The the Hamiltonian which was presented by

The widening attention in the investigation of dissipative systems has its origin in the early days of experimental
works in quantum mechanics 1, 2. Since the combination of the quantum mechanics and damping
concept is a difficult problem, there have been many endeavors to produce different realistic models 3, 4. For
example, the electromagnetic field is modeled with a damped oscillator 9 which is arisen from its classical
analogue. in one of the most fundamental approaches to the damped systems in quantum mechanics, In this
way and after pioneer work of Bateman 1, the Hamiltonian which was presented by Caldirola and Kanai
has provided a vast area of research. The Caldirola–Kanai (CK) Hamiltonian illustrates a damped harmonic
oscillator with a time-dependent mass 10, 11.
On the other hand, the atom-field interaction in its quantum approach has unlocked a wide area of research in quantum optics studies. Jaynes and Cummings suggested a model by which one can study the interaction between a two-level atom with a single-mode cavity field 3. Despite the simplicity of this model, is noteworthy its ability to create some non-classical phenomena observed in the laboratory. Due to these facts, the model has been extended widely. In this regard, we may refer to the generalization of the model to multi-transition Hamiltonians, multi-mode, and multi-atom


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