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IEEE Conference on Decision and Control (CDC)

Averaging simplifies optimal population transfer problems

Symeon Grivopoulos, Bassam A. Bamieh

Published Jan 1, 2006• 0 citations
Publisher

Abstract

We consider energy-optimal population transfer problems for isolated finite dimensional quantum systems. In the limit when the transfer time is large compared to the time scale of the free dynamics, we show how the associated two-point boundary-value problem (TPBVP) can be reduced to a corresponding "averaged" TPBVP where the free dynamics has been averaged over. This approach offers significant computational and conceptual advantages to the solution of such problems

DOI
10.1109/cdc.2006.377174

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We consider energy-optimal population transfer problems for isolated finite dimensional quantum systems. In the limit when the transfer time is large compared to the time scale of the free dynamics, we show how the associated two-point boundary-value problem (TPBVP) can be reduced to a corresponding "averaged" TPBVP where the free dynamics has been averaged over. This approach offers significant computational and conceptual advantages to the solution of such problems
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.
The control time is supposed to be relative large, the averaged system is obtained by using of the averaging theorem in the quantum system. The control field of averaged system is designed by classical optimal control theorem, then analytic expression for the control field is acquired within optimal energy. As a result, the normal iterative solution of a two-point boundary value problem is avoided by the use of averaging optimal control. At last, the method is used to a three-level quantum system. The some analysis are given according to the simulation test results.
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.
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.