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Journal of Physics A Mathematical and General

Geometric phase decomposition for exactly solvable cases with interaction

Rimvydas Aleksiejūnas, V. Ivaška

Published Mar 23, 2000• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/33/12/306

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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.
We introduce the notion of the geometrically equivalent quantum systems (GEQSs) 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 GEQSs. 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 GEQSs includes a system with a T -periodic Hamiltonian. We apply our general results to study the classes of GEQSs that include a system with a cranked Hamiltonian H ( t ) = e -i Kt H 0 e i Kt . 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.
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 H0e 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. 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.
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
We introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and noncyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable state during its evolution in Hilbert space. This is analytically shown for N=2N=2N=2 spins and numerically illustrated for larger chains. This relationship promotes the interaction geometric phase to an indicator of global entanglement in the system, which may constitute a useful tool for quantum tasks based on entanglement as a resource to their performance.
This paper focuses on the geometric phase of entangled states of bipartite systems under bilocal unitary evolution. We investigate the relation between the geometric phase of the system and those of the subsystems. It is shown that (1) the geometric phase of cyclic entangled states with nondegenerate eigenvalues can always be decomposed into a sum of weighted nonmodular pure state phases pertaining to the separable components of the Schmidt decomposition, although the same cannot be said in the noncyclic case, and (2) the geometric phase of the mixed state of one subsystem is generally different from that of the entangled state even if the other subsystem is kept fixed, but the two phases are the same when the evolution operator satisfies conditions where each component in the Schmidt decomposition is parallel transported.