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Physical Review A

Perturbative solution to the single-mode model of superradiance

Lorenzo M. Narducci, V. Bluemel

Published Apr 1, 1975• 15 citations
Publisher

Abstract

We use a perturbative approach to analyze the diffusion equation for the atomic quasiprobability function derived by Narducci, Coulter, and Bowden, and demonstrate explicitly that the first-order correction contributes to the atomic moments only to order $\frac{1}{N}$, where $N$ is proportional to the cooperation number.

DOI
10.1103/physreva.11.1354

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We use a perturbative approach to analyze the diffusion equation for the atomic quasiprobability function derived by Narducci, Coulter, and Bowden, and demonstrate explicitly that the first-order correction contributes to the atomic moments only to order 1N\frac{1}{N}N1​, where NNN is proportional to the cooperation number.
The "superfluorescence equations" deduced in a previous paper are generalized to the case of finite temperature and translated into the atomic coherent-state representation of Arecchi et al. We get three coupled partial differential equations involving the quasiprobability distribution $Q(\ensuremath{\theta},t)$ of the atomic system and two suitable functions $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$ describing the stimulated processes which are relevant in oscillatory superfluorescence. Neglecting $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$, the equation for $Q(\ensuremath{\theta},t)$ reduces to the Fokker-Planck equation of Narducci et al. corrected by the contribution of thermal noise. The stationary solution is given by the unperturbed canonical state, which is shown to be the exact stationary solution of the single-mode model from which the superfluorescence equations have been deduced. These equations prescribe that $\mathfrak{M}(\ensuremath{\theta},t)$ always remain equal and opposite to $\mathcal{F}(\ensuremath{\theta},t)$, so that the superfluorescence equations reduce automatically to two equations. Eliminating $\mathcal{F}(\ensuremath{\theta},t)$ between the two remaining equations, one obtains a non-Markovian equation for $Q(\ensuremath{\theta},t)$ with derivatives of all orders in $\ensuremath{\theta}$. In the case of pure superfluorescence or super-radiance, such an equation reduces to a Markovian Fokker-Planck equation with derivatives of first and second order in $\ensuremath{\theta}$ which exhibits an interesting fourth-order contribution in the diffusion term. In fact, if one excludes a very small region around the north pole of the Bloch sphere, this term is much larger than the second-order contribution in the diffusion term of the equation of Narducci et al. In the case of oscillatory superfluorescence, the presence of memory effects enhances fluctuations, thus forbidding the existence of "classical" time evolutions.
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.
The exact steady-state solution for the reduced atomic density operator for a model system of N identical two-level atoms confined to a single site (the Dicke model) and driven by a totally incoherent broad-band (chaotic) field is derived through the equivalent Fokker-Planck equation of the system. Steady-state values of the atomic observables, higher moments and atomic fluctuations are calculated for an arbitrary N and an arbitrary field strength. It is shown that when the chaotic field is a thermal (black-body) field the atomic system is driven into a steady relative occupation number that obeys the Boltzmann distribution law only for N=1. (The same result for N>1 atoms can be reached only if the cooperative interactions between the atoms are ignored.) In the thermodynamic limit N to infinity there is no critical behaviour in contrast to the resonant coherently driven case. It is also shown that semiclassical (direct) factorisation of the equations of motion does not lead to the correct results for the N to infinity limit of the exact quantum case. Within the exact theory numerical results for finite N (<or=100) are presented for the time-dependent expectation values of the atomic operators, the intensity and the intensity autocorrelation function. Superradiant pulses are emitted by N initially inverted atoms but as the incident intensity is increased this superradiant behaviour is obscured. The steady-state fluorescence spectrum is a single Lorentzian of width proportional to N in the weak-field case. In the strong-field case the spectrum is insensitive to the number of atoms and the result is similar to the one-atom situation. Photon bunching and antibunching effects are investigated for the weak- and strong-field limits. When the chaotic driving field is simply a single-mode thermal field the approach to atomic saturation is delayed in comparison with the case of the broad-band chaotic field.