Coherent states for equally spaced, homogeneous waveguide arrays

TL;DR

This paper develops and compares methods for constructing coherent states in waveguide arrays with different boundary conditions, providing a unified framework for their resolutions of the identity.

Contribution

It introduces new constructions of coherent states for finite and semi-infinite waveguide arrays, extending the Euclidean case to more general boundary conditions.

Findings

Resolutions of the identity are constructed for all cases.

Finite case coherent states are regularized versions of infinite and semi-infinite cases.

Various methods for constructing coherent states on the circle are analyzed.

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.