An exact solution for spontaneous emission of the multiphoton single-atom Dicke Hamiltonian in the dipole and the rotating-wave approximations is derived in two ways: either by a generalisation of a method of solution proposed by Swain, J. Phys., vol.A5, p.1587 (1972), for the one-photon Dicke Hamiltonian or by an exact linearisation of the multiphoton single-atom Hamiltonian to a one-photon single-atom Dicke-type Hamiltonian.

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On alternative Hamiltonians

A. Horzela

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The European Physical Journal Plus

The progressive solutions for the Dicke Hamiltonian

Lina Bao, Feng Pan, Xiaotong Li, Xiao Geng Liang, Jerry Paul Draayer

Published Jul 1, 2015• 3 citations

Publisher

DOI: 10.1140/epjp/i2015-15126-1

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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 $u(1|1)$ and that the Hamiltonian of the Dicke model commutes both with the linear invariant of $\sum_{i} u_{i}(1|1)$ and with the invariant of an additional $su(2)$ algebra. Our methods apply to various coupled Bose-Fermi systems with $u(1|1)$ and more generally with $u(n|m)$ dynamical superalgebras, and may provide efficient tools for studying more complicated examples.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 $u(1|1)$ and that the Hamiltonian of the Dicke model commutes both with the linear invariant of $\sum_{i} u_{i}(1|1)$ and with the invariant of an additional $su(2)$ algebra. Our methods apply to various coupled Bose-Fermi systems with $u(1|1)$ and more generally with $u(n|m)$ dynamical superalgebras, and may provide efficient tools for studying more complicated examples.