Certain Methods of Constructing Controls for Quantum Systems
- PDF / 247,358 Bytes
- 7 Pages / 594 x 792 pts Page_size
- 45 Downloads / 151 Views
CERTAIN METHODS OF CONSTRUCTING CONTROLS FOR QUANTUM SYSTEMS A. N. Pechen’
UDC 517.977.5
Abstract. Recently, various control problems for quantum systems, such as individual atoms, molecules, and electron states at quantum points, have been actively researched. In this paper, we briefly discuss methods of constructing controls for quantum system by means of gradient algorithms, genetic algorithms, and the speed gradient. The violation of the asymptotic stabilizability condition for the problem of generation of unitary operations is proved by the speed gradient method in two-level quantum systems. Keywords and phrases: quantum system control, quantum effects. AMS Subject Classification: 81Q93
1. Introduction. In problems of quantum technologies, i.e., technologies based on the use of quantum effects and individual quantum systems, it is required to develop methods for quantum system control (see [4, 5, 10, 17, 27, 29]). Control can be performed by using a modulated laser pulse, by an incoherent action of the reservoir, by measurements, and so on. The main goal of control is the creation of a desired state of the system under an additional condition such as minimal energy, or minimal duration of control, or generation of a given unitary operation, or stabilization of the mean value of some observable of a quantum system. Various algorithms are used for constructing controls, for example, the gradient ascent pulse engineering (GRAPE) algorithm (see [15]), the Krotov method (see [16, 30]), the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm and its modifications (see [3, 7, 11, 28]), machine learning (see [12]), method of speed gradient (see [2, 9, 20]), genetic algorithms (see [14, 24]), and mixed approaches (see [18]). Below we consider such methods of constructing quantum system controls as gradient methods, genetic algorithms, and the speed gradient method. 2. Problems of quantum systems control. Consider a quantum system with a Hilbert space H. The most general state of the system is described by the density matrix, i.e., a nonnegative operator ρ : H → H, ρ ≥ 0, which has the unit trace, Trρ = 1. Pure states correspond to density matrices that are projectors, i.e., such that ρ2 = ρ. All other density matrices correspond to mixed states. The dynamics of a quantum system isolated from the influence of the environment under the action of coherent control u(t) (for example, a laser pulse) is described by the Schr¨odinger equation: dUt = −i H0 + V u(t) Ut , Ut=0 = I; (1) dt here Ut is a unitary evolution operator, H0 and V are self-adjoint operators in H called the free Hamiltonian and the Hamiltonian of interaction of the system with a coherent control. In this case, the matrix density of the system is transformed by the formula ρt = Ut ρ0 Ut† ,
(2)
where ρ0 is the initial state of the system at the moment of time t = 0. If a quantum system is open, i.e., interacts with a reservoir, then its evolution is described by the following master equation for the density matrix: dρt = −i[H0 + V u(t), ρt ] + L (ρt ). (3)
Data Loading...
