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Physica Scripta

AnSU(2)⊗SU(2) Jaynes–Cummings model with a maximum energy level

A. Martín Ruiz, Alejandro Frank, Luis F. Urrutia

Published Mar 11, 2014• 3 citations
Publisher

Abstract

The SU(2)⊗SU(2) extension of the Jaynes–Cummings model provides a quantum mechanical system with a finite Hilbert space together with a maximum energy level in the spectrum. The model, which remains exactly solvable, is based on the replacement of the bosonic (creation and annihilation) operators by spin operators defined to act within a definite SU(2) irreducible representation, in such a way that the photon field has a finite number of excitations. The usual Heisenberg–Weyl () algebra can be obtained by contraction of the su(2) Lie algebra. We analyze the behavior of both the atomic and field quantum properties, like collapse and revivals, photon antibunching and squeezing, giving special attention to their dependence upon the maximum number of photon excitations.

DOI
10.1088/0031-8949/89/04/045103

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The SU(2)⊗SU(2) extension of the Jaynes–Cummings model provides a quantum mechanical system with a finite Hilbert space together with a maximum energy level in the spectrum. The model, which remains exactly solvable, is based on the replacement of the bosonic (creation and annihilation) operators by spin operators defined to act within a definite SU(2) irreducible representation, in such a way that the photon field has a finite number of excitations. The usual Heisenberg–Weyl () algebra can be obtained by contraction of the su(2) Lie algebra. We analyze the behavior of both the atomic and field quantum properties, like collapse and revivals, photon antibunching and squeezing, giving special attention to their dependence upon the maximum number of photon excitations.
The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics. The relative simplicity of the JCM and the ease with which it can be extended through analytic expressions or numerical computation continues to motivate attention, as evidenced by the growing abundance of publications. We here present an overview of the theory of the JCM and some of the many extensions and generalizations that have appeared.
We introduce a dynamical system that instead of exchanging a single photon as in the atomic system of the usual Jaynes-Cummings model (JCM), it exchanges instead a squeezed coherent photon. Accordingly, the creation and annihilation photon operators in the JCM are replaced by the creation and annihilation operators of squeezed coherent states, respectively. This transformation generates a photon Hamiltonian that modifies the JCM that includes terms usually omitted in the JCM when making the rotating-wave approximation and, in addition, the transformation introduces atom exchange terms absent in the JCM.
Abstract The Jaynes-Cummings (JC) model is presented based on quantized counterpropagating waves, characteristic of a ring resonator. This model, in addition to the Hamiltonian and the operator of the number of excitations, includes an interference operator whose eigenvalues imitate the term cos 2kz of the classical picture of the field. The average number of photons, dispersion, the Mandel parameter Q, and atomic-photon entanglement in states with certain excitations are considered. The collapses and revivals are revealed, which in the standard JC model are present only in states with an indefinite number of excitations. It is shown that photons at the boundary energy levels are antibunched (with Q ≈ –1/2) regardless of the system’s parameters, and photons of other energy levels are grouped to the degree that grows with the total number of photons.
A modification of the Holstein-Primakoff transformation is proposed such that creation and annihilation operators for a bosonic field are rewritten as operators of a SU(2)SU(2)SU(2) algebra. Once it is applied to a quantum Hamiltonian, a subsequent application of the prescription by Lieb to obtain the classical limit for spin operators allows one to write efficiently a classical Hamiltonian for the system. This process is illustrated for the MMM-atom Jaynes-Cummings model.
The formulation of the evolution operator for the multiphoton Jaynes-Cummings interaction in the decomposition form is presented, in terms of the SU(2) rotation transformation proposed by the present author [J. Mod. Opt. 39, 2209 (1992)]. The study of the dynamical evolution of the photon statistics for the case of intensity dependent coupling with m-photon atomic transition exhibits a number of novel features (sub-Poissonian and oscillatory distributions). It is shown that for m=1 the field distribution oscillates exactly in a periodic sequence without the limitation of the large mean photon number n-bar, and that for m=2 the oscillatory distributions emerge at one-quarter and three-quarters of the revival time for the initially high boson excitation regardless of the initial condition of the atomic state.