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

4. Continuously monitored Bose condensates: quasiprobability distributionsthe quantum and semiclassical equations in the closed case was the collapse and revivalsequence, but the homodyne measurement suppresses this in the open system. Hencethere should be a good correspondence between the semiclassical and quantum systems.The mean-field approach has the advantage over the quantum two-mode approximationin that it is tractable and valid for large numbers of atoms.4.3 Atomic Q functionNow we cease analysing the measurement process in terms of distributions that are explicitlydefined as probabilities, and turn to distributions which in general may not qualifyas true probabilities. Useful quasiprobabilities can be derived by expanding the state interms of an overcomplete basis set, such as the set of coherent states, rather than theorthogonal number-state basis set used in Eq. (4.3). The overcompleteness allows a diagonalexpansion that results in a closed evolution equation for the distribution, and whichcontains all the effects of off-diagonal coherences. Thus the quantum state maps entirelyonto the quasiprobability distribution.4.3.1 Definition in terms of Bloch statesThe numerical calculation of quasiprobabilities provides considerable insight into the dynamicson the Bloch sphere. The atomic Q function is defined as〈 ∣ ∣ 〉µ ∣ρc∣µ Q(µ) = ≥ 0 (4.9)πwhere ∣ 〉 µ is the atomic coherent state, or Bloch state. Since Q(µ) is defined to be positiveand normalised, it does qualify as a true probability distribution, however we do not needto interpret it as such for the present application. The Bloch states can be written interms of the Dicke states as[6]∣ µ〉=j∑m=−j⎛⎝2jm + j⎞⎠12µ m+j ∣ 〉 ∣j, m(1 + |µ| 2 ) jz . (4.10)In terms of spherical coordinates, µ = e −iφ tan(θ/2) and〉 N〈Ĵx = sin θ cos φ2〉 N〈Ĵy = sin θ sin φ2〉 N〈Ĵz = − cos θ. (4.11)281