On the Squeezed States for nObservables

Abstract

Three basic properties (eigenstate, orbit and intelligence) of thecanonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observablesXi can be constructed as eigenstates of theirlinear complex combinations or as states which minimize the Robertsonuncertainty relation. When Xi close a Lie algebra L the generalized SS could also be introduced as orbit of Aut (LC). It is shown that for the nilpotent algebrahN the three generalizations are equivalent. Forthe simple su(1, 1) the family of eigenstates ofuK- + vK+(K± being lowering and raising operators)is a family of idealK1 – K2 SS, but it cannotbe represented as an Aut (suC(1, 1) orbit althoughthe SU(1, 1) group related coherent states (CS) with symmetryare contained in it.Eigenstates | z, u, v, w; k ⟩ of general combination of uK- + vK+ + wK3 the three generators Kj of SU(1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed ingreater detail. These are ideal SS for K1,2,3. In the case of the one mode realization of su(1, 1) the nonclassical properties (sub-Poissonian statistics,quadrature squeezing) of the generalized even CS | z, u, v; + ⟩ are demonstrated.The states | z, u, v, w; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.