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arXiv:1210.1021

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

Alain Sarlette, Pierre Rouchon

Published Oct 3, 2012• 0 citations
arXivHTMLPDF

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 Superieure further illustrate the robustness of the method.

DOI
10.48550/arxiv.1210.1021

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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 Superieure further illustrate the robustness of the method.
This paper is concerned with a discrete-time reservoir engineering consisting of a sequence of qubits which are used to manipulate the quantum state of a harmonic oscillator. We consider the controllability problem regarding such a system both in the finite time and infinite time scenarios, with detailed proofs. In more concrete terms, we discuss how the quantum state of the harmonic oscillator evolves step by step, and provide the final state afterkkksteps. The role of entanglement involved in the reservoir is also discussed by demonstrating photon numbers. In terms of infinite time controllability analysis, we take advantage of the approximated Lindblad master equation, to explore under what circumstances can we stabilize coherent states, squeezed states and Schrödinger cat states. Stabilization of certain quantum states as well as the corresponding convergence rates are proved by means of Lyapunov functions. These results shed light on a systematic framework towards controllability analysis in quantum reservoir engineering.
This theoretical proposal investigates how resonant interactions occurring when a harmonic oscillator is fed with a stream of entangled qubits allow us to stabilize squeezed states of the harmonic oscillator. We show that the properties of the squeezed state stabilized by this engineered reservoir, including the squeezing strength, can be tuned at will through the parameters of the "input" qubits, albeit in tradeoff with the convergence rate. We also discuss the influence of the type of entanglement in the input, from a pairwise case to a more widely distributed case. This paper can be read in two ways: either as a proposal to stabilize squeezed states, or as a step towards treating quantum systems with time-entangled reservoir inputs.
We propose an engineered reservoir inducing the relaxation of a cavity field towards non-classical states. It is made up of two-level atoms crossing the cavity one at a time. Each atom-cavity interaction is first dispersive, then resonant, then dispersive again. The reservoir pointer states are those produced by a fictitious Kerr Hamiltonian acting on a coherent field. We thereby stabilize squeezed states and quantum superpositions of multiple coherent components in a cavity having a finite damping time. This robust method could be implemented in state-of-the-art experiments and lead to interesting insights into mesoscopic quantum state superpositions.
We study the state feedback stabilization of a quantum harmonic oscillator near a pre-specified Fock state (photon number state). Such a state feedback controller has been recently implemented on a quantized electromagnetic field in an almost lossless cavity. Such open quantum systems are governed by a controlled discrete-time Markov chain in the unit ball of an infinite dimensional Hilbert space. The control design is based on an unbounded Lyapunov function that is minimized at each time-step by feedback. This ensures (weak-*) convergence of probability measures to a final measure concentrated on the target Fock state with a pre-specified probability. This probability may be made arbitrarily close to 1 by choosing the feedback parameters and the Lyapunov function. They are chosen so that the stochastic flow that describes the Markov process may be shown to be tight (concentrated on a compact set with probability arbitrarily close to 1). Convergence proof uses Prohorov's theorem and specific properties of this Lyapunov function.
Abstract This theoretical proposal investigates how resonant interactions occurring when a harmonic oscillator is fed with a stream of entangled qubits allow us to stabilise squeezed states of the harmonic oscillator. We show that the properties of the squeezed state stabilised by this engineered reservoir, including the squeezing strength, can be tuned at will through the parameters of the ‘input’ qubits, albeit in tradeoff with the convergence rate. We also discuss the influence of the type of entanglement in the input from a pairwise case to a more widely distributed case. This paper can be read either as a proposal to stabilise squeezed states or as a step toward treating quantum systems with time-entangled reservoir inputs.
We analyze a quantum reservoir engineering method, originally introduced by Sarlette et al. [Phys. Rev. Lett. 107, 010402 (2011)], for the stabilization of nonclassical field states in high-quality cavities. We generalize the method to the protection of mesoscopic entangled field states shared by two nondegenerate field modes. The reservoir consists of a stream of atoms consecutively interacting with the cavity. Each individual atom-cavity interaction follows the same time-varying Hamiltonian, combining resonant with nonresonant parts. We gain detailed insight into the competition between the engineered reservoir and decoherence. We show that the operation is quite insensitive to experimental imperfections and that it could thus be implemented in the near future in the context of microwave cavity or circuit quantum electrodynamics.