Efficient method to generate time evolution of the Wigner function for open quantum systems (1212.3406v4)

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.

• The paper introduces a spectral split-operator method using FFT that achieves O(N log N) computational complexity for evolving the Wigner function.
• The paper validates the approach by comparing simulation results with classical Fokker-Planck dynamics, demonstrating environment-induced quantum decoherence.
• The paper successfully applies the method to both single and two-particle systems, paving the way for scalable many-body open quantum simulations.

Efficient Time Evolution of the Wigner Function for Open Quantum Systems

This paper presents an efficient computational method for simulating the dynamical evolution of the Wigner function in open quantum systems using a fast-Fourier transform (FFT)-based approach. The Wigner function is a quasiprobability distribution that provides a bridge between quantum mechanics and classical statistical mechanics, often used to visualize quantum chaos and explore the quantum-to-classical transition. However, evolving the Wigner function in open quantum systems is computationally challenging due to the complexity of the underlying equations, such as Moyal's equation, which can be represented as an infinite partial differential equation or an integral equation.

Method Overview

The authors introduce a spectral split-operator method, leveraging FFT, which operates with computational complexity $O(N \log N)$, where $N$ is the size of the data grid representing the Wigner function. This allows simulations of open system dynamics that were previously computationally prohibitive. The key innovation is in the use of an elegant phase-space formalism for quantum mechanics, combined with efficient computational techniques that take advantage of existing numerical libraries and parallelization capabilities.

Results and Implications

The paper demonstrates the utility of this method through simulations of both single and two-particle systems under varying environmental interactions. For a single particle, comparisons are made between simulation results and the classical Fokker-Planck and Koopman-von Neumann dynamics, illustrating the effects of environment-induced decoherence which leads to the anticipated quantum-to-classical transition. Notably, they showcase that in the asymptotic limits, the decohered quantum dynamics approach solutions of the Fokker-Planck equation, highlighting the role of diffusion induced by environmental interactions.

In two-particle systems, the method handles the complex interactions and mutual decoherence effects, marking a significant advancement in simulating many-body open quantum systems. For instance, when environmental interactions are selectively applied to one of two coupled particles, the resulting dynamics are shown to cause loss of coherence in both, illustrating entanglement's role in quantum decoherence.

Future Directions

This work has promising implications for both theoretical research and practical applications, offering a robust tool for investigating open quantum systems. The authors suggest the potential of extending this method to paper larger systems, wherein the efficiency and scalability of FFT-based methods will be particularly beneficial. Insights from such studies could significantly impact quantum information processing, quantum optics, and the foundational understanding of quantum-to-classical transitions.

In summary, this paper contributes a noteworthy computational technique for simulating the Wigner function in open quantum systems, integrating advanced formalism with practical numerical methods. This approach presents a viable pathway for exploring complex quantum phenomena using limited computational resources.