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Annual Meeting Optical Society of America

Atomic dynamics in a two-photon process

David A. Cardimona, Vassilios I. Kovanis, M. P. Sharma

Published Jan 1, 1990• 0 citations
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Abstract

We investigate the atomic inversion dynamics that occur when an atom interacts with a two-mode quantized radiation field. Our work uses Schrodinger theory with the rotating wave approximation. We adiabatically eliminate all but two of the atomic states in order to obtain analytical and numerical results. By varying the width and average photon number of the field statistics, we have uncovered secondary revivals that arise because of the complicated double sum over the two field modes in the population inversion. The diagonal part of the sum (the part that contains commensurate frequencies) is closely related to the degenerate-mode work of Buck and Sukumar.1 The breaking of the exact periodicity with secondary revivals and eventual erratic behavior result from the off-diagonal terms. Use of Poisson statistics allows exact analytic results to be obtained and also allows comparisons with previous work to be made. However, because the width and mean are intimately related for Poisson statistics, we also use Gaussian statistics so that their effects may be separated. To investigate various possible initial field conditions, we use squeezed states (oscillatory in photon number), displaced number states, and generalized states. The latter interpolate between number states, coherent states, and thermal states.

DOI
10.1364/oam.1990.tue8

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We investigate the atomic inversion dynamics that occur when an atom interacts with a two-mode quantized radiation field. Our work uses Schrodinger theory with the rotating wave approximation. We adiabatically eliminate all but two of the atomic states in order to obtain analytical and numerical results. By varying the width and average photon number of the field statistics, we have uncovered secondary revivals that arise because of the complicated double sum over the two field modes in the population inversion. The diagonal part of the sum (the part that contains commensurate frequencies) is closely related to the degenerate-mode work of Buck and Sukumar.1 The breaking of the exact periodicity with secondary revivals and eventual erratic behavior result from the off-diagonal terms. Use of Poisson statistics allows exact analytic results to be obtained and also allows comparisons with previous work to be made. However, because the width and mean are intimately related for Poisson statistics, we also use Gaussian statistics so that their effects may be separated. To investigate various possible initial field conditions, we use squeezed states (oscillatory in photon number), displaced number states, and generalized states. The latter interpolate between number states, coherent states, and thermal states.
We investigate the atomic inversion dynamics when a multilevel atom interacts with a two-mode quantized radiation field in a Raman-type process. By utilizing the rotating-wave approximation and adiabatically eliminating all but two of the atomic levels, we derive a nonlinear-interaction Hamiltonian for this system. Working in the Schr\"odinger picture, we obtain numerical results that depend on the width and average photon number of the field statistics. Varying these photon statistics allows us to uncover secondary revivals that arise due to the complicated double sum over the two field modes that appear in the solutions. The origin of this phenomenon is described analytically.
The Jaynes–Cummings interaction of a two-level atom with the radiation field is studied when the radiation is initially in a strongly squeezed coherent state. The dynamic response of the atomic inversion shows echoes after each revival when the squeezed coherent state exhibits an oscillatory photon-counting distribution due to the phase-space interference effect. The sensitivity of the dynamic behavior to approximations used in computing the atomic inversion is discussed. Comparison is made with the intensity-dependent interaction model of Buck and Sukumar [ Phys. Lett.81A, 132 ( 1981)]; this model does not exhibit echoes. The mean, variance, and entropy for the photon-number distribution are calculated and found to show behavior similar to that of the atomic inversion.
Photon statistics in some fully quantized models of the interaction of a two-level atom with a single-mode radiation field have been studied using the operator equations of motion. Expressions for the photon number distribution and the mean photon number are presented for various initial conditions. It is found that the mean photon number may show decays and revivals of coherence similar to those of the atomic inversion in the coherent-state Jaynes-Cummings model. Application of these models to the study of multiphoton laser, absorption, and emission processes is also discussed.
The two-photon absorption process is studied by considering the interaction of a quantized single-mode field with an effective two-level atom through intermediate states. The equations of motion of the coupled atom–field probability amplitudes for the effective two-level atom are obtained after adiabatically eliminating the intermediate states. These equations are then solved in the rotating-wave approximation. The time constants associated with the Rabi oscillation, collapse, and revival of the atomic inversion are derived, and the photon statistics are discussed with special reference to photon antibunching.
When the displacement and squeeze parameters for a squeezed coherent state are chosen appropriately, the squeezed state has an oscillatory photon number distribution P(n). Schleich and Wheeler1 have shown that the oscillations arise from the interference between contributions to P(n) from two regions in phase space where an elongated error ellipse, characterizing the squeezed coherent state, overlaps the annuli characterizing the number states. We study the Jaynes-Cummings model for the interaction of a two-level atom with a single mode of the radiation field when the radiation field is initially in such a squeezed coherent state. Revivals of the atomic inversion occur which have a quite different appearance from those produced when the radiation field is initially In a coherent state.2 Each revival is followed by a series of echoes—a phenomenon we call ringing revivals. We discuss the relationship between these echoes and the oscillations in P(n). There is a one-to-one correspondence between echoes and the peaks in P(n); however, interference between the contributions to the inversion from different peaks in P(n) plays an important role in forming the ringing structure of the revivals.
An asymptotic result is derived for the Jaynes-Cummings model of a two-level atom interacting with a quantized single-mode field, which is valid when the field is initially in a coherent state with a large average photon number. It is shown that for certain initial atomic states the joint atom-field wave function factors into an atomic and a field part throughout the interaction, so that each system remains separately in a pure state. The atomic part of the wave function displays a crossing of trajectories in the atom Hilbert space that leads to a unique state for the atom, independent of its initial state, at a specific time t0{\mathit{t}}_{0}t0​ (equal to half the revival time). The field part of the wave function resembles a crescent squeezed state. The well-known collapses and revivals are investigated from this perspective. The collapse appears to be associated with a ``measurement'' of the initial state of the atom with the field as the measuring apparatus. The measurement is not complete for finite average photon number: the system is instead left in a coherent superposition of macroscopically distinct states. At the half-revival time t0{\mathit{t}}_{0}t0​ this superposition of states is entirely in the field part of the state vector, so that the (pure) state of the field at that time is of the form sometimes referred to as a ``Schr\"odinger cat.'' The revivals of the population inversion are seen to be entirely due to the fact that the linear superposition of the two macroscopically distinct field states is coherent (i.e., a pure state), as opposed to an incoherent mixture.