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.
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.
Dissipative phase transitions (DPT) are defined by sudden changes in the physical properties of nonequilibrium open quantum systems and they present characteristics that have no analog in closed and thermal systems. Several methods to detect and characterize DPT have been suggested in the literature, the most famous of which -- the Liouvillian gap -- can be derived from a spectral analysis of the Liouvillian super-operator that governs the complex interplay between coherent and dissipative dynamics. Here, we consider the output current, defined as the average total quantum jumps per unit time between the open quantum system and the environment. We propose that output current fluctuations, and in particular their dynamical correlations, their power spectrum, and their characteristic timescale can provide valuable information about DPT, confirming a dramatic change of behavior at the critical point. We validate our proposal using the dissipative XYZ model and the nonlinear driven-dissipative Kerr model, showing good agreement with previous estimates of the location of the critical point. Compared to previous approaches, our proposal could be already experimentally tested in optical systems, providing a practical method to detect criticality in quantum open systems.
We investigate and characterize the emergence of finite-component dissipative phase transitions (DPTs) in nonlinear photon resonators subject to n -photon driving and dissipation. Exploiting a semiclassical approach, we derive general results on the occurrence of second-order DPTs in this class of systems. We show that for all odd n , no second-order DPT can occur while, for even n , the competition between higher-order nonlinearities determines the nature of the criticality and allows for second-order DPTs to emerge only for n = 2 and n = 4 . As pivotal examples, we study the full quantum dynamics of three- and four-photon driven-dissipative Kerr resonators, confirming the prediction of the semiclassical analysis on the nature of the transitions. The stability of the vacuum and the typical timescales needed to access the different phases are also discussed. We also show a first-order DPT where multiple solutions emerge around zero, low, and high-photon numbers. Our results highlight the crucial role played by s t r o n g and w e a k symmetries in triggering critical behaviors, providing a Liouvillian framework to study the effects of high-order nonlinear processes in driven-dissipative systems, that can be applied to problems in quantum sensing and information processing.
Strong periodic driving of cavity-quantum electrodynamics (QED) in the ultrastrong-coupling regime creates nonequilibrium states whose dissipation is governed by Floquet quasienergies rather than undriven dressed resonances. However, modeling such a regime is a significant theoretical challenge, including a number of subtle problems such as the need to ensure gauge invariance for truncated matter-cavity systems with time-dependent driving. To fill this theoretical gap, we introduce a nonsecular Floquet generalized master equation framework for strongly driven open cavity-QED systems, formulated in the dressed basis of the quantum Rabi model and applicable to structured reservoirs without rotating-wave approximations. Our theory can thus model Floquet-driven dynamics in open ultrastrong-coupling cavity-QED, and demonstrates a wide range of quantum state control. Using strong optical pumping and parametric mechanical modulation, we compute long-time populations, fluorescence spectra, and the Floquet-Liouville eigenspectra, resolving observable resonances into hybridized quasienergy channels and decay rates. By systematically comparing with conventional time-independent dressed-basis generalized master equations, we show that static approaches only reproduce steady-state populations in restricted excitation regimes, and fail for frequency-resolved observables and break down under appropriate Floquet engineering, surprisingly, even for spectrally flat baths. Structured environments, such as Lorentzian-Ohmic reservoirs, further amplify these discrepancies through sideband-selective decay. Our results demonstrate that dissipation in driven ultrastrong cavity-QED is intrinsically quasienergy resolved and we establish Floquet-dissipative theory as an accurate and powerful framework for predicting spectra, controlling decay pathways, and engineering nonequilibrium quantum states and reservoirs.
In open quantum systems, dissipative phase transitions (DPTs) emerge from the interplay between unitary evolution, drive, and dissipation. While second-order DPTs have been predominantly investigated theoretically, first-order DPTs have been observed in single-photon-driven Kerr resonators. We present here an experimental and theoretical analysis of both first and second-order DPTs in a two-photon-driven superconducting Kerr resonator. We characterize the steady state at the critical points, showing squeezing below vacuum and the coexistence of phases with different photon numbers. Through time resolved measurements, we study the dynamics across the critical points and observe hysteresis cycles at the first-order DPT and spontaneous symmetry breaking at the second-order DPT. Extracting the timescales of the critical phenomena reveals slowing down across five orders of magnitude when scaling towards the thermodynamic limit. Our results showcase the engineering of criticality in superconducting circuits, advancing the use of parametric resonators for critically-enhanced quantum information applications.
Periodically driven quantum systems are commonly modeled using master equations constructed in the eigenbasis of an undriven Hamiltonian, implicitly assuming that environmental dissipation couples to static energy transitions even under strong time-periodic driving. The validity of this approximation beyond weak or near-resonant driving remains poorly understood. To address the need for a more self-consistent quantum theory approach, we formulate a nonsecular Floquet--Markov generalized master equation (F-GME) in the quasienergy basis, treating interaction-induced (internal) and drive-induced (external) nonperturbative dressing on an equal footing. We subsequently investigate dissipation in two minimal driven open quantum systems---a harmonically driven two-level system and a harmonically driven coupled-two-level-system---each weakly coupled to a Markovian bath. Comparing the F-GME to a time-independent dressed-basis master equation, we show that even for a flat-bath spectral density and weak dissipation, the two approaches can yield qualitatively different steady-state populations and emission spectra. We resolve dissipation into drive-assisted sideband processes decaying via Floquet extended-space quasienergy channels, and show these channels can hybridize through nonsecular couplings into collective Floquet--Liouville modes governing observable spectral resonances. This analysis demonstrates that time-independent dissipative descriptions can incorrectly weight multiphoton Floquet transitions by collapsing quasienergy-resolved decay pathways into static energy gaps. The F-GME framework provides a systematic diagnostic for identifying regimes where Floquet-consistent dissipation is essential and clarifies the physical origin of discrepancies between commonly used master-equation approaches.
The dissipative quantum phase transitions experienced by a driven optical system are studied in order to understand the underlying light-matter coupling. The primary system under study is composed of a cavity mode coupled to a two-level system with dissipation being introduced through the interaction with a surrounding environment; an external coherent field is included to drive the system out of the ground state. The coupling is given by a generalized Rabi Hamiltonian, where rotating, gr, and counter-rotating, ngr, couplings are both present and can be adjusted independently. A comprehensive study of the different phases the system can exhibit as gr and n are varied is presented. From this, we construct a bridge between two limiting scenarios: (i) the Jaynes-Cummings limit (n = 0), where the system undergoes a phase transition by means of the breakdown of the photon blockade, and (ii) the driven Dicke limit ( = 1 ), where the normal to super-radiant phase transition is found. Novel behaviour encountered in an intermediate regime (1 > n > 0) is discussed. Attention is drawn towards the strong coupling regime, where changes at the one-photon level induce nonlinear effects and behaviour reminiscent of phase transitions is encountered with just a few photons present. A comparison between weak and strong coupling regimes, and, thus, an exploration of the effect of fluctuations over quantum phase transitions of light, is given through a survey of the phases an auxiliary optical system exhibits. This system is composed of two coupled nonlinear cavities that are driven coherently and damped through the interaction with the environment; it is seen to exhibit three phases. Two phases present high-correlation between the cavities and are differentiated by the photon statistics, transitioning from classical to quantum. The third phase is characterized by a highly localized field in one of the cavities. The departure from mean-field results is highlighted and a new way to characterize the possible phases of the system is proposed. Finally, the crucial role of quantum fluctuations is quantified and used to define the phases at hand.