Optimizing the spin qubit performance of lanthanide-based metal–organic frameworks

Abstract

Lanthanide-based spin qubits are intriguing candidates for high-fidelity quantum memories owing to their spin-optical interfaces. Metal–organic frameworks (MOFs) offer promising solid-state platforms to host lanthanide ions because their bottom-up synthesis enables rational optimization of both spin coherence and luminescence. Here, we incorporated Nd³⁺ and Gd³⁺ into a La³⁺-based MOF with various doping levels and examined their qubit performance including the spin relaxation time (T₁) and phase memory time (T_m). Both Nd³⁺ and Gd³⁺ behave as spin qubits with T₁ exceeding 1 ms and T_m approaching 2 μs at 3.2 K at low doping levels. Variable-temperature spin dynamic studies unveiled spin relaxation and decoherence mechanisms, highlighting the critical roles of spin–phonon coupling and spin–spin dipolar coupling. Accordingly, reducing the spin concentration, spin–orbit coupling strength, and ground spin state improves the qubit performance of lanthanide-based MOFs. These optimization strategies serve as guidelines for the future development of solid-state lanthanide qubits targeting quantum information technologies.

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Introduction

Quantum memories are indispensable building units for distributed quantum computing and optical-fiber-based quantum communication.^1–3 They can interconvert the quantum state between a flying qubit, typically a photon, and a stationary qubit that is insensitive to the environment, allowing on-demand storage and extraction of quantum information.^4,5 Paramagnetic lanthanide (rare earth) ions, e.g. Ce³⁺, Nd³⁺, Eu³⁺, Er³⁺, Pr³⁺ and Yb³⁺, are promising candidates for stationary qubits thanks to their spin coherence and narrow and well-defined emission lines.^6–11 When embedded in inorganic crystals such as Y₂O₃ and Y₂SiO₅, they could maintain quantum coherence over 1 ms at approximately 10 mK.^12,13 Meanwhile, the mapping between their emission wavelengths and spin sublevels promotes coherent addressing of single qubits by optically detected magnetic resonance.^5,14 These advantages have inspired the development of lanthanide-based quantum memories that can store photons for over one hour.¹⁵

Spin–photon transduction requires lanthanide qubits with a long spin decoherence time (T₂) and high optical addressing fidelity.¹⁶ While the latter can be improved by integrating lanthanide qubits into nanophotonic cavities,^17,18 the optimization of T₂ demands sophisticated material design. T₂ is upper bound by twice the spin relaxation time (T₁), which often decreases sharply with an increase in temperature for lanthanide qubits stemming from their strong spin–phonon coupling.^19,20 The strength of spin–phonon coupling may be weakened by tweaking the phonon dispersion relations of solid-state matrices.²¹ Nonetheless, this requires fine tuning of the crystal structures of host materials, especially coordination environments of lanthanide ions, which is unfeasible for inorganic solids.²²

Metal–organic frameworks (MOFs) offer tunable solid-state matrices for lanthanide qubits. MOFs are crystalline and microporous materials consisting of metal ions and organic ligands.²³ Their spatial ordering and bottom-up assembly of constituents facilitate the rational design of crystal structures and in turn phonon dispersion relations.^24,25 Meanwhile, lanthanide ions could be incorporated into MOFs with controlled spatial distribution and coordination environments.^26,27 These advantages would allow the strategic suppression of spin relaxation and decoherence.^28,29 Indeed, recent studies on molecular lanthanide qubits have revealed that tuning the electronic configurations and coordination environments of lanthanide ions could enforce electron spins with s-orbital characteristics and strong hyperfine coupling.^30–33 These features give rise to weak spin–orbit coupling and clock-like states, respectively, which improve the qubit performance including T₁ and T₂.

So far, only two lanthanide-based MOFs (Ln-MOFs) have been demonstrated to behave as qubits.^34,35 Both consist of Gd³⁺ ions that exhibit spin coherence well above 10 K, which was attributed to the weak spin–orbit coupling of Gd³⁺. However, the qubit performance of other lanthanide ions in MOFs has not been investigated, and its optimization strategies await elucidation. Herein, we report two new MOFs embedding lanthanide qubits that were synthesized by doping Nd³⁺ and Gd³⁺ into [N(C₂H₅)₄][La(CAN)₂(H₂O)] (CAN²⁻ = chloranilate). The anisotropic 4f³ electron configuration of Nd³⁺ gives rise to a strong spin–orbit coupling and a low ground spin state (S = 3/2), whereas the spherically symmetric 4f⁷ electron configuration of Gd³⁺ leads to a weak spin–orbit coupling and a high ground spin state (S = 7/2). By examining the spin dynamics of these two materials at various temperatures and doping levels by pulse electron paramagnetic resonance (EPR) spectroscopy, we elaborated their spin relaxation and decoherence mechanisms to show that the low spin concentration, weak spin–orbit coupling, and low ground spin state help improve the qubit performance.

Results and discussion

Structures and synthesis

[N(C₂H₅)₄][La(CAN)₂(H₂O)] (La(CAN)), [N(C₂H₅)₄][Nd(CAN)₂(H₂O)] (Nd(CAN)), and [N(C₂H₅)₄][Gd(CAN)₂] (Gd(CAN)) exhibit layered structures in which lanthanide ions and CAN²⁻ ligands form anionic square lattices separated by N(C₂H₅)₄⁺ cations (Fig. S1–S3†).^36,37 The interlayer distances are approximately 1 nm. La(CAN) and Nd(CAN) are isostructural where La³⁺ and Nd³⁺ display a 9-coordinated capped square antiprismatic geometry composed of four bidentate CAN²⁻ ligands and a H₂O molecule (Fig. 1c and d). In contrast, Gd³⁺ in Gd(CAN) exhibits an 8-coordinated square antiprismatic geometry without coordinating H₂O (Fig. 1e).

Fig. 1 (a) Portions of the crystal structure viewed along the crystallographic c axis and (b) the crystallographic b axis for magnetically diluted MOFs exemplified by Nd_xLa_100−x. (c–e) Coordination environments of La³⁺, Nd³⁺, and Gd³⁺. Gray, red, green, cyan, purple, and orange spheres represent C, O, Cl, La, Nd, and Gd atoms, respectively. Hydrogen atoms and N(C₂H₅)₄⁺ cations are omitted for clarity.

Based on the literature, we synthesized La(CAN), Nd(CAN), and Gd(CAN) by heating Ln(NO₃)₃ (Ln³⁺ = La³⁺, Nd³⁺, and Gd³⁺), chloranilic acid, and [N(C₂H₅)₄]Cl in a mixture of N,N-dimethylformamide (DMF) and H₂O at 130 °C for 16 h.³⁶ Powder X-ray diffraction (PXRD) confirmed their structures and crystallinity (Fig. S4†). We further synthesized magnetically diluted MOFs, [N(C₂H₅)₄][Nd_x/100La_1−x/100(CAN)₂(H₂O)] (x = 0.5, 1, 4, and 20; abbreviated as Nd_xLa_100−x) and [N(C₂H₅)₄][Gd_y/100La_1−y/100(CAN)₂(H₂O)_1−y/100] (y = 0.1, 0.5, 1, and 20; abbreviated as Gd_yLa_100−y), using the same procedures with various stoichiometric ratios between Nd³⁺ (or Gd³⁺) and La³⁺ (Fig. 1a and b). PXRD characterization of the doped samples confirmed that doping does not change the structure of the matrix, La(CAN) (Fig. S5†). Inductively coupled plasma atomic emission spectroscopy (ICP-AES) revealed that the actual doping levels of Nd³⁺ and Gd³⁺ are consistent with the designated values (Table S1†). The spatial distributions of these dopants are homogeneous as evidenced by the energy dispersive X-ray spectroscopy (EDS) mapping of Nd₂₀La₈₀ and Gd₂₀La₈₀ (Fig. S6 and S7†).

Continuous wave (CW) EPR spectroscopy

We conducted X-band (9.6 GHz) CW EPR spectroscopic characterization of Nd₄La₉₆ and Gd₁La₉₉ at 8 K and 90 K, respectively. The CW EPR spectrum of Nd₄La₉₆ shows obvious rhombic signals indicating g-anisotropy (Fig. 2a). It can be fitted with the following spin Hamiltonian:where g represents the g-tensor, A represents the hyperfine coupling tensor, B represents the magnetic field, represents the electron spin operator, represents the nuclear spin operator, and μ_B represents the Bohr magneton. The first term describes the Zeeman splitting of the electron spin, and the second term describes the hyperfine interaction between the electron spin and nuclear spin of Nd³⁺. The latter is only relevant for isotopes with non-zero nuclear spins (I = 7/2 for ¹⁴³Nd and ¹⁴⁵Nd with 12.2% and 8.3% natural abundance, respectively). Because hyperfine splitting is not well resolved in y and z directions, A_y and A_z are not included in the fitting. The spectrum can be well reproduced with S = 1/2, g_x = 3.54, g_y = 2.59, and g_z = 1.09 for all isotopes and A_x = 1237 MHz for ¹⁴³Nd and ¹⁴⁵Nd. Thus, the strong spin–orbit coupling of Nd³⁺ causes significant g-anisotropy and large zero-field splitting, the latter of which makes Nd³⁺ embedded in La(CAN) behaving as an effective S = 1/2 spin system. This is consistent with previous studies on Nd³⁺ in other materials.³⁸ The small features at approximately 482 mT that cannot be fitted using eqn (1) may originate from the polycrystalline effect.^38,39

Fig. 2 X-band CW EPR spectra of (a) Nd₄La₉₆ and (b) Gd₁La₉₉ acquired at 8 K and 90 K, respectively. Red lines are fitting curves obtained using EasySpin.⁶⁰ Background signals of the resonator are marked with asterisks (Fig. S8†). The signal marked with an arrow is attributed to the polycrystallinity effect.^38,39

The CW EPR spectrum of Gd₁La₉₉ is shown in Fig. 2b. It can be fitted with the following spin Hamiltonian:

where the first term represents the Zeeman splitting of electron spin with an isotropic g-factor, the second and third represent the second-order zero-field splitting with D and E as principal values, and the last three represent the fourth-order zero-field splitting ( are the extended Stevens operators; B⁰₄, B²₄, and B⁴₄ are the associated coefficients).^35,40 Fitting revealed S = 7/2, g_iso = 2.007, D = 339 MHz, E = 102 MHz, B⁰₄ = −0.27 MHz, B²₄ = 12.7 MHz, and B⁴₄ = 10.8 MHz. Both the g-isotropy and small zero field splitting parameters indicate weak magnetic anisotropy of Gd³⁺ embedded in La(CAN) due to its weak spin–orbit coupling.

Qubit behaviors

DiVincenzo's criteria dictate that a qubit should be a coherently addressable two-level quantum system.⁴ We conducted X-band (9.6 GHz) pulse EPR spectroscopy to investigate the qubit behaviors and spin dynamics of Nd_xLa_100−x (x = 0.5, 1, and 4) and Gd_yLa_100−y (y = 0.1, 0.5, and 1). Echo-detected field sweep (EDFS) spectra of these samples are consistent with their CW EPR spectra (Fig. S9a and S10†). We performed the following experiments under the magnetic field corresponding to g = 2.60 (B = 264.5 mT) and g = 1.99 (B = 347.4 mT) for Nd³⁺- and Gd³⁺-based samples, respectively, where EDFS spectra display maximum intensities. The T₁ and phase memory time (T_m), the latter of which involves all dephasing factors and resembles T₂, were characterized by picket-fence saturation recovery and Hahn echo decay pulse sequences, respectively (Fig. S9a and c†). All samples exhibit quantum coherence at 3.2 K or 3.4 K (Fig. 3a, b, d and e).

Fig. 3 (a) Saturation recovery curves and (b) Hahn echo decay curves of Nd_xLa_100−x collected at 3.4 K. (c) Nutation curves of Nd_0.5La_99.5 collected under various microwave attenuations at 5.2 K. (d) Saturation recovery curves and (e) Hahn echo decay curves of Gd_yLa_100−y collected at 3.2 K. (f) Nutation curves of Gd_0.1La_99.9 collected under various microwave attenuations at 30 K. Black lines in (a), (b) and (e) are mono-exponential decay fitting curves and those in (d) are bi-exponential decay fitting curves.

To verify the coherent addressability of Nd³⁺ and Gd³⁺ in the La(CAN) matrix, we performed nutation experiments for Nd_0.5La_99.5 and Gd_0.1La_99.9 at 5.2 K and 30 K, respectively, at various microwave powers (Fig. S9b†). Rabi oscillations clearly manifest themselves in the time-domain nutation curves (Fig. 3c and f), whose Fourier transforms display two peaks (Fig. S11a and c†). One of the peaks is independent of the microwave power: it remains at 11.7 MHz for Nd_0.5La_99.5 and 14.6 MHz for Gd_0.1La_99.9. Accordingly, this peak is attributed to the Larmor frequency of ¹H nuclear spin as a manifestation of the Hartmann–Hahn effect.⁴¹ The other peak exhibits a linear relationship with the microwave magnetic field strength, which is attributed to the Rabi frequency (Fig. S11b and d†).⁴² Therefore, the electron spins of Nd_xLa_100−x and Gd_yLa_100−y can be coherently manipulated to form arbitrary superpositions of their constituent states, thereby behaving as qubits.

Spin relaxation

Spin relaxation requires energy exchange between the target electron spin and the environment, which is mediated by the surrounding phonon bath and spin bath. The spin–phonon coupling gives rise to spin–lattice relaxation, which may undergo various mechanisms that exhibit different temperature dependencies yet are independent of the spin concentration.⁴³ In contrast, the spin–spin dipolar coupling induces relaxation through spin flip-flop, giving rise to cross relaxation that is typically concentration-dependent.^44,45 Hence, investigating spin relaxation at various temperatures and spin concentrations could distinguish the contributions of spin–phonon coupling and spin–spin dipolar coupling.

The T₁ of Nd_xLa_100−x was measurable below 8.0 K (Fig. S12–S15 and Table S6†). As shown in Fig. 4a, it decreases sharply with an increase in temperature for each sample. For instance, Nd_0.5La_99.5 exhibits a T₁ value of 924 μs at 3.4 K, which drops to 1.48 μs at 8.0 K. Meanwhile, T₁ is weakly dependent on the spin concentration (Fig. 3a and 4a). Nd_0.5La_99.5 and Nd₁La₉₉ display almost identical T₁ values across the temperature range of 3.4 K–8.0 K. Their temperature (T) dependence of T₁ obeys 1/T₁ ∝ T⁷, which is consistent with the Raman relaxation process involving acoustic phonons (Fig. 4b).⁴⁶ This relationship is valid when the Debye temperature (T_D), i.e. the temperature corresponding to the cutoff frequency of acoustic phonons, is much higher than the experimental temperature (Fig. S16; see the detailed discussion in Note S1†). Indeed, through simulation, we found T_D > 80 K for these two materials. This value is comparable with the T_D observed for lanthanide ions in inorganic solids, e.g. Yb-doped Y₂SiO₅ (T_D = 100 K), indicating a rigid lattice of the MOF matrix.⁴⁷

Fig. 4 (a and b) T₁ and 1/T₁vs. T data for Nd_xLa_100−x. Orange and green lines are fits to the data by 1/T₁ = A_RamanT⁷ for Nd_0.5La_99.5 and Nd₁La₉₉, respectively. (c) Fitting of 1/T₁vs. T data for Nd₄La₉₆ with eqn (3). Turquoise and violet-red dashed lines represent contributions from Raman and cross relaxation, respectively, and the wine solid line is their sum. The inset illustrates the cross relaxation caused by flip-flop between saturated spins (right) and nearby thermally populated spins (left). (d and e) T₁ and 1/T₁vs. T data for Gd_yLa_100−y. The wine line is the fit to the data by 1/T₁ = A_RamanT^2.69 for Gd_0.1La_99.9. (f) Fitting of 1/T₁vs. T data for Gd₁La₉₉ with eqn (4). Turquoise and orange dashed lines represent contributions from the Raman and thermally activated relaxation, respectively, and the wine solid line is their sum.

Nd ₄ La ₉₆ exhibits a shorter T₁ value than those of Nd_0.5La_99.5 and Nd₁La₉₉: its T₁ is 417 μs at 3.4 K and it drops to 2.93 μs at 7.2 K (Fig. 4a). The temperature dependence of T₁ slightly deviates from 1/T₁ ∝ T⁷ below 4 K (Fig. 4b). Because this deviation only occurs in the sample with a high doping level, it is likely a manifestation of the cross relaxation rather than the direct process—the former is dependent on the spin concentration yet the latter is not.⁴³ To understand this behavior, we propose a model to describe the cross relaxation taking place in the saturation recovery experiment. It accounts for flip-flop events between saturated spins and adjacent thermally populated spins (Fig. 4c inset). The flip-flop rate scales with the spin–spin dipolar coupling strength, which becomes salient at high spin concentrations.⁴⁸ For Nd³⁺ that behaves as a pseudo S = 1/2 spin system, this model gives rise to a cross relaxation rate that is proportional to the Boltzmann population difference between ground and excited spin sublevels (see the detailed discussion in Note S2†). Accordingly, we fitted the temperature dependence of the T₁ value of Nd₄La₉₆ with the following equation:

where A_cross and A_Raman represent pre-factors of cross relaxation and Raman relaxation, respectively, h represents the Planck constant, ν represents the Larmor frequency of the electron spin, and k_B represents the Boltzmann constant. As shown in Fig. 4c, the cross relaxation rate is significantly smaller than the Raman relaxation rate. Thus, both spin–phonon coupling and spin–spin dipolar coupling contribute to spin relaxation in Nd_xLa_100−x with the former being dominant.

Gd _y La _100−y samples show measurable T₁ values for up to 90 K (Fig. S12, S17–S19 and Tables S7–S9†). The T₁ value of Gd_0.1La_99.9 decreases monotonically with an increase in temperature: it drops from 4.68 ms at 3.2 K to 0.56 μs at 90 K (Fig. 4d). The temperature dependence of T₁ shows a Raman-like relaxation process with 1/T₁ ∝ T^2.69 (Fig. 4e). This small exponent value indicates the participation of both acoustic and optical phonons in Raman relaxation,^46,49 and it is consistent with previously reported values for Gd³⁺-based spin systems.⁵⁰Gd_0.5La_99.5 and Gd₁La₉₉ show comparable T₁ values and relaxation behaviors with those of Gd_0.1La_99.9 above 10 K and 20 K, respectively (Fig. 4d and e). Below these temperatures, T₁ decreases with an increase in spin concentration. At 3.2 K, T₁ drops to 640 μs and 141 μs for samples with 0.5% and 1% doping levels, respectively. These results show that both spin–phonon coupling and spin–spin dipolar coupling contribute to the spin relaxation of Gd_yLa_100−y. In the high-temperature region, spin–phonon coupling dominates regardless of the spin concentration, whereas in the low-temperature region, spin–spin dipolar coupling is salient at a high spin concentration.

Notably, Gd_0.5La_99.5 and Gd₁La₉₉ exhibit reciprocating thermal behaviors below 3.8 K and 4.9 K, respectively, where their T₁ values decrease with a decrease in temperature (Fig. 4d). For instance, upon cooling Gd₁La₉₉, its T₁ increases from 98 μs at 10 K to 213 μs at 4.9 K, and it then decreases to 141 μs at 3.2 K. This trend indicates the presence of a thermally activated relaxation process that reaches the maximum rate below 3.2 K.⁴³ Hence, we fitted this trend using the following equation:

where A_thermal and E_a are the pre-factor and activation energy of thermally activated relaxation, respectively, and n is the exponent of Raman relaxation (Fig. 4f). n was fixed at 2.69 based on the observation from Gd_0.1La_99.9. Fitting revealed that spin relaxation below 6.5 K is dominated by thermally activated relaxation with E_a = 0.47 meV, which corresponds to a temperature of 5.4 K.

Such reciprocating thermal behavior has been observed in another Gd³⁺-based MOF and several single ion magnets whose magnetic relaxation times, acquired by alternative current susceptibility measurements, become shorter at a lower temperature.^51,52 To our knowledge, it has not been observed previously in spin relaxation probed by pulse EPR spectroscopy. This phenomenon was attributed to the phonon bottleneck effect, but its physical interpretation is still unclear.^51,52 Our experiments showed that it involves thermal activation and becomes more salient at a higher spin concentration, so we tentatively assign it as a manifestation of spin–spin dipolar coupling and may be related to electron spin flip-flop involving multiple spin sublevels. We will investigate the nature of the reciprocating thermal behavior of Gd_yLa_100−y in future studies.

Spin decoherence

In theory, T_m may reach 2T₁ (Fig. 5g), but it is typically much shorter than T₁ in molecular systems due to other dephasing processes induced by environmental magnetic noise, e.g. nearby electron spins and nuclear spins, through instantaneous diffusion (ID) and spectral diffusion (SD).⁵³ ID is caused by instantaneous spin rotation during the pulse (Fig. 5k). It scales with the spin concentration but is inherently independent of temperature. The SD takes place during the free evolution time of a pulse sequence. It mainly involves three kinds of events for lanthanide qubits: relaxation-induced stochastic flipping of nearby electron spins (electronic SD, Fig. 5h), direct/indirect electron spin flip-flop (Fig. 5i), and nuclear spin flip-flop (nuclear SD, Fig. 5j).⁴⁸ The nuclear SD is independent of both temperature and electron spin concentration, while the former two are concentration-dependent. The rate of electronic SD scales with that is temperature-dependent,^47,54 whereas the rate of electron spin flip-flop is nearly temperature-independent for Nd³⁺ and Gd³⁺ under our experimental conditions (Fig. S20; see the detailed discussion in Note S3†).^48,55 Thus, conducting variable-temperature and variable-concentration T_m measurements would reveal spin decoherence mechanisms.

Fig. 5 (a and d) T_mvs. T data for Nd_xLa_100−x and Gd_yLa_100−y. (b, c, e and f) Fitting of 1/T_mvs. T data for Nd_0.5La_99.5, Nd₄La₉₆, Gd_0.1La_99.9, and Gd₁La₉₉ with eqn (5). Red, blue, and green dashed lines represent contributions from spin relaxation, electronic SD, and a temperature-independent constant, C, which encompasses contributions from electron spin flip-flop, nuclear SD, and ID. The plum solid line is their sum. (g–k) Spin decoherence mechanisms:⁴⁸ (g) spin relaxation (1/2T₁); (h) electronic SD; (i) direct/indirect electron spin flip-flop; (j) nuclear SD; (k) ID. The pink, cyan, and orange spheres represent the electron spin of interest, the neighbouring electron spin causing decoherence, and the neighbouring nuclear spins, respectively. MW represents microwave. Dashed lines represent dipolar coupling between spins and dashed arrows describe spin flipping events.

We conducted Hahn echo decay experiments with various pulse lengths (32 ns, 64 ns, and 128 ns for the π pulse) to articulate the contribution of ID, which should decrease with an increase in pulse lengths.⁴⁸ While long and selective pulses do not alter the T_m of Nd_0.5La_99.5 and Gd_0.1La_99.9, they significantly improve the T_m of Nd₄La₉₆ and Gd₁La₉₉ and the associated improvements decrease with an increase in temperature (Fig. S21 and S22†). Therefore, ID is negligible at low spin concentrations, whereas it becomes significant at high spin concentrations. In addition, the Hahn echo intensity displays oscillations with the free evolution time, indicating electronic spin echo envelope modulation (ESEEM) from nearby nuclear spins (Fig. 3b and e). Combination-peak ESEEM measurements (Fig. S9d†) of Nd_0.5La_99.5 and Gd_0.1La_99.9 revealed modulation frequencies signifying nuclear spins of ¹H, ¹³C, and ¹³⁹La (Fig. S23†). Hence, these nuclear spins are coupled with the electron spin of Nd³⁺ or Gd³⁺, causing decoherence through nuclear SD.

In the following analysis, we focused on the T_m acquired with the π pulse being 32 ns with the shot repetition time being greater than 5T₁ (see the detailed discussion in Note S4; Fig. S24 and S25, Tables S2 and S3†). The obtained Hahn echo decay curves were fitted by mono-exponential decay function (see the detailed discussion in Note S5; Fig. S26–S33, Table S4†). It decreases with an increase in temperature and an increase in spin concentration for both Nd³⁺ and Gd³⁺ (Fig. 5a and d; Fig. S34–S40, Tables S6–S9†). For instance, the T_m of Nd_0.5La_99.5 drops from 1.46 μs at 3.4 K to 0.28 μs at 8.0 K, and increasing the doping level to 4% reduces the T_m to 0.49 μs at 3.4 K. To understand the decoherence sources, we analyzed the T_m of Nd_xLa_100−x with the following equation:

where the second term describes decoherence caused by the electronic SD, and the last term is a temperature-independent constant that encompasses contributions from the electron spin flip-flop, nuclear SD, and ID. Fitting the temperature dependencies of the T_m of Nd_xLa_100−x with eqn (5) revealed that the electronic SD dominates for all doping levels, whereas C increases significantly with an increase in the spin concentration possibly due to enhanced ID or electron spin flip-flop (Fig. 5b and c; Fig. S41a, Table S5†). Hence, the spin coherence of Nd³⁺ is mainly limited by relaxation-induced spectral diffusion, and reducing the doping level of Nd³⁺ improves its T_m.

Above 10 K, Gd_yLa_100−y exhibits similar decoherence behaviors with Nd_xLa_100−x (Fig. 5d). Analyzing the T_m in the temperature range of 10 K–90 K with eqn (5) revealed the dominant role of electronic SD in decoherence (Fig. 5e and f; Fig. S41b, Table S5†). Below 10 K, the T_m of Gd³⁺ decreases with an increase in the doping level, and it displays distinct temperature dependencies that cannot be fitted by eqn (5) (Fig. 5d). The T_m of Gd_0.1La_99.9 and Gd_0.5La_99.5 increases linearly with a decrease in temperature below 6.8 K and 6.4 K, respectively. Meanwhile, Gd_0.1La_99.9 and Gd₁La₉₉ exhibit plateaus of T_m at 6.8 K–12 K and 3.2 K–4.3 K, respectively. We note that the overall Zeeman splitting between m_s = −7/2 and m_s = −7/2 sublevels of Gd³⁺ is approximately 69 GHz, which corresponds to 3.3 K. Considering the comparability between these temperatures, the unique low-temperature decoherence behaviors may be related to the Boltzmann population of spin sublevels. In addition, the methyl tunnelling within N(C₂H₅)₄⁺ cations in the pores of Gd_yLa_100−y may also contribute to the decoherence behaviors below 10 K.^56–58

Strategies to improve the spin qubit performance of Ln-MOFs

The above analysis reveals critical roles of spin–phonon coupling and spin–spin dipolar coupling in the spin dynamics of Nd_xLa_100−x and Gd_yLa_100−y (Fig. 6a). The spin–phonon coupling causes Raman relaxation that dominates the spin relaxation at relatively high temperatures for both materials. Meanwhile, relaxation-induced spectral diffusion (electronic SD) plays a major role in spin decoherence—although T_m is much shorter than T₁, it is indirectly limited by T₁. The spin–spin dipolar coupling causes decoherence through electronic SD, ID, and electron spin flip-flop, the latter of which also mediates cross relaxation. Our analysis of cross relaxation in T₁ measurements shows that its rate decreases with an increase in temperature, which might explain the unique reciprocating thermal behavior observed in Gd_0.5La_99.5 and Gd₁La₉₉. Overall, both spin–phonon coupling and spin–spin dipolar coupling should be suppressed to improve the key qubit metrics, i.e. T₁ and T_m, of Ln-MOFs.

Fig. 6 (a) Spin relaxation and decoherence mechanisms of Ln-MOFs. Orange-red and blue-green arrows specify influences on T₁ and T_m, respectively. (b–d) Optimization strategies for the qubit performance of Ln-MOFs. The wavy line represents interaction. SD, ID, and SOC represent spectral diffusion, instantaneous diffusion, and spin–orbit coupling, respectively.

Our experiments show that lanthanide qubits exhibit decent T₁ and T_m at low temperatures and spin concentrations (Fig. 6b). Lowering the temperature reduces the number of active phonons that couple with spins, thereby inhibiting spin–lattice relaxation. It also improves T_m by suppressing the electronic SD. Reducing the spin concentration weakens the spin–spin dipolar coupling. Therefore, diluting lanthanide ions in diamagnetic matrices could improve their T₁ and T_m.

Comparisons between Nd_xLa_100−x and Gd_yLa_100−y show that weakening the spin–orbit coupling and reducing the ground spin state also enhance the qubit performance (Fig. 6c and d). The spin–orbit coupling determines the strength of spin–phonon coupling.⁵⁹ The 4f³ electronic configuration of Nd³⁺ gives rise to a strong spin–orbit coupling. As a result, the T₁ of Nd_xLa_100−x declines sharply with an increase in temperature, limiting the operation temperature of these materials below 8 K. In contrast, Gd³⁺ displays weak spin–orbit coupling thanks to its spherically symmetric 4f⁷ electron configuration that quenches the orbital angular momentum. This prompts Gd_yLa_100−y to maintain quantum coherence for up to 90 K. On the other hand, lowering the ground spin state not only reduces the spin–spin dipolar coupling but also avoids decoherence from electron spin flip-flop through multiple spin sublevels. Therefore, to maximize the T₁ and T_m of Ln-MOFs, one should choose appropriate lanthanide ions and optimize their coordination environments to minimize both spin–orbit coupling and the ground spin state. Notably, this is contrary to the design principle of lanthanide-based single-molecule magnets, which demand large spin–orbit coupling to improve the magnetic anisotropy and high ground spin state to enlarge the spin reversal barrier.²² Comparisons between the spin relaxation of lanthanide qubits and the magnetic relaxation of lanthanide-based single-molecule magnets would reveal insights into the opposite design principles of these systems.

Conclusions

In summary, we synthesized two solid-state lanthanide qubits by embedding Nd³⁺ and Gd³⁺ ions into a MOF. Through variable-temperature and variable-concentration spin dynamic characterization of these materials, we articulated their spin relaxation and decoherence mechanisms that reflect the critical influences of the environmental phonon bath and spin bath. Spin–phonon coupling plays a major role in the spin relaxation of both Nd³⁺ and Gd³⁺, and it indirectly limits the T_m through relaxation-induced spectral diffusion. Spin–spin dipolar coupling contributes to cross relaxation and participates in several spin decoherence processes including relaxation-induced spectral diffusion, direct/indirect electron spin flip-flop, and instantaneous diffusion. This analysis highlights the low spin concentration, weak spin–orbit coupling, and low ground spin state as optimization strategies for the T₁ and T_m of lanthanide qubits, providing guidelines for the design of Ln-MOFs as building units of quantum memories.

Author contributions

X. D. designed and conducted experiments. L. S. conceived the idea and oversaw the project. X. D. and L. S. analyzed the data and co-wrote the manuscript.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 22273078) and the Hangzhou Municipal Funding Team of Innovation (TD2022004). We thank Dr Luming Yang, Prof. Mircea Dincă, and Prof. Tijana Rajh for assistance with preliminary experiments and the Instrumentation and Service Center for Molecular Sciences and the Instrumentation and Service Center for Physical Sciences at Westlake University for facility support and technical assistance. X. D. thanks Danyu Gu and Lingyu Xiao for their support with pulse EPR and ICP-AES experiments and Haozhou Sun, Yi Yang, Weibin Ni, and Zhecheng Sun at Westlake University for helpful discussions.

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Footnote

† Electronic supplementary information (ESI) available: Full descriptions of experimental methods, crystal structures, PXRD patterns, ICP-AES results, EDS mapping, and EPR characterization results and analysis. See DOI: https://doi.org/10.1039/d4qi02324b

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