# Robust open-loop stabilization of Fock states by time-varying quantum interactions

**Authors**: Alain Sarlette, Pierre Rouchon, 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, pp. 208-213, August 29 - 31, 2012, Bologna, Italy

A quantum harmonic oscillator (spring subsystem) is stabilized towards a target Fock state by reservoir engineering. This passive and open-loop stabilization works by consecutive and identical Hamiltonian interactions with auxiliary systems, here three-level atoms (the auxiliary ladder subsystem), followed by a partial trace over these auxiliary atoms. A scalar control input governs the interaction, defining which atomic transition in the ladder subsystem is in resonance with the spring subsystem. We use it to build a time-varying interaction with individual atoms, that combines three non-commuting steps. We show that the resulting reservoir robustly stabilizes any initial spring state distributed between 0 and 4n ̄ + 3 quanta of vibrations towards a pure target Fock state of vibration number n ̄. The convergence proof relies on the construction of a strict Lyapunov function for the Kraus map induced by this reservoir setting on the spring subsystem. Simulations with realistic parameters corresponding to the quantum electrodynamics setup at Ecole Normale Supérieure further illustrate the robustness of the method.

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**BibTeX**:

@Proceedings{,

author = {Alain Sarlette, Pierre Rouchon},

editor = {},

title = {Robust open-loop stabilization of Fock states by time-varying quantum interactions},

booktitle = {4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control},

volume = {},

publisher = {},

address = {Bologna},

pages = {208-213},

year = {2012},

abstract = {A quantum harmonic oscillator (spring subsystem) is stabilized towards a target Fock state by reservoir engineering. This passive and open-loop stabilization works by consecutive and identical Hamiltonian interactions with auxiliary systems, here three-level atoms (the auxiliary ladder subsystem), followed by a partial trace over these auxiliary atoms. A scalar control input governs the interaction, defining which atomic transition in the ladder subsystem is in resonance with the spring subsystem. We use it to build a time-varying interaction with individual atoms, that combines three non-commuting steps. We show that the resulting reservoir robustly stabilizes any initial spring state distributed between 0 and 4n ̄ + 3 quanta of vibrations towards a pure target Fock state of vibration number n ̄. The convergence proof relies on the construction of a strict Lyapunov function for the Kraus map induced by this reservoir setting on the spring subsystem. Simulations with realistic parameters corresponding to the quantum electrodynamics setup at Ecole Normale Supérieure further illustrate the robustness of the method.},

keywords = {open-loop control systems, discrete-time, Lyapunov function, stabilization methods, photons, physics, Hilbert spaces}}