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Journal of Chongqing University. English Edition

The Integrable Property and the Topological Property of the Berry's Phase

Aiping Zhang

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

A quantum system with time-dependent Hamiltonian which transports slowly will acquire an undynamic phase factor exp{ iγ n(t) } factor during evolution in addition to a dynamic phase factor. By means of time-dependent gauge transformation, we do a thorough study on undynamic phase which emerge in the quantum system. The results show that the system will gain an non-integrable phase when a periodic evolution doesn't accomplished; however, only when the system achieves a periodic evolution, an integrable phase will be acquire, namely Berry phase, and the phase is invariable under periodic time-dependent gauge transformation, thereby it is observable. Accordingly, we do a general interpretation on the topological property of Berry's phase as an example of NMR.

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In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

InTech eBooks•Feb 24, 2012•16 citations
A quantum system with time-dependent Hamiltonian which transports slowly will acquire an undynamic phase factor exp{ iγ n(t) } factor during evolution in addition to a dynamic phase factor. By means of time-dependent gauge transformation, we do a thorough study on undynamic phase which emerge in the quantum system. The results show that the system will gain an non-integrable phase when a periodic evolution doesn't accomplished; however, only when the system achieves a periodic evolution, an integrable phase will be acquire, namely Berry phase, and the phase is invariable under periodic time-dependent gauge transformation, thereby it is observable. Accordingly, we do a general interpretation on the topological property of Berry's phase as an example of NMR.
We study the dynamics of a quantum system with a time-dependent Hamiltonian which is given by a linear combination of SU(1,1) generators and coupled with a heat bath. An effective Hamiltonian of this dissipative system is derived. With the help of a time-dependent gauge transformation, we obtain the exact solutions of the time-dependent Schrödinger equation, from which the time-evolution operator and the non-adiabatic and adiabatic Berry phases, which depend on time, are calculated explicitly. In the weak dissipation limit, an additional term besides the original Berry phase is found. The additional phase does not have a geometrical meaning due to the dissipation.
The authors discuss certain properties of Berry's phases in two quantum systems with supersymmetrically related Hamiltonians and derive an explicit expression for the difference in their Berry's phases. Moreover, they obtain an expression for a topological quantity which can be interpreted in term of holonomy, and which contains the Berry's phase and is invariant in the two supersymmetrically related quantum systems. They take two examples to illustrate their findings. The first one is the example of the Aharonov-Bohm effect in which the Berry's phases of the two supersymmetrically related quantum systems are equal. The second example is a system with a spin-1/2 particle in a time varying magnetic field. They obtain the stated topological quantity and, in their analysis, they discover that this quantity is corresponding to the essential gauge transformation in such systems.
A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.
Illustration of the geometric and topological properties of Berry phase is often in an obscure and abstract language of fiber bundles. In this article, we demonstrate these properties with a lucid and concrete system whose parameter space is a torus. The instantaneous eigenstate is regarded as a basis. And the corresponding connection and curvature are calculated respectively. Furthermore, we find the magnitude of curvature is exactly the Gaussian curvature, which shows its local property. The topological property is reflected by the integral over the torus due to Gauss-Bonnet theorem. When we study the property of parallel transportation of a vector over a loop, we make a conclusion that the Berry phase is just the angle between the final and initial vectors. And we also illuminate the geometric meaning of gauge transformation, which is just a rotation of basis.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects