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IEEE Journal of Quantum Electronics

Physical interpretation of operator dynamics in the Jaynes-Cummings model

A. Bandilla, H.‐H. Ritze

Published Jul 1, 1988• 14 citations
Publisher

Abstract

By means of simple mathematical methods, typical properties of the quantum dynamics in the Jaynes-Cummings model are discussed. The methods used allow a clear physical interpretation in the sense of an ensemble average and permit an investigation of the influence of the initial photon distribution width on collapse and revivals. Results obtained by a more sophisticated treatment provide corrections which are on the order 1/ square root N/sub 0/ for N/sub 0/>>1 where N/sub 0/ is the average photon number. The time evolution of the electric field strength and the photon distribution are also studied.>

DOI
10.1109/3.972

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Physical Review A•Sep 1, 1996•27 citations
By means of simple mathematical methods, typical properties of the quantum dynamics in the Jaynes-Cummings model are discussed. The methods used allow a clear physical interpretation in the sense of an ensemble average and permit an investigation of the influence of the initial photon distribution width on collapse and revivals. Results obtained by a more sophisticated treatment provide corrections which are on the order 1/ square root N/sub 0/ for N/sub 0/>>1 where N/sub 0/ is the average photon number. The time evolution of the electric field strength and the photon distribution are also studied.>
We study the dynamics of the driven Jaynes-Cummings model where an atom interacts with a single-mode cavity field but is also directly driven by an external radiation field. We show that the dynamic variables can be described in terms of ordinary Jaynes-Cummings dynamic variables with a different initial condition. We investigate the special case of an initial coherent state. We find collapses and revivals in the average number of photons at a much larger time scale than their Jaynes-Cummings model counterparts. We show that these ``super-revivals'' are related to the large-time-scale revivals that occur in the average electric field in the ordinary Jaynes-Cummings model. We also examine the photon statistics in the driven Jaynes-Cummings model and show that the cavity field is sub-Poissonian for certain interaction times.
We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order ααα influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $α= 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.
The driven two-photon Jaynes-Cummings model (JCM) exhibits a collapse and revival phenomenon in its mean photon number on a time scale much larger than the periodic revival time for two-state inversion in the usual two-photon JCM. The time scale of revivals for these oscillations is much larger than the corresponding time scale for the driven one-photon JCM and can be explained with the Hamiltonian of type ${H}_{\ensuremath{\eta}}=\ifmmode\pm\else\textpm\fi{}\sqrt{{(J}_{\ensuremath{\eta}}^{\ifmmode\dagger\else\textdagger\fi{}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}{(J}_{\ensuremath{\eta}}+\ensuremath{\alpha}{P}_{\ensuremath{\eta}}{)}^{2}}$ constructed explicitly for this purpose. The effect of Stark shifts is also studied and it is observed that the dynamics is strongly influenced by such Stark shifts both at early stages as well as at late stages of the evolution but in different ways, and thus Stark shifts play a significant role in determining the dynamical evolution of the system.
The exact time dependence of the density operator and various physical quantities for the motion-quantized Jaynes-Cummings models (MQJCMs) is given using the operator transformation under which it is shown that the MQJCMs have the same dynamics as the ordinary JCMs in the Kerr medium. The quantum collapses and revivals of the atomic motion due to the quantum properties of light fields and the quantization of atomic position and momentum, are predicted. The effects of the initial atomic momentum distribution and the arbitrary intensity-dependent medium on the collapse and revival phenomena of the population inversion, atomic momentum and radiation are formulated in detail. \textcopyright{} 1996 The American Physical Society.