Correspondence between stochastic Wigner trajectories and individual experimental runs

Abstract

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 $P_n$ for a single-mode state and that obtained by discretely binning the individual stochastic realisations of squared mode amplitudes $|\alpha|^2$ of the sampled Wigner distribution $W(\alpha)$, which we denote via $\tilde{P}_n$. We provide an operational definition of $\tilde{P}_n$ 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 $P_n$ and $\tilde{P}_n$ 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 $W(\alpha)$ for the state of interest relative to the scale of oscillations of the Wigner functions for the relevant Fock states.