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

3. Homodyne measurements on a Bose-Einstein condensateFigure 3.5: Unconditional evolution of second-order moments, for N = 100 atoms and Θ=0.In(a) Γ=16and in (b) Γ = 16001400(a)1400(b)12001200Unconditional Evolution1000800600400Unconditional Evolution10008006000 50 100 150 20040020020000 50 100 150 200Time t0Time tthe mean evolution. This change in phase appears to be most rapid over the periods whenthe oscillations are small in amplitude and most likely to suffer random fluctuations.Figures 3.6 and 3.7 show the approximate semiclassical dynamics given by Eq. (3.22).Since the second-order moments behave deterministically in these equations, phase diffusionwill be inhibited and damping will occur for long times. However for short times orweak atom-light coupling, the results should be valid. The results for a weak coupling ofΓ=1.6 × 10 −3 , as shown Fig. 3.6(a), reveal the build up of small tunnelling oscillationswhich vary in amplitude (similar to its quantum counterpart in Fig. 3.3(a)). A strongermeasurement strength (Γ =0.16, as in Fig. 3.6(b)) leads to large regular oscillations,just as in the quantum case (Fig. 3.3(b)). Figure 3.7 gives results for a N = 1000 atomcondensate for the same values of the measurement strength Γ as in Fig. 3.6. The relativeamplitude of the tunnelling oscillations and the rate at which they develop are the sameas for the smaller condensate. This justifies the use of Γ, which contains a factor of N 2 inits definition, as the appropriate normalised measurement strength.The presence of atom-atom interactions increases the phase diffusion, even for quiteweak collisions (Θ ≤ 0.1). Figure 3.8 shows the evolution of 〈 Ĵ x for various atom-atom〉cinteraction strengths Θ, above and below the critical value. The amplitude is also moreirregular, and the Fourier transform of the oscillations (Figs. 3.4(c&d)) no longer shows aclear peak at the expected tunnelling frequency, but a group of peaks, whose amplitudesand positions vary from one run to another, centered on the tunnelling frequency. In74