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The Two-Photon Jaynes-Cummings Model: A Coherent State Approach

Flávio Cordeiro, Constança Providência, Seiya Nishiyama

Published Jan 1, 2008• 2 citations
OpenAlex

Abstract

It is shown, by conventional mean-field technics, that su(2)⊗su(1, 1) coherent states are a good choice for describing the two-photon Jaynes-Cummings model in the sense that results based on mean-field methods for those states are in good agreement with exact results. We also find that this agreement is much improved if the constant of motion of the model is implemented exactly instead of being implemented only in the average. This is done by projecting the su(2) ⊗ su(1, 1) coherent states onto the physical subspaces. It is seen that the Hamiltonian of the model is unbounded from below, i.e., it lacks a ground-state. We also present and study a stabilized version of the model such that the relevant physical features are preserved. Along the article, special attention is given to the description of the super-radiant phase transition exhibited by the model.

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It is shown, by conventional mean-field technics, that su(2)⊗su(1, 1) coherent states are a good choice for describing the two-photon Jaynes-Cummings model in the sense that results based on mean-field methods for those states are in good agreement with exact results. We also find that this agreement is much improved if the constant of motion of the model is implemented exactly instead of being implemented only in the average. This is done by projecting the su(2) ⊗ su(1, 1) coherent states onto the physical subspaces. It is seen that the Hamiltonian of the model is unbounded from below, i.e., it lacks a ground-state. We also present and study a stabilized version of the model such that the relevant physical features are preserved. Along the article, special attention is given to the description of the super-radiant phase transition exhibited by the model.
We study the quantum dynamics of a two-mode two-photon Jaynes–Cummings model of a two-level atom interacting with the correlated two-mode SU(1, 1) coherent states. The collapse and revival patterns in this case are noted to be quite different from those in the case of uncorrelated coherent states and correlated pair-coherent states. As an example of the time evolution of the field statistics we show that violation of the Cauchy–Schwartz inequality by the initial field states may be nullified. We also discuss the evolution of the variance of the two-mode field quadrature operator.
We study the ground-state phase diagrams of two-photon Dicke, one-dimensional Jaynes-Cummings-Hubbard (JCH), and Rabi-Hubbard (RH) models using mean-field, perturbation, quantum Monte Carlo (QMC), and density-matrix renormalization-group methods. We first compare mean-field predictions for the phase diagram of the Dicke model with exact QMC results and find excellent agreement. The phase diagram of the JCH model is then shown to exhibit a single Mott insulator lobe with two excitons per site, a superfluid (SF, superradiant) phase and a large region of instability where the Hamiltonian becomes unbounded. Unlike the one-photon model, there are no higher Mott lobes. Also, unlike the one-photon case, the SF phases above and below the Mott lobe are surprisingly different: Below the Mott lobe, the SF consists of photon pairs as opposed to above the Mott lobe, where it is SF of simple photons. The mean-field phase diagram of the RH model predicts a transition from a normal to a superradiant phase but none is found with QMC.
In this work we have studied the two photon Jaynes-Cummings model to describe the interaction of a three-level atom in the configuration with a single-mode of a cativy-field. We have used the effetive and full microscopic Jamiltonian approach to compare the results obtained in these tow descriptions. We employed the two-photon Jaynes-Cummings model int the full microscopic Hamiltonian context to implement schemes to quantuam entanglement swapping, for approximate and conditional quantuam teleportation for an unknown atomic state, for a superposition of zero- and two photon states. Both schemes were implemented using no measurement in the Bell base states.
The pure-state evolution of the atomic as well as the field states in the resonant two-photon Jaynes-Cummings model with the field initially in the coherent state is studied. Analytic expressions for various quantities involved are obtained under the large-n\ifmmode\bar\else\textasciimacron\fi{} approximation. It is shown that the atom as well as the field are approximately in the pure state for gt=k\ensuremath{\pi}/4, where k is an integer and in a completely pure state for gt=k\ensuremath{\pi}. The field states generated at the times gt=k\ensuremath{\pi}/4 are shown to be a quantum superposition of the coherent states for a particular choice of the relative phase. The interference between these states leads to a spiked photon-number distribution but the field is still Poissonian.
The Jaynes–Cummings model (JCM) is used to give a detailed analysis of the long-time behavior of a two-level atom interacting with a (Glauber) coherent-state field. This work is based on the earlier work of J. H. Eberly and collaborators, who drew attention to the collapse and revival of the Rabi oscillations of the atomic inversion. We use the three components S1, S2, and S3 of the pseudospin vector S to analyze the dynamics of the atom within the rotating-wave approximation and exact resonance. We show that, at the times when the revivals peak, the variables σ = |S| and η = S3/σ become independent of (1) the transition dipole moment of the atom, (2) the intensity of the coherent-state field, and (3) the (common) frequency of the atom and the field. Thus these special values of σ and η are a universal feature of the JCM that should be observed in experiments with any two-level atom in an arbitrarily intense (but resonant) coherent-state field. The derivation of the above results was facilitated by transforming into a rotating coordinate frame in which S1 = 0 so that S2 and S3 are the only dynamical variables. The two-dimensional nature of S in this frame makes it easy to visualize the dynamical changes in the atom. The dynamics of the JCM is clarified by comparison with the closely related Buck–Sukumar model. The relationship of the phenomena discussed here to photon echoes and to quench echoes (in condensed media) is pointed out.