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arXiv:2106.04045

Noise-Resilient Phase Transitions and Limit-Cycles in Coupled Kerr Oscillators

Hadiseh Alaeian, M. Soriente, Khadijeh Najafi, Susanne F. Yelin

Published Jun 8, 2021• 0 citations
arXivHTMLPDF

Abstract

Driven-dissipative quantum many-body systems have been the subject of many studies in recent years. They possess unique, novel classes of dissipation-stabilized quantum many-body phases including the limit cycle. For a long time it has been speculated if such a behavior, a recurring phenomenon in non-linear classical and quantum many-body systems, can be classified as a time crystal. However, the robustness of these periodic dynamics, against quantum fluctuations is an open question. In this work we seek the answer to this question in a canonical yet important system, i.e., a multi-mode cavity with self and cross-Kerr non-linearity, including the fluctuation effects via higher order correlations. Employing the Keldysh path integral, we investigate the Green's function and correlation of the cavity modes in different regions. Furthermore, we extend our analysis beyond the mean-field by explicitly including the effect of two-body correlations via the 2nd-cumulant expansion. Our results shed light on the emergence of dissipative phase transitions in open quantum systems and clearly indicate the robustness of limit-cycle oscillations in the presence of the quantum fluctuations.

DOI
10.3929/ethz-b-000660755

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Driven-dissipative quantum many-body systems have been the subject of many studies in recent years. They possess unique, novel classes of dissipation-stabilized quantum many-body phases including the limit cycle. For a long time it has been speculated if such a behavior, a recurring phenomenon in non-linear classical and quantum many-body systems, can be classified as a time crystal. However, the robustness of these periodic dynamics, against quantum fluctuations is an open question. In this work we seek the answer to this question in a canonical yet important system, i.e., a multi-mode cavity with self and cross-Kerr non-linearity, including the fluctuation effects via higher order correlations. Employing the Keldysh path integral, we investigate the Green's function and correlation of the cavity modes in different regions. Furthermore, we extend our analysis beyond the mean-field by explicitly including the effect of two-body correlations via the 2nd-cumulant expansion. Our results shed light on the emergence of dissipative phase transitions in open quantum systems and clearly indicate the robustness of limit-cycle oscillations in the presence of the quantum fluctuations.
Quantum fluctuations are inherent in open quantum systems and they affect not only the statistical properties of the initial state but also the time evolution of the system. Using a generic minimal model, we show that quantum noise smooths the transition between a stationary and a dynamical phase corresponding to a limit cycle (LC) in the semiclassical mean-field approximation of a generic open quantum system. Employing truncated Wigner approximation, we show that the inherent quantum noise pushes the system to exhibit signatures of LCs for interaction strengths lower than the critical value predicted by the standard mean-field theory, suggesting a noise-induced emergence of temporal ordering. Our work demonstrates that the apparent crossover-like behavior between stationary phases brought by finite-size effects from quantum fluctuations also apply to transitions involving dynamical phases. To demonstrate this on a specific physical system, we consider a transversely pumped atom-cavity setup, wherein LCs have been observed and identified as continuous time crystals. We compare the oscillation frequencies of the LCs in the one-dimensional (1D) and two-dimensional regimes, and find that the frequencies have larger shot-to-shot fluctuations in 1D. This has an important consequence in the effectiveness of entrainment of LCs for a periodically driven pump intensity or light-matter coupling strength.
Driven-dissipative quantum many-body systems have attracted increasing interest in recent years as they lead to novel classes of quantum many-body phenomena. In particular, mean-field calculations predict limit cycle phases, slow oscillations instead of stationary states, in the long-time limit for a number of driven-dissipative quantum many-body systems. Using a cluster mean-field and a self-consistent Mori projector approach, we explore the persistence of such limit cycles as short range quantum correlations are taken into account in a driven-dissipative Heisenberg model.
We study a driven-dissipative model of spins one-half (qubits) on a lattice with nearest-neighbor interactions. Focusing on the role of spatially extended spin-spin correlations in determining the phases of the system, we characterize the spatial structure of the correlations in the steady state, as well as their temporal dynamics. In dimension one we use essentially exact matrix-product-operator simulations on large systems, and pushing these calculations to dimension two, we obtain accurate results on small cylinders. We also employ an approximation scheme based on solving the dynamics of the mean field dressed by the feedback of quantum fluctuations at leading order. This approach allows us to study the effect of correlations in large lattices with over one hundred thousand spins, as the spatial dimension is increased up to five. In dimension two and higher we find two new states that are stabilized by quantum correlations and do not exist in the mean-field limit of the model. One of these is a steady state with mean magnetization values that lie between the two bistable mean-field values and whose correlation functions have properties reminiscent of both. The correlation length of the new phase diverges at a critical point, beyond which we find emerging a new limit cycle state with the magnetization and correlators oscillating periodically in time.
The interplay between many-body interactions and controlled dissipation provides a rich setting for nonequilibrium quantum phases. We explore the superradiant phase transitions in a Rydberg-interacting open Dicke model with cavity loss and local dissipation. We find that the nonlinear feedback from atomic interactions stabilizes the atomic collective excited state and induces resonant collective motion, which can be further converted into robust superradiant oscillations with the inclusion of local dissipation. Beyond the mean-field approximation, we prove this system can support the finite-size local dissipative temporal order, i.e., continuous time crystal, from evidences of the decreasing damping of a finite-frequency Liouvillian mode and the linear-size scaling of quantum fluctuations. The correlation between the atomic colletive dynamics and the cavity photon number makes our findings easy to detect from output optical signal. By bridging interacting atomic matter and driven-dissipative cavity QED, our Rydberg-interacting Dicke model provides a unique spin-1/2 platform for exploring dynamical phase transitions and macroscopic temporal order in open quantum matter.
Understanding the non-Markovian dynamics of dissipative processes induced by memory effects of the environment is a fundamental subject of open quantum systems. Because of the complexity of open quantum systems, e.g., the multiple energy scales involving that of the system, the environment, and their mutual coupling, it is generally a challenging task to characterize the relationship among the parameters of the system dynamics and the reservoir spectra. For the two-level spontaneous emission model within structured environments, it was shown in a recent literature (Opt. Lett. 38, 3650) that a functional relation could be established between the asymptotically non-decaying population and the spectral density of the reservoir as the system undergoes a long-time evolution. It hence renders a distinct perspective to look into the character of long-lived quantum coherence in the corresponding non-Markovian process. This article is devoted to further investigate the phenomena of limit cycle oscillations possibly occurring in such non-Markovian dissipative systems in a long-time evolution. For a two-level system subjected to an environment with Ohmic class spectra, due to the presence of a unique bound-state mode of the system, the evolution trajectory of the given initial states will converge to a limit cycle in the Bloch space. The dependence of the radius and the location of the limit cycle on the spectral density function of the reservoir are manifested by virtue of the described functional relation. For the model subjected to a photonic crystal environment with multiple bands, our studies reveal that, owing to the presence of two or more bound states, the evolution trajectory of the system will converge to a toric curve of a paraboloid in the Bloch space and the phenomena of periodic or quasi-periodic oscillations could exhibit. While the equation of the parabolic curve is fully determined by the initial values of the state vector in the Bloch space, our results reveal that the scope of the evolution trajectory inside the toric curve is related to the spectral density of the reservoir and their quantified relation is distinctly characterized. Finally, the asymptotic dynamics of the correlations of a two-qubit system is discussed when it is subjected locally to the non-Markovian dissipative process.
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.