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Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

Optimal population transfers for a quantum system in the limit of large transfer time

Symeon Grivopoulos, Bassam A. Bamieh

Published Jan 1, 2004• 3 citations
Publisher

Abstract

We consider an optimal population transfer problem for a finite-dimensional quantum system with an energy-like cost. We show that a way to realize a small control limit is as the limit of large transfer time T. In the process we show that, in the large T limit, the optimal control is a sum of terms with the following structure: each term is an exponential with frequency given by a Bohr frequency of the quantum system times a slow varying envelope, that is a function of t/T. The form of these envelopes can be computed by solving an "averaged" two-point boundary value problem. We demonstrate our results with an example.

DOI
10.23919/acc.2004.1383837

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Physical Review A•Jun 1, 1988•890 citations
We consider an optimal population transfer problem for a finite-dimensional quantum system with an energy-like cost. We show that a way to realize a small control limit is as the limit of large transfer time T. In the process we show that, in the large T limit, the optimal control is a sum of terms with the following structure: each term is an exponential with frequency given by a Bohr frequency of the quantum system times a slow varying envelope, that is a function of t/T. The form of these envelopes can be computed by solving an "averaged" two-point boundary value problem. We demonstrate our results with an example.
Transferring a quantum system to a final state with given populations is an important problem with applications to quantum chemistry and atomic physics. In this paper, we consider such transfers that minimize L2the norm of the control. This problem is challenging, both analytically and numerically. With the exception of the simplest cases, there is no general understanding of the nature of optimal controls and trajectories. We find that, by examining the limit of large transfer times, we can uncover such general properties. In particular, for transfer times large with respect to the time scale of the free dynamics of the quantum system, the optimal control is a sum of terms, each being a Bohr frequency sinusoid modulated by a slow amplitude, i.e., a profile that changes considerably only on the scale of the transfer time. Moreover, we show that the optimal trajectory follows a ldquomeanrdquo evolution modulated by the fast free dynamics of the system. The calculation of the ldquomeanrdquo optimal trajectory and the slow control profiles is done via an ldquoaveragedrdquo two-point boundary value problem that we derive and which is much easier to solve than the one expressing the necessary conditions for optimality of the original optimal transfer problem.
This paper considers the optimal control of quantum systems. The controlled quantum systems are described by the probability-density-matrix-based Liouville–von Neumann equation. Using projection operators, the states of the quantum system are decomposed into two sub-spaces, namely the 'main state' space and the 'remaining state' space. Since the control energy is limited, a solution for optimizing the external control force is proposed in which the main state is brought to the desired main state at a certain target time, while the population of the remaining state is simultaneously suppressed in order to diminish its effects on the final population of the main state. The optimization problem is formulated by maximizing a general cost functional of states and control force. An efficient algorithm is developed to solve the optimization problem. Finally, using the hydrogen fluoride (HF) molecular population transfer problem as an illustrative example, the effectiveness of the proposed scheme for a quantum system initially in a mixed state or in a pure state is investigated through numerical simulations.
In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.
We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer. For a single intermediate decaying state, we recover the stimulated Raman adiabatic passage process, which we identify as the global optimum in the limit of infinite control time. We also present analytical solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control under the assumption of finite power. If two or more consecutive states in an NNN-level chain are subject to decay, complete population transfer with finite-power controls is not possible.
The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.