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IEEE Transactions on Automatic Control

Identification of Unknown Parameters for a Class of Two-Level Quantum Systems

Zhengui Xue, Hai Lin, Tong Heng Lee

Published Dec 20, 2012• 13 citations
Publisher

Abstract

This technical note studies the identification of unknown decoherence rates for a class of two-level quantum systems undergoing spontaneous emission. Our previous work shows that estimates of the unknown decoherence rates can be obtained from the ensemble averages by imposing constant control and monitoring a sequence of identical systems continuously. Inspired by the work, this technical note further investigates the identification based on a single quantum system. It is shown that a mean-ergodic solution exists for a single system under constant control and convergence to an ergodic process can be guaranteed by introducing some noise into the control signals. The ensemble average can thus be replaced by the time average of a single system, and the parameter estimate can be obtained from a single implementation.

DOI
10.1109/tac.2012.2235723

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This technical note studies the identification of unknown decoherence rates for a class of two-level quantum systems undergoing spontaneous emission. Our previous work shows that estimates of the unknown decoherence rates can be obtained from the ensemble averages by imposing constant control and monitoring a sequence of identical systems continuously. Inspired by the work, this technical note further investigates the identification based on a single quantum system. It is shown that a mean-ergodic solution exists for a single system under constant control and convergence to an ergodic process can be guaranteed by introducing some noise into the control signals. The ensemble average can thus be replaced by the time average of a single system, and the parameter estimate can be obtained from a single implementation.
To make full use of the amazing characteristics of quantum states, control of open quantum systems is of great significance in practical applications. This paper studies a more practical control problem where the decoherence parameter is not known exactly. The problem is considered in two phases. First, the unknown decoherence strength is estimated based on continuous measurements and an appropriately designed control. Then, Markovian feedback control is designed to deal with the decoherence effect based on the accurate parameter estimation. It is shown that state transfer with a high probability can be achieved by designing the feedback gains and the measurement. Moreover, good state transfer performance can be guaranteed even there exists an estimation error on the decoherence strength. Simulation studies are included to demonstrate the effectiveness of the proposed approach.
An observer-based Hamiltonian identification algorithm for quantum systems has been recently proposed by two of the authors to estimate the dipole moment matrix of a quantum system requiring the measurement of the populations on all states. This could be experimentally difficult to achieve. We propose here an extension to a 3-level quantum system, having access to the population of the ground state only. By a suitable choice of the control field, we show that a continuous measurement of this population, alone, is enough to identify the field coupling parameters (dipole moment). Simulations with 20% of noise on the measured population illustrate the robustness of the proposed estimation algorithm and confirm the convergence analysis.
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Parameter identification of quantum systems is a fundamental task in developing practical quantum technology. In this article, we study the identification of time-varying decoherence rates for open quantum systems. Given the measurement data of local observables, this can be formulated as an optimization problem. We expand the unknown decoherence rates into Fourier series and take the expansion coefficients as optimization variables. We then convert it into a minimax problem and apply a sequential linear programming technique to solve it. Numerical study on a two-qubit quantum system with a time-varying decoherence rate demonstrates the effectiveness of our algorithm.
In this paper, we consider NNN-level quantum angular momentum systems interacting with electromagnetic fields undergoing quantum nondemolition measurements in continuous-time. We suppose unawareness of the initial state and physical parameters, entailing the introduction of an additional state representing the estimated quantum state. The evolution of the quantum state and its estimation is described by a coupled stochastic master equation. Here, we study the asymptotic behavior of such a system in the presence of a feedback controller. We provide sufficient conditions on the feedback controller and on the estimated parameters that guarantee exponential stabilization of the coupled stochastic system toward an eigenstate of the measurement operator. Furthermore, we estimate the corresponding rate of convergence. We also provide parametrized feedback laws satisfying such conditions. Our results show the robustness of the feedback stabilization strategy considered in [W. Liang, N. H. Amini, and P. Mason, SIAM J. Control Optim., 57 (2019), pp. 3939--3960] in the case of imprecise initialization of the estimated state and with respect to the unknown physical parameters.