Generalized definition of coherent states and dynamical groups

A. M. Perelomov

The review is devoted to an analysis of definite overcomplete non-orthogonal state systems that are connected with irreducible representations of Lie groups–the so called systems of generalized coherent states. These systems, which the author is the first to propose, are generalizations of Glauber's coherentstate system and arise in natural fashion in physical problems that have dynamic symmetry. They permit a considerable simplification of the solution of the quantum problem by reducing it to a simpler classical problem. The review deals with the properties of generalized-coherent-state systems connected with the simplest Lie groups.

A. I. Solomon

The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.

Ingemar Bengtsson, Karol Życzkowski

P. Gulshani

Various definitions of the coherent states of the angular momentum are shown to be special cases of the oscillator-like coherent states of the groups SU(2) and SO(3) obtained by Mikhailov on the basis of a generalized Schwinger boson realization of the angular momentum algebra. This realization is then generalized to that of the angular momentum algebra of an asymmetric top by means of a transformation from the Euler angles to the Cayley–Klein parameters. The oscillator-like coherent states of an asymmetric top, analogous to those of Mikhailov, are then constructed. It is, then, shown that Janssen's and Mostowski's definitions of the coherent states of a top are special cases of these.

Lei Ma, Zhong Feng Tang, Yong-de Zhang