Two coupled nonlinear cavities in a driven-dissipative environment

We investigate two coupled nonlinear cavities that are coherently driven in a dissipative environment. We perform semiclassical, numerical, and analytical quantum studies of this dimer model when both cavities are symmetrically driven. In the semiclassical analysis, we find steady-state solutions with different photon occupations in two cavities. Such states can be considered analogs of the closed system double-well symmetry-breaking states. We analyze the occurrence and properties of these localized states in the system parameter space and examine how the symmetry-breaking states, in the form of a bistable pair, are associated with the single-cavity bistable behavior. In a full quantum calculation of the master equation dynamics that includes quantum fluctuations, the symmetry-breaking states and bistability disappear due to the quantum fluctuations. In the quantum trajectory picture, we observe enhanced quantum jumps and switching, which indicate the presence of the underlying semiclassical symmetry-breaking states. Finally, we present a set of analytical solutions for the steady-state correlation functions by using the complex $P$ representation and discuss its regime of validity.

Figure 1

Schematic drawing of our model system of two coupled photonic cavities. Effective photon-photon interactions within each cavity give rise to a Kerr nonlinearity with strength $U$ and intercavity mode overlap contributes to tunneling with strength $J$. The system is coupled to a Markovian bath and photons can decay or leak out with rate $\gamma$. Coherent pumping ($F$) replenishes the photons. We treat the case when drive and dissipation are the same for both cavities. Two coupled cavities are the simplest setting for an array of coupled cavities where the interplay of dissipation, drive, and interaction plays a crucial role that are being investigated in a wider context.

Figure 2

Semiclassical dynamics to reach the steady states in a Bose–Hubbard dimer. For our model, there are nonequilibrium steady states where more photons are localized in one well than the other, a driven-dissipative analog of the symmetry-breaking states of a closed-system double well. Panels (a) and (b) show dynamics where the steady states have equal number of photons in the two cavities: (a) low-density state and (b) high-density state. Panels (c) and (d) show how states with unequal numbers of photons in the two cavities are reached. Symmetry-breaking states come in pairs and are always in a multistable regime. The parameter values are $F/\gamma =2.6,J/\gamma =0.1,U/\gamma =0.6,\mathrm{\Delta }\omega /\gamma =-3$. The four steady states are reached with four different initial values of $(n_1,{\theta }_1,n_2,{\theta }_2)$.

Figure 3

From single-cavity bistability to dimer multistability and driven-dissipative symmetry-breaking states. Panel (a) shows the semiclassical solutions in the limit of a single cavity when there is no coupling ($J=0$). Blue circles and red dots denote stable and unstable states, respectively. For a fixed drive $F/\gamma =2.6$ [denoted by a vertical line in panels (a) and (c)], panel (b) depicts the change in the steady-state photon occupations as the coupling $J/\gamma$ slowly increases—four solutions originate from the two branches of the single-cavity solutions. Panel (c) depicts all the solutions, both stable and unstable, which there are at most nine in some regimes, a maximum of four of which are stable. For $F/\gamma =2.6$, we enter a regime of multiple solutions, two symmetry-preserving states and a pair of symmetry-breaking states. Same parameters are used as in Fig. 2.

Figure 4

Phase diagram for symmetry-breaking states. Panels (a) and (d) show a phase diagram showing the presence of symmetry-breaking states (the region labeled 4) in the tunneling-drive ($J/\gamma -F/\gamma$) plane for $\mathrm{\Delta }\omega /\gamma =-3$ and different values of $U/\gamma =0.6$ and 6, respectively. The area labeled 1 has one symmetry-preserving steady state, 2 has two symmetry-preserving states in a bistable region, while 4 has two symmetry-breaking and two symmetry-preserving states. Panel (b) shows $\mathrm{\Delta }n$ and $\mathrm{\Delta }\theta$ as a function of $J$ for a fixed $F=2.6$. Panel (c) shows $\mathrm{\Delta }n$ and $\mathrm{\Delta }\theta$ as a function of $F/\gamma$ for a fixed $J/\gamma =0.1$. These depict how the symmetry-breaking states appear and disappear as a function of tunneling and drive. We see that there is a critical $J/\gamma$ value beyond which the symmetry-breaking states do not appear. To gain a better insight, panels (e) and (f) depict the phase diagram in the tunneling-detuning ($J/\gamma -\mathrm{\Delta }\mathrm{\Omega }/\gamma$) and tunneling-interaction ($J/\gamma -U/\gamma$) planes, for fixed $F/\gamma =3.5,U/\gamma =0.6$ and $F/\gamma =3.5,\mathrm{\Delta }\mathrm{\Omega }/\gamma =-3$, respectively. The shaded regions contain symmetry-breaking states.

Figure 5

Analysis of quantum trajectory simulations. Panel (a) depicts a histogram of quantum jump statistics for photon occupations for cavity 1 for a sample trajectory in the bistable regime when $F/\gamma =2.6$, plotted along the $y$ axis of panel (b), which gives the multistability diagram for the same parameters as in Fig. 3. Panel (b) also shows the exact quantum steady-state values of $n$ and $g^{\left(2\right)}$ obtained from the Lindblad master equation, showing unique values in the multistable regime and an enhancement of $g^{\left(2\right)}$ due to quantum fluctuations. Panels (c) and (d) show the actual dynamics of $n_1$ and $n_2$ for $F/\gamma =2.6$ showing a sample quantum trajectory run. The magnified region in panel (d) shows that the two cavities can have unequal population during the dynamics which is enhanced when symmetry-breaking states exist. In the space of photon number difference $\mathrm{\Delta }n$, histograms are depicted for (e) $F/\gamma =2$, (f) $F/\gamma =2.6$, and (g) $F/\gamma =3.2$, where the vertical axes are shown on an arbitrary scale. Panel (f) shows enhanced population difference described by a Lorentzian with broad side peaks indicating underlying symmetry-breaking states. On the other hand, panels (e) and (g) is in regions outside multistability and have single central peaks. The number difference statistics thus contain signatures of underlying symmetry-breaking bistability.

Figure 6

A comparison of analytic and numeric solutions for parameters $U/\gamma =4, \mathrm{\Delta }\omega /\gamma =-3, J/\gamma =0.1$. For this set of parameters, our analytic expressions give a very good match with the exact numerics for the first- and second-order correlation functions $n$ and $G^{\left(2\right)}$ shown in panel (a). For small drives $F/\gamma$, the normalized second-order correlation $g^{\left(2\right)}$, however, deviates. Please see the discussions in the text for explanations. Panel (b) shows comparisons for a fixed $F/\gamma =3$ and varying coupling $J/\gamma$. We find that, for small values of $J/\gamma$, the solutions are comparable but deviate slowly as we go to higher $J/\gamma$.