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arXiv:1912.02066 · Physical Review Research

Gaussian trajectory approach to dissipative phase transitions: The case of quadratically driven photonic lattices

Wouter Verstraelen, R. Rota, Vincenzo Savona, Michiel Wouters

Published May 12, 2020• 38 citations
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Abstract

The authors explore the behavior of nonlinear optical lattices in the limit of strong losses by combining the quantum trajectory framework for open quantum systems, with a Gaussian variational ansatz commonly used for closed many body-systems. The paper shows that the photon losses bring the system to the universality class of the thermal Ising model, providing evidence that losses act similarly to temperature.

DOI
10.1103/physrevresearch.2.022037

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The authors explore the behavior of nonlinear optical lattices in the limit of strong losses by combining the quantum trajectory framework for open quantum systems, with a Gaussian variational ansatz commonly used for closed many body-systems. The paper shows that the photon losses bring the system to the universality class of the thermal Ising model, providing evidence that losses act similarly to temperature.
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We study an array of coupled optical cavities in the presence of two-photon driving and dissipation. The system displays a critical behavior similar to that of a quantum Ising model at finite temperature. Using the corner-space renormalization method, we compute the steady-state properties of finite lattices of varying size, both in one and two dimensions. From a finite-size scaling of the average of the photon number parity, we highlight the emergence of a critical point in regimes of small dissipations, belonging to the quantum Ising universality class. For increasing photon loss rates, a departure from this universal behavior signals the onset of a quantum critical regime, where classical fluctuations induced by losses compete with long-range quantum correlations.
We construct a class of variational methods for the study of open quantum systems based on Gaussian ansatzes for the quantum trajectory formalism. Gaussianity in the conjugate position and momentum quadratures is distinguished from Gaussianity in density and phase. We apply these methods to a driven-dissipative Kerr cavity where we study dephasing and the stationary states throughout the bistability regime. Computational cost proves to be similar to the Truncated Wigner Approximation (TWA) method, with at most quadratic scaling in system size. Meanwhile, strong correspondence with the numerically-exact trajectory description is maintained so that these methods contain more information on the ensemble constitution than TWA and can be more robust.
Simulating dissipative quantum many-body systems presents a significant theoretical challenge. Here, the authors develop an efficient nonperturbative framework that is tailored to dissipative impurity problems, combining a superposition of Gaussian states (SGS) variational ansatz with the quantum trajectory approach. They showcase the method with a model of a spinful impurity subject to two-body losses embedded in a fermionic bath, for which they explore the full crossover from weak to strong dissipation. Their approach captures the emergence of a loss-induced Kondo effect under strong dissipation, and reveals an exotic negative conductance phenomenon at intermediate loss rates.
Driven-dissipative Kerr lattices with two-photon driving are experimentally relevant systems known to exhibit a symmetry-breaking phase transition, which belongs to the universality class of the thermal Ising model for the parameter regime studied here. In this work, we perform finite-size scaling of this system as it is quenched to the transition, and the dynamical critical exponent is found to be compatible with $z\ensuremath{\approx}2.18$, corresponding with metropolis dynamics in classical simulations. Furthermore, we show that the Liouvillian gap scales with the same exponent, similar to scaling of the Hamiltonian gap at quantum phase transitions in closed systems.
We adopt a diffusive-Gutzwiller approach to investigate a phase transition in a quadratically driven-dissipative Bose-Hubbard lattice. Diffusive trajectories may lead to lower average entanglement as compared to jump-like trajectories and have a natural tendency to approach coherent states, therefore the method can be less prone to the bias induced by the fully uncorrelated Gutzwiller ansatz. Averaging over trajectories does lead to classical correlations and this allows us to address the correlation length of such 2D lattices of open quantum systems which is the main goal of this work. Under this approximations, we find negligible correlation length in the low density phase and apparently unbounded length grow in the high density phase. Additionally, we show that the effective relaxation times associated to the times scales for synchronisation in the high density phase may also diverge suggesting the vanishing of the Lindbladian gap.