Analytic controllability of quantum-mechanical systems

Erez Abrams

The mechanism governing the evolution of controlled quantum systems is often obscured, making their dynamics hard to interpret. Mitra and Rabitz {[Phys. Rev. A 67, 033407 (2003)]} define mechanism via a perturbative expansion of pathways between eigenstates; the evolution of the system is driven by the constructive and destructive interference of these pathway amplitudes. In this paper, we explore mechanism in controlled single-qubit systems and describe novel analytic methods for computing the mechanism underlying the evolution of a single qubit under a piecewise constant control.

Sonia G. Schirmer, I. C. H. Pullen, A. I. Solomon

Mikhail S. Ananyevskiy

A new approach to synthesize algorithms for selective control of the observables in quantum mechanical systems in the presence of additional constraints during the whole period of control is proposed. Analytic results of achieving the goal of control under some additional assumptions were obtained. It was demonstrated that the error in achieving the goal of control is proportionate to the error in prescribing the initial state of system and the error in realizing the control action. Numerical results for the problem of selective control for energy of hydrogen molecules (H 2 ) with different isotopes are represented. The proposed algorithms are easy to design.

TZYH JONG TARN, J. W. Clark, Garng M. Huang

Dawit Hiluf Hailu

In this chapter, we introduce the application of Sylvester’s formula for systems with degenerate eigenvalues to obtain their analytical solutions. To illustrate its utility, we include two additional methods for analytical solutions: the adiabatic approximation and the Magnus expansion. In quantum mechanics, the Schrödinger equation is a mathematical expression that describes the time evolution of a physical system where quantum effects, such as wave-particle duality, are significant. This equation serves as a framework for analyzing quantum mechanical systems. Similar to how Newton’s laws govern the motion of classical objects, the Schrödinger equation governs the behavior of quantum entities. However, unlike classical motion, quantum mechanics deals with the probabilities of different trajectories. The ability to obtain analytical solutions to such equations provides an in-depth understanding of the dynamics and also allows for the identification of controlling parameters, as analytical functions can readily reveal these insights.

А. Г. Бутковский, Yu. I. Samoilenko