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Physical review. B, Condensed matter

Relaxation of 29- cm−1 phonons in ruby

Ronald Goossens, J. I. Dijkhuis, H. W. de Wijn

Published Dec 1, 1985• 22 citations
Publisher

Abstract

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 ${\mathrm{Cr}}^{3+}$ at 1.5 K. The decay time ${T}_{\mathrm{eff}}$ of the 2\ifmmode \bar{A}\else \={A}\fi{}${(\mathrm{}}^{2}$E) population, which is a measure of the phonon relaxation time, appears to depend on the metastable population ${N}^{\mathrm{*}}$ and the cylinder diameter L as the product ${N}^{\mathrm{*}}$L only, establishing boundary-limited phonon decay. At ${N}^{\mathrm{*}}$L below 5\ifmmode\times\else\texttimes\fi{}${10}^{15}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$, pure spatial diffusion prevails.At higher ${N}^{\mathrm{*}}$L, where the measured ${T}_{\mathrm{eff}}$ 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 ${\mathrm{Cr}}^{3+}$ subject to diagonal exchange with ground-state ${\mathrm{Cr}}^{3+}$. Shifts away from resonance to the extent that the phonons escape the zone by ballistic flight are considered active, which predominantly involve distant ${\mathrm{Cr}}^{3+}$ neighbors.Furthermore, a random distribution of ${\mathrm{Cr}}^{3+}$ is assumed, while the exchange is calibrated with reference to pair spectra. The spin-nonflip transitions of metastable ${\mathrm{Cr}}^{3+}$ that are part of the pairs remain at resonance with the phonons because the exchange splittings of \ifmmode \bar{E}\else \={E}\fi{}${(\mathrm{}}^{2}$E) and 2\ifmmode \bar{A}\else \={A}\fi{}${(\mathrm{}}^{2}$E) 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 ${T}_{\mathrm{eff}}$ with field as well as similar data on the ${R}_{2}$ 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 ${\mathrm{Cr}}^{3+}$ of order 1 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.

DOI
10.1103/physrevb.32.7065

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Diffusion of bottlenecked 29- cm−1 phonons in optically excited ruby

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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 ${R}_{2}$ 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 ${\mathrm{Cr}}^{3+}$ populations in the two-level scheme ($2\overline{A}$,$\overline{E}$) 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.

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Direct relaxation within the optically excited $\overline{E}(^{2}E)$ 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 $\overline{E}$ 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 ${\mathrm{Cr}}^{3+}$ ions, explains the lattice-bath equilibration. Finally, the prospect is discussed of using the $\overline{E}$ 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 ${\mathrm{cm}}^{\ensuremath{-}1}$.

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Trapping of 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons by optically excited ${\mathrm{Cr}}^{3+}$ ions in ruby is studied at phonon occupations of up to order unity. The 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons are supplied by transient excitation of a metallic point contact and subsequent conversion of the injected phonons to resonance. The growth and the decay of the trapped phonon population are markedly non-single-exponential quantities, particularly at higher phonon populations. A quantitative analysis of these effects is made on the basis of phenomenological rate equations of the ion and phonon populations. These equations make explicit allowance for the line shape of the E\ifmmode\bar\else\textasciimacron\fi{}${(}^{2}$E)-2A\ifmmode\bar\else\textasciimacron\fi{}${(}^{2}$E) phonon transition as well as for feeding and loss of resonant phonons by mediation of weakly exchange-coupled ${\mathrm{Cr}}^{3+}$ pairs. At the higher phonon occupations, the transition appears to be dynamically broadened by stimulated phonon emission, which reduces the trapping efficiency.

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J. I. Dijkhuis, H. W. de Wijn

The population ratio $\frac{{N}_{\ensuremath{-}}}{{N}_{+}}$ of the Zeeman states of $\overline{E}(^{2}E)$ 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 $2\overline{A}(^{2}E)$. 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 $^{4}A_{2}$ ground state.

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Spectral transport of nonequilibrium 29- cm−1 phonons in ruby studied by fluorescence line narrowing

M.J. van Dort, J. I. Dijkhuis, H. W. de Wijn

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{}${(}^{2}$E)-2A\ifmmode\bar\else\textasciimacron\fi{}${(}^{2}$E) transition of ${\mathrm{Cr}}^{3+}$ in ruby (${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$:${\mathrm{Cr}}^{3+}$) following injection of a flat spectrum of nonequilibrium phonons into the crystal. Spectral transport within the trapped phonon packet is observed directly from the shape of the phonon spectrum around E\ifmmode\bar\else\textasciimacron\fi{}${(}^{2}$E)-2A\ifmmode\bar\else\textasciimacron\fi{}${(}^{2}$E) as it develops with time. An account is first given in terms of phenomenological rate equations with inclusion of the variations of the spin-phonon coupling through the transition as well as spectral transport. The dynamics of the 29-${\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 ${\mathrm{Cr}}^{3+}$ 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.

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Relaxation of an optically created phonon void in dilute ruby

Paul A. van Walree, A. F. M. Arts, H. W. de Wijn

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{}${(}^{2}$E) 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.

Physical review. B, Condensed matter•Oct 3, 2001•1 citation
The relaxation of resonant 29-cm\ensuremath−1{\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}cm\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+{\mathrm{Cr}}^{3+}Cr3+ at 1.5 K. The decay time Teff{T}_{\mathrm{eff}}Teff​ of the 2\ifmmode \bar{A}\else ={A}\fi{}(2{(\mathrm{}}^{2}(2E) population, which is a measure of the phonon relaxation time, appears to depend on the metastable population N∗{N}^{\mathrm{*}}N∗ and the cylinder diameter L as the product N∗{N}^{\mathrm{*}}N∗L only, establishing boundary-limited phonon decay. At N∗{N}^{\mathrm{*}}N∗L below 5\ifmmode\times\else\texttimes\fi{}1015{10}^{15}1015 cm\ensuremath−2{\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}cm\ensuremath−2, pure spatial diffusion prevails.At higher N∗{N}^{\mathrm{*}}N∗L, where the measured Teff{T}_{\mathrm{eff}}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+{\mathrm{Cr}}^{3+}Cr3+ subject to diagonal exchange with ground-state Cr3+{\mathrm{Cr}}^{3+}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+{\mathrm{Cr}}^{3+}Cr3+ neighbors.Furthermore, a random distribution of Cr3+{\mathrm{Cr}}^{3+}Cr3+ is assumed, while the exchange is calibrated with reference to pair spectra. The spin-nonflip transitions of metastable Cr3+{\mathrm{Cr}}^{3+}Cr3+ that are part of the pairs remain at resonance with the phonons because the exchange splittings of \ifmmode \bar{E}\else ={E}\fi{}(2{(\mathrm{}}^{2}(2E) and 2\ifmmode \bar{A}\else ={A}\fi{}(2{(\mathrm{}}^{2}(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{T}_{\mathrm{eff}}Teff​ with field as well as similar data on the R2{R}_{2}R2​ luminescent intensity taken with continuous optical pumping are consistent with exchange-invoked relaxation and, further, yield effective spectral widths of the 29-cm\ensuremath−1{\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}cm\ensuremath−1 transitions of the weakly coupled metastable Cr3+{\mathrm{Cr}}^{3+}Cr3+ of order 1 cm\ensuremath−1{\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}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∗{N}^{*}N∗ up to ${N}^{*}\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}{10}^{17}/{\mathrm{cm}}^{3}$. At higher N∗{N}^{*}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{R}_{2}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∗{N}^{*}N∗ and ground-state concentration. Rate equations governing the Cr3+{\mathrm{Cr}}^{3+}Cr3+ populations in the two-level scheme (2A‾2\overline{A}2A,E‾\overline{E}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 RRR 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)\overline{E}(^{2}E)E(2E) level of 700-ppm ruby is investigated in magnetic fields up to 65 kG making an angle to the ccc 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∗{N}^{*}N∗ is observed of the equilibrium population of the upper Zeeman level of E‾\overline{E}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+{\mathrm{Cr}}^{3+}Cr3+ ions, explains the lattice-bath equilibration. Finally, the prospect is discussed of using the E‾\overline{E}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 ${\mathrm{cm}}^{\ensuremath{-}1}$.
Trapping of 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons by optically excited Cr3+{\mathrm{Cr}}^{3+}Cr3+ ions in ruby is studied at phonon occupations of up to order unity. The 29-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ phonons are supplied by transient excitation of a metallic point contact and subsequent conversion of the injected phonons to resonance. The growth and the decay of the trapped phonon population are markedly non-single-exponential quantities, particularly at higher phonon populations. A quantitative analysis of these effects is made on the basis of phenomenological rate equations of the ion and phonon populations. These equations make explicit allowance for the line shape of the E\ifmmode\bar\else\textasciimacron\fi{}(2{(}^{2}(2E)-2A\ifmmode\bar\else\textasciimacron\fi{}(2{(}^{2}(2E) phonon transition as well as for feeding and loss of resonant phonons by mediation of weakly exchange-coupled Cr3+{\mathrm{Cr}}^{3+}Cr3+ pairs. At the higher phonon occupations, the transition appears to be dynamically broadened by stimulated phonon emission, which reduces the trapping efficiency.
The population ratio $\frac{{N}_{\ensuremath{-}}}{{N}_{+}}$ of the Zeeman states of E‾(2E)\overline{E}(^{2}E)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)2\overline{A}(^{2}E)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^{4}A_{2}4A2​ ground state.
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{}(2{(}^{2}(2E)-2A\ifmmode\bar\else\textasciimacron\fi{}(2{(}^{2}(2E) transition of Cr3+{\mathrm{Cr}}^{3+}Cr3+ in ruby (Al2{\mathrm{Al}}_{2}Al2​O3{\mathrm{O}}_{3}O3​:Cr3+{\mathrm{Cr}}^{3+}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{}(2{(}^{2}(2E)-2A\ifmmode\bar\else\textasciimacron\fi{}(2{(}^{2}(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+{\mathrm{Cr}}^{3+}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.
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{}(2{(}^{2}(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.