Coherent states of systems with quadratic Hamiltonians

TL;DR

This paper constructs various families of coherent states for one-dimensional systems with time-dependent quadratic Hamiltonians, unifying known states and exploring their properties, including completeness and uncertainty minimization.

Contribution

It introduces a general framework for constructing coherent states for quadratic Hamiltonian systems, encompassing known states and revealing new families distinguished by initial standard deviations.

Findings

Constructed generalized coherent states for time-dependent quadratic Hamiltonians.

Identified different families of coherent states based on initial standard deviations.

Discussed properties like completeness and uncertainty relations of these states.

Abstract

Different families of generalized CS for one-dimensional systems with general time dependent quadratic Hamiltonian are constructed. In principle, all known CS of systems with quadratic Hamiltonian are members of these families. Some of the constructed generalized CS are close enough to the well-known due to Schr\"odinger and Glauber CS of a harmonic oscillator, we call them simply CS. However, even among these CS there exist different families of complete sets of CS. These families differ by values of standard deviations at the initial time instant. According to the values of these initial standard deviations one can identify some of the families with semiclassical CS. We discuss properties of the constructed CS, in particular, completeness relations, minimization of uncertainty relations and so on. As a unknown application of the general construction, we consider different CS of an oscillator with a time dependent frequency.