PHYSICAL REVIEW A 81, 053821 (2010) Cooperative spontaneous emission of N atoms: Many-body eigenstates, the effect of virtual Lamb shift processes, and analogy with radiation of N classical oscillators Anatoly A. Svidzinsky, 1 Jun-Tao Chang, 1 and Marlan O. Scully 1,2 1 Texas A&M University, College Station, Texas 77843, USA 2 Princeton University, Princeton, New Jersey 08544, USA (Received 28 January 2010; published 13 May 2010) We consider collective emission of a single photon from a cloud of N two-level atoms (one excited, N − 1 ground state). For a dense cloud the problem is reduced to finding eigenfunctions and eigenvalues of an integral equation. We discuss an exact analytical solution of this many-atom problem for a spherically symmetric atomic cloud. Some eigenstates decay much faster then the single atom decay rate, while the others undergo very slow decay. We show that virtual processes yield a small effect on the evolution of rapidly decaying states. However, they change the long time dynamics from exponential decay into a power-law behavior which can be observed experimentally. For trapped states virtual processes are much more important yielding additional decay channels which results in a slow decay of the otherwise trapped states. We also show that quantum mechanical treatment of spontaneous emission of weakly excited atomic ensemble is analogous to emission of N classical harmonic oscillators. DOI: 10.1103/PhysRevA.81.053821 PACS number(s): 42.50.Nn, 03.67.Mn, 05.30.Jp I. INTRODUCTION Collective spontaneous emission phenomenon has been a subject of interest since the pioneering work of Dicke in 1954 [1]. In that classical paper, Dicke considered mainly two types of collective radiation phenomena: superradiance and subradiance in a collection of two-level atoms when all atoms are confined inside a volume much smaller than radiation wavelength. Later work generalized Dicke’s description of superradiance to an extended system [2,3]. See especially the experimental and theoretical work of Feld and coworkers [4]. Further studies brought in the concept of superfluorescence [5], which describes the cooperative emission from a system of uncorrelated excited atoms. This process usually starts with normal spontaneous emission but later develops correlation among the system [6]. In the past half century, both types of phenomena were extensively studied theoretically and experimentally. The case of an ensemble of excited nuclei has long been, and continues to be a subject of research interest [7]. From the physical standpoint, cooperative spontaneous emission is an example of a many-body quantum problem of N atoms collectively interacting with an electromagnetic field. Emission from a weakly excited group of atoms is, in some ways, even more interesting than the case of a highly excited system. In the case of a weakly excited ensemble (e.g., one atom out of N is excited) it might be thought that the radiation rate would go as the single atom decay rate γ ; however, the Dicke symmetric state of maximum cooperation radiates at a rate Ŵ N ∝ Nγ . Collective spontaneous radiation is interesting physics and also has potential applications. From the standpoint of applications, superradiance is useful as one of the methods for producing coherent emission without coherent pumping. This is especially important in those regimes, such as x-ray or γ -ray, where there are no effective mirrors which limit the use of ordinary stimulated emission process. On the other hand, with the recent advances of quantum informatics, decoherence-free subspace (DFS) [8] has been proposed to be one of the strategies to combat the effects of decoherence in quantum computation and quantum communication. A collective system of many two-level particles is one of the ideal candidates to realize DFS [8–10]. An ensemble of N two-level atoms with one excitation also plays an important role in quantum memory and quantum networking. Relevant experiments have been carried out by the groups of Lukin [11], Kimble [12], and Vuleti´ c et al. [13]. Cooperative effects of N atoms in a cavity were investigated in 1980s by Cummings [14–16] and the others [17,18]. Buzek [19] studied the dynamics of an excited atom in the presence of N − 1 atoms in the free space and predicted radiation suppression. Dynamics of the system in free space and spatial anisotropy of the emitted radiation have been re-explored in the past few years [20–37]. The problem of cooperative spontaneous emission of N atoms reduces to finding all eigenstates and their decay rates. Once they are determined, evolution of an arbitrary initial state is obtained by expanding the initial condition in terms of the set of the eigenstates. In 1969 Ernst [38] studied such an eigenproblem for a spherical atomic cloud in Weisskopf and Wigner theory disregarding the effect of virtual photons. Later Ressayre and Tallet [39], and Andreev et al. [40] investigated such a problem in various geometries. However, the exchange of virtual photons induces dipole-dipole interaction between atoms [25,41–44]. Recently it was shown that virtual photons modify eigenstates and eigenvalues of the system [26,27] and dramatically change the evolution of the trapped states [32]. However, virtual processes yield a small (yet interesting) effect on the evolution of rapidly decaying states [32]. This question is a subject of recent debate [45–47]. In the present paper we discuss the problem of single- photon cooperative spontaneous emission in details focusing on the issues of current interest. In particular, we clarify the effect of virtual processes and situations when the quantum N -atom problem has a classical analogy with radiation of a system of N harmonic oscillators. 1050-2947/2010/81(5)/053821(15) 053821-1 ©2010 The American Physical Society