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

5. Weak force detection using a double Bose-Einstein condensateeffect of the nonlinearity, and other sources of phase error on the different stages of thescheme.As well as the diffusive effect of collisions, there are also the effects of decoherence toconsider. These could be due to external perturbations of the trapping potential or, if theoptical field is present, to fluctuations in the condensate momentum from the atom-fieldcoupling. The overall effect of the nonlinearity, external perturbations and the back actionon the condensate may be seen in the following master equation, in which the dynamicsof the optical field have been adiabatically eliminated[126]:˙ˆρ p = −i2κ[Ĵ 2 x, ˆρ p ]+iχ|α| 2 [Ĵx, ˆρ p ] − 2χ2 |α| 2γ[Ĵx, [Ĵx, ˆρ p ]] − σ22 [Ĵx, [Ĵx, ˆρ p ]], (5.29)where σ is the size of the external perturbations. This master equation can be generatedfrom the random HamiltonianĤ Ito = ( χ|α| 2 + σ ′ ξ ) Ĵ x + (2κ − i )2 σ′ 2Ĵx 2 (5.30)where σ ′ = σ +(2χ|α|)/( √ γ)andξ = dW/dt is delta-correlated white noise. This Hamiltoniancan generate stochastic differential equations that are to be interpreted in the Itosense 1 . So that ordinary calculus may be used, we use the Stratonovich form:Ĥ Str = σ ′ ξĴx + 2κĴ 2 x. (5.31)We have removed the linear deterministic term since it can be negated by tilting the twowells in proportion to the intensity of the light field.In the detection part of the scheme (stage 2),change in the components of the superposition state:Ĥ Str will induce an extra self-phaseφ ± = ±∆jτ ± σ ′ jW(τ)+2κj 2 τ, (5.32)where W (τ) is the Wiener process. Since the self-phase change due to the nonlinearity isthe same for both components, there is no net effect on the output probability distribution(Eq. (5.10)). The overall phase shift in the fringes is thereforeφ =∆jτ + σ ′ jW(τ). (5.33)Thus the phase shift suffers a random walk, with mean φ(τ) =∆jτ and standard deviationσ φ (τ) =σ ′ j √ τ. This phase uncertainty could be reduced by isolating the trapping1 See Appendix A114