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Classical phase-space approach for coherent matter waves

We investigate a classical phase-space approach of matter-wave propagation based on the Truncated Wigner Equation (TWE). We show that such description is suitable for ideal matter waves in quadratic time-dependent confinement as well as for harmonically trapped Bose Einstein condensates in the Thomas-Fermi regime. In arbitrary interacting regimes, the TWE combined with the moment method yields the low-energy spectrum of a condensate as predicted by independent variational methods. TWE also gives the right breathing mode frequency for long-ranged interactions decaying as $1/r^2$ in 3D and for a contact potential in 2D. Quantum signatures, beyond the TWE, may only be found in the condensate dynamics beyond the regimes of classical phase-space propagation identified here.