Quantum sensing of magnetic fields with molecular spins

The study addresses whether molecular spin ensembles can serve as effective quantum sensors of oscillating magnetic fields using purely microwave-based detection, without optical readout. Quantum sensing leverages coherence and related quantum resources to measure weak signals, with notable prior success using nitrogen-vacancy (NV) centers in diamond for electric and magnetic field sensing and other quantities. While NV centers have demonstrated single-spin sensitivities down to microtesla per root hertz and ensemble sensitivities in the nT Hz−1/2 range (often requiring long, complex dynamical decoupling sequences), molecular spins offer distinct advantages: chemical tunability, the ability to attach selectively to analytes or surfaces, and sufficiently long coherence times even at relatively high spin concentrations. Despite theoretical proposals, experimental demonstrations of quantum sensing protocols with magnetic molecules have been lacking. The purpose of this work is to experimentally implement AC magnetometry with molecular spin ensembles (VO(TPP) and BDPA) embedded in a superconducting coplanar resonator, using Hahn echo and dynamical decoupling protocols synchronized to applied RF fields, and to quantify sensitivity and key protocol dependences (amplitude, phase, symmetry, and additivity).

- Quantum sensing can outperform classical approaches by exploiting quantum coherence and entanglement across applications (magnetic/electric fields, temperature, etc.). NV centers in diamond have achieved single-spin sensitivities around µT Hz−1/2 (scanning tips) and down to tens of nT Hz−1/2 with dynamical decoupling and ODMR; ensembles yield stronger fluorescence but at the expense of reduced coherence at higher spin densities, typically requiring long decoupling sequences (e.g., CPMG, XY, concatenated schemes) to reach nT Hz−1/2 sensitivities. Record sensitivities in the pT Hz−1/2 regime for large ensembles were achieved using auxiliary frequency tones without decoupling. - Molecular spins exhibit room-temperature coherence and Rabi oscillations even at ppm-level concentrations; entanglement and gate proposals have been demonstrated or proposed; ligands enable targeted attachment to analytes or surfaces, making them promising for local sensing of biological and quantum materials. However, prior to this work, experimental quantum sensing with molecular spins had not been reported, despite theoretical proposals and preliminary studies.

Samples and platform: Two molecular spin systems were used: (i) VO(TPP) (oxovanadium(IV) tetraphenylporphyrin) with S = 1/2 and hyperfine interaction with 51V (I = 7/2), prepared as a 2% doped crystalline sample in its diamagnetic analog TIO(TPP); (ii) BDPA organic radical (S = 1/2) diluted in polystyrene at spin concentration ≈ 1×10^15 spins cm−3. Samples were placed on a superconducting YBa2Cu3O7 coplanar resonator (ν0 ≈ 6.91 GHz) on sapphire. Experiments were performed at T = 3 K in a PPMS with an external static magnetic field applied along the resonator axis. Excitation and detection: Microwave (MW) pulses driving electron spin transitions were delivered via the resonator to generate spin echoes; an external RF copper coil (diameter ~5 mm, height ~5 mm, 7 turns) on the resonator surface applied an additional oscillating magnetic field transverse to the static field. A dual-channel AWG generated synchronized MW and RF pulses (independently programmable amplitude, phase, duration, and delays). Readout used heterodyne detection: MW output was downconverted and acquired in time domain with an oscilloscope. Typical pulse parameters: π/2 durations 80–150 ns, π durations 160–300 ns, interpulse delay τ from 0.9 to 1.7 µs; sequence repetition with relaxation T_relax = 15–20 ms. RF frequencies were chosen to synchronize with free precession: ν_RF = n/(2τ) (n integer, typically 0.5–1 MHz). The AWG RF output was calibrated to obtain B1,RF (max VRF ≈ 2.5 V). Protocols and sensing principle: AC magnetometry was implemented using Hahn echo and dynamical decoupling (DD) sequences. For Hahn echo, the RF field B_RF(t) = B1,RF sin(2πν_RF t + φ_RF) was applied during the free precession intervals (τ before and τ after the π pulse), synchronized to ν_RF = n/(2τ). In the small-RF limit, the RF field modulates the Zeeman energy and induces a phase accumulation of the spin precession proportional to the time integral of B_RF(t) over the free precession periods. Echo amplitude and phase at readout encode the amplitude and phase of the applied RF field. Dependences on B1,RF, φ_RF, symmetry (even/odd n), and additivity when RF is applied in first vs second τ were systematically explored. Dynamical decoupling: Two DD sequences were implemented on BDPA, leveraging longer coherence under decoupling: (1) Periodic Dynamical Decoupling (PDD): π/2 followed by a train of π pulses spaced by τ; RF period T = 1/(2τ) and φ_RF = 0°, so RF contributions add constructively after each refocusing. The echo analyzed appears a time τ after the last π pulse. (2) Carr–Purcell–Meiboom–Gill (CPMG/CP): π/2, initial delay τ_rand, then a train of π pulses with spacing 2τ; RF period again set to 1/(2τ) and φ_RF = 0° for synchronization; the echo a time τ after the final π pulse was analyzed. τ values of 1.0, 1.3, and 1.7 µs (ν_RF ≈ 1.0, 0.77, 0.59 MHz) and different numbers of π pulses were investigated. Echo amplitude and phase as functions of B1,RF and φ_RF were measured to extract transduction slopes and sensitivities. Numerical simulations based on the integral phase model were used to interpret symmetry effects.

- Hahn echo sensing (VO(TPP)) shows a clear, approximately linear increase of echo phase with RF amplitude B1,RF, and a decrease of echo amplitude as B1,RF increases. A representative slope of phase versus field was dφ_echo/dB_RF ≈ 9.8×10^−6 T^−1 at ν_RF = 1/(2τ) = 0.42 MHz. - The echo response depends strongly on the RF phase φ_RF: echo amplitude modulates with 180° periodicity in φ_RF, with maxima of effect at 0°, 180°, 360° and minimal effect at 90°, 270°. Echo phase oscillations become faster with increasing B1,RF. - Symmetry with respect to τ matters: choosing ν_RF = n/(2τ), odd n (antisymmetric RF modulation over τ) yields significant phase accumulation and oscillatory phase vs φ_RF, whereas even n (symmetric modulation) yields near-zero net phase accumulation (cancellation). - Additivity across intervals: Applying half periods in the first versus second τ shows opposite-sign contributions consistent with orientation in the xy precession plane; applying both and sweeping the phase demonstrates algebraic additivity of phase integrals from each τ interval. - Sensitivity (Hahn echo, VO(TPP)): With a phase resolution of about 1°, the minimum detectable field is B_min ~ 1×10^−5 T. The corresponding sensitivity is S ≈ 6×10^−6 T Hz^−1/2. Using spin density ρ = 2.3×10^19 spins cm^−3 and an active sensing volume V ≈ 1.75×10^−6 mm^3, the concentration-normalized sensitivity is S_vol ≈ 1.2×10^−9 T Hz^−1/2 µm^3/2. A theoretical lower bound was estimated as S_min ≈ 1.5×10^−6 T Hz^−1/2 and S_min,vol ≈ 3.1×10^−10 T Hz^−1/2 µm^3/2. - Dynamical decoupling (BDPA): Both PDD and CP sequences increase sensitivity by extending the effective interaction time. For matched ν_RF = 1/(2τ) and φ_RF = 0°, sensitivity improves as the number of π pulses increases (notably for 4–5 pulses). Achieved sensitivities are on the order of 10^−6 T Hz^−1/2 with only a few pulses, comparable to ensemble NV-center reports. PDD and CP show similar sensitivities for 2–3 pulses; PDD becomes less sensitive for larger pulse numbers under the present conditions due to dephasing induced by RF coil field inhomogeneity. - Practical considerations: The detection chain (resonator and MW electronics) significantly influences overall sensitivity. The transduction coefficient relating echo phase to B_RF should carry over to isolated spins using the same phase-accumulation protocol. Estimated dipolar fields of a single electron spin at ~5 nm (~7×10^−2 T) are comparable to applied fields here, suggesting feasibility of nanoscale local sensing with single molecular spins.

The experiments demonstrate that ensembles of molecular spins embedded in a microwave resonator can operate as quantum sensors for AC magnetic fields using purely microwave pulse protocols. The dependence of the echo phase on RF amplitude and phase, the role of synchronization and symmetry (odd versus even n), and the additivity across free-precession windows validate the phase-accumulation model and establish control levers for sensing. Sensitivity figures achieved with Hahn echo (few µT per root hertz) and improved under dynamical decoupling (≈10^−6 T Hz^−1/2) compare favorably with ensemble NV-center sensors, despite the absence of optical readout and the use of relatively few pulses. The approach benefits from higher spin concentrations typical of molecular samples and simplified detection. While coil-induced field inhomogeneity currently limits performance at higher pulse counts, further improvements in the resonator and RF/MW chain should boost sensitivity. The method’s robustness and the chemical addressability of molecular spins make them promising local probes for noise spectroscopy, magnetic properties of complex materials, and detection of ESR-silent species, potentially down to the single-molecule scale.

Quantum sensing of oscillating magnetic fields was realized with molecular spin ensembles (VO(TPP) and BDPA) in a superconducting coplanar resonator using Hahn echo and synchronized dynamical decoupling protocols. The echo phase provides a linear, calibrated transduction of RF magnetic field amplitude; protocol synchronization and symmetry govern phase accumulation, and contributions from distinct free-precession windows add algebraically. Experimentally, Hahn echo yields S ≈ 6×10^−6 T Hz^−1/2 (S_vol ≈ 1.2×10^−9 T Hz^−1/2 µm^3/2), while PDD/CP sequences enhance sensitivity to ≈10^−6 T Hz^−1/2 using only 4–5 pulses. The scheme requires no optical readout and is compatible with higher spin densities, simplifying sensing and enabling practical implementations. Future work should optimize the detection chain and field homogeneity to support longer decoupling sequences and further sensitivity gains, extend protocols to multifrequency fields and quadrature detection, and deploy chemically tailored molecular spins as local sensors for nanoscale noise spectroscopy and probing of functional and ESR-silent materials.

- Field inhomogeneity from the RF coil increases with longer decoupling sequences, causing dephasing that reduces echo amplitude and SNR, limiting the usable number of π pulses. - Overall performance is constrained by the microwave resonator and electronics in the detection chain; improvements there could enhance sensitivity. - Experiments were performed at cryogenic temperature (3 K) and within a limited RF frequency band (~0.5–1 MHz); AWG output limited the maximum RF amplitude (VRF ≤ 2.5 V). - Ensemble-based measurements may include inhomogeneities of spin environments; only a fraction (10^2–10^3) of spins participate coherently in the Hahn echo under the reported conditions.