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Journal of Physics A Mathematical and General

Generalized Jaynes–Cummings Hamiltonians by shape-invariant hierarchies and their SUSY partners

Véronique Hussin, Ş. Kuru, J. Negro

Published Aug 18, 2006• 6 citations
Publisher

Abstract

A generalization of the matrix Jaynes–Cummings model in the rotating wave approximation is proposed by means of the shape-invariant hierarchies of scalar factorized Hamiltonians. A class of Darboux transformations (sometimes called SUSY transformations in this context) suitable for these generalized Jaynes–Cummings models is constructed. Finally one example is worked out using the methods developed.

DOI
10.1088/0305-4470/39/36/011

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A generalization of the matrix Jaynes–Cummings model in the rotating wave approximation is proposed by means of the shape-invariant hierarchies of scalar factorized Hamiltonians. A class of Darboux transformations (sometimes called SUSY transformations in this context) suitable for these generalized Jaynes–Cummings models is constructed. Finally one example is worked out using the methods developed.
Abstract The aim of this work is to show how supersymmetric (SUSY) quantum mechanics can be applied to the Jaynes–Cummings (JC) Hamiltonian of quantum optics. These SUSY transformations connect pairs of Jaynes–Cummings Hamiltonians characterized by different detuning parameters as well as Jaynes–Cummings to anti-Jaynes–Cummings Hamiltonians. Therefore, JC Hamiltonians can be classified in hierarchies or sequences which are connected through SUSY transformations. As a byproduct, the symmetries of JC Hamiltonians are found as well as the special case of a sequence of resonant-like interacting systems having the form of a simple shape invariant JC Hamiltonian hierarchy.
A class of shape-invariant bound-state problems which represent transitions in a two-level system introduced earlier are generalized to include arbitrary energy splittings between the two levels. We show that the coupled-channel Hamiltonians obtained correspond to the generalization of the non-resonant Jaynes-Cummings Hamiltonian, widely used in quantized theories of laser. In this general context, we determine the eigenstates, eigenvalues, the time evolution matrix and the population inversion matrix factor.
The aim of this work is to define and investigate the generalized Jaynes-Cummings (JC) Hamiltonian by means of supersymmetric (SUSY) quantum mechanics methods. We have considered trigonometric and hyperbolic Pöschl-Teller (PT) potentials as two natural ingredients to modify the standard JC model. We show that these generalized JC Hamiltonians have a list of characteristics in common with the JC Hamiltonian, for example: matrix intertwining, excitation number symmetry, Rabi frequency, formulas for atomic inversion and classical and revival times. Therefore, we conclude that JC Hamiltonians can be considered as one example of a wider a class of generalized Hamiltonians which can be useful in broader interaction problems.
We demonstrate how supersymmetries of Hamiltonians for coupled Bose-Fermi systems can be used to place the Hamiltonians of the Jaynes-Cummings model and Dicke model under the rotating wave approximation in matrix form and provide explicit analytic solutions for their eigenvalues. We then use this supersymmetry to place the Liouvillians of the associated Markovian open systems in matrix form and provide explicit solutions for their eigenvalues. These results are a consequence of the fact that the Hamiltonian of the Jaynes-Cummings model commutes with the linear Casimir invariant of the superalgebra u(1∣1)u(1|1)u(1∣1) and that the Hamiltonian of the Dicke model commutes both with the linear invariant of ∑iui(1∣1)\sum_{i} u_{i}(1|1)∑i​ui​(1∣1) and with the invariant of an additional su(2)su(2)su(2) algebra. Our methods apply to various coupled Bose-Fermi systems with u(1∣1)u(1|1)u(1∣1) and more generally with u(n∣m)u(n|m)u(n∣m) dynamical superalgebras, and may provide efficient tools for studying more complicated examples.