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

4. Continuously monitored Bose condensates: quasiprobability distributionswhere g =(g1,g2,g3,g4) ′ =(y 1 h 1 ,x 1 h 2 ,y 2 h 3 ,x 2 h 4 ) ′ . Thus the diffusion matrix is positivesemidefinite.Having generated a Fokker-Planck equation, we would then normally write downstochastic Langevin equations for the phase-space variables, so that numerical simulationscan be performed.However the presence of the nonderivative term complicatesmatters, so instead we derive equations of motion for the conditional averages, which willbe sufficient for our present purposes.In the calculations of the evolution of the moment 〈 g(α) 〉 , the first-derivative terms inEq. (4.27) become∫∫dα 4 g(α)dtLF =and the contribution from the second-derivative terms is∫4∑∫dα 4 g(α)dtDF =µ,ν=1dα 4 F L ·∇ α g(α), (4.30)dα 4 −Γα µα ν D Γ µν dt8∂ 2 g(α)∂α µ ∂α ν, (4.31)where we have integrated by parts and ignored any surface terms. The resultant equationsof motion for the conditional moments, up to second order, ared 〈 x 〉 = −Ω 〈 y 〉 dt +2 √ ( 〈x2Γ 〉 − 〈 x 〉 ) 2dW (4.32a)d 〈 y 〉 (= Ω 〈 x 〉 +4κ 〈 xz 〉 − Γ 〈 〉 )y dt +2 √ Γ (〈 xy 〉 − 〈 x 〉〈 y 〉) dW (4.32b)2d 〈 z 〉 (= 4 κ 〈 xy 〉 − Γ 〈 〉 )z dt +2 √ Γ (〈 xz 〉 − 〈 x 〉〈 z 〉) dW (4.32c)2d 〈 x 2〉 = −2Ω 〈 xy 〉 dt +2 √ Γ( 〈x3 〉 − 〈 x 〉〈 x 〉 2 ) dW (4.32d)d 〈 y 2〉 = ( 2Ω 〈 xy 〉 − 8κ 〈 xyz 〉 − Γ (〈 y 2〉 − 〈 z 2〉)) dt +2 √ Γ( 〈xy2 〉 − 〈 x 〉〈 y 〉 2 ) dW(4.32e)d 〈 z 2〉 = ( 8κ 〈 xyz 〉 − Γ (〈 z 2〉 − 〈 y 2〉)) dt +2 √ ( 〈xz2Γ 〉 − 〈 x 〉〈 z 〉 ) 2dW (4.32f)d 〈 xy 〉 (= Ω( 〈 x 2〉 − 〈 y 2〉 )+4κ 〈 y 2 z 〉 − Γ 〈 〉 )xy dt +2 √ Γ (〈 x 2 y 〉 − 〈 x 〉〈 xy 〉) dW2d 〈 xz 〉 =(4.32g)(−Ω 〈 yz 〉 +4κ 〈 x 2 y 〉 − Γ 〈 〉 )xz dt +2 √ Γ (〈 x 2 z 〉 − 〈 x 〉〈 xz 〉) dW (4.32h)2d 〈 yz 〉 = ( Ω 〈 xz 〉 +4κ (〈 xy 2〉 − 〈 xz 2〉) − 2Γ 〈 xz 〉) dt +2 √ Γ (〈 xyz 〉 − 〈 x 〉〈 yz 〉) dW,(4.32i)where the variables x, y and z are as defined in Eq. (2.31). If the expectation of these equationsis calculated, to get the unconditional equations, then they reduce to the equations101