Squeezing in the SU(1,1) Group

Haluk BEKER

G. Satya Prakash, G. S. Agarwal

Squeezed coherent states associated with the SU(1,1) group defined by (\ensuremath{\mu}${\mathit{a}}^{2}$+\ensuremath{\nu}${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}2}$)\ensuremath{\Vert}\ensuremath{\psi}〉=\ensuremath{\lambda}\ensuremath{\Vert}\ensuremath{\psi}〉, \ensuremath{\Vert}\ensuremath{\mu}${\mathrm{\ensuremath{\Vert}}}^{2}$-\ensuremath{\Vert}\ensuremath{\nu}${\mathrm{\ensuremath{\Vert}}}^{2}$=1 are studied in detail. An explicit expression for the quasidistribution Q(\ensuremath{\alpha},${\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}$) in terms of the parabolic cylindrical functions is derived. The nonclassical properties of these states are discussed. The possibility of producing SU(1,1) squeezed coherent states using the down-conversion process is also demonstrated.

D. Han, Y.S. Kim

It is shown that the SU(1,1)-like and SU(2)-like two-photon coherent states can be combined to form a O(3,2)-like two-photon states. Since the O(3,2) group has many subgroups, there are also many new interesting new coherent and squeezed two-photon states. Among them is the two-photon sheared state whose symmetry property is like that for the two-dimensional Euclidean group. There are now two-phonon coherent states which may exhibit symmetries not yet observed for photons, including sheared states. Let us note that both SU(1,1) and S(3,2) are isomorphic to the symplectic groups Sp(2) and Sp(4) respectively, and that symplectic transformations consist of rotations and squeeze transformations.

Stephen Bruce Sontz

Yoshio Kuramoto, Yusuke S. Kato

A. I. Solomon, Jacob Katriel

The authors present the results of an analysis of the squeezing of components of the (conventional) electromagnetic field in quantum group analogues of the Heisenberg-Weyl (HW) coherent state and SU(1,1) squeezed state. They find that squeezing occurs for all finite-q values not equal to unity in the HW q-coherent state, in contrast to the usual case; and also in the SU q (1,1) case, although here less than in the usual (q=1) SU(1,1) squeezed state.