rate $J$. Qubits (circles) are only coupled to the $A$ cavities with strength $g$. The dashed lines show one unit cell of the array and the blue symbols mark a localized plaquette state as discussed in the text ($\pm$ denote the corresponding phases in the wave function); see Eq. (2).

Figure 2

(a) Single-particle dispersion of the lattice in Fig. 1. For ${\delta }_{QB}=J$ (dashed line), all three bands are dispersive. For ${\delta }_{QB}=0$ (solid line), the middle band (blue) becomes flat, corresponding to a set of degenerate single-particle plaquette states. (b) Many-body eigenstates associated with the flat band are constructed from products of plaquettes (see main text). They form an equally spaced multilevel system indexed by the particle number $n=0,\dots ,n_{\max }$ with degeneracies $d_n=(\genfrac{}{}{0ex}{}{2n_{\max }-n}{n})$ and largest filling $n_{\max }=(N+1)/2$. Here we show the case for $N=13$ unit cells.

Figure 3

(a) Excitation number $⟨\nu ⟩=\sum _X⟨{\nu }_X⟩$, with ${\nu }_X=n_X/(3N)$, $n_X=\sum _j{x}_j^{†}{x}_j$ ($X=A,B,Q$ and $x_j=a_j,b_j,{\sigma }_j^{-}$) in the steady state as a function of pump strength $f/\kappa$. Shown are results obtained from projection of the density matrix on the flat band eigenspace for a system with $N=13$ unit cells and open boundary conditions (solid line) and from iTEBD simulations of the infinite system at zero detuning ${\delta }_{QB}=0$ (blue symbols) and finite detuning ${\delta }_{QB}=J$ (circles). The plateau at $⟨\nu ⟩\approx \overline{\nu }\approx 1/12$ is associated with a suppression of number fluctuations $\tilde{K}=[⟨(\sum _X{n}_X{)}^2⟩-(\sum _X⟨n_X⟩{)}^2]/\sum _X⟨n_X⟩$, as shown in (b). The plateau is extended for the difference $⟨\nu ⟩-⟨{\nu }_A⟩$ (asterisks), but almost vanishes in the dispersive case (circles). (c) Probability $p_n$ of finding $n$ excitations in the lattice as calculated within the flat band model for the pump strengths marked with arrows in (a). Other parameters are $g/J=1$, ${\delta }_{AB}/J=0.5$, $\kappa /J=0.05$.

Figure 4

Correlation function of photons emitted by the $B$ sites $g_{0jB}^{(2)}=⟨b_0^{†}{b}_j^{†}{b}_0{b}_j⟩/⟨b_0^{†}{b}_0⟩⟨b_j^{†}{b}_j⟩$ for different drive strengths $f/\kappa$ at fixed detuning ${\delta }_{QB}=0$ (upper panel) and different detunings ${\delta }_{QB}/J$ at fixed drive strength $f/\kappa =0.05$ (lower panel) as calculated with iTEBD. The density-wave oscillations correspond to a period doubling with respect to the unit cell of the underlying lattice. In (b) the drive stays resonant with the top of the middle band. The inset shows the length of the density-wave oscillations $\xi$ obtained from an exponential fit. Arrows mark the corresponding values in the main figure. Other parameters chosen as in Fig. 3.

Figure 5

(a) On-site and nearest-neighbor correlator $g_{00B}^{(2)},g_{01B}^{(2)}$ as a function of $g/J$ for $f/\kappa =0.05$ (blue arrow in Fig. 3). The inset shows the complete spatial dependence of the coherence function for the $g/J$ values marked by arrows in the main figure. (b) Correlation length as a function of $g/J$ obtained from an exponential fit of the correlator. Other parameters chosen as in Fig. 3.