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Journal of Physics B Atomic and Molecular Physics

Cooperative fluorescence from N coherently driven two-level atoms

Daniel F. Walls

Published May 28, 1980• 28 citations
Publisher

Abstract

Considers a system of N coherently driven two-level atoms contained in a volume with dimensions much smaller than a wavelength. The atoms may decay by a completely cooperative mechanism which conserves (J 2 ), alternatively they may decay by a mechanism which fails to conserve (J 2 ). The steady-state behaviour in each case is determined using the single-atom solutions together with two limiting approximations: (a) complete atomic correlation, (b) no atomic correlation. In case (a) a behaviour characteristic of a second-order phase transition is obtained while in case (b) a first-order phase transition results.

DOI
10.1088/0022-3700/13/10/008

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Considers a system of N coherently driven two-level atoms contained in a volume with dimensions much smaller than a wavelength. The atoms may decay by a completely cooperative mechanism which conserves (J 2 ), alternatively they may decay by a mechanism which fails to conserve (J 2 ). The steady-state behaviour in each case is determined using the single-atom solutions together with two limiting approximations: (a) complete atomic correlation, (b) no atomic correlation. In case (a) a behaviour characteristic of a second-order phase transition is obtained while in case (b) a first-order phase transition results.
We consider N two level atoms driven far from equilibrium by a coherent driving field. The atoms may decay by a cooperative (J2 preserving) mechanism or a non-cooperative (J2 non-conserving) mechanism where J is the atomic pseudo angular momentum. We adopt a mean field theory to study the J2 non-conserving interaction. This indicates that the system displays a first order phase transition in several observables, e.g., mean field, spectrum and photon statistics. The two stable branches of the hysteresis curve are shown to be regions of cooperative and non-cooperative behaviour respectively. In the case of cooperative atomic decays we find a solution to the Fokker-Planck equation in the atomic coherent state representation. This predicts a second order phase transition in atomic observables—a cooperative fluorescent critical point.
The cooperative resonance fluorescence steady state is discussed within the context of an operator master equation which conserves total pseudospin. Emphasis throughout is on quantum fluctuations and their significance in relation to a background of factorised dynamics. Atom-atom correlations are shown to play a fundamental role for systems driven beyond the linear regime. Use of the atomic coherent state representation yields a Fokker-Planck description closely allied to the dynamics for a classical angular momentum oscillator. For intense incident fields the quantum-mechanical steady state is understood in terms of diffusion both around and between classical trajectories on the Bloch sphere. In the limit of infinite systems simple closed-form expressions for steady-state features are derived. Coherent and incoherent fluorescent intensities are obtained together with the second-order correlation function for fluorescent light. Specific features are illustrated by numerical results for systems of from two to fifty atoms.
Atomic ensembles strongly interacting with light constitute rich quantum-optical many-body systems, with the potential for observing cooperative effects and dissipative nonequilibrium phase transitions. We theoretically analyze the conditions under which a driven atomic ensemble in free space, characterized by strong dipole-dipole interactions and large spatial extent, can undergo a superradiant phase transition, also known as cooperative resonance fluorescence. In an atomic array, stationary states that conserve the collective pseudospin exhibit completely cooperative decay and undergo a second-order phase transition in the large atom number limit. In contrast, decay mechanisms on longer timescales that fail to conserve pseudospin can lead to discontinuous first-order phase transition at a critical finite atom number, disrupting cooperation despite sharing many similar observable characteristics. A hallmark of the superradiant phase transition is an abrupt shift from total light reflection off the atoms to rapidly increasing transmission, accompanied by significant quantum fluctuations, as a function of light intensity.
Atoms strongly interacting with light constitute rich quantum-optical systems with the potential for observing cooperative effects and dissipative nonequilibrium phase transitions. We analyze the conditions under which a driven atomic array, characterized by strong dipole–dipole interactions and a large spatial extent, can undergo a superradiant phase transition, also known as cooperative resonance fluorescence. We find that the array can exhibit completely cooperative decay that conserves the collective pseudospin, resulting in a second-order quantum phase transition, with a key hallmark being an abrupt shift from total light reflection by the atoms to rapidly increasing transmission and significant quantum fluctuations. We compare the results with decay mechanisms that fail to conserve pseudospin, leading to a discontinuous first-order phase transition at a critical finite atom number.
We discuss from first principles the cooperative decay of a system of two-level atoms, initially prepared in an uncorrelated excited state with population inversion NNN, and we give the conditions under which the superfluorescence effect occurs. Describing the atomic system in terms of collective variables, we derive a master equation for the reduced atomic density operator, which gives rise both to a damping and to a time-dependent frequency shift in the dynamics of collective modes. The coupled equations of motion are solved with a self-consistent approach. It is found that the system goes through a nonexponential decay if the maximum length of the active volume is smaller than a "cooperation range" and larger than a "threshold length," in agreement with the one-mode theory. The radiation burst has a time width proportional to ${N}^{\ensuremath{-}1}$, and its intensity is proportional to N2{N}^{2}N2. Specializing to a pencil-shaped volume, we find that only two atomic modes need to be considered; in this case, the average emitted radiation is all condensed in the two diffraction patterns of the opposite axial modes.
We consider the problem of two-photon cooperative emission in systems of two-level atoms. Two physically distinct regimes are analyzed. First, we investigate the case of a small number of atoms. We study the evolution of two-photon super- and sub-radiant states and associated two-photon spectra. Second, we investigate the problem of a constant density of atoms confined to a spherical volume. We analyze separately the cases in which the radius is small or large in comparison to the resonant wavelength.