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On the interpretation of single stochastic trajectories of the Wigner distribution

Robert J. Lewis-Swan, M. K. Olsen, Karen V. Kheruntsyan

Published Mar 19, 2015• 1 citation
OpenAlex

Abstract

individual stochastic realisations of squared mode amplit udes |�| 2 of the sampled Wigner distribution W(�), which we denote via ˜ Pn. We provide an operational definition of ˜ Pn in terms of the underlying Wigner function, which allows us to explicitly calculate the overlap between the two number distributions and hence quantify the statistical distance between them. We find that there is inde ed a close quantitative correspondence between Pn and ˜ Pn for a wide range of states, justifying the broadly accepted v iew that, for highly occupied modes, individual stochastic realisations of Wigner trajectorie s should approximately correspond to outcomes of single experiments. However, we also find counterexamples for whic h high mode occupation may not be sufficient for such an interpretation; we find instead that a more releva nt and sufficient requirement is the smoothness and broadness of the Wigner function W(�) for the state of interest relative to the scale of oscillatio ns of the Wigner function for a Fock state, W|ni(�).

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individual stochastic realisations of squared mode amplit udes |�| 2 of the sampled Wigner distribution W(�), which we denote via ˜ Pn. We provide an operational definition of ˜ Pn in terms of the underlying Wigner function, which allows us to explicitly calculate the overlap between the two number distributions and hence quantify the statistical distance between them. We find that there is inde ed a close quantitative correspondence between Pn and ˜ Pn for a wide range of states, justifying the broadly accepted v iew that, for highly occupied modes, individual stochastic realisations of Wigner trajectorie s should approximately correspond to outcomes of single experiments. However, we also find counterexamples for whic h high mode occupation may not be sufficient for such an interpretation; we find instead that a more releva nt and sufficient requirement is the smoothness and broadness of the Wigner function W(�) for the state of interest relative to the scale of oscillatio ns of the Wigner function for a Fock state, W|ni(�).
We examine the interpretation of individual phase-space trajectories of the Wigner function as corresponding to possible outcomes of single experimental trials. To this end, we investigate the relation between the true (measured) particle number distribution PnP_nPn​ for a single-mode state and that obtained by discretely binning the individual stochastic realisations of squared mode amplitudes ∣α∣2|α|^2∣α∣2 of the sampled Wigner distribution W(α)W(α)W(α), which we denote via P~n\tilde{P}_nP~n​. We provide an operational definition of P~n\tilde{P}_nP~n​ in terms of the underlying Wigner function, which allows us to explicitly calculate the overlap between the two number distributions and hence quantify the statistical distance between them. We find that there is indeed a close quantitative correspondence between PnP_nPn​ and P~n\tilde{P}_nP~n​ for a wide range of states, justifying the broadly accepted view that, for highly occupied modes, individual stochastic realisations of Wigner trajectories should approximately correspond to outcomes of single experiments. However, we also find counterexamples for which high mode occupation may not be sufficient for such an interpretation, we find instead that a more relevant and sufficient requirement is the smoothness and broadness of the Wigner function W(α)W(α)W(α) for the state of interest relative to the scale of oscillations of the Wigner functions for the relevant Fock states.
We consider the Wigner quasiprobability distribution function of a single mode of an electromagnetic or matter-wave field to address the question of whether a direct stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution function ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ that closely approximates the true particle number probability distribution Pn{P}_{n}Pn​ of the underlying quantum state. By providing an operational definition of the binned distribution ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ in terms of the Wigner function, we explicitly calculate the overlap between ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ and Pn{P}_{n}Pn​ and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence between ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ and Pn{P}_{n}Pn​ for a wide range of quantum states that have a smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for which ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ does not closely approximate Pn{P}_{n}Pn​.
The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used PPP distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the PPP distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the PPP distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.
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-PPP 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.