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arXiv:1509.09028 · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

LQG feedback control of a class of linear non-Markovian quantum systems

Shibei Xue, M. R. James, Valery Ugrinovskii, Ian R. Petersen

Published Jul 1, 2016• 3 citations
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Abstract

In this paper we present a linear quadratic Gaussian (LQG) feedback control strategy for a class of linear non-Markovian quantum systems disturbed by Lorentzian noise. The feedback control law is designed based on the estimated dynamical variables of a whitening quantum filter for an augmented Markovian model of the non-Markovian quantum system. In this augmented Markovian model, an ancillary system plays the role of internal modes of the environment converting white noise into Lorentzian noise and a principal system obeys non-Markovian dynamics due to the direct interaction with the ancillary system. The simulation results show the LQG controller with the whitening filter obtains a better control performance than that with a Markovian filter in the problem of cooling the principal system when the ancillary system is driven by thermal noise.

DOI
10.1109/acc.2016.7526109

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In this paper, we propose a feedback tracking control scheme to reject Lorentzian noise in a linear non-Markovian quantum system. This non-Markovian quantum system is modelled as an augmented linear Markovian quantum system where a principal system represents the plant and an ancillary system represents the internal mode of the non-Markovian environment for generating Lorentzian noise. Based on this augmented system model, a whitening quantum filter can be constructed for continuously monitoring the dynamical variables of the principal system. To reject Lorentzian noise, a linear quadratic Gaussian (LQG) controller, using the estimate of the whitening filter and the dynamical variables of a reference system, can be designed to drive the principal system to track the reference system.
In this paper, we present a feedback tracking control strategy to reject colored noise in linear non-Markovian quantum systems. This non-Markovian system is modeled as an augmented linear Markovian system, which includes a principal system to be controlled and multiancillary systems representing the effect of the internal modes of the non-Markovian environment. The colored noise generated by the multiancillary systems disturbs the principal system through a direct interaction. Based on this model, a whitening quantum filter can be obtained for estimating the dynamical variables of both the principal and multiancillary systems, which can be used for feedback control. To reject the colored noise, a linear quadratic Gaussian controller is designed to drive the dynamical variables of the principal system to track those of a reference system. In examples, we show that colored noise with both rational and nonrational spectra in linear non-Markovian quantum systems can be efficiently rejected using our tracking control strategy.
In this paper we present a Markovian representation approach to constructing quantum filters for a class of non-Markovian quantum systems disturbed by Lorentzian noise. An ancillary system is introduced to convert white noise into Lorentzian noise which is injected into a principal system via a direct interaction. The resulting dynamics of the principal system are non-Markovian, which are driven by the Lorentzian noise. By probing the principal system, a quantum filter for the augmented system can be derived from standard theory, where the conditional state of the principal system can be obtained by tracing out the ancillary system.
In this paper, we utilize a feedback strategy to stabilize a single qubit in a non-Markovian environment with a Lorentzian spectrum. The non-Markovian single qubit is represented in an augmented system model, where an ancillary system is introduced as the internal mode of the environment to generate Lorentzian noise. The density matrix of the single qubit can be estimated by using an augmented-system-based quantum stochastic master equation. Then a feedback controller can be driven by the estimates to stabilize the qubit in a target state. Simulations show the feedback strategy can provide effective control of the non-Markovian single qubit.
In this paper, we present a H ∞ filter for estimating dynamics of a Lorentzian-noise-disturbed non-Markovian cavity system with disturbances. The non-Markovian cavity system can be represented by an augmented system, where the Lorentzian quantum noise is parameterized by an ancillary system. This ancillary system is directly coupled to a principal system of interest and induces the non-Markovian dynamics of the principal system. The noise channels in this augmented model carry disturbances. To reduce the effect of the disturbances on the estimation, we design a H ∞ filter which can be obtained by solving a H ∞ control problem.
This brief presents an augmented Markovian system framework that can be applied to model non-Markovian quantum systems. In this augmented system model, ancillary systems are designed to play the role of internal modes of the non-Markovian environment converting white noise to colored noise. To capture the non-classical mutual influence between a quantum system and non-Markovian environments, direct interactions between a principal system and the ancillary system are introduced. Consequently, we showed that the augmented Markovian formulation can be used to theoretically model the environment for observed non-Markovian behavior in a recent experiment on quantum dots.