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Physical Review Letters

Optimal Identification of Hamiltonian Information by Closed-Loop Laser Control of Quantum Systems

J. M. Geremia, HERSCHEL A. RABITZ

Published Dec 12, 2002• 68 citations
Publisher

Abstract

A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems.

DOI
10.1103/physrevlett.89.263902

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A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems.
In quantum dynamics of many-body systems, to identify the Hamiltonian becomes more difficult very rapidly as the number of degrees of freedom increases. In order to simplify the dynamics and to deduce dynamically relevant Hamiltonian information, it is desirable to control the dynamics to lie within a reduced space. With a judicious choice for the cost functional, the closed loop optimal control experiments can be manipulated efficiently to steer the dynamics to lie within a subspace of the system eigenstates without requiring any prior detailed knowledge about the system Hamiltonian. The procedure is simulated for optimally controlled population transfer experiments in the system of two degrees of freedom. To show the feasibility of steering the dynamics to lie in a specified subspace, the learning algorithms guiding the dynamics are presented along with frequency filtering. The results demonstrate that the optimal control fields derive the system to the desired target state through the desired subspace.
We introduce optimal identification (OI), a collaborative laboratory/computational algorithm for extracting quantum Hamiltonians from experimental data specifically sought to minimize the inversion error. OI incorporates the components of quantum control and inversion by combining ultrafast pulse shaping technology and high throughput experiments with global inversion techniques to actively identify quantum Hamiltonians from tailored observations. The OI concept rests on the general notion that optimal data can be measured under the influence of suitable controls to minimize uncertainty in the extracted Hamiltonian despite data limitations such as finite resolution and noise. As an illustration of the operating principles of OI, the transition dipole moments of a multilevel quantum Hamiltonian are extracted from simulated population transfer experiments. The OI algorithm revealed a simple optimal experiment that determined the Hamiltonian matrix elements to an accuracy two orders of magnitude better than obtained from inverting 500 random data sets. The optimal and nonlinear nature of the algorithm were shown to be capable of reliably identifying the Hamiltonian even when there were more variables than observations. Furthermore, the optimal experiment acted as a tailored filter to prevent the laboratory noise from significantly propagating into the extracted Hamiltonian.
Most quantum systems considered for control by external fields are plagued by a serious lack of complete information about the underlying Hamiltonian. Traditional feedback control techniques are generally not appropriate due to the latter problem, as well as the ultrafast nature of typical quantum dynamics phenomena and the fact that observations of the quantum system will inevitably lead to a disturbance which may often be contradictory to the desired control. In contrast, learning control techniques have a special role to play in the manipulation of quantum dynamics phenomena. The unique capabilities of quantum systems making them amenable to learning control are (a) the ability to have very large numbers of identical systems for submission to control, (b) the high duty cycle of laboratory laser controls, and (c) the ability to observe the impact of trial controls at ultrafast time scales. Various learning algorithms have been proposed to guide this control process. The paper discusses these proposals, as well as some new perspectives.
This work proposes an end-to-end framework for the learning-enabled control of closed quantum systems. The proposed learning technique is the first of its kind to utilize a hierarchical design which layers probing control, quantum state tomography, quantum process tomography, and Hamiltonian learning to identify both the internal and control Hamiltonians. Within this context, a novel quantum process tomography algorithm is presented which involves optimization on the unitary group, i.e., the space of unitary operators, to ensure physically meaningful predictions. Our scalable Hamiltonian learning algorithms have low memory requirements and tunable computational complexity. Once the Hamiltonians are learned, we formalize data-driven model-predictive quantum control (MPQC). This technique utilizes the learned model to compute quantum control parameters in a closed-loop simulation. Then, the optimized control input is given to a physical quantum system in an open-loop fashion. Simulations show model-predictive quantum control to be more efficient than the current state-of-the-art, quantum optimal control, when sequential quadratic programming (SQP) is used to solve each control problem.
In recent years, great strides have been made in controlling quantum phenomena through the use of tailored laser pulses. This paper discusses several issues that lie ahead, drawing on the ability to perform massive numbers of closed-loop laser control experiments. Particular attention is given to the robustness of these experiments, the capability of deducing control mechanisms and the introduction of special closed-loop control techniques to reveal Hamiltonian information. All these prospects point towards an increasing ability to control and understand quantum systems through close-loop laboratory techniques.
This paper investigates the ability of closed loop quantum learning control techniques to meet a posed physical objective while simultaneously steering the dynamics to lie in a specified subspace. Achievement of successful control with reduced space dynamics can have a number of benefits including a more easily understood control mechanism. Judicious choices for the cost functional may be introduced such that the closed loop optimal control experiments can steer the dynamics to lie within a subspace of the system eigenstates without requiring any prior detailed knowledge about the system Hamiltonian. Learning control with reduced space dynamics takes advantage of the expected existence of a multiplicity of fields that can all give acceptable quality control outcomes. The procedure eliminates the hard demands of following a specific dynamical path by only asking that the dynamics reside in a subspace. Additional measurements characterizing the subspace are necessary to monitor the system evolution during the control field learning process. This procedure is simulated for optimally controlled population transfer experiments in systems of one and two degrees of freedom. The results demonstrate that optimal control fields can be found that successfully derive the system to the target state while staying within the desired subspace.