Abstract
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.
Highlights
Many-body systems of interacting photons have come under intense investigation over the last few years, with both circuit QED- and semiconductor heterostructure-based systems [1,2,3,4,5,6]
Such a Kerr cavity provides a model for polariton condensates [48], for which it has been shown that trajectory methods can provide an adequate description [62,63]. Coupled arrays of these cavities are an object of current interest [64], and it is in these systems that the Truncated Wigner Approximation (TWA) method has been proven insufficient [58]. As it is the most straightforward ansatz corresponding to the Gaussian approximation of Section 2, we assume the state to be Gaussian in the quadrature operators Xand Por, equivalently, â and â†, i.e., the states that are colloquially known as Gaussian states [60]
What distinguishes the NΘ-Gaussian method from TWA, is that the NΘ-Gaussian method is able to show the composition of the ensemble: it maintains information of individual trajectories, which is lost in TWA
Summary
Many-body systems of interacting photons have come under intense investigation over the last few years, with both circuit QED- and semiconductor heterostructure-based systems [1,2,3,4,5,6]. After preliminary work by Davies [10], this quantum trajectory formalism was developed towards its current shape [7,11] by a number of different groups [12,13,14,15] It is known under a variety of other names: Monte Carlo wave function method, quantum jump method or stochastic Schrödinger equation. Except for some works on the back-reaction of measurements on quantum many-body systems [40,41], the class of Gaussian states [42] has not received much attention as a variational ansatz for the simulation of quantum trajectories. Code S1 used for the numerical simulations is available as Supplementary Material
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