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Journal of Physics C Solid State Physics

The response of a tunnelling particle to an impulse: the connection with Berry's phase

Stanley Clough

Published Dec 10, 1987• 4 citations
Publisher

Abstract

Clough reconsiders the effect of an impulse on a tunnelling particle after work by J.M.F. Gunn (see ibid., vol.18, p.L357, 1985). He points out that the effect described was a simple consequence of what has become known as Berry's phase. Recent articles demonstrating the existence of such a phase settle the dispute in Clough's favour.

DOI
10.1088/0022-3719/20/34/024

Related papers

Reply to `The response of a tunnelling particle to an impulse: the connection with Berry's phase'

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Journal of Physics C Solid State Physics•Dec 10, 1987•1 citation

Reply to Comment on 'The response of a tunnelling particle to an impulse'

Stanley Clough

Journal of Physics C Solid State Physics•May 20, 1985•4 citations

Certain properties of Berry's phases in supersymmetric quantum mechanics. II

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For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.

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It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order $\\hbar $ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.

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Supersymmetry and non-Abelian geometric phase for a free particle on a circle with point-like interactions

Satoshi Ohya

Though not so widely appreciated in the literature, supersymmetric quantum mechanics provides an ideal playground for studying non-Abelian geometric phase, because supersymmetry always guarantees degeneracies in energy levels. In this paper we first present a simple supersymmetric model for a free particle on a circle with point-like interactions that exhibits N = 2 supersymmetry and doubly degenerate energy levels. We then show that Berry's connection in this model is given by the Wu-Yang-like magnetic monopole in SU (2) Yang-Mills gauge theory. This article is largely based on our recent work [1].

arXiv:1409.8238•Journal of Physics Conference Series•Nov 26, 2014•0 citations
Clough reconsiders the effect of an impulse on a tunnelling particle after work by J.M.F. Gunn (see ibid., vol.18, p.L357, 1985). He points out that the effect described was a simple consequence of what has become known as Berry's phase. Recent articles demonstrating the existence of such a phase settle the dispute in Clough's favour.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order $\\hbar $ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.
Though not so widely appreciated in the literature, supersymmetric quantum mechanics provides an ideal playground for studying non-Abelian geometric phase, because supersymmetry always guarantees degeneracies in energy levels. In this paper we first present a simple supersymmetric model for a free particle on a circle with point-like interactions that exhibits N = 2 supersymmetry and doubly degenerate energy levels. We then show that Berry's connection in this model is given by the Wu-Yang-like magnetic monopole in SU (2) Yang-Mills gauge theory. This article is largely based on our recent work [1].