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In this paper, we investigate some properties through four-level λ -type atom interacting with a single-mode quantized field with multi-photon transitions. We study this system in the presence of detuning parameter, Kerr nonlinearity, and intensity-dependent atom-field coupling in a dissipative cavity (i.e. the field is suffering decay rate). Also, the coupling parameter modulated to be time dependent. The exact solution of this model is given using the Schro ̋dinger equation when the atom and the field are initially prepared in superposition state and coherent state, respectively. We employed the results to calculate some aspects such as atomic population inversion, geometric phase and Husimi Q-function. It is emphasized that the system can be used as a quantum memory.

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