Robertson intelligent states

         D A Trifonov
         Institute for Nuclear Research, 72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria
 

Abstract. The diagonalization of 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 particular case of RIS for
the quadratures of Weyl generators.  Explicit constructions of RIS are considered for operators of
su(1,1), su(2), h_N 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 a_{q,j} (q>0) and of any k power of a_j=a_{1,j} are positive definite
and can be diagonalized by symplectic linear transformations.
 
 
 
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