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

4. Continuously monitored Bose condensates: quasiprobability distributionsand as such is better behaved and less singular than the P function. The Wigner operatorcorrespondences areĉˆρĉ † ˆρˆρĉˆρĉ †⇐⇒⇐⇒⇐⇒⇐⇒(α + 1 )∂2 ∂α ∗ G(α),(α ∗ − 1 )∂G(α),2 ∂α(α − 1 )∂2 ∂α ∗ G(α),(α ∗ + 1 )∂G(α).2 ∂α(4.22a)(4.22b)(4.22c)(4.22d)The evolution equation for the Wigner distribution looks similar to that of the Pdistribution:[ √ΓdG(t, α, α ∗ )=(|α 2 | 2 −|α 1 | 2 − 〈 |α 2 | 2 −|α 1 | 2〉 − ∂2∂α 2 ∂α ∗ 2+ ∂2∂α 1 ∂α ∗ 1)dW + dtL G + dtD G]G, (4.23a)where the differential operators areL G = ∑ (∂ iΩ∂α j 2 α 3−j + iκ ( |α j | 2 −|α 3−j | 2) α j + Γ )8 α j + c.c. (4.23b)D G =j=1,24∑µ,ν=1∂ 2−Γα µ α ν DµνΓ , (4.23c)∂α µ ∂α ν 8except that there are higher-order derivative terms, which we have ignored.There isno diffusion due to atom collisions, but now the conditional part of the master equation(containing dW ) contributes to the diffusion. There are other small differences in the driftterms as well.The common disadvantage of the P and Wigner distributions is that both may gonegative. This can occur when the diffusion matrix is not positive semidefinite; P and Gcannot be regarded as true probability distributions. The Q function does not have thisproblem, since it is by definition positive:〈 ∣ ∣α∣ˆρ c α 〉Q(α) = ≥ 0, (4.24)π98