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Latin America Optics and Photonics Conference

Analytical Solution of the Rabi and Dicke Models

Daniel Braak

Published Jan 1, 2014• 1 citation
Publisher

Abstract

The quantum Rabi model is integrable due to a discrete (parity) symmetry. Qualitative aspects of its spectrum and dynamics are inferred from the analytical solution. The same mathematical technique can be applied to non-integrable generalizations with an arbitrary number of qubits.
DOI
10.1364/laop.2014.ltu3b.5

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The quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum.
The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
The Rabi model is a paradigm for interacting quantum systems.It couples a bosonic mode to the smallest possible quantum model, a two-level system.I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit.A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry.Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
The simplified two-qubit Rabi model is proposed and its analytical solution is presented. There are no level crossings in the spectral graph of the model, which indicates that it is not integrable. The criterion of integrability for the Rabi model proposed by Braak (2011 Phys. Rev. Lett. 107 100401) is also used for the simplified two-qubit Rabi model and the same conclusion, consistent with what the spectral graph shows, can be drawn, which indicates that the criterion remains valid when applied to the two-qubit case. The simplified two-qubit Rabi model is another example of a non-integrable but exactly solvable system except for the generalized Rabi model.
The N=3 Dicke model couples three qubits to a single radiation mode via dipole interaction and constitutes the simplest quantum-optical system allowing for Greenberger-Horne-Zeilinger states. In contrast to the case N=1 (the Rabi model), it is non-integrable if the counter-rotating terms are included. The spectrum is determined analytically, employing the singularity structure of an associated differential equation. While quasi-exact eigenstates known from the Rabi model do not exist, a novel type of spectral degeneracy becomes possible which is not associated with a symmetry of the system.
In this paper, the finite-size Dicke model of arbitrary number of qubits is solved analytically in a unified way within extended coherent states. For the or Dicke models ( k is an integer), the G -function, which is only an energy dependent determinant, is derived in a transparent manner. The regular spectrum is completely and uniquely given by stable zeros of the G -function. The closed-form exceptional eigenvalues are also derived. The level distribution controlled by the pole structure of the G -functions suggests non-integrability for model at any finite coupling in the sense of recent criteria found in the literature. A preliminary application to the exact dynamics of genuine multipartite entanglement in the finite- N Dicke model is presented using the obtained exact solutions.