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1 Non-Abelian Geometric Phase, Floquet Theory, and Periodic Dynamical Invariants

Alí Mostafazadeh

Published Jan 1, 1998• 9 citations
OpenAlex

Abstract

For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field. PACS numbers: 03.65.Bz 1

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1 A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians

For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:

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Classification of cyclic initial states and geometric phase for the spin-j system

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Quantum states which evolve cyclically in their projective Hilbert space give rise to a geometric (or Aharonov-Anandan) phase. An aspect of primary interest is stable cyclic behaviour as realized under a periodic Hamiltonian. The problem has been handled by use of time-dependent transformations treated along the lines of Floquet's theory as well as in terms of exponential operators with a goal to examine the variety of initial states exhibiting cyclic behaviour. A particular case of special cyclic initial states is described which is shown to be important for nuclear magnetic resonance experiments aimed at the study of the effects of the geometric phase. An example of arbitrary spin j in a precessing magnetic field and spin j=1 subject to both axially symmetric quadrupolar interaction and a precessing magnetic field are presented. The invariant (Kobe's) geometric phase is calculated for special cyclic states.

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In the paper a problem of geometric phase decomposition for general evolutions in the Hilbert space is addressed. The decomposition of total phase into dynamical and geometrical parts employs a state representation in the noninteraction picture. The noninteraction picture is introduced with the help of decomposition of the system evolution operator into two parts, the one pertaining to free evolution and the other to interaction. The procedure requires the problem to be an exactly solvable case of the Schrödinger equation. The most common class of such problems includes dipole Hamiltonians, for which the evolution operator can be decomposed into a combination of unitary operators. Geometric phase decomposition in the noninteraction picture can be applied to general noncyclic evolutions, but for the cyclic states it reduces to the Floquet decomposition. Defined this way, geometric phase possesses characteristic features of geometric phase for the free system, but extends some stationary properties to temporal dependences. As a case, the time-dependent relationship between geometric phase and the nonstationarity of a quantum state is illustrated by an example of a spin-½ in a rotating magnetic field. It is shown that geometric phase reaches a maximum when the spin state becomes completely nonstationary.

Journal of Physics A Mathematical and General•Mar 23, 2000•2 citations
For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field. PACS numbers: 03.65.Bz 1
For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field.
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.
We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V(\mathbit{\ensuremath{\lambda}}\left(t\right))$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (noncommuting) geometric phases appearing when the slowly varying parameter $\mathbit{\ensuremath{\lambda}}=\mathbit{\ensuremath{\lambda}}\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
Quantum states which evolve cyclically in their projective Hilbert space give rise to a geometric (or Aharonov-Anandan) phase. An aspect of primary interest is stable cyclic behaviour as realized under a periodic Hamiltonian. The problem has been handled by use of time-dependent transformations treated along the lines of Floquet's theory as well as in terms of exponential operators with a goal to examine the variety of initial states exhibiting cyclic behaviour. A particular case of special cyclic initial states is described which is shown to be important for nuclear magnetic resonance experiments aimed at the study of the effects of the geometric phase. An example of arbitrary spin j in a precessing magnetic field and spin j=1 subject to both axially symmetric quadrupolar interaction and a precessing magnetic field are presented. The invariant (Kobe's) geometric phase is calculated for special cyclic states.
In the paper a problem of geometric phase decomposition for general evolutions in the Hilbert space is addressed. The decomposition of total phase into dynamical and geometrical parts employs a state representation in the noninteraction picture. The noninteraction picture is introduced with the help of decomposition of the system evolution operator into two parts, the one pertaining to free evolution and the other to interaction. The procedure requires the problem to be an exactly solvable case of the Schrödinger equation. The most common class of such problems includes dipole Hamiltonians, for which the evolution operator can be decomposed into a combination of unitary operators. Geometric phase decomposition in the noninteraction picture can be applied to general noncyclic evolutions, but for the cyclic states it reduces to the Floquet decomposition. Defined this way, geometric phase possesses characteristic features of geometric phase for the free system, but extends some stationary properties to temporal dependences. As a case, the time-dependent relationship between geometric phase and the nonstationarity of a quantum state is illustrated by an example of a spin-½ in a rotating magnetic field. It is shown that geometric phase reaches a maximum when the spin state becomes completely nonstationary.