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

4. Continuously monitored Bose condensates: quasiprobability distributionsfrequency Ω, which is given by the overlap between the local modes of each well. This Itostochastic equation describes the state of the system conditioned on a particular historyof measurement results[181, 179]. The value of the measured current, which depends ona classical random variable, feeds back into the master equation via 〈 Ĵ x , resulting in a〉cstochastic state matrix ˆρ c .One ‘classical’ description is to write down equations of motion for diagonal elementsρ m,m of the state matrix in the number state basis:ρ m,m (t + dt) =ρ m,m (t)+ Ωdt (√j(j +1)− m(m +1)(ρm,m+1 (t)+ρ m+1,m (t)) −2√j(j +1)− m(m − 1)(ρm,m−1 (t)+ρ m−1,m (t)))+2 √ ΓdW (m − m) ρ m,m (t), (4.2)〈 ∣ ∣where ρ m,m = x j, m∣ˆρ c j, m 〉 x and m = ∑ m mρ m,m. Now ∣ ∣j, m 〉 are also the eigenstatesxof position, and so this equation gives the stochastic evolution of the conditional positiondistribution. Setting Ω = 0 and identifying x =2q 0 m/N gives the closed equation√ΓNρ c (x, t + dt) =ρ c (x, t)+ dW (x − x) ρ c (x, t), (4.3)q 0where ρ c (x) = ρ m,m .Thus without any tunnelling, the measurement process assignsa classical probability distribution to the variable x (as in [124]).This distribution isconditioned on previous imperfect measurement results and so is stochastic and nonlinear.When the system exists in an eigenstate of position, the stochastic term in Eq. (4.3)disappears and the distribution is stationary. Thus for the measurement to induce backactionfluctuations, the off-diagonal coherences due to tunnelling must be present. Theseperiodically induce an uncertainty in x as the state precesses around the Ĵz axis of theBloch sphere, and it is at these times that measurement-induced fluctuations feed backinto the conditional state.4.2 Stochastic Gross-Pitaevskii equationContinuing our quest to formulate a semiclassical description of the back-action effectsof the measurement, we now derive a modified Gross-Pitaevskii (GP) equation from thestochastic Shrödinger equation (Eq. (3.25)):d ∣ ∣ Ψc (t) 〉 =[−iĤ2dt − Γ (Ĵx − 〈 〉 ) 2 √ Ĵ x2c dt + γ(Ĵx − 〈 〉 ) ] ∣∣ΨcĴ x dW (t) 〉 (4.4)c79