Millihertz Frequency Resolution Achieved Via Geodesic Control and Diamond Sensing

Scientists have long sought methods for precise frequency measurement across a wide spectrum, a challenge often hampered by interference from unwanted signal harmonics. Si-Qi Chen from Shandong University, alongside Qi-Tao Duan and Teng Li, et al., now present a breakthrough in quantum frequency sensing, demonstrating a highly accurate and broadband protocol using geodesic control. Their research, utilising the electron spin of a nitrogen-vacancy centre in diamond, engineers a response focused on a single frequency, effectively eliminating systematic errors caused by harmonics from megahertz to gigahertz frequencies. This innovative approach, bolstered by synchronised readout, achieves millihertz-level resolution even with noisy signals, establishing geodesic control as a viable and practical technique for high-accuracy sensing in real-world applications.

This theoretical framework points toward a fundamentally different route to robust frequency estimation, but its experimental feasibility, resilience to realistic noise, and compatibility with broadband and high-resolution sensing techniques have remained open questions. We quantitatively characterize this frequency selectivity by reconstructing the modulation function and evaluating the fidelity of the control pulses. The Fourier components of the modulation functions under finite-width control pulses are shown in Fig0. The0.87MHz. The Hamiltonian of GD∥is thus given by HGD∥= ΩGD(t) 2 (−cos φGD(t)σz + sin φGD(t)σx).
The Rabi amplitude of πφj pulse is set at ΩGD(t) = π/tπ, and the phase parameter is fixed at φGD(t) = φj = 2πTj/Tscan. In the interaction picture defined by HGD∥, the total sensing Hamiltonian takes the form eH∥= FGD∥(t)B∥(t)σz 2. Governed by the effective Hamiltonian eH∥in Eq. (5), the initial state |ψ∥ 0⟩undergoes a rotation about the z axis by an angel ΦGD∥(t), which depends on the scan frequency ωscan as ΦGD∥(t) = Z t 0 FGD∥(t′)B∥(t′) dt′ ≈ Z t 0 ( bs 2 cos [(ωs −ωscan)t′] + X l bn,l 2 cos [(ωn,l −ωscan)t′] ) dt′. Notably, the accumulated phase ΦGD∥(t) exhibits strong frequency selectivity, i. e., it builds up constructively when the scan frequency matches the target frequency ωscan = ωs, whereas it is strongly suppressed for ωscan ≠ ωs, as illustrated in Fig0.5 (c).

This frequency resonance can be read out. (3)) is applied, with each πφj corresponding to a π rotation about the axis (cos φj, −sin φj, 0) in x-y plane (denoted as GD⊥). Analogous to the parallel case, ΦGD⊥(t) exhibits strong frequency selectivity as well, i. e., it accumulates constructively when ωscan = ∆s, while remaining strongly suppressed for off-resonant detunings. This accumulated phase is read out by projecting the final state, yielding probability of PL = 1 2[1+cos(ΦGD⊥(t))]. More details of geodesic control are provided in the Supplementary Materials.

Geodesic Control Enables Broadband, High-Resolution Sensing for advanced

Experiments revealed that geodesic control shapes the modulation spectrum of the quantum sensor into a tunable single-frequency response, effectively suppressing spurious harmonic resonances. Quantitative characterisation of this frequency selectivity was achieved by reconstructing the modulation spectrum and monitoring sensor-state evolution under controlled multi-frequency noise, directly comparing performance against conventional dynamical decoupling (DD) schemes. Results demonstrate accurate and bias-free frequency estimation in the megahertz regime, even when higher-order harmonic noise is present, a significant advancement for practical applications. The breakthrough delivers extended sensing bandwidth into the gigahertz regime through the combination of geodesic control and heterodyne detection, without compromising accuracy.

Measurements confirm that the NV centre’s electronic ground state, a spin triplet 3A with sublevels |ms = 0⟩ and |ms = ±1⟩, is crucial to the process, where the Hamiltonian is defined as HNV = DS2z + γeB0Sz, with D = 2π × 2.87GHz and γe = 2π × 28GHz/T. A static field of approximately 500 Gauss, applied along the NV axis, introduces a Zeeman splitting of 2γeB0, facilitating the restriction of dynamics to the {|ms = 0⟩, |ms = −1⟩} subspace. Tests prove that geodesic control markedly improves both accuracy and robustness compared to conventional XY and CPMG protocols. This work provides a foundation for advanced metrology and sensing applications, potentially impacting fields like radar, wireless communication, and nanoscale nuclear magnetic resonance.

Geodesic control unlocks robust broadband sensing capabilities

This technique enables bias-free frequency estimation with substantial suppression of harmonic-induced systematic errors across a broad spectral range, extending from megahertz to gigahertz frequencies. The findings establish geodesic control as a practical approach for high-accuracy frequency sensing in realistic environments, offering robustness and broadband operability. This advancement significantly broadens the scope of potential applications, including nanoscale spectroscopy, microwave-field characterisation, and frequency-selective sensing in complex settings. The authors acknowledge limitations related to the influence of noise, which can sometimes generate spurious peaks that complicate accurate frequency estimation. Future research may focus on scaling this approach through multi-sensor and array-based implementations, leveraging the compatibility of geodesic control with existing NV-based platforms.