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IEEE Transactions on Control Systems Technology

Feedback Tracking Control of Non-Markovian Quantum Systems

Shibei Xue, Michael Hush, Ian R. Petersen

Published Oct 21, 2016• 29 citations
Publisher

Abstract

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.

DOI
10.1109/tcst.2016.2614834

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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 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.
In this paper, a quantum filter for estimating the states of a non-Markovian qubit system is presented in an augmented Markovian system framework including both the qubit system of interest and multi-ancillary systems for representing the internal modes of the non-Markovian environment. The colored noise generated by the multi-ancillary systems disturbs the qubit system via a direct interaction. The resulting non-Markovian dynamics of the qubit is determined by a memory kernel function arising from the dynamics of the ancillary system. In principle, colored noise with arbitrary power spectrum can be generated by a combination of Lorentzian noises. Hence, the quantum filter can be constructed for the qubit disturbed by arbitrary colored noise and the conditional state of the qubit system can be obtained by tracing out the multi-ancillary systems.
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.
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.
H2\mathscr{H}_2H2​ Model Reduction for Augmented Model of Linear Non-Markovian Quantum Systems
An augmented system model provides an effective way to model non-Markovian quantum systems, which is useful in filtering and control for this class of systems. However, since a large number of ancillary quantum oscillators representing internal modes of a non-Markovian environment directly interact with the principal system in these models, the dimension of the augmented system may be very large causing significant computational burden in designing filters and controllers. In this context, this paper proposes an H2\mathscr{H}_2H2​ model reduction method for the augmented model of linear non-Markovian quantum systems. We first establish necessary and sufficient conditions for the physical realizability of the augmented model of linear non-Markovian quantum systems, which are more stringent than those for Markovian quantum systems. However, these physical realizability conditions of augmented system model pose non-convex constrains in the optimization problem of model reduction, which makes the problem different from the corresponding classical model reduction problem. To solve the problem, we derive necessary conditions for determining the input matrix in the reduced model, with which a theorem for designing the system matrix of the ancillary system in the reduced system is proved. Building on this, we convert the nonlinear equality constraints into inequality constraints so that a semidefinite programming algorithm can be developed to solve the optimization problem for model reduction. A numerical example of a two-mode linear quantum system driven by three internal modes of a non-Markovian environment validates the effectiveness of our method.