All-metallic magnetic Purcell enhancement in a thermally stable room-temperature maser

Domain: physics
Published: 2025-12-10
Editorial outcome: major_revision

Abstract
Enhancing the spontaneous and stimulated emission rates of magnetic quantum emitters through the Purcell effect is essential for designing high-performance quantum devices based on high quality factor and small mode volume. At room temperature, structures constructed using high-index dielectric materials have been favored due to their ability to effectively confine electromagnetic fields. However, these dielectric resonators are plagued by an inherent sensitivity to thermal variations, which are unavoidable during the optical excitation or readout of the quantum states. Here, we propose to solve this issue with a dielectric-free, all-metallic toroidal split-ring resonator cluster of subwavelength size (≤λm/17), which exhibits a fundamental magnetic mode at approximately 1.45 GHz (λm ~ 207 mm) and a remarkably low mode volume (8.1×10−6λm3). Through experimental investigations, we demonstrate that the proposed resonator exhibits a high Purcell factor (5×106), and observe maser action when paired with a pentacene-based gain medium. We evidence the remarkable stability of the output pulse against thermal variations caused by thousands of consecutive optical excitations, by far surpassing that of masers based on dielectric resonators.

Introduction
Spin-based magnetic transitions have recently attracted a significant amount of attention in the context of quantum information and sensing technologies1–3, due to their potentially long coherence lifetimes, their ability to be tuned using external magnetic fields, and the possibility to optically control level population using spin-selective transitions, even at room temperature. Despite their intrinsic weakness in natural materials4, magnetic transitions can be obtained in particular systems, including rare-earth ions5, quantum dots6, nitrogen-vacancy (NV) centers in diamond7 and organic molecules like pentacene8. Controlling the emission rate of such spin-based magnetic transitions is thus pivotal for a wide range of applications, including quantum computing9,10, quantum sensing11,12, electron-spin resonance spectroscopy13, and active emission devices14.

The Purcell effect, which describes the fundamental back-action of the environment on the dynamics of quantum emitters, is a key phenomenon to control emission rates in quantum electrodynamics15. As shown by Purcell in his seminal paper16, the spontaneous emission rate of quantum emitters can be enhanced when placed in a cavity with a high quality factor (Q factor) and a small mode volume (Vm). This enhancement, characterized by the Purcell factor (Fp = 3λm3Q / 4π2Vm), equivalently contributes to the stimulated emission process17. However, as spin transition frequencies are commonly in the gigahertz range, i.e., with emission wavelengths (λm) on the order of tens of centimeters, leveraging the magnetic Purcell effect comes with the inherent challenge of dealing with the large ratio between λm3 and achievable values of the mode volume Vm. In practice, the physical dimensions of gain media that can be grown experimentally, such as organic crystals or diamond-based samples, occupy physical volumes that are roughly three orders of magnitude smaller than λm3 (i.e., ∝ cm3). To address this mismatch, the resonator geometry must be engineered such that the region of strong magnetic field is confined to a volume only slightly larger than that of the gain medium, with its orientation optimized for alignment with the emitters’ magnetic dipole moments. This squeezing of the magnetic mode maximizes spatial overlap with the active medium, thereby enhancing the efficiency of energy exchange.

To circumvent this issue, many studies have instead focused on increasing the quality factor Q, for example, by using superconducting materials13,18, at the cost of operating under cryogenic conditions. At room temperature, small mode volumes and high Q have been achieved by employing dielectrics with very high permittivities and/or low dielectric losses, such as strontium titanate (SrTiO3)19 and sapphire20,21. These approaches have enabled a substantial enhancement of magnetic stimulated emission, sufficient to achieve maser action. However, dielectric resonators with extremely high permittivities, such as SrTiO3, are inherently sensitive to temperature variations due to their large temperature coefficient of permittivity. Even minor room-temperature variations, or the unavoidable thermal heating coming from optical excitation or readout22, would inevitably shift the mode frequency away from the quantum emitter linewidth. As a result, the system must be continuously retuned, hindering stable operation and limiting the practicality of such dielectric resonators.

In this study, we fix this vexing issue and demonstrate that room-temperature magnetic quantum emission enhancement can be stable over hours in a metamaterial-inspired, all-metallic magnetic resonator with a remarkably small mode volume, five orders of magnitude smaller than λm3. This is obtained with a specially designed multiple three-dimensional split-ring resonator (SRR) cluster, monolithically manufactured by lost-wax metal casting, which is an indirect additive manufacturing technique. We provide direct evidence that this resonator significantly boosts the stimulated emission rate of magnetic quantum emitters, by constructing a room-temperature pentacene maser that is completely free of high-index dielectric materials. We demonstrate maser operation stable in frequency and in output pulse shape over thousands of seconds, which is not achievable in prior dielectric-based designs.

Results
Ultra-strong magnetic Purcell-enhanced stimulated emission
Planar split-ring resonators have been widely used as metal-based resonant magnetic scatterers in metamaterials and metasurfaces23, where they provide a compact way for strong subwavelength magnetic interactions and dipolar emission control24. We have devised here a complex three-dimensional cluster of such resonators comprising four concentric split rings (see the inset in Fig. 1a), whose geometry has been optimized through full-wave finite-element simulations to yield a strong spatial confinement of magnetic energy in its middle region. The SRR cluster is shaped as a torus with an inner radius of 1.52 mm, outer radius of 6.02 mm and a height of 4.5 mm (other geometrical details are shown in Supplementary Fig. S1). To facilitate monolithic fabrication and avoid post-fabrication alignment steps, small metallic connections were added to link adjacent split rings within the cluster, with diameters of 1 mm. These interconnections have been located at adequate positions, so that they do not significantly influence the magnetic resonance. Additionally, the outermost ring is securely fixed to a circular stand via two thin posts having 1-mm diameter. This all-metal monolithic SRR cluster has a fundamental magnetic resonant mode at approximately 1.45 GHz, with a high concentration of magnetic energy at its core (1.5 mm radius × 4.5 mm height), and a magnetic field mostly oriented along the z-axis, as shown in Fig. 1a (the electric field distribution is shown in Supplementary Fig. S2). Assuming that within this region, we align uniformly an ensemble of quantum magnetic emitters along the z-axis, we would get an effective magnetic mode volume Vm = ∫ |H(r)|2 dV / |Hmax(r)|2 of only 0.025 cm3. Remarkably, this estimation is almost 350 thousand times smaller than λm3. Such a resonator has an unloaded Q factor (Qunload) around 700, assuming that it is fabricated in silver and enclosed within a copper cavity to reduce the radiation losses. The corresponding single-mode Purcell factor Fp = 3λm3 / 4π2Vm * Q is estimated to be around 1.89×107, which is of the same order of magnitude as the one reported using strontium titanate, a record-high dielectric with ϵr ~ 32025. Our all-metallic design is also significantly different from metallic structures discussed in other works26–29. In the following, we perform a direct magnetic quantum emission experiment to establish the relevance of our design.

In light of these promising simulation results, we have selected the pentacene-based pulsed maser20 as an ideal test platform, for which we can directly compare the performance obtained with our SRR cluster and with a high-index dielectric resonator. This particular maser is distinguished by its ability to achieve room-temperature masing without requiring an external magnetic field. The maser action predominantly results from the population inversion (N) established between the X and Z sublevels within the first triplet state of pentacene molecules under optical excitation (see Supplementary Fig. S3a)30,31. As depicted in Fig. 1b, the pentacene:p-terphenyl (pc:pt) crystal (1.5 mm radius × 10 mm height) is positioned within the bore of the clustered SRR, which is enclosed inside a copper cavity. The bottom 5 mm of the pc:pt crystal includes a central axial hole of 0.5 mm radius, whereas the top 5 mm remains solid. The resonant frequency of the cavity can be manually fine-tuned to approximately 1.4493 GHz using the top tuning plate. To efficiently pump the crystal, the laser pulses (wavelength at 592 nm, pulse duration ~ 5 ns, repetition rate 1 Hz) from an optical parametric oscillator (OPO, SLI-20 Surelite) are coupled into an optical fiber (core diameter 1 mm) with a coupling efficiency ~ 60%. The other end of the optical fiber protrudes through the cavity bottom and is inserted into the central hole of the crystal. The maser burst following each laser pulse is then captured using a weakly coupled magnetic loop probe and recorded by an oscilloscope (Tektronix MSO64-2.5G). A schematic of the complete experimental setup can be found in Supplementary Fig. S4.

By gradually increasing the pulse energy of the pump laser from 0.8 mJ to 6 mJ, we recorded a total of 1300 maser pulses and analyzed them using a Lotka–Volterra model (L-V model, see “Methods”)32,33. A collection of the peak power in all maser pulses as a function of the laser pulse energy indicates that there is an optical pumping threshold around 2.3 mJ, below which no maser emission can be detected (Fig. 2a). The slight fluctuations in the maser output are a consequence of fluctuations in OPO output beams (see Supplementary Fig. S5). Two maser pulses under excitation energy of 2.5 mJ and 5.9 mJ are given as instances in Fig. 2b and c, respectively. When the excitation energy slightly exceeds the threshold, the maser pulse possesses a single peak in time domain and can be accurately fitted by the L-V model, yielding an initial population inversion N0 ~ 5.2×1013. With pumping energy well above the threshold (N0 ~ 6.8×1013), we observe a weak oscillatory behavior in the maser pulse. This behavior could potentially arise from coherent energy exchange between the spin ensemble and the cavity mode34, or from collective emission phenomena such as superradiance35,36. Further experimental and theoretical investigations will be necessary to determine the precise underlying mechanism of these oscillations.

Excellent spectral robustness against thermal variations
To demonstrate the thermal stability of the proposed clustered SRR, we consecutively recorded the temporal responses of the maser to laser excitation over a period of five hours. The successive laser excitation, operating at a 1-Hz repetition rate with pulse energy around 3 mJ, induced thermal heating in the resonator and the pc:pt crystal, since a given amount of the optical energy is dissipated without contributing to the masing process. As a reference, we built a second maser using a standard hollow cylindrical strontium titanate (SrTiO3) resonator, which is similar in size to our clustered SRR and exhibits a comparable Purcell factor. The SrTiO3 resonator possesses a high Purcell factor (Fp ~ 3.6×107), thanks to its high relative permittivity and low loss tangent (ϵr ~ 320, tan(δ) ≈ 9×10−5)19. The emission spectra of all the pulses, measured under the same conditions from the clustered-SRR- and SrTiO3-based masers, are plotted in Fig. 3a, b, respectively. It is evident that the spectra of the clustered-SRR-based masers exhibit a superior stability in both signal strength and resonant frequency over at least 18,000 shots. This excellent thermal stability is attributed to silver’s high thermal conductivity and moderate thermal expansion coefficient, resulting in a temperature coefficient of resonant frequency of approximately 130 ppm/K. In contrast, the central frequency of the SrTiO3-based maser shifts from 1.449 GHz to 1.453 GHz, resulting from SrTiO3’s large temperature coefficient of resonant frequency (approximately 1700 ppm/K)25,37. We simulated the diffusion of the deposited optical pump energy and the resulting temperature variations in the SrTiO3 resonator using COMSOL. The simulation results are presented in the Supplementary Movie 1 and Fig. S6. More importantly, this frequency shift is accompanied by a variation in the signal strength, which eventually vanishes because of the limited gain linewidth of pentacene molecules (see Supplementary Fig. S3b). We terminated the measurement in the SrTiO3-based maser after 12,000 shots as no maser signal was detected during the final 200 s. After switching off the pump laser, the resonant frequency of the SrTiO3 resonator took approximately two hours to recover back to its initial value (see Supplementary Fig. S7). Apart from the self-termination, another side effect caused by the frequency shift is the transition in the maser operation mechanism: it enters the strong-coupling regime featuring a oscillatory response, and later returns to the Purcell-enhanced regime (see Fig. 3d).

Methods
Design and fabrication of the clustered SRR
The resonator was designed using eigenmode and full-wave frequency-domain solvers in CST Studio Suite for obtaining the optimal magnetic Purcell factor. This design trajectory is concisely summarized in Supplementary Fig. S8. The optimized toroidal SRR cluster was monolithically printed in sterling silver using wax 3D printing and lost-wax metal casting (3D Print India). The 3D model of the clustered SRR is available on the repository Zenodo (see “Data availability”) for precise reproduction of the structure. No post-fabrication alignment was needed thanks to its monolithic structure. The printed SRR cluster was electroplated with 15–20 μm thick silver (Steiger galvanotechnique SA) to reduce the ohmic losses.

Pentacene:p-terphenyl crystal growth
The pentacene:p-terphenyl (concentration around 0.1% mol/mol) is grown using the vertical Bridgman method. Commercially purchased p-terphenyl (A14833, Alfa Aesar) is purified by sublimation (under 10−5 mbar, 150 °C) and then mixed with pentacene (Sigma-Aldrich, 98% purity). After creating a central hole through the crystal, a pyrex rod (diameter 1 mm) is inserted after loading the mixed powders into a doubled-layer Pyrex ampoule (inner diameter 3 mm, Tecglas), which is then sealed under vacuum (≤10−6 mbar). The ampoule has a tapered capillary at the end for self-seeding during the growth process. An in-house Bridgman furnace, which consists of a three-zone tube furnace (EVC 12/450B, 1200 °C, Carbolite Gero) and a vertical motion control system, is used to grow the crystal. The sealed ampoule is suspended in the hot top zone (Tmax ~ 240 °C) for at least 12 h to ensure that the chemicals melt homogeneously. Then, the ampoule is descended towards the colder bottom zone (Tmin ~ 80 °C) at a speed of approximately 0.53 mm/h over 14 days. Thereafter, the motor is stopped and the furnace is gradually cooled down to room temperature to prevent thermal shocks. The as-grown crystal is cut using a diamond saw and gently removed from the Pyrex tubing and the inner rod.

Maser dynamics modelling
To predict and analyze the maser dynamics, we utilize a pair of coupled first-order nonlinear differential equations, namely the Lotka-Volterra model:

Ṅ = −2BNq − γN (1)

q̇ = −kcq + BNq (2)

where N and q are the numbers of population inversion and microwave photons in the cavity. γ is the spin-lattice relaxation rate and kc is the photon decay rate in the cavity, which can be calculated from the loaded Q factor as kc = ω/Qload. B stands for the Einstein coefficient of the stimulated emission, which is directly linked to the resonator’s mode volume as follows32:

B = μ0ℏωγe2T2 / 4Vm (3)

where μ0 is the vacuum permeability, ℏ the reduced Planck constant, γe = 1.76×1011 rad · s−1 · T−1 the electron gyromagnetic ratio, and T2 = 2.1 μs the spin-spin relaxation time of pentacene. Spontaneous emission is omitted from our dynamical model because, at microwave frequencies, its rate is vanishingly small compared to stimulated emission.

For the clustered SRR resonating at ω = 1.4493 GHz · 2π, Qload of the maser, including the pc:pt crystal is determined to be 530 from the S11 parameter measured using a 2-port vector network analyzer (Rohde & Schwarz, 100 kHz ~ 20 GHz) without laser excitation, resulting in kc = 17.2×106 s−1. At room temperature T, the initial value of q equals the thermal photon number in the cavity without optical excitation, which can be calculated by the Bose-Einstein occupation function q0 = (ekBω/kBT − 1)−1 ≈ 4300, where kB is the Boltzmann constant. Given proper values for B, γ, and initial value of N, the coupled equations can be integrated in time to reveal the maser dynamics. From the measured maser pulse power P(t), the microwave photon number as a function of time is given by q(t) = P(t)(1+κ)/ℏωkcκ, where κ = 0.3 is the coupling coefficient between the magnetic probe and the cavity. Using the differential evolution algorithm40, the best agreement between the experimental data and the model is achieved with γ = 1.4±0.4×104 s−1, B = 38.7±2.9×10−8 s−1 while varying N0 from 4.5×1013 to 7.3×1013. We note that the spin-lattice relaxation rate obtained here is slightly higher than the reported value (1.1±0.2×104 s−1)41. This difference might stem from the fact that spin decays from the triplet state back to the singlet ground state is not included in the L-V model. Assuming steady-state operation (q̇ = 0), the threshold population inversion is given by: Nthreshold = kc/B = 4.4×1013. Since the Purcell factor is proportional to 1/Vm, the clustered SRR provides a large Fp (i.e., a large B) that is key to achieving low-threshold, efficient maser operation.

Data availability
The raw experimental data and the 3D model of the clustered SRR are available on the repository Zenodo under the https://doi.org/10.5281/zenodo.10012323.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors present a novel all-metallic resonator with high Purcell factor that has improved temperature stability compared with the strontium titanate dielectric resonators used in previously published pentacene maser work.\nIt is well known that strontium titanate has a very high temperature coefficient of permittivity which leads to shifts in the resonator frequency due to temperature changes. The sources of temperature change in the maser could be environmental but would be dominated by the energy deposited in the pentacene-doped gain medium by the optical pump pulses.\nThe main result of this manuscript is the temperature stability of the maser. However, there is little or no reference to the sensitivity of a the maser to temperature changes, i.e why does the resonant frequency change? Neither the coefficients of thermal expansion of materials, temperature coefficient of permittivity (for the strontium titanate case) nor the temperature coefficient of the zero-field splitting parameters of pentacene are mentioned. Some analysis here would be helpful, by estimating or simulating, by taking the thermal conductivity of the materials into account, how the deposited optical pump energy diffuses through the device. On this note, I noticed that the authors used a polystyrene support material for the strontium titanate dielectric resonator, whereas earlier work used sapphire. These two materials have thermal conductivities that are four orders of magnitude different. Using polystyrene as a support would magnify the effects of the optical pump energy dissipation and so the comparison between strontium titanate and metal may be a little unfair if polystyrene is used.\nIn any case, wouldn't it be trivial, under periodic optical pump pulses, to adjust the resonator frequency to be resonant with the spin transition once a steady state temperature has been reached?\nAlthough the resonator design is interesting, the work reported here is mostly a reproduction of previous work. For this reason I am afraid that I do not think the result is significant enough for publication in Nature Communications.", "reviewer_2": "In the work titled \"All-metallic magnetic Purcell enhancement in a thermally stable room-temperature maser,\" the authors present an innovative approach to enhancing the spontaneous and stimulated emission rates of magnetic quantum emitters through the Purcell effect. This study introduces a dielectric-free, all-metallic toroidal split-ring resonator cluster, demonstrating high Purcell factors and maser action at room temperature. The work contains interesting results that could significantly impact the design of high-performance quantum devices. However, before I can recommend it for publication, the work needs revision on the following issues:\n1. The main concern is the lack of discussion on why the authors need a large Purcell factor for the generation. The role of the Purcell effect in lasers and masers has been debated for many years. Small mode volumes help with the fast exchange of energy between the active media and the resonator. However, enhancing the Purcell effect also enhances all spontaneous effects, including spontaneous emission, leading to an increase in noise. The authors should discuss this and explain why they need a high Purcell factor. Is this useful to boost the effect further? How does the Purcell factor fit into the Lotka-Volterra model? Without this discussion and analysis, the work remains vague.\n2. The authors state in the abstract that “At room temperature, structures constructed using high-index dielectric materials have been favored due to their low absorption losses,” but this is not true for all spectral ranges. For example, at microwave frequencies, metallic structures are usually more favorable due to their vanishing skin depth.\n3. In the abstract, they first say “which are unavoidable during the optical excitation or read-out of the quantum states,” definitely meaning the optical range. However, they then introduce a resonator with a fundamental magnetic mode at 1.45 GHz. There are other examples of this discrepancy. I recommend the authors revise the text to make it more straightforward and concrete.\n4. A similar comment applies to terminology like “the magnetic mode.” What is a magnetic mode? A mode of a magnetic field? Likely, the authors mean something different because they know that an oscillating magnetic field cannot exist without a curl electric field. Terms like magnetic mode and magnetic Purcell should be either revised or explained upon introduction. Additionally, what is “extreme” in “such extreme dielectrics” like sapphire?\n5. The authors formulate the problem as “as spin transition frequencies are commonly in the gigahertz range, i.e., with emission wavelengths (λm) of the order of tens of centimeters, leveraging the magnetic Purcell effect comes with the inherent challenge of dealing with the large ratio between λ^3 and achievable values of mode volume Vm.” First, the concept of the magnetic Purcell effect appears abruptly here without explanation. Second, and more importantly, the problem is not the ratio of the wavelength to the resonator dimension but the fact that in this range, the mode volume is typically close to λ^3.\n6. The authors make this statement: “Indeed, since the latter represents the effective cavity volume over which the magnetic dipoles are aligned with the magnetic field of the mode, it is drastically limited by our ability to squeeze the magnetic mode into a volume sufficiently small to match the one occupied by the emitters,” without explanation. What do the authors mean by “squeeze the magnetic mode”? Take, for example, a piece of coaxial cable. Coax has no cut-off frequency, and its section can be as small as needed. Would such a coaxial resonator create a strong Purcell effect? Why?\n7. I disagree with the statement “At room temperature, small mode volumes and high Q have been achieved by employing dielectrics with very high permittivities” in the context of the importance of the Q-factor. First, metallic high-Q resonators at microwaves are also known. Second, high-index dielectrics not only have high Q-factors but also “squeeze” the wavelength and hence the mode volume.\n8. I cannot fully agree with “However, such extreme dielectrics are inherently plagued by their utmost sensitivity to thermal fluctuations, making them excessively prone to detuning.” What thermal fluctuations are the authors talking about? These have different primary sources of noise. The thermal noise in dielectrics is usually due to dipole relaxation processes and phonon interactions. Metals have free electrons that contribute to Johnson-Nyquist noise. I would say that dielectric resonators will typically have lower thermal fluctuations compared to metallic resonators in the microwave range. The authors have to explain what they mean here and probably support their claims with estimations and/or references.\n9. The statement “Very small room temperature fluctuations, or the unavoidable thermal heating coming from optical excitation or read-out, would inevitably shift the mode frequency away from the quantum emitter linewidth” is also vague. What regime do the authors mean here, good cavity (Qcav >> Qemit) or bad cavity (Qcav << Qemit)? If it is a good cavity, which follows from the narrative, then it is less important what happens to the cavity, as any noise from and fluctuations of the emitter will be more important. Could the authors elaborate on this?\n10. When the authors make statements about the maser stability like “We demonstrate maser operation stable in frequency and in output pulse shape over thousands of seconds, which is not achievable in prior dielectric-based designs,” they should accompany them with references and examples of the state of the art.\n11. The authors should elaborate more on the design of the SRR cluster. Why this geometry? Why does it provide strong confinement of the magnetic energy? Is this number of “almost 350 thousand times smaller than λ^3” theoretically limited? If yes, by what means? If not, would it make sense to boost the magnetic field more?\n12. Did you take into account the realistic loss and skin depth effects in simulations? Does the squeezing of the magnetic (and I believe, electric) field lead to a proportional increase in losses and Johnson-Nyquist noise?\n13. Can the oscillations in Fig.2c be due to other mechanisms beyond the Rabi oscillations? The argument “The Rabi oscillation is also confirmed by the broadened spectrum of the pulse” doesn’t work because any shortening of the pulses in the time domain leads to broadening in the frequency domain. Can this effect, for instance, be caused by collective effects like Dicke or other superradiance?", "reviewer_3": "Review of NCOMMS-23-52768\nThe paper describes a specially designed metallic resonator employed in conjunction with a pentacene-based optically pumped room temperature pulsed maser to facilitate frequency-stable operation with minimal heating effects. The authors show that such heating effects are detrimental in a similar device based on a dielectric resonator made of SrTiO3 single crystal, leading to frequency drifts and the eventual loss of maser signal after approximately 1-2 hours. The main novelty claim of this paper is the use of a unique 3D metallic split-ring resonator with a small mode volume and a relatively large quality factor. This results in a resonator with a relatively high Purcell factor (which is approximately \\(\\lambda^3 \\cdot Q/\\text{mode volume}\\)) that is imperative for facilitating the operation of the maser device.\nI found this paper interesting and informative, but I fail to see significant novelty or impact. My main concern is that the use of small metallic volume resonators is known in the field of electron spin resonance, which seems to undermine the main novel point of the paper. Recent examples of resonators with similar or even smaller mode volumes (in units of \\(\\lambda^3\\)) can be found here: [arXiv:2307.11269](https://arxiv.org/abs/2307.11269) and references therein. Additionally, there is the possibility of using metallic loop-gap resonators of similar or smaller size in conjunction with highly temperature-stable dielectric materials, as discussed in the following references:\n- [NCBI](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6948142/)\n- [Patent](https://patentimages.storage.googleapis.com/77/0b/52/097e05c3bd3d96/WO2013175235A1.pdf)\n- [Wiley](https://onlinelibrary.wiley.com/doi/abs/10.1002/mrm.28595)\nWhile these designs do not necessarily have the exact properties of the resonator presented in the current work, they achieve similar high Purcell factors and use either all-metallic or dielectric materials with negligible temperature dependence of permittivity.\nMoreover, comparing the temperature stability of metallic structures to SrTiO3 seems unfair, as the latter has permittivity drifts in the range of approximately 1% per degree. The same type of maser was realized in the past with a sapphire-based structure that is far more temperature stable.\nAnother major concern is that the paper discusses the use of a pulsed maser device with very limited practical use, if at all. What would happen if one wishes to use a similar resonator for continuous-wave operation, such as a maser based on NV centers in diamond? At some point, the metal itself may heat up and expand.\nLastly, even with SrTiO3, the authors show that the drift is very slow. Is it worth having a worse resonator design (with a lower Purcell factor than SrTiO3) but better temperature stability? Possibly, it would be better to automatically tune back the SrTiO3 resonator by changing the metallic cup on it. What I am trying to say is that the fundamental problem (heating) which the authors claim to exist may have other solutions that would keep the original design with the high Purcell factor. And on a side note, I am not sure this problem is real if one uses more temperature-stable dielectrics.\nOther minor remarks:\n1. The Introduction discusses the Purcell enhancement factor in relation to increasing the spontaneous emission rate. However, the experimental results do not show any change in this rate compared to the usual T1 of the sample (as shown, for example, [here](https://www.nature.com/articles/nature16944)). In my opinion, it would have been better to describe the requirements for achieving maser action through the cooperativity factor. The discussion of changing the spontaneous emission rate seems out of place. Indeed, the Purcell factor is part of the cooperativity factor, but this should be taken in the context of achieving maser threshold (having more gain than losses inside the cavity) and not connected to the spontaneous emission rate.\n2. Page 2 – the expected frequency variations of SrTiO3-based resonators: can you be more quantitative? For SrTiO3, it is clear that there would be relatively large changes, but for sapphire, this is a very small change in permittivity. What about the expected change in resonator frequency due to metal expansion?\n3. Mode volume should refer to the magnetic field in all directions (namely magnitude) in the denominator since it relates to total magnetic energy. What is \\(H_{z,\\text{max}}(r)\\) and how is it dependent on r?\n4. Figure 1 – can you plot the E-field as well?\n5. Page 4 – it should be mentioned clearly that this maser operates only in pulsed mode. What about masers that need a static field? The all-metallic design with its shield is larger than the dielectric design, implying a larger and bulkier magnet.\n6. Page 8 – what is the spin concentration of the sample?"}, "author_response": "Response letter, manuscript number NCOMMS-23-52768\nWe would like to thank all the reviewers for their time and careful review of our manuscript. In the\nfollowing, we provide a point-by-point response to each of the Reviewer's comments.\nReviewer 1:\nR1.0: The authors present a novel all-metallic resonator with high Purcell factor that has\nimproved temperature stability compared with the strontium titanate dielectric resonators\nused in previously published pentacene maser work.\nResponse: We thank the reviewer for his/her careful review and very useful comments on our work,\nthat greatly helped us to improve it and clarify some important points.\nR1.1: It is well known that strontium titanate has a very high temperature coefficient of\npermittivity which leads to shifts in the resonator frequency due to temperature changes. The\nsources of temperature change in the maser could be environmental but would be dominated\nby the energy deposited in the pentacene-doped gain medium by the optical pump pulses.\nResponse: We totally agree with the reviewer that thermal heating, caused by consecutive optical\nexcitation, could happen in the resonator and the pentacene doped gain medium. In our study, the\nmasers constructed using the all-metallic resonator and the SrTiO3 resonator were tested under the\nsame conditions (e.g., optical excitation path, laser pulse energy and repetition rate, etc.), and\ntherefore the heat deposited in the gain media is comparable. However, we don’t observe any\nfrequency shift with the all-metallic resonator, which indicates that the thermal effects that shift the\nfrequency of the X Z transition are negligible, and the maser constructed using theproposed\nresonator is indeed more thermally robust.\nR1.2: The main result of this manuscript is the temperature stability of the maser. However,\nthere is little or no reference to the sensitivity of the maser to temperature changes, i.e why\ndoes the resonant frequency change? Neither the coefficients of thermal expansion of\nmaterials, temperature coefficient of permittivity (for the strontium titanate case) nor the\ntemperature coefficient of the zero-field splitting parameters of pentacene are mentioned.\nSome analysis here would be helpful, by estimating or simulating, by taking the thermal\nconductivity of the materials into account, how the deposited optical pump energy diffuses\nthrough the device. On this note, I noticed that the authors used a polystyrene\nsupport material for the strontium titanate dielectric resonator, whereas earlier work used sapphire.\nThese two materials have thermal conductivities that are four orders of magnitude different.\n1 / 18\nUsing polystyrene as a support would magnify the effects of the optical pump energy\ndissipation and so the comparison between strontium titanate and metal may be a little unfair\nif polystyrene is used.\nResponse: We thank the reviewer for all the relevant remarks that have helped us improve the\nanalysis and presentation of our findings. We are happy to provide quantitative arguments regarding\nthermal sensitivity. The resonant frequency in the SrTiO -based maser changed mainly due to the\n3\nhigh temperature coefficient of the permittivity of SrTiO , which leads to a large temperature\n3\ncoefficient of frequency of approximately 1700 ppm/K1,2.\nAs for the all-metallic SRR cluster, one main potential reason of frequency drifting would be thermal\nexpansion due to thermal heating by the laser. Based on the volumetric thermal expansion coefficient\nof silver around 54×10!\" K!#, the temperature coefficient of resonant frequency of the all-metallic\nSRR cluster is estimated to be approximately 130 ppm/K.\nThe polystyrene support was utilized for both the all-metallic resonator and the SrTiO3 resonator,\nbut for different purposes. For the SrTiO3 resonator, the polystyrene support was employed to reduce\nohmic losses on the bottom copper plate, thereby improving the Q factor. In the case of the all-\nmetallic resonator, the polystyrene support was used solely to elevate the resonator for optimal\npositioning of the magnetic probe. Without this support, the side hole on the copper cylinder—used\nto insert the probe—would be too high for proper alignment with the all-metallic resonator. We\napologize for the lack of clarity on this point in the original manuscript. This information is now\nincluded in Supplementary Fig.S3 of the revised manuscript.\nTo study how the deposited optical pump energy diffuses through the SrTiO3 resonator with different\nsupport materials, we conducted simulations in COMSOL. In the experiment, a laser pulse delivered\nvia an optical fiber had a pulse duration of 5 ns, a repetition rate of 1 Hz, and a pulse energy of 3 mJ;\nthe same parameters were applied in the simulation. As shown in Video_R1.2.1-T vs time-sapphire\nsupport and Video_R1.2.2-T vs time-airfoam support, the laser was active for the first 5 ns of each\nsecond, heating the SrTiO3 resonator. The heat dissipated was concentrated in the region near the\noptical fiber's end. After the laser pulse ended, the accumulated heat gradually dissipated.\nFigure R1.2.1 illustrates the variation in maximum and average temperature of the SrTiO3 resonator\nover five seconds. The results demonstrate that the choice of support material did not result in\nsignificant differences in the temperature variation of the SrTiO3 resonator. While these materials\nmay cause some differences over longer experimental durations, the primary factor contributing to\nthe frequency shift of the SrTiO3 resonator remains its poor temperature coefficient of resonant\nfrequency.\n2 / 18\nFigure R1.2.1 Maximum (a) and average (b) temperature variation of the SrTiO3 resonator placed on sapphire\nsupport (blue solid line) and polystyrene foam support (red dotted line). The pump laser had a repetition rate of\n1Hz, pulse duration 5ns and pulse energy of 3 mJ.\nRevision: We have added the following text in the Results section:\n• This excellent thermal stability is attributed to silver's high thermal conductivity and moderate\nthermal expansion coefficient, resulting in a temperature coefficient of resonant frequency of\napproximately 130 ppm/K. In contrast, the central frequency of the SrTiO based maser shifts\n3\nfrom 1.449 GHz to 1.453 GHz, resulting from SrTiO ’s large temperature coefficient of\n3\nresonant frequency (approximately 1700 ppm/K). We simulated the diffusion of the deposited\noptical pump energy and the resulting temperature variations in the SrTiO resonator using\nWe have added Movie1 and Fig.S6 in the Supplementary materials.\nR1.3: In any case, wouldn't it be trivial, under periodic optical pump pulses, to adjust the\nresonator frequency to be resonant with the spin transition once a steady state temperature\nhas been reached?\nAlthough the resonator design is interesting, the work reported here is mostly a reproduction\nof previous work. For this reason, I am afraid that I do not think the result is significant enough\nfor publication in Nature Communications.\nResponse: The resonant frequency of the SrTiO resonator can be manually tuned using the\n3\nmovable top copper plate. However, achieving and maintaining resonance with precision would\nrequire frequent adjustments and physical access to the copper cavity, making this process\nmaintenance-intensive and potentially impractical for long-term experiments. Alternatively, while\nautomatic retuning with a feedback loop could improve accuracy, it would add substantial complexity\nto the system, impacting simplicity and ease of use. Furthermore, stable operation without frequency\ndrift is critical for certain applications, such as weak magnetic field sensing3.\nIn addition, it is worth noting that the thermal conductivity (𝜅) of metals, such as silver or copper, is\nmore than an order of magnitude higher than that of dielectric materials like SrTiO and Al O .\n3 2 3\n3 / 18\nAccording to Fourier’s law, the temperature increase Δ𝑇 of a material is inversely proportional to 𝜅.\nThis implies that, under comparable heat flux conditions, the steady state temperature of dielectric\nresonators would be higher than that of metallic ones of similar sizes and would be more susceptible\nto thermal fluctuations.\nWhile we recognize the foundational contributions of previous work, our study offers significant\nadvancement through the introduction of a fully metallic structure. This unique design achieves\nexceptional thermal stability and robust three-dimensional magnetic energy confinement at a deep\nsubwavelength scale, resulting in an efficient and maintenance-free platform.\n4 / 18\nReviewer 2:\nR2.0: In the work titled \"All-metallic magnetic Purcell enhancement in a thermally stable room-\ntemperature maser,\" the authors present an innovative approach to enhancing the\nspontaneous and stimulated emission rates of magnetic quantum emitters through the\nPurcell effect. This study introduces a dielectric-free, all-metallic toroidal split-ring resonator\ncluster, demonstrating high Purcell factors and maser action at room temperature. The work\ncontains interesting results that could significantly impact the design of high-performance\nquantum devices. However, before I can recommend it for publication, the work needs\nrevision on the following issues.\nResponse: We thank the reviewer for his/her careful review and very useful comments on our work,\nthat greatly helped us to improve it and clarify some important points.\nR2.1: The main concern is the lack of discussion on why the authors need a large Purcell\nfactor for the generation. The role of the Purcell effect in lasers and masers has been debated\nfor many years. Small mode volumes help with the fast exchange of energy between the active\nmedia and the resonator. However, enhancing the Purcell effect also enhances all\nspontaneous effects, including spontaneous emission, leading to an increase in noise. The\nauthors should discuss this and explain why they need a high Purcell factor. Is this useful to\nboost the effect further? How does the Purcell factor fit into the Lotka-Volterra model?\nWithout this discussion and analysis, the work remains vague.\nResponse: A high Purcell factor plays a pivotal role in achieving robust and efficient maser action at\nroom temperature. By enhancing the interaction between the active medium and the cavity mode, a\nlarge Purcell factor increases the stimulated emission rate and effectively reduces the pumping\nthreshold for maser operation4. While this does raise spontaneous emission and its associated noise,\nimproved efficiency, reduced threshold, and device compactness often outweigh the drawbacks.\nRegarding the Lotka-Volterra model, the Purcell factor indeed enters it directly. The mode volume\n𝑉 , which is relevant to the Purcell factor, can be related to the Einstein coefficient B as follows5:\n$ \nµ ℏωγ'𝑇\n% & ' \n𝐵 = \n4𝑉 \n$ \nWhere µ is the permeability of free space, ℏ is the reduced Planck constant, ω is the maser angular\n% \nfrequency, γ is the Gyromagnetic ratio of electrons and 𝑇 is the spin-spin relaxation time.\n& ' \n5 / 18\nR2.2: The authors state in the abstract that “At room temperature, structures constructed\nusing high-index dielectric materials have been favored due to their low absorption losses,”\nbut this is not true for all spectral ranges. For example, at microwave frequencies, metallic\nstructures are usually more favorable due to their vanishing skin depth.\nResponse: We thank the reviewer for this clarification, which we agree with. We will revise our\nmanuscript to provide a more accurate and nuanced perspective.\nRevision: We have revised the sentence in the abstract from:\n§ “At room temperature, structures constructed using high-index dielectric materials have been\nfavored due to their low absorption losses.”\nto:\n§ “At room temperature, structures constructed using high-index dielectric materials have been\nfavored due to their ability to effectively confine electromagnetic fields.”\nR2.3: In the abstract, they first say “which are unavoidable during the optical excitation or\nread-out of the quantum states,” definitely meaning the optical range. However, they then\nintroduce a resonator with a fundamental magnetic mode at 1.45 GHz. There are other\nexamples of this discrepancy. I recommend the authors revise the text to make it more\nstraightforward and concrete.\nResponse: We appreciate the reviewer’s observation and would like to clarify this point, which is not\na discrepancy. While the resonator itself operates at a microwave frequency (1.45 GHz), the\nunderlying quantum systems involved are often prepared, manipulated, and read out optically. For\nexample, in pentacene-doped p-terphenyl crystals and NV centers in diamond, optical pumping is\nnecessary for achieving population inversion and subsequently generating microwave emission.\nAdditionally, many quantum sensing protocols rely on optical excitation or detection of states whose\ncoherent transitions occur in the microwave regime6.\nWhen we mention “unavoidable thermal fluctuations during the optical excitation or read-out of the\nquantum states,” we are referring to the practical challenges introduced by these optical steps.\nAlthough the masing frequency is in the microwave domain, the necessary optical processes can\nstill introduce thermal loads and fluctuations that affect the system’s overall stability and noise\nperformance.\nR2.4: A similar comment applies to terminology like “the magnetic mode.” What is a magnetic\nmode? A mode of a magnetic field? Likely, the authors mean something different because\nthey know that an oscillating magnetic field cannot exist without a curl electric field. Terms\nlike magnetic mode and magnetic Purcell should be either revised or explained upon\nintroduction. Additionally, what is “extreme” in “such extreme dielectrics” like sapphire?\n6 / 18\nResponse: We apologize for the lack of clarity in our terminology, which we are happy to clarify. What\nwe call a magnetic mode in the context of our masers is an electromagnetic mode whose magnetic\nenergy density strongly dominates the electrical energy density over the volume of the gain medium\n(although as for any electromagnetic eigenmode, the total electric and magnetic energies, integrated\nover space, are equal). As our goal is to enhance the emission rate of magnetic dipole emitters, we\nhave chosen a resonator mode configuration that maximizes the local magnetic field component and\nits alignment to the magnetic quantum emitters. In this context, we refer to it as a “magnetic mode”\nto emphasize the strong magnetic field concentrated in the gain region and aligned with the emitters.\nRegarding the phrase “extreme dielectrics,” our intention was to describe materials like SrTiO₃, which\nhave exceptionally high dielectric constants (ϵ ≈ 320). To avoid confusion, we will rephrase\n(\n“extreme dielectrics” as “dielectrics with exceptionally high dielectric constants” in the revised\nmanuscript.\nRevision: We have revised the sentence in the introduction from:\n§ “However, such extreme dielectrics are inherently plagued by their utmost sensitivity to\nthermal fluctuations, making them excessively prone to detuning.”\nto:\n§ “However, dielectrics with extremely high permittivities are inherently sensitive to\nthermal fluctuations, making them excessively prone to detuning.”\nR2.5a: The authors formulate the problem as “as spin transition frequencies are commonly\nin the gigahertz range, i.e., with emission wavelengths (λm) of the order of tens of centimeters,\nleveraging the magnetic Purcell effect comes with the inherent challenge of dealing with the\nlarge ratio between λ^3 and achievable values of mode volume Vm.” First, the concept of the\nmagnetic Purcell effect appears abruptly here without explanation. Second, and more\nimportantly, the problem is not the ratio of the wavelength to the resonator dimension but the\nfact that in this range, the mode volume is typically close to λ^3. The authors make this\nstatement: “Indeed, since the latter represents the effective cavity volume over which the\nmagnetic dipoles are aligned with the magnetic field of the mode, it is drastically limited by\nour ability to squeeze the magnetic mode into a volume sufficiently small to match the one\noccupied by the emitters,” without explanation. What do the authors mean by “squeeze the\nmagnetic mode”?\nResponse: The magnetic Purcell effect refers to the enhancement of the spontaneous emission rate\nof magnetic dipole transitions when they couple to the magnetic field of a resonant mode. At\nmicrowave frequencies—where spin transitions often reside—the wavelength is much larger\ncompared to the typical size of the gain media that can practically be grown (whether they are organic\ncrystals or diamond-based). The mode volumes that can practically be achieved with standard\nmetallic cavity resonators typically remain close to λ3, and therefore are too big, and the ones\n7 / 18\nachieved with high index dielectrics are thermally sensitive. The point we make is precisely that we\nmanaged to create an all-metallic resonator that is not too big and provides a field that overlaps well\nwith the emitters.\nBy “squeezing the magnetic mode,” we mean engineering the resonator geometry and boundaries\nto reduce the effective volume over which the magnetic field is both significant and well-aligned with\nthe emitter’s magnetic dipole moment. The goal is to confine the field as tightly as possible around\nthe emitters to maximize coupling and hence increase their emission rate.\nR2.5b: Take, for example, a piece of coaxial cable. Coax has no cut-off frequency, and its\nsection can be as small as needed. Would such a coaxial resonator create a strong Purcell\neffect? Why?\nResponse: While a coaxial resonator can have a small cross-sectional area, it still needs a length on\nthe order of λ/2 to make a resonance, and the magnetic field is azimuthal. We would therefore not\nonly need a gain medium with azimuthally oriented magnetic dipoles (which seems extremely\nchallenging in a monocrystal since the emitters align along a definite crystal direction during crystal\ngrowth), but also this gain medium would need to be at least λ/2 long. Therefore, this example does\nnot evade our argument that making an overlap between a resonant magnetic field and the gain\nmedia is challengin with all-metallic structures.\nR2.7: I disagree with the statement “At room temperature, small mode volumes and high Q\nhave been achieved by employing dielectrics with very high permittivities” in the context of\nthe importance of the Q-factor. First, metallic high-Q resonators at microwaves are also\nknown. Second, high-index dielectrics not only have high Q-factors but also “squeeze” the\nwavelength and hence the mode volume.\nResponse: We agree that metallic high-Q microwave resonators are known, but they have never\nbeen successfully employed to make a MASER. Our work is the first to achieve this. This is why they\nare not listed in the state of the art about room-temperature maser resonators.\nThe second part of the referee’s comment (“Second, ...”), we completely agree with. These dielectric\nresonators have been used, they indeed also squeeze the wavelength and the mode volume. Our\npoint is that they come with the drawback of thermal sensitivity (which is our next sentence).\nRevision: We have revised the text and made sure not to make any claim that can be misunderstood.\nR2.8: I cannot fully agree with “However, such extreme dielectrics are inherently plagued by\ntheir utmost sensitivity to thermal fluctuations, making them excessively prone to detuning.”\nWhat thermal fluctuations are the authors talking about? These have different primary\nsources of noise. The thermal noise in dielectrics is usually due to dipole relaxation\n8 / 18\nprocesses and phonon interactions. Metals have free electrons that contribute to Johnson-\nNyquist noise. I would say that dielectric resonators will typically have lower thermal\nfluctuations compared to metallic resonators in the microwave range. The authors have to\nexplain what they mean here and probably support their claims with estimations and/or\nreferences. The statement “Very small room temperature fluctuations, or the unavoidable\nthermal heating coming from optical excitation or read-out, would inevitably shift the mode\nfrequency away from the quantum emitter linewidth” is also vague. What regime do the\nauthors mean here, good cavity (Qcav >> Qemit) or bad cavity (Qcav << Qemit)? If it is a good\ncavity, which follows from the narrative, then it is less important what happens to the cavity,\nas any noise from and fluctuations of the emitter will be more important. Could the authors\nelaborate on this?\nResponse: We apologize for the use of the word “fluctuations” which may have mislead the reviewer\ninto thinking that we were discussing noise-related effects. We are referring to temperature variations\nwithin the dielectric resonator material. These variations can be caused by environmental\ntemperature changes or local heating from processes such as optical pumping. In high-permittivity\ndielectric materials, such as SrTiO3, the temperature coefficient of the permittivity is normally high.\nTherefore, even small temperature changes can lead to noticeable shifts in the resonance frequency,\neffectively “detuning” the resonator.\nThis concern is distinct from the fundamental thermal noise mechanisms in dielectrics (such as dipole\nrelaxation or phonon interactions) or the Johnson-Nyquist noise present in metals. Rather, our point\nis that highly temperature-sensitive resonators may require more stringent thermal stabilization to\nmaintain the desired resonance conditions.\nR2.9: When the authors make statements about the maser stability like “We demonstrate\nmaser operation stable in frequency and in output pulse shape over thousands of seconds,\nwhich is not achievable in prior dielectric-based designs,” they should accompany them with\nreferences and examples of the state of the art.\nResponse: The examples of frequency fluctuation due to laser heating during operation can be found\nin papers:\n1. Breeze, J., Salvadori, E., Sathian, J. et al. Continuous-wave room-temperature diamond\nmaser. Nature 555, 493–496 (2018)\n2. Salvadori, E., Breeze, J., Tan, KJ. et al. Nanosecond time-resolved characterization of a\npentacene-based room-temperature MASER. Sci Rep 7, 41836 (2017).\nRevision: We have added these references in the same text in the revised manuscript.\nR2.10.a: The authors should elaborate more on the design of the SRR cluster. Why this\ngeometry? Why does it provide strong confinement of the magnetic energy?\n9 / 18\nResponse: The chosen geometry is inspired by metamaterial unit cells—split-ring resonators (SRRs)\n— which inherently support sub-wavelength magnetic resonances. We have added a detailed\nexplanation and design steps in the Supplementary Materials (Supplementary Fig. S7).\nRevision: The detailed design procedure of the SRR cluster has been added in the Supplementary\nMaterials (Supplementary Fig. S7).\nR2.10.b: Is this number of “almost 350 thousand times smaller than λ^3” theoretically limited?\nIf yes, by what means? If not, would it make sense to boost the magnetic field more?\nResponse: We do not believe that this number is theoretically limited, however within the constraints\nof operating with silver and the current fabrication methods, it is the best we could achieve.\nBy arranging multiple SRRs in close proximity, their interactions produce electromagnetic modes that\nconcentrate the field into volumes significantly smaller than λ3. The specific figure of “350 thousand\ntimes smaller than λ3” results from an optimization process aimed at striking an effective balance\namong the physical size of the resonator, the magnetic mode volume, and the quality factor. Pushing\nthe mode volume too aggressively below this optimized point typically increases absorption losses,\nlowering the Q too drastically. In practice, the design is therefore limited by absorption losses, which\noccur at the surface of the metal and are strongly dependent on the surface roughness. We employed\nthe more advanced casting techniques to make the best possible prototype with silver: optimization\nmay require new casting methods with less deformation of the geometry upon cooling and better\nroughness.\nWith materials with lower losses, it is possible to beat this limit: a known example is superconducting\ncircuits. There is thus no theoretical bound, but if we do not allow cryogenic temperatures, with silver\nwe are close to the optimum.\nR2.12: Did you take into account the realistic loss and skin depth effects in simulations?\nDoes the squeezing of the magnetic (and I believe, electric) field lead to a proportional increase\nin losses and Johnson-Nyquist noise?\nResponse: Yes, our simulations incorporate the finite conductivity and material parameters of both\nmetallic and dielectric components, allowing us to account for ohmic losses and the associated skin\ndepth limitations. As the electromagnetic field is confined into a smaller volume, its local energy\ndensity increases. This, in principle, can lead to higher ohmic losses and an increase in Johnson-\nNyquist noise, as both are related to the material’s response to the intensified fields. Our optimization\nprocess was therefore not only focused on minimizing mode volume but also on managing these\nlosses to achieve a practical compromise between field confinement and Q factor.\n10 / 18\nR2.13: Can the oscillations in Fig.2c be due to other mechanisms beyond the Rabi oscillations?\nThe argument “The Rabi oscillation is also confirmed by the broadened spectrum of the pulse”\ndoesn’t work because any shortening of the pulses in the time domain leads to broadening\nin the frequency domain. Can this effect, for instance, be caused by collective effects like\nDicke or other superradiance?\nResponse: Yes, the reviewer is right.\nRevision:\n1. We have revised the caption of Fig.2 from\n§ With stronger optical pumping energy, the maser pulse possesses a weak Rabi-\noscillatory behavior which is also reflected in its broadened spectrum.\nto\n§ With higher pump energy, the maser pulse exhibits a weak oscillatory behavior and\ncorresponding spectral broadening.\n2. We have revised the following text in the Results section from:\n§ With pumping energy well above the threshold (𝑁 ∼ 6.8×10#)), we observe a weak\nRabi-oscillatory behavior in the maser pulse, resulting from the coherent energy\nexchange between the spin ensembles in pentacene and the microwave photons in\nthe cavity. The Rabi oscillation is also confirmed by the broadened spectrum of the\npulse caused by the normal mode splitting (Fig.2.c, bottom).\nto\n§ With pumping energy well above the threshold (N ∼ 6.8×10#)), we observe a weak\noscillatory behavior in the maser pulse. This behavior could potentially arise from\ncoherent energy exchange between the spin ensemble and the cavity mode, or from\ncollective emission phenomena such as superradiance. Further experimental and\ntheoretical investigations will be necessary to determine the precise underlying\nmechanism of these oscillations.\n11 / 18\nReviewer 3:\nR3.0: The paper describes a specially designed metallic resonator employed in conjunction\nwith a pentacene-based optically pumped room temperature pulsed maser to facilitate\nfrequency-stable operation with minimal heating effects. The authors show that such heating\neffects are detrimental in a similar device based on a dielectric resonator made of SrTiO3\nsingle crystal, leading to frequency drifts and the eventual loss of maser signal after\napproximately 1-2 hours. The main novelty claim of this paper is the use of a unique 3D\nmetallic split-ring resonator with a small mode volume and a relatively large quality factor.\nThis results in a resonator with a relatively high Purcell factor (which is approximately\n\\(\\lambda^3 \\cdot Q/\\text{mode volume}\\)) that is imperative for facilitating the operation of\nthe maser device.\nResponse: We appreciate the Referee’s constructive and insightful comments, which have prompted\nsignificant revisions that have substantially strengthened our work.\nR3.1: I found this paper interesting and informative, but I fail to see significant novelty or\nimpact. My main concern is that the use of small metallic volume resonators is known in the\nfield of electron spin resonance, which seems to undermine the main novel point of the paper.\nRecent examples of resonators with similar or even smaller mode volumes (in units of\n\\(\\lambda^3\\)) can be found here: [arXiv:2307.11269] (https://arxiv.org/abs/2307.11269) and\nreferences therein. Additionally, there is the possibility of using metallic loop-gap resonators\nof similar or smaller size in conjunction with highly temperature-stable dielectric materials,\nas discussed in the following references:\n- [NCBI](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6948142/)\n-[Patent]\n(https://patentimages.storage.googleapis.com/77/0b/52/097e05c3bd3d96/WO2013175235A1.\npdf)\n- [Wiley](https://onlinelibrary.wiley.com/doi/abs/10.1002/mrm.28595)\nWhile these designs do not necessarily have the exact properties of the resonator presented\nin the current work, they achieve similar high Purcell factors and use either all-metallic or\ndielectric materials with negligible temperature dependence of permittivity.\nResponse: We thank the reviewer for providing these references, which have broadened our\nperspective, and we have added them in the revised manuscript. We would like to address our work’s\nnovelty in two key aspects: three-dimensional magnetic energy confinement and all-metallic strong\nmagnetic Purcell enhancement at subwavelength scales.\n12 / 18\nWhile small metallic volume resonators are well-studied in electron spin resonance, the first three\ncited references (arXiv:2307.11269, NCBI, and Patent) describe quasi-2D structures. However, for\napplications requiring three-dimensional samples or efficient volumetric optical excitation, achieving\nenergy confinement in three dimensions is essential. Our all-metallic resonator fulfills this need,\nproviding robust 3D magnetic energy confinement with subwavelength compactness and thermal\nstability, without additional dielectric materials that can complicate system design.\nIn the last reference [Wiley], the 3D loop-gap ring resonator made from copper, has a simulated\nmode volume around 0.154 cm) and a measured Q factor around 60 at 𝑓 = 2.3 GHz. These\n(\nparameters yield a Purcell factor approximately 6.5×10*, which is considerably lower than that\nachieved by our proposed resonator — 77 times lower than our experimental value and 290 times\nlower than our simulated value.\nRevision: The references are added.\nR3.2: Moreover, comparing the temperature stability of metallic structures to SrTiO3 seems\nunfair, as the latter has permittivity drifts in the range of approximately 1% per degree. The\nsame type of maser was realized in the past with a sapphire-based structure that is far more\ntemperature stable.\nResponse: We compared the temperature stability of our proposed all-metallic structure to the\nSrTiO3 resonator due to their similar sub-wavelength size and comparable magnetic Purcell factor.\nThe sapphire-based structure is indeed more stable thermally but has a much lower dielectric\nconstant, leading to a device around 200 times larger in size, and almost 10 times lower Purcell\nfactor. This requires a higher optical pumping threshold4. In contrast, our proposed all-metallic\nstructure is thermally stable while maintaining compact size and high Purcell factor.\nR3.3: Another major concern is that the paper discusses the use of a pulsed maser device\nwith very limited practical use, if at all. What would happen if one wishes to use a similar\nresonator for continuous-wave operation, such as a maser based on NV centers in diamond?\nAt some point, the metal itself may heat up and expand.\nResponse: In a maser employing NV centers in diamond with a sapphire resonator, a continuous\nlaser pump power of 400 mW resulted in a frequency shift of 25 MHz, attributed to an estimated\nΔ𝑇 = 35°C temperature increase in both the sapphire resonator and diamond caused by laser\n+,!-\"\nheating7. Assuming the toroidal SRR cluster has similar size and is operated under the same\nconditions as the sapphire resonator, the temperature increase Δ𝑇 could be estimated as\n.//\nΔ𝑇 /Δ𝑇 ∼ 𝜅 /𝜅 ≈ 0.1, following the Fourier's law, where 𝜅 and 𝜅 are the thermal\n.// +,!-\" +,!-\" +0 +,!-\" +0\nconductivities of sapphire and silver, respectively. Given that the volumetric thermal expansion\ncoefficient of silver is approximately 54×10!\" K!#, the geometric expansion would be less than\n0.02%. Consequently, the frequency variation in the SRR cluster is expected to be negligible\n13 / 18\ncompared to that observed in the sapphire resonator. We believe that our resonator would still be\nrelevant in these experiments.\nR3.4: Lastly, even with SrTiO3, the authors show that the drift is very slow. Is it worth having\na worse resonator design (with a lower Purcell factor than SrTiO3) but better temperature\nstability? Possibly, it would be better to automatically tune back the SrTiO3 resonator by\nchanging the metallic cup on it. What I am trying to say is that the fundamental problem\n(heating) which the authors claim to exist may have other solutions that would keep the\noriginal design with the high Purcell factor. And on a side note, I am not sure this problem is\nreal if one uses more temperature-stable dielectrics.\nResponse: During the review of the manuscript, we have had discussions with several authors of\nstudies using room-temperature masers, including at the PhD defense of the first author of this\nmanuscript, who confirmed that this is an important problem. Some colleagues asked us for our\nresonators and reproduced our experiment. Others came to our lab to see the maser in action.\nTherefore, we are confident that our study has value.\nWe agree that the drift in SrTiO3 is relatively slow, however, stable operation with robust frequency\nfidelity is essential for certain applications, such as sensors for weak magnetic fields3. In many\ncontext, stable and repeated pulse operation is favored over continuuous operation, because it uses\nmuch less input power. Our all-metallic resonator, though currently exhibiting a somewhat lower\nPurcell factor than theoretical expectations due to fabrication limitations, provides intrinsic thermal\nstability that ensures reliable, long-term performance without requiring ongoing adjustments. We\nanticipate that advancements in fabrication techniques could further enhance the Purcell factor in\nfuture iterations of this design.\nWe acknowledge that adaptive tuning methods, such as adjusting the metallic top plate, could\npotentially address thermal drift in the SrTiO3 resonator. However, they are challenging to implement,\nand as a matter of fact, they have not been demonstrated. On the other hand, temperature-stable\ndielectrics, such as Barium Magnesium Tantalate, could also be considered; however, they typically\nresult in a larger resonator size and, consequently, a greater volume due to lower dielectric constant.\nWe would like to emphasize that our aim is to develop a platform that combines thermal stability with\nsubwavelength compactness and strong Purcell enhancement, comparable to that achieved by ultra-\nhigh-dielectric-constant materials like strontium titanate. Our all-metallic resonator thus offers an\noptimal blend of these qualities in a simple, efficient, and maintenance-free design.\nR3.5: Other minor remarks:\n1. The Introduction discusses the Purcell enhancement factor in relation to increasing the\nspontaneous emission rate. However, the experimental results do not show any change\nin this rate compared to the usual T1 of the sample (as shown, for example, [here]\n14 / 18\n(https://www.nature.com/articles/nature16944)). In my opinion, it would have been better\nto describe the requirements for achieving maser action through the cooperativity factor.\nThe discussion of changing the spontaneous emission rate seems out of place. Indeed,\nthe Purcell factor is part of the cooperativity factor, but this should be taken in the\ncontext of achieving maser threshold (having more gain than losses inside the cavity)\nand not connected to the spontaneous emission rate.\nResponse: We thank the reviewer for this insightful remark. A discussion on cooperativity has been\nincluded in the revised manuscript. As a matter of fact, other experts of the field contacted us during\nthe review process with the request to also report cooperativity.\nRevision: In the discussion section of the revised manuscript, we have added:\n§ “Furthermore, to assess the interaction strength between the emitters and the confined\nEM mode in the resonator, the cooperativity of the maser, constructed with the SRR\ncluster, can be calculated using C = 4g'N/k k , where k = 2/T∗ is the spin dephasing\n1 2 2 '\nrate with the spin dephasing time T∗ ≈ 2.9 µs and g = γ Jµ ℏω/2V is the single spin-\n' 4 % 5\nphoton coupling strength. With N = max(N ) = 7.3×10#) ,\nthe cooperativity is\n%\nestimated to be C ≈ 6.4 , suggesting a relatively weak coupling between the emitters\nand the SRR cluster”.\nR3.6: 2. Page 2 – the expected frequency variations of SrTiO3-based resonators: can you be\nmore quantitative? For SrTiO3, it is clear that there would be relatively large changes, but for\nsapphire, this is a very small change in permittivity. What about the expected change in\nresonator frequency due to metal expansion?\nResponse: The resonant frequency of SrTiO changed mainly due to its high temperature coefficient\n3\nof the permittivity, which leads to a large temperature coefficient of frequency of approximately\n1700 ppm/K1,2. Based on the volumetric thermal expansion coefficient of silver around 54×10!\" K!#,\nthe temperature coefficient of resonant frequency of the all-metallic SRR cluster is estimated to be\napproximately 130 ppm/K. Therefore, the resonant frequency variation in the all-metallic structure\ndue to thermal expansion is negligible, thanks to the good thermal conductivity and very low thermal\ncoefficient of resonant frequency.\nRevision: We have added the following text in the Results section:\n• This excellent thermal stability is attributed to silver's high thermal conductivity and moderate\nthermal expansion coefficient, resulting in a temperature coefficient of resonant frequency of\napproximately 130 ppm/K. In contrast, the central frequency of the SrTiO based maser shifts\n3\nfrom 1.449 GHz to 1.453 GHz, resulting from SrTiO ’s large temperature coefficient of\n3\nresonant frequency (approximately 1700 ppm/K). We simulated the diffusion of the deposited\noptical pump energy and the resulting temperature variations in the SrTiO resonator using\n15 / 18\nWe have added Movie1 and Fig.S6 in the Supplementary materials.\nR.3.7: 3. Mode volume should refer to the magnetic field in all directions (namely magnitude)\nin the denominator since it relates to total magnetic energy. What is \\(H_{z,\\text{max}}(r)\\) and\nhow is it dependent on r?\nResponse: Correct. However, since most of the magnetic field is concentrated in the core region of\nthe resonator and mostly aligned along z axis, the mode volume was approximated using the z-\ncomponent field 𝐇 of the magnetic. R𝐻 (𝑟)R represents the maximum magnitude of the magnetic\n𝐳 7#$%\nfield in the resonator.\nRevision: “Assuming that within this region, we align uniformly an ensemble of quantum magnetic\n|9(;)|!\nemitters along the z-axis, we would get an effective magnetic mode volume V = ∫ dV of\n5 |9&'( (;)|!\nonly 0.025 cm).”\nR3.8: 4. Figure 1 – can you plot the E-field as well?\nResponse: The E-field is shown below in Figure R3.8.1 and is also included in the revised\nsupplementary material as Supplementary Fig. S2.\nFigure R3.8.1 Electric field distribution within the cross section of the toroidal SRR cluster in the yz-plane (a)\nand xy-plane (b).\nRevision: We have added this figure to the revised supplementary material as Supplementary Fig.\nS2.\n16 / 18\nR3.9: 5. Page 4 – it should be mentioned clearly that this maser operates only in pulsed\nmode. What about masers that need a static field? The all-metallic design with its shield is larger\nthan the dielectric design, implying a larger and bulkier magnet.\nResponse: The all-metallic design might shield more static magnetic field than the dielectric design;\nhowever, this influence can be considered negligible due to the use of silver. To verify this, we\nconducted simulations in COMSOL to evaluate the impact of the all-metallic resonator. As shown in\nFigure R3.9.1, the presence of the SRR cluster made in silver has minimal impact on the static\nmagnetic field distribution.\nFigure R3.9.1 Static magnetic field between two electromagnets, simulated in COMSOL. a and b, Homogeneous\ndistributions of the magnetic field in the center of the simulated region without and with the SRR cluster in silver,\nrespectively. The black solid lines depict the magnetic field contours. c, Magnetic field profiles along the x axis at z=0mm\nfrom panel a and b, represented by a solid blue line and a dashed red line, respectively.\nRevisions: In light of these promising simulation results, we have selected the pentacene-based\npulsed maser as an ideal test platform, for which we can directly compare the performance obtained\nwith our SRR cluster and with a high-index dielectric resonator.\nR3.10: 6. Page 8 – what is the spin concentration of the sample?\nResponse: The spin concentration of the sample is approximately 1000 ppm, denoted as 0.1%\nmol/mol in the Method section on Page 8 of the manuscript.\nReferences\n1. Wise, P. L. et al. Structure-microwave property relations in (SrxCa(1Àx))n+1TinO3n+.\nJ. Eur.\nCeram. Soc. (2001).\n2. Geyer, R. G., Riddle, B., Krupka, J. & Boatner, L. A. Microwave dielectric properties of\nsingle-crystal quantum paraelectrics KTaO3 and SrTiO3 at cryogenic temperatures. J. Appl. Phys.\n97, 104111 (2005).\n17 / 18\n3. Wu, H. et al. Enhanced quantum sensing with room-temperature solid-state masers. Sci.\nAdv. 8, eade1613 (2022).\n4. Breeze, J. et al. Enhanced magnetic Purcell effect in room-temperature masers. Nat.\nCommun. 6, 6215 (2015).\n5. Salvadori, E. et al. Nanosecond time-resolved characterization of a pentacene-based room-\ntemperature MASER. Sci. Rep. 7, 41836 (2017).\n6. Anderson, C. P. & Awschalom, D. D. Embracing imperfection for quantum technologies.\nPhysics Today 76, 26–33 (2023).\n7. Breeze, J. D., Salvadori, E., Sathian, J., Alford, N. McN. & Kay, C. W. M. Continuous-wave\nroom-temperature diamond maser. Nature 555, 493–496 (2018).\n18 / 18\nResponse letter – 2nd review, manuscript number NCOMMS-23-52768\nWe would like to thank all the reviewers for their time and careful review of our manuscript. In the\nfollowing, we provide a point-by-point response to each of the Reviewer's comments.\nReviewer 1:\nR 1.1 The authors state that they do not observe any frequency shift using the all-metallic\nresonator. I find this comment confusing since they state later in the rebuttal they estimate\nthe temperature coefficient of frequency to be 130 ppm/K. The shift here being due to thermal\nexpansion of the silver. This is substantial and should be measurable at the frequency of the\nmaser emission at 1.45 GHz.\nResponse: The theoretical temperature coefficient of 130 ppm/K reflects a worst-case scenario,\nassuming a uniform 1 K temperature rise throughout the silver structure. In practice, the actual\ntemperature increase during operation is negligible due to the high thermal conductivity and efficient\nheat dissipation of the metallic resonator. Experimentally, we observed stable maser emission over\n18,000 seconds without any detectable frequency drift, confirming that the real temperature variation\nwas far below the threshold required to produce a measurable shift.\nR 1.2 The authors have produced graphs showing temperature changes in the resonator. No\ndetail on how these graphs were generated is given so I cannot comment further except to\nsay that they authors did not properly address the original concerns.\nResponse: We appreciate the reviewer’s feedback and acknowledge the need for clarity on how\nthermal effects were simulated. To directly address concerns about the thermal stability comparison\nbetween the SrTiO₃ resonator (with polystyrene foam support) and our metallic SRR design, we\nconducted COMSOL Multiphysics simulations to model heat accumulation and dissipation in the\nSrTiO₃ resonator with two support materials: sapphire and polystyrene foam.\nSimulation setup:\n1. Model Geometry (Figure1.2.1):\n- SrTiO₃ resonator: hollow cylindrical structure matching the experimental dimensions.\n- Copper baseplate: circular disk with a central hole.\n- Support: hollow cylinder made of either sapphire or polystyrene foam.\n- Optical fiber end: modeled as a 1 mm-diameter disk positioned at the bore to represent\nthe laser input.\n- Default COMSOL material parameters were used for all components except the\npolystyrene foam, whose properties were obtained from:\nhttps://www.laird.com/sites/default/files/2021-01/RFP-DS-PP%2006242020.pdf .\n1 / 11\nFigure R1.2.1 3D model in the COMSOL simulation\n2. Physics\n- Laser heating: simulated using the Geometrical Optics interface to trace 592 nm laser pulses\n(5 ns duration, 3 mJ energy, Gaussian profile).\n- Heat transfer: modeled via the Heat Transfer in Solids interface to capture transient heat\naccumulation and dissipation.\n3. Simulation protocol:\n- Total duration: 5 s, divided into five 1 s cycles.\n- Within each cycle:\n• First 5 ns: laser excitation and localized heating of the resonator.\n• Remaining time: passive cooling without laser input.\n- Initial conditions:\n• Cycle 1 starts at 293.15 K (room temperature).\n• Each subsequent cycle begins with the final temperature distribution from the previous\ncycle.\nR 1.3 Whilst an improved resonator temperature stability is admirable, I remain unconvinced\nthat the work presented here is significant enough a development to warrant publication in\nNature Communications. As pointed out by Reviewer 2, a patent was published in 2013 (WO\n2013/175235 A1, see Figure 4) that describes using a split-ring resonator, very similar to that\nreported by the authors, suggesting that the idea is not original.\n2 / 11\nhttps://patentimages.storage.googleapis.com/77/0b/52/097e05c3bd3d96/WO2013175235A1.p\ndf\nResponse: We acknowledge that planar split-ring resonators (SRRs) were mentioned in the earlier\npatent WO 2013/175235 A1 as potential cavity architectures for maser applications. However, it does\nnot demonstrate maser operation, Purcell enhancement, or any experimental validation. In general,\nwe wish to point out that it is infinitely less-trivial to devise actual SRRs-based designs than just\nthinking that this might be possible, which is enough in a patent. We also wish to remark that patents\ndo not automatically represent prior art, and may protect technical solutions that were never actually\ndemonstrated and are extremely challenging to implement, providing they actually work.\nIn contrast, our study reports the first experimental realization of a room-temperature maser based\non an all-metallic structure, supported by:\n• comprehensive full-wave electromagnetic design, theory and demonstration\n• quantitative analysis of Purcell enhancement, and\n• evaluation of thermal robustness.\nMoreover, the resonator described in the patent corresponds to a planar structure (~100 µm thick),\nwhile our device is a three-dimensional toroidal SRR cluster that achieves volumetric magnetic\nconfinement. This architecture enables a deep-subwavelength mode volume and large Purcell factor,\nwhile maintaining exceptional thermal stability over thousands of optical excitation cycles. Based on\nour work, we do not believe that planar SRRs can reach sufficient Purcell factors over an appreciable\ngain medium volume: a volumetric design is required.\nThus, although conceptually related, the present work represents a substantial technical and\nexperimental advance, establishing a clear and verifiable step forward in the development of metallic\nresonators for enhancing the emission rates of magnetic quantum emitters. We believe that rejection\nof our paper based on this patent would be scientifically unfounded.\n3 / 11\nReviewer 2:\nR2.0: The authors have addressed all my comments, and I now recommend the paper for\npublication.\nResponse: We sincerely thank the reviewer for their thorough evaluation and positive\nrecommendation for publication.\n4 / 11\nReviewer 3:\nR3.0: The authors have made efforts to address the reviewers’ comments and revise their\nmanuscript accordingly. I see improvements; however, there are still several issues that I\nbelieve require further attention:\nResponse: We thank the reviewer for their constructive feedback and for recognizing the\nimprovements in our revised manuscript. We appreciate the remaining points raised and have\ncarefully addressed each of them in this version.\nR3.1: (R3.3) As noted in my previous report, this work focuses on pulsed maser operation at\na very low repetition rate (1 Hz), which has limited applicability. The authors provided in their\nresponse some estimates regarding the feasibility of supporting continuous-wave (CW)\ndiamond NV-based masers—especially in relation to the resonator—but these were not\nincorporated into the revised paper.\nResponse: We thank the reviewer for highlighting this important point. Our focus on pulsed operation\nat 1 Hz in this work stems from the need to resolve the critical challenge of thermal instability in deep-\nsubwavelength room-temperature masers for enabling more advanced applications. The present\nwork therefore serves as a proof-of-principle demonstration of the resonator’s thermal stability and\nPurcell enhancement under pulsed excitation. We fully agree that continuous-wave operation\nrepresents the next step toward broader applicability. Accordingly, we have now incorporated a\ndiscussion of CW feasibility into the revised manuscript and Supplementary Information, as detailed\nbelow.\nRevision: We have inserted the following passage into the main manuscript:\nIn addition, the geometry of the clustered SRR can be tailored to achieve operation at different\nfrequencies, accommodating other magnetic quantum emitters such as nitrogen–vacancy (NV)\ndefect centers in diamond that are used in continuous-wave (CW) masers. A detailed analysis\nof the thermal stability of CW masers incorporating the SRR cluster is provided in Supplementary\nDiscussion SII B.\nAdditionally, we included the following text in the Supplementary Material:\nSupplementary Discussion SII\nB - Thermal Stability for Continuous-Wave Maser Operation:\nWhile the present work focuses on pulsed operation, the intrinsic thermal properties of the SRR\ncluster make it particularly promising for stable CW applications.\n5 / 11\nTo evaluate the potential for continuous-wave (CW) operation, we analyzed the thermal\nperformance of the silver toroidal SRR cluster and compared it with that of conventional dielectric\nresonators. Consider, for instance, a maser employing NV centers in diamond with a sapphire\nresonator, where continuous laser pumping at 400mW caused a frequency shift of 25MHz due\nto thermal heating. This corresponds to an estimated temperature rise: 𝛥𝑇 = 35∘𝐶.\nAl2O3\nAssuming that the toroidal SRR cluster has a similar size and is operated under the same\nconditions as the sapphire resonator, the temperature increase 𝛥𝑇 could be estimated as\n%&&\n𝛥𝑇 /𝛥𝑇 ∼ 𝜅 /𝜅 ≈ 0.1, following the Fourier's law, where 𝜅 and 𝜅 are the\n%&& '(#)$ '(#)$ '* '(#)$ '*\nthermal conductivities of sapphire and silver, respectively. Given that the volumetric thermal\nexpansion coefficient of silver is approximately 54×10+, K-1, the geometric expansion would be\nless than 0.02%. Consequently, the frequency variation in the SRR cluster is expected to be\nnegligible compared to that observed in the sapphire resonator.\nR3.2: (R2.1) The discussion of the spontaneous emission rate and the Purcell factor is non-\nquantitative. Additionally, the relationship to the Einstein B-coefficient does not seem to be\nincorporated into the revised paper.\nResponse: At microwave frequency, spontaneous emission is negligible compared to stimulated\nemission, since\n'#%\n=\n0123$\n, where 𝐴 and 𝐵 are the Einstein coefficients of spontaneous emission\n/#% 4$ 56 56\nand stimulated emission, respectively, and 𝜈 is the photon frequency1,2. Owing to the 𝜈7 dependence,\nthe spontaneous emission rate is many orders of magnitude smaller than the stimulated emission\nrate in the GHz regime and can therefore be safely neglected in the maser dynamic model.\nFor the Purcell-enhanced stimulated emission, we have the following equations:\n𝑞̇ = −𝑘 𝑞 +𝐵𝑁𝑞\nc\n⎧\n9 ℏ;<#=\n⎪ 𝐵 = & ' #\n>?( \n;\n𝑘 = \n⎨ 4\n@\n)*+,\n⎪ 𝐹 = 7B$ ( ⋅ @)*+,\n⎩ A >C# ?(\nHere, the Einstein coefficient of stimulated emission B scales inversely to the mode volume 𝑉 ,\nand\nD\nthe cavity photon decay rate 𝑘 is inversely proportional to the loaded Q factor 𝑄 .\nThe threshold\n4 (EFG\nof population inversion 𝑁 for maser action is therefore\nH2\n𝑁 = \nI-\n∝\n?( \n∝\n6\n.\nH2\n/ @\n)*+,\nJ(\nHence, achieving a small mode volume and a high loaded Q-factor, corresponding to a large Purcell\nfactor, is essential for low-threshold and efficient maser operation.\n6 / 11\nRevision: We have incorporated the quantitative relation between the Einstein B-coefficient, mode\nvolume, and Purcell factor into the revised manuscript (Methods – Maser Dynamic Modelling) as\nfollows:\n𝐵 stands for the Einstein coefficient of the stimulated emission, which is directly linked to the\nresonator's mode volume as follows:\n𝐵 = \n9&ℏ;<' #=#\n>?( \nwhere 𝜇 is the vacuum permeability, ℏ the reduced Planck constant, 𝛾 = 1.76×1066 𝑟𝑎𝑑 ⋅𝑠+6 ⋅\nK L\n𝑇+6 the electron gyromagnetic ratio, and 𝑇 = 2.1~𝜇𝑠 the spin-spin relaxation time of pentacene.\nits rate is vanishingly small compared to stimulated emission.\nAnd:\nAssuming steady-state operation (𝑞̇ = 0), the threshold population inversion is given by:\n𝑁 = 𝑘 /𝐵 = 4.4×1067. Since the Purcell factor scales as 𝐹 ∝ @)*+,, it follows that\nH2MLN2E(G c A\n?( \n6\n𝑁 ∝ , confirming that an ultrasmall 𝑉 and high 𝑄 (i.e. a large 𝐹 ) are key to\nH2MLN2E(G D (EFG A\nJ.\nachieving low-threshold, efficient maser operation.\nR3.3: (R2.4) The definition of the mode volume used in the text is somewhat controversial in\nthis context. Specifically, normalizing by the maximum of the magnetic field can produce\nartificially low “magnetic mode” volumes if there is a localized magnetic field “hot spot.” In\ncontrast, if the microwave magnetic field is relatively homogeneous across the resonator\nvolume, the chosen definition might be reasonable. However, a more relevant approach could\nbe to define the 50% or 90% mode volume, referring to the region where 50% or 90% of the\nmicrowave magnetic energy is localized. This might explain the discrepancy between the\nmeasured data and the mode volume calculation (a factor of ~3). I believe the authors should\ndiscuss this point in more detail.\nResponse: In our study, the mode volume 𝑉 was calculated within the resonator bore region\nO\noccupied by the gain medium, as the Purcell enhancement depends on the spatial overlap between\nthe resonator’s magnetic field and the emitters’ magnetic dipoles. Magnetic energy outside this\nregion does not contribute to emission enhancement and is therefore excluded from the effective\n𝑉 relevant to maser operation. To verify that the simulated mode volume is not biased by field\nO\nlocalization or numerical artifacts, we performed a statistical analysis of the magnetic energy density\ndistribution inside the bore. The field exhibits moderate uniformity, with a coefficient of variation of\n0.45 and a median-to-mean ratio of 0.94. The ratio of maximum to median magnetic energy density\nis 2.40, indicating that the field maximum is only modestly higher than the typical value and is not\ndominated by an isolated hotspot.\n7 / 11\nFurthermore, the minimal sub-volumes enclosing 50% and 90% of the magnetic energy are 𝑉 =\nPK\n0.010cm7and 𝑉 = 0.025cm7, corresponding to 31% and 78% of the bore volume (𝑉 = 0.032cm7),\nQK RSTU\nrespectively. These results demonstrate that the magnetic energy is broadly distributed across the\ngain region, confirming that the simulated mode volume is not artificially reduced by localized peaks.\nThe simulations provide an idealized lower bound assuming perfect geometry and complete dipole\nalignment, whereas the experimental value naturally reflects structural imperfections, partial filling,\nand non-uniform emitter orientation. The resulting threefold difference between the simulated and\nexperimental mode volumes is therefore reasonable and does not alter the physical interpretation of\nthe Purcell enhancement. Together with the quantitative field-distribution metrics presented above,\nthis agreement confirms that our estimation of 𝑉 is physically meaningful and representative of the\nO\nactual electromagnetic environment experienced by the emitters.\nRevision: We have inserted the following passage into the discussion section of the main manuscript:\nTo confirm that the simulated mode volume is not artificially reduced by local field maxima, we\nquantitatively analyzed the magnetic energy distribution in the resonator bore region. The field\nwas found to be broadly uniform without strong localization, as detailed in Supplementary\nDiscussion SII A.\nAdditionally, we included the following text in the Supplementary Material:\nSupplementary Discussion SII\nA - Statistical Evaluation of the Mode Uniformity in the Resonator Bore Region\nTo evaluate whether the calculated mode volume is affected by localized field peaks or numerical\nartifacts, we performed a statistical analysis of the simulated magnetic-energy density within the\nbore region of the clustered SRR. The mode volume 𝑉 was evaluated only within this region,\nm\nsince the Purcell enhancement depends on the overlap between the resonator’s magnetic field\nand the emitters’ dipoles, while magnetic energy outside the gain medium does not contribute to\nthe emission process.\nThe simulated magnetic energy distribution exhibits moderate uniformity, with a coefficient of\nvariation (CV) of 0.45 and a median-to-mean ratio of 0.94. The ratio of maximum to median\nmagnetic-energy density is 2.40, showing that the field maximum is modestly higher than the\ntypical value and not dominated by a single hotspot. Furthermore, the minimal sub-volumes\nenclosing 50% and 90% of the magnetic energy are 𝑉 = 0.010 cm3 and 𝑉 = 0.025 cm3,\nPK QK\ncorresponding to 31% and 78% of the bore volume ( 𝑉 = 0.032 cm3), respectively. These\nbore\nresults demonstrate that the magnetic energy is broadly distributed across the emitter region.\nThe simulation represents an idealized lower bound assuming perfect geometry and complete\ndipole alignment, whereas the experimental value includes realistic effects such as structural\ndeformation, partial filling, and non-uniform emitter orientation. The resulting threefold difference\n8 / 11\nbetween the simulated (0.025 cm3) and experimental (0.072 cm3) mode volumes is therefore\nreasonable and consistent with realistic experimental conditions. Combined with the statistical\nmetrics presented above, these results confirm that the mode volume estimation used in this\nwork is physically meaningful and representative of the actual electromagnetic environment\nexperienced by the emitters.\nR3.4: (R2.5a) The size of the crystal and the specific dimensions of the device are crucial\ndetails that should be linked more explicitly to the required optimal resonator geometry. This\nappears to be another instance where the authors’ response did not result in changes to the\nmanuscript.\nResponse: To facilitate precise reproduction of our design, the 3D model of the resonator is now\navailable as an STL file on Zenodo (DOI: https://zenodo.org/doi/10.5281/zenodo.10012323), which\ncan be directly used for 3D printing. In addition, we have included the dimensions of the cylindrical\ngain medium in the revised manuscript.\nRevision: We have expanded the introduction of the main manuscript as follows to link the size of\npractically grown gain media to that of a promising resonator:\nHowever, as spin transition frequencies are commonly in the gigahertz range, i.e., with emission\nwavelengths λ on the order of tens of centimeters, leveraging the magnetic Purcell effect comes\nm\nwith the inherent challenge of dealing with the large ratio between λ 3 and achievable values of\nm\nmode volume V . In practice, the physical dimensions of gain media that can be grown\nm\nexperimentally, such as organic crystals or diamond-based samples, occupy physical volumes\nthat are roughly three orders of magnitude smaller than λ 7 (i.e., ∝ cm7 ). To address this\nm\nmismatch, the resonator geometry must be engineered such that the region of strong magnetic\nfield is confined to a volume only slightly larger than that of the gain medium, with its orientation\noptimized for alignment with the emitters’ magnetic dipole moments. This “squeezing” of the\nmagnetic mode maximizes spatial overlap with the active medium, thereby enhancing the\nefficiency of energy exchange.\nAdditionally, we have included the dimensions of the cylindrical gain medium:\nAs depicted in Fig.1.b, the pentacene:p-terphenyl (pc:pt) crystal (1.5 mm radius × 10 mm height)\nis positioned within the bore of the clustered SRR, which is enclosed inside a copper cavity. The\nbottom 5 mm of the pc:pt crystal includes a central axial hole of 0.5 mm radius, whereas the top\n5 mm remains solid.\nR3.5: (R2.8) Here as well, I believe the text of the paper should be modified accordingly,\nreflecting the issues raised in the response.\nRevision: We have replaced “thermal fluctuations” with “thermal variations” in the manuscript and\nrevised the corresponding part of the introduction to clarify this point and avoid ambiguity as follows:\n9 / 11\nHowever, dielectric resonators with extremely high permittivities, such as SrTiO3, are inherently\nsensitive to temperature variations due to their large temperature coefficient of permittivity. Even\nminor room temperature variations, or the unavoidable thermal heating coming from optical\nexcitation or read-out, would inevitably shift the mode frequency away from the quantum emitter\nlinewidth. As a result, the system must be continuously retuned, hindering stable operation and\nlimiting the practicality of such dielectric resonators.\nR3.6: It seems somewhat unusual to discuss frequency stability using MHz-scale data,\npresumably due to the short-pulse operation. Comparing this to the ~10 kHz linewidths of CW\nmaser systems cited in the references is difficult. A more direct comparison or\nacknowledgment of the differences in operational modes would be helpful.\nResponse: We agree that the frequency-stability discussion in our pulsed maser system should not\nbe directly compared to the narrow (~10 kHz) linewidths of continuous-wave (CW) masers. To clarify\nthis distinction, we have added an explicit paragraph in Supplementary Discussion SII B explaining\nthat our measurements characterize pulse-to-pulse spectral stability rather than absolute linewidth,\nand emphasizing the fundamental difference between the transient pulsed regime and the steady-\nstate CW regime.\nRevision: we included the following text in Supplementary Discussion SII B of the Supplementary\nMaterial:\nWhile the present work focuses on pulsed operation, the excellent thermal properties of the SRR\ncluster make it particularly promising for stable continuous-wave (CW) applications. The\nobserved MHz-scale linewidths in our pulsed maser experiments arise from the Fourier limit\nassociated with microsecond pulse durations, rather than from intrinsic decoherence or thermal\ninstability. In contrast, CW masers operate in a steady-state regime that can achieve linewidths\non the order of tens of kilohertz. Although the absolute linewidths are therefore not directly\ncomparable, the measurements presented here demonstrate exceptional pulse-to-pulse spectral\nstability of the SRR-based maser over prolonged operation times.\nR3.7: (R3.1) The “quasi-2D” structures mentioned by the authors in relation to the quoted\nreferences of previous work have heights of approximately 0.5 mm and 1.4 mm—borderline\nfor the intended application. However, there is no fundamental reason why these could not\nbe extended to heights of, for example, 3 mm or more to fit the crystal used in this work. It is\ntherefore unclear why the authors dismiss these structures as unsuitable for the current\napplication.\nResponse: The reviewer is correct that quasi-2D resonator geometries could, in principle, be\nextended vertically. Our toroidal SRR cluster was inspired by planar SRR concepts but represents a\nconceptual three-dimensional evolution developed through detailed electromagnetic design to\nachieve genuine volumetric magnetic confinement. The resulting structure supports a uniform\nmagnetic field across the crystal volume, exhibits deep-subwavelength compactness without\nincorporating any dielectric materials (unlike designs such as that in NCBI\n10 / 11\nhttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC6948142/ which rely on rutile loading), and maintains\nexcellent thermal stability. This three-dimensional architecture differs fundamentally from a simple\ngeometric extrusion and enables strong Purcell enhancement together with stable operation under\nrepeated optical excitation.\nR3.8: 4. (R3.3) The condition C>1 is typically considered indicative of strong (rather than weak)\ncoupling. Overall, while the revised manuscript addresses some of the reviewers’ concerns,\nI believe further clarifications and revisions are needed in the areas detailed above.\nResponse: We thank the reviewer for pointing out the misuse and potential confusion in terminology.\nOur previous use of “relatively weak coupling” was intended to indicate a cooperativity value lower\nthan those reported in other maser systems3,4, rather than to suggest C<1 in the strict formal sense.\nTo avoid confusion, we have revised the manuscript accordingly, replacing “relatively weak coupling”\nwith “strong coupling” to align with the conventional definition for C>1.\nRevision: We have replaced “relatively weak coupling” with “strong coupling” in the discussion\nsection of the manuscript.\nReferences\n1. Einstein, A. Strahlungs-Emission und -Absorption nach der Quantentheorie. Deutsche\nPhysikalische Gesellschaft 18, 318–323 (1916).\n2. Zangwill, A. Modern Electrodynamics. (Cambridge University Press, 2013).\n3. Breeze, J. D., Salvadori, E., Sathian, J., Alford, N. McN. & Kay, C. W. M. Room-temperature\ncavity quantum electrodynamics with strongly coupled Dicke states. Npj Quantum Inf. 3, 40 (2017).\n4. Breeze, J. D., Salvadori, E., Sathian, J., Alford, N. McN. & Kay, C. W. M. Continuous-wave\nroom-temperature diamond maser. Nature 555, 493–496 (2018).\n11 / 11"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Second review\nR 1.1\nThe authors state that they do not observe any frequency shift using the all-metallic resonator. I find this comment confusing since they state later in the rebuttal they estimate the temperature coefficient of frequency to be 130 ppm/K. The shift here being due to thermal expansion of the silver. This is substantial and should be measurable at the frequency of the maser emission at 1.45 GHz.\nR 1.2\nThe authors have produced graphs showing temperature changes in the resonator. No detail on how these graphs were generated is given so I can not comment further except to say that they authors did not properly address the original concerns.\nR 1.3\nWhilst an improved resonator temperature stability is admirable, I remain unconvinced that the work presented here is significant enough a development to warrant publication in Nature Communications. As pointed out by Reviewer 2, a patent was published in 2013 (WO 2013/175235 A1, see Figure 4) that describes using a split-ring resonator, very similar to that reported by the authors, suggesting that the idea is not original.\nhttps://patentimages.storage.googleapis.com/77/0b/52/097e05c3bd3d96/WO2013175235A1.pdf", "reviewer_2": "The authors have addressed all my comments, and I now recommend the paper for publication.", "reviewer_3": "Review of NCOMMS-23-52768\nThe authors have made efforts to address the reviewers’ comments and revise their manuscript accordingly. I see improvements; however, there are still several issues that I believe require further attention:\n1. (R3.3) As noted in my previous report, this work focuses on pulsed maser operation at a very low repetition rate (1 Hz), which has limited applicability. The authors provided in their response some estimates regarding the feasibility of supporting continuous-wave (CW) diamond NV-based masers—especially in relation to the resonator—but these were not incorporated into the revised paper.\n2. (R2.1) The discussion of the spontaneous emission rate and the Purcell factor is non-quantitative. Additionally, the relationship to the Einstein B-coefficient does not seem to be incorporated into the revised paper.\n3. (R2.4) The definition of the mode volume used in the text is somewhat controversial in this context. Specifically, normalizing by the maximum of the magnetic field can produce artificially low “magnetic mode” volumes if there is a localized magnetic field “hot spot.” In contrast, if the microwave magnetic field is relatively homogeneous across the resonator volume, the chosen definition might be reasonable. However, a more relevant approach could be to define the 50% or 90% mode volume, referring to the region where 50% or 90% of the microwave magnetic energy is localized. This might explain the discrepancy between the measured data and the mode volume calculation (a factor of ~3). I believe the authors should discuss this point in more detail.\n4. (R2.5a) The size of the crystal and the specific dimensions of the device are crucial details that should be linked more explicitly to the required optimal resonator geometry. This appears to be another instance where the authors’ response did not result in changes to the manuscript.\n5. (R2.8) Here as well, I believe the text of the paper should be modified accordingly, reflecting the issues raised in the response.\n6. (R2.9) It seems somewhat unusual to discuss frequency stability using MHz-scale data, presumably due to the short-pulse operation. Comparing this to the ~10 kHz linewidths of CW maser systems cited in the references is difficult. A more direct comparison or acknowledgment of the differences in operational modes would be helpful.\n7. (R3.1) The “quasi-2D” structures mentioned by the authors in relation to the quoted references of previous work have heights of approximately 0.5 mm and 1.4 mm—borderline for the intended application. However, there is no fundamental reason why these could not be extended to heights of, for example, 3 mm or more to fit the crystal used in this work. It is therefore unclear why the authors dismiss these structures as unsuitable for the current application.\n8. (R3.3) The condition C>1 is typically considered indicative of strong (rather than weak) coupling.\nOverall, while the revised manuscript addresses some of the reviewers’ concerns, I believe further clarifications and revisions are needed in the areas detailed above."}, "author_response": "Response letter – 2nd review, manuscript number NCOMMS-23-52768\nWe would like to thank all the reviewers for their time and careful review of our manuscript. In the\nfollowing, we provide a point-by-point response to each of the Reviewer's comments.\nReviewer 1:\nR 1.1 The authors state that they do not observe any frequency shift using the all-metallic\nresonator. I find this comment confusing since they state later in the rebuttal they estimate\nthe temperature coefficient of frequency to be 130 ppm/K. The shift here being due to thermal\nexpansion of the silver. This is substantial and should be measurable at the frequency of the\nmaser emission at 1.45 GHz.\nResponse: The theoretical temperature coefficient of 130 ppm/K reflects a worst-case scenario,\nassuming a uniform 1 K temperature rise throughout the silver structure. In practice, the actual\ntemperature increase during operation is negligible due to the high thermal conductivity and efficient\nheat dissipation of the metallic resonator. Experimentally, we observed stable maser emission over\n18,000 seconds without any detectable frequency drift, confirming that the real temperature variation\nwas far below the threshold required to produce a measurable shift.\nR 1.2 The authors have produced graphs showing temperature changes in the resonator. No\ndetail on how these graphs were generated is given so I cannot comment further except to\nsay that they authors did not properly address the original concerns.\nResponse: We appreciate the reviewer’s feedback and acknowledge the need for clarity on how\nthermal effects were simulated. To directly address concerns about the thermal stability comparison\nbetween the SrTiO₃ resonator (with polystyrene foam support) and our metallic SRR design, we\nconducted COMSOL Multiphysics simulations to model heat accumulation and dissipation in the\nSrTiO₃ resonator with two support materials: sapphire and polystyrene foam.\nSimulation setup:\n1. Model Geometry (Figure1.2.1):\n- SrTiO₃ resonator: hollow cylindrical structure matching the experimental dimensions.\n- Copper baseplate: circular disk with a central hole.\n- Support: hollow cylinder made of either sapphire or polystyrene foam.\n- Optical fiber end: modeled as a 1 mm-diameter disk positioned at the bore to represent\nthe laser input.\n- Default COMSOL material parameters were used for all components except the\npolystyrene foam, whose properties were obtained from:\nhttps://www.laird.com/sites/default/files/2021-01/RFP-DS-PP%2006242020.pdf .\n1 / 11\nFigure R1.2.1 3D model in the COMSOL simulation\n2. Physics\n- Laser heating: simulated using the Geometrical Optics interface to trace 592 nm laser pulses\n(5 ns duration, 3 mJ energy, Gaussian profile).\n- Heat transfer: modeled via the Heat Transfer in Solids interface to capture transient heat\naccumulation and dissipation.\n3. Simulation protocol:\n- Total duration: 5 s, divided into five 1 s cycles.\n- Within each cycle:\n• First 5 ns: laser excitation and localized heating of the resonator.\n• Remaining time: passive cooling without laser input.\n- Initial conditions:\n• Cycle 1 starts at 293.15 K (room temperature).\n• Each subsequent cycle begins with the final temperature distribution from the previous\ncycle.\nR 1.3 Whilst an improved resonator temperature stability is admirable, I remain unconvinced\nthat the work presented here is significant enough a development to warrant publication in\nNature Communications. As pointed out by Reviewer 2, a patent was published in 2013 (WO\n2013/175235 A1, see Figure 4) that describes using a split-ring resonator, very similar to that\nreported by the authors, suggesting that the idea is not original.\n2 / 11\nhttps://patentimages.storage.googleapis.com/77/0b/52/097e05c3bd3d96/WO2013175235A1.p\ndf\nResponse: We acknowledge that planar split-ring resonators (SRRs) were mentioned in the earlier\npatent WO 2013/175235 A1 as potential cavity architectures for maser applications. However, it does\nnot demonstrate maser operation, Purcell enhancement, or any experimental validation. In general,\nwe wish to point out that it is infinitely less-trivial to devise actual SRRs-based designs than just\nthinking that this might be possible, which is enough in a patent. We also wish to remark that patents\ndo not automatically represent prior art, and may protect technical solutions that were never actually\ndemonstrated and are extremely challenging to implement, providing they actually work.\nIn contrast, our study reports the first experimental realization of a room-temperature maser based\non an all-metallic structure, supported by:\n• comprehensive full-wave electromagnetic design, theory and demonstration\n• quantitative analysis of Purcell enhancement, and\n• evaluation of thermal robustness.\nMoreover, the resonator described in the patent corresponds to a planar structure (~100 µm thick),\nwhile our device is a three-dimensional toroidal SRR cluster that achieves volumetric magnetic\nconfinement. This architecture enables a deep-subwavelength mode volume and large Purcell factor,\nwhile maintaining exceptional thermal stability over thousands of optical excitation cycles. Based on\nour work, we do not believe that planar SRRs can reach sufficient Purcell factors over an appreciable\ngain medium volume: a volumetric design is required.\nThus, although conceptually related, the present work represents a substantial technical and\nexperimental advance, establishing a clear and verifiable step forward in the development of metallic\nresonators for enhancing the emission rates of magnetic quantum emitters. We believe that rejection\nof our paper based on this patent would be scientifically unfounded.\n3 / 11\nReviewer 2:\nR2.0: The authors have addressed all my comments, and I now recommend the paper for\npublication.\nResponse: We sincerely thank the reviewer for their thorough evaluation and positive\nrecommendation for publication.\n4 / 11\nReviewer 3:\nR3.0: The authors have made efforts to address the reviewers’ comments and revise their\nmanuscript accordingly. I see improvements; however, there are still several issues that I\nbelieve require further attention:\nResponse: We thank the reviewer for their constructive feedback and for recognizing the\nimprovements in our revised manuscript. We appreciate the remaining points raised and have\ncarefully addressed each of them in this version.\nR3.1: (R3.3) As noted in my previous report, this work focuses on pulsed maser operation at\na very low repetition rate (1 Hz), which has limited applicability. The authors provided in their\nresponse some estimates regarding the feasibility of supporting continuous-wave (CW)\ndiamond NV-based masers—especially in relation to the resonator—but these were not\nincorporated into the revised paper.\nResponse: We thank the reviewer for highlighting this important point. Our focus on pulsed operation\nat 1 Hz in this work stems from the need to resolve the critical challenge of thermal instability in deep-\nsubwavelength room-temperature masers for enabling more advanced applications. The present\nwork therefore serves as a proof-of-principle demonstration of the resonator’s thermal stability and\nPurcell enhancement under pulsed excitation. We fully agree that continuous-wave operation\nrepresents the next step toward broader applicability. Accordingly, we have now incorporated a\ndiscussion of CW feasibility into the revised manuscript and Supplementary Information, as detailed\nbelow.\nRevision: We have inserted the following passage into the main manuscript:\nIn addition, the geometry of the clustered SRR can be tailored to achieve operation at different\nfrequencies, accommodating other magnetic quantum emitters such as nitrogen–vacancy (NV)\ndefect centers in diamond that are used in continuous-wave (CW) masers. A detailed analysis\nof the thermal stability of CW masers incorporating the SRR cluster is provided in Supplementary\nDiscussion SII B.\nAdditionally, we included the following text in the Supplementary Material:\nSupplementary Discussion SII\nB - Thermal Stability for Continuous-Wave Maser Operation:\nWhile the present work focuses on pulsed operation, the intrinsic thermal properties of the SRR\ncluster make it particularly promising for stable CW applications.\n5 / 11\nTo evaluate the potential for continuous-wave (CW) operation, we analyzed the thermal\nperformance of the silver toroidal SRR cluster and compared it with that of conventional dielectric\nresonators. Consider, for instance, a maser employing NV centers in diamond with a sapphire\nresonator, where continuous laser pumping at 400mW caused a frequency shift of 25MHz due\nto thermal heating. This corresponds to an estimated temperature rise: 𝛥𝑇 = 35∘𝐶.\nAl2O3\nAssuming that the toroidal SRR cluster has a similar size and is operated under the same\nconditions as the sapphire resonator, the temperature increase 𝛥𝑇 could be estimated as\n%&&\n𝛥𝑇 /𝛥𝑇 ∼ 𝜅 /𝜅 ≈ 0.1, following the Fourier's law, where 𝜅 and 𝜅 are the\n%&& '(#)$ '(#)$ '* '(#)$ '*\nthermal conductivities of sapphire and silver, respectively. Given that the volumetric thermal\nexpansion coefficient of silver is approximately 54×10+, K-1, the geometric expansion would be\nless than 0.02%. Consequently, the frequency variation in the SRR cluster is expected to be\nnegligible compared to that observed in the sapphire resonator.\nR3.2: (R2.1) The discussion of the spontaneous emission rate and the Purcell factor is non-\nquantitative. Additionally, the relationship to the Einstein B-coefficient does not seem to be\nincorporated into the revised paper.\nResponse: At microwave frequency, spontaneous emission is negligible compared to stimulated\nemission, since\n'#%\n=\n0123$\n, where 𝐴 and 𝐵 are the Einstein coefficients of spontaneous emission\n/#% 4$ 56 56\nand stimulated emission, respectively, and 𝜈 is the photon frequency1,2. Owing to the 𝜈7 dependence,\nthe spontaneous emission rate is many orders of magnitude smaller than the stimulated emission\nrate in the GHz regime and can therefore be safely neglected in the maser dynamic model.\nFor the Purcell-enhanced stimulated emission, we have the following equations:\n𝑞̇ = −𝑘 𝑞 +𝐵𝑁𝑞\nc\n⎧\n9 ℏ;<#=\n⎪ 𝐵 = & ' #\n>?( \n;\n𝑘 = \n⎨ 4\n@\n)*+,\n⎪ 𝐹 = 7B$ ( ⋅ @)*+,\n⎩ A >C# ?(\nHere, the Einstein coefficient of stimulated emission B scales inversely to the mode volume 𝑉 ,\nand\nD\nthe cavity photon decay rate 𝑘 is inversely proportional to the loaded Q factor 𝑄 .\nThe threshold\n4 (EFG\nof population inversion 𝑁 for maser action is therefore\nH2\n𝑁 = \nI-\n∝\n?( \n∝\n6\n.\nH2\n/ @\n)*+,\nJ(\nHence, achieving a small mode volume and a high loaded Q-factor, corresponding to a large Purcell\nfactor, is essential for low-threshold and efficient maser operation.\n6 / 11\nRevision: We have incorporated the quantitative relation between the Einstein B-coefficient, mode\nvolume, and Purcell factor into the revised manuscript (Methods – Maser Dynamic Modelling) as\nfollows:\n𝐵 stands for the Einstein coefficient of the stimulated emission, which is directly linked to the\nresonator's mode volume as follows:\n𝐵 = \n9&ℏ;<' #=#\n>?( \nwhere 𝜇 is the vacuum permeability, ℏ the reduced Planck constant, 𝜈 = 1.76×1066 𝑟𝑎𝑑 ⋅𝑠+6 ⋅\nK L\n𝑇+6 the electron gyromagnetic ratio, and 𝑇 = 2.1~𝜇𝑠 the spin-spin relaxation time of pentacene.\nits rate is vanishingly small compared to stimulated emission.\nAnd:\nAssuming steady-state operation (𝑞̇ = 0), the threshold population inversion is given by:\n𝑁 = 𝑘 /𝐵 = 4.4×1067. Since the Purcell factor scales as 𝐹 ∝ @)*+,, it follows that\nH2MLN2E(G c A\n?( \n6\n𝑁 ∝ , confirming that an ultrasmall 𝑉 and high 𝑄 (i.e. a large 𝐹 ) are key to\nH2MLN2E(G D (EFG A\nJ.\nachieving low-threshold, efficient maser operation.\nR3.3: (R2.4) The definition of the mode volume used in the text is somewhat controversial in\nthis context. Specifically, normalizing by the maximum of the magnetic field can produce\nartificially low “magnetic mode” volumes if there is a localized magnetic field “hot spot.” In\ncontrast, if the microwave magnetic field is relatively homogeneous across the resonator\nvolume, the chosen definition might be reasonable. However, a more relevant approach could\nbe to define the 50% or 90% mode volume, referring to the region where 50% or 90% of the\nmicrowave magnetic energy is localized. This might explain the discrepancy between the\nmeasured data and the mode volume calculation (a factor of ~3). I believe the authors should\ndiscuss this point in more detail.\nResponse: In our study, the mode volume 𝑉 was calculated within the resonator bore region\nO\noccupied by the gain medium, as the Purcell enhancement depends on the spatial overlap between\nthe resonator’s magnetic field and the emitters’ magnetic dipoles. Magnetic energy outside this\nregion does not contribute to emission enhancement and is therefore excluded from the effective\n𝑉 relevant to maser operation. To verify that the simulated mode volume is not biased by field\nO\nlocalization or numerical artifacts, we performed a statistical analysis of the magnetic energy density\ndistribution inside the bore. The field exhibits moderate uniformity, with a coefficient of variation of\n0.45 and a median-to-mean ratio of 0.94. The ratio of maximum to median magnetic energy density\nis 2.40, indicating that the field maximum is only modestly higher than the typical value and is not\ndominated by an isolated hotspot.\n7 / 11\nFurthermore, the minimal sub-volumes enclosing 50% and 90% of the magnetic energy are 𝑉 =\nPK\n0.010cm7and 𝑉 = 0.025cm7, corresponding to 31% and 78% of the bore volume (𝑉 = 0.032cm7),\nQK RSTU\nrespectively. These results demonstrate that the magnetic energy is broadly distributed across the\ngain region, confirming that the simulated mode volume is not artificially reduced by localized peaks.\nThe simulations provide an idealized lower bound assuming perfect geometry and complete dipole\nalignment, whereas the experimental value naturally reflects structural imperfections, partial filling,\nand non-uniform emitter orientation. The resulting threefold difference between the simulated and\nexperimental mode volumes is therefore reasonable and does not alter the physical interpretation of\nthe Purcell enhancement. Together with the quantitative field-distribution metrics presented above,\nthis agreement confirms that our estimation of 𝑉 is physically meaningful and representative of the\nO\nactual electromagnetic environment experienced by the emitters.\nRevision: We have inserted the following passage into the discussion section of the main manuscript:\nTo confirm that the simulated mode volume is not artificially reduced by local field maxima, we\nquantitatively analyzed the magnetic energy distribution in the resonator bore region. The field\nwas found to be broadly uniform without strong localization, as detailed in Supplementary\nDiscussion SII A.\nAdditionally, we included the following text in the Supplementary Material:\nSupplementary Discussion SII\nA - Statistical Evaluation of the Mode Uniformity in the Resonator Bore Region\nTo evaluate whether the calculated mode volume is affected by localized field peaks or numerical\nartifacts, we performed a statistical analysis of the simulated magnetic-energy density within the\nbore region of the clustered SRR. The mode volume 𝑉 was evaluated only within this region,\nm\nsince the Purcell enhancement depends on the overlap between the resonator’s magnetic field\nand the emitters’ dipoles, while magnetic energy outside the gain medium does not contribute to\nthe emission process.\nThe simulated magnetic energy distribution exhibits moderate uniformity, with a coefficient of\nvariation (CV) of 0.45 and a median-to-mean ratio of 0.94. The ratio of maximum to median\nmagnetic-energy density is 2.40, showing that the field maximum is modestly higher than the\ntypical value and not dominated by a single hotspot. Furthermore, the minimal sub-volumes\nenclosing 50% and 90% of the magnetic energy are 𝑉 = 0.010 cm3 and 𝑉 = 0.025 cm3,\nPK QK\ncorresponding to 31% and 78% of the bore volume ( 𝑉 = 0.032 cm3), respectively. These\nbore\nresults demonstrate that the magnetic energy is broadly distributed across the emitter region.\nThe simulation represents an idealized lower bound assuming perfect geometry and complete\ndipole alignment, whereas the experimental value includes realistic effects such as structural\ndeformation, partial filling, and non-uniform emitter orientation. The resulting threefold difference\n8 / 11\nbetween the simulated (0.025 cm3) and experimental (0.072 cm3) mode volumes is therefore\nreasonable and consistent with realistic experimental conditions. Combined with the statistical\nmetrics presented above, these results confirm that the mode volume estimation used in this\nwork is physically meaningful and representative of the actual electromagnetic environment\nexperienced by the emitters.\nR3.4: (R2.5a) The size of the crystal and the specific dimensions of the device are crucial\ndetails that should be linked more explicitly to the required optimal resonator geometry. This\nappears to be another instance where the authors’ response did not result in changes to the\nmanuscript.\nResponse: To facilitate precise reproduction of our design, the 3D model of the resonator is now\navailable as an STL file on Zenodo (DOI: https://zenodo.org/doi/10.5281/zenodo.10012323), which\ncan be directly used for 3D printing. In addition, we have included the dimensions of the cylindrical\ngain medium in the revised manuscript.\nRevision: We have expanded the introduction of the main manuscript as follows to link the size of\npractically grown gain media to that of a promising resonator:\nHowever, as spin transition frequencies are commonly in the gigahertz range, i.e., with emission\nwavelengths λ on the order of tens of centimeters, leveraging the magnetic Purcell effect comes\nm\nwith the inherent challenge of dealing with the large ratio between λ 3 and achievable values of\nm\nmode volume V . In practice, the physical dimensions of gain media that can be grown\nm\nexperimentally, such as organic crystals or diamond-based samples, occupy physical volumes\nthat are roughly three orders of magnitude smaller than λ 7 (i.e., ∝ cm7 ). To address this\nm\nmismatch, the resonator geometry must be engineered such that the region of strong magnetic\nfield is confined to a volume only slightly larger than that of the gain medium, with its orientation\noptimized for alignment with the emitters’ magnetic dipole moments. This “squeezing” of the\nmagnetic mode maximizes spatial overlap with the active medium, thereby enhancing the\nefficiency of energy exchange.\nAdditionally, we have included the dimensions of the cylindrical gain medium:\nAs depicted in Fig.1.b, the pentacene:p-terphenyl (pc:pt) crystal (1.5 mm radius × 10 mm height)\nis positioned within the bore of the clustered SRR, which is enclosed inside a copper cavity. The\nbottom 5 mm of the pc:pt crystal includes a central axial hole of 0.5 mm radius, whereas the top\n5 mm remains solid.\nR3.5: (R2.8) Here as well, I believe the text of the paper should be modified accordingly,\nreflecting the issues raised in the response.\nRevision: We have replaced “thermal fluctuations” with “thermal variations” in the manuscript and\nrevised the corresponding part of the introduction to clarify this point and avoid ambiguity as follows:\n9 / 11\nHowever, dielectric resonators with extremely high permittivities, such as SrTiO3, are inherently\nsensitive to temperature variations due to their large temperature coefficient of permittivity. Even\nminor room temperature variations, or the unavoidable thermal heating coming from optical\nexcitation or read-out, would inevitably shift the mode frequency away from the quantum emitter\nlinewidth. As a result, the system must be continuously retuned, hindering stable operation and\nlimiting the practicality of such dielectric resonators.\nR3.6: It seems somewhat unusual to discuss frequency stability using MHz-scale data,\npresumably due to the short-pulse operation. Comparing this to the ~10 kHz linewidths of CW\nmaser systems cited in the references is difficult. A more direct comparison or\nacknowledgment of the differences in operational modes would be helpful.\nResponse: We agree that the frequency-stability discussion in our pulsed maser system should not\nbe directly compared to the narrow (~10 kHz) linewidths of continuous-wave (CW) masers. To clarify\nthis distinction, we have added an explicit paragraph in Supplementary Discussion SII B explaining\nthat our measurements characterize pulse-to-pulse spectral stability rather than absolute linewidth,\nand emphasizing the fundamental difference between the transient pulsed regime and the steady-\nstate CW regime.\nRevision: we included the following text in Supplementary Discussion SII B of the Supplementary\nMaterial:\nWhile the present work focuses on pulsed operation, the excellent thermal properties of the SRR\ncluster make it particularly promising for stable continuous-wave (CW) applications. The\nobserved MHz-scale linewidths in our pulsed maser experiments arise from the Fourier limit\nassociated with microsecond pulse durations, rather than from intrinsic decoherence or thermal\ninstability. In contrast, CW masers operate in a steady-state regime that can achieve linewidths\non the order of tens of kilohertz. Although the absolute linewidths are therefore not directly\ncomparable, the measurements presented here demonstrate exceptional pulse-to-pulse spectral\nstability of the SRR-based maser over prolonged operation times.\nR3.7: (R3.1) The “quasi-2D” structures mentioned by the authors in relation to the quoted\nreferences of previous work have heights of approximately 0.5 mm and 1.4 mm—borderline\nfor the intended application. However, there is no fundamental reason why these could not\nbe extended to heights of, for example, 3 mm or more to fit the crystal used in this work. It is\ntherefore unclear why the authors dismiss these structures as unsuitable for the current\napplication.\nResponse: The reviewer is correct that quasi-2D resonator geometries could, in principle, be\nextended vertically. Our toroidal SRR cluster was inspired by planar SRR concepts but represents a\nconceptual three-dimensional evolution developed through detailed electromagnetic design to\nachieve genuine volumetric magnetic confinement. The resulting structure supports a uniform\nmagnetic field across the crystal volume, exhibits deep-subwavelength compactness without\nincorporating any dielectric materials (unlike designs such as that in NCBI\n10 / 11\nhttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC6948142/ which rely on rutile loading), and maintains\nexcellent thermal stability. This three-dimensional architecture differs fundamentally from a simple\ngeometric extrusion and enables strong Purcell enhancement together with stable operation under\nrepeated optical excitation.\nR3.8: 4. (R3.3) The condition C>1 is typically considered indicative of strong (rather than weak)\ncoupling. Overall, while the revised manuscript addresses some of the reviewers’ concerns,\nI believe further clarifications and revisions are needed in the areas detailed above.\nResponse: We thank the reviewer for pointing out the misuse and potential confusion in terminology.\nOur previous use of “relatively weak coupling” was intended to indicate a cooperativity value lower\nthan those reported in other maser systems3,4, rather than to suggest C<1 in the strict formal sense.\nTo avoid confusion, we have revised the manuscript accordingly, replacing “relatively weak coupling”\nwith “strong coupling” to align with the conventional definition for C>1.\nRevision: We have replaced “relatively weak coupling” with “strong coupling” in the discussion\nsection of the manuscript.\nReferences\n1. Einstein, A. Strahlungs-Emission und -Absorption nach der Quantentheorie. Deutsche\nPhysikalische Gesellschaft 18, 318–323 (1916).\n2. Zangwill, A. Modern Electrodynamics. (Cambridge University Press, 2013).\n3. Breeze, J. D., Salvadori, E., Sathian, J., Alford, N. McN. & Kay, C. W. M. Room-temperature\ncavity quantum electrodynamics with strongly coupled Dicke states. Npj Quantum Inf. 3, 40 (2017).\n4. Breeze, J. D., Salvadori, E., Sathian, J., Alford, N. McN. & Kay, C. W. M. Continuous-wave\nroom-temperature diamond maser. Nature 555, 493–496 (2018).\n11 / 11"}]

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