Thermodynamic universality across dissipative quantum phase transitions

Abstract

We study finite-time driving across second-order dissipative quantum phase transitions described by Lindblad dynamics. We show that the nonadiabatic entropy production, which quantifies deviations from the instantaneous nonequilibrium steady state, exhibits universal power-law scaling with the ramp duration in analogy to the Kibble-Zurek mechanism for closed systems. This establishes the universality of irreversible dissipation induced by driving an open quantum system near criticality. Furthermore, in systems described by bosonic Gaussian states, our scaling laws predict that the nonadiabatic entropy production is independent of driving speed to leading order, revealing a distinctive feature of Gaussian dissipative quantum phase transitions. We validate these analytical predictions in the thermodynamic limit of the driven-dissipative Dicke model and via finite-size scaling in the open Kerr model. Our results establish a general framework for understanding universal nonequilibrium response and thermodynamic irreversibility in critical open quantum systems.

Figures (7)

• Figure 1: a) The Liouvillian spectrum $\{\lambda_i\}$, shown in the complex plane, has a finite Liouvillian gap $\Delta_g$ away from the critical point. The soft-mode eigenvalue is shown in red, and the zero eigenvalue corresponding to the unique steady state $\hat{\pi}_g$ in green. b) At the critical coupling $g_c$, the Liouvillian gap closes. c), d) During a slow, finite-time protocol approaching criticality, the evolving state (blue line in d)) eventually fails to adiabatically follow the instantaneous steady state (red line in d)) and departs from the steady-state manifold. This defines the freeze-out point, $g^* = g(t^*)$, occurring when the intrinsic relaxation time $\tau_R$ exceeds the driving timescale $\tau_D$\ref{['eq:freeze:out:condition']}. The protocol thus separates into a quasiadiabatic and an impulse regime, indicated in grey.
• Figure 2: Example of a dissipative quantum phase transition (DQPT) in the open Dicke model. a) The nonadiabatic entropy production rate is shown as a function of time for linear ramps of the coupling strength $g$ from the normal phase to the critical point $g_c$, with protocol durations $\tau_q$ specified in the legend. Scatter points mark their respective freeze-out points $t^*$, set by \ref{['eq:freeze:out:condition']}, solid lines show the exact result \ref{['eq:sigma:na:dot']}, and black dashed lines correspond to the slow-driving approximation \ref{['eq:nonadiabatic:ep:metric:rate']}. b) Scaling of the nonadiabatic entropy production rate at the onset of the impulse regime $t^*$ as a function of the protocol duration $\tau_q$ for a linear ramp of the coupling strength $g$ from the normal phase to the critical point $g_c$. The scaling exponent is found to be $\dot \Sigma_\mathrm{na}(t^*) \sim \tau_q^y$ with $y = -0.508 \pm 0.001$, in excellent agreement with the theoretical prediction $y = (\alpha-2-\gamma)/(\gamma+1)$ with $\alpha = 2$ and $\gamma =1$. c) The nonadiabatic entropy production as a function of time for different protocol durations $\tau_q$ of a linear ramp of the coupling strength $g$ from the normal phase to the critical point $g_c$. We split the total nonadiabatic entropy production (black squared markers) into two contributions (see Eq. \ref{['eq:na_quasi_impulse']}), stemming from the quasiadiabatic (blue cross markers) and the impulse regime (red dot markers). Parameters: cavity frequency $\omega_c = 1$, atomic frequency $\omega_z = 1$, photon loss rate $\kappa = 0.2$, initial coupling strength $g_0 = g_c - 10^{-2}$, final coupling strength $g_f = g_c - 10^{-15}$.
• Figure 3: Scaling of the soft mode symplectic eigenvalue with the square-root of the inverse Liouvillian gap in the dissipative Dicke model. Parameters: cavity frequency $\omega_c = 1$, atomic frequency $\omega_z = 1$, photon loss rate $\kappa = 0.2$, initial coupling strength $g_0 = g_c - 5\times 10^{-3}$, final coupling strength $g_f = g_c - 10^{-8}$$\tau_q = 10^2\dots 10^5$.
• Figure 4: Left: Complex spectrum of the drift matrix $-W$ as a function of the coupling strength $g$ in the dissipative Dicke model (blue: real part, red: imaginary part). The vertical dashed black line indicates the critical point $g_c$ at which the Liouvillian gap closes. Right: Scaling of the Liouvillian gap with proximity of the coupling strength $g$ to the critical point $g_c$: $\Delta_g \sim \vert g - g_c\vert^\gamma$, in the dissipative Dicke model. From a numerical fit, we obtain $\gamma = 1.009 \pm 0.001$. Parameters: cavity frequency $\omega_c = 1$, atomic frequency $\omega_z = 1$, photon loss rate $\kappa = 0.2$.
• Figure 5: Left: The KMB QFI scales with a power law of the distance to the critical point $\mathcal{I}_g^\mathrm{KMB}\sim \frac{\gamma^2}{2} \vert g - g_c\vert^{-\alpha}$, with $\alpha = 2 \pm 1 \cdot 10^{-4}$, in the Gaussian open Dicke model. Right: Here we show that the slope obtained from a linear fit is $0.500001 \pm 1 \cdot 10^{-6}$, which is in excellent agreement with the predicted $\frac{\gamma^2}{2}$, with $\gamma = 1$. Parameters: cavity frequency $\omega_c = 1$, atomic frequency $\omega_z = 1$, photon loss rate $\kappa = 0.2$.