Identification of an X-Band Clock Transition in Cp′₃Pr^– Enabled by a 4f²5d¹ Configuration

Abstract

Molecular qubits offer an attractive basis for quantum information processing, but challenges remain with regard to sustained coherence. Qubits based on clock transitions offer a method to improve the coherence times. We propose a general strategy for identifying molecules with high-frequency clock transitions in systems where a d electron is coupled to a crystal-field singlet state of an f configuration, resulting in an M_J = ±1/2 ground state with strong hyperfine coupling. Using this approach, a 9.834 GHz clock transition was identified in a molecular Pr complex, [K(crypt)][Cp′₃Pr^II], leading to 3-fold enhancements in T₂ relative to other transitions in the spectrum. This result indicates the promise of the design principles outlined here for the further development of f-element systems for quantum information applications.

The ability to manipulate coherence in a quantum object enables its use as a quantum bit (“qubit”). This is key to the development of new concepts in quantum information science (QIS), including sensing, communication, and computing.¹⁻⁹ High-performance systems based on trapped ions¹⁰⁻¹⁵ and superconducting qubits¹⁶⁻¹⁹ have long coherence times but offer limited scalability and tunability. Recent advances have shown that molecular qubit designs may provide the necessary means for control and functionality, but they are also inherently more coupled to the environment, which tends to diminish performance. Variations of the ligand can provide opportunities to address this shortcoming; however, the principles needed to rationally design new molecular qubits with targeted properties have not been fully developed. Coherence times in molecular qubits can be improved by magnetic dilution^20,21 and elimination of atoms with nuclear moments within the molecule.^22,23 Limitations to both of these approaches make it desirable to explore alternatives.

Coherence times in electron-spin molecular qubits are improved at avoided energy-level crossings where the dependence of the transition frequency (ν) on the magnetic field (B) vanishes (i.e., dν/dB = 0). As such, the transverse relaxation time (T₂) is less sensitive to magnetic noise, engendering resistance to decoherence from nearby magnetic sites²⁴ and other nuclear spins in the molecule.²⁵ Avoided crossings can be generated by nondiagonal perturbations of the spin Hamiltonian such as crystal-field-splitting²⁴ and hyperfine²⁵⁻²⁷ interactions. For lanthanide systems, hyperfine clock qubits have only been realized with La^II (4f⁰5d¹) and Lu^II (4f¹⁴5d¹)²⁵ centers. These systems have closed f shells, which minimize anisotropy, but this is possible only at the beginning and end of the lanthanide series. There is a significant question of whether coherence enhancements can be realized at hyperfine-derived clock transitions in open-f-shell systems.

Here we provide a framework for identifying clock qubit candidates based on open-shell 4fⁿ5d¹ configurations of f elements with even n. Our approach was to identify trivalent 4fⁿ complexes with crystal fields (CFs) that enforce singlet (M_J = 0) ground states (GSs). We hypothesized that reduction to the divalent 4fⁿ5d¹ complex would afford an M_J = ±1/2 GS doublet in the M^II complex. Open-shell-singlet GSs are well-known in actinide chemistry, with precedent in 5f², 5f⁴, and 5f⁸ systems.²⁸ Equatorial ligand fields provide an additional advantage by promoting occupation of a 5d–6s hybrid orbital,²⁹⁻³⁸ where d–s hybridization in this singly-occupied molecular orbital increases Fermi contact and induces massive hyperfine couplings,³⁹ bringing clock transitions to higher frequency. Both of these criteria were met by Cp′₃Pr^III (1) (Cp′ = trimethylsilylcyclopentadienyl), which has a singlet GS, as evidenced in published magnetometry data²⁹ by the limiting value near 0 for χT at low temperatures. These data also indicate a 4f²5d¹ configuration for divalent [K(crypt)][Cp′₃Pr^II] (2) (crypt = [2.2.2]cryptand). Combined with the large hyperfine coupling^40,41 and 100% natural abundance of ¹⁴¹Pr, the Cp′₃Pr system is ideal for testing the hypothesis described above.

1

While the published magnetometry data for 1 were consistent with a singlet GS, in the C_3h pseudosymmetry of the molecule there are in fact three possible singlet states with M_J = 0 and (|+3⟩ ± |−3⟩)/√2, where the latter pair is generated by the B⁶₆ component of the CF. Fortuitously, CF modeling of optical data has been reported for 1,⁴² indicating that the ground state is M_J = 0. We have further confirmed this by fitting⁴³ the reported magnetometry data to the model Hamiltonian in eq 2:

2

with Zeeman, spin–orbit, and CF terms (see section S2.1 in the Supporting Information for a full discussion). The slight differences in CF parameters relative to those previously reported (Table 1) are most likely due to the more simplistic model used for the present treatment. These parameters afforded the state diagram shown in Figure 1, indicating that the GS of this molecule is |J, M_J⟩ = |4, 0⟩, which is separated from |4, ±1⟩ by 190 cm^–1.

Table 1. CF Parameters for 1 and 2 Determined by Fitting Magnetometry Data Reported in Reference (29); The Optical CF Parameters in the First Column Are Taken from Reference (42)^a.

	1 (opt.)	1 (mag.)	2b
B02	–2485	–3129(19)	–810(60)
B04	1323	1980(20)	1230(40)
B06	555	484(11)	484c
B66	–1956	–1760(60)	–1760c
k	–	0.98c	0.98c

^aAll CF values are in units of cm^–1. λ was fixed at 380 cm^–1, which is 90% of the optimized free ion value.⁴³

^bFit includes an empirical TIP component of 7.4 × 10^–4 emu mol^–1. The exchange coupling, j, was set to 10,000 cm^–1.

^cFixed during the fit.

Figure 1.

(top) Best fits of eqs 2 and 3 to the published magnetometry data²⁹ for 1 and 2, respectively. (bottom) Empirical state-energy diagrams for the ground multiplets of 1 (³H₄, left) and 2 (⁴H_7/2, right).

We next set out to model the magnetometry for 2 to assess the effect of the d electron on the state ordering. The reduced high-temperature magnetic moment of 2 relative to 1 is consistent with an LS coupling scheme,^29,44 and eq 2 may be used to model the magnetometry data for 2 with S = 3/2 and L = 5, affording an M_J = ±1/2 GS of ⁴H_7/2 parentage (see section S2.2 for fit details). However, the CF treatment in this model is unrealistic, as V̂_CF operates on all three electrons.

Instead, we elected to model the data by coupling the 4f angular momentum to a second S = 1/2 spin system representing the 5d–6s hybrid electron. This Hamiltonian is shown in eq 3:

3

where V̂_4f indicates the CF term acting only on the 4f electrons and j is a large, positive, isotropic exchange coupling describing the interaction between the 4f and 5d electron spins. This model has the advantage of allowing a direct comparison of 4f CF parameters between 1 and 2. We assumed that the 4f CF was only perturbed by the additional 5d_z² electron, which should impact only the B⁰₂ and B⁰₄ parameters of the 4f CF Hamiltonian (section S2.3). These parameters and a residual temperature-independent paramagnetism (TIP) were the only parameters allowed to vary from those of 1 (section S2.2). The results of this fit to the magnetometry data for 2 are shown in Figure 1, and the best-fit parameters are shown in Table 1. Crucially, the GS from both models can be assigned as |7/2, ±1/2⟩, consistent with a conserved 4f state ordering relative to 1. The reduction of the B⁰₂ and B⁰₄ CF parameters and smaller separation between the ground and first excited state (120 cm^–1 vs 190 cm^–1) indicate that the CF potential felt by the 4f electron is weakened upon addition of the 5d electron. This result is predictable, as the 5d electron itself responds to the ligand field and thus occupies orbitals that generate an antipodal potential with respect to that of the ligands.

Following confirmation of the M_J = ±1/2 GS in 2, we set out to characterize its hyperfine and relaxation behavior using X-band EPR spectroscopy on a magnetically dilute powder of ca. 1% 2 in [K(crypt)][Cp′₃Yb^II] (2_Yb), which is diamagnetic due to a 4f¹⁴ configuration.^30,31 While no spectral features attributable to 2 were observed using continuous-wave (cw) detection, it was possible to collect an echo-detected field-swept (EDFS) spectrum of 2 (Figure 2a). The spectrum was integrated across a range of delay times to suppress distortion due to electron spin echo envelope modulation (ESEEM) from ¹H hyperfine coupling.⁴⁵ This X-band spectrum is highly complex and cannot be immediately attributed to a spin system with pseudoaxial symmetry and hyperfine coupling to a ¹⁴¹Pr nucleus. However, this spectrum could be consistent with 2 in the weak-field limit,³⁹ a possibility that we probed using higher-frequency EPR spectroscopy.

Figure 2.

X-band EPR spectroscopy of 1% 2 in 2_Yb at 5 K. (a) EDFS X-band EPR spectrum of 2 (integrated over ca. 1 μs delay times) together with a simulated spectrum based on the parameters in Table 2. (b) Variable-delay EDFS spectra of 2 showing the more rapid decay of spectral components above about 100 mT. (c) Variable-frequency EDFS spectra integrated over 8 μs delays showing a sharp transition at zero field and 9.834 GHz; spectra were mirrored around zero field for clarity. (d) Empirical state energy diagram for the ⊥ orientation of 2 showing allowed transitions at 9.834 GHz in blue. The clock transition is highlighted in red. (e) Primary echo decay curves at various frequencies and fields from 0 to 100 mT together with results of fitting exponential decays to determine T₂. A decay at 378 mT is shown for comparison.

The 70 GHz spectrum of an undiluted powder of 2 is shown in Figure 3 together with a simulated spectrum. In contrast to the X-band results, this spectrum is pseudoaxial with a clear hyperfine coupling, enabling the modeling of the g⃡ and A⃡ tensors. Analysis of both X-band and the high-frequency EPR spectra of 2 is based on the effective spin Hamiltonian (SH)⁴⁶ in eq 4 using a pseudospin (S = 1/2) system coupled to an I = 5/2 nuclear spin:

4

with electron and nuclear Zeeman contributions and hyperfine terms (see section S5.1).

Figure 3.

Simulation (red) and experimental 70 GHz cw EPR spectrum (blue) of an undiluted powder of 2 at 5 K. The small sharp peak at ca. 2.5 T is a minor g = 2 impurity.

The SH parameters from the 70 GHz spectrum (Table 2) enabled modeling of the X-band spectrum using eq 4. In this case, the spectrum could be modeled using an axial SH, with the g_⊥ and A_⊥ values near the average of g_x,y and A_x,y extracted from the 70 GHz data. The higher apparent symmetry at X-band may be due to the presence of near-neighbor magnetic sites in the concentrated sample, which may be sufficient to desymmetrize the SH. These g values are somewhat different from those predicted for a ⁴H_7/2M_J = ±1/2 GS (g_⊥ = 2.67, g_∥ = 0.67) but are in good agreement with those predicted by PHI⁴³ using the model based on eq 3 (g_⊥ = 2.43, g_∥ = 0.81), supporting our assignment of the GS.

Table 2. Parameters from Simulating Multifrequency EPR to a Rhombic (70 GHz) or Axial (9.834 GHz) Pseudospin Hamiltonian and g Values Predicted by the Two Spin Hamiltonians for an M_J = ±1/2 Doublet.

ν or		⊥		∥
70a	g	2.60	2.21	0.94
	A	4.70a	3.70a	0.87a
9.834a	g	2.43		0.94
	A	4.33a		0.87a
(4H7/2)	g	2.67		0.67
	g	2.42		0.81

^aIn GHz.

The state energy versus magnetic field diagram from this model (Figure 2d) has an avoided energy level crossing generated by a nondiagonal term in the hyperfine coupling with a transition frequency of 9.834 GHz in the perpendicular orientation at low magnetic fields. Variable-frequency EPR spectroscopy shows a dramatic enhancement in signal intensity at 0 mT and 9.834 GHz (Figure 2c), and the primary echo decay shows no evidence of ESEEM. Both of these observations are consistent with a clock transition,⁴⁷ although the vanishing ¹H Larmor frequency at zero field may also contribute to the lack of ESEEM. Intriguingly, all of the features in the range of 0–100 mT have decay constants in the range of 2–3 μs, while the higher-field transitions, which arise from Zeeman splitting, have T₂ times closer to 1 μs (Figure 2b,e). The longest relaxation times do not occur precisely at the zero-field clock transition but rather were observed slightly away at 9.850 GHz and 36 mT (T₂ = 2.94 μs).

The origin of the T₂ enhancement over this wide field range can be explained by the nearly field-independent nature of the state energy of the two |F, M_F⟩ states involved in the relevant transitions (F = J + I is the combined electronic and nuclear angular momentum). In this range, a number of the parent |M_J, M_I⟩ states sequentially approach one another in energy, and mixing between them to form the final |F, M_F⟩ states at low field leads to the extended weak dependence of the state energy on the field. These transitions invariably relax slower than those at higher fields (by a factor of 2–3), consistent with a clocklike enhancement of the relaxation due to the weak dependence of the frequency on the field. We additionally note that the T₁ (longitudinal) relaxation times determined for 2 were also ca. 3 μs at 5 K (Figure S4), which may limit the degree to which T₂ can be enhanced at the clock transition.

The combined magnetometry and EPR results presented above support the fundamental hypothesis that coupling of an open-shell-singlet 4f configuration to an electron in an s–d hybrid orbital results in an M_J = ±1/2 doublet, with large hyperfine couplings that give rise to high-frequency clock transitions with longer T₂ relaxations. This was accomplished by enforcing a low-anisotropy 4f configuration, enabling performance that is normally possible only with the closed-4f-shell configurations at either end of the lanthanide series. Future work will focus on using these design principles to target new molecular hyperfine qubits based on lanthanide and actinide centers with both f and d electrons.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.3c12725.

• Experimental details, discussion of model development for eq 3, additional EPR spectroscopic data, and complete EPR simulation parameters (PDF)

The authors declare no competing financial interest.