Critical dynamical properties of a first-order dissipative phase transition

We theoretically investigate the critical properties of a single driven-dissipative nonlinear photon mode. In a well-defined thermodynamical limit of large excitation numbers, the exact quantum solution describes a first-order phase transition in the regime where semiclassical theory predicts optical bistability. We study the behavior of the complex spectral gap associated with the Liouvillian superoperator of the corresponding master equation. We show that in this limit the Liouvillian gap vanishes exponentially and that the bimodality of the photon Wigner function disappears. The connection between the considered thermodynamical limit of large photon numbers for the single-mode cavity and the thermodynamical limit of many cavities for a driven-dissipative Bose-Hubbard system is discussed.

Figure 1

(a) The rescaled photon density $n/N$, (a) and (b) the second-order correlation function $g^{\left(2\right)}$, and (b) as a function of the rescaled drive amplitude $\tilde{F}$ for $\mathrm{\Delta }=3\gamma$ and $\tilde{U}=\gamma$. The semiclassical prediction also is presented in (a) (SC, dashed line). Different curves correspond to $N=1–3,5,10,25$, and 50.

Figure 2

The photon Wigner function as a function of the real and imaginary parts of the rescaled field $\alpha /\sqrt{N}$ for a drive amplitude $\tilde{F}=1.65\gamma$ (upper row) and $\tilde{F}=1.4\gamma$ (lower row) for $N=1–3$ and 10. Red corresponds to high values, and green corresponds to zero (a different scale is used for the different panels). The other parameters are the same as in Fig. 1.

Figure 3

(a) and (c) The real and (b) and (d) the imaginary parts of the Liouvillian gap $\lambda$ (units of $\gamma$) as a function of the rescaled drive amplitude $\tilde{F}$ (units of $\gamma$) for $\tilde{U}=\gamma$ and two values of the detuning: (a) and (b) $\mathrm{\Delta }=0.8\gamma$ and (c) and (d) $\mathrm{\Delta }=2\gamma$. The different curves correspond to different values of $N=1,5,10,25,50,100$ for $\mathrm{\Delta }=0.8\gamma$ and $N=1–5,10,20,30,40$ for $\mathrm{\Delta }=2\gamma$ (as the curves approach the dashed lines $N$ increases). For the imaginary parts in (b) and (d) all curves drop to zero in a range whose size decreases with increasing $N$ (denoted by the dotted lines). In (a) and (b) the dashed lines correspond to the semiclassical linear-response spectrum ${\lambda }_{\mathrm{LR}}$. In (c) and (d) the dashed lines indicate the edge of the regime where the semiclassical approach predicts bistability.

Figure 4

(a) The relaxation time-scale $-1/\mathrm{Re}\left[\lambda \right]$ (units of ${\gamma }^{-1}$) as a function of the distance from the transition point $\tilde{F}-{\tilde{F}}_c$ (units of $\gamma$) for $\tilde{U}=\gamma$ and $\mathrm{\Delta }=2\gamma$. The different curves correspond from top to bottom to $N=40–10$, and 5. For the same parameters and as a function of $N$: (b) the drive amplitude ${\tilde{F}}_c$ at the transition point (units of $\gamma$), (c) the tunneling time $\tau$ (units of ${\gamma }^{-1}$), (d) the dimensionless fitting parameter $b$, and (e) the fitting parameter $f$ (units of $\gamma$). The dashed-dotted lines in (a) are the power-law fits (4). The dashed lines in (a) and (e) indicate the drive amplitude where the semiclassical approach predicts the edge of the bistable region. The inset in (e) demonstrates the finite-size scaling analysis that is used for the extrapolation to the thermodynamic limit.