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

5. Weak force detection using a double Bose-Einstein condensatecan be detected by measuring the quadrature components of the field. For a dispersiveinteraction that acts on a timescale t I over which the dynamics of the condensate itself isnegligible,Ĥ I = χĴxa † a, (5.15)where χ is the measurement strength and a † , a are the light field creation and annihilationoperators. The quadrature components ˆX = a † + a and Ŷ = i(a† − a) then execute simpleharmonic motion:ˆX(t I ) = cos(χt I Ĵ x ) ˆX(0) + sin(χt I Ĵ x )Ŷ (0) (5.16a)Ŷ (t I ) = cos(χt I Ĵ x )Ŷ (0) − sin(χt IĴx) ˆX(0). (5.16b)After t I , the light field is rapidly damped out and the resulting photocurrent is integratedin order to measure ˆX.If all the light is emptied from the cavity, then thedistribution of the integrated photocurrent for ˆX is the true distribution of ˆX[180, 181].In terms of the Wigner function G, this is given by the marginal distribution:p(x) = 1 4where x = α ∗ + α and y = i(α ∗ − α).∫ ∞−∞dyG(x, y), (5.17)To calculate the Wigner function of the optical field, we must first find its reduceddensity operator. Assuming that field is initially in the coherent state α 0 ,thenwiththecondensate state described by ˆρ c ,ˆρ field = Tr cond {ˆρ} =Tr cond{e −iχt I Ĵxâ↠ˆρ c∣ ∣α0〉〈α0∣ ∣e iχt I Ĵ xââ †}==j∑ 〈 ∣j, m e −iχt ∣ 〉〈 ∣I Ĵxâ↠ˆρ c α 0 α0 e iχt ∣I Ĵxâ↠j, m 〉m=−jj∑m=−jP x (m) ∣ ∣ α′m〉〈α′m∣ ∣ (5.18)where α ′ m = α 0 e −iχt I m and where P x (m) is the probability of measuring the condensatein state ∣ 〉 j, m , as defined in Eq. (5.10). The Wigner characteristic function is then{}χ W (λ, λ ∗ ) = Tr field ˆρ field e λ↠−λâ=j∑m=−jP x (m)χ α′ mW(λ, λ ∗ ), (5.19)111