Exactly solvable time-dependent models in quantum mechanics and their applications
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actly Solvable Time-Dependent Models in Quantum Mechanics and Their Applications A. A. Suzko and G. Giorgadze Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Abstract—Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space–time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed. PACS numbers: 03.65.Nk, 03.65.Vf, 03.67.Lx DOI: 10.1134/S1063779608040059
1. INTRODUCTION Problems of the evolution of dynamical systems are at the center of attention in view of recent achievements in various fields of physics. Many interesting phenomena, such as the molecular Aharonov–Bohm effect [1, 2], geometric phase [3, 4], level crossing [5] identified with the Landau–Zener transitions [6, 7], and dynamical localization of particles in systems with a limited spatial dimension [8–11], have been revealed in atomic and molecular physics, quantum chemistry, quantum optics, and solid state physics. Active investigations into quantum computers (see, e.g., [13–18] and references therein) refresh interest in the geometric phase phenomenon in quantum mechanics. The authors of recent works [19–21] proposed a design of a holonomic quantum computer on the basis of the non-Abelian Berry phase [3]. For this reason, the problem of simulation of dynamical systems with preset properties by quantum-mechanical methods is of current interest. In our opinion, exactly solvable time-independent and time-dependent quantum-mechanical models will be used in fruitful investigations in these scientific areas and will promote the discovery of new properties. As the zeroth approximation for the problem with the Hamiltonian H(t) = h + h1(t), the system with the time-independent Hamiltonian h is considered. The time-dependent part of the Hamiltonian h1 is assumed to be small, h1 h, and is considered as perturbation inducing transitions between the eigenstates of h. However, if h1 is not small and is a periodic function of time, h1(t + T) = h1(t), another approach [22–24] is more appropriate, because neither perturbation theory nor the adiabatic approximation is applicable. According to the group-theoretical represent
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