Non-Gaussian pure states and positive Wigner functions

Non-Gaussian correlations in a pure state are inextricably linked with certain nonclassical features, such as a non-positive-definite Wigner function. In a commonly used simulation technique in ultracold atoms and quantum optics, known as the truncated Wigner method, the quantum dynamics is mapped to stochastic trajectories in phase space, governed by a positive approximation to the true Wigner distribution. The question thus arises: How accurate is this approach in predicting truly nonclassical behavior? In this article, we benchmark the ability of the truncated Wigner phase-space method to reproduce the non-Gaussian statistics of the single-mode anharmonic oscillator. We find that the this method can reliably predict departures from Gaussian statistics over a wide range of particle numbers, whereas the positive-$P$ representation, which involves no approximations, is limited by rapidly growing statistical uncertainty. The truncated Wigner function, furthermore, is able to reproduce the non-Gaussian correlations while satisfying the condition for purity.

Figure 1

Third-order cumulant of the anharmonic oscillator for $N={10}^3$ (upper plot) and $N={10}^6$ particles (lower plot). Dot-dashed line (red), truncated Wigner simulations; dashed line (blue), positive-$P$ simulations; solid line (black), analytic results. $N_p={10}^8$ stochastic paths were used for each simulation, with $\pm \sigma$ estimates of sampling error given by the shading. The insets show the Wigner and analytic results for longer times.

Figure 2

Fourth-order cumulant of the anharmonic oscillator for $N={10}^3$ (upper plot) and $N={10}^6$ particles (lower plot). The symbols and parameters are as in Fig. 1. Note that the positive-$P$ simulation remains within two standard deviations of the analytic curve, even in the upper plot, where the discrepancy temporarily exceeds one standard deviation of the estimated sampling error (between $Nt=5$ and $Nt=6.5)$.