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CRC Press eBooks

Berry’s Phase, Aharonov–Bohm and Sagnac Effects

S. Rajasekar, R. Velusamy

Published Dec 10, 2014• 0 citations
Publisher

Abstract

Consider a quantum mechanical system beginning at a time ti with an eigenstate ψn(x, ti). Suppose its potential is slowly changing. We assume that ∂H/∂t is very small. Then according to adiabatic approximation the wave function ψ of the system is given by ψ(t) = ∑ n an(t)ψn(t) e −i ∫ t where an’s are unknown to be determined, En is the energy eigenvalue of nth state and ∑ runs over all possible states. The phase factor −i ∫ t ti En(t ′) dt′ is called the dynamical phase. In adiabatic approximation the initial state ψn(ti) changes into ψn(t, ti) without combination with other states. Hence, Eq. (5.1) can be written as ψ(t) = eiη(t,ti) ψn(t) . (5.2) Usually, the phase factor η(t, ti) is set to zero as only ψψ∗ is measurable. It has been assumed that η(t, ti) does not give any observable effect.

DOI
10.1201/b17827-9

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•Jan 1, 2019•0 citations
Consider a quantum mechanical system beginning at a time ti with an eigenstate ψn(x, ti). Suppose its potential is slowly changing. We assume that ∂H/∂t is very small. Then according to adiabatic approximation the wave function ψ of the system is given by ψ(t) = ∑ n an(t)ψn(t) e −i ∫ t where an’s are unknown to be determined, En is the energy eigenvalue of nth state and ∑ runs over all possible states. The phase factor −i ∫ t ti En(t ′) dt′ is called the dynamical phase. In adiabatic approximation the initial state ψn(ti) changes into ψn(t, ti) without combination with other states. Hence, Eq. (5.1) can be written as ψ(t) = eiη(t,ti) ψn(t) . (5.2) Usually, the phase factor η(t, ti) is set to zero as only ψψ∗ is measurable. It has been assumed that η(t, ti) does not give any observable effect.
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
This chapter presents the derivation of Berry's phase. Many classical phenomena have corresponding quantum mechanical analogue. Therefore, it is natural to identify a classical analogue of Berry's phase. The concept of Berry's phase has been used in the study of quantized Hall Effect, the spin statistics properties of quasiparticle excitations, the rotation of photon polarization in helical optical fibers, etc. The chapter discusses various applications of Berry's phase in solid state physics. The Berry's phase in one-dimension is usually called Zak phase and is used for a better understanding of electric polarization in dielectrics, protected edge states and Wannier–Stark ladders and integer quantum Hall effect. The chapter discusses the Aharonov–Bohm and Sagnac effects. The Aharonov–Bohm experiment consists of splitting an electron beam, passing around both sides of a solenoid and then recombining the beam. The Sagnac effect has been studied in resonant microcavities and light propagation in a wave guide also.
This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm effect and the Berry phase. The issue includes work presented at two conferences, 50 Years of the Aharonov–Bohm Effect, 11–14 October 2009, Tel Aviv University, Israel, and the Aharonov–Bohm Effect and Berry Phase 50/25 Anniversary, 14–15 December 2009, Bristol University, UK. The Aharonov–Bohm (AB) effect and the Berry phase have many things in common, apart from the happy coincidence that they were discovered 25 years apart, making possible such a joint anniversary. First, they were both discovered in the same place, the H H Wills Physics Laboratory, University of Bristol, UK. Second, they both were there for everybody to see from day one, from the very beginning of quantum mechanics. They were, so to say, right on the surface; there was no need for sophisticated mathematics or involved constructions. Yet it took so long for somebody to see them. The explanation, we dare to speculate, is that they are so unusual, they challenge so much our everyday experience and ingrained intuitions about physics that it was difficult to recognise them. Third, both the AB effect and the Berry phase are quintessentially quantum; they both manifest themselves as topological/geometric quantum phases. The AB effect revolutionized our understanding of the role of potentials in physics. For the first time, it was shown that a particle moving in a field-free region could be affected by a field in a disjoint region. In doing so, it shattered one of the strongest held principles of physics. In the case of the Berry phase, the importance lies in its absolute generality—the topological/geometric phase named after Berry is acquired by all physical systems when the external parameters they depend upon undergo an adiabatic cyclic change. Remarkably this phase depends only on the geometry of the cyclic path in the abstract parameter space, a fact which deepens our appreciation of the underlying geometry of quantum mechanics. The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. First and foremost, they give a deeper understanding of the nature of quantum mechanics itself. Furthermore, we find them in fields as varied as cosmology, particle physics, non-abelian gauge theories, condensed matter, chemical and molecular physics, and laser dynamics. Generalizations of the AB effect to non-abelian gauge theories, such as the Wilson and `t Hooft loops, are important tools for studying confinement and spontaneous symmetry breaking. They explain charge quantization, the quantum Hall effect and the Josephson junction and also play a crucial role in electron microscope holography. Without them an understanding of many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behaviour, would be impossible. Like good wine, the AB effect and the Berry phase become finer and more appreciated with time. Even though they appear in textbooks and encyclopaedias, the number of researchers thinking and publishing papers on these two effects is increasing annually. The AB effect and the Berry phase keep being observed in new systems and with every day that passes novel applications are routinely found. This special issue is just a small snapshot of the state of the art 50 and 25 years, respectively, from their discoveries.
Berry pointed out that the Aharonov-Bohm phase could be considered as a Berry phase.In his argument,he gave up the adiabatic approximation,and the nonstationarity of quantum states was not concerned.In this paper we re-investigate Berry’s proof,and find that his final statement is correct,and in his argument,the adiabatic approximation should be used.In the AB effect,a moving electron is described as a moving wave packet (nonstationary),and the appearance of AB phase originates from the requirement that the evolution of the wave packet must satisfy the Schrdinger equation.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.