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

4. Continuously monitored Bose condensates: quasiprobability distributionsdistribution remains symmetric about the equator of the Bloch sphere, and its behaviouris similar to the previous strong measurement simulations with ∣ ∣ j, 0〉xas the initial state(Fig. 4.2): it periodically splits into two components which move apart, sometimes movingover the poles to come down along a different longitude, which corresponds to the phasejumps in the tunnelling oscillations.When subcritical atom-atom interactions are included (Θ < 1), the Q function evolutionis very similar to the noninteracting case: the distribution gradually moves off thesouth pole of the Bloch sphere, remaining compact as it precesses around the Ĵz axis. Forvery strong self-interactions (as in Fig. 4.11, where Θ = 2.0), the Q function develops therotating asymmetries, but it never (up to time t = 200t 0 ) fully moves off the bottom ofthe sphere, leading only to oscillations of small amplitude.4.4 Quasiprobability distributions using optical coherent statesThe technique of mapping operator equations onto stochastic phase-space equations iscommon in quantum optics.Different phase-space representations are used to deriveFokker-Planck equations from the master equation. This technique is the basis of the secondpart of this thesis, in which quantum features are fully simulated by way of quasiprobabilitydistributions. Here we apply the technique to the conditional master equation (Eq.(3.21)) to get a probabilistic semiclassical description.Exactly what structure in Hilbert space is mapped onto the phase-space equationsdepends on what overcomplete set of states is used to represent ˆρ c . The most commonlyused states are the coherent states ∣ ∣ α〉of the boson field.They have the advantageof producing correspondences between the annihilation (and creation) operators and thephase-space variables that are simple in form 1 . The (conditional) Glauber-Sudarshan Pfunction is defined by∫ˆρ c =P (α) ∣ ∣ α〉〈α∣ ∣d 2 α, (4.18)1 For an overview of the properties of coherent states and the phase-space representations constructedfrom them, see the fourth chapter of [62]93