No abstract

We study the correlated spontaneous emission from a dense spherical cloud of N atoms uniformly excited by absorption of a single photon. We find that the decay of such a state depends on the relation between an effective Rabi frequency Ω ∝ √ N and the time of photon flight through the cloud R/c. If ΩR/c < 1 the state exponentially decays with rate Ω 2 R/c and the state life time is greater then R/c. In the opposite limit ΩR/c 1, the coupled atom-radiation system oscillates between the collective Dicke state (with no-photons) and the atomic ground state (with one photon) with frequency Ω while decaying at a rate c/R.The problem of a single photon absorbed by a cloud of N atoms followed by correlated spontaneous emission is a problem of long standing interest. Dicke [1] first noted that the radiation rate from a small dense cloud is abnormal. In his words "... the greatest radiation intensity anomaly occurs in the transition to the ground state" [2]. In particular he showed that the collective decay rate for the symmetric state with one excitation is Γ N = N γ (as indicated in Fig. 1(a)), where he assumed the atomic volume to have dimensions small compared with the radiation wavelength.In our present work, we are interested in the time evolution of a specially prepared state obtained by absorption of a single photon [3,4,5]. We report novel dynamical oscillations in the evolution of the quantum state of the atom cloud, even without the existence of a cavity. Fig. 1 summarizes the main results of this paper. It is as if the atomic cloud acts to form a new "cavity" with the atom cloud volume V replacing the virtual photon volume V ph defined by the electromagnetic cavity; that is the usual vacuum Rabi frequencyω/( 0 V ) in the present problem, where ℘ is the electric-dipole transition matrix element, ω is the photon energy and 0 is free space permittivity. FIG. 1:Comparison of the different dynamical behavior for the correlated spontaneous emission from an N atomic cloud described by the |+ k 0 state of Fig. 2b. The single atom spontaneous decay life time is taken to be τ0 = 10 ns (γ = 1/2τ0). We assume that atomic density is 10 16 cm −3 and the resonant photon wavelength is λ = 1µm. Plot (a) corresponds to the case when cloud radius is equal to λ/2, hence the number of atoms is Na = 5240, then the state decay time is τa = τ0/Na = 1.9 × 10 −3 ns. In plot (b) the cloud radius is r = 10λ, N b = 4.2 × 10 7 , τ b = 32π 2 r 2 τ0/27N b λ 2 = 2.7 × 10 −4 ns. In plot (c) the radius of the atomic cloud is R = 1 mm which yields Nc = 4.1 × 10 13 , τc = 8R/6c = 4.4 × 10 −3 ns, while the period of oscillations is 2π/Ω = 0.74 × 10 −3 ns.arXiv:0804.3767v1 [physics.optics]

We study emission of a single photon from a spherically symmetric cloud of N atoms (one atom is excited, N-1 are in ground state) and present an exact analytical expression for eigenvalues and eigenstates of this many body problem. We found that some states decay much faster then the single-atom decay rate, while other states are trapped and undergo very slow decay. When size of the atomic cloud is small compared with the radiation wave length we found that the radiation frequency undergoes a large shift.Recent quantum optical experiments and calculations [1, 2] focus on the problem in which a single photon is stored in a gas cloud and then retrieved at a later time. The directionality and spectral content of the cooperatively reemitted photon is then of interest.Furthermore synchrotron radiation experiments involving N nuclei excited by weak γ ray pulse have features in common with the present problem [3]. For example, in such experiments a thin disk of nuclei can easily be prepared in a superposition in which the atoms are all in the ground state together with a small probability of a uniform excitation of the state, similar to Eq. (29), added in. The simplest example of two-atom cooperative decay has been studied in many publications [4]. The N -atom problem has been also investigated by several authors [5]. Time evolution and directionality of the radiation emitted from a system of two-level atoms which are excited by a plane-wave pulse have been discussed in [6].Having motivated our interest in the problem we now turn to the analysis of the correlated spontaneous emission from N atoms in free-space. We consider a system of two level (a and b) atoms, initially one of them is in the excited state a and E a − E b = ℏω. Initially there are no photons. Atoms are located at positions r j (j = 1, ..., N ). In the dipole approximation the interaction of atoms with photons is described by the Hamiltonian(1) whereσ j is the lowering operator for atom j,â k is the photon operator and g k is the atom-photon coupling constant for the k mode. We look for a solution of the Schrödinger equation for the atoms and the field as a superposition of Fock statesStates in the first sum correspond to zero number of photons, while in the second sum the photon occupation number is equal to one and all atoms are in the ground state b. For simplicity we neglect the effects of photon polarization. Substitute of Eq. (2) into the Schrödinger equation yields the following equations for β j (t) and γ k (t) (we put = 1)Integrating Eq. (4) over time gives(5) Substituting this into (3) we obtain equation for β j (t)(6) We proceed by making the Markov approximation a-la Weisskopf and Wigner to obtaiṅwhere for i = jΓ ii = 1, k 0 = ω/c and γ is the single atom spontaneous decay rateπc , V ph is the photon volume. We point out that a rigorous treatment of the problem beyond the rotating wave approximation Hamiltonian (1) also yields Eqs. (7) and (8) [7,8]. Imaginary part of Γ ij

Precision measurement of small separations between two atoms or molecules has been of interest since the early days of science. Here, we discuss a scheme which yields spatial information on a system of two identical atoms placed in a standing wave laser field. The information is extracted from the collective resonance fluorescence spectrum, relying entirely on far-field imaging techniques. Both the interatomic separation and the positions of the two particles can be measured with fractional-wavelength precision over a wide range of distances from about λ/550 to λ/2