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Journal of Physics Conference Series

What hide the Hermite coherent states?

K. Górska, A. Horzela

Published Dec 1, 2023• 0 citations
Publisher

Abstract

Abstract For the single mode the Hermite coherent states are wave packets analogous to the standard harmonic oscillator coherent states but constructed with complex Hermite polynomials substituting traditionally used complex monomials zn . Hermite coherent states unify the standard coherent and squeezed states within one class of quantum states whose representatives share all properties customarily required from the standard coherent states, in particular (over)completeness. Generalization of the Hermite coherent/squeezed states to the multimode case, illustrated on the bipartite states, shows that such states exhibit entanglement which presence or absence depends on the nature of squeezing but also on the choice of description method being used.

DOI
10.1088/1742-6596/2667/1/012008

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Abstract For the single mode the Hermite coherent states are wave packets analogous to the standard harmonic oscillator coherent states but constructed with complex Hermite polynomials substituting traditionally used complex monomials zn . Hermite coherent states unify the standard coherent and squeezed states within one class of quantum states whose representatives share all properties customarily required from the standard coherent states, in particular (over)completeness. Generalization of the Hermite coherent/squeezed states to the multimode case, illustrated on the bipartite states, shows that such states exhibit entanglement which presence or absence depends on the nature of squeezing but also on the choice of description method being used.
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pure Appl. Math. 14, 187 (1961)].
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].
Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.
The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein–Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl–Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.
A general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by 'Gaussian pure states'. The set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup. This construction depends upon a choice of a Borel section which has a certain permissible arbitrariness in its selection; this freedom is attributable to a squeezing of the defining coherent states of the representation, and corresponds in this way to a sort of gauging.