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arXiv:1605.07276 · Physical Review A

Approximate particle number distribution from direct stochastic sampling of the Wigner function

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

Published Sep 9, 2016• 19 citations
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Abstract

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 ${P}_{n}$ 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 ${P}_{n}$ 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 ${P}_{n}$ 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 ${P}_{n}$.

DOI
10.1103/physreva.94.033814

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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 P\ifmmode~\else˜\fin{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}P\ifmmode~\else˜\fi​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 P\ifmmode~\else˜\fin{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}P\ifmmode~\else˜\fi​n​ in terms of the Wigner function, we explicitly calculate the overlap between P\ifmmode~\else˜\fin{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}P\ifmmode~\else˜\fi​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 P\ifmmode~\else˜\fin{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}P\ifmmode~\else˜\fi​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 P\ifmmode~\else˜\fin{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}P\ifmmode~\else˜\fi​n​ does not closely approximate Pn{P}_{n}Pn​.
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.
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 propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.
We define a quasiprobability distribution SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) which describes the quantum statistics of the photon number and phase observables of a single-mode field (or, equivalently, a harmonic oscillator). The properties of SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) are the photon number and phase analogies of the properties of Wigner's original function, which describes the position and momentum observables. For example, the marginals of SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) are the continuous phase and the discrete photon-number probability distributions. We give examples of the SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) representation of various states and show, in particular, that SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) displays the quantum interference associated with Schr\"odinger cat states. We also describe how SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) can be determined from quantities that are, in principle, measurable.
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.
Wigner representation In classical optics (Born and Wolf, 1999), the state of the electromagnetic oscillator is perfectly described by the statistics of the classical amplitude α . The amplitude may be completely fixed (then the field is coherent), or α may fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude?or, equivalently, the statistics of the components position q and momentum p introducing a phase space distribution W ( q , p ). (As explained in Section 3.1, the real and the imaginary part of the complex amplitude α can be regarded as the position and the momentum of the electromagnetic oscillator.) The distribution W ( q , p ) quantifies the probability of finding a particular pair of q and p values in their simultaneous measurement. Knowing the phase space probability distribution, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense the phase space distribution describes the state in classical physics. All this is much more subtle in quantum mechanics. First of all, Heisenberg's uncertainty principle prevents us from observing position momentum simultaneously and precisely. So it seems there is no point in thinking about quantum phase space. But wait! In quantum mechanics we cannot directly observe quantum states either. Nevertheless, we are legitimately entitled to use the concept of states as if they were existing entities (whatever they are). We use their properties to predict the statistics of observations.