Avalanches of phonons are generated following population inversion of the Zeeman-split
E(2E) Kramers doublet in a single crystal of dilute ruby
(Al2O3:Cr3+) at
1.4K. The inversion is achieved in a narrow zone of limited extent by selective pulsed optical pumping, and the phonons are detected via luminescence. The associated acoustic wave emanating from the zone propagates through the crystal, reverberates in the acoustic cavity formed by the crystalline end faces, and engages in further amplification each time it passes through the zone until the inversion becomes exhausted. The systems thus acts as a saser (sound amplification by stimulated emission of radiation). The avalanche is adequately described by coherent Bloch equations governing the acoustic wave and the interacting spin polarization, the losses being due to surface scattering and divergence. This conclusion holds good upon reducing the resonance length of the zone by a gradient magnetic field. Finally, the frequency and angular distributions of the avalanche are measured by using two disjunct zones located at opposite ends of the crystal, one serving as generator and the other as detector, in conjunction with a gradient field and lateral displacement. The frequency spread is found to be compatible with the inhomogeneously broadened transition within
E(2E), while the angular divergence is determined by the geometry of the zone.
Selective pulsed optical excitation is used in 500-at.ppm ruby (Al 2 O 3 :Cr 3+ ) at 1.4 K to prepare complete population inversion of the Zeeman-split Ē( 2 E ) doublet in a zone of limited size. The inversion results in prolific stimulated emission of phonons resonant with the one-phonon transition connecting the doublet states. The phonons are detected via the R 1 luminescence. The angular and spectral distributions of the associated acoustic wave are measured using a geometry with inverted zones at either end of the crystal, one serving as generator and the other as detector. The divergence appears to be governed by the geometry of the zone, while the spectral distribution is, within errors, in keeping with the inhomogeneously broadened phonon transition.
Superfluorescence and amplified spontaneous emission of 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons are observed following inversion of the E\ifmmode\bar\else\textasciimacron\fi{}
(2E)-2A\ifmmode\bar\else\textasciimacron\fi{}
(2E) acoustic transition of
Cr3+ in ruby. The conditions for both of these manifestations of phonon avalanches are selected via the
Cr3+ concentration. Superfluorescence is described in terms of a pendulum equation of the acoustic Bloch vector. Rate equations of the level and phonon populations account for amplified spontaneous emission.
The population ratio $\frac{{N}_{\ensuremath{-}}}{{N}_{+}}$ of the Zeeman states of
E(2E) has been measured over five decades of metastable-state concentrations ${N}_{\ensuremath{-}}$ in 2500-ppm ruby at 28 kG and 1.5 K, i.e., in the absence of thermalization between ${\overline{E}}_{\ifmmode\pm\else\textpm\fi{}}$. Upon raising the pumping, $\frac{{N}_{\ensuremath{-}}}{{N}_{+}}$ falls below the value corresponding to optical feeding only due to optically induced two-phonon processes via
2A(2E). Rate equations account for the findings under the assumption of marginal (ballistic regime) or moderate (spatially diffusive regime) trapping of these 29-${\mathrm{cm}}^{\ensuremath{-}1}$ phonons. At high pumping, $\frac{{N}_{\ensuremath{-}}}{{N}_{+}}$ further drops to unity, reflecting disturbance of the Boltzmann equilibrium within the
4A2 ground state.
The relaxation of resonant 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons trapped in a laser-excited cylinder of submillimeter diameter is investigated under the conditions of modulated pumping in ruby of 130, 700, and 2500 at. ppm
Cr3+ at 1.5 K. The decay time
Teff of the 2\ifmmode \bar{A}\else \={A}\fi{}
(2E) population, which is a measure of the phonon relaxation time, appears to depend on the metastable population
N∗ and the cylinder diameter L as the product
N∗L only, establishing boundary-limited phonon decay. At
N∗L below 5\ifmmode\times\else\texttimes\fi{}
1015 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$, pure spatial diffusion prevails.At higher
N∗L, where the measured
Teff levels off at about 1 \ensuremath{\mu}s, dependent on the ground-state concentration, the phonon relaxation is adequately accounted for by the assumption of enhancement by mediation of frequency-shifting Orbach processes at metastable
Cr3+ subject to diagonal exchange with ground-state
Cr3+. Shifts away from resonance to the extent that the phonons escape the zone by ballistic flight are considered active, which predominantly involve distant
Cr3+ neighbors.Furthermore, a random distribution of
Cr3+ is assumed, while the exchange is calibrated with reference to pair spectra. The spin-nonflip transitions of metastable
Cr3+ that are part of the pairs remain at resonance with the phonons because the exchange splittings of \ifmmode \bar{E}\else \={E}\fi{}
(2E) and 2\ifmmode \bar{A}\else \={A}\fi{}
(2E) are comparable. This is borne out by a separate calculation based on a perturbative molecular field. In addition, experiments have been performed as a function of an external magnetic field up to 4 T parallel to the c axis to disclose the effects of successive separation of the spin-flip and spin-nonflip transitions. The resultant complex development of
Teff with field as well as similar data on the
R2 luminescent intensity taken with continuous optical pumping are consistent with exchange-invoked relaxation and, further, yield effective spectral widths of the 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ transitions of the weakly coupled metastable
Cr3+ of order 1 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.