Hyperfine interactions and coherent spin dynamics of isotopically purified ¹⁶⁷Er³⁺ in polycrystalline Y₂O₃

Rare-earth (RE) ions in solids possess shielded 4f intra-shell optical and spin transitions with narrow linewidths and long coherence times [1–3], making them ideal candidates for applications such as quantum memory [1, 4] in a quantum network [5], microwave-to-optical quantum transduction [6] and quantum sensing [7]. Among RE ions, erbium (Er³⁺) in particular, constitutes a unique system that simultaneously possesses optical fiber-compatible telecom C-band optical transition (1535 nm), high electron g-factor (14.6 Er³⁺:Y₂SiO₅) [8] with long nuclear spin coherence time of 1.3 s at 1 K (¹⁶⁷Er³⁺: Y₂SiO₅) [3]. ¹⁶⁷Er³⁺ with nuclear spin I = 7/2 exhibits strong anisotropic hyperfine interactions, resulting in electron-nuclear hybridized states in low symmetry sites with reduced sensitivity to magnetic field noise at zero field—known as the zero-first-order Zeeman (ZEFOZ) transitions [9] as well as a large zero field transition frequencies in the microwave range. Narrow optical homogenous and inhomogeneous linewidth [10, 11] have been demonstrated in ¹⁶⁷Er³⁺ doped yttrium orthosilicate (YSO) [3]. Comparable to YSO, yttrium oxide (Y₂O₃) is another low magnetic noise host with a similar ionic radius of Y³⁺ to Er³⁺ (1.04 vs 1.03 Å) and can be synthesized in diverse range of material platform compatible with nanoscale integration [12] including thin films [13, 14], nanoparticles [15, 16], and transparent polycrystalline ceramics [11]. This combination makes the characterization of spin Hamiltonian of ¹⁶⁷Er³⁺:Y₂O₃ crucial to further develop this quantum platform by exploiting its long-coherence-time hyperfine transitions as well as performing coherent control of ¹⁶⁷Er³⁺ spins as a resource for quantum technology applications.

We report X-band (9.5 GHz) electron paramagnetic resonance (EPR) (continuous-wave and pulsed) spectroscopic studies of the atomic environment and dynamics of the Er³⁺ spins in Y₂O₃ polycrystalline matrix. We investigate the electronic structure of Er³⁺ using continuous-wave (CW) EPR and characterize Er³⁺ local structure variations in different crystalline environments (C₂ and C_3i) in cubic Y₂O₃ by determining the Zeeman, hyperfine and quadrupole interactions. We compare X-band EPR with low field C-band (5 GHz) EPR to verify the spin Hamiltonian in a regime where the Zeeman energy is comparable to hyperfine energy. The hyperfine transitions in ¹⁶⁷Er³⁺ introduce mixed states that show a relatively slow spin relaxation, which can be detected at temperatures of 4.0–6.0 K. We measure a long spin coherence time of 24.4 μs at 4 K for lowest g-factor transition. The limits for coherence relaxation times given by spin-lattice relaxation (characterized as T₁) is 5 ms at 4.0 K. Our study provides a comprehensive spin spectroscopy of ¹⁶⁷Er³⁺ in Y₂O₃, which shows significant prospects of this platform for future applicable quantum technologies.

The Er³⁺ ion contains 4f¹¹ electrons which result in a total spin quantum number of S = 3/2, an orbital quantum number of 6 and total angular momentum of J = 15/2. The ground level ⁴ I_15/2 is 16-fold degenerate and in the presence of crystal field of low symmetry it can split into 8 (J + ½) Kramer doublets. The energy of microwave radiation generated in conventional X-band EPR experiments (9.5 GHz) is generally sufficient to probe only the lowest-lying doublet of the ⁴ I_15/2 manifold of trivalent erbium. As a result, common studies of Er³⁺ in metal oxides consider only the lowest lying Kramer doublet with an effective spin S = 1/2 system to account for the features of Er³⁺ EPR spectra:

$$\begin{equation}H={\mu }_{\mathrm{B}}{B}_{0}\cdot {\mathbf{g}}_{\mathrm{e}}\cdot {S}_{\mathrm{e}}+{S}_{\mathrm{e}}\cdot \mathbf{A}\cdot {I}_{\mathrm{n}}+{I}_{\mathrm{n}}\cdot \mathbf{Q}\cdot {I}_{\mathrm{n}},\end{equation} \tag{ 1 }$$

where the first Zeeman term represents the energy of the electronic spin in a magnetic field B₀ and the second term accounts for the hyperfine interaction with nuclear spins and the third term is nuclear quadrupole interaction, μ_B is the Bohr magneton and g, A and Q are interaction tensors, which reflect local structure particularities of the paramagnetic complex. EPR spectrum of Y₂O₃ doped with natural abundance of Er³⁺ is shown in figure 1(a). The sample has a nominal Er³⁺ doping of 20 parts per million (ppm), and was prepared by sintering 40 nm-sized nanocrystals into polycrystalline ceramics according to the method reported in reference [11]. For X-band EPR measurement, the sample was grinded to microparticles to increase the filling factor of the microwave cavity. Figure 1(a) inset shows the crystal structure of cubic Y₂O₃ with annotated different local sites available for the substitution of Y³⁺ with Er³⁺. Two possible sites are: a lower symmetry C₂ site whose substitution should result in a rhombic EPR signal and a higher symmetry trigonal C_3i site whose substitution should result in an axial EPR signal. Indeed, we clearly observe four main transitions, three for a rhombic signal (g_z = 12.20, g_y = 4.89 and g_x = 1.64) and a main signal for trigonal symmetry site (g_⊥ = 3.32). Upon close inspection of the lowest rhombic component using small modulation amplitude, we find that the lowest signal at g = 12.20 is composed of two signals, one with g = 12.20, which is approximately 5 times more intense than the one with g = 12.08 (figure 1(b)).

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The two signals have different saturation properties suggesting their association with different sites (inset, figure 1(b)). Simulation of the experimental spectra with even isotope ¹⁶⁶Er³⁺ (33.5% natural abundance) using the above listed parameters for the parallel component of the axially symmetric signal (C_3i) perfectly matched observed transitions and signal's intensity. The average g_avg factors for both C₂ and C_3i systems were found to be 6.243, suggesting Γ₇ ground state (g_ave = 6.0) rather than Γ₆ (g_ave = 6.8) usually found in Er³⁺ doped samples [17–19].

The g_z component simulations using a natural distribution of Er³⁺ isotopes fit well with the observed hyperfine structure (HFS) in Er³⁺ spectra. ¹⁶⁷Er³⁺ isotope with a nuclear spin I = 7/2 and 23% abundance resulted in an approximately symmetrical HFS pattern of 8 satellite lines with 3.7% amplitude of the central line and with hyperfine splitting A_z = 1288 MHz, A_y = 518 MHz and A_x = 120 MHz for C₂ symmetry and A_∥ = 1260 MHz and A_⊥ = 350 MHz for C_3i. However, g_z component of a rhombic C₂ signal and g_∥ component of a C_3i signal in the odd nuclear isotope ¹⁶⁷Er³⁺ were slightly shifted from that associated with even isotopes and found to be 12.28 and 12.20, respectively (figure 1(b)). This indicates that odd isotope ¹⁶⁷Er³⁺ experiences slightly different environmental effects caused by a different effective crystal field.

To examine this phenomenon and to elucidate how these hyperfine states affect spin electronic structure and relaxation dynamics, we synthesized an isotopically purified ¹⁶⁷Er³⁺ doped Y₂O₃ sample with 20 ppm doping and observed a spectrum with a high density of hyperfine transitions indicative of nuclear spin for ¹⁶⁷Er³⁺. The lowest field components g_z and g_∥ components are composed of eight lines, each as expected from I = 7/2 (figure 2(a)). Importantly, the central line associated with transitions of even isotopes at g = 12.20 was significantly reduced, testifying to the isotopic purity of the sample. The g_y component for C₂ state of ¹⁶⁷Er³⁺, however, does not show expected eight-line pattern, but instead 16 major lines of different intensities were observed. This spectrum could not be deconvoluted to the set of octets if one would assume existence of two nonidentical sites leading to two different hyperfine structures previously found in reference [21]. Our spectrum has very irregular line positions and intensities with separations that can be closely simulated with simple I = 7/2 nuclear hyperfine interactions A along with strong nuclear quadrupole interaction Q. Nuclear quadrupole interactions impose additional torque acting on ¹⁶⁷Er³⁺ nuclei and affect quantization directions, thereby shifting energy levels and magnetic resonance transitions while also leading to forbidden transitions. Similarly, g_⊥ component for C_3i state is now composed of more than 8 lines each, suggesting the presence of forbidden transitions. A similar observation was made for ¹⁶⁷Er³⁺ in Y₂SiO₅ [8, 22] where forbidden transitions with intensity comparable to allowed transitions were described with a traceless nuclear quadrupole interaction (Q).

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The spin Hamiltonian was fitted to the resonance fields for hyperfine transitions obtained from the experimental spectrum. We also assumed that g, A and Q tensors have coincident principal axes and can be simultaneously diagonalized as limited information about the orientation of the tensor can be derived from a powder spectrum alone for an anisotropic spin Hamiltonian. The low field transitions could be successfully fitted with g_z , A_z components of C₂ (g_z = 12.28, A_z = 1280 MHz) and g_∥ of C_3i sites (g_∥ = 12.20, A_∥ = 1268 MHz), in both peak positions and their intensity (see supporting information S1). Intermediate field transitions were fitted to obtain g_y , A_y , Q [Q_x , Q_y , Q_z ] terms of the C₂ site (g_y = 4.78, A_y = 480 MHz, Q = [−50, 10, 40] MHz) and g_⊥, A_⊥, Q [Q_∥, Q_⊥] terms of the C_3i site (g_⊥ = 3.28, A_⊥ = 310 MHz, Q = [12.5, −25] MHz). The C₂ g_x component at g _x = 1.64 had broad transitions with small intensity which limited the number of hyperfine transitions to be resolved. Therefore, A_x was estimated from the difference in splitting of the lowest and highest field hyperfine transitions observed in the experimental spectrum (g_x = 1.64, A_x = 212 MHz). The simulated spectrum for C₂ site (red) and C_3i site (blue) generated using the fitted spin Hamiltonian shows a close agreement to experimental spectrum in terms of peak position, shape as well as the relative intensity (figures 2(a) and (b)). Tables 1 and 2 summarize the principal values of g tensor for Er³⁺ (I = 0) and g, A, Q for ¹⁶⁷Er³⁺ (I = 7/2), respectively.

The predictions from an anisotropic ¹⁶⁷Er³⁺ spin Hamiltonian fitted with the X-band EPR have been shown to exhibit disagreement with the low field EPR measurements in low symmetry hosts such as Y₂SiO₅ [21]. Therefore, to validate the fitted spin Hamiltonian at a lower field, we performed CW-EPR on the ¹⁶⁷Er³⁺:Y₂O₃ sample with a C-band microwave (resonance frequency of 4.96 GHz) loop-gap cavity. The experimental spectra (figure 3) show close agreement with simulated spectrum from C₂ and C_3i sites and suggest a good estimate of EPR parameters. The fitted g values are also in close agreement with previous optical and EPR measurements [23–25].

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We sent π/2 and π pulses and swept the magnetic field to obtain two pulse-field swept echo, (2P-FSE), (π/2−τ−π−τ − echo, τ = 250 ns) spectrum of ¹⁶⁷Er³⁺ in X-band. Figure 4 shows EPR transition from hyperfine split levels which show similarities to the CW EPR spectrum of the same sample (supplement figure S2). The signal is temperature dependent, disappearing at 15 K, which is indicative of EPR signals of Er³⁺ ions. Four different regions of discrete multiline transitions are observed in addition to a broad underlined absorption that suggests existence of unresolved hyperfine structures of strongly interacting ¹⁶⁷Er³⁺ ions. The signal intensity ratio for C_3i:C₂ states in the 2P-FSE spectrum is higher compared to their intensity ratio in CW EPR, suggesting different spin dynamics of C_3i and C₂ states. Moreover, the 2P-FSE spectrum features are strongly dependent on the delay time τ between π/2 and π pulses, revealing individual dynamic features of C₂, C_3i and strongly interacting ¹⁶⁷Er³⁺ sites. The strongly interacting sites show the fastest decay. Figure 4(a) shows the change of the 2P-FSE spectra with the change of the delay time τ between two pulses. As τ increases from 250 to 400 ns, the first signal to lose the intensity is the broad feature under the discrete transitions that we associate with strongly interacting ¹⁶⁷Er³⁺ sites. In the same time frame, discrete transitions become even more intense, possibly because of their diminished relaxation due to the decay of the strongly interacting species. At later times both C₂ and C_3i sites lose their intensity, and at τ = 1.5 μs only one pronounced transition remains, corresponding to the energetically highest m = +7/2 hyperfine state of C_3i.

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While all low field spectral features for ¹⁶⁷Er³⁺ in rhombic crystalline environment C₂ vanish for the pulse delay of 1.5 μs, figure 4(b) shows decay of the spectral features for high field g = 1.64 C₂ component with increasing delay time τ. These features, although weak, persists for up to τ delay time of 3 μs, suggesting their long coherence time. Temperature dependence of this signal is consistent with the temperature behavior of Er³⁺ signals at g = 4.8 and 12.2 which also disappear at 15 K. The g = 1.64 transition exhibited very slow stretched exponential echo decay with T₂ = 24.4 μs measured at 4.3 K which was superimposed with slow nuclear modulation (figure 4(c)). The Fourier transform of electron spin envelope echo modulations ESEEM (obtained using 3P-ESEEM, π/2-τ-π/2-T-π/2 − τ − echo, with 16 step phase cycling) shown in figure 4(d) reveals two modulation frequencies, one stronger component of 0.45 MHz consistent with modulation of ¹⁶⁷Er³⁺ and one 0.80 MHz consistent with modulation of ⁸⁹Y suggesting that the interaction with ¹⁶⁷Er³⁺ ions from C_3i sites and to a lesser extent weakly coupled ⁸⁹Y 'matrix' nuclei contribute as a source of decoherence. However, due to the weak signal intensity, we were not able to measure spin lattice relaxation T₁ of the transition with g = 1.64.

Transitions with higher g factors (g = 12.2, 4.78, 3.28) exhibit stronger sensitivity to magnetic noise leading to the fast echo decay with a shorter T₂ time of approximately (0.5–0.9 μs) (figure 4(e) for g = 4.78 & table 3). The g = 4.78 transition could be effectively dynamically decoupled from the sources of decoherence by applying a single XY16 [26] sequence, prolonging its T₂ for an order of magnitude to 3.9 μs. Extending of coherence time obtained by applying dynamic decoupling sequences with multiple rotation axes such as XY16 demonstrates the presence of spectral diffusion as a source of fast spin decoherence. However, applying multiple sequences of XY16 only shortens spin coherence time T₂ due to pulse error. Only after the sign of the signal was inverted by applying a π pulse between two XY16 sequences, we obtained further enhancement of the spin coherence to 6.5 μs. The signal obtained after dynamic decoupling reveals slow modulations reminiscent of those obtained in 3P ESEEM measurements of g = 1.64 transition, suggesting that the same ions participate in relaxation of spins with g = 4.78.

Instantaneous diffusion [27] arising from the dipole interaction of resonant ¹⁶⁷Er³⁺ spins during inversion pulses is another mechanism of spin decoherence. We investigated instantaneous diffusion by applying the Hahn echo (π/2 − τ − Θ − τ − echo) sequence with various spin-flip angles. Figure 4(f) shows the dependence of 1/T₂ of C₂ g = 4.89 measured as a function of different spin-flip probability (sin²Θ/2, where Θ is spin-flip angle). We obtain a linear dependence of the 1/T₂ on the average spin flip probability with the slope of 0.8 MHz, suggesting an effective concentration of resonant spins with density 1.38 × 10¹⁷ spin/cm³ (equivalent dopant density of ∼5 ppm) with g = 4.78 contributing to instantaneous diffusion. The instantaneous diffusion determined for the spins in C_3i symmetry (figure 4(f)) shows very similar dependence of 1/T₂ on the spin flip probability sin²Θ/2 (slope of 0.7 MHz) corresponding to an estimated density of 2.53 × 10¹⁷ spins/cm³ with g = 3.28.

The electron spins in highly symmetric C_3i sites also experience spectral diffusion from spin flips of neighboring spins that could be mitigated through XY16 dynamic decoupling pulse sequences. Applying the multiple XY16 sequences separated by a sign inversion π pulse, extends coherence time T₂ of C_3i states up to 6 μs measured at 4.2 K. However, decoupling schemes do not extend the coherence time up to the T₁ limits given by spin-lattice relaxation. Spin lattice relaxation times were determined through three pulse inversion recovery sequence (π − T − π/2 − τ − π − τ − echo with 4 step phase cycling) and were found to be biexponential with larger fraction of the slow component (table 3). Determined slow component of T₁ is long for ¹⁶⁷Er³⁺ in C_3i states, reaching 0.6 ms at 4.2 K. As T₁ spin-lattice relaxation bounds coherence time, long T₁ times suggests orders of magnitude improvement of electron spin coherence times by eliminating spectral and instantaneous diffusion. The spin lattice relaxation times of C₂ states are even longer ranging from 0.7 ms for g = 4.78, to 5 ms for g = 12.20 at 4.2 K. The T₁ times for ¹⁶⁷Er³⁺ are significantly longer than that of natural isotopic abundance Er³⁺ sample with three orders of magnitude longer T₁ for C₂ g = 12.2, an order of magnitude for C₂ g = 4.8 and for C_3i g = 3.3 (table 4). These results suggest that isotope inhomogeneity in natural isotope promotes spin-lattice decays.

We have comprehensively characterized the spin Hamiltonian of ground state ¹⁶⁷Er³⁺ in both C₂ and C_3i sites of Y₂O₃ using X-band EPR. The spin Hamiltonian exhibited an anisotropic g, A matrix with a strong nuclear quadrupole interaction Q leading to forbidden EPR transitions and showed good agreement with the 5 GHz CW-EPR spectra. The measured strong anisotropic hyperfine interaction of ¹⁶⁷Er³⁺ in both C₂ and C_3i sites of Y₂O₃ suggests the presence of zero first-order Zeeman (ZEFOZ) transitions at low fields with possibly long spin coherence times [9, 28]. The spin dynamics of ¹⁶⁷Er³⁺ studied using X-band pulsed EPR show a long coherence time reaching 24.4 μs for g = 1.64 transition at 4 K. The spin coherence time was found to be limited by spectral diffusion with and instantaneous diffusion for both C₂ and C_3i sites. With lower doping concentration the spin coherence time could be further improved at 4 K. Addressing low field ZEFOZ transitions and utilizing pulsed sequences aimed at decoupling dipolar interaction and spectral diffusion [29] could be future steps in maximizing the coherence of this system for enabling quantum technologies.

We acknowledge Oleg Poluektov and Jens Niklas for D-Band EPR measurements.

The manuscript was written through contributions of all authors. JY and HZ prepared the samples; TR and LS performed X-band EPR spectroscopy. SG & TZ performed 5 GHz EPR spectroscopy. TR, LS, SG and TZ analyzed and simulated EPR spectra. All authors have given approval to the final version of the manuscript.

This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. S Gupta and T Zhong acknowledge the support from Department of Defense, Army Research Office Grant No. W911NF2010296, and National Science Foundation (NSF) Faculty Early Career Development Program (CAREER) Grant (No. 1944715). Work performed at the Center for Nanoscale Materials, a US Department of Energy Office of Science User Facility, was supported by the US DOE, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

The authors declare no competing financial interests.

CW-EPR, continuous-wave electron paramagnetic resonance; ZEFOZ, zero-first-order Zeeman; ESEEM, electron spin echo envelope modulation.

The data that support the findings of this study are available upon reasonable request from the authors.