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Chinese Control Conference (CCC)

The controllability analysis of a discrete-time reservoir engineering system

Weiming Guo, Ye Xia, Zibo Miao

Published Jul 25, 2022• 0 citations
Publisher

Abstract

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 after$k$steps. 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.

DOI
10.23919/ccc55666.2022.9902525

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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.
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.
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 and analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator, simplifying a previous proposal to alleviate implementation constraints. It approximately stabilizes periodic grid states introduced in 2001 by Gottesman, Kitaev and Preskill (GKP), with applications for quantum error correction and quantum metrology. We obtain explicit estimates for the energy of the solutions of the Lindblad master equation. We estimate the convergence rate to the codespace when stabilizing a GKP qubit, and numerically study the effect of noise. We then present simulations illustrating how a modification of parameters allows preparing states of metrological interest in steady-state.
In this paper, we continue our investigation on controlling the state of a quantum harmonic oscillator, by coupling it to a reservoir composed of a sequence of qubits. Specifically, we show that sending qubits separable from each other but initialised at different states in pairs can stabilise the oscillator at squeezed states. However, only if entanglement is allowed in the reservoir qubit can we stabilise the oscillator at a wider set of squeezed states. This thus provides a proof for the necessity of involving entanglement in the reservoir qubits input to the oscillator, as regard to the stabilisation of quantum states in the proposed system setting. On the other hand, this system setup can be in turn used to estimate the coupling strength between the oscillator and reservoir qubits. We further demonstrate that entanglement in the reservoir input qubits contributes to the corresponding quantum Fisher information. From this point of view, entanglement is proved to play an indispensable role in the improvement of estimation precision in quantum metrology.
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.
Quantum reservoir engineering leverages dissipative processes to achieve desired behaviour, with applications ranging from entanglement generation to quantum error correction. Therein, a structured environment acts as an entropy sink for the system and no time-dependent control over the system is required. We develop a theoretical framework for active reservoir engineering, where time-dependent control over a quantum system is used to manipulate its environment. In this case, the system may act as an entropy sink for the environment. Our framework captures the dynamical interplay between system and environment, and provides an intuitive picture of how finite-size effects and system-environment correlations allow for manipulating the environment by repeated initialisation of the quantum system. We illustrate our results with two examples: a superconducting qubit coupled to an environment of two-level systems and a semiconducting quantum dot coupled to nuclear spins. In both scenarios, we find qualitative agreement with previous experimental results, illustrating how active control can unlock new functionalities in open quantum systems.