Dynamical recurrence and the quantum control of coupled oscillators

Abstract

Controllability -- the possibility of performing any target dynamics by applying a set of available operations -- is a fundamental requirement for the practical use of any physical system. For finite-dimensional systems, as for instance spin systems, precise criterions to establish controllability, such as the so called rank criterion, are well known. However most physical systems require a description in terms of an infinite-dimensional Hilbert space whose controllability properties are poorly understood. Here, we investigate infinite-dimensional bosonic quantum systems -- encompassing quantum light, ensembles of bosonic atoms, motional degrees of freedom of ions, and nano-mechanical oscillators -- governed by quadratic Hamiltonians (such that their evolution is analogous to coupled harmonic oscillators). After having highlighted the intimate connection between controllability and recurrence in the Hilbert space, we prove that, for coupled oscillators, a simple extra condition has to be fulfilled to extend the rank criterion to infinite dimensional quadratic systems. Further, we present a useful application of our finding, by proving indirect controllability of a chain of harmonic oscillators.

5 pages (+ 1 page of supplemental material), 1 figure; v2: title changed and minor replacements in the rest of the paper

References in corpus (5)

• Strong dispersive coupling of a high finesse cavity to a micromechanical membrane
• Coupled quantized mechanical oscillators
• Local controllability of quantum networks
• Optimal Control Theory for Continuous Variable Quantum Gates
• Reconstructing the quantum state of oscillator networks with a single qubit

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• Locally optimal symplectic control of multimode Gaussian states
• Controllability in tunable chains of coupled harmonic oscillators
• Open loop linear control of quadratic Hamiltonians with applications