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The Journal of Chemical Physics

Incorporating physical implementation concerns into closed loop quantum control experiments

J. M. Geremia, Wusheng Zhu, HERSCHEL A. RABITZ

Published Dec 22, 2000• 111 citations
Publisher

Abstract

In quantum control experiments, it is desirable to build features into the field that address physical concerns such as simplicity, robustness, dynamical coherence, power expenditure, etc. With a judicious choice for the cost functional, it is possible to incorporate such secondary features into the field, often without altering the experimental procedure or apparatus. Through simulated closed-loop population transfer experiments, we demonstrate the benefit of carefully designed cost functionals. As specific examples, we address two common physical concerns: removing extraneous structure from the control pulse and finding robust fields. Removing unnecessary field components is critical if information about the mechanism is to be interpreted from the structure of the optimal pulse. Robust fields produce a stable outcome despite noise in the field and, perhaps, environmental inhomogeneities in the quantum system as is typical of condensed phase experiments.

DOI
10.1063/1.1326905

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In quantum control experiments, it is desirable to build features into the field that address physical concerns such as simplicity, robustness, dynamical coherence, power expenditure, etc. With a judicious choice for the cost functional, it is possible to incorporate such secondary features into the field, often without altering the experimental procedure or apparatus. Through simulated closed-loop population transfer experiments, we demonstrate the benefit of carefully designed cost functionals. As specific examples, we address two common physical concerns: removing extraneous structure from the control pulse and finding robust fields. Removing unnecessary field components is critical if information about the mechanism is to be interpreted from the structure of the optimal pulse. Robust fields produce a stable outcome despite noise in the field and, perhaps, environmental inhomogeneities in the quantum system as is typical of condensed phase experiments.
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The examples of quantum-state manipulation and coherent excitation discussed in this monograph present idealizations of actual quantum systems, simplifications that allow straightforward theoretical description. As one moves beyond the models of isolated atoms, few essential states, and transform-limited pulses to deal with more realistic models that can describe experimental reality, the basic tools described hitherto require elaboration and extension. Theoretical treatments of large molecules and chemical reactions involving laser-induced changes rely upon numerical simulation more than on analytic solutions. This final chapter discusses two of the themes applicable to that work: control theory and optimization. Control theory Classical control theory, as followed by mathematicians and engineers, deals with procedures for manipulating the input (the “controls”) of a dynamically changing system to obtain a desired output of the system. In a closed-loop control system some device measures the output and, using a feedback loop, alters the input (via a control element ) to bring the output closer to conformity with a goal. These techniques typically find application in control of experiments but they also work for theoretical modeling. An open-loop control system is one without such feedback; the controls are adjusted in accord with some established plan. Design of a suitable control mechanism (a control function ratioing output to input) must ensure that the system is stable (i.e. a finite input signal produces a finite output signal) and controllable (i.e. it is possible to obtain the desired output).
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.
Advances in quantum control theory and optimization techniques have made possible a new phenomena of quantum engineering. Chemical reactions can now be dynamically manipulated using first-principles quantum chemical models, in which a laser field, rather than a catalyst, serve as photonic reagents to selectively break a chemical bond and yield products via intramolecular rearrangement. A field pulse may also be designed to perform ultrasensitive spectroscopy, where compounds with very similar chemical properties are differentiated through their unique interaction with resonant and off-resonant photons. Despite significant progress in theoretical and laboratory quantum control, however, quantum engineering remains principally challenging due to manifestation of noise and uncertainties associated with the input and system parameters. Specifically, a small magnitude of variation in the field and Hamiltonian parameters would cause an otherwise optimal field to deviate from controlling desired quantum state transitions and reaching a particular objective. Thus, an accurate analysis of robustness of controlled quantum dynamics followed by a robust control framework is essential in achieving quantum control in a practical setting. In this work, theoretical foundations for quantum control robustness analysis are presented from both a distributional perspective - in terms of moments of the transition amplitude, interferences, and transition probability - and a worst-case perspective. In addition, an evolutionary control strategy for achieving robust control via open- and closed-loop approach are described with its mechanism of convergence elucidated using the robustness analysis method.
High fidelity coherent control of quantum systems is critical to building quantum devices and quantum computers. We provide a general optimal control framework for designing control sequences that account for hardware control distortions while maintaining robustness to environmental noise. We demonstrate the utility of our algorithm by presenting examples of robust quantum gates optimized in the presence of nonlinear distortions. We show that nonlinear classical controllers do not necessarily incur additional computational cost to pulse optimization, enabling more powerful quantum devices.