Introduction

Quantum sensing, i.e., the exploitation of genuine quantum properties such as quantum coherence or entanglement to probe tiny physical quantities, can overcome fundamental limitations or outperform classical sensing1. Its advantages have been proved by measuring a large variety of physical quantities, such as electric or magnetic fields, temperature, pressure, rotation angle or speed, time (frequency) and acceleration1. This has brought to applications in photonics2, nuclear3,4 and electron spin resonance5,6,7,8,9 among others. Spins are natural candidates to perform quantum sensing of magnetic fields. Recently, Nitrogen-Vacancy (NV) centers in diamonds, both as ensembles (bulk collections) as well as single (isolated) spins have shown remarkable performances as quantum sensors6. Sensitivities on the order of μT Hz−1/2 have been reported for single NV centers scanning probe microscopy tips6,10,11, while typical values from few units and up to tens of nT Hz−1/2 are reported for single spins using Optically Detected Magnetic Resonance (ODMR) protocols based on Dynamical Decoupling microwave (MW) sequences12,13,14. Using ensembles has the advantage to give the parallel average of the response of N (identical) copies of single spins15 and larger fluorescence signals to be measured in ODMR13,15. However, increasing the spin concentration reduces the memory time giving an upper bound for sensitivity, which can be overcome only with very specific and ad-hoc microwave decoupling schemes15. Typical sensitivity values of the order of units or tens of nT Hz−1/2 have been achieved on bulk sensors by means of ODMR combined with long (tens to hundred/s of pulses) Dynamical Decoupling sequences such as Carr-Purcell-Meiboom-Gill or XY87,13,16,17, concatenated Dynamical Decoupling18, or advanced protocols compensating sample-specific inhomogeneities and disorder15,19. Record sensitivity of pT Hz−1/2 with a sensor of 1013 spins and a volume of 4 ⋅ 10−2 mm3, has been achieved using a different protocol based on an auxiliary frequency tone, without Dynamical Decoupling20. Typical concentration sensitivity (i.e. sensitivity normalized over the square root of the spin density13,19) values reported for NV centers are on the order of few units up to ten nT Hz−1/2 μm3/2 15,19.

Molecular spins have shown sufficiently long coherence times to observe Rabi oscillations even at room temperature and with relatively high spin concentrations (typically from 0.01 up to few %, corresponding to ≈1 up to 100 ppm)21,22,23,24,25,26,27. Quantum correlation -entanglement- between molecular spins has been observed in static configurations28,29,30 while schemes to entangle spins by exploiting excited molecular levels have been proposed31,32. Due to these intrinsic quantum properties, molecular spins are attracting much interest for the implementation of quantum gates and algorithms33,34,35,36,37,38,39. The possibility to tailor the external organic ligands21,34,40, and to selectively link/attach spin centers to a desired target, being this either an analyte in a supramolecular structure or a functional surface, is also well documented41,42,43. As a matter of facts, this turns out to be one of the key characteristic of molecular spins44. Molecular spins provide a valid alternative to study biological systems as well as functional surfaces such as topological 2D materials or superconductors41,42. Despite this peculiarity may actually open promising avenues for quantum technologies45, except for some interest and curiosity44,46 and theoretical proposals47,48, there are no experimental reports concerning the realization of quantum sensing schemes with magnetic molecules yet.

In this work we consider two different classes of molecular spins, i) single transition metal ions, namely an oxovanadium (IV) tetraphenyl porphyrin (VO(TPP) for short), and ii) organic radicals, namely the α, γ-bisdiphenylene-β-phenylally (BDPA for short), as two prototypical systems for testing the sensing of AC magnetic fields under Dynamical Decoupling-based magnetometry. The applied Radiofrequency (RF) fields (here with frequency values between ≈ 0.5 and 1 MHz and variable amplitude and phase) are synchronized to MW pulse sequences such as Hahn echo and Dynamical Decoupling49. The sensing principle is based on the phase accumulation induced by the RF field during the free spin precession time of the protocols1 (see Methods, Section “Sensing Principle”). The variation of the phase of the echo signal measured at the end of the protocol can be related to the amplitude and the phase of the RF field applied, demonstrating the potential of molecular spin ensembles for quantum sensing. Moreover, the possibility to perform sensing of magnetic fields and the unique peculiarities of molecular spins, pave the way for the implementation of noise spectroscopy50,51,52 or for using spins as local probes, once integrated into hosts, such as metallic or superconducting surfaces.

Results

We place the magnetic sample on a superconducting coplanar resonator (ν0 ≈ 6.91 GHz) as in Fig. 1.a, and we carry-out our experiments at the temperature of 3K (see Methods, Section “Samples and resonator” for details). The static magnetic field is applied along the longitudinal axis of the resonator and MW pulses are used to drive the molecular spin ensemble through the resonator to obtain a spin echo53,54,55. A RF copper coil is placed on the surface of the resonator and it is used to apply an additional RF modulation on the sample.

Fig. 1: Quantum Sensing of magnetic fields with Molecular Spins.
figure 1

a Sketch of the coplanar resonator used in the experiments with the position of the sample (purple) and the radiofrequency (RF) coil (orange) used. Red, blue and green arrows show the orientations of the static magnetic field (B0), the microwave (MW) field (B1,MW), and the RF field (B1,RF), respectively. b Molecular structure of VO(TPP) and sketch of BDPA used in this work. VO(TPP) molecule is adapted from53 under CC Licence 4.0, while BDPA molecule is reproduced from55 with permission. c Timing diagram of the protocol used for quantum sensing. The MW sequence is the Hahn echo and the RF modulation is synchronized with its free precession time, τ. d Sketch of the precession of the spins in the rotating reference frame during the application of protocol shown in c, with (bottom) or without (top) the application of the RF field. The different precession plane when the RF modulation is added is shown in yellow.

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Quantum sensing with Hahn echo sequence

We first test the Hahn echo-based protocol shown in Fig. 1c on the VO(TPP) sample in several different experiments designed to show the dependences of the echo phase on the amplitude, on the phase or on the integral of the applied RF modulation (see Methods, Section “Sensing Principle”). We first fix n = 1 and ϕRF = 0° in the RF modulation (see Eq. (3)). The RF amplitude B1,RF is increased step-by-step and the Hahn echo signal is acquired for each field amplitude value. The Hahn echo amplitude and phase extracted as a function of the amplitude of the RF magnetic field are shown in Fig. 2a–c. The echo amplitude decreases as the RF amplitude is increased, while the echo phase linearly increases with the RF amplitude. This latter result is quite similar to what observed on Nitrogen-Vacancy centers by using similar protocols13,49,56, and it nicely shows the effect of the phase accumulation induced by the RF field on the spin precession, according to Eq. (3). Here the echo amplitude never vanishes and the phase exhibits a linear dependence on the amplitude of the RF field. Similar results are observed also if the interpulse delay is changed between 900 ns and 1700 ns (See Supplementary Fig. 7).

Fig. 2: Hahn Echo-based Quantum Sensing of amplitude and phase of AC magnetic fields.
figure 2

a–c Dependence of Hahn echo on the amplitude of the RF magnetic field for the VO(TPP) sample. a 2D map of the echo signal measured in time domain for different amplitudes of RF magnetic field. Echo amplitude (b) and phase (c) extracted from the map in a as a function of the RF amplitude. The phase of the output echo, which is wrapped between −90° and 90° for better clarity, linearly increases with the RF amplitude. dashed line is a linear fit. d–f Dependence of Hahn echo on the phase of the RF magnetic field for the VO(TPP) sample. d 2D map of the echo signal measured in time domain for different phase values of RF magnetic field. Echo amplitude (e) and phase (f) extracted from the map in d as a function of the RF phase.

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Secondly, we investigate the effect of the phase of the RF field, ϕRF. In the Bloch sphere (rotating reference frame), this operation corresponds to set a different orientation of the RF magnetic field in the xy plane with respect to the direction of the microwave pulses (which are along the +x axis, since ϕπ/2 = ϕπ = 0° 57). This introduces an initial phase delay between the Hahn echo sequence and the RF modulation itself, changing the position at which the maximum of the magnetic field occurs during the free precession time. Results for a full 0–360° phase sweep for different B1,RF values are shown in Fig. 2d–f. A modulation of the echo amplitude with period of 180° is visible as a function of the RF phase (Fig. 2d, e). The effect of the applied modulation is maximum for ϕRF = 0°, 180° and 360°, while is completely negligible for ϕRF = 90° and 270°, where the echo amplitude is equal to the B1,RF = 0T case. The echo results to be sensitive to the orientation of the RF field in the precession plane. This effect is more evident in the echo phase, which undergoes a larger variation with respect to the B1,RF = 0T case as B1,RF is increased. The slope of the phase signal as a function of ϕRF increases with B1,RF showing that faster phase oscillations between −90° and +90° are achieved for larger RF field amplitudes. Similar oscillation are visible also if the interpulse delay is changed between 800 ns and 1900 ns (see Supplementary Fig. 7). We explain these results according to Eq. (3). From a mathematical point of view, changing the RF phase is equivalent to change of the extremes of the integration interval by an amount t = − ϕRF/(2πνRF). This implies that the integral goes from a maximum value for ϕRF = 0° down to a minimum for ϕRF = 90° (RF modulation antisymmetric over τ, integral equal to zero, no effective phase accumulation) and, then, up to another maximum for ϕRF = 180°. These results suggest that the echo phase signal is sensitive also to the integral of the RF modulation applied during the free precession time.

The next set of experiments further assess the dependence of the echo signal on the integral of the phase accumulation, according to Eq. (3). We first change the symmetry of the RF modulation during τ, i.e. the factor n of νRF (see Eq. (3)) for fixed B1,RF = 1.8 mT. In this way, the modulation signal becomes symmetric (even n) or antisymmetric (odd n) over the duration τ, as in Fig. 3a. Results as a function of ϕRF are shown in Fig. 3b, c for VO(TPP) sample (here we have τ = 1200 ns, corresponding to νRF = n 0.83 MHz). The echo amplitude shows a modulation for n = 1, 3 (0.83 and 2.5 MHz, respectively), similar to the one in Fig. 2e, while a negligible modulation is found for n = 2, 4 (1.7 and 3.3 MHz, respectively). The amplitude modulation for n = 3 is smaller with respect to the one for n = 1, suggesting that the RF signal has lower effect. The result is more evident in the echo phase signal, where periodical oscillations are clearly visible only for odd n and the signal is unperturbed for even n. In addition, the echo phase for n = 1 shows faster oscillations as a function of ϕRF with respect to n = 3, corroborating the different effect on the same total precession time. These results are further corroborated by the simulation performed according to Eq. (3) (Fig. 3d). In particular the phase accumulation predicted is zero for odd n only for ϕRF = 90°, 270°, while is always zero for all even n. Also the relative scaling between n = 1 and n = 3 is predicted, highlighting the different effectiveness of the two modulations.

Fig. 3: Hahn’s Echo-based quantum sensing of symmetry and additivity of AC magnetic fields.
figure 3

a–d Investigation of the effect of the symmetry of the RF field for VO(TPP) sample. a Sketch of the protocol used, echo amplitude (b) and phase (c) as a function of the RF phase ϕRF. d Simulation of the phase accumulation carried out by means of Eq. (3). e, f Investigation of the sum of RF modulations for VO(TPP) sample. e Sketch of the protocol used. f Echo phase measured as a function of the RF phase, ϕRF.

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We further show how the phase accumulation builds up during the free precession time 2τ by applying half period of RF modulation only during the first τ interval, only during the second one or, alternatively, the same half period of modulation in both τ intervals, as sketched in Fig. 3e. Sweeping the RF phase only during the first or the second τ interval (red trace, first, and green trace, second, respectively) at fixed BRF = 1.8 mT gives phase oscillation as in Fig. 2 but with slower changes (≈doubled period) with respect to the full RF modulation applied (gray trace, RF sin in the legend). The nearly opposite sign further confirms the different orientation of the modulation in the xy precession plane. Finally, if the RF modulation is applied on both τ intervals (BRF = 1.8 mT) and the phase of the first one is swept (for fixed ϕRF = 0° on the second one) the behavior of the echo phase results from the difference of the two modulation integrals over τ. This can be realized by plotting the difference between the full RF modulation (gray trace) and the first one (red trace), which gives the contribution “removed” from the total RF modulation due to the effect of the first one (black dashed trace, to be compared with light-blue one). These results corroborate the algebraic additivity of the modulations over the two different τ, as expected from Eq. (3). Results similar to the ones reported in Figs. 2 and 3 hold also for the BDPA sample (see Supplementary Section 2.4 and Supplementary Figs. 8 and 9), suggesting that the protocols tested can be successfully extended to other molecular spins.

Quantum sensing with Dynamical Decoupling protocols

Dynamical Decoupling protocols58 can be exploited for quantum sensing as well1,49,59. These, essentially, increase the spin lifetime and also give longer time for the RF field to act on spins during free precession, enhancing the sensitivity. To this end, we extend the protocol of Fig. 1 to Dynamical Decoupling sequences and we apply it on the BDPA organic radical, since its memory time is found to largely increase through a Carr-Purcell-Meiboom-Gill sequence performed through the coplanar resonator (see Supplementary Fig. 6). Here, we notice that applying a Dynamical Decoupling protocol implicitly acts as bandpass frequency filter to the environmental field (noise) acting on spins1,49,59. This implies that, in order to be effective, the applied RF modulation must match the band of the filter (i.e. being synchronized with the protocol and not suppressed by the response of the filter itself). In other words, the sensitivity strongly depends upon the specific choice of parameters used in the Dynamical Decoupling (frequency ν = 1/τ, phase of the RF modulation and number of π pulses)13,16.

We first investigate a Periodic Dynamical Decoupling (PDD)49. Here, a first π/2 pulse is followed by a train of π pulses equally-spaced in time by a delay τ. The RF field has period T = 1/(2τ) and ϕRF = 0°. In this way the RF modulation builds always up after the spin refocusing given by each π pulse. The echo considered and analyzed in this protocol is the one appearing at a time τ after the last π pulse sent. Results are shown in Fig. 4b. The echo amplitude normalized over each corresponding zero-modulation value decreases as the number of π pulses increases due to the longer time available to the modulation to act on spins.

Fig. 4: Enhancing quantum sensing of magnetic fields using Dynamical Decoupling.
figure 4

a, c Sketch of the Dynamical Decoupling protocols used in this work: the Periodic Dynamical Decoupling (PDD, a) and the Carr-Purcell-Meiboom-Gill sequence (CP, c). Normalized echo amplitude detected as a function of the RF field amplitude for different number of π pulses used in the protocols: the case of Periodic Dynamical Decoupling (b) and of Carr-Purcell-Meiboom-Gill sequences (d) are shown, respectively.

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Next, we consider a Carr-Purcell-Meiboom-Gill protocol (hereafter, CP)1,49. Here a first π/2 pulse is followed by an initial interpulse delay τ and, then, by a train of π pulses equally-spaced by 2τ. Again, we considered the echo appearing at a time τ after the last π pulse sent, and the RF modulation has a period T = 1/(2τ) and ϕRF = 0° in order to synchronize with every spin refocusing. We have investigated this dependency for different interpulse delays (τ = 1, 1.3, 1.7 μs, corresponding to νRF = 1, 0.77, 0.59 MHz, respectively) and number of π pulses, and we report in Fig. 4d the results obtained for τ = 1.7 μs. Additional results are reported in Supplementary Section 2.5 and Supplementary Fig. 10. The echo amplitude normalized over each corresponding zero modulation value decreases as the number of π pulses is increased. However, for larger number of π pulses applied, the echo signal gets smaller, and the addition of the RF modulation results in a strong decrease of its signal and of the signal-to-noise ratio, preventing from an accurate measure of both echo amplitude and phase under our experimental conditions. We attribute this effect to the increasing inhomogeneity of the magnetic field introduced by the RF coil, which induces strong dephasing during spin precession.

Discussion

Our results demonstrate that the echo signal given by the spins can be used to probe an oscillating monochromatic magnetic field applied perpendicularly to the static one.

We first discuss the magnetic field sensitivity achieved under our experimental conditions. To this end, we consider the linear dependence of the echo phase shown in Fig. 2c. This data allow us to estimate a transduction coefficient, \(\frac{d{\phi }_{{{{\rm{echo}}}}}}{d{B}_{{{{\rm{RF}}}}}}\), between the echo phase and the RF magnetic field amplitude (dashed line) through

$$\frac{d{\phi }_{{{{\rm{echo}}}}}}{d{V}_{{{{\rm{RF}}}}}}=\frac{d{\phi }_{{{{\rm{echo}}}}}}{d{B}_{{{{\rm{RF}}}}}}\frac{d{B}_{{{{\rm{RF}}}}}}{d{V}_{{{{\rm{RF}}}}}},$$
(1)

where the relation between the amplitudes of RF magnetic field and RF voltage given by the Arbitrary Waveform Generator (AWG), \(\frac{d{B}_{{{{\rm{RF}}}}}}{d{V}_{{{{\rm{RF}}}}}}\), is known from independent calibrations (see Supplementary Section 1.2). Eq. (1) refers to the “slope detection" case defined in1. From the data of Fig. 2c (dashed line) we can get \(\frac{d{B}_{{{{\rm{RF}}}}}}{d{\phi }_{{{{\rm{echo}}}}}}={(\frac{d{\phi }_{{{{\rm{echo}}}}}}{d{B}_{{{{\rm{RF}}}}}})}^{-1}=9.8\cdot 1{0}^{-6}\,{{{\rm{T}}}}\,\cdot\,{}^{\circ }{}^{-1}\) for VO(TPP) when the Hahn echo sequence is used to probe an RF magnetic field with frequency νRF = 1/(2τ) = 0.42 MHz. Under our experimental conditions and with our experimental set up, the phase resolution (i.e. the minimum phase variation of the echo signal which can be detected) is ≈1° 57, which implies a minimum detectable field amplitude of \({B}_{\min }=9.8\cdot 1{0}^{-6}\,{{{\rm{T}}}}\,\cdot\,{}^{\circ}{}^{-1}\,\cdot\,{1}^{\circ }=9.8\cdot 1{0}^{-6}\,\)T. This value is consistent also with the estimation obtained from the analysys of Allan deviation (see Supplementary Section 3.3 and Supplementary Fig. 13). To get a direct comparison of the sensitivity achieved for similar experiments performed on spin ensembles, we evaluate the sensitivity as the minimum detectable field for unitary Signal-to-Noise ratio and unitary acquisition bandwidth60. This reads as S ≈ 6 ⋅ 10−6 T Hz−1/2 (see Supplementary Section 3.2), which well compares with some reports in the literature56,61. In our experiments the effective number of spins is of the order of ≈1014 spins, while the number of spins which are taking part to the Hahn echo signal is ≈1012 − 1013 (see Supplementary Section 3.1). This gives the size (in absolute number of spins) of the sensor which is coherently experiencing the RF modulation applied. Taking the values of the spin concentration (ρ = 2.3 ⋅ 1019 spin cm−3 22,62) we can estimate an active sensing volume Vs = 1.75 ⋅ 10−3 mm3 for VO(TPP) and obtain a concentration sensitivity \({S}_{{{{\rm{vol}}}}}=S/\sqrt{\rho }\,\approx\, 1.2\cdot 1{0}^{-9}\,{{{\rm{T}}}}\,{{{{\rm{Hz}}}}}^{-1/2}\,\mu {m}^{3/2}\). This value is consistent with the theoretical limit of sensitivity expected from the derivation in1 (see Supplementary Section 3.4), for which the lower bound to sensitivity results to be \({S}_{\min }\approx 1.5\cdot 1{0}^{-6}\,{{{\rm{T}}}}\,{{{{\rm{Hz}}}}}^{-1/2}\), corresponding to an average concentration sensitivity Smin,vol ≈ 3.1 ⋅ 10−10 T Hz−1/2 μm3/2.

Likewise, we estimate the sensitivity for the Dynamical Decoupling experiments by using the same definition as above. To take into account the different behavior of BDPA as a function of B1,RF, we use the maximum of the first derivative of the phase as a function of B1,RF in the estimation of the minimum magnetic field. This ensures that the definition is equivalent to the one used for VO(TPP) if the phase depends linearly by the RF magnetic field, while the maximum signal variation is considered if the data are in the non-linear regime (see Supplementary Section 2.4). Our results are summarized in Fig. 5. Overall, for a given τ, increasing the number of π pulses (vertical alignment of points) increases the sensitivity to the RF magnetic field in all cases. PDD protocol results to have similar sensitivity with respect to CP sequences for 2 and 3 π pulses, while becomes less sensitive for larger number of pulses. This matches well with the different total precession time available for sensing the RF field. Our estimated sensitivity values (10−9 T Hz−1/2) compare well with the values reported in the literature for ensembles of NV centers in diamond7,13,15,16,18,19 and, for instance, they outperform what recently reported for defects in hexagonal Boron Nitride (hBN) layers63 or spin-vacancy nanodiamonds64,65. It is worth mentioning that, in spite of the fact that optical detection can be pushed to the single event (i.e. single photon), the detection of photons typically has low yield. This in part compensates the advantage given by the longer memory time of NV center in diamond as compared to that of molecular spins. However, it is also remarkable that the sensitivity here reported for molecular spins can be reached with a relatively low number (4-5) of applied π pulses, without optical detection47, and by using samples with higher spin concentration with respect to the typical values used in literature for defects in solids. This suggests that there are still margins of improvement for the sensitivity obtained with molecular spins.

Fig. 5: Sensitivity of quantum sensing protocols.
figure 5

Best sensitivity (a) estimated from the sensing experiments with Dynamical Decoupling protocols on BDPA sample and its corresponding Sensitivity per unit of concentration (b). For PDD the colors indicates the different number of π pulses used from 1 (red) up to 5 (blue). Point for Hahn echo (red, corresponding to CP1 and PDD1) is added for comparison.

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To get more insights on our results, we first note that the performances obtained in this type of experiments depend not only on the sensor itself, but also by the whole detection chain, including the microwave resonator and microwave electronics used to read out the echo signal, which can be potentially improved in future experiments. On the other hand, the RF field B1,RF is applied over the whole volume in such a way that each spin is locally experiencing the same RF field. This implies that the transduction coefficient estimated with Eq. (1) is the same for all spins, and it can be expected to hold also for isolated spins using the phase accumulation method and the same experimental protocol shown in this work. To compare B1,RF in our experiments with local fields, let us consider the magnetic field generated by dipole-like spin center with a magnetic moment equal to Bohr’s magneton (e.g. a electronic spin S = 1/2). The field magnitude at distance of 5 nm from the axis is BS = 7 ⋅ 10−6 T which is comparable to what we used in our experiments. This suggests that a local magnetic field or magnetic moment would be detectable in an experiment in which a single sensing molecule is used at nanoscale distance47. This rough estimation suggests us a strategy for realistic sensing experiments at the molecular scale, in which single molecular sensors are attached and interact with magnetic analytes or to functional magnetic surfaces through a selective organic thread. For instance, we expect that electromagnetic noise locally generated by molecular process can be detectable by an electron spin such as a radical or a VO(TPP) spin center. Experiments can be performed on diluted ensembles of units containing the analyte bringing spin label and repeating the quantum sensing protocol (e.g. DD) at different frequencies in such a way that noise spectrum can by reconstructed by Fourier analysis. Such noise spectroscopy has been developed in the context of nuclear magnetic resonance50,51,66 and also proposed for NV centers52. Finally, another potential benefit of the introduction of a local molecular spin sensors relies on the possibility to probe the magnetic properties of Electron Spin Resonance-silent ions or of moieties with large zero field splitting, which cannot be directly or easily studied by Spin Resonance techniques67.

In conclusion, we have realized quantum sensing protocols using ensembles of molecular spins as active element embedded into a planar microwave resonator. To assess the performances of our sensor, we implemented the detection of RF magnetic fields within spin echo (continuous Dynamical Decoupling) magnetometry schemes. The sensitivity achieved with Hahn echo sequence is S ≈ 10−8 − 10−6 T Hz−1/2, corresponding to concentration sensitivity Svol ≈ 10−10 T Hz−1/2 μm3/2. Extending our experiments with Dynamical Decoupling protocols, such as the Period Dynamical Decoupling and Carr-Purcell-Meigboom-Gill, we can improve the sensitivity up to S ≈ 10−10 − 10−9 T Hz−1/2 (Svol ≈ 10−11 T Hz−1/2 μm3/2) with a relatively low (4–5) number of π pulses. This sensitivity compares well with the standard ones reported for NV centers and our concentration sensitivity results to be even slightly better due to the different spin density of our sensors. Our scheme is based on the resonant readout of a spin echo and does not require optical (fluorescence) readout, which usually has the drawback of having low photon conversion efficiency. Moreover, similar sensitivity values are achieved with rather short sequences (few pulses, to be compared with tens or hundreds) making the sensing protocol easier to be implemented. Our results can find application in sensing the amplitude or the phase of periodic signals (here, RF magnetic fields) or in noise spectroscopy to detect local processes using molecular (electronic) spin labels. Our results can be extended to oscillating fields containing multiple Fourier components thus to B1,RF(t) with arbitrary time dependence1,49. We finally notice that the phase additivity over the free spin precession time might be useful in designing protocols for the Revival of Silent Echo (ROSE) of spin ensembles68,69 or for the realization quadrature detection70.

Methods

Sensing principle

Sensing of DC or AC fields requires different protocols. DC magnetometry with spins is based on the slightly change of the resonant condition given by the addition of the field to be probed, which (vector) sums to the static one. Typical experiments are performed by measuring the change in the oscillations of the Ramsey fringes or of the Rabi oscillations induced by the presence of the additional DC field1. Conversely, AC magnetometry1, which is considered in our work, can be applied to probe oscillating magnetic fields in a relatively large frequency band, and it can be further extended also to the multifrequency case. AC magnetometry is also more suitable for probing slow-varying fields or the local environment surrounding spins.

The full prototypical Hamiltonian of a quantum sensing experiment is

$$H={H}_{0}+{H}_{{{{\rm{control}}}}}+{H}_{{{{\rm{V}}}}},$$
(2)

where H0 is the Hamiltonian describing the system acting as a sensor, Hcontrol is the protocol used on the sensor to drive it and measure its response, and HV is the Hamiltonian describing the physical quantity to be probed. In our experimental case, H0 is the Hamiltonian describing the VO(TPP)53 or the BDPA55 spin ensemble, while Hcontrol is the microwave pulse sequence (Hahn echo or Dynamical Decoupling) used to induce the spin free precession and to get a spin echo. HV describes the interaction induced by the RF oscillating magnetic field B1,RF acting during the free spin precession, which is the signal to be measured. In our scheme (see Fig. 1), the microwave magnetic field B1,MW lies on a plane perpendicular to the direction of the static one B0, allowing for the perpendicular excitation of the spin transitions71. The direction of the RF field is perpendicular to the static one, as for typical magnetic field sensing protocols12,13,14,15. On resonance and in absence of RF excitation, the effect of the microwave sequence is to induce the free spin precession, which can be visualized as a spin dephasing in the xy plane of the rotating reference frame, as in Fig. 1d71. The RF modulation acts only when the microwave field is off and (vector) sums to the static magnetic field modulating the resonance condition and the direction around which spin precession occurs. In the rotating reference frame this modulates the orientation of the precession plane with respect to the readout one, which remains the same as in absence of RF modulation, leading to a change in both amplitude and phase of the echo, as in Fig. 1d. If the RF frequency is synchronized with the MW sequence, these effects build up during the free spin precession, leading to an extra phase accumulation in the preceeding spins and to a macroscopic change (reduction) of the echo amplitude, which can be used to estimate the RF field13,14.

To better understand the phase accumulation given by the RF modulation we focus on the simplest protocol used in this work, i.e. the Hahn echo sequence, which is implemented by giving a first π/2 pulse followed by an interpulse delay τ and, then, by a π pulse. An electron spin echo is found after a delay τ from the π pulse, as in Fig. 1c. The RF excitation is a monochromatic field in the form B1,RF(t) = B1,RF sin(2πνRFt + ϕRF) with amplitude B1,RF, frequency νRF and additional phase term ϕRF. The RF modulation is synchronized with the period of the free precession time by choosing νRF = n/(2τ) (with n as an integer, non negative, number). Under the conditions in which the RF field is small with respect to the static one, the RF field modulates the Zeeman energy of spins and can be taken as HV = g μBBRF(t), which gives a corresponding angular frequency \({\omega }_{{{{\rm{V}}}}}=\frac{{H}_{{{{\rm{V}}}}}}{\hslash }=\frac{g\,{\mu }_{{{{\rm{B}}}}}{B}_{{{{\rm{RF}}}}}(t)}{\hslash }\). The total phase accumulation after a single Hahn echo sequence is13,14:

$$\begin{array}{l}{\phi }_{{{{\rm{RF}}}}}=\int\nolimits_{0}^{2\tau }{\omega }_{V}dt=\int\nolimits_{0}^{2\tau }\frac{g{\mu }_{{{{\rm{B}}}}}}{\hslash }{B}_{{{{\rm{RF}}}}}(t)dt\\\qquad\quad=\,\int\nolimits_{0}^{2\tau }\frac{g{\mu }_{{{{\rm{B}}}}}}{\hslash }{B}_{1,{{{\rm{RF}}}}}sin(2\pi {\nu }_{{{{\rm{RF}}}}}t+{\phi }_{{{{\rm{RF}}}}})dt\\ \qquad\quad=\,2\int\nolimits_{0}^{\tau }\frac{g{\mu }_{{{{\rm{B}}}}}}{\hslash }{B}_{1,{{{\rm{RF}}}}}sin(2\pi {\nu }_{{{{\rm{RF}}}}}t+{\phi }_{{{{\rm{RF}}}}})dt\\\qquad\quad=\,\frac{2g{\mu }_{{{{\rm{B}}}}}}{\hslash }{B}_{1,{{{\rm{RF}}}}}\int\nolimits_{0}^{\tau }sin(2\pi {\nu }_{{{{\rm{RF}}}}}t+{\phi }_{{{{\rm{RF}}}}})dt\propto {B}_{1,{{{\rm{RF}}}}}\end{array}$$
(3)

The phase accumulated by the echo signal is proportional to the amplitude of the applied magnetic field B1,RF and it also depends by the integral of the RF modulation over the free precession time of spins, i.e. on the synchronization (phase and duration) of B1,RF with respect to the sequence.

Samples and resonator

We use two different molecular spins systems (Fig. 1b). The first one is a 2% doped crystalline sample of VO(TPP) in its isostructural diamagnetic analog, TiO(TPP). Each molecule has an electronic spin S = 1/2 ground state and an additional hyperfine splitting given by the interaction with the I = 7/2 nuclear spin of the 51V ion (natural abundance: 99.75%). The magnetic properties and the electron spin resonance spectroscopy of this molecule have been previously reported in22. The latter one is a BDPA organic radical diluted into a Polystirene matrix with spin concentration of ρ ≈ 1 ⋅ 1015 spin cm−3, similar to the one reported in55. Each molecule has an unpaired electron giving an electronic spin S = 1/2 center. We place the sample on a superconducting coplanar resonator (ν0 ≈ 6.91 GHz) made out of superconducting YBa2Cu3O7 (YBCO) films on a Sapphire substrate (Fig. 1a), on which we have already integrated molecular spin ensembles72,73 and implemented microwave sequences for spin manipulation53,55. The Continuous Wave (CW) and Pulsed Wave (PW) microwave spectroscopy of VO(TPP) through the resonator has been previously reported in53,57, while the one for BDPA has been previously reported in55 (see also Supplementary Section 2.1). The sample and the resonator are cooled-down to 3K into a commercial Quantum Design Physical Properties Measurement System (QD PPMS), which is also used to apply the external static magnetic field53,54,55. A RF copper coil (diameter = 5 mm, height = 5 mm and 7 windings) is placed on the surface of the resonator as in Fig. 1a, the way that the magnetic sample is at its central axis.

Set up for the generation of sensing protocols

The experimental setup is essentially the microwave heterodyne spectrometer previously reported in53,57 in which one channel of the waveform generator is used to generate the microwave excitation tone through frequency upconversion. The other channel of the waveform generator is used to generate a RF excitation, which is routed directly to the copper coil with a dedicated RF coaxial line. In this way, the setup can generate synchronized MW and RF pulse sequences in which each pulse parameter (amplitude, duration, phase, inter-pulse delay) can be tailored (see Supplementary Section 1.1 and Supplementary Fig. 1). The readout is performed at MW frequency and in time domain with an oscilloscope, after the output MW signal has been downconverted with a second mixer. The generation of the two excitations and acquisition are controlled by home-written Python scripts. Typical pulse parameters used in our experiments are: pulse durations between tπ/2 = 80 ns and tπ/2 = 150 ns for the π/2 pulse and between tπ = 160 ns and tπ = 300 ns for the π pulse, and interpulse dely τ = 900 ns up to τ = 1700 ns. Each sequence is followed by a relaxation time trelax = 15–20 ms to avoid sample saturation. The RF excitation amplitude can be as high as VRF = 2.5 V, limited by maximum output voltage available at AWG port, and a preliminary independent calibration procedure has been performed to convert VRF into its corresponding magnetic field B1,RF (see Supplementary Section 1.2).