We represent two groups of excited atoms which radiate at slightly different frequencies by means of a pair of inverted harmonic oscillators coupled to the radiation field. The radiation emitted spontaneously by these oscillators is amplified exponentially and the field generated by each exerts a strong dynamical influence on the other. For the regime in which the amplification rate is large compared to the frequency difference of the uncoupled oscillators, the indirect coupling via the field tends to lock the two systems in phase so that they radiate in unison. When the amplification rate is small compared to the frequency difference, however, each oscillator undergoes, in addition to its own spontaneous oscillation, a partially coherent oscillation forced at the frequency of the other oscillator. These mutually induced oscillations give rise to intensity beats in the radiated field. Such beats have predetermined phases which are independent of the random initial phases of the oscillator amplitudes. The beats can be present and detectable in statistical terms even before a single photon has been emitted on the average.
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.
This paer is concerned with the spontaneous-emision rate of two two-level atoms which have different resonance frequencies and whose spatial separation is much smaller than the corresponding mean resonance wavelength. The total radiation rate of such an atom pair is calculated completely quantum mechanically as a function of time. Explicit results are given for various values of the frequeny difference and the following initial states of the atomic system: (a) both atoms in the excited state, (b) one taom in the excited state the other in the ground state, (c) a symmetric combination of those states, (d) an antisymmetric combination of those states, and product states where each atom is in an equal superposition of its ground and excited states, for which the expectation values of the dipole moments are nonzero and may be considered to be osicillating (e) in phase, (f) in opposite phae, and (g) in random phase. Oscillations are found to appear in the radiation rate in all cases when the frequency difference becomes larger than a certain critical value, viz., the mean half-width of the atomic resonance lines. The oscillations are especially pronounced for the initial atomic states (c), (d), (e), and (f), which exhibit superradiance and radiation trapping. For example, for atoms in the antisymmetric state (d), for which the radiation from two identical atoms is zero (completely self-trapped), a pair of nonidentical atoms begins radiating and even oscillates into the superradiant region at times.
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.
Photon statistics are obtained from exact closed-form solutions for both spontaneous and stimulated coherent emissions by a large number of two-level atoms in a Dicke state. Depending on the initial excitation level of the atoms, three types of photon statistics are obtained for the emitted radiation field. Moments of the photon number of the emitted field are calculated in the case of initial atomic coherent states as well as Dicke states, and in the case of n-photon stimulating pulses as well as coherent stimulating pulses. In the particular case of an initial "superradiant" Dicke state stimulated by a coherent-radiation field, the emitted radiation pulse is also fully coherent. In other cases the photon distributions undergo large fluctuations.
Even with incoherent driving, coherence effects may show up in the spontaneous emissions of an atom. We derive a general master equation and predict that for both the three-level V and \ensuremath{\Lambda} systems the intensity correlation function
g(2)(\ensuremath{\tau}) may exhibit pronounced oscillations with frequency \ensuremath{\delta}\ensuremath{\omega} if the level separation \ensuremath{\Elzxh}\ensuremath{\delta}\ensuremath{\omega} is small. We obtain a marked difference in the beat amplitude for a single \ensuremath{\Lambda} system as compared to a beam or gas. If \ensuremath{\delta}\ensuremath{\omega} is much smaller than the natural linewidth we find dark and light periods in the spontaneous emission.
The two-time second-order correlation function of the electric field spontaneously emitted by a system of two non-identical two-level atoms with different transition frequencies and different natural linewidths is calculated from the viewpoint of quantum beats and photon anticorrelation effects. Our analytical solution shows that the intensity correlation exhibits quantum beats. The quantum beats are shown to appear even if the two atoms are independent. For identical atoms, beats appear only if dipole–dipole interaction between the atoms is included, and they are then strongly dependent on the direction of observation with respect to the line connecting the two atoms. The anticorrelation effect between the photons emitted in different directions is also discussed. It is shown that for a single-time correlation function the condition for anticorrelation is independent of interatomic interactions. For a two-time correlation function, interatomic interactions reduce the anticorrelation effect.