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Aharonov–Bohm effect in phase space

  • Regular Article - Quantum Optics
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Abstract

The Aharonov–Bohm effect is a genuine quantum effect typically characterized by a measurable phase shift in the wave function for a charged particle that encircles an electromagnetic field located in a region inaccessible to the particle. However, this definition is not possible in the majority of the phase-space descriptions since they are based on quasiprobability distributions. In this work, we characterize for the first time the Aharonov–Bohm effect within two different formalisms of quantum mechanics. One of them is the phase-space formalism relying on the canonical commutation relations and Weyl transform. In this framework, the aim is to obtain a consistent description of the quantum system by means of the quasiprobability Wigner function. The other one is the Segal–Bargmann formalism, which we mathematically describe and connect with quantum mechanics by means of the commutation relations of the creation and annihilation operators. After an introduction of both formalisms, we study the Aharonov–Bohm effect within them for two specific cases: one determined by a non-zero electric potential, and another determined by a non-zero magnetic vector potential. Subsequently, we obtain a more general description of the Aharonov–Bohm effect that encompasses the two previous cases and that we prove to be equivalent to the well-known description of this effect in the usual quantum mechanics formalism in configuration space. Finally, we delve into the Aharonov–Bohm effect, employing a density operator to depict states with positional and momentum uncertainty, showcasing its manifestation through distinctive interference patterns in the temporal evolution of Wigner functions under an electric potential, and emphasizing the intrinsically quantum nature of this phenomenon.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]

Notes

  1. It must be emphasized that plane waves are a pathological case when normalized. However, we will use them since they allow us to simplify the discussion of the Aharonov–Bohm effect.

  2. It is supposed that the activation and disconnection processes will not affect the particle evolution.

  3. We should remember that the Segal–Bargmann space has been built with dimensionless variables, so the wave function must be expressed in terms of these variables.

  4. This property may actually be posed as the commutation relation \([{\hat{P}}-qA,{\hat{M}}]\), which implies that it is fulfilled in all formalisms.

  5. We do not include the condition \(\text {Tr}[{\hat{\rho }}]=1\) because it is related to the norm of the state. Once again, we are taking a non-normalizable state as a simple limit of a normalizable one.

  6. Note that we have removed a global factor \(1/2\pi \) since it is irrelevant due to the non-normalizability of the Wigner function.

  7. When \(q\Delta \varphi \cdot \uptau =\pi /2+2\pi n\) with \(n\in {\mathbb {Z}}\) and \(t>\uptau \), the maximum contours of the oscillation background become minimum and vice versa.

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Acknowledgements

This work was partially supported by the MICINN (Spain) project PID2019-107394GB-I00/ AEI/10.13039/501100011033 (AEI/FEDER, UE) and PID2022-139841NB-I00, COST (European Cooperation in Science and Technology) Actions CA21106 and CA21136. JARC acknowledges support by Institut Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2022, with the support of the P2IO Laboratory of Excellence (program “Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038) and the P2I axis of the Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA, APPEC, IAS, OSUPS, and the “IN2P3 master projet UCMN”.

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Authors and Affiliations

  1. Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de Ciencias 1, 28040, Madrid, Spain

    Jose A. R. Cembranos, David García-López & Zoe G. del Toro

  2. Institute of Particle and Cosmos Physics (IPARCOS), Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de Ciencias 1, 28040, Madrid, Spain

    Jose A. R. Cembranos

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  1. Jose A. R. Cembranos
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All authors contributed equally to the study conception and design. Material preparation and analysis were performed by all authors. The first draft of the manuscript was written cooperatively, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to David García-López.

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Cembranos, J.A.R., García-López, D. & del Toro, Z.G. Aharonov–Bohm effect in phase space. Eur. Phys. J. D 78, 16 (2024). https://doi.org/10.1140/epjd/s10053-024-00804-y

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