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 ${\mathrm{cm}}^{\ensuremath{-}1}$.
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.
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 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.
Phonon avalanches resonant with the optically inverted Zeeman-split $\ifmmode \bar{E}\else \={E}\fi{}{(}^{2}E)$ doublet of
Cr3+ have been measured and analyzed in a single crystal of 500-at. ppm ruby
(Al2O3:Cr3+) with a geometry adapted to the study of coherence. A set of coherent Bloch equations governing the interacting acoustic wave and spin polarizations is found to provide an excellent description of the results, and to be far superior to incoherent rate equations for the phonon and level populations. The dephasing time conforms with the width of the inhomogeneously broadened transition connecting the $\ifmmode \bar{E}\else \={E}\fi{}{(}^{2}E)$ states, which indicates that dephasing primarily occurs by the spread in frequencies.
The magnetization of ruby in a magnetic field at 300\ifmmode^\circ\else\textdegree\fi{}K is changed when the ruby is optically pumped with linearly polarized radiation from a
Q-switched ruby laser. With a magnetic field parallel to the trigonal axis, laser light polarized parallel to this axis induces transitions from the spin \ifmmode\pm\else\textpm\fi{}\textonehalf{} levels of the
4A2 ground state to the levels of the
E(2E) excited state. The change in
Mz is linearly proportional to
H except near the anticrossing points of the
4A2 spin levels at 2.07 and 4.14 kG. At these field strengths, there is an enhancement of the magnetization caused by state mixing of the
4A2 wave functions. When
H has a small component perpendicular to the
z axis, a magnetization is detected parallel to this component in the vicinity of 4.14 kG. The effect requires a long ground-state relaxation time. In a separate experiment, the relaxation of
Mz was found to vary from 0.13 \ensuremath{\mu}sec in zero magnetic field to a constant value of 0.57 \ensuremath{\mu}sec for fields above 60 G.
A local frequency-selective depletion of phonons is created in dilute ruby at 1.8 K by optical pumping into the Zeeman-split metastable \ifmmode \bar{E}\else \={E}\fi{}
(2E) doublet. This phonon ``void,'' which is in dynamic balance with the spin system, is observed to recover to thermal equilibrium by an inward flow of phonons supplied from regions beyond the void, and ultimately from the helium bath. Coupled rate equations of the level populations in the void as well as the phonon density throughout the crystal account for these effects. In particular, a quartic frequency dependence of the rate for phonon transport across the crystal boundary is found, in accordance with diffuse surface scattering.