Non-Abelian geometric phase, Floquet theory and periodic dynamical invariants

2001

Self Cite

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.

“…[10,12], a similar analysis can be performed for the degenerate eigenvalues λ n . This yields an expression for the non-Abelian cyclic geometric phase, namely…”

Section: The Spectrum Of H[r]mentioning

confidence: 99%

“…Yet its evolution operator satisfies [U (T ), I(0)] = 0. This is actually a necessary and sufficient condition for the vectors |λ n , a;R(0) to perform cyclic evolutions, [10].…”

Section: (T) := X[r(t)] and X[r] Is Any Hermitian Operator That Commumentioning

confidence: 99%

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

2001

Self Cite

“…respectively, [9,7]. Note that E n (t), A n (t), ∆ n (t) are Hermitian matrices and u n (t) is unitary.…”

Section: Dynamical Invariantsmentioning

confidence: 99%

Supersymmetric dynamical invariants

2001

Self Cite

We address the problem of identifying the (nonstationary) quantum systems that admit supersymmetric dynamical invariants. In particular, we give a general expression for the bosonic and fermionic partner Hamiltonians. Due to the supersymmetric nature of the dynamical invariant the solutions of the time-dependent Schrödinger equation for the partner Hamiltonians can be easily mapped to one another. We use this observation to obtain a class of exactly solvable time-dependent Schrödinger equations. As applications of our method, we construct classes of exactly solvable time-dependent generalized harmonic oscillators and spin Hamiltonians.

“…The cyclic geometric phase can be conveniently discussed within the framework of the theory of dynamical invariants of Lewis and Riesenfeld [15]. The application of dynamical invariants in the study of the cyclic geometric phases has been considered by Morales [16] and Monteoliva, Korsch and Núñes [17] for the Abelian case and by the present author [18] for the non-Abelian case.…”

mentioning

confidence: 99%

“…In section 4, we derive an expression for the evolution operator and discuss its gauge invariance. In section 5, we give our definition of the noncyclic geometric phase and explore its relationship to the cyclic geometric phase [20,18]. In section 6, we restrict to the Abelian case and compare our definition of the noncyclic Abelian geometric phase with the earlier definitions [5].…”

mentioning

confidence: 99%

Noncyclic geometric phase and its non-Abelian generalization

1999

Self Cite

We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin 1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.