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Physical review. B, Solid state

Coherent radiation from thin films

Y. C. Lee, P. S. Lee

Published Jul 15, 1974• 42 citations
Publisher

Abstract

The spontaneous coherent radiation from thin crystal films of $n$ layers is studied. By treating each layer as a unit and by employing the method of multiple-time-scale expansion, exact solutions for such systems, with no restriction on the value $\frac{{\ensuremath{\lambda}}_{0}}{d}$, are found. Here ${\ensuremath{\lambda}}_{0}$ is the characteristic radiation wavelength and $d$ is the average interparticle spacing. In the limit $\frac{{\ensuremath{\lambda}}_{0}}{d}\ensuremath{\gg}1$, it is seen that there exist correlated states of the film which are exact analog of Dicke's superradiant states. The decay rate for the superradiant states is roughly enhanced by a factor of $n{(\frac{{\ensuremath{\lambda}}_{0}}{d})}^{2}$ compared to the single-atom decay rate ${\ensuremath{\gamma}}_{s}$.

DOI
10.1103/physrevb.10.344

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The spontaneous coherent radiation from thin crystal films of nnn layers is studied. By treating each layer as a unit and by employing the method of multiple-time-scale expansion, exact solutions for such systems, with no restriction on the value \ensuremathλ0d\frac{{\ensuremath{\lambda}}_{0}}{d}d\ensuremathλ0​​, are found. Here \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ is the characteristic radiation wavelength and ddd is the average interparticle spacing. In the limit \ensuremathλ0d\ensuremath≫1\frac{{\ensuremath{\lambda}}_{0}}{d}\ensuremath{\gg}1d\ensuremathλ0​​\ensuremath≫1, it is seen that there exist correlated states of the film which are exact analog of Dicke's superradiant states. The decay rate for the superradiant states is roughly enhanced by a factor of n(\ensuremathλ0d)2n{(\frac{{\ensuremath{\lambda}}_{0}}{d})}^{2}n(d\ensuremathλ0​​)2 compared to the single-atom decay rate \ensuremathγs{\ensuremath{\gamma}}_{s}\ensuremathγs​.
The theory of superradiance of excitons in thin crystal films is presented. While the broken crystal symmetry in the direction normal to the film plane renders the radiative decay of the exciton possible, the remaining crystal symmetry on the film leads to phase coherence between the exciton and the photon fields everywhere on the film. Consequently, a trading between the degrees of freedom of the radiation field and those of the atoms is shown to be responsible for the predicted superradiative enhancement of the exciton decay by a factor ( lambda /d) 2 >>1 in the optical region.
The radiative frequency shift of an exciton in a thin semiconductor film, like its radiative level width, is shown to be superradiatively enhanced. Unlike the latter, however, a finite frequency shift can only be obtained after proper renormalization for the correlated system. The shift is found to be inversely proportional to the square of the factor k0{\mathit{k}}_{0}k0​d and proportional to the film thickness T, k0{\mathit{k}}_{0}k0​=Eqn{\mathit{E}}_{\mathbf{q}\mathit{n}}Eqn​/\ensuremath{\Elzxh}c, Eqn{\mathit{E}}_{\mathbf{q}\mathit{n}}Eqn​ being the exciton energy gap and d the lattice constant of the semiconductor. Therefore, the coherent frequency shift can be observed experimentally if one varies the thickness, or the exciton energy gap Eqn{\mathit{E}}_{\mathbf{q}\mathit{n}}Eqn​ by imposing high pressure.
The basic features of coherent spontaneous emission, including its dependence on the square of the number of participating emitters, are reviewed. Analogies with magnetic resonance phenomena are exploited, and we explain the need to go beyond such analogies in order to treat optical emission. Some recent results in the theory of optical superradiance are discussed. The radiative decay process is treated as a quantum-classical interpolation problem, and explicit solutions are given. In addition to its time dependence, we give the angular distribution of radiated power and the expected intensity fluctuations, and we estimate the maximum number of emitters which can cooperate in a superradiant event. Dicke's “coherence brightening” criterion is discussed from the point of view of the present paper. A “failure” of the criterion is exhibited and explained. Two of the purely quantum aspects of superradiance are isolated and discussed briefly.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1{A}_{1}A1​, A2{A}_{2}A2​, whose separation r21{r}_{21}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{A}_{1}A1​ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1{O}_{1}O1​, O2{O}_{2}O2​. The atoms exhibit superradiant behavior, whereas O1{O}_{1}O1​, O2{O}_{2}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{A}_{2}A2​ and A2{A}_{2}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.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
Fully excited two-level atoms separated by less than the transition wavelength cooperatively emit light in a short burst, a phenomenon called superradiance by R. Dicke in 1954 [Phys. Rev. 93, 99 (1954)]. The burst is characterized by a maximum intensity scaling with the square of the number of atoms NNN and a temporal width reduced by NNN compared to the single atom spontaneous decay time. Both effects are usually attributed to a synchronization of the electric dipole moments of the atoms occurring during the process of light emission. Contrary to this explanation, it was recently shown by use of a quantum path description that the peak intensity results from the quantum correlations among the atoms when occupying symmetric Dicke states. Here we investigate from this perspective the temporal evolution of the ensemble, starting in the small sample limit, i.e., when the atoms have mutual separations much smaller than the transition wavelength $\ensuremath{\lambda}$ and pass down the ladder of symmetric Dicke states. In addition, we explore the temporal evolution for the case of distant noninteracting atoms with mutual separations much larger than $\ensuremath{\lambda}$. We show that in this case a similar superradiant burst of the emitted radiation is observed if the quantum correlations of the atoms are generated by conditional photon measurements retaining the atomic ensemble within or close to the symmetric subspace.