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arXiv:cond-mat/9903255 · Physical review. B, Condensed matter

Mesoscopic cooperative emission from a disordered system

Tigran V. Shahbazyan, M. É. Raǐkh, Zeev Valy Vardeny

Published May 15, 2000• 46 citations
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Abstract

We study theoretically the cooperative light emission from a system of $N\ensuremath{\gg}1$ classical oscillators confined within a volume with spatial scale L much smaller than the radiation wavelength ${\ensuremath{\lambda}}_{0}=2\ensuremath{\pi}c/{\ensuremath{\omega}}_{0}.$ We assume that the oscillator frequencies are randomly distributed around a central frequency ${\ensuremath{\omega}}_{0}$ with some characteristic width $\ensuremath{\Omega}\ensuremath{\ll}{\ensuremath{\omega}}_{0}.$ In the absence of disorder, that is, $\ensuremath{\Omega}=0,$ the cooperative emission spectrum is composed of a narrow subradiant peak superimposed on a wide superradiant band. When $\ensuremath{\Omega}\ensuremath{\ne}0,$ we demonstrate that if N is large enough, the subradiant peak is not simply broadened by the disorder but rather splits into a system of random narrow peaks. We estimate the spectral width of these peaks as a function of N, L, $\ensuremath{\Omega},$ and ${\ensuremath{\lambda}}_{0}.$ We also estimate the amplitude of this mesoscopic structure in the emission spectrum.

DOI
10.1103/physrevb.61.13266

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We study theoretically the cooperative light emission from a system of N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1 classical oscillators confined within a volume with spatial scale L much smaller than the radiation wavelength \ensuremathλ0=2\ensuremathπc/\ensuremathω0.{\ensuremath{\lambda}}_{0}=2\ensuremath{\pi}c/{\ensuremath{\omega}}_{0}.\ensuremathλ0​=2\ensuremathπc/\ensuremathω0​. We assume that the oscillator frequencies are randomly distributed around a central frequency \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ with some characteristic width \ensuremathΩ\ensuremath≪\ensuremathω0.\ensuremath{\Omega}\ensuremath{\ll}{\ensuremath{\omega}}_{0}.\ensuremathΩ\ensuremath≪\ensuremathω0​. In the absence of disorder, that is, \ensuremathΩ=0,\ensuremath{\Omega}=0,\ensuremathΩ=0, the cooperative emission spectrum is composed of a narrow subradiant peak superimposed on a wide superradiant band. When \ensuremathΩ\ensuremath≠0,\ensuremath{\Omega}\ensuremath{\ne}0,\ensuremathΩ\ensuremath=0, we demonstrate that if N is large enough, the subradiant peak is not simply broadened by the disorder but rather splits into a system of random narrow peaks. We estimate the spectral width of these peaks as a function of N, L, \ensuremathΩ,\ensuremath{\Omega},\ensuremathΩ, and \ensuremathλ0.{\ensuremath{\lambda}}_{0}.\ensuremathλ0​. We also estimate the amplitude of this mesoscopic structure in the emission spectrum.
The effect of disorder on the collective emission from a system of classical oscillators is studied theoretically. Three types of disorder are considered: random orientation of dipole moments, finite spread in frequencies of the individual oscillators (diagonal disorder), and dipole- dipole interaction (off-diagonal disorder). We found that for sufficiently high concentration of oscillators the diagonal disorder does not completely destroy the collective character of the emission. We show that the eigenmodes of a cooperative system with disorder comprise a large number of oscillators and, due to cooperative character of the emission, their lifetime is much longer than the lifetime of an individual oscillator. Consequently, the cooperative emission spectrum of the system is not simply broadened by the disorder, but represents a superposition of relatively narrow peaks.
We analyze a one-dimensional ring structure composed of many two-level systems, in the limit where only one excitation is present. The two-level systems are coupled to a common environment, where the excitation can be lost, which induces super- and subradiant behavior, an example of cooperative quantum coherent effect. We consider time-independent random fluctuations of the excitation energies. This static disorder, also called inhomogeneous broadening in literature, induces Anderson localization and is able to quench superradiance. We identify two different regimes: (i) weak opening, in which superradiance is quenched at the same critical disorder at which the states of the closed system localize; (ii) strong opening, with a critical disorder strength proportional to both the system size and the degree of opening, displaying robustness of cooperativity to disorder. Relevance to photosynthetic complexes is discussed.
We give a general nonperturbative treatment of cooperative emission in systems of NNN two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to N2{N}^{2}N2, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length LLL of the active volume is much larger than a suitable threshold length LT{L}_{T}LT​ (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length LLL is smaller or of the same order of a suitable cooperation length Lc{L}_{c}Lc​ (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to NNN). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to NNN, so that superfluorescence effects disappear.
We study the collective emission of a disordered array of N excited two-level atoms into a one-dimensional photonic waveguide. In the perfectly ordered case, where atoms are spaced by exact integer multiples of the wavelength, the system exhibits the characteristic superradiant burst with a peak emission rate scaling as N 2 . Using large-scale semiclassical simulations, we find that this key signature of superradiance remains asymptotically robust under strong spatial and spectral disorder, but also exhibits subtle finite-size scaling toward this limit. To explain our observations, we provide an analytical variational estimate for the maximal decay rate, which tightly bounds the numerical results and reveals how disorder shapes the collective decay. Specifically, we find that even in the presence of strong disorder, the spins tend to self-organize spontaneously according to their locations, which overall optimizes constructive interference effects and explains the emergence of mirror-asymmetric correlations in superradiant decay. These findings resolve important open questions regarding the existence and nature of superradiance in strongly disordered arrays and offer valuable insights for understanding collective quantum optical phenomena in realistic systems.
In the collective photon emission from atomic clouds both the atomic transition frequency and the decay rate are modified compared to a single isolated atom, leading to the effects of superradiance and subradiance. In this article, we analyze the properties of the Euclidean random matrix associated with the radiative dynamics of a cold atomic cloud, previously investigated in the contexts of photon localization and Dicke super- and subradiance. We present evidence of a phase transition, surprisingly controlled by the cooperativeness parameter, rather than the spatial density or the diagonal disorder. The numerical results corroborate the occurrence of such a phase transition at a critical value of the cooperativeness parameter, above which the lower edge of the spectrum vanishes, exhibiting a macroscopic accumulation of eigenvalues. Independent evaluations based on the two phenomena provide the same value for the critical cooperativeness parameter.
When atoms are coupled to a common electromagnetic environment, the exchange of photons through dipole-dipole interactions leads to the emergence of cooperative effects. As a particular example, superradiance arises from spontaneous emission when this exchange leads to constructive interference of the emitted photons. Here, we introduce an integrated method for studying cooperative radiation in many-body systems. This method, which allows to study extended systems with arbitrarily large number of particles, can be formulated by an effective, nonlinear, two-atom master equation that describes the dynamics using a closed form which treats single- and many-body terms on an equal footing. We apply this method to a homogeneous gas of initially inverted two-level atoms, and demonstrate the appearance of both superradiance and subradiance, identifying a many-body coherence term as the source of these cooperative effects. We describe the many-body induced broadening -- which is analytically found to scale with the optical depth of the system -- and light shifts, and distinguish spontaneous effects from induced ones. In addition, we theoretically predict the time-dependence of subradiance, and the phase change of the radiated field during the cooperative decay.