Our gradiometric quadrature detection scheme is shown in Figure 1A , where two commercially available dual-axis vector magnetometers (Osborne et al., 2018) are placed orthogonally in the x-y plane. If the sensors, denoted 1 and 2, are positioned symmetrically around a magnetic dipole initialized along − x^ and undergoing Larmor precession clockwise in the plane at positive angular frequency ω ₀, we can write the following (ideal) expressions for the oscillating magnetic field sensed by the four magnetometer channels as a function of time t:

Here, r is the offset distance of the center of the sensing volumes from the magnetic dipole (the sample is modeled as a uniformly magnetized sphere); T ₂ is the characteristic exponential decay time of the precession signal; µ ₀ is the vacuum permeability constant. Note the factor-of-two difference between pairs of measured field components; this is the result of a dipole-field geometry as measured by point-like sensors. The initial magnetic-moment amplitude m ₀ equals the sample volume times the magnetization of the sample. Note that each measured magnetic-field component may be positive or negative, since vector rather than scalar magnetometers are used. For a spherical 1mL sample of water polarized at 1T and room temperature (typically 22°C in our lab), one can estimate magnetic-field values (at the beginning of the measurement assuming no relaxation losses) on the order of 100pT, given the experimental offset distance of 17.5mm. This agrees with experimental data ( Figure 2 ); a complete calculation is provided in Supplementary Material . We note that, fundamentally, Equations 1, 2 should contain T2* rather than T ₂, as we do not employ CPMG or other dynamic decoupling sequences to suppress effects of magnetic-field inhomogeneities at the position of the sample during measurement. However, because the studied leaf samples have intrinsically high relaxation rates (T ₂ of 150–300ms, see Section 3) which are larger than contributions due to inhomogeneity of the solenoid ( Figure S8 ), such simplification is warranted in our case.

We see from Equations 1, 2 and the geometry in Figure 1A that by subtracting the measured fields along the x- and y-axes, respectively, the signals along each axis add—leading to a signal enhancement of 1.5, assuming identical sensor response—while common-mode noise is canceled. Furthermore, the two gradiometric signals have a relative phase of π/2—i.e., are in-quadrature—which becomes useful for signal processing in the frequency domain via Fourier transform. This is the basis of what we have termed the “gradiometric quadrature” detection scheme, used in the work reported here to achieve enhanced (by a factor of ∼3 compared to single-channel measurement) proton signals in ex vivo tree leaves.

The experimental setup, contained in a portable instrument rack, is depicted schematically in Figure 1B . Thermal polarization of nuclear spins is created using a 1T Halbach magnet with 15mm bore, which defines the maximum possible diameter of measured samples. After adequate polarization-buildup time in the magnet (5–10s), rapid (∼100ms) mechanical shuttling of the sample into the magnetic shield (Twinleaf MS-1LF) is performed using an Arduino-controlled stepper motor driving a plastic gear rack, to which a nonmagnetic sample holder is attached. Shuttling occurs inside a double-layer piercing solenoid wrapped around a plexiglass tube and reaching from the top of the Halbach magnet through the magnetic shield. At the center of the shield, a 3D-printed frame of ABS plastic contains three pairs of Helmholtz coils (radius 33mm) for creation of magnetic-field pulses along the x-, y-, or z-axes to manipulate the nuclear-spin state, as well as two atomic magnetometers (QuSpin QZFM Gen-2) for detection of nuclear-spin signals ( Figure 1A ). These zero-field sensors can operate in ambient magnetic fields of up to tens of nT, which is readily achieved with the magnetic shield, with the aid of built-in compensation coils. We introduced the piercing solenoid specifically to be able to separate the background field on the sensors from the precession field on the sample. A field on the order of 10µT may be generated inside the piercing solenoid without compromising sensor operation, as field leakage outside the solenoid is usually below 1%. The range of achievable precession frequencies is ultimately limited by sensor bandwidth—below 500Hz for the atomic magnetometers used in this work.

Figure 1B shows the typical relaxometry protocol for the experiments reported here. After spin polarization and shuttling, immediate application of a π/2-pulse in the y-direction rotates the bulk z-magnetization into the x-axis, where it subsequently precesses about the leading z-field of amplitude B ₀ at an angular frequency ω ₀ given by the proton gyromagnetic ratio γ1H according to ω0=γ1HB0 , where γ1H/2π≈ 43Hz/µT . This Larmor precession gives rise to a free-induction-decay (FID) signal which is acquired by the magnetometers. For proton spins (positive sign of the gyromagnetic ratio), precession is “left-handed”, occurring clockwise about the applied magnetic field (Levitt, 1997).

Experimental timing and control as well as detector readout were implemented in Labview using NI TTL-pulse and data-acquisition cards. To maximize the SNR of the measured FID signal, a number of experimental parameters were iteratively optimized. These include: polarization time, solenoid field (proton precession frequency), shuttling time and speed/acceleration, pulse duration and amplitude, and acquisition time. Various calibration data, along with photos and further details of the apparatus, may be found in Supplementary Material .

During acquisition of an FID according to the protocol in Figure 1B , the analog voltage outputs of all four magnetometer channels are recorded at a user-defined sampling rate, usually 2kHz, for subsequent analysis. Although the sensors are always calibrated (i.e., ambient magnetic fields internally compensated) with the sample in the measurement position prior to each experiment, application of the magnetic-field pulse drives the sensors out of their sensitive range for some tens of milliseconds. Thus, initial data points must be discarded in post-processing, resulting in an effective linear phase shift of the recorded oscillating signal. Because each magnetometer channel provides a vector measurement—sensitive to the sign of the magnetic field, in contrast to a scalar sensor—the handedness of spin precession may be deduced from a single-channel time trace, if it is possible to reconstruct the true phase of the FID. However, the quadrature detection scheme allows us to determine the handedness without having to reconstruct the phase.

As an illustration of the analysis procedure, Figure 2 shows calibration data from a liquid water sample at a proton precession frequency of 10Hz, obtained by supplying a current of 54µA to the piercing solenoid. The first 25ms of data, corresponding to the first 50 points of the FID sampled at 2kHz, have been discarded to remove post-pulse artifacts which would otherwise adversely affect the spectral lineshape and baseline (the calibrated magnetometers have a dynamic range of approximately 5nT; the applied pulse of 30µT prior to acquisition temporarily pushes them out of range). Plots show the average of multiple (here, four) scans, where a linear trend has been removed from each individual raw time trace prior to averaging. This detrending of time-domain signals mitigates the effect of low-frequency magnetic-field drifts in the lab environment, which may negatively impact spectral baselines in the frequency domain. Initially, we convert from voltage to magnetic-field units to obtain four time series associated with the four magnetometer channels, which we denote x ₁, y ₁, x ₂, and y ₂, following Equations 1, 2. The two gradiometric channels are subsequently constructed as x = x ₁ − x ₂ and y = y ₁ − y ₂. By examining the single-channel and gradiometric time traces in the context of the detection geometry ( Figure 1A ), we can confirm that the spin signal was initialized along − x^ and is precessing clockwise in the x-y plane, consistent with a precession field along +z^ . Furthermore, reduction of higher-frequency noise via gradiometry is visible.

The gradiometric-quadrature channel can be constructed from the two gradiometric channels as x+iy (see Supplementary Material for more details). In our experimental geometry, the time-dependent quadrature signal may be thought of as a vector rotating clockwise in the complex plane defined by a real x-axis and imaginary y-axis. Assuming perfect quadrature geometry, when a standard Fourier transform is performed to convert the signal into the frequency domain, the resulting spectrum should contain a resonance only at +10Hz and not at −10Hz.

In the time domain, the processed complex gradiometric-quadrature signal is written as

where x(t) and y (t) are the gradiometric signals, and ϕ is an overall phase selected such that the real part of the frequency spectrum has an absorptive peak (see below) (Keeler, 2010). The numerical coefficients a and b are defined so that a + b = 2, and may be adjusted to account for possible differences in gain between the two channels—usually a few percent or less, based on suppression of the residual negative-frequency peak. Prior to the Fourier transform, zeros may be added to the end of the time series defined by Equation 3 (zero-filling) in order to increase the spectral resolution.

All frequency spectra throughout this manuscript are plotted such that the y-axis has units of pT/Hz or fT/Hz, depending on the signal strength. These units arise from the discrete Fourier transform (DFT) used to convert data from the time domain to the frequency domain and subsequent data processing, as explained in detail in Supplementary Material .

The exact SNR enhancements attainable by the gradiometric quadrature method depend strongly on performance of the individual sensors, which may vary between experiments, as well as non-common-mode systematic noise. In our experience with water calibration samples, compared to single-channel spectra, SNR could be enhanced up to 75% via gradiometry alone and up to 300% via the gradiometric quadrature method (depending on sensor performance). This indicates not only that gradiometry is effective in terms of noise suppression, but also that the quadrature approach is more beneficial than simply summing the positive-frequency and negative-frequency peaks in a traditional “mirrored” non-quadrature spectrum. See Supplementary Material for quadrature simulations, gradiometer sensitivity data, and further details about phasing of quadrature signals. We expect that gradiometric quadrature detection can be especially advantageous for situations in which sensitive magnetometers operate in unshielded environments—if the contribution from common-mode noise dominates the contribution from uncorrelated noise at the positions of two sensors, total measurement noise will be significantly suppressed.

A description of all 19 tree-leaf samples included in our study is provided in Table 1 . For identification purposes, samples from each genus were numbered in order of preparation/measurement date. We note that, due to seasonal availability, evergreen (spruce, pine, and yew) leaves were collected from late February to early April, while deciduous (hazel, cherry, beech, and oak) leaves were collected from early April to late May. Sample collection occurred as close as possible to the time of first measurement. All samples were sourced from the Eurasian arboretum of the Mainz Botanical Garden, which informed the specific choice of species and/or cultivar, although we purposely selected a diverse range of common genera.

During harvest, a branchlet containing sufficient leaf coverage was cut from the branch tip of the tree donor and immediately placed in a glass beaker partially filled with distilled water such that the cut end of the branchlet was submerged, as shown in the photo inset of Figure 3 . Sample preparation in the laboratory proceeded as follows. Leaves or needles were gently removed from the branchlet, thoroughly dried with a clean tissue to remove any excess moisture, and packed into a pristine glass shell vial (BGB SV2ML) with outer diameter 11.6mm and interior volume 3.3cm³. Spruce, pine, and yew needles were placed lengthwise vertically into the vial, whereas the hazel, cherry, beech, and oak leaves had to be rolled or folded. Care was taken to avoid tearing or otherwise damaging leaves during sample preparation, thereby preserving the original structure of the plant tissue, while fitting as many leaves or needles as possible into the tubular vial ( Figure 3 photo inset). Due to leaf geometry, it was not possible to achieve completely uniform density of leaf material in the vial, particularly for larger deciduous leaves. Each individual vial and its plastic cap were weighed with a digital scale before and after leaf insertion to extract the total leaf mass. Vials were capped prior to and throughout each 13h experiment in the relaxometry setup, to prevent dehydration of the sample. After each experiment, the capped vial was removed from the setup, uncapped to release any trapped water vapor, recapped and weighed again. We found that post-measurement mass was always within a few mg of pre-measurement mass ( Table 1 ), suggesting that leaf dehydration during the experiment was negligible, and that little to no excess water vapor had been contained inside the vial.

For the dehydration investigation, one evergreen genus (spruce) and one deciduous genus (oak) were selected; each sample was measured three different times, on the dates indicated in Table 1 . Between experiments, the sample vial was stored uncapped in an on-site plant growth chamber (poly klima PK-520, 22°C, 12/12 h light/dark cycle, no humidity control). Because the vial was open only at one end, uniform dehydration of the leaf content could not be ensured. However, visible inspection of leaf color as well as the usual weighing procedure indicated that water loss had occurred during each storage period. It was critical to ensure that no condensation collected inside the vials at any stage of dehydration or measurement preparation, as this could introduce an additional spurious water-proton signal (sharp peak due to free water) not originating from water protons contained in the leaves themselves.

Each leaf measurement was conducted under identical experimental conditions to produce a proton-NMR spectrum as in Figure 3 . A resonance frequency of around 285Hz, corresponding to a 7µT precession field with 1.5mA applied to the piercing solenoid, was selected to avoid lower-frequency noise while remaining with the sensitive bandwidth of the magnetometers (see Supplementary Material ). In the plotted spectrum, linear background was removed in the region 260–320Hz and a Lorentzian fit (three-parameter Lorentzian function with constant term) was performed using the lorentzfit script in Matlab. As expected, the experimental lineshape is well-fitted by a Lorentzian, as it results from an exponentially decaying signal. Phasing of the quadrature signal was optimized by minimizing the root mean square error (RMSE) of the fit. Using the calculated fit parameters and errors thereon, the fit amplitude and linewidth (FWHM, full width at half maximum) were extracted. Typical fit amplitudes were on order 1 to 10fT/Hz—at least three orders of magnitude smaller than for pure water samples of similar volume—after averaging over 4096 scans (repetitions of the measurement protocol). The relaxation time T ₂ is related to the FWHM Δ as 1/(πΔ). Reported SNR values were obtained from the ratio of fit amplitude to the standard deviation of spectral noise in the region 265–275Hz. To avoid artificial broadening of the spectral line, no apodization was used; instead, a 300Hz low-pass filter was applied to each spectrum for lineshape correction of the averaged signal. Measured proton-signal amplitudes and T ₂ times are recorded for all 35 leaf experiments in Table 1 . Complete fitted spectra, along with example analysis code and further details of the analysis protocol, may be found in Supplementary Material .

In principle, multiple mechanisms may affect the spectral linewidth. These include magnetic-field gradients at the location of the sample, combined effects of different relaxation mechanisms, and possible contributions from non-water protons. Due to the significant differences in linewidth between liquid and leaf samples, we conclude that field inhomogeneity at the location of the sample is negligible, and therefore we report T ₂ rather than T2* times in this work. As described in Section 2.2, the dead time required before data acquisition places a lower limit of 25ms on measurable T ₂ times, which is acceptable for studies of free or loosely bound water protons. Because experimental parameters remained unchanged over the course of the leaf study and all data underwent an identical analysis procedure, direct comparison of signal amplitudes and T ₂ times is possible. Comparing the leaf results to typical water calibration data, we find that T ₂ times in fresh leaf samples are an order of magnitude shorter than in liquid water samples of similar volume.