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arXiv:2112.01673

Coherent states for equally spaced, homogeneous waveguide arrays

J A Guerrero, H. M. Moya-Cessa

Published Dec 3, 2021• 1 citation
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Abstract

Coherent states for equally spaced, homogeneous waveguide arrays are defined, in the infinite, semiinfinite and finite cases, and resolutions of the identity are constructed, using different methods. In the infinite case, which corresponds to Euclidean coherent states, a resolution of the identity with coherent states on the circle and involving a nonlocal inner product is reviewed. In the semiinfinite case, which corresponds to London coherent states, various construction are given (restricting to the circle with a non-local scalar product, rescaling the coherent states, modifying them, or using a non-tight frame). In the finite case, a construction in terms of coherent states on the circle is given, and this construction is shown to be a regularization of the infinite and semiinfinite cases.

DOI
10.48550/arxiv.2112.01673

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Coherent states for equally spaced, homogeneous waveguide arrays are defined, in the infinite, semiinfinite and finite cases, and resolutions of the identity are constructed, using different methods. In the infinite case, which corresponds to Euclidean coherent states, a resolution of the identity with coherent states on the circle and involving a nonlocal inner product is reviewed. In the semiinfinite case, which corresponds to London coherent states, various construction are given (restricting to the circle with a non-local scalar product, rescaling the coherent states, modifying them, or using a non-tight frame). In the finite case, a construction in terms of coherent states on the circle is given, and this construction is shown to be a regularization of the infinite and semiinfinite cases.
Perelomov coherent states for equally spaced, infinite homogeneous waveguide arrays with Euclidean E(2) symmetry are defined, and new resolutions of the identity are constructed in Cartesian and polar coordinates. The key point to construct these resolutions of the identity is the fact that coherent states satisfy Helmholtz equation (in coherent states labels) an thus a non-local scalar product with a convolution kernel can be introduced which is invariant under the Euclidean group. It is also shown that these coherent states for the Eucliean E(2) group have a simple and natural physical realization in these waveguide arrays.
Perelomov coherent states for equally spaced, infinite homogeneous waveguide arrays with Euclidean [Formula: see text] symmetry are defined and a new resolution of the identity is obtained. The key point to construct this novel resolution of the identity is the fact that coherent states satisfy the Helmholtz equation (in coherent states labels), and thus every coherent state belongs to a one-parameter family uniquely determined by the Cauchy initial data of the coherent state in a one-dimensional Cauchy set. For this reason we call Cauchy coherent states to these initial data. The novel, nonlocal resolution of the identity in terms of Cauchy coherent states is provided using frame theory. It is also shown that Perelomov coherent states for the Euclidean [Formula: see text] group have a simple and natural physical realization in these waveguide arrays.
We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product × S 1 . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization over a finite periodic chain where the phase space was M × M . Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.
We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization over a finite periodic chain where the phase space was Z_M x Z_M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.
A regular coherent state (CS) is a special type of quantum state for boson particles placed in a single site. The defining feature of the CS is that it is an eigenmode of the annihilation operator. The construction easily generalizes to the case of a finite number of sites. However, the challenge is altogether different when one considers an infinite array of sites. In this work we demonstrate a mathematically rigorous construction that resolves the latter case. The resulting nonlocal coherent states (NCS) are simultaneous eigenmodes for all of the infinitely many annihilation operators acting in the infinite array’s Fock space. Our construction fundamentally relies on Dirichlet series-based analysis and number theoretic arguments.
As a substantial generalization of the technique for constructing canonical and the related nonlinear and q -deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.