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Title: | Numerical representation of quantum states in the positive-P and Wigner representations |

Author(s): | M. K. Olsen, A. S. Bradley |

As Contributors: | Ashton Bradley |

Journal ref.: | Optics Communications 282, 3924-3929 |

DOI: | http://dx.doi.org/10.1016/j.optcom.2009.06.033 |

Date: | 2009-10-01 |

Numerical stochastic integration is a powerful tool for the investigation of quantum dynamics in inter- acting many-body systems. As with all numerical integration of differential equations, the initial condi- tions of the system being investigated must be specified. With application to quantum optics in mind, we show how various commonly considered quantum states can be numerically simulated by the use of widely available Gaussian and uniform random number generators. We note that the same methods can also be applied to computational studies of Boseâ€“Einstein condensates, and give some examples of how this can be done.

## Ashton Bradley on 2018-01-24

- The discussion in section 3.2 on thermal state sampling is incorrect. See the erratum for details: https://doi.org/10.1016/j.optcom.2016.02.068

- Equation (18) should read

\[

P(\alpha,\alpha^+)=\frac{1}{4\pi^2}\langle \tfrac{1}{2}(\alpha+(\alpha^+)^*)|\rho|\tfrac{1}{2}(\alpha+(\alpha^+)^*)\rangle e^{-|\alpha-(\alpha^+)^*|^2/4}

\]

- The transform in equation (33) should be precisely equation (19), without the switch $\mu\longleftrightarrow\gamma$. The inversion (36) is then consistent as published.

- A library of functions is available in Julia: https://github.com/AshtonSBradley/PhaseSpaceTools.jl