Few-Mode Geometric Description of a Driven-Dissipative Phase Transition in an Open Quantum System

By example of the nonlinear Kerr mode driven by a laser, we show that hysteresis phenomena in systems featuring a driven-dissipative phase transition can be accurately described in terms of just two collective, dissipative Liouvillian eigenmodes. The key quantities are just two components of a non-Abelian geometric connection, even though a single parameter is driven. This powerful geometric approach considerably simplifies the description of driven-dissipative phase transitions, extending the range of computationally accessible parameter regimes, and providing a new starting point for both experimental studies and analytical insights.

Figure 1

Single-mode cavity (frequency ${\omega }_c$, loss rate $\kappa$) pumped by a laser field (amplitude $F$, frequency ${\omega }_p$). Hysteretic behavior of the cavity occupation number $n$ under a cyclic sweeping of $\delta ={\omega }_p-{\omega }_c$ is sketched by brown symbols inside circles. The red and blue color map illustrates the sweeping range ${\omega }_p^{\min }\le {\omega }_p\le {\omega }_p^{\max }$.

Figure 2

(a) ${\gamma }_q$ for few values of $q$. (Dashed line) $t_c=10/\kappa$ is the duration of a step used in a staircase protocol sketched in Figs. 3 and 3. Gray area indicates the critical region of the width $\approx 21.8\kappa$. (b) (Solid line) The cavity occupation number $n=⟨b^{†}b⟩$ for the quantum steady state vs detuning $\delta$. (Dashed lines) The mean-field solution $|⟨b⟩{|}^2$ shown in the same range of $\delta$. The parameter values $U=-0.5$ and $F=4$ in the units of $\kappa$.

Figure 3

Hysteretic behavior of the cavity occupation number $n$ [(c),(d)] under sweeping the detuning parameter $\delta$ through the critical region in a stepwise manner or using a linear ramp with the same inclination [(a),(b)]. The duration of steps used in the former protocol is always the same, $t_c=10/\kappa$ [see also the dashed line in Fig. 2]. [Filled circles (triangles)] Values of $n$ calculated at multiples of $t_c$ at which $\delta$ abruptly changes for a forward (backward) sweep. (Blue lines) Corresponding numerical results for the linear ramp with arrows indicating the sweep direction. The number of steps used in the calculations for the forward and backward sweep was taken to be $N=11$ [(a),(c)] and $N=51$ [(b),(d)]. The values of $\kappa$, $U$, and $F$ are the same as those used in Fig. 2.

Figure 4

(a) Sarandy-Lidar geometrical connection $A_{10}(\delta )$ in the gauge $\mathrm{tr}[b^{†}b{\rho }_1(\delta )]=1$ and the gauge-invariant function $f(\delta )$. (b) Hysteretic area $\mathcal{A}$ (solid line) as a function of $v^{-1}$. (Insets) The cavity occupation number $n$ of the metastable states for the staircase protocol [see Figs. 3 and 3] with the number of steps $N\approx 1.4\times {10}^3$ (left inset) and $N\approx 1.2\times {10}^5$ (right inset) in either direction. (Dashed line) Quasiadiabatic approximation for $\mathcal{A}\propto v$ becomes valid only for extremely slow sweeping velocities (gray region).