Generalized SU(1,1) Coherent States

The SU(1,1) Lie algebra is of great interest in quantum optics because it can characterize many kinds of quantum optics systems. In particular, the bosonic realizations of SU(1,1) Lie algebra describe the degenerate and non-degenerate parametric amplifiers. And the linear dissipative process can also be described by this algebra. So, in the past decade the single-, two- and multi-mode bosonic realizations of the SU(1,1) Lie algebra have been receiving a lot of attention. Meanwhile, the associated SU(1,1) generalized coherent states have been derived and their some non-classical properties have also been studying. Similar to these works, by introducing new two- and multi-mode bosonic realizations of SU(1,1) Lie algebra, I derive the corresponding two- and multi-mode SU(1,1) coherent states and discuss their non-classical properties respectively.In section 1 and section 2, I present a simple overview of the bosonic realization of the SU(1,1) Lie algebra, single-mode SU(1,1) coherent states and their main non-classical properties. In section 3, I introduce a new kind of two-mode bosonic realization of SU(1,1) Lie algebra, on the basis of which the two-mode SU(1,1) coherent states in the two-mode Fock space are derived. These two-mode SU(1,1) coherent states, which are called as uncorrelated two-mode SU(1,1) coherent states, include three special cases. For these states, I study the mean photon number distribution and their non-classical properties, which are photon antibunching, violations of Cauchy-Schwarz inequality and two-mode squeezing. Insection 4, I extend uncorrelated two-mode SU(1,1) coherent states to the multi-mode SU(1,1) coherent states. These states can be reduced to the single- and uncorrelated two-mode SU(1,1) coherent states under various special conditions. Meanwhile, each mode in multi-mode SU(1,1) coherent states will remain it's properties as single-mode SU(1,1) coherent states. So, the multi-mode SU(1,1) coherent states can be considered as the direct product of single-mode SU(1,1) coherent states. They can be called as uncorrelated multi-mode SU(1,1) coherent states.