Representations of Coherent and Squeezed States in an Extended Two-parameters Fock Space

Recently a $f$-deformed Fock space which is spanned by $|n>_{\lambda}$ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by $|n>_{\lambda}$ bases, to a new one, spanned by an extended two-parameters bases $|n>_{\lambda_{1},\lambda_{2}}$. These bases are now the eigen-states of a non-Hermitian Hamiltonian $H_{\lambda_{1},\lambda_{2}}=a^{\dagger}_{\lambda_{1},\lambda_{2}}a+1/2$, where $a^{\dagger}_{\lambda_{1},\lambda_{2}}=a^{\dagger}+\lambda_{1}a + \lambda_{2}$ and $a$ are respectively, the deformed creation and ordinary bosonic annihilation operators. The bases $|n>_{\lambda_{1},\lambda_{2}}$ are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states, in our two-parameters Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states, numerically.