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Physical Review A

Assuring robustness to noise in optimal quantum control experiments

Andreas F. Bartelt, Matthias Roth, Manjusha Mehendale, HERSCHEL A. RABITZ

Published Jun 15, 2005• 22 citations
Publisher

Abstract

Closed-loop optimal quantum control experiments operate in the inherent presence of laser noise. In many applications, attaining high quality results [i.e., a high signal-to-noise $(S∕N)$ ratio for the optimized objective] is as important as producing a high control yield. Enhancement of the $S∕N$ ratio will typically be in competition with the mean signal, however, the latter competition can be balanced by biasing the optimization experiments towards higher mean yields while retaining a good $S∕N$ ratio. Other strategies can also direct the optimization to reduce the standard deviation of the statistical signal distribution. The ability to enhance the $S∕N$ ratio through an optimized choice of the control is demonstrated for two condensed phase model systems: second harmonic generation in a nonlinear optical crystal and stimulated emission pumping in a dye solution.

DOI
10.1103/physreva.71.063806

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Closed-loop optimal quantum control experiments operate in the inherent presence of laser noise. In many applications, attaining high quality results [i.e., a high signal-to-noise (S∕N)(S∕N)(S∕N) ratio for the optimized objective] is as important as producing a high control yield. Enhancement of the S∕NS∕NS∕N ratio will typically be in competition with the mean signal, however, the latter competition can be balanced by biasing the optimization experiments towards higher mean yields while retaining a good S∕NS∕NS∕N ratio. Other strategies can also direct the optimization to reduce the standard deviation of the statistical signal distribution. The ability to enhance the S∕NS∕NS∕N ratio through an optimized choice of the control is demonstrated for two condensed phase model systems: second harmonic generation in a nonlinear optical crystal and stimulated emission pumping in a dye solution.
Optimal control operates in the presence of noise. Enhancement of the signal to noise ratio through optimized control is demonstrated for two condensed phase systems: the concept was demonstrated for second harmonic generation (SHG) in a nonlinear optical crystal and stimulated emission pumping (SEP) in a dye solution. Management of inherent control noise in the manipulation of quantum system is a subject of basic concern in virtually all applications of manipulating quantum dynamics phenomena. The cases of SHG and SEP in this work serve as illustrations of this capability. For control lasers operating at a kHz repetition rate, readily available electronics can handle this data stream.
The impact of control field fluctuations on the optimal manipulation of quantum dynamics phenomena is investigated. The quantum system is driven by an optimal control field, with the physical focus on the evolving expectation value of an observable operator. A relationship is shown to exist between the system dynamics and the control field fluctuations, wherein the process of seeking optimal performance assures an inherent degree of system robustness to such fluctuations. The presence of significant field fluctuations breaks down the evolution of the observable expectation value into a sequence of partially coherent robust steps. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations by taking advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness are discussed in the light of recent experiments and numerical simulations on the optimal control of quantum phenomena. The analysis in this paper bodes well for the future success of closed-loop quantum optimal control experiments, even in the presence of reasonable levels of field fluctuations.
Despite significant progress in theoretical and laboratory quantum control, engineering quantum systems remains principally challenging due to manifestation of noise and uncertainties associated with the field and Hamiltonian parameters. In this paper, we extend and generalize the asymptotic quantum control robustness analysis method, which provides more accurate estimates of quantum control objective moments than standard leading-order techniques, to diverse quantum observables, gates, and moments thereof and also introduce the Pontryagin maximum principle for quantum robust control. In addition, we present a Pareto optimization framework for achieving robust control via evolutionary open-loop (model-based) and closed-loop (model-free) approaches with the mechanisms of robustness and convergence described using asymptotic quantum control robustness analysis. In the open-loop approach, a multiobjective genetic algorithm is used to obtain Pareto solutions in terms of the expectation and variance of the transition probability under Hamiltonian parameter uncertainty. The set of numerically determined solutions can then be used as a starting population for model-free learning control in a feedback loop. The closed-loop approach utilizes a real-coded genetic algorithm with adaptive exploration and exploitation operators in order to preserve solution diversity and dynamically optimize the transition probability in the presence of field noise. Together, these methods provide a foundation for high-fidelity adaptive feedback control of quantum systems wherein open-loop control predictions are iteratively improved based on data from closed-loop experiments.
Quantum control theory is a rapidly evolving research field, which has developed over the last three decades. Quantum optics has practical importance in quantum technology and provides a promising means of implementing quantum information and computing device. \nIn quantum control, it is difficult to acquire information about quantum states without destroying them since microscopic quantum systems have many unique characteristics such as entanglement and coherence which do not occur in classical mechanical system. Therefore, \nthe Lyapunov-based control methodology is used to first construct an artificial closed-loop controller and then an open-loop law is derived by simulation of the artificial closed-loop system. \n \nThis work considers the stabilization of laser cavity as the main integral part of quantum optical systems through squeezing the output beam of the cavity. As a comprehensive example of this type of system, quantum optomechanical sensors are investigated. To this end, a nonlinear model \nof quantum optomechanical sensors is first extended incorporating various noises. Then, linear quadratic Gaussian (LQG) control method is used to tackle the problem of mode-squeezing in optomechanical sensors. \nCoherent feedback quantum control is synthesized by incorporating both shot noise and back-action noise to attenuate the output noise well below the shot noise level (Two waves are said to be coherent if they have a constant relative phase). In the second phase of this work, due to entanglement of the system with critical uncertainties and technical limitations such as laser noise and detector imprecision, robust H [infinity] method is employed for the robust stabilization and robust performance of the system in practice. In H [infinity] methods, a control designer expresses the control problem as a mathematical optimization problem and then finds the controller that solves this. The effectiveness of the proposed control strategy in squeezing the cavity output beam is demonstrated by simulation.
The impact of control field fluctuations on the optimal manipulation of quantum dynamics phenomena is investigated. The presence of significant field fluctuations is shown to break down the evolution into a sequence of partially coherent robust steps. Robustness occurs because the optimization process reduces sensitivity to noise-driven quantum system fluctuations. This process takes advantage of the observable expectation value being bilinear in the evolution operator and its adjoint. The consequences of this inherent robustness bodes well for the future success of closed loop quantum optimal control experiments.
This paper investigates the impact of control field noise on the optimal manipulation of quantum dynamics. Simulations are performed on several multilevel quantum systems with the goal of population transfer in the presence of significant control noise. The noise enters as run-to-run variations in the control amplitude and phase with the observation being an ensemble average over many runs as is commonly done in the laboratory. A genetic algorithm with an improved elitism operator is used to find the optimal field that either fights against or cooperates with control field noise. When seeking a high control yield it is possible to find fields that successfully fight with the noise while attaining good quality stable results. When seeking modest control yields, fields can be found which are optimally shaped to cooperate with the noise and thereby drive the dynamics more efficiently. In general, noise reduces the coherence of the dynamics, but the results indicate that population transfer objectives can be met by appropriately either fighting or cooperating with noise, even when it is intense.