The intelligent states. I. Group-theoretic study and the computation of matrix elements

Abstract

In this first of a series of papers, a group-theoretic study is presented of the quasi-intelligent states which are a generalization of the intelligent states satisfying equality in the Heisenberg uncertainty relation ΔJ12ΔJ22? (1/4) ‖〈J3〉‖2. A method based on the knowledge of a certain generating function is given for the calculation of matrix elements of polynomials in the infinitesimal generators of the rotation group between quasi-intelligent states. Examples of such computations are also included to exhibit the improvement and efficiency of the present methods.

Phase-diffusion pattern in quantum-nondemolition systems

“…(63) and (67), and is usually fixed by normalization as in the above equations, where |A p | 2 = (cosh Θ) −1 which is equal to 1 for Θ = 0 [56], implying an infinite initial squeezing ζ of the system (64). The expression in Eq.…”

“…where |θ, φ are the atomic coherent states [56,57] given by an expansion over the Wigner-Dicke states [44] as…”

The Uncertainty Way of Generalization of Coherent States

“…where N 1,2 are normalization constants. In slightly different notations the operators J ′ 3 , J ′ ± were introduced by Rashid [48]. An important physical property of the states |z, u, v; X 1 , X 2 is that they can exhibit arbitrary strong squeezing of the variances of X 1 and X 2 when the parameter v tend to ±u, i.e.…”

“…The canonical SS |α, u, v are p-q ideal SS, while the group-related CS |τ ; j and |ξ; k are not. Explicitly the families of |z, u, v; X 1 , X 2 are constructed for the generators K i -K j and J i -J j of SU (1, 1) [7,30,9] and SU (2) [47,48,9] (in [47,48] with no reference to the inequality (22)). It is worth noting an important application of the K i -K j and J i -J j optimal US (intelligent states) in the quantum interferometry: the SU (1, 1) and SU (2) optimal US which are not group-related CS can greatly improve the sensitivity of the SU (2) and SU (1, 1) interferometers as shown by Brif and Mann [33].…”

Generalized coherent and squeezed states based on the h(1)⊕su(2) algebra

“…On the contrary of the preceding example where the HUR is never redundant (because x and p are canonical), here the commutator of J 1 and J 2 is not a multiple of the identity and then J 3 may be equal to zero for some special cases. Some of these cases have been discussed elsewhere [6,13,14,15]. Here we give the general solution of the equation It would be better to work with the operators J ± = J 1 ± iJ 2 instead of J 1 and J 2 .…”