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arXiv:quant-ph/0112061 · Journal of Mathematical Physics

Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians

Clive Emary, R. F. Bishop

Published Aug 1, 2002• 45 citations
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Abstract

We present a new method for finding isolated exact solutions of a class of nonadiabatic Hamiltonians of relevance to quantum optics and allied areas. Central to our approach is the use of Bogoliubov transformations of the bosonic fields in the models. We demonstrate the simplicity and efficiency of this method by applying it to the Rabi Hamiltonian.

DOI
10.1063/1.1490406

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We present a new method for finding isolated exact solutions of a class of nonadiabatic Hamiltonians of relevance to quantum optics and allied areas. Central to our approach is the use of Bogoliubov transformations of the bosonic fields in the models. We demonstrate the simplicity and efficiency of this method by applying it to the Rabi Hamiltonian.
The two-photon Rabi Hamiltonian is a simple model describing the interaction of light with matter, with the interaction being mediated by the exchange of two photons. Although this model is exactly soluble in the rotating-wave approximation, we work with the full Hamiltonian, maintaining the non-integrability of the model. We demonstrate that, despite this non-integrability, there exist isolated, exact solutions for this model analogous to the so-called Juddian solutions found for the single-photon Rabi Hamiltonian. In so doing we use a Bogoliubov transformation of the field mode, as described by the present authors in an earlier publication.
Within extended coherent states, a recent exact solution to the quantum Rabi model (QRM) [D. Braak, Phys. Rev. Lett. 107, 100401 (2011)] can be recovered in an alternative simpler and more physical way, without use of any extra conditions. In the same framework, the two-photon QRM is solved exactly by treating extended squeezed states on an equal footing. Concise transcendental functions responsible for the exact solutions are derived. The isolated Juddian solutions are also analytically obtained in terms of degeneracy. Both the extended coherent states and squeezed states employed here are essentially Fock states in the space of the corresponding Bogoliubov operators, which result in free-particle number operators. The present approach can be summarized concisely in a unified way and easily extended to various spin-boson systems with multiple levels, even multiple modes.
The quantum Rabi model can be solved exactly by the Bargmann transformation from real coordinate to complex variable recently [Phys. Rev. Lett. \textbf{107}, 100401 (2011)]. By the extended coherent states, we recover this solution in an alternative simpler and perhaps more physical way without uses of any extra conditions, like Bargmann conditions. In the same framework, the two-photon Rabi model are solved exactly by extended squeeze states. Transcendental functions have been derived with the similar form as those in one-photon model. Both extended coherent states and squeeze states are essentially Fock states in the space of the corresponding Bogoliubov operators. The present approach could be easily extended to study the exact solvability or integrability of various spin-boson systems with multi-level, even multi-mode.
We use a non-gauge-invariant modification of the exact Hamiltonian to obtain a new Hamiltonian-like operator for a simple exactly solvable boson model. The eigenvalues of the new operator are close to those of the original Hamiltonian. We make a one-body approximation of the new two-body operator in the spirit of the Bogoliubov approximation. Because only the number operator appears, the c-number approximation is not required individually for the creation or annihilation operators in the ground state. For the simple model, the results using the new approximation are closer to the exact results than the usual Bogoliubov results over a wide range of parameters. The improvement increases dramatically as the model interaction strength increases.