The reachable set of single-mode quadratic Hamiltonians

Uther Shackerley-Bennett, Alexander Pitchford, Marco G. Genoni, Alessio Serafini, Daniel K. Burgarth

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

Open-loop controllability in quantum mechanics refers to finding conditions on time-varying Hamiltonians such that a full group of unitary transformations can be enacted with them. For compact groups controllability is well understood and is dealt with using the Lie algebra rank criterion. Gaussian systems, however, evolve under Hamiltonians generating the non-compact symplectic group, rendering the rank criterion necessary but no longer sufficient. In this setting it is possible to satisfy the rank criterion without the ability to enact all symplectic transformations. We refer to such systems as 'unstable' and explore the set of symplectic transformations that remain reachable. We provide a partial analytical characterisation for the reachable set of a single-mode unstable system. From this it is proven that no orthogonal-symplectic operations ('energy-preserving' or 'passive' in the literature) may be reached with such controls. We then apply numerical optimal control algorithms to demonstrate a complete characterisation of the set in specific cases. These results suggest approaches to the long-standing open problem of controllability in n modes.

Original languageEnglish
Article number155203
Pages (from-to)1-22
Number of pages22
JournalJournal of Physics A: Mathematical and Theoretical
Volume50
Issue number15
DOIs
Publication statusPublished - 14 Mar 2017
Externally publishedYes

Keywords

  • control theory
  • symplectic group
  • gaussian states

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