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arXiv:1805.01851 · Applied Sciences

Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems

Wouter Verstraelen, Michiel Wouters

Published Aug 21, 2018• 33 citations
arXivHTMLPDF

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.

DOI
10.3390/app8091427

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•Dec 11, 2019•1 citation
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.
We present a framework for simulating the open dynamics of spin-boson systems by combining variational non-Gaussian states with a quantum trajectories approach. We apply this method to a generic spin-boson Hamiltonian that has both Tavis-Cummings and Holstein type couplings and which has broad applications to a variety of quantum simulation platforms, polaritonic physics, and quantum chemistry. Additionally, we discuss how the recently developed truncated Wigner approximation for open quantum systems can be applied to the same Hamiltonian. We benchmark the performance of both methods and identify the regimes where each method is best suited. Finally, we discuss strategies to improve each technique.
In this paper, it is proposed to improve the quality of a variational description of a closed quantum system by adding fictitious dissipation that reduces the entanglement. The proposal is implemented for a small Bose-Hubbard chain in the mean field regime, which shows chaotic behavior and associated fast growth of quantum fluctuations. For appropriately chosen dissipation, good agreement with the truncated Wigner approximation is found. This shows that, when chaos leads to the breakdown of a variational method, adding fictitious dissipation can be a cure.
Phase-space approaches such as the truncated Wigner approximation (TWA) provide an efficient semiclassical framework for performing approximate simulations of the dynamics of open quantum many-body systems outside the reach of exact numerical methods but beyond the mean-field level. For bosonic systems, TWA is known to be equivalent to a Keldysh path-integral formulation truncated at second order in the so-called quantum fluctuations. This semiclassical approach provides an alternative transparent route towards approximate stochastic equations of motion, which can be efficiently solved. Here we establish the corresponding path-integral formulation for interacting open spin-1/21/21/2 systems using the continuous SU(2)\mathrm{SU}(2)SU(2) phase space. We show, in particular, that a consistent treatment of dissipation requires correctly mapping operator products onto the curved spin phase space, leading to stochastic equations that coincide with those obtained from the continuous TWA formulation and thus reproduce the exact dynamics of a single dissipative spin. Our results provide a unified field-theoretic foundation for the TWA to dissipative spin dynamics and offer a systematic starting point for extensions beyond the semiclassical approximation.
Many-body quantum-optical systems, where a collection of emitters interacts through a common electromagnetic reservoir, exhibit rich out-of-equilibrium behavior and hold promise for applications in quantum technologies. However, exact numerical simulations of their dynamics quickly become unfeasible due to the exponential growth of the Hilbert space with system size. Semiclassical, phase-space approaches -- such as the Truncated Wigner approximation (TWA) -- provide computationally efficient alternatives by capturing leading-order quantum fluctuations. In this paper, we present a comprehensive overview of how to tackle problems in many-body quantum optics using phase-space methods. We derive the exact partial differential equation governing many-body dissipative evolution in any phase-space representation and discuss the approximations that yield the dissipative TWA proposed by Mink and Fleischhauer [SciPost Phys. 15, 233 (2023)]. We find that PPP and QQQ distributions are generally suboptimal for many-body quantum optics. Additionally, we extend the formalism to calculate multi-time correlation functions, thereby broadening the scope of phase-space simulations of open spin systems to include coherence and spectral properties, as well as directional correlations of collectively radiating emitters. These developments provide valuable tools for investigating exotic light sources driven by collective dissipation, driven-dissipative phase transitions, and a wealth of many-body phenomena arising in state-of-the-art experimental platforms.
This thesis is devoted to the theoretical study of driven-dissipative many-body quantum systems with a particular focus on advanced computational methods. After reviewing several state-of-the-art numerical methods to determine the steady-state and dynamics of open quantum systems described by a Lindblad master equation for the density matrix, this thesis manuscript presents two original methods: (i) an optimal unraveling of stochastic quantum trajectories for disordered systems; (ii) a variational neural network method. The stochastic methods have been applied to investigate the driven-dissipative Bose-Hubbard lattice model that can for instance describe lattices of coupled nonlinear electromagnetic resonators such those obtained with semiconductor microcavities or superconducting quantum circuits. Our study has shown that a critical slowing down of the dynamics occurs for some parameters in dimensions higher than one. The phenomena is linked to a first-order dissipative phase transition and to lattice-level collective bistability. Furthermore, we reveal the role of disorder on the spatial correlation properties of these phenomena. In the last part of the thesis, we present the strong connection between fundamental concepts of machine learning and variational approaches in quantum physics. We present a novel computational method for open quantum systems based on neural network ans\"{a}tze that is compatible with a stochastic minimisation of a cost function through Monte Carlo sampling. As a first application, we have applied this method to the dissipative transverse quantum Ising model. Exciting perspectives and possible developments are also drawn in the concluding remarks.