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arXiv:2205.02622 · Physical Review A

Diverging current fluctuations in critical Kerr resonators

Michael J. Kewming, Mark T. Mitchison, Gabriel Teixeira Landi

Published Sep 19, 2022• 22 citations

Abstract

The parametrically pumped Kerr model describes a driven-dissipative nonlinear cavity, whose nonequilibrium phase diagram features both continuous and discontinuous quantum phase transitions. We consider the consequences of these critical phenomena for the fluctuations of the photocurrent obtained via continuous weak measurements on the cavity. Considering both direct photodetection and homodyne detection schemes, we find that the current fluctuations diverge exponentially at the discontinuous phase transition. However, we find strikingly different current fluctuations for these two detection schemes near the continuous transition, a behavior which is explained by the complementary information revealed by measurements in different bases. To obtain these results, we develop formulas to efficiently compute the diffusion coefficient---which characterizes the long-time current fluctuations---directly from the quantum master equation, thus connecting the formalisms of full counting statistics and stochastic quantum trajectories. Our findings highlight the rich features of current fluctuations near nonequilibrium phase transitions in quantum-optical systems.

DOI: 10.1103/physreva.106.033707

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The parametrically pumped Kerr model describes a driven-dissipative nonlinear cavity, whose nonequilibrium phase diagram features both continuous and discontinuous quantum phase transitions. We consider the consequences of these critical phenomena for the fluctuations of the photocurrent obtained via continuous weak measurements on the cavity. Considering both direct photodetection and homodyne detection schemes, we find that the current fluctuations diverge exponentially at the discontinuous phase transition. However, we find strikingly different current fluctuations for these two detection schemes near the continuous transition, a behavior which is explained by the complementary information revealed by measurements in different bases. To obtain these results, we develop formulas to efficiently compute the diffusion coefficient---which characterizes the long-time current fluctuations---directly from the quantum master equation, thus connecting the formalisms of full counting statistics and stochastic quantum trajectories. Our findings highlight the rich features of current fluctuations near nonequilibrium phase transitions in quantum-optical systems.

Understanding nonequilibrium transport and fluctuations driven by nonclassical light in nonlinear quantum optical systems is challenging. Here, we formulate a driven quantum master equation in the rotating frame combined with full counting statistics, retaining the drive-induced frequency shifts in the system-reservoir interactions and providing a unified description of drive-assisted incoherent transitions based on the rotated system dressed-basis. Applied to a Kerr resonator and a nonlinear Jaynes-Cummings model, the approach reveals pronounced multiphoton-resonant enhancement of the drive input energy current, with significant high peaks under two-photon driving compared to the single-photon case. The two-photon resonance relations are analytically obtained. Resonance structure in the nonlinear Jaynes-Cummings model nonlinearly relies on qubit-photon couplings, with additional dressed-state branches. A low-energy-state approximation attributes the resonant current enhancement to dressed-state hybridization and efficient activation of incoherent energy exchange processes. Beyond the average current, two-photon-driving induced energy exchange picture near resonance also substantially modifies the second-order current fluctuation and higher-order current cumulant, while increasing the time-normalized signal-to-noise ratio. Adding two-photon loss introduces extra incoherent photon-pair exchange pathways and further reshapes the current and its fluctuations. We hope these results may deepen interpretation of photon driving and quantum dissipation cooperatively governing nonequilibrium energy transport and fluctuations.

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

\textit{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

\textit{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.

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.

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,

{Z}_{2}

symmetry breaking,

\mathcal{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.