Geometric phases, symmetries of dynamical invariants and exact solution of the Schrödinger equation

Abstract

We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T -periodic, the corresponding class of GEQS includes a system with a T -periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian H(t) = e −iKt H 0 e iKt . We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator.

Invariant Quantum States of Quadratic Hamiltonians

“…The important applications of the Lewis-Riesenfeld and Malkin-Man'ko invariants include, e.g., the "inverse engineering" of quadratic Hamiltonians: a search of "shortcuts to adiabaticity" in different kinds of traps [57][58][59][60]. Other applications are related to the theory of the geometric (Berry) phase [61][62][63][64][65][66][67], invariants of non-Hermitian Hamiltonians [68][69][70], and open quantum systems [71][72][73][74].…”

Quantum systems simulatability through classical networks

“…where This kind of mapping was studied in previous works from a pure mathematical point of view for applications to differential equations in complex variables with singular operators [7][8][9]. For a physical point of view the same kind of mapping was presented in [10][11][12] in order to solve particular quantum systems.…”

Fast generation of three-dimensional entanglement between two spatially separated atoms via invariant-based shortcut

“…In order to construct the invariant-based shortcut for generating three-dimensional entanglement, we need to find out the Hermitian invariant operator I(t), which satisfies i ∂I(t) ∂t = [H 0 (t), I(t)]. Since H 0 (t) possesses SU(2) dynamical symmetry, I(t) can be easily given by [35,36]…”