A theory of phase-matched cascade correlated emission is developed based on a source field approach. An ensemble of three-level atoms is prepared in a phased state for which the probability to have two excitations is negligibly small. The field intensity radiated on each of the transitions is calculated. The radiation on the upper transition is isotropic and unpolarized, but the phase-matched component on the lower transition can be directional and polarized. Moreover, for sufficiently high optical densities, the emission on the lower transition can be superradiant. In addition to the field intensities, the joint probability for emission on both transitions is calculated, exhibiting correlations in both the directions of emission and polarization of the fields.
We present a theory of cascade emission from an array of
N fixed, ``three-level'' atoms. The total angular momentum of the ground state of each atom is
J=0, that of the intermediate state is
J=1, and that of the upper state is
J=0. The atoms are prepared in a spatially phased superposition of their ground and upper states, collectively sharing a single excitation. We calculate the time-integrated, joint probability distribution for radiation to be emitted in a given direction with a given polarization on each transition and use this joint probability distribution to calculate the time-integrated intensity emitted on each transition. The dipole-dipole interaction between the atoms is taken into account. Analytic expressions are obtained for two atoms and the calculation is then extended formally to an ensemble of
N atoms at arbitrary positions. As expected, the radiation emitted on the upper transition is unpolarized and isotropic. However, somewhat surprisingly, we find that for excitation by counterpropagating fields and for nearly equal ground to intermediate and intermediate to upper level transition frequencies, the radiation on the lower transition is also isotropic and unpolarized, provided the atoms are prepared in a fully symmetric state. We also establish conditions for which there can be a phase-matched enhancement of the joint probability density for specific directions of emissions on the upper and lower transitions.
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.
Expressions are derived for the intensity correlations or degrees of second-order coherence within and between two light beams of different frequency generated by cascade emission from a three-level atom. The atoms are assumed to be excited by two monochromatic coherent light beams in resonance with the upper and lower pairs of levels that constitute the triplet. The two emitted beams individually show photon antibunching and the correlation between the beams shows another complementary kind of non-classical effect. It is suggested that measurements on the system described here would provide more detailed information on quantum effects in cascade emission than previous observations with direct excitation to the uppermost atomic level by electron bombardment.
Scully and coworkers [M. O. Scully, E. S. Fry, C. H. R. Oii, and K. W\'odkiewicz, Phys. Rev. Lett. 96, 010501 (2006)] demonstrated that there is directional, phase-matched emission following the excitation of an ensemble of atoms by a single-photon pulse. While the phase-matched emission intensity is proportional to the the number of atoms, for optically thin samples the total energy emitted in the phase-matched direction is much less than that radiated in other directions. Moreover, even for optically thin samples, it is necessary to take into account effects related to cooperative decay if energy is to be conserved in the overall emission process. An analytic calculation is presented to show explicitly how cooperative decay reduces the incoherent emission and restores energy conservation in this low-density limit.
The superradiant emission properties from an atomic ensemble with cascade level configuration is numerically simulated. The correlated spontaneous emissions (signal then idler fields) are purely stochastic processes which are initiated by quantum fluctuations. We utilize the positive-
P phase-space method to investigate the dynamics of the atoms and counterpropagating emissions. The light field intensities are calculated, and the signal-idler correlation function is studied for different optical depths of the atomic ensemble. A shorter correlation time scale for a denser atomic ensemble implies a broader spectral window needed to store or retrieve the idler pulse.