Section: New Results

Deterministic submanifolds and analytic solution of the quantum stochastic differential master equation describing a monitored qubit

Participants: A. Sarlette, P. Rouchon

In the paper [18], we study the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of 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 that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.

This paper was motivated by a striking experimental observation of Ph.Campagne-Ibarcq (group of Benjamin Huard - now at ENS Lyon and still collaborator). It appears to be actually quite general, and to generalize to higher-dimensional systems than the qubit. We are working on this extension, time permitting (as we have no student support currently), to publish a complete story about relevant experimental systems where the QSDE can be modeled in a very low-dimensional manifold.