Decoherence characterization and quantum memory design in ¹⁶⁷Er³⁺:Y₂SiO₅

Abstract

Erbium-doped crystals are promising candidates for fiber-compatible quantum memories due to their optical transition in the telecommunication C-band. Here, we investigate the optical decoherence dynamics of isotopically enriched {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:{\mathrm{Y}}_2\mathrm{S}\mathrm{i}{\mathrm{O}}_5 (YSO) at sub-Kelvin temperatures and low magnetic fields (<1 T). We observed an optical homogeneous linwidth of 100 Hz at 100 mK and 0.2 T – significantly narrower than previous values obtained at 7 T in a sample with the same {}^{167}\mathrm{E}{\mathrm{r}}^{3+} concentration. We analyze the dependence of coherence properties on magnetic field orientation and temperature, identifying spectral diffusion resulted from the host spin bath as key decoherence mechanisms. We further identified that superhyperfine interactions between the {}^{167}\mathrm{E}{\mathrm{r}}^{3+} and the host bath spins pose challenges for long-term storage protocols requiring precise spectral tailoring. To address this issue, we propose a strategy that combined the Stark modulation memory scheme with noiseless photon echo storage protocol, which is expected to achieve spin-wave quantum storage with lower noise and higher efficiency.

Graphical abstract

Keywords

quantum communication / quantum interface / quantum memory / rare earth ions

1 Introduction

Quantum memories are fundamental components for quantum communication and distributed quantum computing, enabling the reversible storage and retrieval of quantum states [1-4]. A practical quantum memory system must satisfy several stringent requirements, including long storage times, high storage efficiency and multimode capacity [5, 6]. Among various candidate platforms, rare-earth-ion (REI) doped crystals have been extensively investigated over the past two decades for quantum memory application due to their exceptionally long optical and spin coherence times at cryogenic temperatures [7-12]. Their intrinsic optical inhomogeneous broadening also enables broadband and multimode storage [13-17].

Among REI systems, erbium ions (Er{}^{3+}) are especially attractive for fiber-based quantum networks due to their optical transitions near 1.5 μm, within the telecommunication C-band [18-21]. When doped at low concentrations into {\mathrm{Y}}_2\mathrm{S}\mathrm{i}{\mathrm{O}}_5 (YSO), a host known for its low magnetic noise [22, 23], Er{}^{3+} exhibit outstanding coherence properties, featuring an optical coherence time of 4.4 ms and a nuclear spin coherence time of 1.3 s [24, 25]. Furthermore, efficient entanglement distribution using \mathrm{E}{\mathrm{r}}^{3+}:YSO has been experimentally demonstrated, revealing non-classical correlations between distinct temporal modes [26]. These results were achieved under stringent conditions – namely, liquid helium temperatures and high magnetic field (6–7 T). However, recent studies suggest that further lowering the operating temperature can reduce the requirement for such strong magnetic fields [27, 28].

In this work, we investigate the feasibility of Er{}^{3+}:YSO quantum memory operation at sub-Kelvin temperatures under relatively low magnetic field (< 1 T). Using spectral hole burning (SHB) measurements, we demonstrate that strong superhyperfine interactions emerge in this regime, imposing constraints on memory performance for protocols based on spectral tailoring. To address this, we propose a modified noiseless photon echo (NLPE) protocol that enhances compatibility with backward retrieve and cavity-enhanced architectures [29-32]. This approach provides a promising path toward high-fidelity, low-field quantum memory operation in integrated photonic platforms.

2 Crystals and setup

Erbium ions (\mathrm{E}{\mathrm{r}}^{3+}) in YSO exhibit an effective electronic spin \mathit{S} = 1/2, with only the lowest electronic doublet thermally populated at sub-Kelvin temperatures. For the isotope {}^{167}\mathrm{E}{\mathrm{r}}^{3+}, which possess a nonzero nuclear spin (\mathit{I} = 7/2), hyperfine interactions further split each electronic levels into eight hyperfine sublevels. The spin Hamiltonian describing the ground {}^4{\mathrm{I}}_{15/2}({\mathrm{Z}}_1) and excited {}^4{\mathrm{I}}_{13/2}({\mathrm{Y}}_1) states of {}^{167}\mathrm{E}{\mathrm{r}}^{3+} in YSO is [33]

(1)H={\mu }_e\mathit{B}\cdot \mathit{g}\cdot \mathit{S}+\mathit{I}\cdot \mathit{A}\cdot \mathit{S}+\mathit{I}\cdot \mathit{Q}\cdot \mathit{I}-{\mu }_n{\mathrm{g}}_n\mathit{B}\cdot \mathit{I},

where {\mu }_e\approx 14 GHz/T and {\mu }_n\approx 7.62 MHz/T are the electronic Bohr magneton and the nuclear magneton, and \mathit{g}, \mathit{A} and \mathit{Q} are the Landé g, hyperfine and effective electric-quadrupole tensor respectively while \mathit{B} is the magnetic field vector. The parameters of this Hamiltonian have been characterized in prior studies [33-37]. A schematic of the energy level structure of {}^{167}\mathrm{E}{\mathrm{r}}^{3+} in YSO is shown in Fig. 1(a). Due to the difference in the \mathit{g} tensor of the ground {}^4{\mathrm{I}}_{15/2}({\mathrm{Z}}_1) and excited {}^4{\mathrm{I}}_{13/2}({\mathrm{Y}}_1) states, the combination of Zeeman and hyperfine interactions leads to a set of resolved optical absorption lines corresponding to transitions with \mathrm{\Delta }{M}_{\mathrm{I}} = 0, ±1, ±2,··· under an applied magnetic field.

In the YSO host, \mathrm{E}{\mathrm{r}}^{3+} ions substitute {\mathrm{Y}}^{3+} ions at two crystallographically inequivalent sites of C_1 symmetry, known as “site 1” and “site 2”, each with distinct transition frequencies. Each crystallographically site consists of a pair of magnetically inequivalent subclasses, related by a rotation about the crystal’s C_2 (or b) axis. When the magnetic field is oriented either perpendicular or parallel to the b-axis of the crystal, transitions associated with these magnetically inequivalent sub classes remain degenerate. This degeneracy effectively doubles the number of ions contributing to a given transition and enhancing the optical depth – an essential parameter for quantum memory performance. This enhancement is particularly important at low \mathrm{E}{\mathrm{r}}^{3+} doping concentrations, which are used to suppress ion-ion interactions that cause spectral diffusion and coherence degradation.

In current experiments, we used a YSO crystal doped with 50 ppm isotopically enriched {}^{167}Er{}^{3+} (Scientific materials Corp.). The sample, with dimensions of 5 mm\times5.5 mm\times3 mm along the D_1, D_2 and b axes, respectively, was mounted in a dilution fridge (LD400, Bluefors) to reach sub-kelvin temperatures. A vector superconducting magnet (American Magnetics Inc.) was equipped, providing an external magnetic field in the D_1−D_2 plane to maintain the magnetic equivalence of \mathrm{E}{\mathrm{r}}^{3+} subclasses. The magnetic angle \theta corresponding to the D_1 axis is defined in Fig. 1(b).

An external cavity diode laser (Toptica) with erbium-doped fiber amplifier and an acousto-optic modulator (AOM) generated both monochromatic and frequency-chirped optical pulses. Light was directed along the b-axis of the sample, with its polarization adjusted to maximize the optical depth of the \mathrm{\Delta }{M}_{\mathrm{I}} = 0 transition. For two pulse photon echo (PE) measurements, the echo signal was isolated from the strong excitation pulses (\sim29.2 W/cm{}^2) with an additional AOM and detected by an avalanche photodetector. For SHB spectroscopy, the probe field was detected using an InGaAs photodetector. The timing sequences used in these experiments are illustrated in Fig. 1(c).

3 Results

3.1 Optical coherence measurement

In the PE experiment, the observed echo intensity as a function of the time separation between the two excitation pulses \tau was analyzed using the Mims decay model [38]:

(2)I(\tau )=I_0\mathrm{e}\mathrm{x}\mathrm{p}\left\{-2{\left({\displaystyle \frac{2\tau }{T_M}}\right)}^m\right\},

where I_0 is the initial echo intensity, and the stretch factor m characterizes the shape of the decay curve. The phase memory time T_M is related to the homogeneous linewidth {\mathrm{\Gamma }}_h by {\mathrm{\Gamma }}_h=1/(\pi T_M) [39].

Figure 1(d) presents the dependence of {\mathrm{\Gamma }}_h on the magnetic field orientation at a fixed field strength of |\mathit{B}| = 0.2 T, measured at a temperature of 0.1 K. Two maxima in {\mathrm{\Gamma }}_h are observed near \theta = 26{}^{\circ } and 82{}^{\circ }, which correspond to the local minimum in the g-factor value of the Z_1 state for the two crystallographically inequivalent sites [24, 34]. These linewidth maxima suggest enhanced mutual \mathrm{E}{\mathrm{r}}^{3+}−\mathrm{E}{\mathrm{r}}^{3+} spin flip−flops processes at orientations with low g-factors, where reduced spin polarization rate leads to higher flip−flop rate.

In contrast, the narrowest {\mathrm{\Gamma }}_h\approx 100 Hz occurs near \theta = 140{}^{\circ } for both sites. This indicate strongly suppressed \mathrm{E}{\mathrm{r}}^{3+} spin flip−flop interactions at orientations with higher g-factors and increased spin polarization. The observed 100 Hz linewidth is significantly narrower than the previously reported value of {\mathrm{\Gamma }}_h\approx 245 Hz (T_M = 1.3 ms) for {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O} with the same \mathrm{E}{\mathrm{r}}^{3+} concentration, measured at site 2 under a 7 T magnetic field [25]. Notably, the narrower linewidth in our experiment was achieved under a substantially lower magnetic field of only 0.2 T. This improvement is primarily attributed to the lower operating temperature, which enhances spin polarization and suppresses decoherence from flip−flop interactions.

To further investigate the impact of temperature on \mathrm{E}{\mathrm{r}}^{3+} optical decoherence, we measured the temperature dependence of the homogeneous linewidth {\mathrm{\Gamma }}_h(T) at a fixed magnetic field of |\mathit{B}| = 0.2 T oriented along \theta = 140{}^{\circ }. As shown in Fig. 2, both crystallographic sites of \mathrm{E}{\mathrm{r}}^{3+} exhibit similar temperature-dependent behavior in their optical homogeneous broadening. At a constant magnetic field, the {\mathrm{\Gamma }}_h(T) dependence can be approximated by the following model [39]:

(3){\mathrm{\Gamma }}_h(T)={\mathrm{\Gamma }}_0+{\alpha }_{\mathrm{T}\mathrm{L}\mathrm{S}}\cdot T+{\mathrm{\Gamma }}_{\mathrm{S}\mathrm{D}}\cdot {\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left({\displaystyle \frac{g_{env}{\mu }_{e}B}{2k_{\mathrm{B}}T}}\right),

where the first term {\mathrm{\Gamma }}_0 is temperature-independent linewidth. The second term describes the two-level systems (TLS) which scale linearly with temperature. The final term accounts for spectral diffusion, with g_{env} representing the effective magnetic moment of the environmental spin bath that cause spin flips and frequency fluctuations [24, 40].

Since the measurements were conducted at T < 1.5 K, phonon-related decoherence mechanisms, such as two-phonon Raman and Orbach processes, were neglected. This approximation is justified because the thermal energy satisfies k_{B}T\ll \mathrm{\Delta }E, where k_B is the Boltzmann constant and \mathrm{\Delta }E = 39.37 (26.60) \mathrm{c}{\mathrm{m}}^{-1} is the energy separation to the next electronic level {}^4{\mathrm{I}}_{15/2}({\mathrm{Z}}_2) \mathrm{\Delta }E for site 1 (site 2) [41]. Additionally, at this field orientation, \mathrm{E}{\mathrm{r}}^{3+} ions in both sites exhibit a large ground-state g factor, resulting in substantial electronic Zeeman splittings and highly polarized spins. Consequently, decoherence contributions from thermally driven spin fluctuations are negligible under these conditions [42].

The solid curves in Fig. 2 shows fits to Eq. (3), yielding the linewidth parameters (for both site 1 and site 2) as follows: {\mathrm{\Gamma }}_0 = 75 \pm 5 (80 \pm 1) Hz, {\mathrm{\Gamma }}_{\mathrm{S}\mathrm{D}} = 880 \pm 60 (1010 \pm 50) Hz, {\alpha }_{\mathrm{T}\mathrm{L}\mathrm{S}} = 340 \pm 40 (270 \pm 40) Hz/T, respectively. Notably, the stretch factor m from the Mims decay [Eq. (2)] increases from approximately 1.0 to over 1.5 as the temperature rises from 0.1 K to 1.4 K, as shown in the inset of Fig. 2. This behavior further supports the conclusion that spectral diffusion dominates the homogeneous linewidth broadening at sub-Kelvin temperatures.

3.2 Spectral hole burning and super-hyperfine interaction

We then evaluate the quantum memory potential of this system at the current temperature and magnetic field regime using the atomic frequency comb (AFC) protocol, a widely adopted scheme in rare-earth-based quantum memories. The theoretical forward retrieval efficiency for an ideal infinite AFC comb with square-shaped teeth is given by [43]

(4){\eta }_{AFC}={\tilde{d}}^2{\mathrm{e}}^{-\tilde{d}}\mathrm{s}\mathrm{i}\mathrm{n}{\mathrm{c}}^2(\pi /F){\mathrm{e}}^{-d_0},

where \tilde{d} = d/F is the effective optical depth (OD), F = \mathrm{\Delta }/\delta is the comb finesse, and d_0 is the background absorption. High memory efficiency requires large OD, high finesse, and minimal background absorption, all of which depend critically on precise spectral tailoring.

To investigate spectral preparation, we performed SHB on site 2 ions using a single-frequency pump with durations t_p = 50, 500 and 5000 μs at |\mathit{B}| = 0.5 T and T = 0.1 K. Figure 3(a) shows the resulting transmission spectra. Longer pump times improved pumping efficiency but also introduced significant broadening and side hole formation, attributed to laser instability and super-hyperfine interactions. Narrow hole linewidths are crucial for spin-wave AFC memories, as the minimum rephasing interval 1/\mathrm{\Delta } is constrained by the comb teeth width.

In \mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O}, super-hyperfine broadening of \mathrm{E}{\mathrm{r}}^{3+} ions arises from their interactions with neighboring nuclear spins, primarily from {}^{89}{\mathrm{Y}}^{3+} (I = 1/2), as described by the Hamiltonian [44]:

(5)H_{SH}={\displaystyle \sum _j^{N_Y}\left\{-{\mu }_Y\cdot \left(\mathit{B}-{\displaystyle \frac{{\mu }_0}{4\pi {\mathit{r}}_j^3}}({\mathit{\mu }}_{\mathrm{E}\mathrm{r}}^{g,e}-3{\displaystyle \frac{({\mathit{\mu }}_{\mathrm{E}\mathrm{r}}^{g,e}\cdot {\mathit{r}}_j)\cdot {\mathit{r}}_j}{{\mathit{r}}_j^2}})\right)\right\},}

where N_Y is the number of {}^{89}{\mathrm{Y}}^{3+} being considered, {\mu }_Y = 2.1 MHz/T is the nuclear magnetic moment for {}^{89}{\mathrm{Y}}^{3+}, {\mu }_0 is the vacuum permeability, and {\mu }_{\mathrm{E}\mathrm{r}}^{g,e} denotes the erbium electron gyromagnetic ratios for the ground or excited states. Each neighboring {}^{89}{\mathrm{Y}}^{3+} ion splits the \mathrm{E}{\mathrm{r}}^{3+} hyperfine sublevels of {}^4{\mathrm{I}}_{15/2}({\mathrm{Z}}_1) and {}^4{\mathrm{I}}_{13/2}({\mathrm{Y}}_1) in two, resulting sidebands in the SHB spectra. Contributions from {}^{29}\mathrm{S}{\mathrm{i}}^{4+} (I = 1/2, 4.7% abundance) are a also visible, with a magnetic moment of 8.5 MHz/T but a much smaller spectral weight. Figure 3(b) shows that the sideband frequency shifts increase linearly with the magnetic field, at rates of 2.1 MHz/T and 8.5 MHz/T. Figure 3(c) displays typical hole spectra, the fitting of which yielding central hole widths of 300–500 kHz, limiting the achievable AFC storage time to μs. Similarly, we estimated the d\approx 0.5 and F\approx 0.2 from the hole burning result in Fig. 3(c) and calculated the storage efficiency using Equation 4, which is merely 2%. For this calculation, we set the comb space \mathrm{\Delta } to be 0.63 MHz, equal to the side-hole splitting [18].

3.3 Stark modulated NLPE protocol

Compared to the AFC protocol, the NLPE scheme offers improved suppression of both coherent and incoherent noise during retrieval. A key feature of NLPE is the use of two backward-propagating control pulses, which cancel the first echo that would otherwise occur in forward retrieval [31]. However, NLPE also faces intrinsic challenges that may limit its efficiency in practical memory implementations. For example, the use of a reverse control pulse, as in the ROSE scheme [45], can inhibit backward retrieval. Additionally, NLPE’s reliance on spatial phase mismatching for noise-suppression can be incompatible with standing-wave cavity configurations [46].

As an alternative, here we propose an optimized NLPE strategy that replaces spatial phase mismatch with Stark-modulation to suppress the first-order echo [47-49]. This approach requires an ensemble with inversion symmetry and a weak mixed transition moment – conditions that are met by \mathrm{E}{\mathrm{r}}^{3+} ions at site 2 in YSO [50]. Furthermore, a reliable initialization protocol for {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O} that has already been proposed and experimentally demonstrated [51].

Here we illustrate our scheme based on a NLPE spin-wave memory as shown in Fig. 4. The optical transitions used are depicted in Fig. 4(a), with the corresponding pulse sequence in Fig. 4(b). At time t_1, an electric field pulse is applied for a duration T_E, inducing a relative phase shift {\mathrm{e}}^{\mathrm{i}4\pi \mathrm{\Omega }{T}_E} between the two subsites, where \mathrm{\Omega } denotes the Stark shift magnitude. When \mathrm{\Omega }{T}_E=1/4, the primary echo at t_4=t_3+t_2-t_0 is suppressed. The corresponding phase-matching condition for echo cancellation is [45]

(6){\mathit{k}}_4={\mathit{k}}_0-{\mathit{k}}_2-{\mathit{k}}_3,

where {\mathit{k}}_i represents the wavevector of each input pulse at time t_i. This configuration relaxes the requirement for counter-propagating control pulses, i.e., {\mathit{k}}_2,{\mathit{k}}_3=-{\mathit{k}}_0 is no longer necessary. After the fourth optical control pulse (for t > t_6), a second electric field pulse is applied to reverse the induced phase, thereby recovering the collective excitation at

(7)t_{echo}=t_6+t_5-t_3-t_2+t_0

with the spatial phase-matching condition:

(8){\mathit{k}}_{\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{o}}={\mathit{k}}_0-{\mathit{k}}_2-{\mathit{k}}_3+{\mathit{k}}_5+{\mathit{k}}_6.

In the forward retrieval process, the present storage scheme eliminates the need for a backward-propagating control \pi pulse, resulting in forward retrieval efficiency {\eta }_f\propto d^2{\mathrm{e}}^{-d} [52]. Importantly, the electric field control approach introduces no additional optical noise and effectively suppress coherent noise in the quiet region [53]. This is particularly advantageous for integrated photonic platforms, such as waveguides, where control and signal fields propagate in the same direction and angular separation is not feasible. In such cases, the electric field can mitigate noise caused by imperfect \pi pulses, enhancing the fidelity of quantum storage. To achieve higher efficiency via backward retrieval for samples with higher optical density, one can reverse {\mathit{k}}_5 or {\mathit{k}}_6, leading to following efficiency [52],

(9){\eta }_{\mathit{b}}=(1-{\mathrm{e}}^{-d}{)}^2{\mathrm{e}}^{\frac{{\mathrm{\Gamma }}_{12}^2(t_6-t_2{)}^2}{2\ln 2/{\pi }^2}}{\mathrm{e}}^{\frac{{\mathrm{\Gamma }}_{34}^2(t_5-t_3{)}^2}{2\ln 2/{\pi }^2}}{\mathrm{e}}^{-2{\gamma }_h(t_5-t_3)},

where the first item defines the efficiency of a back-retrieval echo. {\mathrm{\Gamma }}_{ij} is the inhomogeneous broadening of the spin transitions from |i⟩ state to |j⟩ state. In 50 ppm {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O} system, we approximately consider {\mathrm{\Gamma }}_{ij} = 130 kHz [51]. If an intense control light can be applied, a Rabi frequency of several MHz for the control pulse can be achieved. The last term {\gamma }_h, represents the effective optical decoherence. Its contribution can be neglected on the timescale considered here, as {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O} exhibits a millisecond long optical coherence time. Therefore, the retrieval efficiency in the backward configuration can be obtained, as indicated by the solid line in Fig. 4(c) with t_5-t_3=1 μs, t_6-t_2=2 μs. A reduction in the inhomogeneous linewidth {\mathrm{\Gamma }}_{ij} can significantly improve the memory efficiency, as illustrated by the dashed line in Fig. 4(c). Such a reduction may be achieved by employing a crystal with a lower doping concentration, as demonstrated in Yb:YSO [54].

4 Conclusion

In conclusion, we characterized the optical coherence times of {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O} at both crystallographic sites of the \mathrm{E}{\mathrm{r}}^{3+} dopants at sub-Kelvin temperatures. We demonstrated that, in this temperature regime, moderate magnetic fields can substantially suppress optical decoherence, achieving homogeneous linewidths as narrow as 100 Hz at 0.2 T. Detailed angular and temperature-dependent coherence measurements indicate that the dominant decoherence mechanisms are Er−Er spin flip−flop processes and thermally activated spectral diffusion. The experimental conditions used here notably eliminate the need for multi-Tesla fields previously required to suppress such effects. Nevertheless, spectral hole burning measurements reveal that the resolution of spectral features is fundamentally limited by super-hyperfine interactions, thereby complicating precise spectral tailoring. To address this challenge, we developed an optimized NLPE protocol. Since this strategy is, in principle, compatible with backward retrieval and cavity-enhanced configurations, it consequently offers a higher theoretical limit for storage efficiency. These advancements significantly enhance the viability of {}^{167}\mathrm{E}{\mathrm{r}}^{3+}:\mathrm{Y}\mathrm{S}\mathrm{O} as a platform for integrated solid-state quantum memories, particularly in applications involving on-chip waveguides and fiber-based quantum networks.

References

[1] M. Gündoğan , J. S. Sidhu , V. Henderson , L. Mazzarella , J. Wolters , D. K. Oi , and M. Krutzik , Proposal for space-borne quantum memories for global quantum networking, npj Quantum Inf. 7, 128 (2021)

[2] K. Azuma , S. E. Economou , D. Elkouss , P. Hilaire , L. Jiang , H. K. Lo , and I. Tzitrin , Quantum repeaters: From quantum networks to the quantum internet, Rev. Mod. Phys. 95(4), 045006 (2023)

[3] K. Heshami , D. G. England , P. C. Humphreys , P. J. Bustard , V. M. Acosta , J. Nunn , and B. J. Sussman , Quantum memories: Emerging applications and recent advances, J. Mod. Opt. 63(20), 2005 (2016)

[4] S. H. Wei , B. Jing , X. Y. Zhang , J. Y. Liao , C. Z. Yuan , B. Y. Fan , C. Lyu , D. L. Zhou , Y. Wang , G. W. Deng , H. Z. Song , D. Oblak , G. C. Guo , and Q. Zhou , Towards real-world quantum networks: A review, Laser Photonics Rev. 16(3), 2100219 (2022)

[5] N. Sangouard , C. Simon , H. de Riedmatten , and N. Gisin , Quantum repeaters based on atomic ensembles and linear optics, Rev. Mod. Phys. 83(1), 33 (2011)

[6] Z. Q. Zhou , C. Liu , C. F. Li , G. C. Guo , D. Oblak , M. Lei , A. Faraon , M. Mazzera , and H. de Riedmatten , Photonic integrated quantum memory in rare-earth doped solids, Laser Photonics Rev. 17(10), 2300257 (2023)

[7] C. Thiel , T. Böttger , and R. Cone , Rare-earth-doped materials for applications in quantum information storage and signal processing, J. Lumin. 131(3), 353 (2011)

[8] M. Guo , S. Liu , W. Sun , M. Ren , F. Wang , and M. Zhong , Rare-earth quantum memories: The experimental status quo, Front. Phys. (Beijing) 18(2), 21303 (2023)

[9] M. Zhong , M. P. Hedges , R. L. Ahlefeldt , J. G. Bartholomew , S. E. Beavan , S. M. Wittig , J. J. Longdell , and M. J. Sellars , Optically addressable nuclear spins in a solid with a six-hour coherence time, Nature 517(7533), 177 (2015)

[10] F. Wang , M. Ren , W. Sun , M. Guo , M. J. Sellars , R. L. Ahlefeldt , J. G. Bartholomew , J. Yao , S. Liu , and M. Zhong , Nuclear spins in a solid exceeding 10-hour coherence times for ultra-long term quantum storage, PRX Quantum 6(1), 010302 (2025)

[11] J. O’Sullivan,J. Travesedo,L. Pallegoix,Z. W. Huang,A. May,B. Yavkin,P. Hogan,S. Lin,R. Liu,T. Chaneliere, ., Individual solid-state nuclear spin qubits with coherence exceeding seconds, arXiv: 2024)

[12] M. Le Dantec , M. Rančić , S. Lin , E. Billaud , V. Ranjan , D. Flanigan , S. Bertaina , T. Chanelière , P. Goldner , A. Erb , R. B. Liu , D. Estève , D. Vion , E. Flurin , and P. Bertet , Twenty-three-millisecond electron spin coherence of erbium ions in a natural-abundance crystal, Sci. Adv. 7(51), eabj9786 (2021)

[13] E. Saglamyurek,N. Sinclair,J. Jin,J. A. Slater,D. Oblak,F. Bussieres,M. George,R. Ricken,W. Sohler,W. Tittel, Broadband waveguide quantum memory for entangled photons, Nature 469(7331), 512 (2011)

[14] S. H. Wei,B. Jing,X. Y. Zhang,J. Y. Liao,H. Li,L. X. You,Z. Wang,Y. Wang,G. W. Deng,H. Z. Song, ., Quantum storage of 1650 modes of single photons at telecom wavelength, npj Quantum Inf. 10, 19 (2024)

[15] M. Businger , L. Nicolas , T. S. Mejia , A. Ferrier , P. Goldner , and M. Afzelius , Non-classical correlations over 1250 modes between telecom photons and 979-nm photons stored in ¹⁷¹Yb³⁺:Y₂SiO₅, Nat. Commun. 13(1), 6438 (2022)

[16] A. Ortu , J. V. Rakonjac , A. Holzäpfel , A. Seri , S. Grandi , M. Mazzera , H. de Riedmatten , and M. Afzelius , Multimode capacity of atomic-frequency comb quantum memories, Quantum Sci. Technol. 7(3), 035024 (2022)

[17] D. Lago-Rivera , S. Grandi , J. V. Rakonjac , A. Seri , and H. de Riedmatten , Telecom-heralded entanglement between mode solid-state quantum memories, Nature 594(7861), 37 (2021)

[18] M. H. Jiang , W. Xue , Q. He , Y. Y. An , X. Zheng , W. J. Xu , Y. B. Xie , Y. Lu , S. Zhu , and X. S. Ma , Quantum storage of entangled photons at telecom wavelengths in a crystal, Nat. Commun. 14(1), 6995 (2023)

[19] Y. Y. An , Q. He , W. Xue , M. H. Jiang , C. Yang , Y. Q. Lu , S. Zhu , and X. S. Ma , Quantum teleportation from telecom photons to erbium-ion ensembles, Phys. Rev. Lett. 135(1), 010804 (2025)

[20] X. Zhang , B. Zhang , S. Wei , H. Li , J. Liao , T. Zhou , G. Deng , Y. Wang , H. Song , L. You , B. Fan , Y. Fan , F. Chen , G. Guo , and Q. Zhou , Integrated spectrally multiplexed light−matter interface at telecom band, Photon. Res. 12(12), 2821 (2024)

[21] P. Stevenson , C. M. Phenicie , I. Gray , S. P. Horvath , S. Welinski , A. M. Ferrenti , A. Ferrier , P. Goldner , S. Das , R. Ramesh , R. J. Cava , N. P. de Leon , and J. D. Thompson , Erbium-implanted materials for quantum communication applications, Phys. Rev. B 105(22), 224106 (2022)

[22] D. C. Liu , P. Y. Li , T. X. Zhu , L. Zheng , J. Y. Huang , Z. Q. Zhou , C. F. Li , and G. C. Guo , On-demand storage of photonic qubits at telecom wavelengths, Phys. Rev. Lett. 129(21), 210501 (2022)

[23] B. Lauritzen , J. Minář , H. de Riedmatten , M. Afzelius , N. Sangouard , C. Simon , and N. Gisin , Telecommunication wavelength solid-state memory at the single photon level, Phys. Rev. Lett. 104(8), 080502 (2010)

[24] T. Böttger , C. W. Thiel , R. L. Cone , and Y. Sun , Effects of magnetic field orientation on optical decoherence in Er³⁺:Y₂SiO₅, Phys. Rev. B 79(11), 115104 (2009)

[25] M. Rančić , M. P. Hedges , R. L. Ahlefeldt , and M. J. Sellars , Coherence time of over a second in a telecom-compatible quantum memory storage material, Nat. Phys. 14(1), 50 (2018)

[26] J. Stuart,K. Smith,M. Hedges,R. Ahlefeldt,M. Sellars, Progress towards efficient 4-level photon echo memories, arXiv: 2024)

[27] J. Y. Huang , P. Y. Li , Z. Q. Zhou , C. F. Li , and G. C. Guo , Extending the spin coherence lifetimes of ¹⁶⁷Er³⁺:Y₂SiO₅ at subkelvin temperatures, Phys. Rev. B 105(24), 245134 (2022)

[28] Z. Li,Y. Lei,T. Kling,M. Hosseini, Efficient storage of multidimensional telecom photons in a solid-state quantum memory, Optica Quantum 3(3), 295 (2025)

[29] T. X. Zhu , M. X. Su , C. Liu , Y. P. Liu , C. F. Wang , P. X. Liu , Y. J. Han , Z. Q. Zhou , C. F. Li , and G. C. Guo , Integrated spin-wave quantum memory, Natl. Sci. Rev. 11(11), nwae161 (2024)

[30] Y. P. Liu,Z. W. Ou,T. X. Zhu,M. X. Su,C. Liu,Y. J. Han,Z. Q. Zhou,C. F. Li,G. C. Guo, A millisecond integrated quantum memory for photonic qubits, Sci. Adv. 11(13), eadu5264 (2025)

[31] Y. Z. Ma , M. Jin , D. L. Chen , Z. Q. Zhou , C. F. Li , and G. C. Guo , Elimination of noise in optically rephased photon echoes, Nat. Commun. 12(1), 4378 (2021)

[32] M. Jin,Y. Z. Ma,Z. Q. Zhou,C. F. Li,G. C. Guo, A faithful solid-state spin-wave quantum memory for polarization qubits, Sci. Bull. (Beijing) 67(7), 676 (2022)

[33] O. Guillot-Noël , P. Goldner , Y. L. Du , E. Baldit , P. Monnier , and K. Bencheikh , Hyperfine interaction of Er³⁺ ions in Y₂SiO₅: An electron paramagnetic resonance spectroscopy study, Phys. Rev. B 74(21), 214409 (2006)

[34] N. L. Jobbitt , J. P. R. Wells , M. F. Reid , S. P. Horvath , P. Goldner , and A. Ferrier , Prediction of optical polarization and high-field hyperfine structure via a parametrized crystal-field model for low-symmetry centers in Er³⁺-doped Y₂SiO₅, Phys. Rev. B 104(15), 155121 (2021)

[35] Y. Sun , T. Böttger , C. W. Thiel , and R. L. Cone , Magnetic g tensors for the 4I15/2 and 4I13/2 states of Er³⁺:Y₂SiO₅, Phys. Rev. B 77(8), 085124 (2008)

[36] Y. H. Chen , X. Fernandez-Gonzalvo , S. P. Horvath , J. V. Rakonjac , and J. J. Longdell , Hyperfine interactions of Er³⁺ ions in Y₂SiO₅: Electron paramagnetic resonance in a tunable microwave cavity, Phys. Rev. B 97(2), 024419 (2018)

[37] S. P. Horvath , J. V. Rakonjac , Y. H. Chen , J. J. Longdell , P. Goldner , J. P. R. Wells , and M. F. Reid , Extending phenomenological crystal-field methods to C1 point-group symmetry: Characterization of the optically excited hyperfine structure of ¹⁶⁷Er³⁺:Y₂SiO₅, Phys. Rev. Lett. 123(5), 057401 (2019)

[38] W. B. Mims, Phase memory in electron spin echoes, lattice relaxation effects in CaWO₄:Er, Ce, Mn, Phys. Rev. 168(2), 370 (1968)

[39] T. Böttger , C. W. Thiel , Y. Sun , and R. L. Cone , Optical decoherence and spectral diffusion at 1.5 µm in Er³⁺:Y₂SiO₅ versus magnetic field, temperature, and Er³⁺ concentration, Phys. Rev. B 73(7), 075101 (2006)

[40] N. Kukharchyk,D. Sholokhov,A. Kalachev,P. Bushev, Enhancement of optical coherence in ¹⁶⁷Er³⁺:Y₂SiO₅ crystal at sub-kelvin temperatures, arXiv: 2019)

[41] T. Böttger , Y. Sun , C. W. Thiel , and R. L. Cone , Spectroscopy and dynamics of Er³⁺:Y₂SiO₅ at 1.5 µm, Phys. Rev. B 74(7), 075107 (2006)

[42] S. Probst , H. Rotzinger , A. V. Ustinov , and P. A. Bushev , Microwave multimode memory with an erbium spin ensemble, Phys. Rev. B 92(1), 014421 (2015)

[43] M. Afzelius , C. Simon , H. de Riedmatten , and N. Gisin , Multimode quantum memory based on atomic frequency combs, Phys. Rev. A 79(5), 052329 (2009)

[44] B. Car , L. Veissier , A. Louchet-Chauvet , J. L. Le Gouët , and T. Chanelière , Selective optical addressing of nuclear spins through superhyperfine interaction in rare-earth doped solids, Phys. Rev. Lett. 120(19), 197401 (2018)

[45] V. Damon , M. Bonarota , A. Louchet-Chauvet , T. Chanelière , and J. L. Le Gouët , Revival of silenced echo and quantum memory for light, New J. Phys. 13(9), 093031 (2011)

[46] M. Afzelius and C. Simon , Impedance-matched cavity quantum memory, Phys. Rev. A 82(2), 022310 (2010)

[47] A. Arcangeli , A. Ferrier , and P. Goldner , Stark echo modulation for quantum memories, Phys. Rev. A 93(6), 062303 (2016)

[48] S. P. Horvath , M. K. Alqedra , A. Kinos , A. Walther , J. M. Dahlström , S. Kröll , and L. Rippe , Noise-free on-demand atomic frequency comb quantum memory, Phys. Rev. Res. 3(2), 023099 (2021)

[49] D. L. McAuslan , P. M. Ledingham , W. R. Naylor , S. E. Beavan , M. P. Hedges , M. J. Sellars , and J. J. Longdell , Photonecho quantum memories in inhomogeneously broadened two-level atoms, Phys. Rev. A 84(2), 022309 (2011)

[50] R. L. Ahlefeldt , A. Lyasota , J. Smith , J. Ren , and M. J. Sellars , Effect of a hybrid transition moment on Stark-modulated photon echoes in Er³⁺:Y₂SiO₅, Phys. Rev. B 108(19), 195122 (2023)

[51] J. S. Stuart , M. Hedges , R. Ahlefeldt , and M. Sellars , Initialization protocol for efficient quantum memories using resolved hyperfine structure, Phys. Rev. Res. 3(3), L032054 (2021)

[52] N. Sangouard , C. Simon , M. Afzelius , and N. Gisin , Analysis of a quantum memory for photons based on controlled reversible inhomogeneous broadening, Phys. Rev. A 75(3), 032327 (2007)

[53] M. K. Alqedra , S. P. Horvath , A. Kinos , A. Walther , S. Kröll , and L. Rippe , Stark control of solid-state quantum memory with spin-wave storage, Phys. Rev. A 109(1), 012607 (2024)

[54] T. S. Mejia,L. Nicolas,A. G. Rodriguez,P. Goldner,M. Afzelius, Broadband and long-duration optical memory in Yb:YSO, arXiv: 2025)

RIGHTS & PERMISSIONS

Higher Education Press