Quantum magnetic J-oscillators

Domain: neuroscience
Published: 2026-01-29
Editorial outcome: major_revision

Abstract
Zero-field nuclear magnetic resonance (NMR) offers magnet-free access to nuclear spin-spin (scalar J) couplings, which define an intrinsic, molecule-specific frequency scale. However, the transient nature of zero-field NMR signals constrain spectral resolution and frequency stability. Here we introduce quantum J-oscillators that exploit J-couplings in molecules to produce phase-coherent continuous oscillations. Operated in zero magnetic field and driven by digital feedback, they generate sub-hertz to a few tens of hertz frequencies. In a proof-of-principle experiment on [15N]-acetonitrile, the oscillator achieves a 340 μHz linewidth over 3600 s, more than two orders of magnitude narrower than in conventional zero-field NMR. This methodology may facilitate precision measurements of J-coupling constants and enables discrimination of molecules whose zero-field NMR spectra are otherwise difficult to resolve. In addition, the combination of strongly coupled spin systems and programmable feedback turns J-oscillators into a compact table top platform for exploring nonlinear spin dynamics, including chaos and dynamical phase transitions. By uniting high-resolution spectroscopy and controllable quantum dynamics in a single, magnet-free setup, J-oscillators open new opportunities for applications where ultra precise frequency references or molecular fingerprints are required.

Introduction
Masers and lasers have revolutionized science and technology, finding applications in fields as diverse as telecommunications, medical diagnostics, astronomy, precision measurements, and fundamental physics1–7. Both technologies harness coherent electromagnetic radiation amplified through the process of stimulated emission8, a phenomenon that traditionally requires achieving population inversion between quantized energy states (i.e., a higher-energy state has to be more populated than a lower-energy state)9.

Conventional masers have been realized across diverse physical systems, including atomic beams1,10, gases11,12, and solid-state systems13–17. These devices typically operate in the GHz frequency range, achieving population inversion via various mechanisms. Recent advances extended these principles to the so-called “rasers” generating kHz-to-MHz frequencies using nuclear spins18. To create the required population inversion, such low-frequency rasers rely on hyperpolarization approaches such as spin-exchange optical pumping12, dynamic nuclear polarization19, photochemical polarization transfers20, and parahydrogen-based techniques18,21–24. Additionally, their emission amplification is typically triggered by “radiation damping”25 stemming from inductive coupling of polarized nuclear spins with detection coils. However, these rasers exhibit intrinsic limitations as they operate on Zeeman-split levels. Since the frequencies of these levels depend on the applied (bias) magnetic field, they are susceptible to magnetic field drifts, limiting their long-term stability and reproducibility.

In this work, we introduce zero-field quantum oscillators operating at frequencies from near-DC to tens of hertz, overcoming limitations of conventional rasers that rely on Zeeman-split levels. Unlike those, our oscillators function without a bias field, exploiting intrinsic nuclear spin-spin scalar interactions (J-couplings) within molecules26,27. Importantly, they utilize Δm=0 transitions27, with the quantization axis along the measurement axis (Fig. 1A, B). Because these transition frequencies depend primarily on molecular J-coupling constants27, the zero-field quantum oscillators achieve significantly improved frequency stability: coherent operation of up to one hour is demonstrated (Fig. 1D). The magnetic oscillation occurs along the same axis as the feedback axis due to Δm=0 transitions27, unlike the precessing magnetization with Δm=±1 of conventional rasers. Such an oscillator requires only a population imbalance—not a strict population inversion—achieved in situ by bubbling parahydrogen (para-H2) into a liquid containing the activated SABRE catalyst (SABRE = signal amplification by reversible exchange), see Fig. 1C28,29. The catalyst enables spontaneous polarization transfer in situ at zero field, creating J-transition population imbalances in target molecules30,31. Details of the para-H2 gas handling are given in Supplementary Methods.

Results
A central challenge in implementing zero-field quantum oscillators arises from the absence of radiation damping. In ultralow-field raser experiments—also lacking radiation damping—researchers have relied on external feedback loops32–34. Typically, these feedback loops detect precessing magnetization using optically pumped magnetometers (OPMs)35, and subsequently feed signals back into the sample through coils orthogonal to the measurement axis36. However, such a feedback scheme cannot be directly adopted for zero-field J-oscillators. This is because coherent amplification of the Δm=0 transitions requires that the feedback magnetic field be applied along the same measurement axis (y-axis,)27.

To resolve this, we developed an external feedback loop, implemented via software control without the need for specialized hardware modifications, as schematically depicted in Fig. 1A. In our implementation, an OPM detects the y-cartesian component of the magnetic field generated by the sample (BOPM). This signal is processed digitally, allowing precise control of both a tunable external feedback delay (τ) and feedback gain (Gext). The processed feedback signal is reapplied to the sample (Bext, along the y-axis) using a piercing solenoid37. The piercing solenoid is the solenoid that goes all the way through the μ-metal shield. It is designed to generate a magnetic field at the sample while producing no field at the OPM37.

Stability of J-oscillations
The J-oscillator was initially tested on a model system consisting of 5% [15N]-acetonitrile ([15N]-ACN) dissolved in acetonitrile (ACN) solvent29,31. The feedback configuration enables spontaneous emergence of the quantum oscillator, as demonstrated in Fig. 1D with a feedback delay of τ=222 ms and gain Gext=+30, requiring no pulse excitation. A zoomed-in view in Fig. 1E highlights the initial noisy background from the OPM sensor before the J-oscillator starts to emerge, while Fig. 1F illustrates the subsequent dynamic steady-state response. Likely, the electronic noise from the feedback loop triggers initial transitions. The signal from these spontaneous transitions is amplified by the feedback loop, and the phase is tuned for positive reinforcement. The resulting amplified field (Bext) drives the transitions, pushing the population imbalance away from the SABRE-pumped hyperpolarized steady state38. With high enough external feedback gain, this SABRE-pumped population imbalance can be temporarily inverted (see Supplementary Fig. 1). Deviations from the hyperpolarized steady state manifest experimentally as transient bursts. Such bursts represent the so-called “overshooting,” where the feedback-amplified transition intensities rise beyond sustainable levels21,36. The SABRE pumping and nuclear spin relaxation processes oppose the feedback amplification, dampening coherences and partially restoring the hyperpolarized steady state. Over successive faded bursts, the system stabilizes into what we call the “dynamic steady state” under the feedback, in which the combined effects of SABRE-pumping (replenishing population imbalances), the feedback-amplified transitions (driven by Bext), and relaxation come into a balance.

The J-oscillator operating on the molecular J-transition was continuously recorded for 3600 s on a naturally abundant [15N]-acetonitrile ([15N]-ACN, 0.36%) obtained by setting τ=222 ms and Gext=+200. Fast Fourier transformation (FFT) (Fig. 1H) revealed a sharp, delta-function-like peak with a full-width-at-half-maximum (FWHM) of 340 μHz (Fig. 1H, inset). This linewidth is approximately the inverse of the measurement time, indicating minimal frequency drift of the experimentally observed quantum oscillation. It was later found that this linewidth has contributions of various frequencies due to a slow J-coupling drift (see the discussion below).

In contrast, the linewidth obtained from conventional zero-field NMR spectra does not improve with increased measurement time, as it is fundamentally limited by nuclear spin relaxation. For instance, [15N]-ACN has a FWHM of 32 mHz for the same transition (Fig. 1H). Similarly to high-field rasers24, the reduced linewidth achieved with the J-oscillators facilitates resolving closely located resonance lines.

The reproducibility of the J-oscillator’s frequency response is confirmed in Fig. 2. We repeatedly ran the J-oscillator (τ=222 ms, Gext=+200) on a naturally abundant acetonitrile sample for 10 min, then performed free-decay (FD) measurements: a 1 nT magnetic excitation at 3.374 Hz was applied during 10 s of bubbling to selectively excite the 2-J transition, followed by 40 s of signal detection. Figure 2B, C show the FFTs of the steady-state J-oscillations and the FD signals, respectively. The J-oscillator consistently produces narrow peaks at around the same frequencies, albeit with a small upward drift (~300 μHz per hour).

To further test the reproducibility, three independently prepared identical samples were run with the sequence eleven times per sample (Fig. 2A). Frequencies were extracted from the time-domain signals by fitting a sinusoid for the J-oscillator or a decaying sinusoid for the FD measurements (Supplementary Fig. 3 for an example of the fits). The results (Fig. 2D–F) show a slow, monotonic increase in the J-coupling constant, the exact nature of which is not fully understood. However, this effect is likely attributable to gradual changes in sample composition39, primarily solvent evaporation during continuous bubbling (which, in turn, slowly alters the exchange dynamics with the SABRE complex). A similar drift was observed in the J-oscillator frequency during continuous hour-long monitoring without restarting the J-oscillator (see Supplementary Fig. 4, panel A). Despite this drift, the J-oscillator (orange traces in Fig. 2D–F) reproducibly returns to the same frequency response upon each restart, tracking the increase in J with much higher precision compared to the FD measurements (blue traces in Fig. 2D-F), and remaining consistent across all three samples.

A systematic millihertz-level offset in the J-oscillation frequency with respect to J-coupling extracted from the FD measurement was observed; however, the resulting frequency offset was stable and reproducible across runs and samples. A detailed theoretical analysis of this offset will be presented elsewhere.

While both J-oscillators and conventional zero-field J-spectra demonstrate the same scaling behavior of SNR with total acquisition time (SNR ∝ √T, see Supplementary Fig. 4, panel B), the error in determining transition frequency using the J-oscillator approach decreases more rapidly with the acquisition time due to the above-mentioned linewidth narrowing. Therefore, standard deviation in measuring precision in frequency space using J-oscillators approaches the Cramér-Rao lower bound40.

On-demand spectral editing
For a negative external feedback gain (Gext=−20, Fig. 3C), the 1-J quantum oscillator is sustained for feedback delays ranging from 60 to 275 ms (minimal delay in our system is 60 ms). Under the same negative feedback gain conditions, the 2-J quantum oscillator emerges only within a narrower delay range of 60–140 ms. For positive external feedback gain (Gext=+20, Fig. 3D), the 1-J oscillator emerges spontaneously at delays ranging from 325 to 400 ms (400 ms being the maximum sampled delay in our system). Meanwhile, the 2-J oscillator is sustained under delays spanning from 160 ms to 290 ms. Thus, the emergence of 1-J and 2-J quantum oscillators exhibits different dependencies on the externally applied feedback delays.

This behavior is due to the frequency-dependent phase shifts introduced by the delays. To gain more insight, the delay (τ) is converted to the corresponding feedback phase lag (φ) at the operating J-transition frequency f, calculated as φ=2πfτ as presented in Supplementary Table 1. Combining the results from both plots, it was found that the 1-J quantum oscillator is sustained across the range of phase lags from around 3.4 to 6.1 radians. Similarly, the 2-J oscillator remains active within a comparable phase lag interval, between approx. 3.4 and 6.1 radians. These phase intervals are symmetric around 3π/2, spanning roughly ±0.86·(π/2) around this central value.

The dependence of the J-oscillator operation on phase can be understood by decomposing the feedback field mathematically into two components relative to the field generated by the sample. The first component aligns with the signal (“in-phase,” phase lag 0) or is opposite to it (“anti-phase,” phase lag π). The second component, the “quadrature” component, is shifted by a quarter cycle (phase lag ±π/2) relative to the signal. Only this quadrature component effectively contributes to amplifying the selected J-transitions. To quantify this amplifying contribution, we define an “effective external feedback gain,” represented as Gext sin φ41. For sustained quantum oscillations to emerge, the absolute value of this effective gain must exceed a specific threshold (discussed below). Moreover, the sign of the effective gain must match the direction of the hyperpolarized population imbalances: in the current case, a negative effective gain is required, corresponding to a population inversion between the selected J-transitions. Consequently, the dependence of quantum oscillator emergence on the externally imposed phase lag provides a convenient method to selectively amplify individual J-transitions at specific frequencies, demonstrating the capability for on-demand spectral editing (see Fig. 4).

Threshold dynamics
Figure 3E, F show the dependence of the steady-state and the initial burst amplitudes on the external feedback gain (Gext) for quantum oscillators operating selectively at the 1-J (Fig. 3E, τ=160 ms) and 2-J (Fig. 3F, τ=222 ms) transitions, respectively. These delays ensure that the feedback corresponds to a phase lag near 3π/2 for each peak, resulting in a maximal contribution of the quadrature component (j sin φj = 1). In both plots, the steady-state amplitudes initially increase with feedback gain but decline at higher gains, mirroring the conventional maser/laser dynamics where output peaks at an optimized resonator quality factor42. The fact that higher gains result in lower SS amplitudes is also indicated by the minima observed in Fig. 3C (1-J at 160 ms) and 3D (2-J at 220 ms). These minima correspond to conditions with the feedback phase lag of approximately 3π/2, where the effective feedback gain has the biggest amplitude. At higher gains, the maximal amplitude of the first burst plateaus as the hyperpolarization-driven population imbalance is converted into coherence.

The observed J-oscillations in Fig. 3E, F are affected by two contributions: the passive magnetic intrinsic gain of the system Gint, and the actively applied external feedback gain Gext. The intrinsic gain Gint, which is analogous to the conventional gain definition in laser physics9, quantifies the ability of the system to amplify magnetic fields. Specifically, the intrinsic gain Gint is defined as the ratio between the amplitude of the magnetic field produced by the sample (as measured by the OPM) and the amplitude of the externally applied AC field that acts on the sample (see “Methods” section for details). For sustained quantum oscillations to emerge, the total gain from combined internal and external contributions must exceed unity:

|Gext + Gint| > 1: (1)

For the cases shown in Fig. 3E, F, the threshold external gains Gth ext required to initiate sustained oscillations, extracted from the numerical simulation illustrated in figures, are approximately 7.2 for the 1-J transition and 5.8 for the 2-J transition, respectively, as highlighted by the shaded areas in the respective figures. Additionally, simulations of the exact same modeled spin systems yield intrinsic gain values Gint of approximately 0.138 and 0.172 for the 1-J and 2-J transitions, respectively (see Supplementary Fig. 5 and “Methods” section for detailed numerical procedures). Overall, these extracted values meet the theoretical threshold condition defined by Eq. (1).

By tuning the external feedback gain, it is possible to achieve the J-oscillator behavior under challenging conditions such as low molar polarization (product of molecular concentration and nuclear polarization) or rapid relaxation, where intrinsic gain alone would be insufficient to initiate the emergence of the oscillator18. Increasing the external feedback gain (Gext) effectively lowers the required polarization threshold, allowing the observation of the quantum oscillator behavior for various molecules, see the section below. The minimum SNR required for self-oscillations is therefore set by the maximum practically applicable Gext (see the “Methods” section), which is limited by solenoid-sensor cross-talk: at large feedback, leakage noise is fed back into the loop, causing instability. This solenoid leakage originates mainly from imperfect coil winding, and improved coil construction can mitigate the problem. Conversely, when the external feedback gain is limited, only the J-transitions with sufficiently large intrinsic gain can surpass the threshold condition. Exploiting this property enables selective excitation of individual resonance lines within spectra containing densely overlapping peaks (see Fig. 4).

Generality across various molecules
Figure 4 presents spectra of J-oscillators alongside their corresponding zero-field NMR spectra for various illustrative chemical systems. Unless otherwise noted, the J-oscillators were acquired for 10 min (except for the trace in Fig. 4D collected with Gext=−1000 and Fig. 4H, I). The corresponding FFT spectra were normalized using division by the total number of data points (Convention I, see Supplementary Notes 2), allowing for the direct comparison between quantum oscillator-based and zero-field NMR spectra, despite differences in acquisition times. The sample preparation is discussed in the “Methods” section.

Acetonitrile (ACN) is revisited first. Figure 3A shows J-oscillations generated by naturally abundant acetonitrile ([15N]-ACN, 0.36%). Despite roughly ten-fold isotopic dilution compared to the previously discussed samples (5%), we successfully achieved sustained oscillations by compensating the reduced intrinsic gains with increased external feedback gain. Additionally, isotopically labeled ACN molecules with 1% abundance of 15N were tested: specifically, [1-13C, 15N]-ACN and [2-13C, 15N]-ACN (Fig. 4B, C).

The approach also works for more complex spin systems. A longer-chain nitrile, [U-13C, 15N]-butyronitrile (Fig. 4I) and several heterocycles—[15N]-pyridine, 4-amino[15N]-pyridine, [15N3]-metronidazole, and [15N2]-imidazole (Fig. 4D–G)—were tested. They all show complicated zero-field spectra, yet individual lines could still be “picked out” and driven into clean, millihertz-wide J-oscillations by optimizing the feedback parameters. The J-oscillators operating on [15N]-pyridine under a higher feedback gain (Gext=−3000) are given in Supplementary Fig. 6. Under specific feedback conditions, near-DC J-oscillations were observed in butyronitrile, and sub-0.5 Hz J-oscillations were seen in metronidazole and 4-amino-pyridine (Supplementary Figs. 7, 8). Whether they represent extremely low-frequency J-transitions or some nuclear-paramagnetic effect warrants further investigations.

Finally, we established J-oscillators in the biologically relevant probe [1-13C]-pyruvate (Fig. 4H). Due to the poor hyperpolarization at elevated temperature conditions (heat up by the OPM), together with the broad Hz-scale FWHM, the conventional zero-field NMR spectrum of this molecule (blue trace) exhibited poor SNR (~4). But the resulting J-oscillator (orange trace) showed significantly improved SNR (~70 for 2-J) and a narrower linewidth. A frequency shift of both transitions was detected because the feedback loop coupled the two transitions and “drew” the peaks toward each other41. One can envision J-oscillator experiments with cell cultures aimed at monitoring continuous pyruvate-to-lactate production—something hard to achieve by conventional zero-field NMR.

To illustrate the analytical power of the J-oscillator method, we investigated a series of mixtures that all contain 50 mM pyridine, 50 mM 4-aminopyridine and 3 mM of the Ir-based catalyst, but differed in the 14N/15N isotopic ratio of the two substrates. Conventional zero-field NMR spectra of these samples show that both molecules have their main peaks overlapping at around 15 Hz (Supplementary Fig. 9). In contrast, the J-oscillator approach yields well-resolved peaks for each molecule. Figure 5 shows the resulting spectra, all obtained using an identical feedback delay (τ=115 ms) but varying feedback gain. For the mixture in which both substrates were 100% 15N-labeled, two narrow resonances with approximately equal intensities emerge at moderate feedback gains. Comparison with the entire set of data reveals that the lower-frequency peak originates from 4-amino-[15N]-pyridine ([15N]-FamPy), whereas the higher-frequency line stems from pyridine (Py). The opposite 1:2 intensity ratio is observed when the enrichment levels are swapped (see Supplementary Fig. 10). Furthermore, as illustrated in Supplementary Fig. 11, adjusting the feedback delay allows the J-oscillation peaks of individual molecules to be selectively excited and distinguished within the same sample.

The experiments also show several interesting dynamical effects. First, whenever the feedback gain is tuned such that only one molecule oscillates, its resonance drifts to higher frequencies as the gain is increased. Second, when both oscillators are excited, resonances from the two molecules push each other apart, and the frequency separation increases with gain. These shifts do not reflect changes in the intrinsic J-couplings; they arise because the feedback field loop couples the spin dynamics of the two molecules. A detailed quantitative treatment is beyond the scope of the present work and will be reported elsewhere. Note that the shifts are reproducible and can be calibrated through careful tuning of the feedback parameters so that the intrinsic J-couplings can be extracted.

The results suggest that the J-oscillator method can assist in analyzing complex mixtures containing structurally similar molecules, particularly in situations where conventional zero-field NMR does not provide sufficient resolution to distinguish signals24. Since the method relies on SABRE hyperpolarization, it is only applicable to molecules with sufficient affinity to the SABRE catalyst, thereby restricting the range of detectable species. Nonetheless, this limitation can be mitigated in the future by employing alternative hyperpolarization schemes such as SABRE-relay, Overhauser-DNP, or even thermally polarized samples under continuous flow, to broaden the scope of molecules accessible to this method.

Toward many-body non-linear spin dynamics
As the applied feedback gain is further increased, a rich variety of nonlinear dynamics—including the emergence of multiple lines and chaos—arises from the imposed feedback that couples different J-transitions. These phenomena highlight the unique potential of zero-field J-oscillators as convenient platforms for systematic exploration of nonlinear spin dynamics.

A key advantage of the zero-field J-oscillators (compared to Zeeman Rb-Xe co-magnetometers36 or high-field rasers)18 is their potential to operate as engineered multi-mode systems. Such behavior can be achieved by applying a static bias field, which is known to split a single J-transition into multiple transitions43. Existing theoretical work predicts that the multi-mode systems of this type, when subject to active feedback, could undergo transitions in dynamical behavior ranging from mode collapse to frequency combs formation and chaos, depending on the separation between the transitions44. This mechanism suggests a potential route for future studies of nonlinear spin dynamics using zero-field J-oscillators. In contrast, high-field rasers oscillate at the nuclear Larmor frequency, and additional modes arise from intrinsic J-coupled multiplets, these spacings are set by molecular parameters and are therefore not readily tunable.

Digital signal processing provides an opportunity for future extensions of the feedback scheme. Advanced digital filtering algorithms could mitigate experimental imperfections; for example, OPM temperature drift could be suppressed by incorporating a DC-blocking filter. In addition, more complicated feedback protocols based on derivatives or nonlinear functions of the detected signal may provide access to complex dynamical regimes, such as chaos or time-crystalline-like behavior in liquid-state molecular systems45. These possibilities remain to be explored experimentally in future work.

Methods
External feedback loop apparatus
The hardware components of the feedback loop system are as follows. The signal produced by the sample is measured using a commercially available optically pumped magnetometer (model QuSpin Gen-2) with a sensitivity of around 15 fT/√Hz in a 3–100 Hz band and a calibration factor of 1000/2.7 pT/V. The analog output from the OPM is digitized with a National Instruments (NI) analog input card (model NI 9239) at a sampling rate of 2000 Hz. This digitized signal is transmitted to the computer via the NI Measurement & Automation Explorer interface. A Python-based real-time processing algorithm running on the computer set the delay and gain parameters to the acquired signal. The processed signal is then converted back to an analog form using an analog output (model NI 9263) at a sampling rate of 2000 Hz as well and delivered to a piercing solenoid, which generates the feedback magnetic field.

Sample preparation
Samples were prepared in a nitrogen atmosphere and transferred into a spherical NMR tube (outer diameter 10.5 mm, inner diameter 8.5 mm)31 that can be adapted to the para-H2 gas-handling setup. Samples were prepared of a common Ir-precursor [Ir(IMes)(COD)Cl] dissolved in acetonitrile or methanol, c.f. detailed sample composition is provided in Supplementary Table 2. The following substrates and solvents were purchased from Sigma Aldrich and used without further purification: (A) acetonitrile (0.36% [15N]-ACN and 99.64% [14N]-ACN), (B) 1% [2-13C, 15N]-ACN in 99% [14N]-ACN, and (C) 1% [1-13C, 15N]-ACN in 99% [14N]-ACN; (D) [15N]-pyridine, (E) 4-amino[15N]-pyridine, (F) [15N3]-metronidazole, (G) [15N2]-imidazole, (H) [1-13C]-pyruvate, and (I) [U-13C, 15N]-butyronitrile, see Figure 4. An additional co-substrate29,46,47 was introduced to stabilize hyperpolarization transfer efficiency of samples (A, B, C, H, and I), which are listed in Supplementary Table 2.

Programmable feedback algorithm
Variable definitions and constraints. Apart from Gext and τ, the following variables are defined for the implementation of the feedback loop:
• Rbuf, Wbuf (Size Nb): Cyclic buffers storing the history of input (read) and output (written) samples, respectively; both are initialized with zero values.
• Rc, Wout (Size N): Arrays for the current voltage samples read in from NI 9239 (AI) and to be written to NI 9263 (AO); both are initialized with zero values.
• D: The set feedback delay in samples.
• Dp = τ × fs: The hardware propagation delay in samples.
• Ni: The number of zero-valued initial samples sent to the AO at the start of the experiment.
• i: Index pointer for the current reading and writing position.

To ensure smooth operation of the loop and to prevent exceeding the available range of the cyclic buffers, the buffer and memory sizes must satisfy the conditions Db ≥ Nc ≥ Ni and Nb ≥ max(Nc + (D − Ni), Nc + Dp). In practice, we typically set Nb = 4D and ensure Nc ≥ Ni + 20fs (where fs is in kHz) to avoid potential hardware overheads.

The active feedback protocol. The feedback loop runs continuously, with the analog output (AO) and analog input (AI) modules sharing the same clock and start trigger. Since the loop requires memory of previously written samples to generate delayed outputs, we begin by writing Ni zero-valued samples to the AO module. Given that the samples produce zero signal initially, we assume that all prior reads and writes are also zero, meaning the buffers Wbuf and Rbuf are initialized with zero values. During operation, while the AO module is still outputting its current buffer, the AI module simultaneously acquires new Nc samples from the OPM output. These samples are processed according to Algorithm 1 and written into the AO memory buffer before the previous data in AO memory are fully run out. This continuous overlap between acquisition, processing, and output ensures that the AO buffer never becomes empty, allowing uninterrupted feedback loop execution.

The programmable external active feedback algorithm is presented in Algorithm 1. It is essential to feed back only the sample-generated signals back to the sample, excluding the feedback field produced by the solenoid itself or amplified OPM noise. Although the piercing solenoid is designed to achieve this by producing a magnetic field at the sample location while generating no measurable field at the OPM. In practice, the solenoid is not perfectly isolated, and a small portion of its field leaks outside. We mitigate the effect of this leakage by a digital compensation step (Step 2 in Algorithm 1) which subtracts the expected leakage from the OPM readout based on the previously written output and a calibrated leakage factor Cl. The solenoid leakage factor Cl is defined as the ratio between the voltage measured by the OPM due to solenoid field leakage (via the NI 9263) and the input voltage applied to the solenoid (via the NI 9239). The programmable feedback delay is subsequently implemented by retrieving the processed signal from the cyclic buffer Rbuf at an index offset by D samples relative to the current write position (Step 5 in Algorithm 1).

Calibration. Due to the frequency-dependent response of the OPM, leakage is calibrated at 5 Hz (proximal to our <20 Hz region of interest), while high-frequency leakage (>4 Hz) is suppressed via moving-average filters (Step 3 in Algorithm 1). Standard measurements employ a 10 ms window; pyruvate data utilized a 40 ms window applied sequentially three times. The corresponding filter amplitude responses are shown in Supplementary Fig. 12. With active feedback, the OPM sensitivity obtained under different moving-average settings is presented in Supplementary Figs. 13–14. The OPM noise floor remains constant across increasing gains, confirming that the J-oscillator sensitivity is not limited by amplifier noise.

Calibration of the leakage factor Cl and propagation delay τpd utilizes a synchronized 5 Hz sine test with shared AO/AI clocks and start trigger. A test signal drives the solenoid (from NI 9263, R0 = 430 kΩ) while the OPM response is recorded (by NI 9239). The propagation delay τpd is determined from the signal onset lag, and Cl is derived from the output-to-input amplitude ratio.

In our case, driving the solenoid with 10.9 nT amplitude sinusoid at 5 Hz produced 8.6 pT, 5 Hz sinusoid at the OPM, giving Cl ≃ 8.6 pT / 10.9 nT ≃ 7.9 × 10−4. The compensation algorithm gives an additional ~200× suppression (verified via independent Helmholtz coil injection and residual measurement). The maximal stable external feedback gain is bounded by the inverse compensated leakage, |Gext,max| ≲ 1 / (Cl × 200) ≃ 2.5 × 105. In practice, |Gext| is restricted to <20% of this limit to prevent leakage-induced instability.

To achieve the intended feedback delay τ, the set delay in the program must be adjusted to account for the intrinsic hardware propagation delay and the additional delay introduced by the digital filter. As a result, the set delay is D = τ × fs − Dp − Df, where Dp is the hardware propagation delay and Df is the delay caused by the filter. Standard experiments used a 10 ms window (order 1, Df = 5 ms), while pyruvate measurements required stronger smoothing (60 ms window, order 3). For example, measuring ACN (2-J transition) with target τ=222 ms, hardware delay Dp = 12 ms, and filter delay Df = 5 ms, the set delay is D=(222−12−5)×fs. We also choose Ni = 160fs, Nc = 100fs, and Nb = 4D to satisfy the buffer requirements.

Theory of J-oscillators
The dynamics of the J-oscillator signal can be simulated using a master equation:

d/dt ρb(t) = −i[Hb0 + Vextb(t), ρb(t)] + Rbbρb(t) + Pb(t), (2)

where ρb(t) is the density operator of the molecule, Hb0 represents the J-coupling Hamiltonian, Vextb(t) represent the coupling of the molecule with the delayed feedback filed, Rbb accounts for relaxation processes, including intramolecular dipolar interactions and intermolecular paramagnetic effects modeled via fluctuating random fields48,49 and Pb represents SABRE pumping.

Specifically, the feedback field couples to the sample magnetization along the detection axis,

Vextb(t) = −Gext · BOPM(t − τ) · Σi γi Ii,y, (3)

where BOPM(t) represent the field generated by the sample as measured by the OPM. The operator Σi Ii,y denotes the angular momentum operator of the i-th nuclear spin along the detection axis, and γi is its corresponding gyromagnetic ratio.

The forms of the Hamiltonians, relaxation superoperators, and pumping operator are provided in the Supplementary Notes 1. The master equation was solved numerically using a second-order Strang splitting scheme50, which separates coherent evolution from dissipative dynamics. Full implementation details are given in the Supplementary Notes 1.

The intrinsic magnetic gain of samples
To quantify the intrinsic magnetic gain of the sample, Gint, we add the interaction between the sample and a weak magnetic field applied via the piercing solenoid. The applied field H(t) (magnetic field intensity) oscillates at frequency f, the system dynamics is determined by,

d/dt ρb(t) = −i[Hb0 + Vextb(t), ρb(t)] + Rbbρb(t) + Pb(t), (4)

where the perturbation Hamiltonian (for [15N]-acetonitrile) is,

Vextb(t) = −μ H(t) · (γ15N S15N,y + γ1H K1H,y), (5)

In the dynamic steady state, the magnetization M(t) of the sample oscillates at the same frequency f as the applied field H(t). This magnetization generates a detectable magnetic flux at the OPM. When the applied field amplitude is sufficiently small (i.e., when the perturbation from the applied field is weak compared to the system’s relaxation rate), the ratio between the amplitudes of the measured magnetic flux (at the sensor) and the applied magnetic flux becomes independent of the applied field amplitude. Instead, it depends solely on the frequency f.

We define the internal magnetic gain Gint(f) as this frequency-dependent response,

Gint(f) = (r3/3d3) · (|M(t)| / |H(t)|) = |χ(f)|, (6)

where r is the radius of the sample and d is the distance between the sensor and sample, |χ(f)| is a frequency-dependent complex magnetic susceptibility.

Simulation shows maximal internal gains at f=J and f=2J, with internal gains Gint(J) ≈ 13.8% and Gint(2J) ≈ 17.2%, respectively (Supplementary Fig. 5). As discussed, the threshold external feedback gain (Gth ext) required for maser oscillations to spontaneously emerge corresponds to the inverse of the on-resonance internal gains, Gth ext = 1/Gint(fj). For 1-J and 2-J transitions, this yields Gth ext ≈ 7.2 and Gth ext ≈ 5.8 respectively, consistent with the results in Fig. 3E, F.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "See attached review in PDF format. I expressed some criticism and suggested changes to improve the manuscript. Overall, I recommend accepting the manuscript for publication after minor revisions.", "reviewer_2": "The authors report a novel and useful approach to generating a steady state signal that is insensitive to temporal variations in the Zeeman interaction. They base their work on driving the scalar coupling in zero magnetic field.\nThe work is well described and could be published as is, however there are a few points they may wish to consider.\nThere are a few places where a steady state response is somewhat confused with a coherence. An example is where a linewidth is reported for a steady state response. Since this is steady state, the response will appear for all time (provided that the spin polarizer continues to function). I would change the language and make clear that the reported linewidth is artificial and is not an indication of a sensor sensitivity. The same factor makes the use of the method for time-crystals somewhat less interesting.\nThe use of an amplifier in the feedback loop to drive the response is interesting, though it has been done before including by Laukien (to reduce radiation damping). Is there any analysis on the effects of noise from the addition of the amplifier? There is also the settling in period which provides information on the implementation.\nI found the theory description to be very clear and useful. I would have put it earlier since to me the steady state description in the modified Bloch equations provides a concise description of the dynamics, but I am not suggesting any changes.\nFinally, a tiny point that I have never understood what precisely is meant by a “piercing solenoid.” I don’t find the description in their ref 33 to be clear. Is it just a solenoid that goes through the magnetic shield, or ...? Might the authors take the opportunity to clarify this?", "reviewer_3": "There is no doubt that the article describes an interesting physical concept of a low-field J-oscillator. Nuclear spin order is continuously provided through parahydrogen gas, coupled to the nuclear spin system of interest using the SABRE technique, and electronic feedback is used to generate a continuous oscillating magnetization whose frequency is closely related to the J-couplings in the solution-state nuclear spin system. The results are interesting, although not particularly surprising: NMR feedback oscillations have been observed almost since the birth of NMR in the 1940’s, with particularly extensive investigations by researchers such as the Zürich group of Blum and co-workers in the 1970’s-1980’s (not cited in the article).\nThe authors claim that the J-oscillator described here has “exhibits superior long-term frequency stability .. in comparison to conventional NMR rasers”. That might be true but I don’t see that the evidence for “improved long-term frequency stability” anywhere in the reported results. Since the oscillator is a combination of electronics and nuclear spin dynamics, one would expect the electronic parameters, performance, and noise figure, to play a role both in the oscillation frequency and also its stability. But those possible issues do not seem to be considered at all.\nFigure 4 is particularly instructive. It shows the Fourier transform of the RASER signal for mixtures of compounds as a function of electronic feedback gain. The frequencies shift with a change in feedback gain, which reinforces the point made above. In the accompanying text, the authors write that “In the sample that contained 100 % [15N]-pyridine and 50 % 4-amino-[15N]-pyridine, the right peak is about twice as strong as the left one, giving the expected 2:1 ratio that reflects upon its isotopic composition”. After some trouble I can identify the spectra in Fig.4 that the authors refer to, and I can also see that the peak amplitudes are roughly in the indicated ratio - although there is no quantitation given, and this ratio looks as if it shifts around depending on the feedback gain (glossed over by the authors). The authors then say “The opposite 1:2 intensity ratio is observed when the enrichment levels are swapped”. However, when I try to find the corresponding spectra “when the enrichment levels are swapped”, I do not see a swap of the 1:2 intensity ratio. Instead it looks to me as if one peak is simply absent in that case. Am I missing something or are the authors reporting what they would like to see, rather than what is actually observed? None of this inspires confidence for the putative analytical applications advanced by the authors.\nThe authors report that in one case the Fourier transform of the continuous oscillation gives a linewidth of 337 microHz, whereas the ordinary linewidth without feedback, and limited by spin relaxation, is 37 mHz. From this they conclude that the resolution has been enhanced. However, I would like to know how reproducible is the oscillation frequency? If the oscillations are restarted, is the same frequency observed, within a standard deviation of 337 microHz? Or does it jitter within a standard deviation of 37 mHz (which is what I would expect)? Such statistics are not reported. Nevertheless statements are given which suggest that authors expect applications based on a greatly improved frequency resolution. I do not think that such statements are warranted based on the data presented so far.\nThe conclusion section on page 12 reads more-or-less like a marketing document. The future is rosy. There is no discussion of limitations, issues that need resolving, or possible obstacles. In short, the presentation is quite uninformative and unscientific.\nIn summary:\n* What are the noteworthy results?\n-> demonstration of feedback oscillations for nuclear spins in very low magnetic field\n* Will the work be of significance to the field and related fields? How does it compare to the established literature? If the work is not original, please provide relevant references.\n-> The feedback device and observations are original and interesting. I have doubts that the observations are significant, but that is a very subjective view.\n* Does the work support the conclusions and claims, or is additional evidence needed?\n-> In my view the work does not support some of the claims of increased stability, resolution, etc.\n* Are there any flaws in the data analysis, interpretation and conclusions? Do these prohibit publication or require revision?\n-> There are some flaws in the conclusions and discussion of the results, as detailed above.\n* Is the methodology sound? Does the work meet the expected standards in your field?\n-> the experimental methodology is sound and meets high standards. However some of the claims made are unsupported and the presentation is uncritical and unscientific in some places\nIs there enough detail provided in the methods for the work to be reproduced?\n-> yes, I think so."}, "author_response": "Authors’ response: We appreciate the referees’ constructive criticism and recommendation to accept our manuscript for publication after a minor revision.\nThe authors demonstrate an intriguing approach to stimulate nuclear magnetic oscillations at zero field via an external magnetic feedback loop and nuclear hyperpolarization. These oscillations exhibit exceptionally long coherence times and ultra-low frequencies associated with the intrinsic J-couplings within the excited molecule. The proposed method is an ultra-low-frequency analogy of the RASER effect, originally observed at high magnetic fields. An interesting novelty here is the capability to recreate “radiation damping” – a key feedback phenomenon for the high-frequency RASER. The authors effectively mimic this by returning the phase-shifted signal back to the sample at various gains using a low-Q solenoid coil. By tuning the gain and the phase shift, the authors have managed to excite multiple RASER frequencies in a variety of 15N-labelled compounds.\nThis work is well written, provides sufficient evidence, and has the potential to lead to further investigations of the effect. However, I am not fully convinced by the proposed applications and find a few items questionable:\n1) Time-crystals. Despite the catchy name, it is unclear how time-crystal behaviour would be derived from J-oscillations. They disappear immediately after the feedback is turned off as well as match the external feedback frequency. Therefore, authors should provide stronger arguments how time-crystals would manifest using J-oscillations in the liquid-state.\nAuthors’ response: We thank the reviewer for this comment and agree that our initial wording may have overstated the connection to time-crystal; we have accordingly softened the wording in the revised manuscript. Our intention is not to claim the direct realization of a time crystal, but rather to present a versatile platform for studying nonlinear dynamical phenomena in liquid-state molecular spin systems, which can include time-crystalline-like behavior under certain conditions. For instance, when a bias field is applied, the system exhibits distinct dynamical regimes as a function of the applied field and feedback gain, such as fixed points, limit cycles, and chaotic behavior [43]. In the regime of fixed points, the oscillation frequency remains locked to the specific values. Similar behavior can also happen under non-linear feedback schemes.\nChanges made to manuscript: the abstract was updated to remove the mentioning of time crystals. The only sentence mentioning this topic now reads as follows “These capabilities facilitate controlled studies of complex phenomena, ranging from chaos to the time crystalline-like behavior in liquid-state molecular systems.”\n2) J-coupling determination. The proposed method can measure J-oscillations with high precision but the absolute value drifts depending on external gain, sample composition. Authors suggest, however, that numerical simulation may help in such cases but do not provide a confidence limit for the determined Jcoupling values. An additional table with the determined J-coupling values and errors would be highly recommended.\nAuthors’ response: We appreciate the referee’s constructive criticism. We have now performed additional experiments, substantially expanded the discussion and added a new Figure (Fig. 2). As shown for [15N]-acetonitrile, the extracted J-coupling exhibits a gradual drift (blue traces in Fig. 2D–F) which we attribute primarily to slow changes in sample composition, most likely due to solvent evaporation during continuous bubbling, leading to subtle variations in exchange dynamics with the SABRE complex. Despite this drift, the J-oscillator yields highly stable and reproducible frequencies, remaining well within 300 µHz across the three independent samples for the same run number, where the sample composition is similar. Since the values of J-couplings for other molecules are expected to drift as well, we do not aim to provide a table with extracted values at this stage. Additionally, the J-oscillators indeed exhibit a systematic frequency offset compared to the J-coupling values extracted from the conventional zero-field measurements. This offset depends on the feedback gain and delay parameters used in the feedback loop. A detailed quantitative analysis of this dependence will be addressed in our future work.\nChanges made to manuscript: Multiple changes throughout the text and a new figure added containing new experimental data.\n3) Analysis of mixtures. Although the example on a binary mixture is demonstrated, it is also evident that the spin-dynamics and observed frequency is a function of the feedback.\nAuthors’ response: we agree as we also made this observation.\nFurthermore, as hyperpolarization by SABRE is only applicable to molecules with good affinity to the catalyst it may also limit the possible scope of detectable molecules. Authors probably anticipate these challenges but should warn the reader.\nAuthor’s response: Indeed, this work fundamentally relies on SABRE hyperpolarization to achieve the required population inversion. While this dependence limits the current method to molecules with good affinity to the SABRE catalyst, the scope can be extended in the future by employing alternative hyperpolarization techniques such as SABRE-relay, Overhauser-DNP, or even thermally polarized systems (e.g., delivered via liquid flow). It is worth noting that zero-field NMR as a field has evolved from its origins employing hyperpolarization methods such as PHIP and SABRE (alongside developments using thermal polarization) and continues to expand to encompass all available hyperpolarization strategies.\nChanges made to manuscript: We added the sentences in the section 2.5 “Additionally, since the method relies on SABRE hyperpolarization, it is only applicable to molecules with sufficient affinity to the SABRE catalyst, thereby restricting the range of detectable species. Nonetheless, this limitation can be mitigated in the future by employing alternative hyperpolarization schemes such as SABRE-relay, Overhauser-DNP, or even thermally polarized samples under continuous flow, to broaden the scope of molecules accessible to this method.”\nOther comments:\n4) Conventional RASER. Authors refer to few articles on the RASER effect but it is hard to understand what falls under the “convention”. I suggest using nomenclature of high- and ultra-low frequency RASER which is less ambiguous.\nAuthors’ response: We appreciate the reviewer’s suggestion. However, we prefer to use our terminology, J-oscillators, rather than adopting or extending the “RASER” nomenclature. The term RASER (radio-frequency amplification by stimulated emission of radiation) imply radiation processes, while virtually all NMR systems operate in the near-field regime without radiation propagation. Thus, using laser-like terminology is scientifically inaccurate in this context. We nevertheless clearly acknowledge the relation between J-oscillators and previously reported RASER phenomena in the manuscript. By conventional we mean reported RASER systems relying on Zeeman transitions.\n5) Abstract. Bold formatting of “J” should be removed.\nAuthors’ response: The bold formatting of “J ” in the abstract has been removed as suggested.\n6) References 9, 12, 28, 30, 37, 38, 39, 43 are missing bibliographic information or have formatting errors. Please fix.\nAuthors’ response: The References has been checked and properly formatted.\n7) Figure A4. It seems that 4 energy levels are missing in K = 1/2 space. I suggest marking all levels in K = 1/2 space with double lines for clarity.\nAuthors’ response: We have now added double lines in K=1/2 space in the now Figure A9.\n8) Figure A11, A inset. It seems that the red bar is upside down compared to figure 4.\nAuthors’ response: The red bar in Figures A10 and A11 has been corrected to ensure consistency with now Figure 5.\n9) Authors have used a spherical sample geometry presumably to ensure uniform magnetic field flux inside the sample. However, during the bubbling with hydrogen gas, the sample geometry could be distorted, especially at flow rates much higher than 20 mL/min. The explanation for setting such low flow rate would clarify the text.\nAuthors’ response: In our setup, the spherical region is filled up to around 90 %, and the liquid remains effectively confined within this volume during continuous bubbling up to 100 scc/m. Even at higher flow rates, where the liquid is extruded to the cylindrical part of the sample, the homogeneous field region of the solenoid is sufficiently large to avoid any field gradients across the sample. A relatively low flow rate of 20 scc/m was intentionally used to minimize liquid evaporation.\nChanges made to manuscript: We have added a sentence to the Methods section: “A low flow rate was chosen to slow down sample evaporation.”\n10) Algorithm 1. The sequential code provided implies that while reading a portion of the incoming signal NI9263 card cannot output (write) any signal to the solenoid. Does that mean that the external drive is windowed and not continuous? The size of NC should be provided as well.\nAuthors’ response: We thank the reviewer for the comment. The feedback loop operates in a continuous manner. Both the analog output (AO) and the analog input (AI) modules are synchronized. During operation, new samples are acquired, processed, and written to the AO card buffer while output from the current buffer is still ongoing, ensuring uninterrupted signal generation.\nChanges made to the manuscript: We have added a detailed discussion in Section 4.4, including explicit buffer size requirements and some examples in the revised text.\n11) Page 11, “In contrast, high-field rasers lack this flexibility, as their modes spacing cannot be easily tuned (e.g. intrinsic J-couplings are fixed)”. This is a confusing comment since J-couplings are intrinsically fixed for zero-field raser studied as well. Please rewrite.\nAuthors’ response: We thank the reviewer for the comment.\nChanges made to the manuscript: We have revised the text and now it reads:\n“…this can be achieved by applying a static bias field, which splits a single J-transition into multiple transitions and allows continuous tuning of their frequency separations simply by adjusting the applied field (to be published elsewhere). Such tunability can enable the systematic exploration of phase transitions in the system’s dynamics—from mode collapse, to frequency combs, to chaos—as the bias field is varied. In contrast, high-field rasers oscillate at the nuclear Larmor frequency, and additional modes arise from intrinsic J-coupled multiplets, these spacings are set by molecular parameters and are therefore not readily tunable.”\n12) Equation 7. I recommend modifying HRF notation as it can be confused with standard notation for the Hamiltonian associated with radio-frequency pulses. In addition, changing BNj(t) to BSj(t) would add consistency since “S” is used to denote the second spin species already.\nAuthors’ response: We thank the reviewer for the helpful suggestion.\nChanges made to the manuscript: $H_{RF} $ was replaced with $H_{fluc}$ and $B_{Sj}(t) $ was updated to $B_{Kj}(t)$ for consistency.\n13) Page 8, 4 paragraph, “(…) significantly improved SNR (…)”. Please specify the SNR achieved by the J-oscillations.\nAuthors’ response: We thank the reviewer for the comment. We have specified the achieved SNR in the revised text (Sec. 2.5), indicating that the 2J oscillations for [13C]-pyruvate exhibit an SNR of approximately 70.\nDespite my expressed criticism, I find this work acceptable for publication after minor revisions.\nAuthors’ response: We thank the referee for the proposal to publish our work as is.\nThere are a few places where a steady state response is somewhat confused with a coherence. An example is where a linewidth is reported for a steady state response. Since this is steady state, the response will appear for all time (provided that the spin polarizer continues to function). I would change the language and make clear that the reported linewidth is artificial and is not an indication of a sensor sensitivity.\nAuthors’ response: We thank the reviewer for this comment. We agree that the observed signal corresponds to something different from a conventional free-decay coherence. However, what we call a “dynamic steady state” corresponds to a self-sustained coherence generated under external feedback, rather than being an artificial feature.\nAs discussed in Section 2.2 and shown in Fig. 2D–F, the J‑oscillator reproducibly returns to the same frequency response upon each restart. The J‑oscillation frequency exhibits a small, systematic offset relative to the value measured with conventional zero-field NMR. This offset is stable and reproducible across repeated runs and among three independently prepared samples, demonstrating that it strongly correlates with the measured J‑coupling of the sample. In fact, the linewidth narrowing does not persist indefinitely with measurement time, as slow drifts in the frequency of a running J-oscillator are also observed (see Fig. A2A). We have clarified these points in the revised text.\nChanges made to the manuscript: multiple sentences in the text and the SI.\nThe same factor makes the use of the method for time-crystals somewhat less interesting.\nAuthors’ response: We thank the reviewer for the comment. As addressed in our response to Reviewer #1 above, we do not claim direct observation of a time crystal. In fact, we mean that nonlinear spin dynamics of liquid-state molecular spin systems can exhibit time‑crystalline-like behavior.\nChanges made to the manuscript: mentioning time crystals was removed from the abstract.\nThe use of an amplifier in the feedback loop to drive the response is interesting, though it has been done before including by Laukien (to reduce radiation damping). Is there any analysis on the effects of noise from the addition of the amplifier? There is also the settling-in period which provides information on the implementation.\nAuthors’ response: We appreciate the reviewer’s comment. In our system, the piercing solenoid configuration is designed so that the solenoid field is localized inside and produces no field at the OPM sensor location. Therefore, any noise generated by the amplifier and fed into the solenoid will not couple to the OPM sensor. To verify this, we measured the OPM’s noise floor as we varied the external feedback gain: the noise floor remains essentially unchanged with increasing gain (see Fig. A5, A6, and updated discussion in Section 4.4).\nNow we also provide a demonstration in Fig. 1E, showing a zoomed-in view of the settling-in period (where only background OPM noise under the active feedback is present) before the J-oscillation starts to emerge. Details of the feedback initiation have been added to Section 4.4. Specifically, at the start of the feedback loop, we assume that the prior read and write data samples are zero, consistent with the fact that no detectable signal is initially present.\nChanges made to the manuscript: In addition to the new data and figures added to the text, references 32-33 for the effect of external feedback on the coil Q-factor have been cited.\nI found the theory description to be very clear and useful. I would have put it earlier since to me the steady state description in the modified Bloch equations provides a concise description of the dynamics, but I am not suggesting any changes.\nAuthors’ response: We appreciate finding the theory section useful.\nChanges made to the manuscript: none.\nFinally, a tiny point that I have never understood what precisely is meant by a “piercing solenoid.” I don’t find the description in their ref 33 to be clear. Is it just a solenoid that goes through the magnetic shield, or ...? Might the authors take the opportunity to clarify this?\nAuthors’ response: “Piercing” means that a solenoid goes through the magnetic shield. Your understanding is exactly right and we explain it better in the text now. Such solenoid is specifically designed to generate a magnetic field at the sample position while producing (ideally) no field at the OPM sensor. We have also updated the discussion in Section 4.4 to include the algorithm used to mitigate the effect of any small residual field that may leak to the OPM.\nAuthors’ response: more likely, the reviewer meant the works of Brum (not Blum) and we cited the following works (Refs. [16-17]).\nThe authors claim that the J-oscillator described here has “exhibits superior long-term frequency stability .. in comparison to conventional NMR rasers”. That might be true but I don’t see that the evidence for “improved long-term frequency stability” anywhere in the reported results.\nAuthors’ response: We thank the reviewer for this insightful comment, and we have now clarified the relevant points in the revised manuscript.\nIn contrast to high-field NMR rasers, whose self-oscillation frequencies depend on nuclear Zeeman transitions and are therefore highly sensitive to the stability of the external magnetic field (e.g., temperature-driven drifts of the magnet), the J-oscillators operate on molecular J-coupling constants, which are intrinsic molecular properties and therefore exhibit much better frequency stability. Compared to Fig. 1H, high-field rasers show substantially greater frequency fluctuations under much shorter monitoring durations (see Fig. 2 and 3 in [40] Fleischer et al., Approaching the Ultimate Limit in Measurement Precision with RASER NMR, Appl. Magn. Reson., 2023).\nSince the oscillator is a combination of electronics and nuclear spin dynamics, one would expect the electronic parameters, performance, and noise figure, to play a role both in the oscillation frequency and also its stability. But those possible issues do not seem to be considered at all.\nAuthors’ response: the feedback gain and delay parameters indeed shift the J-oscillation frequency albeit in a predictable manner. The two parameters are digitally controlled, ensuring high robustness against analog noise in the feedback loop.\nThe main bottleneck of the performance of the J-oscillator is the piercing solenoid leakage field which couples the amplifier noise into the feedback loop. We have updated Section 4.4 to discuss and address this issue and revised description of the Algorithm 1 to include the mitigation procedure. This has been experimentally verified — e.g., Figs. A5 and A6 demonstrate that the OPM noise floor remains unchanged with increasing feedback gain. Other problems like additional signal propagation delays in the feedback loop are also addressed in Section 4.4.\nFinally, the stability and reproducibility of the J-oscillator have been systematically tested using three independently prepared, identical samples of naturally abundant acetonitrile. Each sample was measured repeatedly (11 runs per sample), and in every case the J-oscillator frequency reproducibly tracked the gradual increase of the J-coupling constant with much smaller fluctuations compared to the corresponding free-decay (FD) measurements (see Fig. 2D–2F). The small systematic offset between the J-oscillation and the J-coupling value extracted the FD measurements is stable and reproducible across all runs and samples, confirming the high long-term frequency stability of the J-oscillator. A detailed theoretical analysis of this offset will be presented in future work.\nFigure 4 is particularly instructive. It shows the Fourier transform of the RASER signal for mixtures of compounds as a function of electronic feedback gain. The frequencies shift with a change in feedback gain, which reinforces the point made above. In the accompanying text, the authors write that “In the sample that contained 100 % [15N]-pyridine and 50 % 4-amino-[15N]-pyridine, the right peak is about twice as strong as the left one, giving the expected 2:1 ratio that reflects upon its isotopic composition”. After some trouble I can identify the spectra in Fig.4 that the authors refer to, and I can also see that the peak amplitudes are roughly in the indicated ratio - although there is no quantitation given, and this ratio looks as if it shifts around depending on the feedback gain (glossed over by the authors). The authors then say “The opposite 1:2 intensity ratio is observed when the enrichment levels are swapped”. However, when I try to find the corresponding spectra “when the enrichment levels are swapped”, I do not see a swap of the 1:2 intensity ratio. Instead it looks to me as if one peak is simply absent in that case. Am I missing something or are the authors reporting what they would like to see, rather than what is actually observed? None of this inspires confidence for the putative analytical applications advanced by the authors.\nAuthors’ response: we now make the text clearer and provide a reference to the SI (Fig. A16) where we indeed see the opposite ratio with larger applied gains.\nThe authors report that in one case the Fourier transform of the continuous oscillation gives a linewidth of 337 microHz, whereas the ordinary linewidth without feedback, and limited by spin relaxation, is 37 mHz. From this they conclude that the resolution has been enhanced. However, I would like to know how reproducible is the oscillation frequency? If the oscillations are restarted, is the same frequency observed, within a standard deviation of 337 microHz? Or does it jitter within a standard deviation of 37 mHz (which is what I would expect)? Such statistics are not reported. Nevertheless statements are given which suggest that authors expect applications based on a greatly improved frequency resolution. I do not think that such statements are warranted based on the data presented so far.\nAuthors’ response: We thank the reviewer for raising this question. We have now clarified this point and added relevant discussion in the revised manuscript (Section 2.2). Unlike systems whose linewidth is dominated by spontaneous emission (the Schawlow–Townes limit), the J-oscillator frequencies show excellent reproducibility upon repeated restarts. As shown in now Fig. 2D–2F, the frequency offset between the J-oscillator and the value measured by a conventional zero-field NMR is also stable and reproducible across repeated runs and independently prepared samples. When comparing the same n-th run among three identical acetonitrile samples, the measured oscillation frequencies agree within 300 µHz, confirming that the frequency reproducibility is comparable to the observed linewidth and far better than the 32-37 mHz linewidths (naturally abundant sample has a slightly narrower linewidth) of the free-decay signal.\nThe conclusion section on page 12 reads more-or-less like a marketing document. The future is rosy. There is no discussion of limitations, issues that need resolving, or possible obstacles. In short, the presentation is quite uninformative and unscientific.\nAuthors’ response: We now deleted the conclusions section as per the journal’s requirement but clearly highlighted open questions in each subsection.\nChanges made to the manuscript: Multiple changes in the text to discuss limitations/open questions.\nIn summary:\n* What are the noteworthy results? ->\ndemonstration of feedback oscillations for nuclear spins in very low magnetic field\n* Will the work be of significance to the field and related fields? How does it compare to the established literature? If the work is not original, please provide relevant references. ->\nThe feedback device and observations are original and interesting. I have doubts that the observations are significant, but that is a very subjective view.\n* Does the work support the conclusions and claims, or is additional evidence needed? ->\nIn my view the work does not support some of the claims of increased stability, resolution, etc.\nAuthors’ response: Updated manuscript clearly supports claims made about stability and resolution. Additionally, we clearly demonstrate that the ratio of the observed J-oscillator peak intensities in the case of molecular mixture corresponds to the proportion of chemicals (Fig. 5 and Fig. A16).\n* Are there any flaws in the data analysis, interpretation and conclusions? Do these prohibit publication or require revision? ->\nThere are some flaws in the conclusions and discussion of the results, as detailed above.\nAuthors’ response: See above.\n* Is the methodology sound? Does the work meet the expected standards in your field? ->\nthe experimental methodology is sound and meets high standards. However some of the claims made are unsupported and the presentation is uncritical and unscientific in some places\nAuthors’ response: Updated manuscript fully supports claims that have been made.\nIs there enough detail provided in the methods for the work to be reproduced? -> yes, I think so."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed my previous concerns and made changes to the manuscript accordingly. The current version is noticeably written with improved clarity. I have no further comments and recommend accepting this manuscript.", "reviewer_3": "The resubmitted manuscript is more detailed and has less marketing-style hype and more rigorous content than the original submission. However, in my view, the improved content has made it more clear that, although the described experiments are fascinating and are of much interest, some of the claimed achievements are not supported by the evidence. In particular, the abstract makes two claims, of relevance to analytical applications:\n(1) [The advance] “may facilitate precision measurements of J-coupling constants” and\n(2) “allows distinguishing mixtures of molecules whose zero-field NMR spectra would otherwise be hard to separate.”\nWith respect to (1), more detail on the stability and reproducibility of the oscillations is now given in figure 2. This study does support their contention of good reproducibility. However their abstract claim of “precision measurement of J-coupling constants” is undermined by their admission that “A systematic milli-hertz-level offset in the J-oscillation frequency [..] was observed [..] A detailed theoretical analysis of this offset will be presented elsewhere.” This unexplained frequency deviation is evident in the figure. In other words, the authors have no idea, and no evidence, as to whether the observed oscillation frequency, however stable it is, does provide an authentic measurement of the J-coupling, or not. The authors cite some theory of the observed frequency deviation which “will be presented elsewhere”. In the absence of this, there is no evidence at all that the device could be used for highly precise J-coupling measurements.\nWith respect to (2), I do not see any evidence in the manuscript of distinguishing mixtures of molecules, in cases where the resolution of peaks is not already achieved by conventional means. Indeed, it looks to me as if the introduced feedback simply magnifies the largest of the already resolved peaks - which is exactly as one would expect by non-linear feedback amplification. The authors seem to have believed their own hype in claiming that the observation of one feedback-narrowed peak translates into a real and useful gain in spectral resolution - which would require that unresolved peaks become resolved somehow. That does not happen.\nI think that the authors should step back completely from their overhyped claims of enhanced useful spectral resolution. I think that the work is of sufficient wide interest to publish in Nature Communications, but only if the unrealistic claims of possible analytical applications are stripped out or highly qualified."}, "author_response": "The authors’ response: As discussed in the manuscript, frequency of the J-oscillators is determined by several factors. It depends on the parameters of the feedback loop as well as intrinsic spin-dynamical properties of the sample, including J-couplings and the relaxation time scales of the relevant transitions. Crucially, the developed theory shows (to be published elsewhere), and numerical simulations confirm, that the oscillation frequency is independent of the feedback gain and, therefore, independent of the signal intensity or polarization level of the sample, see the Supplementary Information for review only. A simple calibration procedure allows the J-coupling constant to be extracted from measurements performed at several feedback delays.\nOur intention is not to withhold this theory, but to present it in a dedicated publication where the full derivation and validation will be given. To avoid overstating the present results, we therefore adopted cautious wording in the abstract (e.g., “This MAY facilitate precision measurements of J-coupling constants …”) when referring to precision measurements.\nChanges made to the manuscript: None.\nWith respect to (2), I do not see any evidence in the manuscript of distinguishing mixtures of molecules, in cases where the resolution of peaks is not already achieved by conventional means. Indeed, it looks to me as if the introduced feedback simply magnifies the largest of the already resolved peaks - which is exactly as one would expect by non-linear feedback amplification. The authors seem to have believed their own hype in claiming that the observation of one feedback-narrowed peak translates into a real and useful gain in spectral resolution - which would require that unresolved peaks become resolved somehow. That does not happen.\nI think that the authors should step back completely from their overhyped claims of enhanced useful spectral resolution. I think that the work is of sufficient wide interest to publish in Nature Communications, but only if the unrealistic claims of possible analytical applications are stripped out or highly qualified.\nThe authors’ response: To clarify our claims regarding mixture analysis, we have revised Fig. 5 and Fig. A16 to include the corresponding conventional zero-field NMR spectra, shown as shaded traces for direct visual comparison. In both figures, the conventional spectra are broad, with overlapping features that do not provide sufficient resolution to distinguish the individual components of the mixture. In contrast, the J-oscillator method yields narrow, well-separated oscillation features whose identities can be assigned even when the zero-field spectra lack adequate clarity.\nWe have also added Fig. A17 which demonstrates that adjusting the feedback-delay parameter provides an additional degree of control: by selecting appropriate delays, the J-oscillation peaks associated with individual complexes can be selectively excited within the same sample. This tunability further assists in disentangling contributions from structurally similar molecules.\nChanges made to the manuscript:\n1. Fig. 5 and Fig. A16 have been updated to include the corresponding conventional zero-field NMR spectra for direct visual comparison.\n2. A new Fig. A17 has been added to demonstrate how adjusting the feedback-delay parameter provides additional control for mixture analysis.\n3. The relevant sentence in the manuscript has been revised to read:\n“The results suggest that the J-oscillator method can assist in analyzing complex mixtures containing structurally similar molecules, particularly in situations where conventional zero-field NMR does not provide sufficient resolution to distinguish signals [24].”"}, {"round": 3, "reviewer_comments": {"reviewer_3": "This second resubmission has further improved the manuscript. It is now clearer and the supporting evidence for the claims made is now clearly laid out. I am still not completely convinced myself by the claims of enhanced resolution, etc., but the main point is that the arguments are now made clearly, without reference to material that is not presented, and free from unsupported marketing claims, and with supporting evidence in plain sight, so that the scientific community is able to debate the issues and challenge the conclusions, if necessary. I judge that these criteria have now been fulfilled and that the paper should be published.\nThe only very small change I suggest is that when the figure captions states that a spectrum “has been scaled by a factor of 10” the wording should be changed to “has been scaled up by a factor of 10” (assuming that this is the correct interpretation)."}, "author_response": "The authors’ response: We thank the reviewer for their positive reassessment.\nThe authors’ response: We have revised all relevant figure captions to read “scaled up by a factor of 10”."}]

Decision letter