Maser Threshold Characterization by Resonator Q-Factor Tuning

Whereas the laser is nowadays an ubiquitous technology, applications for its microwave analogue, the maser, remain highly specialized, despite the excellent low-noise microwave amplification properties. The widespread application of masers is typically limited by the need of cryogenic temperatures. The recent realization of a continuous-wave room-temperature maser, using NV$^-$ centers in diamond, is a first step towards establishing the maser as a potential platform for microwave research and development, yet its design is far from optimal. Here, we design and construct an optimized setup able to characterize the operating space of a maser using NV$^-$ centers. We focus on the interplay of two key parameters for emission of microwave photons: the quality factor of the microwave resonator and the degree of spin level-inversion. We characterize the performance of the maser as a function of these two parameters, identifying the parameter space of operation and highlighting the requirements for maximal continuous microwave emission.


INTRODUCTION
The first maser system was realized using ammonia molecules in the gas phase 1 and applications for signal amplifiers, frequency standards or as spectrometers were subsequently proposed.However, the need for cryogenic and/or high vacuum environments restricted miniaturization and integration towards more general applications.Maser research and development was mainly focused on low-noise microwave receiving systems for deep-space antenna networks 2 , and several other maser systems based on ruby 3 , atomic hydrogen 4 or Rydberg atoms 5 were realized.These systems were all still subject to the same restrictions and the focus on fundamental research into masers declined.The field was reinvigorated upon the realization of a room-temperature pulsed maser in an optically pumped crystal of pentacenedoped p-terphenyl, placed inside a high quality factor microwave resonator [6][7][8] .Here, masing was achieved not only a) Electronic mail: c.zollitsch@ucl.ac.uk b) Electronic mail: christopher.kay@uni-saarland.dewithout the need of cryogenics or a high-vacuum environment, but also with easily accessible optical pump rates.Shortly afterwards, a proposal for a continuous-wave roomtemperature maser in optically spin polarized, negatively charged nitrogen vacancy (NV − ) centers in diamond 9 and its consecutive experimental realization 10 followed.The excellent low-noise amplification properties of the maser have been demonstrated in recent work [11][12][13] and has found application in enhanced quantum sensing of molecular spin ensembles 14 .
ħ ω M The light blue sapphire ring is held by two Teflon holders.On the top is the iris coupled waveguide port.The coupling to the resonator is adjusted by a Teflon screw with a metal ring at its tip.The 532 nm optical pump laser is aligned along the symmetry axis of the cylindrical resonator, while the static magnetic field B 0 is perpendicular to it.Finally, the diamond sample is placed inside the resonator.b Loaded Q-factor as a function of the resonator coupling, showing the full range of coupling, with the diamond sample in the resonator.
The search for other solid-state maser materials, such as SiC 15 has continued, and an application in quantum technology by creating a maser system based on Floquet states in Xe atoms 16 has also been reported.
With the increased focus on room-temperature solid-state masers a quantitative experimental characterization of the parameter space of operation supplements the current endeavours to optimize the performance of such systems.Understanding the behaviour of the maser performance, defined by the level-inversion of the spin ensemble and the loaded Qfactor Q L of the resonator, can lead to higher output power maser systems.To guarantee reproducibility and maximise the power output of maser-based technologies, a complete understanding of the system parameter space, including the minimal requirements to surpass the masing threshold, is required.
Here, we present an experimental setup that we exploited to investigate the maser performance of a NV − spin ensemble hosted in diamond as a function of the resonator quality factor and the degree of spin level-inversion.In the resulting maser threshold diagram, we can clearly identify the threshold for maser action, thereby obtaining a set of experimental boundary conditions for the optimal operation of a maser system.Additionally, our optimized setup yields the highest continuous-wave maser output power reported to date.

RESULTS AND DISCUSSION
Experimental setup: We use a cylindrical dielectric ring resonator made of sapphire to deliver/detect resonant microwaves to/from the NV − centers contained in the diamond host.A key parameter for the continuous maser emission is the loaded Q-factor, Q L , of the resonator, which is defined by 1 /Q L = 1 /Q int + 1 /Q ext with the internal Q-factor, Q int , and the external Q-factor, Q ext .Although sapphire dielectric resonators exhibit low dielectric losses 17 , radiative losses typically dominate and prevent high internal quality factors.Hence, to suppress radiative losses the resonator is placed inside a metal cavity.The cavity design is further constrained by two conditions: (i) it has to fit between the poles of our electromagnet system, which gives the static magnetic field B 0 used to tune the energy levels of the NV − spins via the Zeeman interaction, and (ii) the resonance frequency of the cavity containing the sapphire ring is required to be within the 9 − 10 GHz (Xband) frequency range, to allow a fast pre-characterization of the NV − spin ensemble by conventional electron spin resonance (ESR).To this end, we designed a cylindrical cavity made of oxygen-free high-thermal conductivity (OFHC) copper, plated with thin layers of (first) silver and (second) gold to prevent oxidation of the metal surfaces, thereby minimising additional resistive losses.
Figure 1 (a) shows a 3D schematic of the fully assembled cavity cut at a symmetry plane to reveal the interior.The cavity has sample entries at opposite ends of the cylinder symmetry axis for sample and optical access.In the center of the cavity the resonator is held in place by two wire-frame Teflon holders.The holders are machined to have minimal volume in order to minimize additional dielectric losses.Microwave power is coupled in and out via a single waveguide iris port on the top.This port represents the external quality factor Q ext and together with Q int the resonator coupling k = Q int /Q ext is defined.k is controlled via a Teflon screw with a metal ring at its tip.Changing the coverage of the iris by the metal ring allows a continuous change 18 from over-coupled to under-coupled: the regimes where Q L is dominated by external losses or intrinsic losses, respectively.Figure 1  (b) shows the achievable Q L as a function of k for our resonator-cavity system, loaded with the diamond sample.The resonator parameters are extracted from microwave reflection measurements, using a vector network analyzer.By fitting the microwave reflection as a function of frequency with a Lorentzian model function the parameters for resonance frequency, resonator coupling and internal quality factor are determined.A description of the model function can be found in the methods section.We use the TE 01δ mode for our experiments, where the electric field is mostly contained in the sapphire ring and the magnetic field is mostly focused in the bore.Without a sample, the sapphire resonator has a resonance frequency ω res /2π = 9.25 GHz and the Q L = 42, 500 when fully under-coupled.We define the resonator as fully under-coupled when k = 0.0027, where the Iris is no longer covered by the metal tip of the Teflon screw.
Here, Q L remains unchanged upon further extraction of the screw.For detailed cavity and resonator dimensions see Supplementary Note 4 19 .
Maser working principles: The process of continuous emission of microwave photons from the NV − centers is schematically shown in Fig. 2 (a) to (c).The diamond hosting the NV − centers is placed inside a high-Q resonator, which is highly under-coupled (k ≪ 1).The resonator may be pictured in analogue to its optical counterpart the laser, with one perfectly reflective mirror and one weakly transmitting mirror.The latter represents the iris coupled single microwave port on the microwave cavity.
The applied static magnetic field B 0 lifts the degeneracy of the |± 1⟩ states and, tunes the |−1⟩ state energetically below the |0⟩ state such that the splitting hω M is resonant with the microwave resonator frequency.The experiment is performed only on one sub-set of NV − centers which are aligned with the external magnetic field B 0 .This provides the shown level structure, having the largest Zeeman splitting of the energy levels and consequently the largest initial population difference at Boltzmann equilibrium.By illuminating the NV − centers continuously with a 532 nm laser, the spin populations which are initially at Boltzmann equilibrium are predominately pumped into the |0⟩ state 20 , resulting in a levelinversion (see Fig. 2 (a)).A description of the optical spin polarization process is found in Supplementary Note 2 19 .Finally, the laser polarization is required to be aligned along the NV − defect axis to achieve most efficient pump rates 21,22 .
To trigger a collective stimulated emission, an initial photon with hω M is required.This is provided either by an externally applied seeding photon, due to spontaneous emission or thermal photons (see Fig. 2 (b)).From this point an avalanche of stimulated photons is created, forming a coherent microwave field inside the resonator (see Fig. 2 (c)).If the laser pump rate is sufficient to maintain the level inversion and the resonator loaded quality factor is high enough to support a large enough coherent microwave field, continuous microwave emission is achieved.
-60 -80 -100 -120 NV defect axis alignment: In our experimental setup, the diamond sample is held inside a quartz ESR tube, supported between two additional quartz tubes fitted inside the first one.The tube is inserted into the cavity and positioned such that the diamond is located at the center of the sapphire ring where the magnetic component of the microwave field is largest.The cavity with sample is mounted between the poles of a electromagnet system.We connect either a conventional ESR spectrometer, a vector network analyzer or a spectrum analyzer to the microwave port of the cavity, to perform low-power microwave spectroscopy of the NV − spin transitions or to study the maser emission.A goniometer is attached to the sample tube, allowing a precise rotation of the quartz tube containing the diamond with respect to the static magnetic field.Conventional ESR as a function of B 0 and rotation angle is performed to find an orientation of the diamond where the defect axis of a sub-set of NV − centers is mostly parallel to the applied magnetic field.For such an orientation the NV − spins feature an energy level scheme of the electronic ground state as schematically shown in Fig. 2 (a) and the states |0⟩ and |±1⟩ can be considered pure.This can be characterized by the frequency/magnetic field splitting between the low-field (|0⟩ → |+1⟩) and high-field (|0⟩ → |−1⟩) transitions corresponding to twice the zero-field splitting D. Away from this alignment the states become mixed, resulting in a smaller splitting than 2D between the two allowed transitions 23,24 and a smaller maximal achievable spin polarization.Without laser illumination, we find a maximal splitting of about 205 mT or 5.762 GHz, agreeing well with twice the zero-field splitting, D, of NV − centers in diamond.

Maser emission spectrum characterization:
Having optimized the orientation of the NV − centers, we characterize the performance of our maser setup by analyzing the microwave emission spectrum.Figure 3 (a) shows the color encoded maser emission power |A M | as a function of frequency and static magnetic field for a laser pump rate w L = 430 s −1 and with the resonator fully under-coupled, Q L ≈ Q int , which gives a loaded quality factor Q L of 33, 500.The three bright lines represent the maser emission of the three 14 N hyperfine transitions of the NV − centers oriented along B 0 .In this configuration, we achieve a maximum maser emission power of −56.5 dBm.The maser emission shows a finite frequencymagnetic field dispersion, where the maser power is maximal in the center of the line.This is illustrated in Fig. 3 (b) which depicts the maser emission power as a function of frequency for two fixed magnetic fields.The dispersion results from the hybridization of the microwave resonator mode and the resonant NV − transition.The resonator resonance frequency ω res /2π = 9.12 GHz lies at the center of the middle maser emission line.The inverted spin population causes a dispersive shift of the resonator frequency to lower frequencies for the lower magnetic field emission line and respectively to a shift to higher frequencies for the higher magnetic field emission line.The maser dispersion can be described via a Tavis-Cummings model with an inverted spin polarisation, where the frequency range covered by the maser emission lines increases with increasing inversion.
In order to determine the dependency of the maser output on the laser pump rate w L and the loaded quality factor Q L , both were varied systematically.The latter is dependent on the resonator coupling k which is controlled by the iris screw.Figure 4 (a) shows the peak maser power |A M | of the central maser line as a function of w L (from 165 s −1 to 695 s −1 ) and Q L (from 14, 000 to 33, 500).Note, that after each change of w L we wait 45 min before starting measurements to allow the diamond to reach a thermal equilibrium (see Supplementary Note 1 for details 19 ).For low loaded quality factors, i.e. an over-coupled resonator, there is no microwave emission for all laser pump rates studied and the signal amplitude is represented by the noise floor of the spectrum analyzer.Note that critical resonator coupling (k = 1) is achieved at Q L ≈ 20, 000, marking the transition between over-and undercoupling 25 .With increasing Q L a weak microwave emission is observed for moderate to high laser pump rates.Here, the emission spectrum is broad and governed by the amplification of thermal photons residing in the resonator 9 .In this region, the resonator losses are still too high to allow the build-up of sufficient stimulated photons for continuous masing.For Q L > 26, 000 and w L > 400 s −1 the rate of stimulated emission exceeds the losses in the resonator and the spin system, and continuous masing is established.
Here, T 1 is the effective longitudinal relaxation time of the NV − spins, g 0 gives the strength of the magnetic dipole coupling between a single spin and a single microwave photon 27 , N is the absolute number of spins per hyperfine transition and per NV − defect axis and κ 0 and γ are the resonator and spin loss rates (HWHM), respectively.We estimate g 0 /2π by finite element simulations of the magnetic field profile inside the sapphire ring resonator 28 to an average value of 0.244 Hz and determine γ/2π and N via lowpower microwave spectroscopy of the low-field transition (|0⟩ → |+1⟩) to 530.8 kHz ± 7.7 kHz and 2.32 × 10 13 , respectively.A detailed derivation of these system parameters is found in Supplementary Note 3 19 .The resonator loss rate is defined through Q L as κ 0 = ω res /2Q L , which we control by changing the resonator coupling.The scaling factor η modifies the optical pump rate defined for a two-level system within the framework of the Tavis-Cummings model to take all seven energy levels involved in the pumping of a single NV − spin into account.We extract η = 14.05 from the calculated spin-level inversion as a function of w L 9,10 , by solving the set of optical pump rate equations in the steady-state 29 .For T 1 we explicitly take the influence of an optical pump into account.In addition to polarizing spins, the pump leads to excessive heating of the diamond, which decreases the T 1 time 30 .We determine T 1 for each w L , finding 5.2 ms for low w L and a minimum of 1.5 ms for the highest w L , where a detailed description of the T 1 dependence on w L is given in Supplementary Note 1 19 .The white dashed line in Fig. 4 (a) gives the masing threshold w th as a function of the loaded quality factor and is in excellent agreement with the experimentally found threshold of our maser, when acknowledging the reduction of the relaxation time due to high pump rates.For comparison, we include the expected threshold for a fixed T 1 = 5.2 ms (grey dashed line), demonstrating the significant influence of laser heating.

CONCLUSION
To summarize, our characterization setup allows the resonator coupling to be continuously and precisely adjusted between over-and under-coupled in conjunction with the optical pump rate.This permits the detailed study of the performance of the maser as a function of the rate of stimulated emission and the degree of level-inversion.Control of these parameters enabled the first experimental verification of the maser threshold equation over a wide parameter space.Thus, the regions of microwave emission below the detection limit, thermal photon amplification and continuous masing could be identified in a NV − diamond maser.
Our results highlight an efficient operation of the maser is in the highly under-coupled regime.Clearly, this limits the maser output power as only a small fraction of the microwaves in the weakly coupled resonator can exit.For applications, a higher output is essential and, hence, a larger coupling to the resonator is required.The resulting threshold diagram suggests that either higher pump rates or a resonator with a higher Q int will enable a larger coupling of the resonator.However, higher pump rates not only require bulky laser systems but cause the sample to heat up.We demonstrate that the T 1 relaxation time is shortened, thus reducing the spin inversion.Indeed, it is clear for our results that pump rates can be reduced to a level where small (e.g.< 1 W) laser diodes can be employed instead.Therefore, increasing the Q factor of the resonator is a more viable approach, although it is necessary to point out that this will decrease the bandwidth 11 .Finally, the optimized design yielded an increase of the maximal maser output of more than three orders of magnitude compared to the initial report on a NV − -based maser from −90.3 dBm to −54.1 dBm 10 .We attribute this mainly to an improved heat management of the diamond sample to limit a reduction in T 1 time and consequently limit the increase of the masing threshold.Thus it provides not only a blueprint for solid-state based maser systems and but also sets a benchmark for future characterization and optimization studies.

METHODS
Diamond Sample: Our diamond sample is of rhombic shape with its long axis having about 5 mm, its short axis having about 4 mm and a thickness of 1 mm.The diamond consists of natural abundance carbon and we estimate the total number of NV − to 2.78 × 10 14 or 0.16 ppm (see Supplementary Note 3 for details on the number of spins estimate 19 ).At room temperature and no optical pump the NV − feature a T 2 = 25 µs, determined by pulsed ESR.
Laser Pumping: We use a 532 nm Coherent Verdi V-5 laser to optically pump the spin population of the NV − centers.The laser features a spot size of about 4 mm and hits the diamond sample on its flat edge, an area of about 4 mm × 1 mm.We determine the laser pump rate as w L = σ P pump /A pump hω pump (Ref.10), with the one-photon absorption cross-section for NV − centers σ = 3.1 × 10 − 21 m 2 , the pump laser power P pump , the laser spot area A pump and the laser frequency ω pump .
Microwave Spectroscopy: We study the microwave emission from the maser with a Keysight N9020B MXA 10 Hz to 44 GHz spectrum analyzer.To improve the SNR we preamplify the maser signal, using a Mini Circuits Low Noise Amplifier ZX60-06183LN+ prior detection by the spectrum analyzer.The low power microwave spectrocopy was carried out using an Anritsu vector network analyzer MS46122B 1 MHz to 43.5 GHz.For conventional ESR measurements a Bruker EMXplus spectrometer is used.
Resonator Coupling: We determine the microwave resonator coupling k by measuring the microwave reflection as a function of frequency with a VNA, far detuned from the NV − centers.We fit the magnitude of the reflection scattering parameter |S 11 | of the resonator for different positions of the Iris screw, using a Lorentzian model function derived from an equivalent circuit model, which depends on k 31 .
In addition, we can extract the internal quality factor Q int and via the relation for the resonator coupling k = Q int /Q ext the external Q-factor Q ext and hence the loaded quality factor Q L , as well as the resonator frequency ω res and its loss rate κ 0 .
Supplementary Material: Maser Threshold Characterization by Resonator Q-Factor Tuning

SUPPLEMENTARY NOTE 1: TEMPERATURE INCREASE OF THE DIAMOND DUE TO OPTICAL PUMPING
The study of the maser output power as a function of optical pump rate requires considerable laser powers.The highest pump rate w L = 695 s −1 is achieved by a laser power of 1.05 W. Such high powers lead to excess heating of the diamond, which cannot be dissipated fully in our measurement setup.As described in the main text, the diamond sample is held inside a quartz ESR tube (5 mm outer diameter (OD)), supported between two additional quartz tubes (4 mm OD) fitted tightly inside the first one.Through the ESR tubes the diamond is indirectly thermally connected to the sapphire ring as well as the tube holders on the cavity body, which allows a finite thermal transport of the excess heating from the laser.In addition, we introduce a weak nitrogen gas flow into the cavity body to achieve an additional cooling effect.The nitrogen gas is connected to an inlet on the waveguide and flows via the Iris into the cavity and out via gaps in the ESR tube holders.Together, a finite thermal equilibrium in the diamond will establish after some time.For the experiments presented in the main text, we wait 45 minutes before a start of a measurement, after changing the laser power.The longitudinal relaxation time T 1 exhibits a strong temperature dependence S30 .For temperatures above room-temperature the relaxation rate 1 /T 1 predominately increases with T 5 .As the maser threshold w th is indirect proportional to T 1 (see main text), the threshold will increase due to the effects of laser heating.
To determine the temperature increase in our maser characterization setup, we first determine the change of the zero-field splitting ∆D as a function of optical pump rate, via low power microwave spectroscopy of the low-and high-field NV − spin transitions, and is shown in Supplementary Figure 1 (a).With increasing pump rate the zero-field splitting is decreasing.We define the room-temperature value of D = 2.88 GHz as the reference for the relative shift ∆D.Due to this temperature dependence, NV centers are highly sensitive thermometers.The temperature dependence can be described by a second order polynomial ?, Finally, we can find the corresponding T 1,eff times for each optical pump rate used in the maser characterization.Supplementary Figure 1 (c) shows the calculated temperature dependence of T 1 (solid red line) together with the T 1,eff for each pump rate used in the experiment (symbols), ranging from 165 s −1 to 695 s −1 .We calculate the relaxation time following the formalism presented in S30 Optical spin polarization scheme.Schematic of the electronic energy levels of a NV − spin ensemble.The triplet states 3 A 2 and 3 E are considered to be quantized along a principal axis of the zero-field splitting tensor.A continuous optical pump via a 532 nm laser can achieve a majority population of the 3 A 2 |0⟩ state, due to the spin selective intersystem crossing in the excited 3 E triplet.
The coefficients are A 1 = 0.06 s −1 , A 2 = 2.1 × 10 s −1 and A 3 = 2.2 × 10 −11 s −1 K −5 , ∆ = 1.17 × 10 −20 J is the dominant local vibrational energy and k b is the Boltzman constant.linewidth function of the homogeneously broadened spin packets in the sample for which we assume a Lorentzian lineshape ? )   where the spin-spin relaxation time is given by T 2 and (ω − ω res ) gives the detuning from the resonance frequency of the spin transition ω ESR .We estimate the rate of excitation via an averaged B 1,avg field strength, determined by finite element simulations of the sapphire ring resonator S28 to B 1,avg = 1.5 × 10 −7 T. On resonance and with T 2 = 25 µs, determined by conventional pulsed ESR, we estimate an excitation rate of 93.0 s −1 .At maximal laser pump rates, T 1 = 1.5 ms (see previous section) and the rate of relaxation results in 45.9 s −1 .The population difference n is calculated for an optical pump rate of 695 s −1 , following the rate equation formalism presented in S29.With the rate of excitation being about twice as high as the relaxation rate the spins are partially saturated.Comparing the calculated spin populations with and without a microwave drive we find that the microwave drive is reducing the population difference of the |0⟩ → |+1⟩ transition by 19.1 %.We scale the extracted value for the collective coupling g eff to remove the effects of saturation and yield g eff /2π = 390 kHz ± 3.1 kHz.
From the finite element simulations of the resonator we are able to determine an average single spin-photon coupling rate g 0 /2π = 0.244 Hz.Assuming a homogeneous B 1 -field over the diamond sample, and consequently a homogeneous distribution of g 0 , the collective coupling is given by g eff = g 0 √ S z , with S z being the population difference between the two spin states.With the un-saturated value for g eff we find S z = 2.57 × 10 12 .To find the total number of spins for a given hyperfine transition and single NV − defect axis, we scale S z up to 100 % spin polarization.At a pump rate of 695 s −1 we calculate a normalized population difference of 0.07 between (|0⟩ → |+1⟩).Together, we find a total number of spins per hyperfine transition and per NV − defect axis to N = 3.67 × 10 13 .To get a good agreement of the threshold equation with the experimental data, we require to adjust the number of spins by a factor of 0.63, leading to a final number of spins of 2.32 × 10 13 .An accurate determination of the absolute number of spins in a given sample is typically challenging and often only accurate to the order of magnitude.Having estimated N within a third to arrive at an excellent agreement with the experimental data corroborates our method to determine N.

SUPPLEMENTARY NOTE 4: MICROWAVE CAVITY AND RESONATOR DESIGN DETAILS
As described in the main text, the cavity's purpose is to suppress the radiation losses of the dielectric sapphire ring resonator.The dimensions are chosen such that the quality factor of the resonator is as large as possible, while still being able to be fitted inside of the poles of a normal conducting magnet system.Supplementary Figure 4 shows the technical drawing of the cavity, including all relevant dimensions.The body of the cavity is made of OFHC copper and is gold platted to protect the copper surfaces from oxidation.Prior gold platting a silver adhesion layer 1-2 µm thickness was put on the copper followed by a 0.2-0.3µm gold layer.Since the conductivity of gold is lower than of pristine OFHC copper the thickness of the gold layer was kept minimal.The cavity is coupled via an iris to a standard X-band waveguide.The sapphire dielectric ring resonator is of cylindrical shape with an outer diameter of 10 mm, an inner diameter of 5.1 mm and a height of 6 mm.We determine the mode volume of the resonator to V m = 5.1 × 10 −8 m 3 , via finite element simulations S28 .Together with the volume of the diamond of about V dia = 10 −8 m 3 the filling factor of the resonator with diamond sample is given by η = V dia /V m = 0.2.

FIG. 1 .
FIG.1.Schematic cross-section of the microwave cavityresonator setup.a Microwave cavity, holding the sapphire ring resonator.The schematic shows half of the cavity to reveal the interior.The light blue sapphire ring is held by two Teflon holders.On the top is the iris coupled waveguide port.The coupling to the resonator is adjusted by a Teflon screw with a metal ring at its tip.The 532 nm optical pump laser is aligned along the symmetry axis of the cylindrical resonator, while the static magnetic field B 0 is perpendicular to it.Finally, the diamond sample is placed inside the resonator.b Loaded Q-factor as a function of the resonator coupling, showing the full range of coupling, with the diamond sample in the resonator.

FIG. 3 .
FIG. 3. Maser emission spectrum.a Maser emission power |A M |, in logarithmic units, as a function of frequency and static magnetic field at w L = 430 s −1 and Q L of about 33, 500.b |A M | as a function of frequency for two different magnetic field values, indicated by the red and green dashed lines in a.

FIG. 4 .
FIG. 4. Maser threshold diagram.a Maximal maser emission power |A M |, in logarithmic units, of the central hyperfine maser transition line as a function of laser pump rate w L and the loaded quality factor Q L of the resonator.The white and grey dashed lines represent the theoretical threshold for masing, with and without temperature effects, respectively.b |A M | as a function of Q L for two w L , indicated by the purple and red dashed lines in a. c |A M | as a function of w L at two Q L , indicated by the blue and green dashed lines in a.
) with coefficients a 1 = 3.279 × 10 −2 MHzK −1 and a 2 = −1.787× 10 −4 MHzK −2 .With the constant offset ∆D cal = 5.9 MHz we calibrate the model to produce a zero shift at room-temperature.With the model function we can relate the measured ∆D to a temperature, as shown in Supplementary Figure 1 (b).

1 .
Temperature dependence of zero-field splitting and relaxation time.a Color encoded microwave reflection |S 11 | as a function of VNA frequency and static magnetic field, at a pump rate of 695 s −1 .b Effective loss rate κ eff of the resonator -spin ensemble hybrid system as a function of the static magnetic field.The orange solid line is a fit of Eq. (S4) to the data. w

4 .
Microwave Cavity Technical Drawing.The technical drawing shows the dimensions of the gold platted copper cavity. )