Increased Crystal Field Drives Intermediate Coupling and Minimizes Decoherence in Tetravalent Praseodymium Qubits

Abstract

Crystal field (CF) control of rare-earth (RE) ions has been employed to minimize decoherence in qubits and to enhance the effective barrier of single-molecule magnets. The CF approach has been focused on the effects of symmetry on dynamic magnetic properties. Herein, the magnitude of the CF is increased via control of the RE oxidation state. The enhanced 4f metal–ligand covalency in Pr⁴⁺ gives rise to CF energy scales that compete with the spin–orbit coupling of Pr⁴⁺ and thereby shifts the paradigm from the ionic ζ_SOC ≫ V_CF limit, used to describe trivalent RE-ion, to an intermediate coupling (IC) regime. We examine Pr⁴⁺-doped perovskite oxide lattices (BaSnO₃ and BaZrO₃). These systems are defined by IC which quenches orbital angular momentum. Therefore, the single-ion spin–orbit coupled states in Pr⁴⁺ can be chemically tuned. We demonstrate a relatively large hyperfine interaction of A_iso = 1800 MHz for Pr⁴⁺, coherent manipulation of the spin with Q_M = 2Ω_RT_m, reaching up to ∼400 for 0.1Pr:BSO at T = 5 K, and significant improvement of the temperature at which T_m is limited by T₁ (T* = 60 K) compared to other RE ion qubits.

Introduction

In the field of quantum information science (QIS), the fundamental unit of a quantum computer is the quantum bit or qubit, which can be placed into an arbitrary superposition of two states.¹ Several candidates have been proposed to exhibit a two-state quantum system that can be coherently manipulated, including superconducting circuits,² trapped ions,³ topological states in condensed matter,⁴ and electron and nuclear spins in solids.⁵⁻⁷ Interfacing between different components of a computer by using hybrid quantum systems composed of an ensemble of electron spins has been proposed as a promising route for quantum memories operating in the microwave regime.⁸ Such memories are possible by exploiting the ability to coherently manipulate the electron spins, usually implemented using various magnetic impurities, as evidenced by nitrogen-vacancy centers in diamonds, phosphorous defects in silicon,⁷ and double-vacancy sites in silicon carbide.^5,6 Building on top of these approaches, an attractive design is to incorporate nuclear spins interacting with the electron spins via hyperfine interactions, which can offer an extra resource for storage by transfer of polarization between electron and nuclear spins and the ability to have low error rates.⁹ Furthermore, the incorporation of nuclear spins also offers the ability to scale the number of qubits by utilizing the multitude of transitions that results from the hyperfine interaction.¹⁰ Within this framework, rare-earth (RE) ions have been extensively studied because of their excellent coherence properties and wealth of naturally abundant nuclear spins.¹¹

Paramagnetic RE ions exhibit unquenched orbital angular momentum from the atomic-like 4f states possessing electron and nuclear spins and accessible optical transitions. These properties make them attractive materials to generate a versatile quantum interface by bringing together optical and microwave addressability. As a result, a hybrid quantum system can be achieved to develop efficient and faithful microwave-optical conversion, entanglement storage, and light–matter teleportation in the telecom wavelength. In the trivalent oxidation state (RE³⁺), the core-like 4f orbitals of the RE ions are only weakly perturbed by the crystal field (CF) and minimally split the otherwise 2J + 1 fold-degenerate free-ion ground-state (GS) ^2S+1L_J. This electronic structure results in rich physics and has been used to design new quantum materials with emergent phenomena.¹²⁻¹⁴ Therefore, in RE ions, the CF states can act as qubit states.¹¹ Among the RE ions, Pr³⁺,¹⁵ Nd³⁺,¹⁶ Er³⁺,¹¹ and Yb^3+17,18 have been of primary interest. Recently, synthetic chemistry has been proposed to offer tunability of quantum states by engineering the ligand field experienced by the electron spin in the form of molecular qubits.¹⁹⁻²¹ Besides tuning the ligand field, synthetic chemistry also offers the ability to engineer the electronic structure of the RE ion by providing control over the formal oxidation state of the metal center, as evidenced in recently explored reduced RE molecular complexes of La²⁺ and Lu²⁺. In these systems, a single unpaired electron resides in an orbital with a mixed 5d/6s character rather than the 4f orbital, giving rise to a clock transition and enhanced coherence.^22,23

While 3+ is the most stable oxidation state for RE ions, synthetic chemistry has pushed the boundaries of RE ions by accessing the unusually high 4+ oxidation state in Ce, Pr, and Tb.²⁴⁻²⁸ Recently, we showed that Pr⁴⁺ ions exhibit an unusually large CF energy scale, almost an order of magnitude greater than its 3+ counterparts, and established that the traditional ionic paradigm, used to describe Ln³⁺ ions, breaks down for Pr⁴⁺ due to hybridization of the Pr-4f electrons with the ligand valence electrons (analogous to transition metals as shown in Figure 1a).²⁹ Following this observation, it is enticing to use high-valent RE ions like Pr⁴⁺ as an alternative or compliment to Ln³⁺ systems to build novel quantum architectures with long-lived quantum memories.

Figure 1.

(a) Energy splitting diagram of a 4f¹ ion from ζ_SOC ≫ V_CF limit (left) and ζ_SOC ≪ V_CF limit (right) with the IC regime in the middle. (b) Composition of the GS wavefunction in Pr⁴⁺ as a function of B₄⁰. (c) Evolution of the eigen energies of the CF states and g_av in Pr⁴⁺ as the paradigm shifts from ζ_SOC ≫ V_CF limit to the ζ_SOC ≪ V_CF limit. The black line in (b) and (c) shows the position of BaPrO₃ based on our CF calculations.

A key ingredient in determining the coherence time of an electron spin is the spin–lattice relaxation time, T₁, that is strongly temperature-dependent.³⁰ The relaxation dynamics of T₁ arises from the interplay of direct, Raman, and Orbach processes, which enable the exchange of energy between the spin system and lattice bath when the electronic energy levels (CF states for RE ions) are modulated.³¹ Suppressing the efficacy of the Raman and Orbach processes driven by acoustic and optical phonons, respectively, is key to establishing long coherence times. The intuitive approach to minimize spin–phonon coupling is engineering of lattice vibrational modes and can be achieved by judicious choice of the host lattice/ligand architecture.³²⁻³⁴ However, the chemical properties that drive T₁ remain an open question and demonstrate that there is a rich chemical space still to be explored for QIS applications.

Alternative strategies for RE ions include CF control usually achieved by careful choice and control of symmetry and use of ions with an S-like electronic GS, where the vanishing orbital angular momentum (μ_orb/μ_spin ≈ 0) suppresses spin–phonon coupling. Within this framework, Pr⁴⁺ ions are advantageous, given that the first CF excited state is at ∼2000 cm^–1 compared to a few hundreds of cm^–1 observed for other Ln³⁺ ions.³⁵ Furthermore, the large CF energy scale competes with spin–orbit coupling (SOC), which mixes the excited-state SOC multiplet into the GS and thereby minimizes the orbital momentum (as observed from X-ray magnetic circular dichroism measurements: μ_orb/μ_spin ≈ 1.8 for Pr⁴⁺ compared to μ_orb/μ_spin ≈ 8 for Ce³⁺).³⁵ Pr⁴⁺ also has a very small g value (g_av < 0.8), one of the smallest among the RE ions, which offers the ability to suppress decoherence from spectral diffusion (SD) due to magnetic dipolar interactions. As a bonus, Pr has a very large nuclear spin, ¹⁴¹Pr (100% natural abundance with I = 5/2), which can further be used for coherent manipulation via hyperfine interactions.

In this work, we use the anomalously large CF splitting of Pr⁴⁺ to avoid electronic excitations overlapping with the vibrational density of states. This electronic manifold is achieved by using chemical design principles to stabilize Pr in its unusually high 4+ oxidation state by a judicious choice of oxide host lattices.²⁸ The evolution of single-ion GS wavefunction and properties of Pr⁴⁺ as the paradigm shifts from ζ_SOC ≫ V_CF to ζ_SOC ≪ V_CF is considered using a toy model to evaluate the unique ability to tune the spin–orbit coupled single-ion states in Pr⁴⁺-based systems. We demonstrate the single-ion electronic structure of Pr⁴⁺ in a six-coordinate perovskite lattice, BaPrO₃, by using a combination of thermo-magnetic measurements and CF theory. Coherence studies on Pr⁴⁺ doped in BaZrO₃ and BaSnO₃ host lattices using a combination of continuous-wave (CW) and pulsed X-band EPR measurements lends credibility to our design strategy and establishes Pr⁴⁺ as a potential candidate for QIS applications.

Results and Discussion

Crystalline powder samples of BaPrO₃, Pr:BaSnO₃ (2% doping: 2Pr:BSO; 0.1% doping: 0.1Pr:BSO), and Pr:BaZrO₃ (2% doping: 2Pr:BZO; 0.1% doping: 0.1Pr:BZO) were synthesized using traditional solid-state reactions (see Supporting Information), and phase purity was confirmed using powder X-ray diffraction (Figure S1). The parent compound, BaPrO₃, crystallizes in an orthorhombic Pnma space group^36,37 different from the host materials (BaMO₃; M = Sn, Zr) which crystallize in a cubic, ideal perovskite Pm3̅m structure, as shown in Figure 2.³⁸ The orthorhombic distortion in BaPrO₃ is due to cooperative buckling of the corner sharing octahedra with respect to each other, resulting in reduction of local symmetry at the B site from m3̅m (O_h) in the host materials to 1̅ in the parent compound. However, the PrO₆ octahedra in BaPrO₃ are very close to a perfect O_h with only small changes in the nearest-neighbor oxygen coordination. Therefore, BaPrO₃ is an ideal model compound to study the single-ion electronic structure of Pr⁴⁺ in the PrO₆ moiety and to understand the microscopic origins of coherent spin dynamics of ¹⁴¹Pr⁴⁺ ions in BaMO₃ (M = Zr, Sn) host lattices. It should be noted here that BaMO₃ host lattices were chosen to provide stabilization of the 4+ oxidation state and to obtain phase pure compounds. Experiments to design host lattices with minimal nuclear spins in the surrounding bath (in materials such as La₂M₂O₇ (M = Sn, Zr)) were not fruitful as phase segregation was observed.

Figure 2.

(a) Crystal structure representation of BaPrO₃. (b) Crystal structure representation of the host lattice BaSnO₃.

As reported by our group and others,^35,39 Pr⁴⁺ exhibits an unusually large CF splitting which competes with the SOC, yielding drastically different single-ion properties than expected in the ζ_SOC ≫ V_CF limit as shown in Figure 1a, and requires an intermediate coupling (IC) scheme to describe the GS properties.^35,39,40 Pr⁴⁺ is a Kramers ion with a 4f¹ electron configuration and couples the electron spin, S = 1/2, and orbital angular momentum, L = 3, to give rise to a J = 5/2 GS (²F_5/2) and a J = 7/2 excited state (²F_7/2) in the |j, m_j⟩ basis. The GS Kramers doublet (KD) is given by , where α² ∼ 1/6. In this framework, the CF Hamiltonian (Ĥ_CF) is diagonalized only within the ²F_5/2 SOC manifold as is the case for traditional trivalent Ce³⁺ systems.

Since j is not a good quantum number in the IC regime, the |m_l, m_s⟩ basis can be used to describe the electronic structure of Pr⁴⁺. In the |m_l, m_s⟩ basis, the O_h CF splits the seven 4f orbitals to GS a_2u and excited triply degenerate t_1u and t_2u states. In the presence of SOC, the seven 4f orbitals mix, yielding seven KD. In the |m_l, m_s⟩ basis, the nature of the GS KD is given as . The first two components of Γ₇^LS KD (m_l = −3, −2) are derived from , , states in the |j, m_j⟩ basis, while the last components (m_l = 1, 2) are derived from , states. Within this framework of IC, the Ĥ_CF is diagonalized using the entire set of 14 |m_l, m_s⟩ states.

To understand the implications of the IC scheme, the evolution of the single-ion properties of Pr⁴⁺ is studied in the model Hamiltonian:

1

where B_n^m are the fourth- and sixth-order terms and Ô_n are the corresponding Stevens operator equivalents,⁴¹ constrained by the O_h symmetry of an isolated PrO₆ unit. The eigen energies, GS wavefunction composition, and g_av as a function of B₄⁰ for fixed values of B₆ are calculated as shown in Figure 1b,c (see also Supporting Information). With B₆⁰ = −0.004, for small values of B₄, the system can be considered as a traditional trivalent lanthanide where the ζ_SOC ≫ V_CF limit still applies. The eigen states split as expected for the O_h CF (Figure 1c). As B₄⁰ increases, the paradigm shifts from ζ_SOC ≫ V_CF limit to ζ_SOC ≪ V_CF. For nonphysically large values of B₄, the eigen states relax to three sets of 4f orbitals a_2u, t_2u, and t_1u as expected in the ζ_SOC ≪ V_CF limit. Looking at the composition of the wavefunction (Figure 1b), it is evident that with an increase in CF energy scale, m_l = +1, −3 states begin to decrease from the original Γ₇ KD as the system relaxes to m_l = ±2 states corresponding to the a_2u GS. The shift in paradigm between the two limits significantly impacts the single-ion electronic structure, evident from the consistent change in g_av of the system. Within this framework, among the RE ions, Pr⁴⁺ offers the unique ability to tune the spin–orbit coupled wavefunction for a given symmetry by its ability to access the IC regime. It is important to note that the quenching of the orbital angular momentum is a product of the IC regime (Figure 1a–c). In either the CF or SOC limits, the orbital angular momentum is partially or completely recovered.

BaPrO₃ exhibits a magnetic transition at T_N ∼ 11 K observed in χ(T), as shown in Figure 3a.³⁷ Curie–Weiss analysis in the 10 < T < 40 K range yields θ_CW ∼ −35 K and μ_eff^CW = 0.75(2) μ_B, which is significantly lower than the expected value for a free f¹ ion (2.54 μ_B). All CF excitations for Pr⁴⁺ in BaPrO₃ (measured using optical spectroscopy) have been reported previously with E¹ = 0, E² ≈ 249.5, E³ ≈ 252, E⁴ ≈ 389, E⁵ ≈ 655, E⁶ ≈ 657, and E⁷ ≈ 818 meV.⁴⁷ As expected, Pr⁴⁺ exhibits an unusually large V_CF energy scale comparable to the ζ_SOC ≈ 112 meV,⁴⁸ and therefore, the single-ion properties must be modeled in the IC regime as described earlier. The single-ion CF Hamiltonian for Pr⁴⁺ can be written in a truncated symmetry basis, as shown in eq 1. The CF Hamiltonian, Ĥ_CF, is then fit to the observed eigen energies from optical measurements, their corresponding degeneracies, and magnetic susceptibility data at μ_oH = 0.1 T above 40 K (T > 40 K was chosen to avoid short-range correlations) as shown in Figure 3a with a fixed value of ζ_SOC ≈ 112 meV. This analysis yields a set of new KD’s with the GS wavefunction expressed as with g_av^CF = 0.68 comparable to the value extracted from the CW fits. The resulting model reproduces magnetization at T = 50 K, as shown in Figure 3b. This analysis clearly shows that the GS of Pr⁴⁺ deviates significantly from the expected V_CF ≪ ζ_SOC.

Figure 3.

(a) Magnetic susceptibility (χ(T)) and inverse susceptibility (χ(T)⁻¹) data of BaPrO₃ measured under μ₀H = 0.1 T plotted together with the CF model and a Curie–Weiss analysis in the temperature range 10 < T < 40 K. (b) Isothermal magnetization M(H) at T = 50 K for BaPrO₃ plotted together with the CF model. T = 50 K was chosen so that BaPrO₃ is well above the ordering temperature and free from short-range correlations.

Having established the single-ion electronic structure of Pr⁴⁺ in a perovskite ABO₃ lattice, the relaxation and coherent spin dynamics of electron and nuclear spins of ¹⁴¹Pr⁴⁺:BaMO₃ (M = Zr, Sn) were investigated using CW and pulsed EPR at the X-band (f = 9.4 GHz and B₀ < 1.8 T). Figure 4a shows the CW-EPR spectra for 2Pr:BSO and 2Pr:BZO measured at T = 5 K, revealing a six-line pattern due to the unpaired electron and its hyperfine coupling with the I = 5/2 ¹⁴¹Pr isotope. The EPR spectrum was analyzed using an effective spin Hamiltonian of eq 2 describing a lone S = 1/2 electron coupled to an I = 5/2 nuclear spin:

2

where the first two terms denote electron and nuclear Zeeman interactions, g̃ is the g-tensor, the third term represents the electron–nuclear hyperfine interaction, and à is the hyperfine coupling tensor. The X-band EPR simulations (using the MATLAB toolbox EasySpin)⁴⁹ also shown in Figure 4a yield a g_iso^EPR ≈ 0.57 comparable to g_iso and a large hyperfine interaction of A_iso ≈ 1771 MHz. At zero field, where the hyperfine interaction is the strongest, the nuclear spin and electron spin couple, yielding two states with total angular momentum F = I ± S = 5/2 ± 1/2 = 2 and 3 separated by , as shown in Figure 4c. Figure 4c also shows the six allowed EPR transitions (red lines) expected for ¹⁴¹Pr⁴⁺ and the corresponding forbidden transitions (gray lines) calculated with the parameters extracted for 2Pr:BSO. 2Pr:BZO yields very similar results with g_iso^EPR≈ 0.63 and A_iso ≈ 1789 MHz as shown in Figure 4a and are tabulated in Table S6.

Figure 4.

(a) CW X-band EPR of 2Pr:BSO measured at T = 5 K and the corresponding simulation. (b) echo-detected field-swept (EDFS) spectra of 2Pr:BSO and 2Pr:BZO measured at T = 5 K and the corresponding simulation. (c) Simulated energy diagrams of Pr⁴⁺ for the B₀∥Z orientations of the applied magnetic field calculated with parameters extracted from 2Pr:BSO. Given that only isotropic values were used for calculations, there was no orientation dependence for the energy diagrams.

Probing the spin dynamics of ¹⁴¹Pr⁴⁺ with pulsed EPR methods, the echo-detected field-swept (EDFS) spectrum of 2Pr:BSO was recorded by monitoring the integrated spin echo intensity as a function of applied dc field using the two-pulse echo sequence (π/2 – τ – π – τ – echo) with τ = 120 ns and is shown in Figure 4b. The spectrum reveals six broad transitions consistent with CW-EPR. Modeling the spectrum with the parameters extracted from CW-EPR yields good agreement with the experimental data (Figure 4b). 2Pr:BZO yields very similar results in good agreement with CW data as shown in Figure 4b and are tabulated in Table S6. Electron spin relaxation, characterized by the spin–lattice relaxation time constant, T₁, is commonly caused by spin–phonon coupling and, in most cases, limits the coherence times of the electron spin. Furthermore, T₁ also affects decoherence indirectly, where the spin flips of neighboring electron spins lead to SD of the central spin.¹⁸ Therefore, the T₁ of ¹⁴¹Pr⁴⁺ ions in the temperature range 5–60 K was studied using the inversion-recovery method (π – τ_r – π/2 – τ_e – π – τ_e – echo), where τ_r is swept, as shown in Figure 5a. These experiments, at an applied field of B₀ = 592.7 mT (2Pr:BSO) and 530.8 mT (2Pr:BZO), focus on the field of the largest intensity echo. The resulting saturation recovery traces were fit with a standard stretched mono-exponential function (see Supporting Information), and the extracted T₁ values are plotted in Figure 5c as a function of temperature. At low temperatures (<12 K), T₁^2 Pr:BSO shows weak temperature dependence, reaching a maximum of ∼13 ms at 5 K, while T₁ shows a strong temperature dependence, reaching a maximum of ∼0.7 ms at 5 K, comparable to other oxide host lattices.^17,18 In either case, the electron spin–lattice relaxation rate in the low-temperature regime is inversely proportional to temperature which can be attributed to a direct one-phonon process.¹⁸ At high temperatures (>12 K), T₁ begins to precipitously decrease, reaching a value of T₁^2 Pr:BSO = 4.65 μs and T₁ = 0.31 μs at T = 30 K. In order to understand the effects of dipolar magnetic interactions, further diluted samples with a Pr⁴⁺ concentration of ∼0.1% were analyzed. Dilution to ∼0.1% improves T₁, reaching a maximum of T₁^0.1 Pr:BSO ≈ 33 and T₁ ≈ 16 ms for the BSO and BZO host lattices at T = 5 K, respectively. 0.1% dilution improved T₁ only for T < 12 K, as shown in Figure 5c, consistent with direct processes from spin–spin-based dipolar interactions being the dominant decoherence mechanism. For T > 12 K, T₁ for 0.1% dilutions overlaps with 2% dilutions for both host lattices, indicating that decoherence is not limited by dipolar interactions with additional decoherence mechanisms coming in to play.

Figure 5.

(a) Inversion recovery traces and the corresponding fits used to extract T₁ for 0.1Pr:BSO measured at different temperatures. (b) Echo decays and the corresponding fits used to extract T_m for 0.1Pr:BSO measured at different temperatures. (c) T₁ and T_m as a function of temperature for different Pr⁴⁺ systems studied. (d) Frequency domain data of 3P ESEEM on 2Pr:BSO measured at T = 5 K and μ_0H = 597.2 mT and the corresponding simulation showing the Lamour frequency of ¹³⁵Ba at the field measured. (e) Specific heat data measured for 2Pr:BSO at T < 1 K and μ_0H = 7 and 14 T showing the Schottky anomaly corresponding to hyperfine interaction in Pr⁴⁺ and the corresponding fits. (f) Spin–lattice relaxation rate of 0.1Pr:BSO as a function of temperature and the fits corresponding Raman and local mode-based decoherence mechanisms.

In this higher temperature regime, two-phonon processes characterized via a combination of resonant (Orbach), nonresonant (Raman), and local modes dominate.⁵⁰ The Orbach process dominates when the temperature is sufficient to excite phonons that resonate with an excited state (in this case CF states). Given the first CF excited state is ∼2000 cm^–1, Orbach processes should have little effect on the relaxation. Therefore, the temperature dependence of T₁^0.1 Pr:BSO for T > 10 K was fit to a combination of Raman and local modes based on a general description of the two-phonon process which takes into account the maximum phonon energy k_bθ_D, where θ_D is the characteristic Debye temperature (Figure 5f and further details in Supporting Information). The data suggests a θ_D = 180 K consistent with the IR-phonon spectra of BaSnO₃ which identifies the lowest optical phonon mode at 135 cm^–1 (195 K).⁵¹

Having now established the bounds on spin coherence lifetimes from spin–lattice relaxation, the lifetimes of the coherent superposition state of the qubit, parameterized by phase memory time T_m, were measured through two-pulse Hahn echo measurements. The echo intensity was measured as a function of 2τ as shown in Figure 5b and clearly shows an exponentially decaying signal. T_m was extracted by fitting the data to a standard mono-exponential function, yielding T_m^2 Pr:BSO = 3.1 and T_m = 2.3 μs at T = 5 K. T_m^2 Pr:BSO follows T₁ with weak temperature dependence, while T_m^2 Pr:BZO shows weaker temperature dependence compared to T₁ in the temperature range T < 12 K. Similar to T₁, T_m improved on dilution to 0.1%, reaching a maximum of T_m^0.1 Pr:BSO = 18 and T_m = 2.5 μs at T = 5 K for both host lattices and begins to decrease at higher temperatures, reaching T_m^0.1 Pr:BSO = 0.26 and T_m = 0.1 μs at T = 60 K with minimal effects of dilution for T ≳ 12. T_m is bound by T₁ above T* = 40 and 60 K for 0.1Pr:BZO and 0.1Pr:BSO, respectively. Note that at low temperatures, the echo decay curves show strong electron spin-echo envelope modulation (ESEEM) from Ba nuclear spins (vide infra). The maximum T_m value extracted for ¹⁴¹Pr⁴⁺ is one of the largest among the RE ions (Table 1). Furthermore, coherent spin dynamics are detectable up to T* = 60 K in 0.1Pr:BSO, which is greater than all RE ions except Gd³⁺. This phenomenon is attributed to the high-energy first electronic state in both Pr⁴⁺ and Gd³⁺. However, in Gd³⁺ the excited state is a SOC multiplet, whereas in Pr⁴⁺, the excited state is purely CF derived (Table 1). We note here that relaxation measurements at other hyperfine transitions also yield very similar results (see Table S5 and Figures S4 S5).

Table 1. Comparison of Parameters Extracted for Pr⁴⁺ from This Work with Other 4f-Based RE Ions^a.

RE ion	host lattice	excited state (cm–1)	AJ (MHz)	gav	T (K)	T1 (ms)	Tm (μs)	T* (K)	ref
Pr4+	BaSnO3	∼2000	∼1800	∼0.58	5	∼33	∼18	∼60	this work
Ce3+	CaWO4	53–135	N/A	∼1.4	5	∼24.7	∼14.2	∼18	(42)
Nd3+	Y2SiO5	∼77	∼530	∼2.6	5	∼30	∼106	∼7	(16)
Gd3+	CaWO4	∼33,000d	∼15	∼2	6	∼8	∼5.4	∼70	(43)
Gd3+	trensalb	∼33,000d		∼2	5	∼0.03	∼12	>30	(44)
Gd3+	sTPATCN	∼33,000d		∼2	7	∼0.36	∼18.3	>50	(45)
Er3+	CaWO4	∼19	∼125	∼5	3.5		∼7 (50)e		(11)
Yb3+c	Y2SiO5	∼115	∼500	∼4	5	∼7	∼10 (73)e	∼9	(18)
Yb3+c	trensalb	∼500	∼600	∼3.8	5	∼0.4	∼0.5	20	(46)

^aT* corresponds to the temperature at which T_m is limited by T₁ given by T_m/T₁ = 1.

^bMolecular complex.

^cValue reported for ¹⁷¹Yb.

^dFor Gd³⁺, the excited state corresponds to the SOC multiplet and not a CF excited state. Given the ⁸S_7/2 GS, the CF splitting is usually very small. This minimizes mixing from the excited ⁶P_7/2 and ⁶D_7/2 SOC multiplets and has been proposed to suppress decoherence due to the vanishing orbital angular momentum.

^eThe values in the parenthesis correspond to the maximum value reported but measured at a lower temperature of 2–3 K. For a more direct comparison, the table is constructed with values reported at the base temperature of this study at T = 5 K.

While the two host lattices BZO and BSO are isostructural, they exhibit very different T₁ relaxation rates—particularly in the T < 12 K regime, indicating that an additional decoherence mechanism besides the SD from neighboring spin-flips is active. In order to understand the decoherence mechanism, a three-pulse stimulated echo technique (π/2 – τ – π/2 – T_W – π/2 – τ – echo) and T = 5 K is used. Fitting the echo decay to a combination of SD linewidth (Γ_SD) and relaxation time, a T₁^SD = 0.7 and 1.8 ms are obtained for 2Pr:BSO and 2Pr:BZO, respectively. The extracted T₁ values are half of those obtained from the inversion recovery measurements. This difference indicates that besides the spin-flip process, spin-flip flops from neighboring nuclear spins are active as well. The fast Fourier transform (FFT) of the time domain data of 2Pr:BSO measured with τ = 120 ns clearly shows a peak ≈2.8 MHz, corresponding to the Larmour frequency of ¹³⁵Ba (I = 3/2) in the field measured (Figure 5d). The data is well simulated by coupling between an S = 1/2 electron and the ¹³⁵Ba nuclei, yielding a hyperfine coupling of A_iso^135_Ba = 0.8 MHz, as shown in Figure 5d.

Additionally, Hyperfine Sublevel Correlations (HYSCORE) spectroscopy further resolves the coupling to the surrounding Ba nuclei. Both host lattices show two sets of two sharp peaks, which can be simulated by coupling to both ¹³⁵Ba and ¹³⁷Ba (I = 3/2) nuclei, yielding A_∥^Ba = 0.8 MHz and A_⊥ = 1.8 MHz for both nuclei and quadrupolar contributions of Q^Ba = 3.5 for both nuclei as shown in Figure 6a,b. The key difference between the host lattices is the presence of nuclear spin bearing ^117,119Sn nuclei (I = 1/2) in BSO and ⁹¹Zr nuclei (I = 5/2) in BZO in the surrounding bath of ¹⁴¹Pr⁴⁺. It is possible that the large nuclear spin of ⁹¹Zr explains the faster T₁ relaxation rate for BZO compared to that for BSO, leading to a faster decoherence.

Figure 6.

(a) HYSCORE data (top) and simulation (bottom) for 0.1Pr:BSO measured at μ_0H = 901 mT (b) HYSCORE data (top) and simulation (bottom) for 2Pr:BZO measured at μ_0H = 530.8 mT. All data was measured at T = 5 K.

One of the key properties of ¹⁴¹Pr⁴⁺ is the unusually large hyperfine interaction of ∼2000 MHz. Among the RE elements, holmium metal exhibits the largest hyperfine interaction with A_J^Ho = 6500 MHz, followed by praseodymium metal with A_J = 4500 MHz.⁵² Such large hyperfine coupling interactions have been attributed to the polarization of conduction electrons by s–d mixing minimizing SOC.⁵² A similar argument has been invoked for a very large hyperfine interaction of A_iso^Lu2+ ≈ 3500 MHz observed in Lu²⁺ molecular complexes where the spin bearing d orbitals undergoes symmetry-allowed mixing with s orbitals minimizing SOC.²² However, in the solid-state, RE ions, when doped in wide band-gap host lattices, exhibit a significantly reduced hyperfine interaction, as evidenced for Ho³⁺:LiYF₄ with A_iso ≈ 800 MHz⁵³ or in molecules like Ho³⁺ polyoxometalates with A_iso ≈ 700 MHz.^54,55 This large reduction of the hyperfine interaction necessitates the need to understand the origin of the very large hyperfine coupling in ¹⁴¹Pr⁴⁺ and a comparison of the hyperfine interaction of 4f^1 141Pr⁴⁺ with 4f^2 141Pr³⁺.

However, since Pr³⁺ is a non-Kramers ion, it is often EPR silent, at least in the X-band. Therefore, specific-heat measurements were employed. The heat-capacity C_N arising from a discrete set of 2I + 1 hyperfine energy levels W_m (m = I, ..., –I) occurs as a Schottky anomaly with a peak or maximum at a temperature T ≃ ⟨ΔW⟩_av/k, where ⟨ΔW⟩_av is the mean spacing of the energy levels and k is the Boltzmann constant. The Hamiltonian for the hyperfine Schottky contribution can be written as⁵⁶

3

where A is the hyperfine interaction constant, I_z is the expectation values of I, μ_eff is the saturated magnetic moment, g_J is the landau g-factor, and P is the quadrupolar contribution. P for ¹⁴¹Pr is usually three to four orders of magnitude smaller than A and therefore can be neglected. Simply, eq 3 can be written as a function of A and μ_eff. By fitting the observed Schottky anomaly in the specific heat data arising from thermal depopulation of the hyperfine spin levels, experimental values of A can be extracted.

Heat-capacity measurements on Pr³⁺:LnCl₃ yield A^{Pr3+ : LnCl₃} ≃ 1089 MHz, significantly less than the Pr metal as expected.⁵⁷ Following a similar approach, the heat capacity of 2Pr:BSO at μ₀H = 7 and 14 T was measured (Figure 5e). Below 1 K, an upturn in specific heat is observed, which is attributed to nuclear Schottky contribution. By fitting the data to eq 3, an A^Pr4+ ≃ 1800 MHz and a moment of ∼ 0.6 μ_B are extracted, which are consistent with X-band EPR (see Supporting Information for full fitting details). The A^Pr4+ value obtained is almost twofold greater than the value reported for Pr³⁺ : LnCl₃ or other oxide host lattices like Pr³⁺:Y₂O₃¹⁵(A^{Pr3+ : Y₂O₃} ≃ 800 MHz, measured using spectra hall burning). On increasing the oxidation state from Pr³⁺ to Pr⁴⁺, two antagonistic effects compete to drive the observed hyperfine interaction with the increased nuclear charge, leading to a larger value which is diminished by the enhanced 4f metal–ligand covalency. In this system, the increase in nuclear charge appears to dominate and lead to the significant enhancement in hyperfine interaction.⁵⁸

Given the relatively long T_m, coherent spin manipulations can be performed, as demonstrated by the observation of Rabi oscillations for 0.1Pr:BSO in transient nutation experiments (Figure 7a). The damping oscillations were fit with the “on-resonance” transient nutation following

4

where τ_R is the damping time and Ω_R is the Rabi frequency. The corresponding fits are shown in Figure 7a, yielding τ_R ≈ 0.2 μs at 0 dB microwave power (which is significantly less than the phase memory time T_m^0.1 Pr:BSO due to homogeneous and inhomogeneous broadening mechanisms). The FFT of the time domain data yields Ω_R, consistent with the values obtained by fitting to eq 4 (Figure 7b). The linear relationship between Ω_R and relative amplitude B₁ as shown in Figure 7c establishes with certainty the provenance of the observed nutations as Rabi oscillations opposed to coherence transfer from the central spin to the dense bath of nuclear spins in the surroundings. The number of Rabi oscillations given by N_c = τ_R(c)Ω_R with N_c ≈ 15 compares well with the value reported for ¹⁶⁷Er³⁺ doped in CaWO₄. The qubit figure of merit given by Q_M = 2Ω_RT_m reaches up to ∼400 for 0.1Pr:BSO at T = 5 K, the same order of magnitude as other RE qubits except for Er³⁺ which is in the order of 10⁴.

Figure 7.

(a) Rabi oscillations for 0.1Pr:BSO measured at T = 5 K at different microwave attenuation. The black lines correspond to fits to the data as described in the text. (b) Frequency domain data for the nutation experiment. (c) B₁ dependence of the Rabi frequency, Ω_R. The solid line is a linear fit, emphasizing the relationship B₁, where P is the microwave power.

Conclusions

In conclusion, we show that Pr⁴⁺ offers the ability to chemically tune the spin–orbit coupled single-ion states as the paradigm shifts from ζ_SOC ≫ V_CF limit to ζ_SOC ≪ V_CF limit. CW X-band EPR measurements of Pr:BSO and Pr:BZO and CF analysis of the parent material establish the unique single-ion electronic structure of Pr⁴⁺ with a very small g_av ≈ 0.6 and a large hyperfine interaction of A_iso ≈ 1800 MHz. Building on these results, pulsed X-band measurements show the possibility for coherent manipulation of the Pr⁴⁺ ion with coherence times, reaching a maximum of T₁ = 33 ms and T_m = 18 μs, with spin dynamics detectable up to T* = 60 K. Therefore, in this tetravalent RE qubit, we have demonstrated long phase memory times exceeding most trivalent RE qubit systems via an alternative approach by employing the large CF energy scale of Pr⁴⁺ with a vanishing orbital angular momentum via control of the metal oxidation state. Additionally, our work establishes the IC regime as a potential avenue for designing new RE, actinide, heavy (4d and 5d) transition-metal, and main-group^59,60-based quantum materials, both in solid-state and molecular systems.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.3c02820.

• Complete experimental details and further analysis of the data (PDF)

• CW data for BSO (TXT)

• CW data for BZO (TXT)

• EDFS data for BSO (TXT)

• EDFS data for BZO (TXT)

• MvH data for BaPrO₃ (TXT)

• MvH calculated for BaPrO₃ (TXT)

• MvT fit for BaPrO₃ (TXT)

• MvT ZFC data for BaPrO₃ (TXT)

• MvT FC data for BaPrO₃ (TXT)

• Nutation 0 db (TXT)

• Nutation 3 db (TXT)

• Nutation 6 db (TXT)

• Nutation FFT 0 db (TXT)

• Nutation FFT 3 db (TXT)

• Nutation FFT 6 db (TXT)

The authors declare no competing financial interest.