Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Invariants and geometric phase for systems with non-Hermitian time-dependent Hamiltonians

Xiao-Chun Gao, Jing-Bo Xu, Tiezheng Qian

Published Oct 1, 1992• 46 citations
Publisher

Abstract

In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. It is then used to study the nonadiabatic cyclic evolution and the Aharonov-Anandan phase. It is shown that the study of noncyclic evolution can be reduced to the study of cyclic evolution. The two-level dissipative system and the classical time-dependent harmonic oscillator are discussed as illustrative examples.

DOI
10.1103/physreva.46.3626

Related papers

Geometric phase and the generalized invariant formulation

Xiao-Chun Gao, Jing‐Bo Xu, Tiezheng Qian

An alternative concept of the basic invariants is introduced. The Lewis-Riesenfeld invariant theory is extended to obtain a generalized invariant formulation. The formulation is then used to establish four facts: (i) Any invariant for a quantum system can be constructed in terms of the basic invariants. (ii) It is possible to introduce a solution-generating technique by making use of the basic invariants. (iii) The path integral in the generalized invariant formulation reduces to an ordinary integral. (iv) The study of noncyclic evolution of a quantum system reduces explicitly to the study of the cyclic evolution. Finally, phase factors and general solution for the driven generalized time-dependent harmonic oscillator are studied as an illustrative example.

Physical Review A•Dec 1, 1991•152 citations

Non-Hermitian Hamiltonian beyond PT symmetry for time-dependent S U ( 1 , 1 ) and S U ( 2 ) systems — Exact solution and geometric phase in pseudo-invariant theory

Nadjat Amaouche, Maroua Sekhri, Rahma Zerimeche +2 more

In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.

Physics Open•Nov 21, 2022•4 citations

Exact Solution of the Bloch Equation with Time-Dependent Coefficients and Aharonov-Anandan Phase

Jing‐Bo Xu, Lan Wang, Xiao-Chun Gao

In this letter, the matrix invariant of the Bloch equation with time-dependent coefficients is found, the time evolution of the system is discussed and the exact solution of the system is obtained. We then proceed to study the cyclic evolution of the system, obtaining the Aharonov-Anandan phase. Finally, it is shown that the study of noncyclic evolution of the system can be reduced to the cyclic evolution.

Chinese Physics Letters•Mar 1, 1993•0 citations

Time evolution, cyclic solutions and geometric phases for the generalized time-dependent harmonic oscillator

Qiong-Gui Lin

The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.

arXiv:quant-ph/0402159•Journal of Physics A Mathematical and General•Jan 9, 2004•5 citations

Time Evolution of a Particle with Spin-j in an Oscillating Magnetic Field and Geometric Phases

Yu You

Quantum invariant theory is an effective method to study geometric phase problems. By making use of the Lewis Riesenfeld invariant theory, a particle with spin j in an oscillating magnetic field is studied and the exact solutions for two different cyclic periods are found. It is then used to obtain two different geometric phases corresponding to two different cyclic periods. It is quite clear from our derivation that the geometric phases corresponding to two different cyclic periods are closely associated with the initial states of the system.

Bulletin of Science and Technology•Jan 1, 2000•0 citations

BLOCH THEOREM FOR THE EVOLUTION OF STATES IN THE CYCLIC QUANTUM SYSTEMS AND THE UNIFICATION OF RESONANT GEOMETRIC PHASES

Bo‐Zang Li, Degang Zhang, WU JIAN-HUA +2 more

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.

Acta Physica Sinica•Jan 1, 1997•5 citations
In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. It is then used to study the nonadiabatic cyclic evolution and the Aharonov-Anandan phase. It is shown that the study of noncyclic evolution can be reduced to the study of cyclic evolution. The two-level dissipative system and the classical time-dependent harmonic oscillator are discussed as illustrative examples.
An alternative concept of the basic invariants is introduced. The Lewis-Riesenfeld invariant theory is extended to obtain a generalized invariant formulation. The formulation is then used to establish four facts: (i) Any invariant for a quantum system can be constructed in terms of the basic invariants. (ii) It is possible to introduce a solution-generating technique by making use of the basic invariants. (iii) The path integral in the generalized invariant formulation reduces to an ordinary integral. (iv) The study of noncyclic evolution of a quantum system reduces explicitly to the study of the cyclic evolution. Finally, phase factors and general solution for the driven generalized time-dependent harmonic oscillator are studied as an illustrative example.
In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
Quantum invariant theory is an effective method to study geometric phase problems. By making use of the Lewis Riesenfeld invariant theory, a particle with spin j in an oscillating magnetic field is studied and the exact solutions for two different cyclic periods are found. It is then used to obtain two different geometric phases corresponding to two different cyclic periods. It is quite clear from our derivation that the geometric phases corresponding to two different cyclic periods are closely associated with the initial states of the system.
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.