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

4. Continuously monitored Bose condensates: quasiprobability distributionsand leads to the operator correspondences[174]ĉˆρ ⇐⇒ αP (α), (4.19a)(ĉ † ˆρ ⇐⇒ α ∗ − ∂ )P (α),(4.19b)∂α(ˆρĉ ⇐⇒ α −∂ )∂α ∗ P (α),(4.19c)ˆρĉ † ⇐⇒ α ∗ P (α). (4.19d)Applying these operator correspondences to Eq. (3.21) results in[ ( √ΓdP (t, α, α ∗ )= |α 2 | 2 −|α 1 | 2 − 〈 |α 2 | 2 −|α 1 | 2〉 − 1 ( ∂α 2 −∂ α 12 ∂α 2 ∂α 1+ ∂∂α ∗ α ∗ 2 −∂ )) ]2 ∂α ∗ α ∗ 1 dW + dtL P + dtD P P, (4.20a)1where the differential operators areL P = ∑ ( (∂ iΩ∂αj=1,2 j 2 α 3−j + iκ |α j | 2 −|α 3−j | 2 + 1 )α j + Γ )2 8 α j + c.c. (4.20b)( )4∑ ∂ 2 −ΓDΓµνD P =− iκDκ µνα µ α ν ,(4.20c)∂α µ ∂α ν 8 2µ,ν=1and where α 3 = α ∗ 1 and α 4 = α ∗ 2 . The diffusion matrices are⎡⎤⎡1 −1 −1 11 0 −1 0−1 1 1 −10 −1 0 1D Γ =, D κ =⎢ −1 1 1 −1 ⎥⎢ −1 0 1 0⎣⎦⎣1 −1 −1 10 1 0 −1⎤. (4.20d)⎥⎦The first-derivative terms in Eq. (4.20a) govern the deterministic behaviour of the phasespacevariables, causing a drift in the P distribution. The diffusive behaviour, due to thesecond-derivative terms D P , consists of a collisional contribution D κ and a decoherencecontribution D Γ .One disadvantage of the P function is that it does not always exist for a given ˆρ c .Analternative representation uses the Wigner function, which does not have this limitation.The Wigner function is a Gaussian convolution over the P function:G(α) = 2 ∫P (β)e −2|β−α|2 d 2 β (4.21)π97