Matrix diagonalization and exact solution of the k-photon Jaynes–Cummings model

Abstract

We study and exactly solve the two-photon and k-photon Jaynes-Cummings models by using a novelty algebraic method. This algebraic method is based on the Pauli matrices realization and the tilting transformation of the SU (2) group and let us diagonalize the Hamiltonian of these models by properly choosing the coherent state parameters of the transformation. Finally, we explicitly obtain the energy spectrum and eigenfunctions for each model.

Sp(4, R) algebraic approach of the most general Hamiltonian of a two-level system in two-dimensional geometry

“…Thus, this group provides a unified framework of both bound and scattering states and therefore, a unified treatment of the various approaches to the solution of problems which are described by the su(2) and su(1, 1) algebras. The 10 operators of equation (26) belonging to the Sp(4, R) group obey the following commutation relations…”

“…In addition to the Jaynes-Cummings model, there are other equally important models in quantum optics like the Rabi model [19,20], the Dicke model [21] (also called Tavis-Cummings model [22]), the E ǫ Jahn-Teller Hamiltonian [23,24], the modified Jaynes-Cummings model [25], among others [26]. Some of these models are particular cases of a general Hamiltonian which is related to the Sp(4, R) group, as can be seen in reference [27].…”

Berry phase of the Tavis-Cummings model with three modes of oscillation

“…which are the expression of the coherent state parameters in the reference [28]. Now, if we impose that n a ≫ 1 and use the following relationship 1…”

“…This model has been studied in terms of the Holstein-Primakoff transformation [16], quantum inverse methods [17,18], and polynomially deformed su(2) algebras [19]. In general, the Jaynes-Cummings model and the Tavis-Cummings model are still under study as can be seen in the references [20][21][22][23][24][25][26][27][28][29].…”

Algebraic approach and Berry phase of a Hamiltonian with a general $SU(1,1)$ symmetry

“…Since its introduction, it has been extensively studied in several quantum systems [31][32][33]. We recently have applied the theory of the SU (1, 1) and SU (2) groups to obtain the energy spectrum, eigenfunctions and the Berry phase of some of these Quantum Optics models [34][35][36][37][38]. In reference [39], an algebraic method based on the Sp(4, R) group (and contains the SU (1, 1) and SU (2) groups) was introduced to solve exactly the interaction part of the most general Hamiltonian of a two-level system in two-dimensional geometry.…”