Constructing Squeezed States of Light with Associated Hermite Polynomials

TL;DR

This paper introduces a novel class of light states called associated Hermite polynomial squeezed states, which are superpositions of photon-number states with unique properties, expanding the landscape of quantum optical states.

Contribution

It presents a new method to construct squeezed states using associated Hermite polynomials, revealing states with distinct photon-number superpositions not previously documented.

Findings

States are superpositions of photon-number states with coefficients from associated Hermite polynomials.

Includes states with only odd-photon number components, complementing traditional squeezed states.

Provides a mathematical framework for generating and analyzing these new quantum states.

Abstract

A new class of states of light is introduced that is complementary to the well-known squeezed states. The construction is based on the general solution of the three-term recurrence relation that arises from the saturation of the Schr\"odinger inequality for the quadratures of a single-mode quantized electromagnetic field. The new squeezed states are found to be linear superpositions of the photon-number states whose coefficients are determined by the associated Hermite polynomials. These results do not seem to have been noticed before in the literature. As an example, the new class of squeezed states includes superpositions characterized by odd-photon number states only, so they represent the counterpart of the prototypical squeezed-vacuum state which consists entirely of even-photon number states.