This paper presents a first mathematical convergence analysis of a Fock states feedback stabilization scheme via single-photon corrections. This measurement-based feedback has been developed and experimentally tested in 2012 by the cavity quantum ele
ctrodynamics group of Serge Haroche and Jean-Michel Raimond. Here, we consider the infinite-dimensional Markov model corresponding to the ideal set-up where detection errors and feedback delays have been disregarded. In this ideal context, we show that any goal Fock state can be stabilized by a Lyapunov-based feedback for any initial quantum state belonging to the dense subset of finite rank density operators with support in a finite photon-number sub-space. Closed-loop simulations illustrate the performance of the feedback law.
At the quantum level, feedback-loops have to take into account measurement back-action. We present here the structure of the Markovian models including such back-action and sketch two stabilization methods: measurement-based feedback where an open qu
antum system is stabilized by a classical controller; coherent or autonomous feedback where a quantum system is stabilized by a quantum controller with decoherence (reservoir engineering). We begin to explain these models and methods for the photon box experiments realized in the group of Serge Haroche (Nobel Prize 2012). We present then these models and methods for general open quantum systems.
The present work establishes necessary and sufficient conditions for a nonlinear system with two inputs to be described by a specific triangular form. Except for some regularity conditions, such triangular form is flat. This may lead to the discovery
of new flat systems. The proof relies on well-known results for driftless systems with two controls (the chained form) and on geometric tools from exterior differential systems. The paper also illustrates the application of its results on an academic example and on a reduced order model of an induction motor.
