K-Quantum Nonlinear Coherent States Formulation, Realization and Nonclassical Effects

K -Quantum Nonlinear Coherent States: Formulation, Realization and Nonclassical Effects

The coherent state (CS) introduced in 1963 [1, 2] has become a useful and necessary tool for treating ideal boson fields subjected to external pumping sources. Although any quantum state can be described in terms of CS’s thanks to their resolution of unity, the CS’s themselves are classical states in which nonclassical effects such as antibunching, squeezing, etc. cannot occur. The conventional squeezed states [3] have been generalized to different types of higher-order ones [4-7], still for ideal bosonic particles. Elementary excitations in matter, on the other hand, are quasi-particles often obeying neither Bose-Einstein nor Fermi-Dirac statistics. Recently, the notion of nonlinear coherent state (NCS) has emerged to adequately deal with such quasi-particles, with commutation relations deformed from the usual boson/fermion ones. q-deformed oscillators [8, 9] were found as a particular case of the general f -oscillators [10], which possess in themselves a kinematic nonlinearity causing orbit-dependent oscillation frequencies. The NCS [11-15] is defined as the right-hand eigenstate j»;fi of the nonboson operator A = af (n)0,

A j »; fi = » j »;fi; (1)

with n = a⁺a; a the bosonic annihilation operator, » a complex eigenvalue and f an arbitrary nonlinear operator-valued function of n: An island where NCS’s live is single trapped ions driven by lasers [11, 16-19].

In this paper we introduce the K-quantum nonlinear coherent state (KNCS) which reduces to the usual NCS defined by Eq. (1) when K = 1. In Section II the formulation is given for the KNCS together with their mathematical properties. Section III is devoted to a physical scheme of generation for KNCS’s. Their nonclassical properties are studied in detail in Section IV. The conclusion is the final section.

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