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International Conference on Submillimeter Waves and Their Applications

Criteria for the Observation of Dicke Superradiance in the Submillimeter Wavelength Region

Charles M. Bowden

Published Jun 1, 1974• 0 citations
Publisher

Abstract

The phenomena of Dicke superradiance is discussed from the standpoint of the Fokker-Planck differential equation representation derived from the master equation in the atomic coherent states. Results calculated from the solution of this equation together with a self-consistency argument first put forth by Arecchi and Courtens are used to present characteristics of and criteria for its detection.

DOI
10.1109/icswa.1974.9333740

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The phenomena of Dicke superradiance is discussed from the standpoint of the Fokker-Planck differential equation representation derived from the master equation in the atomic coherent states. Results calculated from the solution of this equation together with a self-consistency argument first put forth by Arecchi and Courtens are used to present characteristics of and criteria for its detection.
The results of Haake and Glauber concerning the solution of the superradiant master equation of Bonifacio et al., are compared with those of Degiorgio and Ghielmetti. This is accomplished by using the atomic coherent-state representation of Arecchi et al., which provides a convenient mathematical formalism for the comparison. The relationship of these results to the recently presented exact Fokker-Planck-equation representation of Narducci, Coulter, and Bowden is discussed. This representation is used to prove the equivalence of the main conclusions derived by Degiorgio and Ghielmetti to a special case of the results of Haake and Glauber. In addition, it is shown that the expectation values calculated by Haake and Glauber are particular solutions of a coupled hierarchy of equations of motion constructed from the exact Fokker-Planck equation, neglecting the noise and diffusion terms.
Here we present an approach to the problem of superradiance in dense atomic samples, when dipolar interactions arise between atoms. Our treatment consists of the sequential use of the Holstein-Primakoff and mean-field approximations, from which we derive master equations for the strong and weak couplings of the sample with the reservoir. We find, in both cases, that the radiation emission presents remarkable features, with characteristic emission times much shorter and intensities much higher than those of Dicke superradiance. In particular, for strong sample-reservoir coupling, a whole comb of superpulses occurs within an envelope with the above-mentioned characteristic emission times much shorter and intensities much higher than those of Dicke superradiance.
We discuss the problem of calculating the superradiant behavior in samples small compared to a radiation wavelength. Difficulties associated with the analysis of Dicke are considered in the framework of quantum electrodynamics and equivalently from a classical point of view. The evolution of superradiance in a small sphere is followed numerically with attention to spatial variation of the Bloch dipole density. Coherence is maintained except in the region ${(\mathrm{kR})}^{\ensuremath{-}1}|cos\ensuremath{\theta}|\ensuremath{\lesssim}2$ which is at the Bloch equator, where it suddenly disappears. Away from the equator, the spatial variations of dipole density oscillate with amplitude $\ensuremath{\sim}{(\mathrm{kR})}^{2}sin\ensuremath{\theta}$ and frequency $\ensuremath{\sim}{(\mathrm{kR})}^{\ensuremath{-}3}|cos\ensuremath{\theta}|$. All these results agree with the predictions of a theoretical discussion previously published.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
The super-radiant master equation (SME) of Bonifacio et al. is analyzed using the coherent-atomic-state representation. We have succeeded in deriving an exact Fokker-Planck equation for the density function corresponding to the reduced atomic density operator in the diagonal atomic-state representation. A solution to the Fokker-Planck equation has been provided in an elementary fashion for arbitrary atomic states which are sufficiently removed from the state of complete inversion at time zero. The general solution for arbitrary initial conditions (including the initial state of complete inversion) has been obtained using the method of eigen-function expansions and the final result expressed in terms of an integral over the initial density function. The moments of the collective atomic operators are also discussed.
Superradiance is an extensive cooperative spontaneous emission phenomenon, exhibited by some atomic collective-spin states. However, distinct initial states differ in their decay dynamics. Dicke states ∣j,m⟩|j,m\rangle∣j,m⟩ with distinct numbers of excitations n=m+jn=m+jn=m+j, driven by vacuum fluctuations, have their peak emission intensity shifted in time. Rotating Dicke states relative to the decay axis introduces an interesting parity structure that affects the pulse profile and photon correlations. Squeezed-bath prepared states undergo a squeezing-controlled crossover to the rotated Dicke states, making the emission character dependent on the amount of squeezing transferred from light to the atomic state. For semiclassical states with a macroscopic dipole moment, like the atomic coherent state, the emission intensity depends on their polarization. We present detailed results on the superradiant dynamics of a representative selection of states expanded in Dicke states to highlight the initial state as an independent dynamical control parameter. For large-NNN, we are able to predict fairly accurately the pulse profile in each case using the mean-field approximation, an approach based on the Fokker--Planck equation. We also present comparative results on the intensity correlation function, quantify the coherent and incoherent contributions of the emission, and contrast between small and large ensembles.