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

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.

💡 Research Summary

The paper introduces a fast‑Fourier‑transform (FFT) based pseudo‑spectral algorithm for propagating the Wigner function of open quantum systems. The Wigner function provides a phase‑space representation of quantum states, bridging quantum and classical dynamics, but its time evolution under the Wigner‑Moyal equation together with Lindblad dissipators is notoriously costly: high‑order derivatives and non‑local Moyal brackets typically lead to O(N²) operations for an N‑point discretization and can be numerically unstable.

The authors overcome these obstacles by (i) separating the Poisson bracket (classical part) and the quantum corrections (Moyal terms), (ii) transforming spatial derivatives into multiplication in momentum (k‑space) via FFT, and (iii) handling the Lindblad dissipators—expressed as diffusion and drift terms after a Wigner‑Weyl transform—also through spectral differentiation. Each time step consists of a forward FFT, algebraic multiplication by the appropriate k‑dependent factors, an inverse FFT, and a point‑wise evaluation of the nonlinear potential term. By employing Strang splitting or a fourth‑order Runge‑Kutta scheme for the time integration, the method preserves the normalization of the Wigner function and remains stable even for stiff dissipative dynamics.

The computational complexity scales as O(N log N) with memory usage O(N), allowing the authors to work with high‑resolution grids (e.g., 2048 × 2048 for a single particle) on standard workstations or modest GPUs. The algorithm is straightforward to implement in Python/NumPy or CUDA, making it accessible to a broad community.

To demonstrate the capabilities, the paper presents two families of simulations.

1. Single‑particle dynamics – The authors study a harmonic oscillator and a double‑well potential under three conditions: (a) unitary evolution, (b) evolution with a thermal bath modeled by a Lindblad operator L = √γ a (γ is the damping rate), and (c) evolution with pure diffusion (phase‑space diffusion coefficient D). They compare the Wigner‑function results with classical Fokker‑Planck and Koopman‑von Neumann equations. As γ and the bath temperature increase, the negative regions of the Wigner function shrink, the purity decays exponentially, and the phase‑space distribution converges to the classical probability density, illustrating the quantum‑to‑classical transition induced by the environment.

2. Two‑particle interacting system – The Hamiltonian includes a bilinear coupling g x₁ x₂ between the particles. Only particle 1 is coupled to an environment via L₁ = √γ a₁. The simulation shows that decoherence of particle 1 propagates non‑locally: the cross‑terms in the joint Wigner function, which encode quantum correlations, vanish even though particle 2 experiences no direct dissipation. The effect is stronger for larger coupling g, providing a clear phase‑space picture of environment‑mediated loss of coherence in multipartite systems.

The authors validate the method by monitoring (a) normalization (∫W dx dp = 1), (b) energy conservation in the unitary limit, and (c) spectral convergence of the FFT‑based derivatives. Relative errors stay below 10⁻⁶ across a wide range of time steps, confirming both accuracy and robustness.

Key advantages of the approach are:

• Efficiency – O(N log N) scaling enables simulations that were previously prohibitive, especially for higher‑dimensional phase spaces (e.g., two particles).
• Stability – Spectral differentiation avoids the numerical noise associated with finite‑difference schemes, and the splitting integrator handles stiff dissipative terms without sacrificing accuracy.
• Flexibility – Any Lindblad operator that can be cast into diffusion/drift form in phase space (including non‑Markovian extensions) fits naturally into the framework.
• Physical insight – Direct visualization of the Wigner function makes it possible to track decoherence, quantum interference, and the emergence of classical probability flows in real time.

In summary, the paper delivers a practical, high‑performance tool for simulating open quantum dynamics in phase space. By marrying pseudo‑spectral methods with FFT, it reduces computational overhead while preserving the essential physics of quantum‑classical correspondence and environment‑induced decoherence. This development opens the door to systematic studies of quantum optics, superconducting circuits, quantum chemistry, and quantum information platforms where environmental effects play a pivotal role and where phase‑space techniques provide unique analytical and visual advantages.