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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Dissipative phase transition in systems with two-photon driving and dissipation near the critical point

Valentin Yu. Mylnikov, S. O. Potashin, Grigorii S. Sokolovskii, Nikita S. Averkiev

Published Jun 21, 2021• 1 citation
Publisher

Abstract

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.

DOI
10.1109/cleo/europe-eqec52157.2021.9592659

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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.
Nowadays, dissipative phase transition (DPT) attracts a great deal of interest from the physics and quantum optics community. It is characterized by abrupt changes of the physical observables on the parameters of the system, such as a dissipation and drive rates as well as coupling constants. The increased attention to the DPT roots in significant development of a new research area of quantum optics, known as reservoir engineering, which allows to obtain a wide range of controllable nonequilibrium quantum systems. A prime example of such a system, which we will study here, is an optical oscillator with two-photon drive and two-photon dissipation (Fig 1(a)). The immediate practical application of such a model is the qubit platform based on the quantum Schrödinger cat states which is applicable for universal quantum computation [1]. However, the quantum and dissipative nature of such a system significantly complicates its consideration beyond the mean-field description. In statistical physics, one can apply the well-developed Landau theory, which is used for a wide range of equilibrium phenomena. Here, we rigorously demonstrate that the nonequilibrium DPT that arises in an optical oscillator with two-photon drive and two-photon dissipation can be mapped onto a nonlinear classical oscillator in a colored-noise environment in regions distant from the critical point. Furthermore, using Fokker-Plank approach, we develop a description of the quantum critical region near the critical point, where strong quantum fluctuations destroy mean-field treatment.
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.
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.
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.
We study a minimal model that has a driven-dissipative quantum phase transition, namely, a Kerr nonlinear oscillator subject to driving and dissipation. Using mean-field theory, exact diagonalization, and the Keldysh formalism, we analyze the critical phenomena in this system, showing which aspects can be captured by each approach and how the approaches complement each other. Then critical scaling and finite-size scaling are calculated analytically using the quantum Langevin equation. The physics contained in this simple model is surprisingly rich: it includes a continuous phase transition, Z2{Z}_{2}Z2​ symmetry breaking, PT\mathcal{PT}PT symmetry, state squeezing, and critical fluctuations. Due to its simplicity and solvability, this model can serve as a paradigm for exploration of open quantum many-body physics.