Sacha Welinski

Alexey Tiranov

Moritz Businger

Alban Ferrier

Mikael Afzelius

Philippe Goldner
email: philippe.goldner@chimieparistech.psl.eu

N Vamivakas

S.-Y Lee

J Wrachtrup

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 spinspin interactions and spin diffusion, leads to substantial spin polarization in a paramagnetic rare earth doped crystal, 171 Yb 3+ :Y2SiO5. 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.

Systems with both spin and optical transitions offer improved functionalities for quantum technologies 1 . They allow storage and entanglement of photonic quantum states for quantum communications 2,3 , interfacing processing nodes with optical networks for distributed quantum computing 4 or efficient detection for quantum sensing 5 . Key parameters in these centers are the coherence lifetimes of optical and spin transitions, which affect storage time, operation fidelity and sensitivity. In the solid state, major sources of perturbation to quantum states are due to magnetic fluctuations in the centers environment that couple to transitions through magnetic dipole-dipole interactions. This magnetic noise is often due to flips of electron or nuclear spins carried by the host material atoms 6 , impurities or defects 7 , or the centers of interest themselves 8,9 . Considerable work has been devoted to engineer these ensembles, known as the spin bath, in order to reduce their detrimental effect on coherence lifetimes. This includes isotope purification to eliminate elements with non-zero spins 10,11 and application of high magnetic fields 12,13 and/or very low temperatures 14 to freeze spins in their lower energy state. Another approach is to decouple the optically addressable spins from the bath by using clock transitions that are insensitive to magnetic field fluctuations in first order [15][16][17] or by filtering out bath fluctuations by dynamical techniques 18,19 . Although they can be very efficient, these methods may be complex to implement.

). As shown in this Article, DEOP allows coherence lifetime extension by creating specific, highly out of equilibrium, population distributions among spin levels, a feature difficult to obtain by using magnetic field, low temperature, or by optical pumping alone.

We demonstrate DEOP in a rare earth (RE) doped crystal, 171 Yb 3+ :Y 2 SiO 5 (YSO). These materials, in which optical and spin coherence lifetime can reach up to ms and hours at low temperatures 6,12,20 , are actively investigated for spin based quantum photonic applications ranging from quantum memories 2,21-24 , processors 25 and single photon sources 26,27 to optical-microwave transducers 28,29 . 171 Yb 3+ :YSO, in which long coherence lifetimes have been shown for both optical and spin transitions at zero magnetic field 17 , is particularly promising in this area [30][31][32][33] . Using DEOP, we obtained spin polarizations > 90%, a much stronger effect than previously reported in ruby 34 . Thanks to its spin ensemble tailoring capabilities, we then used DEOP to reduce or enhance specific spin-spin interactions, and in particular extend 171 Yb 3+ optical coherence lifetimes T 2 to about 800 µs, a 2.5-fold increase compared to thermal equilibrium. This is the longest optical T 2 reported for any paramagnetic solid state system at zero or verylow magnetic fields, which can be especially interesting for interfacing with superconducting qubits and resonators. DEOP uses a counter intuitive scheme that exploits interactions to ultimately control them and should be effective in b -6 -4 -2 0 2 4 6 Frequency (GHz) 0 0.5 1 1.5 2 2.5 Abs. Coef. (cm -1

)

 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 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

k ∞ 4g ≡ p +,∞ B ≈ 1 2 + Cβ f f /β o (1) R P ≈ R(2 + Cβ f f /β o ), (2)
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 R o ≫ W 1 , 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, T 2,o = 278 ± 20 µs or Γ h,o = 1/πT 2,o = 1.1 ± 0.1 kHz. 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:

Γ h,o = 1 2πT 1,4g + 1 2πT 1,1e + Γ φ . (3)
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 and3d. 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.

III. ACKNOWLEDGEMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 71272, the IMTO Cancer AVIESAN (Cancer Plan, C16027HS, MALT) and FNS Research Project No 172590.

IV. AUTHOR CONTRIBUTIONS S.W. and A.F. grew the sample, S.W. and P.G. conceived and performed the experiments, except for the additional spin coherence measurements which were performed by M. B. and A. T.. S.W., P.G., M.A., A.T. and M.B analyzed the results, and S.W. and P.G. wrote the manuscript with inputs from all co-authors. P.G. and M.A. provided overall oversight of the project.

V. METHODS

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

C = 2 π Γ Γ 0 α α 0 , (4)
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:

α(ν) = α 0 i=1..4,j=1..4 k ig β ij g(ν, ν 0,ij , Γ 0 ) + α 1 g(ν, ν 1 , Γ 1 ). ( 5
)
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 I: Branching ratios for 171 Yb 3+ :Y2SiO5 in site 2 and light polarized along D2 axis using spectral holeburning technique 36 .

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.

Spectra obtained after DEOP were fitted by varying k ig coefficients while keeping Γ 0,1 to the previous values. In some cases, adjustments of α 0 (± 6 %) and α 1 (± 10 %) were needed due to small changes in experimental conditions (light polarization, beam alignment with respect to sample). Fig. 5b displays the fit of the spectra in Fig. 1c of the main text resulting in k 1g , k 2g , k 3g , k 4g = 1.67 ± 0.30, 1.28 ± 0.36, 1.01 ± 0.12, 0.04 ± 0.04.

VII. DEOP MODELING 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 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 resonant spin flip-flops mediated by the magnetic dipole-dipole interaction. This interaction is assumed to be fast enough on the time scale of the experiment so that all B-spins have the same probability p +,- B of occupying the upper or lower states. The rate equation for an A-spin i is:

dp + A,i dt = -(R 0 + R)p + A,i + R(1 -p + A,i ) - j,B W ij p + A,i (1 -p + B,j ) -W ij (1 -p + A,i )p + B,j , (6)
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:

dp + A dt = -(R 0 + R)p + A + R(1 -p + A ) -   j,B W ij   p + A (1 -p + B ) -(1 -p + A )p + B = -(R 0 + R)p + A + R(1 -p + A ) -W 1 p + A (1 -p + B ) -(1 -p + A )p + B = -(R 0 + R)p + A + R(1 -p + A ) -W 1 p + A -p + B . (7)
Similarly we have for B-spins:

dp + B dt = -Rp + B + R(1 -p + B ) -   i,A W ij   p + B (1 -p + A ) -(1 -p + B )p + A = -Rp + B + R(1 -p + B ) -W 2 p + B (1 -p + A ) -(1 -p + B )p + A = -Rp + B + R(1 -p + B ) -W 2 p + B -p + A . (8)
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:

0 = -(R 0 + R)p +,∞ A + R(1 -p +,∞ A ) -W 1 p +,∞ A -p +,∞ B (9) 0 = -Rp +,∞ B + R(1 -p +,∞ B ) -W 2 p +,∞ B -p +,∞ A , (10)
which gives:

p +,∞ A = R(2R + W 1 + W 2 ) 2R(R o + W 1 ) + W 2 (R o + 2R) + 4R 2 (11) p +,∞ B = R(R o + 2R + W 1 + W 2 ) 2R(R o + W 1 ) + W 2 (R o + 2R) + 4R 2 . ( 12
)
We further assume R o ≫ R, W 2 and W 1 ≫ R, W 2 , because in our system the SLR R 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: p +,∞ A ≈ W 1 R o + W 1 2 + R o W 1 R(R o + W 1 ) C -1 (13)
≈ β f f 2 + Cβ f f /β 0 (14) p +,∞ B ≈ 2 + R o W 1 R(R o + W 1 ) C -1 (15)
≈ 1 2 + Cβ f f /β o . ( 16
)
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 = R(2 + Cβ f f /β o ). ( 17
)
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:

|1 = (| ↑⇑ -| ↓⇓ )/ √ 2, |2 = (| ↑⇑ + | ↓⇓ )/ √ 2, |3 = (| ↑⇓ -| ↓⇑ )/ √ 2, |4 = (| ↑⇓ + | ↓⇑ )/ √ 2.
All the spin transitions, in this situation, are connected purely by S 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.

R c = α D coth( hν ef f 2k B T ) + α R π 2 0 q 8 e -θ D T sinq dq (1 -e -θ D T sinq ) 2 (θ 2 E -θ 2 D sin 2 q) 2 (18)
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

VIII. OPTICAL COHERENCE

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 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:

T -1 2,s = R f f k ′ 3g k ′ 4g + πΓ φ , (19)
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. 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 17: Absorption spectrum after 20 s OP for the laser set at +0.22 GHz (black line) and fit (red line). Corresponding normalized populations are: k1 = 3.84 ± 0.02, k2 = 0.04 ± 0.02, k3 = 0.12 ± 0.12, k4 = 0.12 ± 0.04. Blue line: absorption spectrum at thermal equilibrium.

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.

FIG. 1 :

 1 FIG. 1: Diffusion Enhanced Optical Pumping (DEOP) in 171 Yb 3+ :Y2SiO5. a DEOP mechanism: optically pumped spins (red circles) initially surrounded by spins in thermal equilibrium (blue circles) gradually polarize neighboring and further apart spins (yellow circles) through flip-flop interactions. b Energy diagram showing the 171 Yb 3+ hyperfine structure. 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 ).

FIG. 2 :

 2 FIG.2: Dynamics of DEOP and spin-lattice relaxation. a Absorption profile around the |4g ↔ |1e transition after 0 s, 1 s, 3 s, 5 s, 7 s, and every 5 s from 10 s to 40 s of OP at -0.17 GHz (light blue arrow). 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 processes31 . 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.

FIG. 3 :

 3 FIG. 3: Optical coherence lifetime under DEOP. a Absorption spectra after 0, 1 and 10 s of OP at -1.44 GHz. Inset: normalized ground state populations k1g, k2g, k3g, k4g = 0.02 ± 0.02, 2.30 ± 0.12, 0.02 ± 0.02, 1.70 ± 0.06. b Absorption spectra after 0 , 1 and 10 s of OP at +2.67 GHz. Inset: normalized ground state populations k1g, k2g, k3g, k4g = 0.02 ± 0.02, 0.02 ± 0.02, 1.95 ± 0.10, 2.05 ± 0.08. c Photon echo decays under DEOP for the |4g ↔ |1e transition measured at -4.6 GHz (arrow in a,b). Decay colors and numbers correspond to spectra in a,b, solid lines are exponential fits to the data. d Optical coherence times T2,o deduced from fits in c (color-coded). All error bars correspond to a 95% confidence interval.

FIG. 4 :FIG. 5 :

 45 FIG. 4: a. Energy diagram of the hyperfine structure of 171 Yb 3+ in Y2SiO5 in site 2 for 2 F 7/2 (0) and 2 F 5/2 (0). The different colors correspond to the optical transitions connecting the same ground state spin level. b. Absorption spectrum at 2 K without prior OP. The different optical transitions constituting the whole absorption spectrum are shown. Their color correspond to the arrow ones in the energy diagram. The residual absorption line from I = 0 Yb 3+ isotopes is observed. The green dashed line represents the fitted curve α(ν) defined in Eq. (5).a b

FIG. 6 :FIG. 7 :

 67 FIG. 6: Qualitative mechanism for DEOP.

FIG. 8 :

 8 FIG.8: Decay of an anti-hole in the |4g ↔ |1e line (crosses) and two-exponential fit (solid line).

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 :

  -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.

FIG. 9 :FIG. 10 :

 910 FIG. 9: Same as in Fig.4a,b but with the absorption spectrum recorded after 0.5 s OP at -1.44 GHz.

FIG. 11 :FIG. 12 :FIG. 13 :

 111213 FIG.11: Same as Fig.9, but after 2 s OP at -1.44 GHz.

FIG. 14 :FIG. 15 :

 1415 FIG.14: Same as Fig.9, but after 1 s OP at +2.67 GHz.

FIG. 16 :

 16 FIG. 16: Spin coherence times T2,s of |4g ↔ |3g transition for various initial populations in |3g and |4g spin states (circles) and fit to a model of coherence limited by population lifetimes (red line, see text).

  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.

  FIG.17: Absorption spectrum after 20 s OP for the laser set at +0.22 GHz (black line) and fit (red line). Corresponding normalized populations are: k1 = 3.84 ± 0.02, k2 = 0.04 ± 0.02, k3 = 0.12 ± 0.12, k4 = 0.12 ± 0.04. Blue line: absorption spectrum at thermal equilibrium.

  Table VI A and36 ). 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 experiments31 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.

	|1e |2e |3e |4e
	|1g 0.15 0.06 0.08 0.71
	|2g 0.06 0.19 0.71 0.04
	|3g 0.07 0.71 0.16 0.06
	|4g 0.72 0.04 0.05 0.19

TABLE

  .

	OP condition	Absorption spectrum	k1	k2	k3	k4
	No prior OP	Figure 4	1	1	1	1
	0.5 s OP laser at -1.44 GHz	Figure 9	0.82 ± 0.06 1.36 ± 0.08 0.56 ± 0.08 1.26 ± 0.06
	1 s OP laser at -1.44 GHz	Figure 10	0.61 ± 0.06 1.62 ± 0.08 0.39 ± 0.06 1.39 ± 0.06
	2 s OP laser at -1.44 GHz	Figure 11	0.27 ± 0.04 2.01 ± 0.08 0.13 ± 0.12 1.59 ± 0.06
	5 s OP laser at -1.44 GHz	Figure 12	0.06 ± 0.04 2.24 ± 0.08 0.05 ± 0.05 1.70 ± 0.08
	10 s OP laser at -1.44 GHz	Figure 13	0.02 ± 0.02 2.30 ± 0.12 0.02 ± 0.02 1.70 ± 0.06
	1 s OP laser at +2.67 GHz	Figure 14	0.50 ± 0.02 0.10 ± 0.02 1.64 ± 0.08 1.76 ± 0.04
	10 s OP laser at +2.67 GHz	Figure 15	0.

02 ± 0.02 0.02 ± 0.02 1.95 ± 0.10 2.05 ± 0.08

TABLE II :

 II Normalized ground state populations corresponding to the different OP conditions.


S Zhang

N Galland

N Lučić

R Le Targat

A Ferrier

P Goldner

B Fang

Y Le Coq

S Seidelin
email: signe.seidelin@neel.cnrs.fr

Inhomogeneous response of an ion ensemble from mechanical stress

Keywords: numbers: 42.50.Wk, 42.50.Ct., 76.30.Kg

Material strain has recently received growing attention as a complementary resource to control the energy levels of quantum emitters embedded inside a solid-state environment. Some rare-earth ion dopants provide an optical transition which simultaneously has a narrow linewidth and is highly sensitive to strain. In such systems, the technique of spectral hole burning, in which a transparent window is burnt within the large inhomogeneous profile, allows to benefit from the narrow features, which are also sensitive to strain, while working with large ensembles of ions. However, working with ensembles may give rise to inhomogeneous responses among different ions. We investigate experimentally how the shape of a narrow spectral hole is modified due to external mechanical strain, in particular, the hole broadening as a function of the geometry of the crystal sites and the crystalline axis along which the stress is applied. Studying these effects are essential in order to optimize the existing applications of rare-earth doped crystals in fields which already profit from the more well-established coherence properties of these dopants such as frequency metrology and quantum information processing, or even suggest novel applications of these materials, for example as robust devices for force-sensing or highly sensitive accelerometers.

The extraordinary coherence properties of rare-earth ion dopants in a host crystal have led to their description in terms of a "frozen ion gaz" as they constitute the solid state emitters that come the closest to isolated ions trapped with electromagnetic fields inside a vacuum chamber in terms of coherence properties [START_REF] Yano | Ultralong optical dephasing time in Eu 3+ :Y2SiO5[END_REF][START_REF] Equall | Ultraslow optical dephasing in Eu 3+ :Y2SiO5[END_REF][START_REF] Thiel | Rare-earthdoped materials for applications in quantum information storage and signal processing[END_REF][START_REF] Goldner | Rare Earth-Doped Crystals for Quantum Information Processing[END_REF]. However, the impact of the crystalline matrix cannot be entirely neglected, as illustrated for instance by the large inhomogeneous absorption profile of the ion ensemble, arising from a non-uniform distribution of static strain due to the doping which slightly distorts the crystalline lattice locally. The technique of spectral hole burning has been shown to surmount such inhomogeneous effects, as ions with a particular frequency are selectively pumped to an hyperfine, dark state. The width of the created spectral hole, despite the fact that a large number of ions participate in the signal, reflects the single-ion characteristics, and as long as the crystal is not subject to external perturbations, these characteristics are preserved in time. This has been extensively exploited for instance in ultrahigh-precision laser stabilisation and spectroscopy [START_REF] Julsgaard | Understanding laser stabilization using spectral hole burning[END_REF][START_REF] Thorpe | Frequency stabilization to 6 × 10 -16 via spectral-hole burning[END_REF][START_REF] Gobron | Dispersive heterodyne probing method for laser frequency stabilization based on spectral hole burning in rare-earth doped crystals[END_REF], in which a laser is frequency stabilized by locking it to a narrow spectral hole. However, if the crystal containing a spectral hole is subject to modifications in its physical environment (for instance, external stress), the ions constituting the spectral hole may react differently according to their local environment, and in addition to a global displacement of the spectral hole, a deformation can occur. In order to fully benefit from solid state systems based on ensembles, such effects must be studied and understood.

In this work, we are particularly interested in the 580 nm optical transition in Eu 3+ ions in an Y 2 SiO 5 (YSO) host matrix, as it is both ultra-narrow linewidth, potentially down to 122 Hz [START_REF] Equall | Ultraslow optical dephasing in Eu 3+ :Y2SiO5[END_REF], and is sensitive to mechanical strain, allowing for an efficient strain coupling between dopants and crystalline matrix. Earlier experiments have demonstrated such strain coupling in the context of optomechanical systems, for instance using quantum dots [START_REF] Yeo | Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system[END_REF] or NV centers in diamond [START_REF] Teissier | Strain Coupling of a Nitrogen-Vacancy Center Spin to a Diamond Mechanical Oscillator[END_REF][START_REF] Ovartchaiyapong | Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator[END_REF][START_REF] Macquarrie | Cooling a mechanical resonator with nitrogenvacancy centres using a room temperature excited state spin-strain interaction[END_REF], but these emitters either exhibited a large linewidth or low strain sensitivity. Generally speaking, optical transitions exhibit relatively high strain sensitivity, but in most solid state emitters, such transitions exhibit very broad (in the GHz range) linewidths, with, at best, zero-phonon lines in the MHz range. On the contrary, the electronic structure of trivalent lanthanide ions, in which the optical transitions inside the 4f shell are protected by 5s and 5p fully populated shells, allows for record narrow transitions, as well as the high strain sensitivity that is typical for an optical transition. Thus, when viewing the strain sensitivity as a resource, this kind of system represents the best of both worlds, as witnessed by proposals for creating strain-engineered quantum hybrid systems based on this system [START_REF] Mølmer | Dispersive coupling between light and a rare-earth-ion-doped mechanical resonator[END_REF][START_REF] Seidelin | Rapid cooling of a strain-coupled oscillator by an optical phase-shift measurement[END_REF]. On the other hand, depending on the application, this high sensitivity can also be a nuisance, of which the effect needs to be minimized. For instance, vibrations of the sample give rise to an acceleration of the crystal which leads to a compression, in turn creating fluctuations in strain [START_REF] Louchet-Chauvet | Piezospectroscopic measurement of high-frequency vibrations in a pulse-tube cryostat[END_REF], which can limit the performance of the system. Such vibrations often occur due to the operation of a cryostat, and represents a recurring challenge for many applications with this type of dopants. Therefore, whether a resource or a nuisance, an in-depth study of the response of Eu 3+ in a YSO matrix to strain seems valuable in a variety of contexts.

We study the 7 F 0 → 5 D 0 transition shown in figure 1 a). More precisely, the YSO crystal possesses two locations within the unit cell where Eu 3+ can substitute for Y 3+ , referred to as site 1 and 2, with vacuum wavelengths of 580.04 nm and 580.21 nm, respectively. These are the only two non-equivalent substitution sites in the crystalline matrix, both exhibiting a C 1 symmetry. Each state is composed by 3 hyperfine states, with mutual separations in the 30-100 MHz range [START_REF] Yano | Ultralong optical dephasing time in Eu 3+ :Y2SiO5[END_REF]. As illustrated in figure 1 a), the spectral holes are formed by pumping the ions resonantly from the 7 F 0 to the 5 D 0 state. From there, they decay radiatively, mainly to the 7 F 1 state [START_REF] Könz | Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu 3+ : Y2SiO5[END_REF], before decaying non-radiatively to the three hyperfine states in the 7 F 0 manifold. Optical pumping prevents population build-up in the hyperfine level with which the pump beam is resonant, creating a transparent window in the inhomogeneous profile at this exact frequency. A spectral hole represents approximately 10 13 Eu 3+ ions.

We recently determined the frequency shift of this transition by studying the linear displacement in frequency of a spectral hole when applying external stress to the crystal [START_REF] Galland | Coq Mechanical tunability of an ultra-narrow spectral feature with uniaxial stress[END_REF]. Here, we will instead focus on how the shape of the spectral hole evolves according to the applied stress, for these two different crystal sites, as this provides insight into the microscopic environment of the ion dopants constituting the spectral hole. As the direction parallel to which the stress is applied also plays an important role, in the overall frequency shift as well as in the broadening, we investigate the application of stress In both schematics and photo, an electrically grounded metal structure surrounding the crystal is visible, which prevents a build-up of static charges.

along two non-equivalent axes: either parallel to the D 1 or to the D 2 dielectric axes, see figure 1 b). For maximum absorption, the pump beam is propagated along the crystalline b-axis (which coincides with the dielectric D 3 axis) and polarized linearly along the D 1 axis [START_REF] Könz | Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu 3+ : Y2SiO5[END_REF][START_REF] Ferrier | Variations in the oscillator strength of the 7 F0 -5 D0 transition in single crystals[END_REF].

The physical setup is similar to the one described in detail elsewhere [START_REF] Gobron | Dispersive heterodyne probing method for laser frequency stabilization based on spectral hole burning in rare-earth doped crystals[END_REF][START_REF] Galland | Coq Mechanical tunability of an ultra-narrow spectral feature with uniaxial stress[END_REF]. In brief, we use 2 diode lasers (master and slave) at 1160 nm. The master laser is locked to a frequency stabilization cavity by the Pound Drever Hall method allowing for a frequency instability below 10 -14 for time constants of 1-100 s (corresponding to a few Hz linewidth). The slave laser is locked to the master laser. They are both subsequently frequency doubled to reach 580 nm, with an output intensity of approximately 5 mW. The slave laser possesses the same stability as the master, but its frequency is continuously tunable in a range of several GHz. In addition, acousto-optic modulators allow us to scan the frequency across the spectral structures with a range of approximately 1 MHz. An absolute frequency measurement is provided by referencing the signal to a frequency comb. To monitor the shape of the spectral hole, we use an avalanche photodiode to record the laser intensity after absorption of the crystal, normalized by the laser intensity send directly to the detector. Polarizing beamsplitters combined with λ/2 waveplates allows for adjusting the intensity. The crystal is maintained at 3.15 K in a closed-loop cryostat positioned on an active vibration isolation platform.

In order to quantitatively study the response of a spectral structure to mechanical stress, a calibrated force in a given direction must be applied at cryogenic temperature. We use a staircase-like structure (see figure 2), with a vertical position being controlled by a cryo-compatible motorized platform, providing a way of placing a vari- able number of weights on the top of the crystal inside the science chamber in the cryostat (see photo in figure 2). From the mass of the weights, we have access to an absolute calibrated force. We start the sequence by burning a spectral hole with no weights on the crystal, and we then add the number of weights corresponding to the desired stress.

During the initial stages of the experiment, we observed irreversible effects on the spectral hole while operating the translation stage for depositing the weights. More precisely, by operating the piezo translation stage, even without depositing weights on the crystal, the hole would exhibit a splitting, and moreover, the spectral hole would not return to its initial shape, even after the piezo operation has ended (note however that no permanent micro-structural modification of the crystal was observed). Examples of this behavior is illustrated in figure 3, where we show an example of the shape of a spectral hole after having operated the piezo stage for a given excursion distance. This behavior can be understood as the energy transitions in Eu 3+ are also sensitive to electric fields [START_REF] Thorpe | Shifts of optical frequency references based on spectralhole burning in Eu 3+ :Y2SiO5[END_REF][START_REF] Macfarlane | Optical Measurement of the Effect of Electric Fields on the Nuclear Spin Coherence of Rare-Earth Ions in Solids[END_REF][START_REF] Li | Slow-light-based optical frequency shifter[END_REF], and when operating the motorized translation stage (which works by stick-slip mechanism of a piezo on a fine threaded screw), accumulated stray charges can generate irreversible displacements or deterioration of the spectral hole. Moreover, silver lacquer on the crystal used for optimal thermal contact seemed to exert an additional stress on the crystal while applying weights, modifying the spectral hole in a manner depending of the amount and position of the silver lacquer in contact with the crystal. The solution we adopted consists in removing the silver lacquer and replacing it with a metal shield positioned on the top of, and extending down The effect of stress on the shape of the spectral hole, for crystal site 2, with stress applied along the D1 axis. The solid line shows the initial hole without stress. The dotted line corresponds to the case in which the stress is applied without depositing a crystalline powder at the contact surfaces, and the dashed line to the case in which a powder has been used. The stress also gives rise to a global shift in frequency of the spectral structures, but this shift has been subtracted in order to center the curves in frequency.

around, the crystal, but without direct contact with the vertical sides of the crystal. In addition we surrounded the entire piezo translation stage with a metal cover (visible in the photo in figure 2.) Both the crystal-and piezo stage cover were subsequently grounded. The weights for generating stress, instead of being deposited directly on the crystal, were then put on the crystal metal shield. These modifications resulted in the disappearing of any effects on shape of the spectral hole due to operating the piezo translation stage.

However, when subsequently adding weights to the crystal, instead of a pure shift in frequency of the spectral hole, we observed a clear splitting in the spectra. As such splitting is not expected to appear as long as the stress is applied perpendicular to the b-axis [START_REF] Bartholomew | Investigation of the scalability of rareearth-ion quantum hardware[END_REF] (which is our case, as we apply stress parallel to either D 1 or D 2 ), we suspected that the stress was not perfectly parallel to the axis chosen. One potential explanation was that the weights were not completely centered relative on the crystal. To improve on this aspect, an x -y translation stage was added for centering the objects on the crystal, and the splitting seemed to decrease slightly, but was not eliminated. We suspected that the contact surface between the weights and crystal, despite all the care taken, was not sufficiently uniform as to create a strain field perfectly perpendicular to the b-axis inside the crystal. In order to homogenize the stress, we used a ≈ 1 mm layer of YSO powder with a grain size of the order of micrometers at the two contact surfaces of the crystal. That is, we deposited the powder on the top of the crystal (between the crystal and the metal shield), and below the crystal (between the crystal and the cryostat platform). When repeating the experiment, we observed that the splitting is no longer visible for the given pressures. These effects are illustrated in figure 4, in which we have applied stress parallel to the D 1 -axis. The solid line corresponds to the initial spectral hole with a width below 5 kHz. When applying stress, before homogenizing the stress with the crystalline powder, we obtain spectra exhibiting large splittings, as illustrated by the dotted line, which appear to be eliminated by the use of the crystal powder (dashed line).

We do observe, however, a persistent broadening of the spectral hole with applied stress, an effect we wish to investigate quantitatively in the following. First, in order to verify that the strain does not exhibit a significant variation across the crystal, after burning a spectral hole and adding strain, we probe the spectral hole in 5 different locations (center, left, right, top and bottom) of the crystal, with a beam diameter of 2 mm. The spectral hole appears to have the same shape in the different location, and the same global shift in frequency (with a variation much smaller than the width of the spectral hole), eliminating the possibility of a strong strain gradient across the crystal. In order to study the residual broadening systematically, we record the shape of the spectral hole as a function of stress applied (corresponding to 1 to 5 weights added to the crystal) with a 2.5 millimeter beam diameter, centered in the middle of the crystal. The experiment is repeated for both crystal sites (S1 and S2) and for adding the stresses parallel to the D 1 and D 2 axes. The result of this study is shown in figure 5. By denoting the initial width of the spectral hole, before applying stress, Γ 0 , and the width measured for a given stress Γ, the broadening is given by (Γ-Γ 0 )/(Γ 0 ). For the data in figure 5, the initial hole widths are the following: Γ 0 (D1S1)=5.4 kHz, Γ 0 (D1S2)=4.0 kHz, Γ 0 (D2S1)=3.8 kHz and Γ 0 (D2S2)=4.4 kHz. We checked that the width of the initial hole does not change (to within the uncertainty of the determination of the width of the hole) in the temperature range between 3,1 K and 4,2 K. As our actual temperature fluctuations are at the 1 mK level, the width of the initial hole should therefore remain constant during the measurement sequence. This is further confirmed by removing all weights at the end of the measurement sequence, and observing that the hole regains its initial shape.

Moreover, in the case that the broadening arises from a residual inhomogeneity in the amplitude of the stress field in the crystal across the optical spot, the broadening should depend directly on the strain sensitivity, which is different for different crystal sites and axes parallel to which the stress is applied. That is, the same gradient in strain should give rise to a larger broadening for a more sensitive site and direction of stress. Thus, in order to compare the different measurements, we have also normalized the broadening according to the sensitivity. We previously determined these values [START_REF] Galland | Coq Mechanical tunability of an ultra-narrow spectral feature with uniaxial stress[END_REF]: κ D1S1 = 46 ± 17 The effect of the stress on the width of a spectral hole, for different directions parallel to which the stress is applied (D1 and D2) and crystal sites (S1 and S2). The broadening has been normalized by the strain sensitivity, in order to plot (Γ -Γ0)/(Γ0|κDiSj|/κD1S1), where Γ0 is the initial width of the unperturbed hole. The dashed lines are linear fits to the data. Hz Pa -1 ; κ D1S2 = 137 ± 16 Hz Pa -1 ; κ D2S1 = -19 ± 10 Hz Pa -1 ; and κ D2S2 = -213 ± 13 Hz Pa -1 , where we have used the notation κ DiSj where Di corresponds to the direction parallel to which the strain is applied, and Sj the crystal site (i, j =1 or 2). Using these values, we choose to normalize the broadening to the sensitivity of the crystal site 1, with a stress applied parallel to the D 1 axis. That is, we plot (Γ -Γ 0 )/(Γ 0 |κ DiSj |/κ D1S1 ). The results are shown in figure 5. Here, the error-bars account for the reproducibility of the experiment (4 to 6 measurements per data point), and solid lines correspond to linear fits to the data. By allowing for a quadratic component, we notice a small improvement of the quality of the fit, but the linear contribution is clearly dominant, so in what follows, we will consider this component only.

In the simple case where the broadening were due uniquely to an inhomogeneity in the amplitude of the crystal strain, we would expect all linear fits to exhibit the same slope to within the error-bars, as we have normalized the broadening to the strain sensitivity. However, we observe a significant variations in slopes: α D1S1 = 9.5 ± 4.6 kPa -1 , α D1S2 = 3.6 ± 2.1 kPa -1 ; α D2S1 = 18.5 ± 0.2 kPa -1 , and α D2S2 = 3.1 ± 0.1 kPa -1 . The uncertainties on the slopes account for the repeatability of the experiment. Even taking into account the uncertainty on both the κ DiSj and the α DiSj coefficients, α D2S1 cannot be brought to overlap with α D1S2 and α D2S2 . We cannot exclude a small inhomogeneity along the axis of propagation of the probe beam (along the baxis). However, a gradient in the amplitude of strain in this direction would also give rise to a broadening being identical (when normalized by the sensitivity) among sites and stress-directions. In order to explain the variations of the slopes due to the inhomogeneity in the strain, one would require a component of the force to be in a direction perpendicular to the applied stress-direction. This is thus unlikely to account for the differences observed.

Instead, we may, at least partially, attribute the observed effect to an inhomogeneity of the different ions' responses on a microscopic scale, as suggested by other authors for a different system [START_REF] Reeves | Measurement of the effect of uniaxial stress on the spectrum of CaF2:P r 3+ :D-using spectral hole burning[END_REF]. In our case, the ions forming the spectral hole all have a slightly different electromagnetic environment, coming from a nonuniform distribution of static strain within the crystal which stems essentially from local distortion of crystal lattice due to doping, which also accounts for the large inhomogeneous profile. Although the ions contributing to the spectral hole all have the same absorption frequency (to within the width of the hole), this frequency is both a combination of the exact hyperfine states between which they have transitioned, which are separated by several tens of MHz (all three hyperfine levels are populated at 4 K prior to optical pumping), combined with the local static crystal strain. That is, contributing to a single spectral hole, there will be ions with internal static strain varying both in amplitude and direction, and they are therefore likely to react differently according to the externally applied strain. Although the strain sensitivities cited above have been shown to be independent on the frequency position in the inhomogeneous profile [START_REF] Galland | Coq Mechanical tunability of an ultra-narrow spectral feature with uniaxial stress[END_REF], this explanation can still be valid: In each position of the profile, a spectral hole will be formed by ions with different local strain environment (reacting differently to strain), but the global strain sensitivity corresponds to an average effect of the ensemble of the ions, which can still be identical to that of a second position in the inhomogeneous profile.

Finally, another type of inhomogeneity on a microscopic level could stem from the fact that transitions among different hyperfine states could react differently to strain as a direct consequence of atomic structure, but comparing with other systems, we believe the corresponding variation may be negligible. For instance, for the NV -in diamond, the strain-sensitivity of the optical 638 nm transition is 1000 Hz Pa -1 [START_REF] Davies | Optical studies of the 1.945 eV vibronic band in diamond[END_REF] while for the ground-state spin-levels, the sensitivity only amounts to 0.01 Hz Pa -1 [10], thus 10 5 times smaller. Of course, dedicated measurements will be required in order to confirm whether this also applies to the case of rare-earth ion dopants, but the absence of distinct components in the profile of the spectral hole subject to stress also supports this assumption.

In conclusion, we have discussed how the shape of a narrow spectral hole is sensitive to stray charges, and we have shown how to eliminate these detrimental effects. We have also demonstrated that adding a fine layer of crystalline powder at the contact interfaces of a crystal significantly reduces inhomogeneities, both in direction and amplitude, in the strain field generated in the crystal due to added weights. This is a prerequisite for studying the inhomogeneities of the ion dopants on a microscopic level. The fact that the broadening, normalized by the strain-sensitivity, is distinct for different crystal sites and directions for the stress, indicate that the various ions participating in the spectral structure react differently according to their local strain environment. Future work includes repeating the experiments with a much smaller number of ions to see how this influences the inhomogeneous effects. Ideally, one would study the response to strain of a single ion, but currently, the smallest number of Eu 3+ ions in an YSO matrix detected is approximately 10 [START_REF] Casabone | Cavity-enhanced spectroscopy of a few-ion ensemble in Eu 3+ :Y2O3[END_REF], made possible by enhancing the signal by means of a micro-cavity, and thus currently incompatible with the application of an external, calibrated stress. However, by using a protocol of hyperfine class selection already demonstrated for rare-earth ion dopants [START_REF] Lauritzen | Spectroscopic investigations of Eu3+:Y2SiO5 for quantum memory applications[END_REF] it should be possible to assess to which extend different classes react differently to strain. Theoretical calculations of the frequency shift of an ion dopant as a function of its local strain environment would also be of great value to better understand inhomogeneous effects resulting in spectral hole broadening, but due to the lack of symmetry of the crystal and the substitution sites, such calculations are extremely challenging. Even if the exact positions of the individual atoms can be calculated under strain, the calculation of Eu 3+ energy levels under strain would still have to be performed. This is currently out of reach of the best crystal field models. At present, it is therefore not possible to propose a model describing the results, thus the measurements seem to be the most efficient avenue for investigating these effects.

Figure 1 :

 1 Figure1: Details on ion level structure and crystal axes. In a) the energy diagram of the Eu 3+ ions, where each electronic level is separated in 3 hyperfine states. When excited at resonance at 580 nm, the ions decay radiatively primarily to 7 F1 and then non-radiatively to one of the hyperfine states. In b) we show the Y2SiO5 crystal, which is cut parallel to the 3 dielectric axes, D1, D2 and D3, as well as its dimensions.

Figure 2 :

 2 Figure 2: Experimental device for applying a calibrated stress to the crystal. Left: A schematics of the ladder structure which allows us to deposit a given number of weights on the top of the crystal (illustrated by the light blue square), by vertically displacing the structure (z-axis translation stage), and to center the weights (by using the x, y-axis). Right: A photo of the device, mounted inside the cryostat science chamber.In both schematics and photo, an electrically grounded metal structure surrounding the crystal is visible, which prevents a build-up of static charges.

Figure 3 :

 3 Figure3: The effect on a spectral hole of operating the piezo translation stage near the crystal, for crystal site 1 (without applying weights) prior to electrical shielding. The solid line corresponds to the initial spectral hole, and the dashed and dotted line to the hole after having operated the piezo one and two steps, respectively. The deformations of the spectral holes persist in time, despite ended operation of the piezo translation stage.

Figure 4 :

 4 Figure 4:The effect of stress on the shape of the spectral hole, for crystal site 2, with stress applied along the D1 axis. The solid line shows the initial hole without stress. The dotted line corresponds to the case in which the stress is applied without depositing a crystalline powder at the contact surfaces, and the dashed line to the case in which a powder has been used. The stress also gives rise to a global shift in frequency of the spectral structures, but this shift has been subtracted in order to center the curves in frequency.

Figure 5 :

 5 Figure5: The effect of the stress on the width of a spectral hole, for different directions parallel to which the stress is applied (D1 and D2) and crystal sites (S1 and S2). The broadening has been normalized by the strain sensitivity, in order to plot (Γ -Γ0)/(Γ0|κDiSj|/κD1S1), where Γ0 is the initial width of the unperturbed hole. The dashed lines are linear fits to the data.

  The project has been supported by the European Union's Horizon 2020 research and innovation program under grant agreement No 712721 (NanOQTech). It has also received support from the Ville de Paris Emergence Program, the Région Ile de France DIM C'nano and SIRTEQ, the LABEX Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01) within the Program "Investissements d'Avenir" operated by the French National Research Agency (ANR), and the EMPIR 15SIB03 OC18 and from the EMPIR program co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program.