Deterministic Submanifolds and Analytic Solution of the Stochastic Differential Equation Describing a Continuously Measured Qubit

Wednesday, April 13, 2016 - 3:30pm - 4:30pm
Keller 3-180
Pierre Rouchon (Mines-ParisTech)
In this joint work with A. Sarlette, P. Campagne-Ibarcq, P. Six, L. Bretheau, M. Mirrahimi, and B. Huard, we investigate the stochastic differential equation (SDE) associated to a qubit subject to Hamiltonian evolution as well as unmonitored and continuously monitored deco herence channels. The latter imply a stochastic evolution of the three Bloch coordinates whose associated probability distribution we characterize. We show that for a set of typical experimental settings, the qubit state remains in fact confined to a deterministically evolving surface inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface. For a general diffusive stochastic master equation, we relate the existenceof such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show e.g. that the considered qubit cases (dispersive measurements of the energy or of the fluorescence) are essentially the only ones featuring deterministic submanifolds.
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