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

4. Continuously monitored Bose condensates: quasiprobability distributionsFortunately, these two requirements for a semiclassical description are precisely satisfiedin the limit of large atom number (j ≫ 1), since in this limit all ordering-dependentterms, including the nonclassical diffusion terms, disappear. The result is a conditionalprobability distribution. It is conditioned upon the continuous, but imperfect, monitoringof the moment 〈 x 〉 = 〈 |α 2 | 2 −|α 1 | 2〉 /2:[√dF (t, α, α ∗ ()= Γ |α2 | 2 −|α 1 | 2 − 〈 |α 2 | 2 −|α 1 | 2〉) dW + dtL + dtD]F, (4.27a)where the differential operators areLF = ∇ α · F L= ∑D =j=1,24∑µ,ν=1∂∂α j( iΩ2 α 3−j + iκ ( |α j | 2 −|α 3−j | 2) α j + Γ 8 α j∂ 2 −Γα µ α ν DµνΓ∂α µ ∂α ν 8)F + c.c.(4.27b)(4.27c)and where, as before, the diffusion matrix is⎡⎤1 −1 −1 1−1 1 1 −1D Γ =. (4.27d)⎢ −1 1 1 −1 ⎥⎣⎦1 −1 −1 1That the diffusion matrix in D is positive semidefinite can be checked by rewriting it interms of the real variables x i = R(α i ), y i = I(α i ). The result is⎡y 1 2 −x 1 y 1 −y 1 y 2 x 2 y 14∑ ∂ 2 −Γα µ α ν DµνΓ 4∑ Γ ∂ 2−x=1 y 1 x 2 1 x 1 y 2 −x 1 x 2∂αµ,ν=1 µ ∂α ν 88 ∂xµ,ν=1 µ ∂x ν ⎢ −y⎣ 1 y 2 x 1 y 2 y2 2 −x 2 y 2⎤,⎥⎦x 2 y 1 −x 1 x 2 −x 2 y 2 x 2 2(4.28)where x µ =(x 1 ,y 1 ,x 2 ,y 2 ) µ . For an arbitrary vector h =(h 1 ,h 2 ,h 3 ,h 4 ) ′ ,⎡⎤ ⎡⎤y 1 2 −x 1 y 1 −y 1 y 2 x 2 y 11 −1 −1 1h ′ −x 1 y 1 x 2 1 x 1 y 2 −x 1 x 2⎢ −y⎣ 1 y 2 x 1 y 2 y2 2 h = g ′ −1 1 1 −1g−x 2 y 2 ⎥ ⎢ −1 1 1 −1 ⎥⎦ ⎣⎦x 2 y 1 −x 1 x 2 −x 2 y 2 x 2 21 −1 −1 1= 1 ((g1 − g 2 ) 2 +(g 4 − g 3 ) 2 +(g 1 − g 2 + g 4 − g 3 ) 2)2≥ 0 ∀g, (4.29)100