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Journal of Physics A Mathematical and Theoretical

Wigner functions and coherent states for the quantum mechanics on a circle

K. Kowalski, Karol Ławniczak

Published Jun 10, 2021• 9 citations
Publisher

Abstract

Abstract The Wigner functions for the coherent states of a particle on a circle are discussed. The nontrivial analytic forms of these functions are derived. The classicality of the circular coherent states existing in the literature as well as the new ones constructed by means of the Fourier transformation of the Gaussian is compared based on negativity of the Wigner function.

DOI
10.1088/1751-8121/ac019d

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Abstract The Wigner functions for the coherent states of a particle on a circle are discussed. The nontrivial analytic forms of these functions are derived. The classicality of the circular coherent states existing in the literature as well as the new ones constructed by means of the Fourier transformation of the Gaussian is compared based on negativity of the Wigner function.
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Abstract We extend former results for coherent states on the circle in the literature in two ways. On the one hand, we show that expectation values of fractional powers of momentum operators can be computed exactly analytically by means of Kummer’s confluent hypergeometric functions. Earlier, these expectation values have only been obtained by using suitable estimates. On the other hand, we consider the Zak transformation not only to map harmonic oscillator coherent states to coherent states on the circle as it has been discussed before, but we also use the properties of the Zak transformation to derive a relation between matrix elements with respect to coherent states in <?CDATA L2(R){L}_{2}({\mathbb{R}})L2​(R)?> L 2 ( R ) and L 2(S 1). This provides an alternative way for computing semiclassical matrix elements for coherent states on the circle. In certain aspects, this method simplifies the semiclassical computations in particular if one is only interested in the classical limit, that is the zeroth order term in the semiclassical expansion.
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We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
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