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arXiv:2303.03355 · Quantum

Dissipative phase transitions in n -photon driven quantum nonlinear resonators

Fabrizio Minganti, Vincenzo Savona, Alberto Biella

Published Nov 7, 2023• 31 citations
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Abstract

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.

DOI
10.22331/q-2023-11-07-1170

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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.
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.
We investigate nonequilibrium criticality and chaos in a two-photon-driven optomechanical system. The parametric drive preserves a discrete Z2\mathbb{Z}_2Z2​ symmetry of the optical field, while radiation-pressure coupling transfers the resulting nonlinear dynamics to a mechanical oscillator. Combining semiclassical stability analysis, exact Liouvillian spectra, and stochastic quantum trajectories, we show that this driven-dissipative optomechanical model supports both first- and second-order dissipative phase transitions. At negative detuning a second-order transition yields spontaneous breaking of the cavity-parity symmetry in the thermodynamic limit. At positive detuning the same symmetry breaking coexists with a first-order transition, signaled by metastability and by an additional symmetric Liouvillian mode. At stronger pump power the mean-field dynamics loses all stable fixed points and develops limit cycles and chaotic attractors with positive Lyapunov exponent. Quantum trajectories in this regime display chaotic-like motion, enhanced steady-state entropy, and delocalization over many entropic Liouvillian modes. These results establish two-photon-driven optomechanics as a platform where dissipative criticality, symmetry breaking, and quantum signatures of chaos emerge within the same experimentally accessible setting.
Dissipative phase transitions (DPTs) are traditionally characterized through the spectrum of a time-independent Liouvillian superoperator. However, this definition does not apply to time-periodic (Floquet) systems that cannot be exactly recast as time-independent problems. Here, we develop a general framework to characterize DPTs in time-periodic open quantum systems through 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 shift the critical point and significantly change the time scales associated with the transition. We then investigate DPTs in the driven quantum Rabi model and its time-independent approximation, the driven Jaynes-Cummings model, finding 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, due to light-matter decoupling. Our approach sets the stage for the study of dissipative criticality in a broad class of time-dependent open quantum systems.
Recently, there has been considerable interest in studying quantum sensing near the critical points of a system evolving unitarily and undergoing a phase transition. However, there are many quantum systems in which dissipation is very important. The quantum phase transition in such systems results as a result of competition between nonlinearity, drive, and dissipation. These quantum phase transitions belong to the class of nonequilibrium phase transitions. In this work, we present a detailed study of quantum sensing of the system parameters near the critical points of a dissipative phase transition. We illustrate the key results by considering a coherently driven nonlinear Kerr medium interacting with environment. The first-order quantum phase transition is characterized by the closing of the gap in eigenvalues of the underlying Liouvillian. The steady-state sensing away from the critical point has standard quantum limit scaling with respect to the effective system size, whereas Heisenberg scaling is found at the critical point. The latter is subject to a critical slowing down. We have also examined sensing of system parameters using the metastable states and find better sensing performance at intermediate timescales, i.e., when the system has not yet settled in steady state. Detailed results on the scaling behavior in different time domains are given.
Excited-state quantum phase transitions (ESQPTs) generalize the concept of quantum phase transitions to excited levels. Here we investigate how such ESQPTs can emerge in Kerr nonlinear oscillators, where a bosonic mode with a Kerr nonlinearity is subject to both coherent (one-photon) and parametric (two-photon) driving. We determine the structure and properties of the complete phase diagram by defining proper dimensionless parameters. Rich phases and phase transitions exist in the phase diagram, including phases with two or four critical energies above the ground-state level and first-order quantum phase transitions. We characterize the ESQPTs by singularities in the density of states, which show universal logarithmic divergence or discontinuity at the critical energies. Furthermore, we find that observables such as the photon number also exhibit singular behaviors at the critical energies and obey scaling laws. These results may extend our understanding of the driven Kerr oscillator and ESQPTs in finite-component systems.
Driven-dissipative phase transition in an open quantum systems nowdays is among the most rapidly developing fields of quantum optics. An example of such a system is depicted in Fig. 1 (a) , which shows a superconducting microwave cavity undergoing two-photon driving at g rate as well as two-photon dissipation at γ rate [1] . In this system, the presence of the two-photon dissipation does not destroy, but stabilizes the quantum state, also known as Schrödinger cat state, and it is possible to prepare a dynamically protected qubit on its basis [2] . Numerical methods are commonly applied to study such quantum systems. However, their use typically does not allow to uncover physical phenomena underlying the evolution of the system. In the limit of a large number of photons, a qualitative picture of the occurring phenomena can be obtained using the so-called semiclassical approximation. Nevertheless, this approach completely neglects the quantum fluctuations, which can play a significant role, even in the case of large average number of photons in the system.