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arXiv:1711.04034

Coherent states in a magnetic field and their generalizations

V. V. Dodonov

Published Nov 10, 2017• 1 citation
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Abstract

This is a brief review of various families of coherent and squeezed states (and their generalizations) for a charged particle in a magnetic field, that have been constructed for the past 50 years. Although the main attention is paid to the Gaussian states, various families of non-Gaussian states are also discussed, and the list of relevant references is provided.

DOI
10.48550/arxiv.1711.04034

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This is a brief review of various families of coherent and squeezed states (and their generalizations) for a charged particle in a magnetic field, that have been constructed for the past 50 years. Although the main attention is paid to the Gaussian states, various families of non-Gaussian states are also discussed, and the list of relevant references is provided.
For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iqadIfagaqcaiGacoga% caGGVbGaai4CaiabfA6agjabgUcaRiqadMfagaqcaiGacohacaGGPb% GaaiOBaiabfA6agbaa!4894! X^Φ=X^cos⁡Φ+Y^sin⁡Φ\hat X_\Phi = \hat X\cos \Phi + \hat Y\sin \Phi X^Φ​=X^cosΦ+Y^sinΦ , % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aaaaa!3A82! X^\hat XX^ and Ŷ being the coordinates of the Larmor circle center and Φ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Besides, for arbitrary gauge of the vector potential we derive the symmetry operators of translations and rotations.
Seventy five years ago, three remarkable papers by Schrödinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called `nonclassical states' (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the `standard citation package' used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and `nonclassical' states commenced. The 1990s are described in a more `squeezed' manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.
In this work, we review two formalisms of coherent states for the case of a particle in a magnetic field. We focus our revision on both pioneering (Feldman and Kahn 1970 Phys. Rev. B 1 4584) and recent (Kowalski and Rembieliński 2005 J. Phys. A: Math. Gen. 38 8247 ) formulations of coherent states for this problem. We introduce a general application, the Husimi function, which takes into account collective and environmental effects, and which can be derived in a variety of ways. However, we calculate this function as the expected value of the density operator in a basis of coherent states.