Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
The earliest studies on the subject of spontaneous emission from an excited atom via interaction with the vacuum modes of the radiation field go back to the early days of quantum mechanics. In view of the fundamental nature of the vacuum fluctuations, it was long thought that spontaneous emission could not be controlled. However, in 1946 Purcell noted that the spontaneous emission rate of the atom inside a resonant cavity can be substantially enhanced (or suppressed) over its free-space value. This is due to the dramatic change of the density of modes of the vacuum field in a cavity. The enhancement and suppression of spontaneous emission in resonant and off-resonant cavities were observed experimentally in the 1980s. It was also realized in the 1950s and 1960s that coherent driving fields can dramatically change the characteristics of the spontaneous emission. The Autler-Townes doublet and Mollow three-peak spectrum are the examples of such behavior. A dramatic narrowing of the spectral peaks were discovered in driven multilevel atomic systems in the early 1990s. The role of atomic coherence in quenching spontaneous emission fluctuations in the context of laser physics was studied in the 1980s, via the correlated spontaneous emission laser (CEL). In this case atomic coherence results in the correlation of two spontaneous emission noise events so that the spontaneous emission quantum noise is eliminated in the relative linewidth. Later, it was “discovered” that the spontaneous emission can be suppressed through quantum interference between decay channels. Properly stated, this was rediscovered since it was first discovered and explained by Lamb in the 1940s. If the dipoles of two transitions from one upper level to two lower levels are parallel, the spontaneous emission in an ordinary vacuum can be cancelled. In a nonuniform vacuum, e.g., a cavity, the quenching of spontaneous emission can be achieved, even for the orthogonal dipoles.
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.
Coherent spontaneous emission from a system of
N two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number"
r is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\ensuremath{\sim}r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ensuremath{\ll}r$), the emitted radiation may be represented approximately by a coherent state.
We present a numerical study of the spatial coherence of light that is radiated from a dilute ensemble of atoms. The spatial coherence is established as a result of the collective (cooperative) coupling of the atoms to the light and is qualitatively different from the coherence of a laser. Specifically, the coherence in collective spontaneous emission does not rely on population inversion and stimulated emission, is governed by antiphasing of the dipoles (subradiance), and the key figure of merit for the observed coherence is $\ensuremath{\approx}N/(L/\ensuremath{\lambda})$. Here,
N is the number of atoms in the ensemble,
L is the size of the sample, and $\ensuremath{\lambda}$ is the wavelength of the emitted light.