The intelligent spin states are defined as those states which satisfy the Heisenberg equality for the spin operators: Delta J x 2 Delta J y 2 = mod (J z ) mod 2 . The 2j+1 states which behave intelligently in each angular momentum space of spin j are found explicitly. For this purpose the Radcliffe states are used, it is shown that only the real and the pure imaginary Radcliffe states are intelligent. These intelligent states also satisfy the quartic consistency condition. The result, however, does not disagree in principle with the recent claim of Kolodziejczyk and Ryter (1974) that mod mu )= mod 0) is the only state which minimizes the uncertainty product because minimum uncertainty does not necessarily imply intelligence.
Generalized coherent states associated with SU(1,1) Lie algebra are reviewed. A state is called intelligent if it satisfies the strict equality in the Heisenberg uncertainty relation. The eigenvalue problem satisfied by intelligent states (IS) is solved. The IS associated with SU(1,1) Lie algebra are investigated. We have constructed some realizations for our results of IS, and some applications are discussed. Some nonclassical properties such as Glauber second-order correlation function, photon number distribution and squeezing are investigated.
Ordinary intelligent states (OISs) hold equality in the Heisenberg uncertainty relation involving two noncommuting observables
A,B, whereas generalized intelligent states (GISs) do so in the more generalized uncertainty relation, the Schr\"odinger-Robertson inequality. In general, OISs form a subset of GISs. However, if there exists a unitary evolution
U that transforms the operators
A,B to a new pair of operators in a rotation form, it is shown that an arbitrary GIS can be generated by applying the rotation operator
U to a certain OIS. In this sense, the set of OISs is unitarily equivalent to the set of GISs. It is the case, for example, with the su(2) and the su(1,1) algebras which have been extensively studied, particularly in quantum optics. When these algebras are represented by two bosonic operators (nondegenerate case), or by a single bosonic operator (degenerate case), the rotation, or pseudorotation, operator
U corresponds to phase shift, beam splitting, or parametric amplification, depending on two observables
A,B.
The diagonalization of the uncertainty matrix and the minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), and sp(N,R) algebras. Unlike the group-related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators (q>0) and of any k power of are positive definite and can be diagonalized by symplectic linear transformations.