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

4. Continuously monitored Bose condensates: quasiprobability distributionsIn this geometry, the maximal and minimal eigenstates of Ĵ z are localised at the north(θ = π) and south (θ = 0) poles, respectively. The maximal and minimal eigenstates ofĴ x and Ĵy are localised around points that lie on the equator (θ = π/2). Soanumberstate with all the atoms in one well has a Q function centred at either (θ, φ) =(π/2, 0) or(θ, φ) =(π/2,π).The Bloch states can be obtained by a rotation (θ, φ) around the sphere of the lowestweightDicke state ∣ ∣ j, −j〉z [6]: ∣ ∣µ〉= Rθ,φ∣ ∣j, −j〉z , (4.12)whereR θ,φ = eĴx sin φ−Ĵ y cos θ = e ζĴ + −ζ ∗Ĵ −, (4.13)for ζ = 1 2 θ exp(−iφ). The step operators Ĵ+ and J − are the atomic equivalents of theannihilation and creation operators, in terms of which,Ĵ x = 1 ) (Ĵ+ +2 Ĵ− , (4.14a)Ĵ y = i ) (Ĵ− −2 Ĵ+ , (4.14b)Ĵ z = 1 [Ĵ+ Ĵ−], . (4.14c)2Useful Bloch state relations include[110]: the inner product〈λ|µ〉=(1 + λ ∗ µ) 2j(1 + |λ| 2 ) j (1 + |µ| 2 ) j , (4.15a)the completeness identity1=2j +1π∫〉〈 ∣∣ µ µ d 2 µ(1 + |µ| 2 ) 2 , (4.15b)the matrix elements〈λ∣ ∣Ĵ+∣ ∣ µ 〉 = 2jλ∗ (1 + λ ∗ µ) 2j−1(1 + |λ| 2 ) j (1 + |µ| 2 ) j = ( 〈µ ∣ ∣Ĵ−∣ ∣ λ 〉) ∗, (4.15c)and the integral2j +1π∫d 2 µ ∗m µ n (2j − m)!m!µ=(1 + |µ| 2 2j+2) (2j)!δ m,n 0