Coherence Time Extension by Large Scale Optical Spin Polarization in a Rare-Earth Doped Crystal

Optically addressable spins are actively investigated in quantum communication, processing and sensing. Optical and spin coherence lifetimes, which determine quantum operation fidelity and storage time, are often limited by spin-spin interactions, which can be decreased by polarizing spins in their lower energy state using large magnetic fields and/or mK range temperatures. Here, we show that optical pumping of a small fraction of ions with a fixed frequency laser, coupled with spin-spin interactions and spin diffusion, leads to substantial spin polarization in a paramagnetic rare earth doped crystal, $^{171}$Yb$^{3+}$:YSO. Indeed, up to more than 90 % spin polarizations have been achieved at 2 K and zero magnetic field. Using this spin polarization mechanism, we furthermore demonstrate an increase in optical coherence lifetime from 0.3 ms to 0.8 ms, due to a strong decrease in spin-spin interactions. This effect opens the way to new schemes for obtaining long optical and spin coherence lifetimes in various solid-state systems such as ensembles of rare earth ions or color centers in diamond, which is of interest for a broad range of quantum technologies.

c Absorption spectrum without (purple) and after 20 s of optical pumping (orange). The OP is along the |4g → |1e transition (red arrow in b,c). d Ground state spin populations kig, normalized by thermal equilibrium values, over the volume addressed by the laser without OP (top, purple, kig = 1) and after OP (bottom, orange, k1g, k2g, k3g, k4g = 1.67 ± 0.30, 1.28 ± 0.36, 1.01 ± 0.12, 0.04 ± 0.04). e Enlarged regions 1 and 2 in c showing a narrow hole at the laser frequency (|4g → |1e ) and a corresponding antihole (|2g → |3e ). other materials. It paves the way to applications of concentrated, optically active, spin ensembles such as multimode optical or microwave quantum memories and high sensitivity magnetic sensing.

I. RESULTS
Experiments were performed using a 10 ppm 171 Yb 3+ :Y 2 SiO 5 (YSO) single crystal sample (see Methods). 171 Yb 3+ has 1/2 electron and nuclear spins and the corresponding ground ( 2 F 7/2 ) and excited state ( 2 F 5/2 ) hyperfine structures, for ions in site 2, are presented in Fig. 1b. The optical transition is centered at 978.854 nm (vac.). Due to anisotropic Zeeman and hyperfine interactions, all hyperfine levels are non-degenerate and their states show completely symmetric superposition of electron and nuclear spin projections. This results in levels that are insensitive to magnetic field fluctuations at first order under zero external magnetic field. Coherence lifetimes of all transitions are thus significantly enhanced for very low magnetic fields, reaching up to 4 ms and 180 µs for spin and optical transitions at 3 K 17 .

A. Optical pumping
Diffusion Enhanced Optical Pumping (DEOP) was studied at 2 K. A narrow linewidth (about 1 MHz) laser excited the 2 F 7/2 → 2 F 5/2 transition for a few 10s of seconds. The laser was then blocked for a few ms to let the excited state population relax to the ground state and finally, with a reduced power, shone again on the sample and frequency scanned to determine 171 Yb 3+ absorption spectrum (see Methods). In many crystals, RE spins can be optically pumped at low temperature since excitation to the optical state and subsequent decay often result in population transfer between ground state spin levels. Since the laser linewidth is usually much narrower than the RE optical inhomogeneous linewidth, optical pumping creates spectral holes and anti-holes in transmission spectra, i.e. regions of low and high absorption that can be as narrow as twice the optical homogeneous linewidth 20,35 .
In 171 Yb 3+ :YSO, spectral hole burning is not the only phenomenon that occurs. Indeed, as shown in Fig. 1c, after pumping for 20 s the |4g ↔ |1e transition with a 1 MHz linewidth laser, the whole 550 MHz inhomogeneously broadened line vanished. This means that essentially all spins in the sample volume addressed by the laser that were initially in the |4g state (≈ 2 × 10 14 spins) have been transferred to other spin states, despite only 0.4 % of these spins being optically excited. The fraction of excited ions is determined from the overlap between the absorption spectrum and the laser lineshape (see Methods). An analysis of the absorption spectrum, based on previously determined energies and branching ratios of transitions between ground and excited state spin levels, allowed us to accurately determine ground state spin level populations ( 31,36 and see SI for details). This analysis shows that only 4 % of the initial population is left in |4g (Fig. 1d). We also note in Fig. 1c,e that holes and anti-holes can be observed, although with a low contrast, in the spectrum. They reveal the narrow homogeneous linewidth of the optical transition. The frequency positions of the holes and anti-holes correspond to the pattern expected under spectral hole burning 31 , ruling out spurious effects like large laser drifts during optical pumping.
As explained in the introduction, we attribute this large spin polarization to energy exchanges by flip-flop between 171 Yb 3+ ground state spins. This process leads to a diffusion of the population imbalance imposed by the optical pumping of a small fraction of the spins. RE ions are randomly distributed over the volume of the crystal and their optical frequencies, determined by local strains, are not expected to be correlated with their location 37 . Ions resonant at a given optical frequency are therefore distributed over the volume addressed by the laser, which results, under optical pumping, in a macroscopic spatial spin population gradient and in turn population diffusion, as illustrated in Fig. 1a and SI. The observation of a decrease of the overall optical absorption also indicates the absence of strong correlation between optical and spin transition frequencies.
An important goal of this study was to quantify how the degree of polarization depends on the fraction of optically pumped ions, as well as the characteristic time required to reach that polarization. To this end we varied the optical pumping frequency across the |4g ↔ |1e transition and recorded absorption spectra for different pumping durations τ P . Fig. 2a shows the region of the absorption spectrum corresponding to the |4g ↔ |1e transition, centered at zero frequency detuning, with a smaller contribution from the |3g ↔ |1e transition at +0.65 GHz. The laser frequency is set at -0.17 GHz, as shown by the hole that appears on spectra recorded for pump durations 1 ≤ τ P ≤ 25 s. When τ P is increased, the whole inhomogeneously broadened absorption decreases, without change in shape, and reaches a plateau after about τ P = 30 s. From the peak absorption coefficient, measured at 0 GHz in Fig. 2a, we deduce k 4g , which is level |4g population normalized by its value without pumping, i.e. at thermal equilibrium (see Methods and SI). The same experiment was repeated for different laser frequencies, shown by the arrows on Fig. 2a, which effectively reduce the fraction of optically pumped spins. The corresponding variations of k 4g are displayed in Fig.  2b.
As the laser moves away from the peak absorption of the |4g ↔ |1e transition, k 4g decreases more slowly as a function of τ P and plateaus at a higher value. We found that it could be well fitted by an exponential expression of the form k 4g = exp(−R P τ P ) + k ∞ 4g . Fig. 2c shows the variation of the steady-state population k ∞ 4g and the polarization rate R P as a function of the fraction C of optically pumped ions, determined from the overlap between the absorption spectrum and laser lineshape (see Methods). The variation of R P and k ∞ 4g can be understood in the following way: when C decreases, each pumped spin has to polarize a larger number of non-pumped spins to reach a given k 4g value; this slows down the overall spin diffusion and decreases the polarization rate R P . The degree of achievable spin polarization is limited by the interaction of individual spins with the phonon bath, so-called spin lattice relaxation (SLR), which will counteract DEOP. Hence, the steady-state population k ∞ 4g is determined by the balance between SLR and spin diffusion rates, such that a smaller R P (and thus C) implies a larger k ∞ 4g . At the highest fraction of pumped ions, C = 5.8 × 10 −3 , k ∞ 4g = 0.08 ± 0.02 and 1/R P = 3 ± 0.6 s, which increases to k ∞ 4g = 0.74 ± 0.02 and 1/R P = 91 ± 27 s for C = 2 × 10 −4 , the lowest value investigated. Even by pumping such a small fraction of ions, 25% of |4g spins are transferred to another level, showing the efficiency of DEOP in this system.
The data in Fig. 2c suggest that R P depends linearly on C whereas k ∞ 4g varies as 1/C. This can be accounted . The whole inhomogeneously broadened absorption decreases when pumping duration increases. The laser creates a narrow spectral hole at -0.17 GHz clearly seen for pump duration between 1 and 7 s. The small side hole at +0.5 GHz originates from the |3g level and should therefore appears as an antihole. This is explained by a fast relaxation between the |4g and |3g levels by flip-flop processes 31 . The absorption centered at +0.66 MHz corresponds to the |3g ↔ |1e transition. b Normalized level |4g population k4g as a function of OP duration for different laser frequencies shown in a by color-coded arrows. Solid lines are exponential fits to the data (see text). c Level |4g normalized steady state populations (k ∞ 4g ) and polarization rate (RP ) as a function of the fraction of pumped ions (see text). Solid lines correspond to fits using a spin 1/2 model and rate equations. Inset: Spin 1/2 model scheme (see text). d Spin lattice relaxation time as a function of temperature deduced from absorption recovery after OP is stopped. Solid lines are fits using direct and two-phonon processes (see text). All error bars correspond to a 95% confidence interval.
for by a simple model that treats 171 Yb 3+ ions as an ensemble of 1/2 spins divided into two groups: the A-spins are optically pumped to their ground state; the B-spins are not pumped but are expected to polarize to their lower state through DEOP (Fig. 2c, inset). C is therefore the ratio between A and B-spin concentrations. We use rate equations to describe the individual relaxations as well as the flip-flop processes between A and B-spins (see SI for details). They can be solved analytically, leading to where p + B is B-spins upper state population and R the spin-lattice relaxation rate. These expressions have indeed the correct dependence on C with respect to experimental observations. β f f = W 1 /(W 1 + R o ) and β o = R/R o , where R o is A-spins effective optical pumping rate (Fig. 2c, inset). W 1 is the relaxation rate of A-spins by flip-flop with B-spins and W 2 = CW 1 is the relaxation rate of B-spins by flip-flop with A spins.
As shown in Fig. 2c, reasonable agreement was obtained when fitting experimental k ∞ 4g and R P using Eqs. (1) and (2), which indicates that a two-level system can be indeed used to model 171 Yb 3+ under these DEOP conditions. Theoretical flip-flop rates show that this is possible due to the fast flip-flops that occur within the |1g -|2g and |3g -|4g pair of levels (see SI). In this case, each pair of levels can be grouped and considered as one level, leading to an effective 1/2 system. Flip-flops within others pairs of levels, like |4g -|2g , are much slower. W 1 corresponds to these slow rates, as they are found to be the limiting interaction for B spins polarization. With the additional assumption Fig. 2c fits give W 1 = 57 ± 5 s −1 and R = (1.4 ± 0.4) ×10 −2 s −1 . We estimate R o = 384 s −1 from excited state lifetime and optical branching ratios, and W 1 = 13 s −1 from narrow hole decays (see SI). These qualitative  agreements support our simple spin 1/2 level-rate equation approach. However, we expect that when transitions connecting different ground state levels are simultaneously pumped (see section I B), a more complex 4-level-modeling is necessary. We finally recorded the |4g ↔ |1e absorption spectra at different delays after DEOP. As in the previous experiments, the line shape did not change while the initial absorption was gradually recovered, and the peak absorption coefficient allowed us to monitor level |4g population over all the spins in the volume addressed by the laser. Since flip-flops do not change overall level populations, the recovery rate R c , obtained by an exponential fit to the data, corresponds to the SLR rate. This is confirmed by R c temperature dependence shown in Fig. 2d which can be well modeled by a sum of a direct process and 2-phonon processes, with parameters consistent with previous studies at higher magnetic field 32 , as detailed in SI. At 2 K, the temperature used for DEOP experiments, R c = 1/(72 s) = 1.4 × 10 −2 s −1 , in qualitative agreement with the fitted value 2R = (2.8 ± 0.8) × 10 −2 s −1 .

B. Optical coherence
We next investigated optical coherence lifetimes, T 2,o under DEOP. This was motivated by several studies that have shown that flipping ground state spins of paramagnetic RE can be a major source of magnetic noise and therefore cause dephasing to RE optical and spin transitions 8,32 . This can be reduced by inducing strong spin polarization under large magnetic field and/or ultra-low temperatures, broadband optical pumping or using excited state spins 13,14,29,38,39 . As shown in the previous section, DEOP also induces large scale spin polarization and could therefore achieve similar effects.
Optical coherence lifetimes were measured for the |4g ↔ |1e transition under several DEOP conditions. In a first series of measurements, DEOP was performed with a laser set at -1.44 GHz in the spectrum displayed in Fig.  3a. At this frequency, some ions are pumped along the |1g ↔ |1e transition and others along the |3g ↔ |2e one because of the overlap between these inhomogeneously broadened lines. This results in progressively pumping away the populations of the |1g and |3g levels when the DEOP duration is increased (Fig. 3a). After 10 s, nearly all the initial populations of these two levels have been transferred to |4g and |2g . A second DEOP configuration was studied, setting the laser frequency at +2.67 GHz (Fig. 3c). In this case, the optical excitation is resonant with the four optical transitions |1g ↔ |3e , |1g ↔ |4e , |2g ↔ |3e and |2g ↔ |4e . As a result, the states |1g and |2g are now emptied, and |3g and |4g filled, when DEOP is applied (Fig. 3b). These two configurations highlight the versatility of DEOP that allow polarizing spins in different levels and not only in the lowest energy one, as would result from using large magnetic fields or low temperatures.
For each laser frequency and DEOP duration, the optical coherence time of the |4g ↔ |1e transition was measured with photon echoes (see Methods). The echo decays obtained by varying the delay between the excitation and rephasing pulses are displayed in Fig. 3c. Decay rates show large variations as a function of DEOP conditions and corresponding coherence lifetimes T 2,o , obtained by single exponential fits, are gathered in Fig. 3d. Without DEOP, It reaches 782 ± 30 µs after 10 s of DEOP that empties the |1g and |3g levels (Fig. 3a,d). The corresponding homogeneous linewidth is Γ h,o = 407 ± 15 Hz. This is the narrowest homogeneous linewidth reported at zero magnetic field for any RE, with the exception of non-Kramers Eu 3+ :Y 2 SiO 5 in which linewidths < 290 Hz have been measured 40 .
When DEOP is used to empty |1g and |2g levels, a very different result is observed: T 2,o is strongly reduced, down to 84 ±8 µs, equivalent to a homogeneous linewidth of Γ h,o = 3.8 ± 0.4 kHz. This is a factor of ten difference as compared to the first DEOP configuration, and about 3.5 times the value obtained without pumping. To the best of our knowledge, this is the first demonstration of changes, and especially significant enhancement, in coherence lifetime induced by optical pumping. This is especially significant for systems that should be used at low magnetic field, to take advantage of magnetic insensitive transitions 15 , as here in 171 Yb 3+ :Y 2 SiO 5 , or when constraints from other devices such as superconducting resonators are relevant.
Contributions to the |4g ↔ |1e homogeneous linewidth can be expressed in terms of levels |4g and |1e populations lifetimes T 1 and pure dephasing Γ φ as: The excited-state lifetime T 1,1e can be taken to be simply its radiative lifetime, T 1,1e = T 1,o = 1.3 ms, hence independent of DEOP, as SLR rates on the same order than in the ground state (close hyperfine and crystal field splittings 30,31 ) and the spin flop-flop rates are negligible due to the low concentration of excited ions. For the ground state we also disregard SLR contributions to T 1,4g , as the estimated SLR lifetime is ≈ 2/R c = 144 s −1 . However, the flip-flop rates can contribute to Γ h,o both directly through the T 1,4g lifetime and indirectly through the dephasing term Γ φ . The contribution to Γ φ is then a spectral diffusion process, where flip-flops in the 171 Yb 3+ spin bath create a time-varying magnetic field noise on the optically probed ion. Both the direct and indirect flip-flop contributions are expected to change through DEOP. Our calculations of the flip-flop rates between ions in different hyperfine states show that the highest rates are due to flip flops in between ions in |1g and |2g , and |3g and |4g , respectively (see SI). Hence, we expect that the flip flop rate will strongly decrease when ions are pumped into states |2g and |4g using DEOP, as in Fig. 3a. Conversely we expect the flip flop rate to strongly increase when ions are pumped into states |3g and |4g using DEOP, as in Fig. 3b. This qualitatively explains the change in coherence time due to DEOP, as seen in Figs 3c and 3d. However, these data are not sufficient to distinguish between the direct (lifetime) and indirect (spectral diffusion) contributions to the coherence time.
To this end, we also performed spin coherence measurements on the |3g ↔ |4g transition at 655 MHz, as described in the SI. For this, we polarized a large fraction of the spins into either the |1g and |2g states, or the |3g and |4g states, respectively. In both cases the populations in these two levels were essentially the same. An indirect spectral diffusion contribution to the spin coherence lifetime would be roughly equal in both cases, as the flip-flop rates are expected to be the same for both cases (see flip-flop calculations in SI). However, the direct lifetime contribution to the probed states |3g and |4g would strongly decrease when spins are polarized into |1g and |2g states, as the spin flip-flop probability of spins in both the |3g and |4g states would be reduced. Indeed, we observed a strong change in spin coherence lifetime for the two cases, going from 0.2 to 2.5 ms as spins are polarized into |1g and |2g states.
The spin coherence measurements indicate that direct flip-flop lifetimes significantly contribute to the optical coherence lifetimes in 10 ppm doped 171 Yb 3+ :Y 2 SiO 5 . Strong spin polarization using DEOP into selected hyperfine states can strongly reduce this contribution, as well as indirect contributions, to optical and spin dephasing. The data in Fig. 3d show that the DEOP effect is saturated for the longest pumping time, which suggests that direct and indirect contributions have been largely quenched. This allows to estimate the dephasing contribution to Γ φ independent from DEOP to 285 Hz. Presumably, 171 Yb 3+ in site 1, which represent 50% of the total Yb 3+ concentration, cause a significant part of this broadening. It could be reduced to a large extent by using DEOP on these ions using e.g. a second laser. In this case, the remaining dephasing would be interactions with 89 Y 3+ . Since the latter are very slow 8 , it could be possible to reach the T 2,o = 2T 1,o limit.
Although this qualitative analysis can account for the general trends observed, a more detailed modeling and additional experiments are needed to precisely evaluate the processes affecting T 2,o . In particular, all ground state flip-flops should be included in simulations and their rates determined using spectral hole burning or other techniques. Further theoretical calculations of flip-flop rates and frequency shifts caused by spin flips could also be very useful.

II. DISCUSSION
The spin 1/2 rate equation model (Eqs. 1-2) is convenient to estimate parameters for efficient DEOP, i.e. low remaining population in B-spin upper level, p +,∞ B . First, low values of β o = R/R o are required and therefore small SLR rate R and/or strong optical pumping, i.e. larger R o . The latter may be limited, as in our case, by the spontaneous emission rate and branching ratios, which in turn can be increased using optical nano-cavities 26,27,41 . Small R values can be achieved by lowering magnetic field and temperature 8 . This is the case in our experiments, running at 2 K and zero magnetic field, giving β o ≈ 3.6 × 10 −5 and polarizations over 90%. However, SLR increases quickly with temperature or magnetic field, and Fig. 2d modeling predicts that at 4 K and zero field DEOP polarizes only 11% of the spins. Large β f f is also favorable and corresponds to strong flip-flops, i.e. large W 1 . This can be obtained with high spin concentration n 0 and lower inhomogeneous linewidth since W 1 ∝ n 2 0 /Γ inh,spin 8 . In 171 Yb 3+ :Y 2 SiO 5 , it is worth noting that the spin linewidth is especially narrow at zero magnetic field, Γ inh,spin = 1 MHz 17 , which increases flip-flop rates and gives β f f ≈ 0.13. Finally, pumping a larger fraction of ions will obviously result in stronger polarization by increasing C. However, as demonstrated in our experiments, low C values can still provide strong polarization. The case when more than one transition is optically pumped, as investigated in the 'Optical Coherence' section is more difficult to analyze with a simple 1/2 spin model. Non-pumped spins will interact with several classes of optically pumped spins that are polarized in different levels. This can lead to high polarization, as we observed, but also to opposite population changes and therefore remaining populations in some pumped levels. In this respect, isolated optical transitions, like the |4g ↔ |1e in 171 Yb 3+ :Y 2 SiO 5 simplify pumping schemes and will appear for spin splittings at least comparable to the optical inhomogeneous linewidth. While this can often be obtained with a high enough magnetic field, it may also increase SLR through direct processes and lower flip-flop rates by increasing spin inhomogeneous linewidth 39,42 , effects that both reduce DEOP.
This study suggests that efficient DEOP could be observed in other paramagnetic RE or transition metal ions doped materials. DEOP was observed in Cr 3+ :Al 2 O 3 34 and can also be seen in Nd 3+ :YVO 4 as shown in Fig. 5 of Ref. 43 , although it was not recognized as such in this work. It could especially be observed in other candidates of interest for applications in quantum technologies including Er 3+ , although this ion suffers from inefficient optical pumping which increases the requirement on low SLR 44 . Paramagnetic RE with non zero nuclear spins that show ground state splittings of a few GHz at zero magnetic field such as 167 Er 3+ , 145 Nd 3+ or 173 Yb 3+ could also behave similarly to 171 Yb 3+ for DEOP. Finally, other concentrated spin systems with optical transitions, such as NV − centers in diamond, could also show DEOP.
Using DEOP, we managed to reduce by 250% the optical homogeneous linewidths, to get a value of Γ h,o = 407 ± 15 Hz, which is the narrowest homogeneous linewidth reported at zero magnetic field for any RE, except for Eu 3+ :Y 2 SiO 5 40 . However, Eu 3+ only possesses nuclear degrees of freedom and its ground state nuclear hyperfine structure spans only 60 to 160 MHz in this crystal, depending on the isotope 45 . This is about 20 times less than 171 Yb 3+ (3 GHz), which in addition has 3 × 10 6 times stronger spin transition dipole moments 17 . 171 Yb 3+ is therefore much better suited for interactions with microwave photons while showing comparable optical coherence lifetimes.
As it is the case for 171 Yb 3+ :Y 2 SiO 5 , DEOP could extend optical and/or spin coherence lifetimes of other solid state systems by reducing spin-spin interactions. It therefore allows keeping a high concentration of active species with low dephasing. This is a particularly important point for ensemble based quantum devices, like absorptive quantum memories that require high optical absorption 46 . In addition, various configurations of populated levels can be in principle obtained, allowing to select the best configurations for e.g. strong optical and spin transitions, long coherence lifetimes, long lived shelving states etc. DEOP can also provide large scale spin initialization prior to processing and/or spectral tailoring. As an example, we achieved 96 ± 1% polarization in the single |4g level of 171 Yb 3+ :Y 2 SiO 5 (see SI). Finally, in the case of systems with different sites for optically adressable spins, a common feature in rare-earth doped crystals, it can also lower the perturbations from the unused centers.
In conclusion, we have observed large-scale spin polarization under laser excitation at fixed frequency in a rare earth doped crystal, 171 Yb 3+ :Y 2 SiO 5 . This is explained by a combination of optical pumping and spin diffusion by flip-flops that results in > 90% polarization for all spins in the sample volume addressed by the laser. The efficiency and versatility of the process is furthermore demonstrated by significantly increasing and decreasing optical coherence lifetimes T 2,o , depending on the pumping conditions. The longest T 2,o recorded, ≈ 800µs, is the longest recorded for a paramagnetic RE at zero magnetic field and is comparable to values for non-paramagnetic RE. Given the other favorable optical and spin properties of 171 Yb 3+ :Y 2 SiO 5 , our results open the way to new designs for broadband and efficient quantum memories for light 36 or optical to microwave transducers. We expect this process to be efficient in other rare earth doped crystals and concentrated systems of optically addressable spins like color centers in diamond. It could be used to tailor spin baths and therefore extend coherence lifetimes, or initialize spins on a large scale, topics which are central to many quantum technologies.

A. Sample
The Y 2 SiO 5 single crystal was doped with 10 ppm of 171 Yb 3+ (94% isotopic purity, see SI) and grown by the Czochralski technique. It was cut along the extinction axis b, D 1 and D 2 , with light propagating along the b axis and polarized along D 2 for maximal absorption. The length of the sample along b was 9.4 mm. Y 2 SiO 5 has a monoclinic structure and belongs to the C 6 2h space group. Yb 3+ can substitute Y 3+ ions equally in their two sites of C 1 point symmetry.

B. Experimental setup and optical pumping
The sample was placed inside a liquid helium bath cryostat at 2 K. Excitation was provided by a tunable single mode diode laser (Toptica DL 100) with a spectral width of 1 MHz. The beam on the sample was weakly focused on the sample with a diameter of 1 mm. All experiments were performed in transmission mode. Spectra were calibrated by recording signal from a Toptica FPI 100 Fabry-Perot interferometer (1 GHz free spectral range). An acousto-optic modulator (AOM, AA Optoelectronics MT80) in single pass configuration was used to gate the laser. The detector was an amplified Si photodiode (Thorlabs PDA150A). The power during optical pumping and scans was 7 mW and 0.4 mW, with frequency scans performed at a rate of 3 GHz/ms. A delay of 10 ms was kept between optical pumping and scanning to let the excited state population decay to the ground state. To probe absorption recovery, scans were performed at different delays after 20 s of optical pumping and at different temperatures.
Fraction of pumped ions are calculated from absorption spectrum using the formula where α and α 0 are absorption coefficients at the laser frequency and peak of the line, Γ 0 the full width at half maximum and Γ the pumped region spectral width. We have Γ 0 = 550 MHz 17 and Γ ≈ 5 MHz. The latter value corresponds to the hole observed on the absorption spectra and takes into account the laser linewidth (1 MHz), drift and other effects like power broadening. In Fig. 2a, the absorption lineshape does not change with pumping duration and the fraction of pumped ions is small. The peak absorption coefficient measured at 0 GHz and normalized by its value without OP, i.e. at thermal equilibrium, is therefore equivalent to k 4g .

C. Optical coherence measurement
For those measurements, a second AOM was added to enhance gating and avoid optical pumping during photon echo sequences. The beam was focused by a 100 mm focal length lens with a power of 7 mW. The photon echo was measured using a standard Hahn echo sequence (π/2 − τ − π − τ − echo) with durations of 1 and 2 µs for the π/2 and π pulses. Due to the laser jitter, the echo amplitude significantly fluctuated for τ > 300 µs. To overcome this issue, for a given delay τ , 50 successive echo sequences were recorded and only the strongest echo was kept. We checked that the echo sequences themselves did not cause optical pumping. Echo pulse power was also varied to look for instantaneous spectral diffusion, which was not observed.
Coherence enhancement by optically induced electron spin polarization in a rare-earth doped single crystal -Supplementary information VI. OPTICAL PUMPING

A. Populations
Since we are using an isotopically purified sample, the main contribution to the absorption spectra are composed of the 16 different optical transitions belonging to the S = 1/2, I = 1/2 171 Yb 3+ ions. A weak contribution of the S = 1/2, I = 0 nuclear spin isotopes (≈ 6% of all Yb 3+ ions), is observed at 0 GHz, see Fig. 4a. All spectra were fitted using the expression: in which the first term on the right hand side corresponds to 171 Yb 3+ and the second term to I = 0 isotopes. In Eq. (5), α(ν) and α 0,1 are absorption coefficients, ν the frequency, i (j) ground (excited) state labels and β ij the branching ratio between levels i and j (see Table VI A and 36 ). g(ν, ν k , Γ k ) are area-normalized Lorentzian functions with full width at half maximum Γ k and center frequency ν k ( g(ν, ν k , Γ k )dν = 1). Center frequencies were determined from hole burning experiments 31 and correspond to the scheme in Fig. 4. The ground state populations k ig verify 0 ≤ k ig ≤ 4 and k ig = 4. These are normalized to their thermal equilibrium values, which for our working temperatures are k ig,eq = 1, to a good approximation.  The absorption spectrum recorded at 2 K without prior optical pumping (OP) was first used to determine Γ 0,1 and α 0,1 . In this case, the 171 Yb 3+ population is equally distributed into the four hyperfine ground states and k ig = k ig,eq = 1. Best fit values were Γ 0 = 572 ± 40 MHz, Γ 1 = 540 ± 100 MHz, α 0 = 1.37 ± 0.16 cm −1 and α 1 = 0.32 ± 0.08 cm −1 . Experimental and fitted spectra are shown in Fig. 4b.

A. Qualitative mechanism
The cartoon shown in Fig. 6 explains the mechanism of Diffusion Enhanced Optical Pumping using a simplified system. Here, the ground state is composed of two spin levels and we consider only one optically excited state. Optical pumping (OP) is applied along the transition connecting the higher ground state spin level to the excited state. Ions in the excited state can relax towards the lower spin state, which results in spin polarization. However, since the optical transition is inhomogeneously broaden, the laser is resonant only with a subgroup of all ions. Relaxation of individual spins by interaction with the lattice is assumed to be very slow. At thermal equilibrium, both ground state spin levels are equally populated. DEOP can be described in the following way: an optically pumped spin initially in the higher spin state is transferred to the lower state. This spin can flip-flop with a non-pumped neighbor in the higher spin state. The pumped spin goes back to the higher state, while the neighbor goes to the lower state. If that happens, the pumped spin is transferred again to the lower state, resulting in two spins in the lower state. The neighboring spin can further flip-flop with another non-pumped spin in the higher state, after which the previous sequence can repeat, eventually leading to three spins in the lower state. In this way, the whole system of pumped and non-pumped spins can be completely polarized. Qualitatively this requires that the optical pump rate is higher than the flip-flop rate, such that the optically pumped spin spends most of their time in their lower state, while the flip-flop rate should be higher than the spin lattice relaxation rate.

B. Rate equation modeling of DEOP
Here, we use the same energy level scheme as described above for 171 Yb 3+ spins. Therefore, instead of the 4 levels resulting from the low-symmetry anisotropic hyperfine interaction, we consider a S = 1/2 spin system. The A-spins are optically pumped and assumed to be non-interacting with each other because of their low concentration. This is justified because the pump laser is much narrower than the optical inhomogeneous broadening. However, they interact with the non-pumped B-spins via magnetic dipole-dipole interactions, see Fig. 7. For A-spins, the optical pumping is resonant with the transition connecting the upper ground state level and the excited state. The effective pumping rate to the lower level is noted R 0 . It takes into account the optical excitation rate R L from |+ to |e , the excited state radiative population lifetime T 1,o and the branching ratio β for |e ↔ |− transition. In our experimental conditions, R L ≫ 1/T 1,o and R 0 = β/T 1,o . The population in the upper and lower ground states of A-and B-spins can also relax to each other through spin-lattice relaxation (SLR) at the same rate rate R. Although each A-spin has presumably a different environment in terms of distances and directions of neighboring B-spins, we do not take it into account and consider an average probability p +,− A for A-spins to be in the upper or lower state. The B-spins are not pumped, and, since they are in higher concentration, they interact with each other through where W ij is the flip-flop rate between A-spin i and B-spin j. We used p − X,i = 1 − p + Xi . According to the above assumptions, subscripts i and j can be dropped for probabilities and j,B W ij does not depend on i. Eq. (6) is then written as: Similarly we have for B-spins: Since W 1 and W 2 are summed respectively on the B-and A-spins, we have C = W 2 /W 1 , where C is the fraction of pumped spins.
The steady state solution of (7) and (8) is obtained by setting the time derivatives on the left hand side to zero: which gives: We further assume R o ≫ R, W 2 and W 1 ≫ R, W 2 , because in our system the SLR R rate is small, the optical pumping rate R 0 is strong, and only a small fraction of ions is pumped (C = W 2 /W 1 ≪ 1), as discussed in the main text. We introduce β o = R/(R o + R) ≈ R/R 0 and β f f = W 1 /(R o + W 1 ) and obtain: The rates at which the steady states are reached can also be obtained from the system of equations (7) and (8). Under the assumption R, W 2 ≪ R o , W 1 , there is a fast component with a rate R o + R and a slow one, called the polarization rate in the main text, with a rate: R P is the rate that is determined from the experiments of Fig. 2b, main text.

C. Spin flip-flops
For simplicity we assume the Zeeman g-and hyperfine A-tensors are diagonal in the same basis and have anisotropic form g x = g y = g z and A x = A y = A z . This assumption is well justified for certain solid-state systems (for example 171 Yb 3+ :Y 2 SiO 5 crystal 31 ). In this case, at zero magnetic field, the wavefunctions are given only by the hyperfine tensor, which makes all the levels to be non-degenerate: All the spin transitions, in this situation, are connected purely by S x (|1 ↔ |4 and |2 ↔ |3 ), S y (|1 ↔ |3 and |2 ↔ |4 ) or S z (|1 ↔ |2 and |3 ↔ |4 ) spin 1/2 operators. This strongly simplifies the expression for the dipole-dipole interaction H dd that will contain only corresponding operators. In this case the flip-flop rate estimated by Fermi golden rule | i, f |H dd |f, i | 2 will be proportional to the corresponding element of the g-tensor 39 for |1 ↔ |2 and |3 ↔ |4 : ∝ g 4 z for |1 ↔ |3 and |2 ↔ |4 : ∝ g 4 y , for |1 ↔ |4 and |2 ↔ |3 : ∝ g 4 x . The anisotropy of the g-tensor can lead to dramatically different relaxation times for different transitions. In the case of 171 Yb 3+ in site 2 of Y 2 SiO 5 , g z = 6.06, g y = 1.5, g x = 0.13 30 . This predicts a few orders of magnitude variation for different transitions, flip-flops within the |1 ↔ |2 and |3 ↔ |4 pairs of levels being much faster than all the other ones.

D. SLR modeling
The recovery time R c of the |4g ↔ |1e optical line after the OP has been stopped corresponds to spin lattice relaxations. Indeed spin flip-flops do not change overall populations in the case of no optical pumping. R c has been measured for different temperatures and its variations are shown on Fig 2d in the main text. Those variations can be modeled by considering the one-phonon direct process and two-phonons processes 48 : In this equation, the same parameters than in a previous study 32 , θ D = 100 K and θ E = 337 K, have been used for the two-phonon part. The direct process term uses the average splitting between |4g and the three other ground state spin levels, ν ef f = 2.06 GHz. The fitted coefficients are α D = 3.9 × 10 −4 s −1 and α R = 0.9 × 10 18 s −1 .K 4 . The latter value is reasonably close to the one determined in 32 .

E. Narrow hole decays
In order to compare the W 1 value extracted from the fit in Fig. 2c (main text) to experimental flip-flop rates, we investigated the dynamics of a narrow anti-hole in the |4g ↔ |1e transition. It was obtained by burning a hole close to the center of the |2g ↔ |4e transition at +2.5 GHz (see Fig. 1c, main text) for 10 ms, a duration short enough to avoid DEOP. The anti-hole height was measured at varying delays after the hole burning. Each measurement was preceded by an initialization sequence of 50 pulses scanned over 10 GHz to prevent accumulating populations. Fig.  8 shows the experimental data together with a two-exponential fit, giving rates of 500 and 13 s −1 . The larger rate is attributed to the fast |4g ↔ |3g flip-flops and the other one, which correspond to W 1 in the rate equation model, to the intermediate |4g ↔ |2g flip-flops.

A. Populations
For echo measurements under DEOP (Fig. 3, main text), normalized populations (k 1g ,k 2g ,k 3g ,k 4g ) were determined from absorption spectra, as described in section VI A. The spectra and fits are shown in figures 9 to 15, and the corresponding k ig values presented in Table II.

B. Spin coherence
We performed another set of measurements to investigate the effect of the optical pumping on the spin coherence at 3 K. For this, we measured the spin coherence time T 2,s through optical detection of a spin echo in a Hahn sequence using Raman heterodyne scattering (RHS), see 17 . All spin echo measurements were carried out on the |4g ↔ |3g transition (655 MHz) of site 2 by varying total population in |3g and |4g states for different optical pumping conditions.
In a first set of measurements, the optical pumping was performed with the laser set between |4g ↔ |1e and |3g ↔ |1e optical transitions to polarize the spin ensemble into |1g and |2g spin states. The optical pumping, in this set of measurements, is done by scanning the laser over the inhomogeneous broadening during 500 ms to speed up the pumping process and polarize larger spin population. The second laser was used to detect the spin echo signal through RHS detection and was set to |4g ↔ |1e transition. The populations in each state were estimated using separate optical absorption measurements utilizing the previously measured optical branching ratio table (see section VI A). As a result, we observe a substantial increase of the spin coherence time up to 2.5 ms for the strongest polarization of the spin ensemble (Fig. 16). We note that similar values were measured previously in this sample 17 .
On a second stage, the optical pumping was performed with a laser set between the |1g ↔ |4e and the |2g ↔ |3e optical transitions to inverse the polarization and have higher population of |3g and |4g spin states. Additionally the transition |3g ↔ |2e was weakly driven to initialize the spin ensemble to create initial spin polarization for the RHS generation. As a result, a reduction of the spin coherence time up to 0.2 ms was measured (Fig. 16).
We attribute this behavior to the modification of the spin flip-flop process on |3g ↔ |4g transition, directly limiting the spin coherence time through |3g and |4g population lifetimes. Indeed, in contrast to the optical coherence study, the increase of the spin coherence was measured when pumping into both |3g and |4g . In this case, the optical pumping reduces the flip-flop rate on |3g ↔ |4g transition, by proportionally increasing the cross relaxation between |1g ↔ |2g ground states. Assuming that the flip-flop probabilities on |1g ↔ |2g and |3g ↔ |4g transition are the same, such population distribution doesn't lead to an overall reduction of the magnetic field noise created by the flipflops, which could potentially explain the coherence increase. In this situation, the increase of the spin coherence on |3g ↔ |4g transition can be explained by the increase of |3g and |4g population lifetimes induced by the reduction of flip-flops between |3g and |4g spin states.
To estimate the spin flip-flop rate we apply the simple coherence time model from the main text (Eq. (3)). For this we assume that the pure dephasing term is constant for different pumping conditions and the lifetime of the ground states is limited by the flip-flop process on |4g ↔ |3g transition. The last assumption results in quadratic dependence on the population in these two spin states: where k ′ 3g and k ′ 4g are the populations after applying the π/2 microwave pulse for the spin echo measurement. Since the first π/2 microwave pulse will average initial populations between |3g and |4g spin states we can write k ′ 3g k ′ 4g = ((k 3g + k 4g )/2) 2 . Initial populations k 3g , k 4g for various optical pumping conditions were measured by fitting the absorption profile taken before applying the microwave sequence. By fitting the measured coherence times T 2,s (Fig. 16) using Eq. (19) we estimate the flip-flop rate to be 0.39 ms for equal population of all spin states, with spin coherence time limit of (πΓ φ ) −1 = 7.2 ms.
The estimated equilibrium flip-flop process will limit the optical coherence to 0.8 ms which is more than two times bigger than the optical coherence time of 0.3 ms measured without optical pumping. This can be explained by a stronger sensitivity of the optical transition frequency to magnetic field fluctuations coming from the crystalline spin bath, which are modified by the optical pumping.

A. Polarization into a single hyperfine level
Here the laser is set at +0.22 GHz on the absorption spectrum shown on Fig. 17. At this frequency, the laser is resonant with some ions in |2 g , |3 g and |4 g , through the optical transitions |2 g ↔ |2 e , |3 g ↔ |3 e , |3 g ↔ |4 e , |4 g ↔ |3 e and |4 g ↔ |4 e (see Fig. 4b). After 20s of OP duration, the absorption spectrum shown in Fig.  17 is obtained. Corresponding normalized populations are k 1 = 3.84 ± 0.02, k 2 = 0.04 ± 0.02, k 3 = 0.12 ± 0.12, k 4 = 0.12 ± 0.04, which means that 96 ± 1 % of the total population in the volume addressed by the laser has been stored into the |1 g state. As expected, we can see that the |1 g ↔ |4 e absorption has reached a value almost four times larger than at thermal equilibrium.