On the squeezed states for n observables

Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of arbitrary n observables. The SS for n observables Xi can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty 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 algebra hN the three generalizations are equivalent. For the simple su(1, 1) the family of eigenstates of uK− + vK+ (K± being lowering and raising operators) is a family of ideal K1-K2 SS, but it cannot be represented as an Aut(suC(1, 1)) orbit although the SU(1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates |z, u, v, w; k〉 of general combination uK− + vK+ + wK3 of 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 in greater 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 = 14 , 3 4〉 can exhibit strong both linear and quadratic squeezing.