1. INTRODUCTION

The phenomena of slow, stopped, and fast light in materials have attracted a lot of attention over the last three decades [1–5]. The terms slow and fast light are used for significantly altered group velocities ${v_g}$ of light pulses in a medium. The group velocity is defined by the propagation time of a light pulse through a medium and is related to the refractive index, described as follows:

In particular, large positive values of $\partial n/\partial \omega$ result in a significant reduction of the group velocity and negative values lead to superluminal (fast light); even negative group velocities may be observed. For the generation of slow light a variety of phenomena have been employed, including specifically, electromagnetically induced transparency [5,7,8], coherent population oscillations [9–11], and spectral hole-burning [12–14].

Ruby ($\alpha - {{\rm Al}_2}{{\rm O}_3}$ doped with ${{\rm Cr}^{3 +}}$) has served as an archetypal material for the spectroscopy of transition metal ion doped insulators for more than a century. Figure 1 is a schematic energy level diagram for the ${^4{{\rm A}}_2}$ ground state and the ${^2{\rm E}}$ lowest-excited state, with the transitions used in the present article indicated. The so-called R-lines (${^4{{\rm A}}_2}\to {^2{\rm E}}$ transitions) are, to a very good approximation, simple spin-flip excitations within the ${t}_2^3$ electronic configuration and hence are in a first order approximation independent on the ligand field strength, i.e., they do not couple strongly to the environment. As a consequence, the transitions display very narrow inhomogeneous line widths, given that they are transitions within a d-electron system.

 

Fig. 1. Energy level diagram for the ${\rm R}$-lines (${^4{{\rm A}}_2}\to {^2{\rm E}}$ transitions). The ${\rm g}$-factors for an external magnetic field parallel the $c$-axis, ${\rm B}||{\rm c}$, are indicated. Excited state wavenumbers are given as vacuum values at 2 K. ${\sigma}$-polarized $({\textbf E} \bot {\rm c)}$ transitions are shown for ${\rm B}||{\rm c}$.

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Chromium replaces aluminum in corundum (${\alpha} - {{\rm Al}_2}{{\rm O}_3}$) in its trigonal ${C_3}$ site, and the combined effect of the trigonal field and spin-orbit coupling split the ${^4{{\rm A}}_2}$ and ${^2{E}}$ states by 0.38 (11.4 GHz) and ${29}\;{{\rm cm}^{- 1}}$, respectively [15]. The first excited state level is at ${14418}\;{{\rm cm}^{- 1}}$ at 2 K (vacuum value, 693.579 nm; 432.2398 THz). In a magnetic field with ${\rm B}||{\rm c}$ the ${^4{{\rm A}}_2}$ ground state spin levels $| {\pm 3/2} \rangle$, $| {\pm 1/2} \rangle$ and the $\bar E ({^2{\rm E}})$ and $2\bar A({^2{\rm E}})$ excited states split linearly with the field with $g$-factors of 1.984 [16], ${2.445}\;{\pm}\;{0.001}$ [17], and 1.6 [18], respectively. The splitting of these levels in ${\rm B}||{\rm c} = {59}\;{\rm mT}$ (the magnetic field that was applied in the present work) is shown on the right- hand side of Fig. 1. The observed $\sigma$-polarized $({\textbf E} \bot {\rm c)}$ transitions that are relevant for the present article are also indicated. It is noted here that the absorbance for the ${{\rm R}_1}(\pm {3/2})$ transitions is ${\sim}{20 - 30}\%$ larger in ${\sigma}$- (or ${\alpha}$-) polarization than for the corresponding ${{\rm R}_1}(\pm {1/2})$ transitions (see Fig. 1), in accordance with peak absorption cross sections determined in early work [19].

The theoretically expected ratio of the square of the transition dipole moments, $| {\langle {^2E, \pm 1/2,u \mp} |\bar P| {^4{A_2}, \pm 3/2} \rangle} |^2/$${| {\langle {^2E, \pm 1/2,u \mp} |\bar P| {^4{A_2}, \pm 1/2} \rangle} |^2} = 3/2$, is also in reasonable agreement with the experimental result [20].

The R-lines around 694 nm in the deep-red region of the visible spectrum were investigated as early as 1867 by Becquerel [21]. This work was followed by that of Deutschbein, [22] who reported their Zeeman splitting. Then in the 1950s ligand field theory was fully developed around the ruby system [15,23]. In a groundbreaking development, Maiman successfully employed the R-lines to demonstrate the first laser action 60 years ago (May 16, 1960) [24]. In other classical work, McCumber and Sturge reported on the line shift and width of the R-lines in ruby in 1963 [25]. This theoretical analysis has been applied to many other systems over the last six decades. Then in the early 1970s Szabo presented, for the first time, laser-based fluorescence line narrowing [26] and transient spectral hole-burning measurements [27] in the solid state by using the ruby R-lines. Subsequently, Szabo, with colleagues, continued to make important contributions to the understanding of the R-lines in ruby [28–35]. In particular, it was shown that the hole width for the ${{\rm R}_1}(- {1/2})$ transition (see Fig. 1) for low ${{\rm Cr}^{3 +}}$ ruby (23 ppm) in ${\rm B}||{\rm c} = {0.35}\;{\rm T}$ is given by indirect Cr–Cr electron spin flips, yielding a narrow hole of ${\sim}{20}\;{\rm kHz}$ FWHM, and flipping of the ${^{27}{\rm Al}}$ nuclear spins in the bulk and in the frozen core, leading to a broader hole of ${\sim}{1}\;{\rm MHz}$ FWHM [36]. The amplitude of the broad hole increases with (i) lengthening the burn pulse time, (ii) increasing the time delay between the burn and the probe pulse, and, importantly (iii) increasing the Rabi frequency.

Another important contribution was the report on photon-echo measurements of the ${{\rm R}_1}(- {3/2})$ transition in the superhyperfine limit conducted in a high magnetic field by Macfarlane et al. [37]. In the superhyperfine limit, the optical line width is purely limited by the flipping of  ${^{27}{\rm Al}}$ nuclear spins because Cr–Cr electron spin flipping is suppressed as predominantly only the $| {- 3/2} \rangle$ level is populated at 2 K. In this limit, Szabo also reported spectral hole widths as narrow as 17.9 kHz for the ${{\rm R}_1}(- {1/2})$ line when using a Rabi frequency of ${\sim}{10}\;{\rm kHz}$ in 50 µs long burn pulses [34].

At the beginning of this century, coherent population oscillation-based slow light in ruby at room temperature was purported [10]. However, this work has come under scrutiny and it has been conclusively shown that the observed delays can also be explained by pulse distortion in a saturable absorber [38–41]. In contrast, we have recently reported slow and fast light effects based on transient spectral hole-burning in the ${{\rm R}_1}$-line at low temperatures [42,43]. We have also reported the generation of ultraslow solitons by self-induced transparency [44,45]. The latter could quantitatively be modeled by simulations applying the optical Bloch equations [44]. For the hole-burning-caused slow light, the linear filter theory as presented in Refs. [46,47] was applied. It is noted here that the linear filter theory was developed from the perspective of persistent spectral holes where the application of very short probe pulses does not alter the transmission spectrum of the sample. In the case of the transient hole-burning experiments described here, very weak probe pulses need to be applied to minimize any change of excited and ground state populations, which may otherwise affect the delay and width of the transmitted probe pulse. In brief, the linear filter theory predicts the spectral and temporal shape of a pulse propagating through a medium whose (generally complex) index of refraction is not altered by light [47]. If the amplitude of the probe pulse is sufficiently low [35], the amplitude of the pulse ${E_{\rm{out}}}$ after the medium with a spectral hole can be described by

The transmitted pulse intensity as a function of time can then be calculated by the Fourier back transform into the time domain according to

These equations were implemented into a MATLAB code with empirical values for all parameters.

The present article is a significant extension of our previous experiments on slow light in ruby based on transient spectral hole-burning. In particular, we take advantage of the slow spin-lattice relaxation time (${\gt}{500}\;{\rm ms}$) in ruby at low temperatures and low magnetic fields ${\rm B}||{\rm c}$. This slow relaxation time allows for optical pumping of the population of the ground state spin levels, i.e., optical spin polarization, by pre-pumping the system before the hole-burning based slow light experiment is conducted. It was shown some time ago by photon echo experiments that the optical pumping of spin levels in the ${^4{{\rm A}}_2}$ ground state leads to a narrowing of the spectral line width because the Cr–Cr electron spin flips are suppressed [30,48]. This is in principle the same effect as is achieved by applying a strong magnetic field at low temperatures. In addition to generating a higher optical density, such narrowing of line widths and thus spectral holes enhances slow light effects.

2. EXPERIMENT

A pale pink ruby crystal with a 30 ppm ${{\rm Cr}^{3 +}}$ (${\rm Cr}/{\rm Cr} + {\rm Al}$) concentration was grown along the $c$-axis by the Czochralski method, as has been described in detail before [49]. In brief, pre-melted high purity $(\gt\! {4}\;{\rm N})\;{{\rm Al}_2}{{\rm O}_3}$ and a small ruby crystal were melted together by induction-heating in an iridium crucible that was insulated by zirconia and alumina. The crystal was pulled in a nitrogen atmosphere with 0.3-vol % ${{\rm CO}_2}$ content on a sapphire seed crystal that was [1] aligned. This specific atmosphere leads to an oxygen activity of ${\sim}{2} \times {{10}^{- 4}}$ bar at the growth temperature, which in turn results in a uniform ${{\rm Cr}^{3 +}}$ distribution with a coefficient close to unity [49]. The crystal was cut parallel to the $c$-axis to a thickness of 7.1 mm and the two faces were polished to high optical quality. This allows experiments in ${\sigma}$-polarization, i.e., with the electric vector ${\textbf E}$ of the pumping and probing light pulses perpendicular to the crystal ${c}$-axis.

For the experiments in this article two lasers were used, as indicated schematically in Fig. 2(a). First, a MOGLabs Cateye laser (https://www.moglabs.com) served as the pre-pump light source to modify the spin level population in the ${^4{{\rm A}}_2}$ state by optical pumping. The beam of this laser was pulsed by an electromechanical shutter (Thorlabs SHB025T diaphragm shutter), controlled by a Tektronix AFG3102 arbitrary function generator (AFG), with its frequency ${{\omega}_1}$ scanned over a range of $\pm {\Omega}$ by changing the tilt angle of the filter in the cateye laser via the piezo stack at 40 Hz. Second, a TOPTICA DL100 external cavity diode laser (ECDL), controlled by a TOPTICA DigiLock 110 controlyzer module, was used to measure the spectrum across the region of the ${{\rm R}_1}$ lines with and without pre-pump by the cateye laser. The scan control was facilitated by applying a triangular voltage ramp generated by a second Tektronix AFG3102 via the Toptica SC110 scan control interface. The second laser was also used for burning holes and then probing the pulse delay at a frequency ${{\omega}_2}$ with and without the pre-pump pulse. The TOPTICA ECDL laser was pulsed by using an 80 MHz Isomet 1205C-1 acousto-optic modulator (AOM) with a 222 A-1 RF driver that was gated by a signal from a third Tektronix AFG3102. The three AFGs were synchronously triggered. The pulse sequence for the two ECDLs for the particular case of slow light experiments is shown schematically in Fig. 2(b). The two laser beams were combined by a non-polarizing beam splitter cube and tightly focused with a 150 mm lens onto the 7.1 mm ruby sample that was mounted between two copper plates with 1 mm apertures on the cold finger of a closed-cycle refrigerator (Janis Sumitomo SH-4.5). The 1 mm apertures helped to achieve good overlap of the two beams within the sample. The crystal was mounted so that the $c$-axis was perpendicular to the light propagation direction and parallel to the magnetic field direction. Typically, the two ECDLs were run at powers of 10 mW and 0.45 mW during the spin polarizing (pre-pump) and hole-burning pulses, respectively. The magnetic field of ${\rm B}||{\rm c} = {59}\;{\rm mT}$ was generated by two very low-cost neodymium permanent magnets of 25 mm diameter and 1 mm thickness (https://magnet.com.au/neodymium-disc-25mm-x-1mm.html) that were mounted on the sample holder and spaced by 12.5 mm, i.e., in the Helmholtz arrangement. Photodiode detectors with fixed (Thorlabs PDA10A) and variable (Thorlabs PDA36A) gain were employed to detect the light after the sample for the slow light experiments and the spectral scans, respectively, and the signal was processed by a LeCroy WaveSurfer 422 digital oscilloscope. The wavelengths of the two lasers were monitored by a Bristol 721 spectrum analyzer. Relative frequency excursions in the scan of the Toptica ECDL were calibrated by 1.5 GHz and 300 MHz Fabry–Perot etalons (1.5 GHz, Thorlabs SA200-5B; 300 MHz, Coherent Model 216).

 

Fig. 2. (a) Schematic diagram of experimental setup. (b) Pulse sequence for the slow light experiments.

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3. RESULTS AND DISCUSSION

Absorption spectra of the 7.1 mm thick 30 ppm ruby crystal measured in zero field and ${\rm B}||{\rm c} = {59}\;{\rm mT}$ are compared in Fig. 3. The pronounced transitions are due to the 52 isotope that has a natural abundance of 83.8%. Corresponding transitions from the 50 (4.3%), 53 (9.5%), and 54 (2.4%) isotopes are indicated for the zero field spectrum. It is noted here that the zero field spitting (ZFS) of the ${^4{{\rm A}}_2}$ ground state is constant within the experimental accuracy for the four isotopes, in accordance with expectations as the isotope shift is purely a “vibrational” effect. The zero point energies are slightly reduced with increasing ${{\rm Cr}^{3 +}}$ isotope mass. This reduction is less for the ${^2{E}}$ excited state than for the ${^4{{\rm A}}_2}$ ground state since the excited state phonon/vibrational frequencies are in general slightly lower. So electronic origins are expected to shift to higher energy in accord with the following observation: The ${{\rm R}_1}(\pm {3/2})$ transitions of the 50, 53, and 54 isotopes are shifted by ${-}{7.54}$, ${+}{3.48}$, and ${+}{6.84}\;{\rm GHz}$, respectively, compared with the transition for the 52 isotope. In the following we focus on the spectroscopy of the dominant (${\sim}{84}\%$) ${^{52}{\rm Cr}}$ isotope. The 2-K width (full width at half ${\rm maximum} = {\rm FWHM}$) of the transitions in Fig. 3 is determined by inhomogeneous broadening and is ${\sim}{1.4}\;{\rm GHz}$ in zero field. Such a value is typical for a Czochralski-grown ruby with a 20–30 ppm ${{\rm Cr}^{3 +}}$ concentration.

 

Fig. 3. $\sigma$-polarized 2-K absorption spectra in the region of the ${{\rm R}_1}$ line of the 7.1 mm thick, 30 ppm ruby sample in ${\rm B}||{\rm c} = {59}\;{\rm mT}$ (solid line) and zero field (dashed line). Corresponding transitions by the other three stable isotopes are labeled for the zero field spectrum. The x-axis shows the relative frequency excursion of the scanning laser where zero is set at 432.23984 THz (${14417.969}\;{{\rm cm}^{- 1}}$, 693.57901 nm, vacuum values).

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The expected splittings in a magnetic field ${\rm B}||{\rm c}$ for the ${{\rm R}_1}(\pm {3/2})$ and ${{\rm R}_1}(\pm {1/2})$ lines are given by

 

Fig. 4. $\sigma$-polarized absorption spectra of 30 ppm ruby in the region of the ${{\rm R}_1}$-lines in ${\rm B}||{\rm c} = {59}\;{\rm mT}$ with (red solid lines) and without (blue dashed lines) optical pumping at ${{\omega}_1}\;{\pm}\;{\Omega}$ for 400 ms with a repetition rate of 2 Hz. The difference spectra $\Delta{\rm A} = {\rm log}({{\rm I}_{\rm{np}}}/{{\rm I}_{\rm{wp}}})$, where ${{\rm I}_{\rm{np}}}$ and ${{\rm I}_{\rm{wp}}}$ are the transmitted light intensities without and with pump pulse, respectively, are also shown (green dash-dotted line). In (a) and (b) the pump laser was scanned during the 400-ms pulse at 40 Hz across the ${{\rm R}_1}(\pm {1/2})$ and ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ lines, respectively (as indicated by the red double arrow).

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Hence the magnetic splitting factor for the ${{\rm R}_1}(\pm {3/2})$ and ${{\rm R}_1}(\pm {1/2})$ transitions is

For ${\rm B}||{\rm c} = {59}\;{\rm mT}$ the splitting is not large enough to resolve the ${{\rm R}_1}(+ {1/2})$ and ${{\rm R}_1}(- {1/2})$ transitions but a broadening is clearly observed. We have also measured the spectrum in a stronger magnetic field ${\rm B}||{\rm c} = {235}\;{\rm mT}$ (not illustrated here), generated by 5 mm thick permanent magnets of 25 mm diameter. The ratio of the excited state and ground state $g$-factors is given as follows:

The observed ratio in ${\rm B}||{\rm c} = {235}\;{\rm mT}$ yields $g_{||}^{\textit{ex}} = {2.45}\;{\pm}\;{0.01}$, in excellent agreement with the literature value [17].

The effect of 400 ms pump pulses, scanned by $\pm {\Omega}$ at 40 Hz repetition rate across the ${{\rm R}_1}(+ {1/2})$, ${{\rm R}_1}(- {1/2})$, or, ${{\rm R}_1}(+ {3/2})$, ${{\rm R}_1}(- {3/2})$ transitions, is displayed in Fig. 4. Relatively strong spin polarization is achieved in both cases. For ${{\rm R}_1}(+ {1/2})$ and ${{\rm R}_1}(- {1/2})$ pumping for 400 ms at a 2 Hz repetition rate, the $| {\pm 1/2} \rangle$ levels get depleted by ${\sim}{85}\%$ (measured at a delay of 20 ms after the pump pulse). From Fig. 4(a) it follows that the $| {+ 3/2} \rangle$ level receives about seven times more of this population re-shelving than the $| {- 3/2} \rangle$, i.e., about 85% of the depleted $| {\pm 1/2} \rangle$ population ends up in the $| {+ 3/2} \rangle$ state. When the pump laser is scanned across the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ transitions, depleting the $| {+ 3/2} \rangle$ and $| {- 3/2} \rangle$ levels, the $| {+ 1/2} \rangle$ and $| {- 1/2} \rangle$ population increases by about ${\sim}{95}\%$ compared with the non-pumped system.

Cross relaxation (Cr–Cr electron spin flip-flops) in the ${^4{{\rm A}}_2}$ ground state is a pronounced feature in ruby even at concentrations as low as 20 ppm. This effect was noted early on in the initial papers on hole-burning [27,50], and some of the kinetics have been explored more recently [51]. For example, cross relaxation between resonant and non-resonant ions leads to a transient decrease and an increase of the entire inhomogeneous width of the ${{\rm R}_1}(\pm {1/2})$ and ${{\rm R}_1}(\pm {3/2})$ lines, respectively, when the ${{\rm R}_1}(\pm {1/2})$ transition is subject to transient hole-burning and vice versa. Interestingly, cross relaxation also happens between the ${^{52}{\rm Cr}}$ ion and the ${^{53}{\rm Cr}}$ ion despite the fact that the electronic structure of the latter is subject to hyperfine splitting.

Cross relaxation between the spins of ${^{52}{\rm Cr}}$ and the other isotopes is clearly visible in Fig. 4. Importantly, in a magnetic field ${\rm B}||{\rm c}$ only Cr–Cr spin flips ${{\rm M}_{S}}\leftrightarrow {{\rm M}_S} + {1}$ (with ${{\rm M}_S} = - {3/2}$, ${-}{1/2}$, ${+}{1/2}$) can occur. Cross relaxation does not alter the population distribution within the spin levels, i.e., the spin temperature ${T_S}$ of the ground state, because for every spin flip there is a corresponding spin flop at the same energy. So, thermodynamic equilibrium according to the lattice temperature can only be restored by spin-lattice relaxation.

The dependence of the spectrum on delay time after pump pulses of 1 s, which are scanned across the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ lines at 40 Hz during the pulse period, is depicted in Fig. 5(a). The inset summarizes the time dependence of the absolute value of the integrated change in absorbance, $\Delta A$, and hence provides a direct measure of the spin-lattice relaxation time. The time dependences display single exponential behavior, yielding spin-lattice relaxation times of ${T_1}(+ {3/2})\;= {1.09}\;{\pm}\;{0.03}\;{\rm s}$ and ${T_1}(- {3/2})\;= {417}\;{\pm}\;{10}\;{\rm ms}$ for the $| {+ 3/2} \rangle$ and ${-}{3/2}$ levels, respectively, at 2 K. For comparison, corresponding data is shown in zero field with ${{\rm R}_1}(\pm {3/2})$ pumping in Fig. 5(b), but in this case the evolution in time of the difference spectrum $\Delta{\rm A} = {\rm log}({{\rm I}_{\rm{np}}}/{{\rm I}_{\rm{wp}}})$ is directly illustrated. The inset of Fig. 5(b) also shows the time dependence of the integrated change of absorbance for the ${{\rm R}_1}(\pm {1/2})$ line. A fit to both data sets yields values for the spin-lattice relaxation time of ${T_1}\;(\pm {3/2})= \;{T_1}\;(\pm {1/2})= {503}\;{\pm}\;{7}\;{\rm ms}$ at 2 K. We have also conducted the corresponding experiment with a ${{\rm R}_1}(\pm {3/2})$ pump pulse (not illustrated here) and obtained the same spin-lattice relaxation time at 2 K (${496}\;{\pm}\;{9}\;{\rm ms}$) within the experimental error.

 

Fig. 5. Time resolved spectra in the region of the ${{\rm R}_1}$-transitions at 2 K after pumping the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ transitions for 1 s in (a) ${\rm B}||{\rm c} = {59}\;{\rm mT}$ and (b) zero field. In (a) and (b) the absorbance, A, and the change in absorbance, $\Delta {\rm A}$, as a function of delay time, are illustrated, respectively. The insets show the time dependences of the change in the integrated areas (normalized for b). The scan/readout delays are indicated in the two panels in units of ${\rm s}$.

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The higher $| {+ 3/2} \rangle$ population upon ${{\rm R}_1}(+ {1/2}),\;{{\rm R}_1}(- {1/2})$ pumping (see also Fig. 4 and text above) is caused by the longer spin-lattice relaxation time for this level in low magnetic fields; possible relaxation pathways are ${\Delta{\rm M}_s} = {1}$ and ${\Delta{\rm M}_s} = {2}$, i.e., $| {+ 3/2} \rangle \to | {+ 1/2} \rangle$ and $| {+ 3/2} \rangle \to | {- 1/2} \rangle$. Spin-lattice relaxation rates as a function of temperature as well as the applied external magnetic field and its direction have been investigated more extensively in ruby than in any other system [52–62]. This interest stemmed from the demonstration of ruby as a maser material in 1958, and since 1960 the ruby maser has been used in the Deep-Space Network [63]. From the vast literature on spin-lattice relaxation in the ${^4{{\rm A}}_2}$ ground state of ruby it is concluded that at liquid helium temperatures the one phonon (direct process) is the active mechanism [64]. Using the Debye approximation for the density of phonon states in the one phonon relaxation mechanism, and neglecting the difference between longitudinal and transverse sound velocities ($v$), following dependence on the energy gap $\Delta{E} = {\rm h}\nu$ (where ${\nu}$ is the phonon frequency) and temperature, $T$, is expected for the spin-lattice relaxation time ${T_1}$:

In Eq. (11) $\rho$ is the density of the crystal and ${k_B}$ the Boltzmann constant. The matrix element $\langle {{\psi _1}} |\sum {V_n^m} | {{\psi _2}} \rangle$ is purely electronic, with the operator being an expansion of the potential in terms of spherical harmonics $Y_n^m$, and hence the relative energy/frequency and temperature dependences are given by the first term of the equation. Expanding the hyperbolic cotangent via a Taylor series and keeping only the first term, i.e., ${\rm coth}({\Delta}E/{2}{k_B}T) \approx {2}{k_B}T/{\Delta}E$, results in the following:

The $| {+ 3/2} \rangle$ spin-level is closer to the $| {- 1/2}\rangle$ level in ${\rm B}||{\rm c} = {59}\;{\rm mT}$. This can be seen in Table 1, a compilation of the energy differences between spin levels in ${^4{{\rm A}}_2}$ ground state.

Table 1. Spin-Lattice Relaxation Pathways and Associated Energy Differences in ${^4{\rm A}_2}$ for ${\rm B}||{\rm c} = {59}\;{\rm mT}$

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Misu [65,66] measured the ${\Delta{\rm M}_S} = {2}$ ($| {+ 3/2} \rangle \to | {- 1/2} \rangle$, $| {+ 1/2} \rangle \to | {- 3/2} \rangle$) and ${\Delta{\rm M}_S} = {1}$ ($| {+ 3/2} \rangle \to | {+ 1/2} \rangle$, $| {- 1/2} \rangle \to | {- 3/2} \rangle$) spin relaxation rates at 4.2 K in high magnetic fields ${\rm B}||{\rm c} = {4.7}\;{\rm T}$ to 12.8 T where energy levels are almost equidistantly spaced (which is not the case in low fields such as ${\rm B}||{\rm c} = {59}\;{\rm mT}$ in the current work). In high fields, ${\Delta{\rm M}_S} = {2}$ transitions occur at approximately twice the energy as ${\Delta{\rm M}_S} = {1}$ transitions. The spin transition rates were reported to vary quadratically with the magnetic field ${\rm B}||{\rm c}$ with values of $k({\Delta{\rm M}_S} = {2}){{\rm /B||c}^2} = ({7.66}\;{\pm}\;{0.15}) \times {{10}^{- 4}}\;{{\rm ms}^{- 1}}/{{\rm T}^2}$ and $k({\Delta{\rm M}_S} = {1}){{\rm /B||c}^2} = ({7.65}\;{\pm}\;{0.2}) \times {{10}^{- 5}}\;{{\rm ms}^{- 1}}/{{\rm T}^2}$. Hence, the ${\Delta{\rm M}_s} = {2}$ transition probability is about 2.5 higher than for the ${\Delta{\rm M}_s} = {1}$ transitions when the energy difference is the same. We assume that this factor of 2.5, given by the ratio of the electronic matrix elements, ${| {\langle {+ 3/2} |\sum {V_n^m} | {- 1/2} \rangle} |^2}/{| {\langle {+ 3/2} |\sum {V_n^m} | {+ 1/2} \rangle} |^2}$, is also valid at low fields as in ${\rm B}||{\rm c}$ electronic states are not mixed and the condition of $\Delta{E}/{k_B}T \ll {1}$ is better met in the present experiments. Using the factor of 2.5 and the quadratic energy gap dependence, the following ratio between the relaxation for the $| {+ 3/2} \rangle$ and $| {- 3/2} \rangle$ levels can be calculated using the tabulated energy differences of Table 1:

This is in good agreement with the ratio of 2.6 as obtained from the experimentally determined rates noted above. A value of 2.74 results when Eq. (11) is applied, i.e., the approximation to derive Eq. (12) seems to be valid.

The hole-burning spectra in Fig. 6 are for the ${{\rm R}_1}(\pm {1/2})$, and ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$, transitions with and without a prior pump pulse in the ${{\rm R}_1}(- {3/2})$, ${{\rm R}_1}(+ {3/2})$ or ${{\rm R}_1}(+ {1/2})$, ${{\rm R}_1}(- {1/2})$ transitions, respectively (as per Fig. 4). Measured hole widths (FWHM) are summarized in Table 2.

 

Fig. 6. Deep spectral holes in the ${{\rm R}_1}$-transitions of 30 ppm ruby readout with a delay of 1 ms after hole-burning at 2 K and in ${\rm B}||{\rm c} = {59}\;{\rm mT}$ with (red trace) and without (blue trace) pre-pumping the spin levels of the ${^4{{\rm A}}_2}$ ground state. For the ${{\rm R}_1}(\pm {1/2})$ spectrum the ${{\rm R}_1}(+ {3/2})$, ${{\rm R}_1}(- {3/2})$ transitions were pumped, as shown in Fig. 4. For the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ experiment the ${{\rm R}_1}(+ {1/2})$, ${{\rm R}_1}(- {1/2})$ transitions were pre-pumped.

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As is well documented, the homogenous line width of an optical transition is given by the following:

Table 2. Observed Spectral Hole Widths (FWHM-Gaussian)^a

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In Eq. (16), $\vec \mu$ is the transition dipole moment and $\vec E$ the electric vector of the radiation field, and for ruby, ${T_2} \approx T_2^*$ since ${T_1} \gg T_2^*$. In the experiments in this article, relatively high laser power densities were used and hence holes are significantly broadened by the power broadening term ${\Omega ^2}{T_1}{T_2}$. For example, Szabo and Kaarli [33] have shown that holes in the ${{\rm R}_1}(- {1/2})$ transition in a 23 ppm ruby and in ${\rm B}||{\rm c} = {350}$ ${\rm mT}$ broaden from ${\sim}{20}\;{\rm kHz}$ to ${\sim}{1}\;{\rm MHz}$ by increasing the Rabi frequency from ${\sim}{10}\;{\rm kHz}$ to 100 kHz and the period of the burn pulse from 50 to 300 µs. In the present experiments the Rabi frequency was well ${\gt}{100}\;{\rm kHz}$. For example, assuming a dephasing time ${T_2}$ of the order of magnitude of 1 ms, the experimentally determined excited state lifetime ${T_1} = {3.79}\;{\rm {ms}}$, and a Rabi frequency of 500 kHz, the power broadening leads to an increase of ${{\Gamma}_{\rm{hom}}}$ by a factor of ${\sim}{30}$.

It is important to note here that for an optically thin sample and a very low Rabi frequency, i.e., negligible power broadening, spectral holes are only twice the homogeneous line width in the limit of zero hole depth. Typically, the low limits of hole widths are deduced by extrapolation to zero hole depth/fluence from a series of measurements with different fluences. Slightly off-resonant optical centers get bleached out through the wings of the line shapes. In particular, the Lorentzian line shape has extensive wings, and saturated holes will be about five to eight times broader than shallow holes, even if low intensities would be used to avoid power broadening [67]. In optically thick samples (${\rm OD} \gt {1}$) the situation is even more complex but may be modeled numerically. The holes are narrowed upon pre-pumping the spin levels in the ground state since Cr–Cr electron spin flips are less likely to occur and hence the dephasing time is lengthened. Indirect spin flip-flops, i.e., flip-flops that do not involve the resonant ions, can be a major cause of observed hole widths in ruby samples, with chromium(III) concentrations as low as 20 ppm and certainly in low magnetic fields. The effect on the dephasing time is inversely proportional to the root mean square value of the magnetic field fluctuations due to spin flips (nuclear and electronic), as is expressed in the following:

As mentioned above, Szabo and Heber [30] reported this effect some time ago by utilizing the ${\rm R}_2$ line for the pumping of spin levels in the ${^4{{\rm A}}_2}$ ground state. They observed an order of magnitude increase of the dephasing time in the ${\rm R}_1$ line by photon-echo experiments [30]. For example, the photon-echo decay time for the ${{\rm R}_1}(- {3/2})$ line increased from 0.45 µs to 5.4 µs upon pumping the ${{\rm R}_2}(+ {1/2})$ and ${{\rm R}_1}(- {1/2})$ transitions simultaneously. The ${{\rm R}_1}(\pm {1/2})$ transitions are less susceptible than the ${{\rm R}_1}(\pm {3/2})$ lines to magnetic fluctuations in their environment, as can be seen from the magnetic splitting factor in Eq. (9). This is qualitatively reflected in the hole widths summarized in Table 2.

The ${\sim}{5}\;{\rm MHz}$ hole width of the ${{\rm R}_1}(\pm {1/2})$ transitions upon spin polarization would be given mostly by power and saturation broadening effects, but with some finite contributions from residual Cr–Cr spin flipping and a 1 MHz contribution from flips of the ${^{27}{\rm Al}}$ nuclear spins. Considering the results of Fig. 6 and Table 2, it follows that the ${{\rm R}_1}(+ {3/2})$ hole width is significantly broader (35 MHz) than the corresponding ${{\rm R}_1}(- {3/2})$ hole width (15 MHz) without electronic spin flip suppression by optical pumping of the spin levels. This indicates that the $| {+ 3/2} \rangle$ level is more affected by electronic spin flip-flops than the $| {- 3/2} \rangle$ level. A similar result was obtained by Szabo and Heber [30], who observed that $| {- 3/2} \rangle$ spins are more affected by $| {+ 3/2} \rangle$ to $| {+ 1/2} \rangle$ than $| {- 3/2} \rangle$ to $| {- 1/2} \rangle$ flips, which they interpreted as indicating a spin spatial correlation. More work is required to understand these effects.

Hole decay in 30 ppm ruby samples is governed by rapid direct cross relaxation to non-resonant ions and the excited state lifetime, as documented by Ref. [51]. Importantly, cross relaxation beween resonant and non-resonant ions prevents the buildup of holes based on the storage in the ground state spin levels.

 

Fig. 7. Probe pulse transmission in the ${{\rm R}_1}$-lines of 30 ppm ruby at 2 K and in ${\rm B}||{\rm c} = {59}\;{\rm mT}$ with (red solid trace) and without (blue solid trace) pre-pumping ${^4{{\rm A}}_2}$ spin levels. The reference pulse in the absence of a spectral hole-burning pulse is also shown for comparison with (black solid line) and without (green solid line) pre-pumping the spin polarization. The dashed green line is the reference pulse scaled to the delayed pulse with spectral hole-burning (both without pre-pump pulse). For the ${{\rm R}_1}(\pm {1/2})$ pulse the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ were pumped, as illustrated in Fig. 4. For the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ experiment the ${{\rm R}_1}(+ {1/2})$, ${{\rm R}_1}(- {1/2})$ transitions were pumped, as per Fig. 4. The insets show the hole-burning curves (transmitted light intensity) with (red trace) and without (blue trace) pre-pump pulse. The 10-µs delayed probe pulse is also visible in these graphs (sharp peak).

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Table 3. Summary of Slow Light Data and Experimental Parameters^a

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Slow light experiments are summarized in Fig. 7 and Table 3. The plots of Fig. 7 illustrates that the delay of the probe pulse after spectral hole-burning can be significantly increased by pre-pumping the spin levels of the ${^4{{\rm A}}_2}$ ground state. The biggest relative change upon pre-pumping the ground state spin levels is observed for slow light in the ${{\rm R}_1}(+ {3/2})$ transition upon depleting the $| {\pm 1/2} \rangle$ levels, although the slowest light is observed for the ${{\rm R}_1}(\pm {1/2})$ line upon depleting the $| {+ 3/2} \rangle$ and $| {- 3/2} \rangle$ spin levels by pre-pumping. The latter shows a delay of the transmitted pulse by ${\sim}{93}\;{\rm ns}$, which corresponds to a group velocity of $c/{3940} = {76}\;{\rm km/s}$ where $c$ is the velocity of light in vacuum. The slow light caused by the rapid dispersion of the refractive index by the deep spectral hole is retarded because of the combined effect of higher optical density (absorbance), and hence deeper spectral holes, and the narrowing of spectral holes caused by the suppression of indirect electron-spin flip-flops (non-resonant cross relaxation). The latter leads to fluctuating magnetic fields at the excited chromium centers, as was discussed above. Calculated results employing the linear filter theory summarized in the Introduction are also shown in Table 3, and a comparison between simulated and experimental data is illustrated in Fig. 8 for the case of slow light in the ${{\rm R}_1}(\pm {1/2})$ transitions with and without depleting the $| {+ 3/2} \rangle$ and $| {- 3/2} \rangle$ levels. The simulations employed the experimental values for initial absorbance A, hole depth ${\Delta{\rm A}_{\rm{HB}}}$, hole width ${{\Gamma}_{\rm{hole}}}$, probe pulse period, center frequency, and inhomogeneous line width. The transmission spectrum $T(w)$ was simulated using the values of A, ${\Delta{\rm A}_{\rm{HB}}}$, and ${{\Gamma}_{\rm{hole}}}$ in these calculations.

 

Fig. 8. A comparison of simulated and experimental pulse shapes transmitted through a ${{\rm R}_1}(\pm {1/2})$ hole with and without pre-pumping (pp) the ${^4{{\rm A}}_2}$ ground state spin levels via the ${{\rm R}_1}(+ {3/2})$ and ${{\rm R}_1}(- {3/2})$ transitions.

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Whereas the calculated delays are in reasonable agreement with the experiment, the calculated widths of the transmitted pulse seem to be somewhat underestimated. It is noted here that the linear filter theory is not dependent on uniformity of the absorption/refraction along the optical pathway and hence such non-uniformity cannot explain the deviation. The larger experimental widths may be caused by the frequency jitter of the laser. Also, the probe pulse intensity most likely affects the excited state and ground state populations and hence the “filter” is subject to change during the probe pulse. It is also possible that the simulation of the transmission spectrum $T(w)$ is not accurate enough. In addition, the current linear filter theory does not take into account possible polarization changes during the propagation of the light pulse through the medium. However, good semi-quantitative agreement is obtained and hence the theory is useful as a spectroscopic tool and for predicting slow light effects.

4. CONCLUSION

The present article demonstrates that slow light generated by transient spectral hole-burning in ruby with a low ${{\rm Cr}^{3 +}}$ concentration can be controlled by optical spin polarization via pre-pumping the spin levels of the ${^4{{\rm A}}_2}$ ground state. Spin-lattice relaxation rates appear to be quadratically dependent on energy gaps and hence a slow relaxation time of ${\sim}{1}\;{\rm s}$ is observed for the $| {+ 3/2} \rangle$ spin level in a magnetic field of ${\rm B}||{\rm c} = {59}\;{\rm mT}$. This is because the $| {+ 3/2} \rangle$ level and $| {- 1/2} \rangle$ spin levels of the ${^4{{\rm A}}_2}$ ground state are relatively close in such a field, resulting in a decrease of the dominant ${\Delta{\rm M}_S} = {2}$ transition rate. The optical spin polarization renders higher optical densities and narrower spectral holes. In turn, both these effects slow down the spectral hole-caused slow light. In other words, slow light is controlled by the light of two pump pulses of slightly different colors. The first, a pre-pump pulse, polarizes the spin. The second, a hole-burning pulse, creates a transient hole that renders a rapidly varying refractive index in the system resulting in slow light at that frequency. Our results suggest that a simple linear filter model can semi-quantitatively describe the shape and group delay of slow light pulses, which are based on transient spectral holes. However, to achieve an even better agreement with the experiment, and thus advance the application of slow light as a quantitative spectroscopic tool, more elaborate modeling may be needed. Such modeling could take into account contributing factors such as the finite intensity of the probe pulse, which in turn changes ground and excited state populations, and coherent effects such as optical nutation, saturation of the hole shape, etc.