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

B. Quasiprobability distributions using squeezed-state expansionsξ m that would minimise the spread in phase space. But not only do we not know thisoptimum value, we also have no guarantee that it exists as a constant independent of thesystem dynamics. In fact, we would expect the optimum value ξ m to be correlated toα (for the single-mode case the phase of ξ m must be such as to minimise the amplitudefluctuations). So we need a phase-space representation in which ξ is a variable:∫ˆρ = dαdα ∗ dξdξ ∗ P (α, α ∗ ,ξ,ξ ∗ ) ∣ ∣α, ξ 〉〈 α, ξ ∣ (B.11)To obtain evolution equations for ξ, we need to rewrite the effect of the creation andannihilations on the squeezed state in terms of partial derivatives with respect to ξ, inthe manner of Eq. (B.7). To prevent any higher order derivative terms from appearingin the evolution equation for the P distribution, we need direct operator correspondencesfor â 2 and â †2 . One may consider doing this via the squeezing operator S(ξ,ξ ∗ ). Thesimplest way to differentiate this operator (because [ â 2 , â †2] does not commute with â 2 orâ †2 )appearstobeL(ξ,ξ ∗ )S(ξ,ξ ∗ ) =(= 1 2ξ ∗ ∂∂ξ ∗ + ξ ∂ )S(ξ,ξ ∗ )∂ξ(ξ ∗ â 2 − ξâ †2) S(ξ,ξ ∗ ),(B.12)which is not useful, since we cannot generate operator correspondences separately for â 2and â †2 . The exponent of S(ξ,ξ ∗ ) can be factored to find ∂ ∂ξ S andlike[156]∂∂ξ ∗ S using resultse ξ∗Â−ξ† = e −ξ† e − ln cosh r ˆBe ξ∗ ,(B.13)where  = 1 2â2 and ˆB =â † â + 1 2. But this does not appear to lead to anything moreuseful.Alternatively the phase of ξ can be locked into that of α to ensure an expansion inamplitude-squeezed states:ˆρ =∫dαdα ∗ drP (α, α ∗ ,r) ∣ ∣ α, r〉〈α, r∣ ∣.(B.14)This reduces the phase-space dimensionality, but has complications of its own: sinceS(ξ,ξ ∗ ) −→ S(α, α ∗ ,r), the correspondences for â and â † must be rederived.Thus there are many ways in which generalised phase-space representations couldbe generated. However, finding one which gives exact phase-space equations (regardlessof whether these give improved statistics) is not a trivial matter, without having thesimplicity of coherent states.175