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Science China Mathematics

Bloch theorem for the evolution of states in a cyclic quantum system and a new type of quantum phases

BZ Li, Yan Fl, JH Wu, J.-Q. Liang

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

It is found that in a quantum system with cyclic Hamiltonian the time-dependent Schrodinger equation admits the Bloch type solutions which have the form of Bloch function with respect to the time. These solutions constitute an orthonormal basis of the solution space and lead to a new type of quantum phases. The new phases, called Bloch phases by the authors, are the special Aharonov-Anandan phases, and reduce to the Berry phases under the adiabatic approximation if the degeneracy of instantaneous eigenenergy is time-independent. The above conclusions are proved and calculation for the evolution of spin magnetic moments in a magnetic field precessing with constant angular velocity is exemplified.

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Journal of Physics A Mathematical and General•Aug 14, 1998•19 citations
It is found that in a quantum system with cyclic Hamiltonian the time-dependent Schrodinger equation admits the Bloch type solutions which have the form of Bloch function with respect to the time. These solutions constitute an orthonormal basis of the solution space and lead to a new type of quantum phases. The new phases, called Bloch phases by the authors, are the special Aharonov-Anandan phases, and reduce to the Berry phases under the adiabatic approximation if the degeneracy of instantaneous eigenenergy is time-independent. The above conclusions are proved and calculation for the evolution of spin magnetic moments in a magnetic field precessing with constant angular velocity is exemplified.
The Bloch theorem holds also for the evolution of states in the cyclic quantum systems in which the Hamiltonian varies cyclically with time.In light of the theorem a new type of geometric phases——Bloch phases——is defined.In this paper it is shown that the resonant-(i.e.,acquired by certain states after evolving a cycle)geometric phases so far discovered can all be unified into the Bloch phases.That is,the Bloch phases are identical with the Pancharatnam phases,Aharonov-Anandan phases and Lewis-Riesenfeld phases,and reduce to the Berry phases in adiabatic approximation.To this end,the equivalent alternation of defining the former three types of quantum phases and the generalization of Lewis-Riesenfeld phases and Berry phases to the degenerate case are made.In addition,two methods are given for efficiently searching for the Bloch phases.
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.
Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.