Phase Space Representations and Stochastic Simulations of Dissipative Phase Transitions in Nonlinear Quantum Systems

Abstract

The class of generalized P representations allow a phase space description of quantum mechanics which, for certain systems, uniquely maps to a computationally efficient description via stochastic differential equations (SDEs). Recent interest in dissipative phase transitions in many-body open quantum systems has opened up P representations as a potential candidate for efficiently simulating such phenomena, since the exponential growth of the Hilbert space with system size quickly renders standard methods unfeasible. We provide an overview of the various P representations and their associated problems, and discuss ways of dealing with numerical issues that arise. A numerical infrastructure is developed to efficiently simulate SDEs, and this is directly applied to the study of phase transition phenomena in real nonlinear quantum systems. We first study the bistability and first-order phase transition in the single Kerr oscillator. We then extend the positive-P representation to a dimer of coupled Kerr oscillators, and study the emergent spatial-symmetry-breaking phase transition. Our work on the Kerr dimer extends very recent studies into high excitation number regimes unreachable using standard density matrix approaches.