A new method, based on fluorescence line narrowing, is presented for the detection of near-resonant trapped phonons with an extremely high spectral resolution. The method is applied to 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons resonant with the inhomogeneous E\ifmmode\bar\else\textasciimacron\fi{}
(2E)-2A\ifmmode\bar\else\textasciimacron\fi{}
(2E) transition of
Cr3+ in ruby (
Al2O3:
Cr3+) following injection of a flat spectrum of nonequilibrium phonons into the crystal. Spectral transport within the trapped phonon packet is observed directly from the shape of the phonon spectrum around E\ifmmode\bar\else\textasciimacron\fi{}
(2E)-2A\ifmmode\bar\else\textasciimacron\fi{}
(2E) as it develops with time. An account is first given in terms of phenomenological rate equations with inclusion of the variations of the spin-phonon coupling through the transition as well as spectral transport. The dynamics of the 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons in frequency and space is understood more quantitatively from Monte Carlo simulations following the life history of the phonons. Generally good agreement is found with an earlier model based on weakly exchange-coupled
Cr3+ pairs, in which the phonon frequency, and as a result the mean free path, are modified by one-site resonant Raman processes, and diffusive motion is terminated by ballistic escape out of the hot zone.
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 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.
We introduce a technique utilizing the evanescent field of an optical waveguide to examine the dynamics of nonequilibrium high-frequency phonons resonant with optically excited centers in zones of submicrometer dimension. The technique is used to create a nearly saturated zone of excited
Cr3+ ions at the surface of a 700-at. ppm ruby crystal at 1.5 K by optical pumping through the broad bands. At the same time, the optical pumping serves to generate resonant 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons. The decay of the combined spin-phonon system, measured via the
R2 luminescence following interruption of the optical pumping, is explained by elastic spatial diffusion in parallel with spectral wipeout. The time constant for loss by spatial diffusion is derived rigorously from rate equations. Spectral wipeout is treated according to the model developed by Goossens et al. based on one-site inelastic scattering by weakly exchange-coupled
Cr3+ pairs, but modified to include the appreciable elastic diffusion remaining after wipeout under the present extreme conditions. Spectral wipeout turns out to predominate at high excited-state concentration. The optical feeding into 2A\ifmmode\bar\else\textasciimacron\fi{}
(2E) is found to result mainly from excited-state absorption starting from E\ifmmode\bar\else\textasciimacron\fi{}
(2E), despite the low probability of the latter at the light wavelength used (514.5 nm).
Spatial and, additionally, spectral diffusion of 29-${\mathrm{cm}}^{\ensuremath{-}1}$ phonons trapped in an excited cylindrical region with diameter \ensuremath{\sim}0.1 mm has been investigated under continuous optical pumping of ruby. For 130- and 700-ppm ruby, phonon decay strictly limited by spatial diffusion is reflected in a quadratic dependence of the bottlenecking factor $\ensuremath{\sigma}$ on the excited-state concentration
N∗ up to ${N}^{*}\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}{10}^{17}/{\mathrm{cm}}^{3}$. At higher
N∗, attainable in 700- and 2500-ppm ruby, the observed reduction of $\ensuremath{\sigma}$ relative to the second-power dependence, as well as the increased width of the dependence of $\ensuremath{\sigma}$ on magnetic field, reveal that spectral diffusion from inhomogeneous packet to packet is operative. In the spatially diffusive regime, evidence for Zeeman-induced spectral diffusion is found in a characteristic narrowing of the magnetic-field dependence of the
R2 fluorescence upon raising $\ensuremath{\sigma}$. At full separation of the Zeeman transitions $2\overline{A}(^{2}E)\ensuremath{\leftrightarrow}\overline{E}(^{2}E)$, $\ensuremath{\sigma}$ is reduced to 15-40% relative to zero field, depending on
N∗ and ground-state concentration. Rate equations governing the
Cr3+ populations in the two-level scheme (
2A,
E) with inclusion of spatial diffusion and anharmonic decay of the phonons account for the second-power dependence. Diffusive phonon decay turns out to dominate from ${N}^{*}R\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{13}/{\mathrm{cm}}^{2}$ up to at least ${N}^{*}{R}^{2}\ensuremath{\sim}{10}^{14}/\mathrm{cm}$, with
R the characteristic dimension of the excited region. Spectral diffusion, treated as a perturbation, satisfactorily fits the data at higher $\ensuremath{\sigma}$, yielding a diffusive step of \ensuremath{\sim}2% of the linewidth. At low bottlenecking, the functional form of the magnetic field dependence of $\ensuremath{\sigma}$, including its width and level at full separation, is closely described by use of rate equations, taking the line shape to be Gaussian-type, and assuming the phonons associated with spin flip and those with non spin flip to belong to distinct classes.
Direct relaxation within the optically excited
E(2E) level of 700-ppm ruby is investigated in magnetic fields up to 65 kG making an angle to the
c axis. At low pump intensities, the experimental relaxation rates are found to be in excellent agreement with the theory by Blume et al., predicting a fifth-power dependence on field and a ${cos}^{3}\ensuremath{\theta}{sin}^{2}\ensuremath{\theta}$ dependence on orientation. The observed proportionality constant of 4.7 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}19}$ ${\mathrm{G}}^{\ensuremath{-}5}$${\mathrm{s}}^{\ensuremath{-}1}$ is close to the theoretical estimate based on static experiments in ruby. When increasing optical pumping, the phonon-bath conduction is demonstrated to become the limiting process due to resonant trapping of the phonons. Following pulsed excitation, a prolonging of the decay time by a factor up to 13 is attained at excited-state concentrations ${N}^{*}\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}{10}^{17}/{\mathrm{cm}}^{3}$. In stationary experiments, a nonlinear increase with
N∗ is observed of the equilibrium population of the upper Zeeman level of
E to over a decade above the nonbottlenecked value, corresponding to bottlenecking factors of over \ensuremath{\sim}200. Under the assumption of a constant phonon lifetime of the order of the flight time through the illuminated region, rate equations appear to consistently account for the experiments. A Gaussian-type resonance line, fully homogenized supposedly by magnetic interactions with the surrounding Al nuclei and
Cr3+ ions, explains the lattice-bath equilibration. Finally, the prospect is discussed of using the
E direct transition in ruby as a tunable phonon generator and detector with bandwidth 0.002 ${\mathrm{cm}}^{\ensuremath{-}1}$ in the practical frequency range of 2-15 ${\mathrm{cm}}^{\ensuremath{-}1}$.
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.