Efficient method to generate time evolution of the Wigner function for open quantum systems

Abstract

The Wigner function is a useful tool for exploring the transition between quantum and classical dynamics, as well as the behavior of quantum chaotic systems. Evolving the Wigner function for open systems has proved challenging however; a variety of methods have been devised but suffer from being cumbersome and resource intensive. Here we present an efficient fast-Fourier method for evolving the Wigner function, that has a complexity of $O(N\log N)$ where $N$ is the size of the array storing the Wigner function. The efficiency, stability, and simplicity of this method allows us to simulate open system dynamics previously thought to be prohibitively expensive. As a demonstration we simulate the dynamics of both one-particle and two-particle systems under various environmental interactions. For a single particle we also compare the resulting evolution with that of the classical Fokker-Planck and Koopman-von Neumann equations, and show that the environmental interactions induce the quantum-to-classical transition as expected. In the case of two interacting particles we show that an environment interacting with one of the particles leads to the loss of coherence of the other.

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• Evolution of hybrid quantum-classical wavefunctions
• Dirac open quantum system dynamics: formulations and simulations
• Quantum Dynamics in Phase Space using Projected von Neumann Bases
• Time evolution of coupled spin systems in a generalized Wigner representation
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• The wave operator representation of quantum and classical dynamics
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• Operational Dynamical Modeling of spin 1/2 relativistic particles: the Dirac equation and its classical limit
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• Free to Harmonic Unitary Transformations in Quantum and Koopman Dynamics
• Exponential unitary integrators for nonseparable quantum Hamiltonians
• Memory Effect in Upper Bound of Heat Flux Induced by Quantum Fluctuations
• Lindblad Superoperators from Wigner's Phase Space Continuity Equation
• Wigner's Phase Space Current for Variable Beam Splitters -Seeing Beam Splitters in a New Light-
• Classical Propagation in the Quantum Inverted Oscillator
• Double-Bracket Master Equations: Phase-Space Representation and Classical Limit
• Extended Hilbert Phase Space and Dissipative Quantum Systems
• Towards the resolution of a quantized chaotic phase space: The interplay of dynamics with noise