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Mathematical Models and Methods in Applied Sciences

CONTROL OF QUANTUM SYSTEMS

R. B. PÉREZ, V. Protopopescu, J.L. Muñoz-Cobo

Published Mar 1, 1999• 2 citations
Publisher

Abstract

We propose a new control method for systems whose evolution is described by Schrödinger's equation (quantum dynamics). The goal of the control is to induce modifications of observable quantities — with possible effects at mesoscopic or macroscopic levels — by modifying the potential at the microscopic level. We illustrate the feasibility of the approach on a harmonic oscillator system.

DOI
10.1142/s0218202599000166

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We propose a new control method for systems whose evolution is described by Schrödinger's equation (quantum dynamics). The goal of the control is to induce modifications of observable quantities — with possible effects at mesoscopic or macroscopic levels — by modifying the potential at the microscopic level. We illustrate the feasibility of the approach on a harmonic oscillator system.
This paper aims at finding the real-valued dynamics that is equivalent to the Schrödinger equation and then implementing quantum control by making use of the well developed classical control theory. Firstly, pure state identification approaches are presented for two-level, three-level and n-level systems, respectively. Secondly, based on the discussions on the pure state identification, real-valued dynamics that are equivalent to the Schrödinger equations are deduced for both two-level and three-level systems. Finally, a control strategy based on Lyapunov approach is proposed by making use of the obtained real-valued dynamics. Different from the existing Lyapunov control based on the Schrödinger equation, the proposed control strategy can achieve state convergence to its goal state without any constraints on the internal Hamiltonian. Simulation results are included to demonstrate the effectiveness of the approach.
Abstract. The fundamental mathematical definitions of the controlled feedback Markov dynamics of quantum-mechanical systems are introduced with regard to the dynamical reduction and filtering of quantum states in the course of quantum measurement in either discrete or continuous real time. The concept of sufficient coordinates for the description of a posteriori quantum states in a given class is introduced and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schrődinger equation. 1.
Summary form only given. The development of methods to manipulate and measure the quantum mechanical state of a physical system represents one of the great challenges of modern science. Examples of systems in which quantum control is sought or has been accomplished include trapped atomic ions, the electromagnetic field, Rydberg electrons, nuclear spins and vibrations in molecules, and chemical reactions. For many years a paradigm for quantum coherent evolution has been the occurrence of tunneling in macroscopic and mesoscopic double-well potentials. Much theoretical work has been done on this model system, but a series of fundamental questions remain to be addressed in well-controlled experiments, notably whether coherent evolution can be enforced in the presence of a noisy environment, by suitable perturbations of the state of the tunneling particle. The challenge is to find simple physical systems, which can be made to interact with a well characterized environment in a controlled manner.
We illustrate a reverse Von Neumann measurement scheme in which a geometric phase induced on a quantum harmonic oscillator is measured using a microscopic qubit as a probe. We show how such a phase, generated by a cyclic evolution in the phase space of the harmonic oscillator, can be kicked back on the qubit, which plays the role of a quantum interferometer. We also extend our study to finite-temperature dissipative Markovian dynamics and discuss potential implementations in micro- and nanomechanical devices coupled to an effective two-level system.
The previously proposed method to obtain classical limits of quantized fields using von Neumann - lattice coherent states as basis states for the projection form of the Schr\"odinger equation is modified in order to take into account the reaction of the radiation field to the charge in a compact manner. The modification is based on the work of Glauber on the time evolution of coherent states. To illustrate the method, a charged harmonic oscillator with and without initial external electromagnetic field is investigated.