Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

Appendix BQuasiprobability distributions usingsqueezed-state expansionsAs the numerical results from Ch. 7 indicate, the use of the coherent-state positive-Prepresentation in evaporative cooling simulations leads to phase-space excursions when theatoms are strongly interacting. Large phase-space excursions indicate that the positive-Pfunction is diffusing rapidly and that it may be developing finite boundary terms, leadingto the breakdown of the method.These phase-space excursions arise because the system is evolving in a state whichis not compactly described by a coherent-state expansion. Moments such as the Fourieratom density n(k) =a 1 (k)a ∗ 2 (k) becoming negative in single trajectories indicates that thestate description requires superpositions of coherent states.As a possible alternative basis set for phase-space representations, we now consider theset of squeezed states.B.1 Squeezed statesA general squeezed state for a singe mode is defined as:〉 ∣ 〉∣ ξ,α = D(α)S(ξ) ∣0 ,(B.1)in which the vacuum state is squeezed by S(ξ) and then displaced by D(α). The squeezingand displacement operators are defined respectively asS(ξ) = e 1 2 (ξ∗ â 2 −ξâ †2 )(B.2a)D(α) = e α↠−α ∗â, (B.2b)172