Schr\"{o}dinger Intelligent States and Linear and Quadratic Amplitude Squeezing

Abstract

A complete set of solutions |z,u,v>_{sa} of the eigenvalue equation (ua^2+va^{dagger 2})|z,u,v> = z|z,u,v> ([a,a^{dagger}]=1) are constructed and discussed. These and only these states minimize the Schr\"{o}dinger uncertainty inequality for the squared amplitude (s.a.) quadratures. Some general properties of Schr\"{o}dinger intelligent states (SIS) |z,u,v> for any two observables X, Y are discussed, the sets of even and odd s.a. SIS |z,u,v;+,-> being studied in greater detail. The set of s.a. SIS contain all even and odd coherent states (CS) of Dodonov, Malkin and Man'ko, the Perelomov SU(1,1) CS and the squeezed Hermite polynomial states of Bergou, Hillery and Yu. The even and odd SIS can exhibit very strong both linear and quadratic squeezing (even simultaneously) and super- and subpoissonian statistics as well. A simple sufficient condition for superpoissonian statistics is obtained and the diagonalization of the amplitude and s.a. uncertainty matrices in any pure or mixed state by linear canonical transformations is proven.