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

4. Continuously monitored Bose condensates: quasiprobability distributionsbe rather broad, leading to large sampling errors in any numerical simulation of the correspondingphase-space equations. A more natural set of states to work with is that ofthe atomic coherent states ∣ ∣µ 〉 , defined in Sec. 4.3.1. Distribution functions defined interms of these states are constrained to remain on the Bloch sphere. The disadvantagein using a quasiprobability distribution that better reflects the structure of the masterequation, is that the governing equations are more complicated and not so amenable tosimple analysis.In order to obtain an equation for the atomic Q function, the following matrix elementsare needed[148]:〈µ∣ ∣ ρĴ+∣∣µ 〉 =〈µ∣ ∣ ρĴ−∣∣µ 〉 =1 ∂(1 + |µ| 2 ) 2j ∂µ(1(1 + |µ| 2 ) 2j 2jµ − µ 2 ∂∂µQ(µ) (4.34a))Q(µ),(4.34b)whereQ(µ) = ( 1+|µ| 2) 2j 〈µ∣ ∣ρ∣ ∣µ〉. (4.35)We can transform the conditional master equation (Eq. (3.21)) into a differential equationfor Q(µ) using the correspondences:as well as the conjugates of these.The resultant equation isˆρ c Ĵ + ⇐⇒ ∂ Q, (4.36a)∂µ(ˆρ c Ĵ − ⇐⇒ 2(j +1)µ − ∂ )∂µ µ2 Q, (4.36b)( )∂ˆρ c Ĵ z ⇐⇒∂µ µ − (j +1) Q, (4.36c)[ ( √ΓdWdQ(t, µ, µ ∗ )= (j +1)(µ + µ ∗ )+ 1 ∂ (1 − µ2 )2 ∂µ+ 1 ∂(2 ∂µ ∗ 1 − µ ∗2) − 2 〈 〉 ) ]Ĵ x + dt (S + L Q + D Q ) Q, (4.37a)103