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

3. Homodyne measurements on a Bose-Einstein condensateAs with the unconditional evolution, we can adiabatically eliminate the cavity fieldvariables from Eq. (3.18), by substituting in for the density operator the expansion Eq.(3.10). Tracing over the field variables gives˙ˆρ c = ˙ρ 0 +˙ρ 2[ ] ]= S(ρ 0 + ρ 2 )+iχ|α 0 | 2 Ĵ x ,ρ 0 + ρ 1 + iχα 0[Ĵx ,ρ † 1 − ρ 1+ √ γ dW )(ρ 1 + ρ † 1dt− (ρ 0 + ρ 2 )Tr(ρ 1 + ρ † 1 ) + γO(ɛ 3 0 ). (3.20)For γ ∼ ɛ −20 ≫ 1, we can replace ρ 1 by Eq. (3.13) to give]]2dˆρ c = −iΩ[Ĵz , ˆρ c dt − i2κ[Ĵ x , ˆρ c dt + Γ ]][Ĵx ,[Ĵx , ˆρ c dt2+ √ )Γ(Ĵx ˆρ c +ˆρ c Ĵ x − 2 〈J x 〉 c ˆρ c dW + dtγO(ɛ 3 0 ), (3.21)where we have adjusted the initial cavity detuning δ and tilted the wells to remove unimportantterms. The adiabatic elimination in both the conditional and unconditional casesis equivalent to making the replacement â −→ ΓĴx/γ. Thus the environmental decoherencein the light field is transferred directly to a decoherence in the atomic operators, andconstitutes the back action of the homodyne measurement on the condensate.Just as in the unconditional evolution, we can derive equations of motion for theconditional moments from Eq. (3.21).Even when κ = 0, the stochastic terms in theequations always couple to higher order moments. To get a closed set of equations, we canfactorise each third-order moment into a second-order moment multiplied by 〈 Ĵ x〉. For aninitial number state, we get the following closed set:d 〈 〉 〉 √ ( 〈Ĵ 〉 〉 )Ĵ x = −Ω〈Ĵy dt +2 Γ2 2x −〈Ĵx dW (3.22a)d 〈 (〉Ĵ y = Ω 〈 〉 Γ 〉 )Ĵ x −〈Ĵy dt +2 √ ( 1 〉Γ〈ˆΛ〉 ) −〈Ĵx〉〈Ĵy dW (3.22b)22d 〈 〉 Γ 〉 √ 〉Ĵ z = −〈Ĵz dt − 2 Γ〈Ĵx〉〈Ĵz dW(3.22c)2d 〈 Ĵx〉 2 = −Ω〈ˆΛ〉 dt (3.22d)d 〈 Ĵy2 〉 (= Ω 〈ˆΛ〉 ( 〈Ĵ 〉 〈Ĵ 〉 ))− Γ2y −2z dt(3.22e)d 〈 Ĵ 2 z〉= Γ( 〈Ĵ2y〉−〈Ĵ2z〉 ) dt (3.22f)d 〈ˆΛ〉 =(2Ω( 〈 Ĵx2 〉 〈Ĵ 〉−2 Γy ) −〈ˆΛ〉 ) dt.(3.22g)2Because the third-order correlations are now factorised, the second-order moments evolvedeterministically. This means that the tunnelling oscillations do not show a slow phase68