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Optical Engineering

Theoretical aspects of the quantum electrodynamics of two-level atoms in high-Q cavities

Amitabh Joshi, S. V. Lawande

Published Jun 1, 1994• 1 citation
Publisher

Abstract

The Jaynes-Cummings model provides a useful means of studying several quantum electrodynamic effects in matter-radiation interaction. One such effect is the phenomenon of collapses and revivals of Rabi oscillations. Various aspects of this phenomenon are studied in terms of the dynamical evolution of the atom as well as the statistical properties of the field. In particular, the sensitivity of collapses and revivals on the initial photon statistics, the cavity field damping mechanism is highlighted. For this purpose the basic model of Jaynes and Cummings has been extended and modified. Some multiatom effects are also discussed.

DOI
10.1117/12.168237

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The Jaynes-Cummings model provides a useful means of studying several quantum electrodynamic effects in matter-radiation interaction. One such effect is the phenomenon of collapses and revivals of Rabi oscillations. Various aspects of this phenomenon are studied in terms of the dynamical evolution of the atom as well as the statistical properties of the field. In particular, the sensitivity of collapses and revivals on the initial photon statistics, the cavity field damping mechanism is highlighted. For this purpose the basic model of Jaynes and Cummings has been extended and modified. Some multiatom effects are also discussed.
A study of multiatom effects, i.e., dipole-dipole interactions and phase-interrupting collisions between atoms in the Jaynes-Cummings model, is presented here. For this purpose we consider two identical two-level atoms interacting with a single-mode field in a lossless cavity and study the effect of the dipole-dipole interaction between these two atoms on the collapse-and-revival phenomenon of Rabi oscillations by calculating the analytical expression of the excitation probability in the coherent field. We also consider another situation in which a single atom interacting with a single-mode field in a damped cavity undergoes phase-interrupting collisions. The effect of phase-interrupting collisions along with the usual cavity damping on the collapse-and-revival phenomenon has been explicitly calculated using the dressed-atom approach under the secular approximation.
Solutions of the basic equations of a simplified model of the cavity quantum electrodynamics are presented under the condition that the single-photon Rabi frequency is much larger than the cavity decay rate. Such solutions are used to calculate the quantum-statistical properties of the field and atomic observables under a variety of initial conditions involving the states of the field and atom. Effects of increasing cavity damping and of the addition of thermal photons on collapse and revival phenomena are discussed. Phase-sensitive aspects of the cavity field are also treated. Quantum effects are shown to be most easily isolated by sending an atom initially prepared in a dressed state for such a state does not evolve further under the influence of a classical field. The appearance of squeezing in the cavity field is demonstrated. The squeezing is most prominent for a coherently prepared atom passing through an empty cavity. The quantum features of the complex dipole moment and their detectability are also discussed in detail.
The collapse and revival of quantum states appear in diverse areas of physics. In quantum optics the occurrence of such a phenomena in the evolution of an atomic state, interacting with a light field initially in a coherent state, was predicted by using the Jaynes-Cummings model (JCM), and subsequently demonstrated experimentally. In this paper we revisit the JCM with the Monte-Carlo wave function approach and investigate the time evolution of the photon emission rate of the atom in a cavity. Analytical and numerical quantum trajectory calculations show that the cavity and the initial field statistics strongly influence the photon emission dynamics. A coherent field indeed gives rise to a collapse and revival behavior that mirrors atomic state evolution. However, there are differences between the two. The emission rate for a field in a Fock number state exhibits a sinusoidal oscillation, and there exists a quiescent period for a thermal field. These properties are quite different from those in free space. It is also shown that the fluctuation in photon emission is much less than that of the atomic population.
The position as well as the width of the atomic wave packet passing through an optical cavity may affect the matter-field interaction. As a result, the internal dynamics of a two-level atom, such as the Rabi oscillations or the collapse and revival phenomenon, may be strongly modified with respect to the standard Jaynes-Cummings model. In particular, for a sufficiently large spread of the atomic position, the atomic population inversion displays the characteristic ringing behavior of the Bessel function J0{\mathit{J}}_{0}J0​ differently from the usual full Rabi oscillation.
The dynamical properties of a two-level atom interacting with a single non-decaying mode of an electromagnetic field in a binomial state are studied. The statistical aspects of the field, such as intensity-intensity correlation and squeezing, are also investigated. The binomial state reduces to a pure number state and a pure coherent state in different limits. Hence it enables us to study how the sinusoidal Rabi oscillations in a pure number state develop to give rise to the phenomenon of collapse and revival which has been studied extensively in the coherent-state field. In addition, the binomial state exhibits squeezing for certain values of parameters, but it is not a minimum-uncertainty-product state.
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.