A model radiating system consisting of two atoms in close proximity is analyzed. This system demonstrates the influence of spatial coherence on the spectrum of the radiation field. Explicit expressions for the degree of coherence, the source spectrum, and the spectrum of the radiation field are derived. The results are discussed in terms of Wolf's work [Phys. Rev. Lett. 56, 1370 (1986)] on this effect, which can be considered in terms of a multiple-atom analog of the effects of radiation reaction on a single atom, i.e., spontaneous decay and the Lamb shift.
Not long ago Wolf discovered a modification of the spectrum of radiation due to the spatial coherence of the source.1 His analysis used a model correlation function that demonstrated the effect, without discussing any physical mechanism by which the correlations may arise. This paper presents an analysis of one such physical mechanism, namely, the radiation reaction. Following earlier work by Varada and Agarwal,2 an explicit calculation, performed in the harmonic oscillator approximation, for two radiating atoms in equilibrium with a thermal field reveals a relationship between this effect and the well-known shift of atomic oscillator frequencies due to Lamb.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms
A1,
A2, whose separation
r21 can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only
A1 inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators
O1,
O2. The atoms exhibit superradiant behavior, whereas
O1,
O2 tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that
A2 and
A2 radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
Emil Wolf discovered how the spatial coherence characteristics of the source affect the spectrum of the radiation in the far zone. In particular, the spatial coherence of the source can result either in red or blue shifts in the measured spectrum. His predictions have been verified in a large number of different classes of systems. Wolf and coworkers usually assume a given form of source correlations and study its consequence. In this paper we consider the microscopic origin of spatial coherence and radiation from a system of atoms. We discuss how the radiation is different from that produced from an independent system of atoms. We show that the process of radiation itself is responsible for the creation of spatial correlations within the source. We present different features of the spectrum and other statistical properties of the radiation, which show strong dependence on the spatial correlations. We show the existence of a new type of two-photon resonance that arises as a result of such spatial correlations. We further show how the spatial coherence of the field can be used in the context of radiation generated by nonlinear optical processes.We conclude by demonstrating the universality ofWolf shifts and its application in the context of pulse propagation in a dispersive medium.
Wolf discovered how the spatial coherence characteristics of the source affect the spectrum of the radiation in the far zone. In particular the spatial coherence of the source can result either in red or blue shifts in the measured spectrum.His predictions have been verified in a large number of different classes of systems. Wolf and coworkers usually assume a given form of source correlations and study its consequence. In this paper we consider microscopic origin of spatial coherence and radiation from a system of atoms. We discuss how the radiation is different from that produced from an independent system of atoms. We show that the process of radiation itself is responsible for the creation of spatial correlations within the source. We present different features of the spectrum and other statistical properties of the radiation, which show strong dependence on the spatial correlations. We show the existence of a new type of two-photon resonance that arises as a result of such spatial correlations. We further show how the spatial coherence of the field can be used in the context of radiation generated by nonlinear optical processes. We conclude by demonstrating the universality of Wolf shifts and its application in the context of pulse propagation in a dispersive medium.
Interaction with a multimode quantized radiation field of an assembly of
N two-level atoms contained in a small volume is considered for the special cases of (i) highly excited, (ii) weakly excited, and (iii) superradiant atomic assemblies, and solutions of equations of motion are obtained in closed form. For (i) and (ii) we show that, if the atomic assembly is in the Dicke state
∣r,m>, the spontaneously emitted radiation is Lorentzian, and its spectral width and the numerical value of the Lamb shift are
r+∣m∣ times that for a single excited atom. For (i) the sign of Lamb shift is opposite to that for (ii) and for a single excited atom. Change in the frequency shift gives chirping in the pulse radiation emitted spontaneously by an assembly of atoms. We obtain the reduced density operator for a single mode and for the complete spontaneous emitted radiation, and discuss its characteristic function, weight function in diagonal representation, coherence functions, and normalized variance. We find that the emitted radiation is chaotic if all atoms are excited or if the assembly is weakly excited and is in thermal equilibrium. For a superradiant assembly, the radiation is coherent; it is only amplitude coherent if the initial atomic density operator is nondiagonal. We also study stimulated emission and scattering for the cases (i) and (ii). For case (iii) these phenomena do not take place.
The problem of the dynamics and the spectrum of spontaneous radiation is solved for a system of two atoms in one-dimensional space. In order to single out, to the maximum possible degree, phenomena associated with the influence of spatially separated atoms on each other via the radiation field, the present analysis is performed precisely for the one-dimensional case. As a result, two effects are revealed and considered in detail: (i) the existence of stable (metastable) entangled superposition states at specific distances between the atoms and (ii) a considerable distinction between the spectra of photons emitted in two opposite directions from the system where only one of the atoms is initially excited. The possibilities of observing these effects are discussed.