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Physical Review Letters

Quantum Particle on a Rotating Loop: Topological Quenching due to a Coriolis-Aharonov-Bohm Effect

Stefan Weigert

Published Aug 21, 1995• 8 citations
Publisher

Abstract

A particle is assumed to move along a one-dimensional loop such as an ellipse that rotates in a plane. Because of the centrifugal force the particle is subjected to a symmetric double-well potential. Classically, the Coriolis force does not affect the motion of the particle, whereas the corresponding term in the Lagrangian influences the properties of the quantum system: its ground state turns out to be degenerate for a discrete set of angular velocities. The analogy between a constant magnetic field and a uniform rotation is used to propose, in addition, a variant of the Aharonov-Bohm experiment, which can be performed also with neutral particles.

DOI
10.1103/physrevlett.75.1435

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A particle is assumed to move along a one-dimensional loop such as an ellipse that rotates in a plane. Because of the centrifugal force the particle is subjected to a symmetric double-well potential. Classically, the Coriolis force does not affect the motion of the particle, whereas the corresponding term in the Lagrangian influences the properties of the quantum system: its ground state turns out to be degenerate for a discrete set of angular velocities. The analogy between a constant magnetic field and a uniform rotation is used to propose, in addition, a variant of the Aharonov-Bohm experiment, which can be performed also with neutral particles.
The motion of a particle along a one-dimensional closed curve in a plane is considered. The only restriction on the shape of the loop is that it must be invariant under a twofold rotation about an axis perpendicular to the plane of motion. Along the curve a symmetric double-well potential is present leading to a twofold degeneracy of the classical ground state. In quantum mechanics, this degeneracy is lifted: the energies of the ground state and the first excited state are separated from each other by a slight difference \ensuremath{\Delta}E, the tunnel splitting. Although a magnetic field perpendicular to the plane of the loop does not influence the classical motion of the charged particle, the quantum-mechanical separation of levels turns out to be a function of its strength B. The dependence of \ensuremath{\Delta}E on the field B is oscillatory: for specific discrete values Bn{\mathit{B}}_{\mathit{n}}Bn​ the splitting drops to zero, indicating a twofold degeneracy of the ground state. This result is obtained within the path-integral formulation of quantum mechanics; in particular, the semiclassical instanton method is used. The origin of the quenched splitting is intuitively obvious: it is due to the fact that the configuration space of the system is not simply connected, thus allowing for destructive interference of quantum-mechanical amplitudes. From an abstract point of view this phenomenon can be traced back to the existence of a topological term in the Lagrangian and a nonsimply connected configuration space. In principle, it should be possible to observe the splitting in appropriately fabricated mesoscopic rings consisting of normally conducting metal.
The Aharonov-Casher (AC) effect for quantum motion of a neutral magnetized particle in the electric field is believed to be a topological effect closely related to the known Aharonov-Bohm (AB) effect. We study how it depends on the spin of the particle involved. Duality of the AB and AC effects is demonstrated to exist only for two extreme spin (and magnetic moment) projections. Classical consideration confirms the conclusion. Motion of a classical magnetized particle with generally oriented magnetic moment in the AC field appears to be subjected to both forces and torques. Only for two special orientations of the magnetic moment (same as in the quantum case) the motion is effectively free and similar to the AB motion of a charged particle. Thus, the AC effect is not really topological. Presence of higher multipoles totally destroys the possible AB-AC duality.
We present an application of the affine covariant integral quantization (ACIQ) (Adv. Oper. Theory, 5, 2020; Adv. Oper. Theory, 7, 2022) to quantum mechanics on the punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), which comprises translations, rotations, and dilations of the plane. Due to the topology of the punctured plane, our quantization procedure gives rise to an affine vector potential. This potential can be interpreted as the Aharonov-Bohm (AB) gauge field produced by an infinite solenoid. This observation supports a reinterpretation of the AB effect: it emerges from the topological constraint imposed by the impenetrable coil rather than from an externally applied classical gauge field. In addition to this gauge structure, ACIQ also generates a repulsive, centrifugal-like scalar potential, a feature already encountered when applying ACIQ to motion on the half-line, whose phase space is the open half-plane. These results provide a new perspective on the AB effect, highlighting the central roles of topology and symmetry in quantum mechanics.
The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.
In this paper, we apply covariant affine integral quantization to study motion in the 2D punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), representing transformations of the plane composed of translations, rotations, and dilations. Near the punctured point, we observe that the topology influencing quantum fluctuations gives rise to an affine vector potential, which can be interpreted as the Aharonov-Bohm (AB) gauge potential generated by an infinite coil. This finding suggests that the AB effect originates from the topology enforced by the impenetrable coil, rather than from a classical gauge potential. Our results offer a novel perspective on the AB effect, emphasizing the fundamental role of topology in quantum mechanics.
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.