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Physical Review A

Optimal control of a three-level quantum system by laser fields plus von Neumann measurements

Dominique Sugny, Cyrill Kontz

Published Jun 24, 2008• 62 citations
Publisher

Abstract

We investigate the control of a three-level quantum system by laser fields assisted by von Neumann measurements. We consider a system which is not completely controllable by unitary evolution but which becomes controllable if particular measurements are used. The optimal control is defined from a cost functional which takes into account the measurements. The cost corresponds either to the minimization of the duration of the control or to the minimization of the energy of the laser field. Using the Pontryagin maximum principle, we determine the optimal control which steers the system from a given initial state toward a desired target state. This allows one to determine which observable has to be chosen for the measurement and the time at which the measurement has to be performed.

DOI
10.1103/physreva.77.063420

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We investigate the control of a three-level quantum system by laser fields assisted by von Neumann measurements. We consider a system which is not completely controllable by unitary evolution but which becomes controllable if particular measurements are used. The optimal control is defined from a cost functional which takes into account the measurements. The cost corresponds either to the minimization of the duration of the control or to the minimization of the energy of the laser field. Using the Pontryagin maximum principle, we determine the optimal control which steers the system from a given initial state toward a desired target state. This allows one to determine which observable has to be chosen for the measurement and the time at which the measurement has to be performed.
A general scheme is presented for controlling quantum systems using evolution driven by nonselective von Neumann measurements, with or without an additional tailored electromagnetic field. As an example, a two-level quantum system controlled by nonselective quantum measurements is considered. The control goal is to find optimal system observables such that consecutive nonselective measurement of these observables transforms the system from a given initial state into a state which maximizes the expected value of a target operator (the objective). A complete analytical solution is found including explicit expressions for the optimal measured observables and for the maximal objective value given any target operator, any initial system density matrix, and any number of measurements. As an illustration, upper bounds on measurement-induced population transfer between the ground and the excited states for any number of measurements are found. The anti-Zeno effect is recovered in the limit of an infinite number of measurements. In this limit the system becomes completely controllable. The results establish the degree of control attainable by a finite number of measurements.
In this paper we have synthesized optimal control of multi-level quantum system by representing unitary operator in terms of the projection operators of the Hamiltonian of the system. We have solved the steering problem for multi-level quantum systems minimizing energy cost functional of the control. The optimal control problem of the time evolution of quantum spin of Pauli two-level system subjected to an external field with the minimum energy function is also illustrated and formulated in terms of the quantum spin up and spin down states of the Pauli two-level system.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.
We study a quantum spin system acting on a single quantum bit. The evolution of this system is governed by the Schrodinger equation which takes the form of a right-invariant system on the special unitary group SU(2) with two control inputs. Using a suitable version of Pontryagin's Principle which is tailor-made for control problems on Lie groups, the optimal controls are derived in two cases: the energy-optimal case (in which the control effort is minimized for a specified end time) and the time-optimal case (in which the control duration is minimized for given constraints on the size of the controls).
We propose an optimal control strategy to generate maximally entangled states in bipartite quantum systems. Leveraging the Pontryagin's Principle, we derive time-dependent control fields that maximize the entanglement measure, specifically concurrence, within minimal time while adhering to input constraints. Our formulation addresses the Liouville-von Neumann dynamics of the reduced density matrix under unitary evolution. The strategy is numerically validated through simulations, demonstrating its ability to drive the system from an initial perturbed separable state to a maximally entangled target state. The results showcase the effectiveness of switching control fields in optimizing entanglement, with potential applications in quantum technologies, including communication, computation, and sensing.
This paper addresses the optimal control of quantum coherence in multi-level systems, modeled by the Lindblad master equation, which captures both unitary evolution and environmental dissipation. We develop an energy minimization framework to control the evolution of a qutrit (three-level) quantum system while preserving coherence between states. The control problem is formulated using Pontryagin's Minimum Principle in the form of Gamkrelidze, incorporating state constraints to ensure coherence remains within desired bounds. Our approach accounts for Markovian decoherence, demonstrating that the Lindblad operator is non-unital, which reflects the irreversible decay processes inherent in the system. The results provide insights into effectively maintaining quantum coherence in the presence of dissipation.