# Squeezed Fourier Meets Toeplitz Algebras

## TL;DR

This paper explores new methods of quantum state squeezing using external fields, presenting exactly solvable cases and linking the problem to Toeplitz algebra, offering a novel algebraic approach.

## Contribution

It introduces a new algebraic framework based on Toeplitz algebras for analyzing quantum squeezing, with explicit solutions for symmetric evolution intervals.

## Key findings

- Identified conditions for simultaneous squeezing of quantum states.
- Derived explicit external field profiles for specific evolution intervals.
- Connected quantum squeezing dynamics with Toeplitz algebra structures.

## Abstract

We look for new steps on the dynamical operations that may squeeze simultaneously some families of quantum states, independently of their initial shape, induced by softly acting external fields which might produce the squeezing of the canonical observables $q,p$ of charged particles. Also, we present some exactly solvable cases of the problem which appear in the symmetric evolution intervals permitting to find explicitly the time dependence of the external fields needed to generate the required evolution operators. Curiously, our results are interrelated with a simple, non--trivial, anti--commuting algebra of Toeplitz which describes the problem more easily than the frequently used Ermakov--Milne invariants.