Floquet Dissipative Phase Transitions

Abstract

Dissipative phase transitions (DPTs) are traditionally characterized through the spectral properties of a time-independent Liouvillian superoperator. However, this definition cannot be applied to time-periodic (Floquet) systems that cannot be exactly recast into equivalent time-independent problems. In this article, we develop a general framework to characterize DPTs in time-periodic open quantum systems by analyzing the spectrum of the Floquet propagator. We first study driven-dissipative Kerr resonators, known to display a DPT, showing that counter-rotating terms in the drive induce a shift in the critical point and a significant change in the time scales associated with the transition. We then investigate DPTs in the driven quantum Rabi model and in its time-independent approximated counterpart, the driven Jaynes-Cummings model. We find that the Rabi model exhibits distinct critical features as the ultrastrong coupling regime is approached. Moreover, our Floquet analysis unveils the disappearance of the DPT in the deep strong coupling regime of the quantum Rabi model due to light-matter decoupling. Our rigorous approach sets the stage for the study of dissipative criticality in a broad class of time-dependent open quantum systems.

Figures (5)

• Figure 1: Spectral signatures of DPTs in time-independent and Floquet open systems. (a,b) Liouvillian picture, where criticality is marked by the closure of the Liouvillian gap, $\mathrm{Re}(\lambda_1)\to 0$. (c-f) Floquet picture, where the one-period propagator eigenvalues $\varepsilon_j$ lie within the unit circle. (c,d) Criticality is determined through the closure of the gap $\mathrm{Re}(\varepsilon)\to 1$. (e,f) With weak symmetry breaking (here $\mathbb{Z}_2$), the propagator splits into sectors ($P_+$, $P_-$), and the sector-resolved leading eigenvalues approach the corresponding symmetry roots of unity, here $\Re(\varepsilon_{1, 0}) \to +1$ and $\Re(\varepsilon_{0, 1})\to -1$.
• Figure 2: Standard vs Floquet DPT in a single-photon driven, dissipative Kerr oscillator. (a,b) Comparison of the steady-state photon number $\expval*{\hat{a}^\dagger\hat{a}}_\mathrm{ss} / N$ between the RWA approximation and the Floquet picture as a function of $\tilde{F}_1/\kappa$, for $N=1$ and $N=2.5$, respectively. (c) Analysis of the Floquet spectral gap in the vicinity of the DPT as a function of $\tilde{F}_1 / \kappa$. (d) Scaling of $\min(|\varepsilon_1 - \varepsilon_0|)$ with the thermodynamic limit of the system. Used parameters: $\omega_0 = 50 \kappa$ (solid blue and dashed red), $\omega_0 = 100 \kappa$ (dash-dotted yellow), $\Delta = -80 \kappa$, $U = 10 \kappa$.
• Figure 3: Standard vs Floquet DPT in a two-photon driven, dissipative Kerr oscillator. (a) Comparison between the steady-state photon number $\expval*{\hat{a}^\dagger \hat{a}}_{\rm ss}/N$ as a function of $\Delta/\kappa$ in the RWA approximation and within the time-dependent picture. (b) Analysis of the Floquet spectral gap in both symmetry sectors as a function of $\Delta/\kappa$. Used parameters: $\omega_0 = 100 \kappa$, $U = 10 \kappa$, $\eta = \kappa / 2$, $F_2 = 40 \kappa$, and $N = 1$.
• Figure 4: Driven-dissipative JCM (a,b) and QRM (c,d) from weak to ultrastrong coupling. Panels (a,c) show the steady-state output field field versus coupling $g$ and rescaled drive $\tilde{F}$, while panels (b,d) report the corresponding Floquet gap $|\varepsilon_1-\varepsilon_0|$. Used parameters are $\omega_\mathrm{c} = \omega_{\rm q} = \omega_{\rm d} = 50 \kappa$.
• Figure 5: Driven-dissipative QRM in the deep-strong-coupling regime. (a) Steady-state output field versus $g$ and $\tilde{F}$. (b) Floquet gap $|\varepsilon_1-\varepsilon_0|$, showing suppression of criticality as $g$ increases. (c) Output field scaling at fixed coupling, consistent with a quadratic dependence on $\tilde{F}$ as expected for a driven-dissipative harmonic oscillator. Parameters are the same as in \ref{['fig:jc_qrm_weak_to_usc']}.