A near-quantum-limited diamond maser amplifier operating at millikelvin temperatures

The diamond sample used in this work is identical to those described in previous works (Refs. [17, 16]). It was synthesized by the temperature gradient method under high-pressure and high-temperature (HPHT) conditions of $5.5\,\text{GPa}$ and $1350\,^{\circ}\text{C}$ using an Fe-Ni-Co alloy as the metal solvent and natural diamond grains as the carbon source. A $(100)$ plate obtained by laser-cutting and polishing was irradiated with $3\leavevmode\nobreak\ \text{MeV}$ electrons at a fluence of $4\times 10^{17}\,\mathrm{e/cm^{2}}$ at room temperature, followed by annealing at $850\leavevmode\nobreak\ ^{\circ}\text{C}$ for 2 hours in vacuum. Subsequently, it was further irradiated with $2\,\text{MeV}$ electrons at $700\,^{\circ}\text{C}$ to a fluence of $5\times 10^{17}\,\mathrm{e/cm^{2}}$ and annealed at $1000\,^{\circ}\text{C}$ for 2 hours in vacuum. The final dimensions of the diamond ($3\times 1.5\times 0.5\,\mathrm{mm}^{3}$) used in this study were obtained from the central part of the plate via laser-cutting and polishing. Following electron irradiation and annealing, the concentrations of P1 centers and negatively-charged nitrogen-vacancy (NV^-) centers were measured to be approximately $16\,\mathrm{ppm}$ and $2\,\mathrm{ppm}$, respectively.

The microwave resonator used in this work is a loop-gap type [16]. We determined its internal and external quality factors using the circle-fit method [30] after subtracting background amplitude oscillations to achieve a flat baseline. The results are shown in Fig. S1 for both $Q_{\text{ext}}\approx 250$ and $Q_{\text{ext}}\approx 730$. The internal quality factor $Q_{\text{int}}$ consistently stays around $3000$. The resonator frequency, $\omega_{r}$, is very sensitive to the distances of the two gaps. Consequently, after each disassembly and reassembly of the resonator and its enclosure to adjust $Q_{\text{ext}}$, $\omega_{r}$ varied over $6.59\,\text{GHz}$ to $6.86\,\text{GHz}$ upon each cooldown.

The detailed wiring and configuration inside the dilution refrigerator (Bluefors, LD-400) are illustrated in Fig. S2. Cryogenic attenuators (XMA, 2082-624X-CRYO series), represented as gray round rectangles, are inserted in each input line at every temperature stage to attenuate thermal noise from the microwave tone injected from room temperature. The heavily attenuated “Probe” line and moderately attenuated “Pump” line are combined at a directional coupler (Marki Microwave, C20-0226) on the mixing chamber stage. The labels “LPF” and “IRF” represent low-pass filters (Marki Microwave, FLP-0750) and homemade infrared filters using iron-loaded epoxy Eccosorb CR110, respectively. A HEMT (High Electron Mobility Transistor) amplifier (Low Noise Factory, LNA-LNC4_8C) was installed on the $4\,\text{K}$ plate.

The calibrated noise source [18] is used for the noise temperature measurements. While the noise temperature is heated maximally up to $\approx 4.5\,\text{K}$, the base temperature of the mixing chamber plate slightly increases up to $\approx 19\,\text{mK}$.

Two microwave switches (Radiall, R573423600) were used to check the total attenuation of the coaxial cables to and from the noise source, which turned out to be negligible. The microwave switch (Analog Devices Inc., HMC547ALP3E), depicted inside the dashed box in Fig. S2, was installed and used only for pulse electron spin resonance (pulse-ESR) measurements to protect the HEMT. Further details are described in the Supplementary Text.

The probe tone was fed into the fridge through the line labeled “Probe”, while the pump tone was sent through the line “Pump”. To minimize the impact of the pump tone on the HEMT and the following receiver circuits, a cancellation microwave tone was introduced through a dedicated microwave input line labeled “Cancellation” (see also Fig. S3). When operating the maser amplifier, a microwave tone was generated by a signal generator (PSG; Keysight Technologies, E8267D) for both pump and cancellation. This tone was then split by a power splitter (Mini-Circuits, ZC2PD-01263-S+), as shown in Fig. S3A. On one branch, the cancellation tone was adjusted using a voltage-controlled phase shifter (Qorvo, CMD297P34) and a voltage-controlled variable attenuator (Pulsar Microwave, AAT-22-479A/7S). The amplitude and phase of this cancellation tone were adjusted to destructively interfere with the reflected pump tone through a directional coupler placed after the two circulators (Quinstar, QCY-G040082AS), resulting in suppression of the pump power by at least $50\,\text{dB}$.

The reflection coefficient $r(\omega)$ of the resonator was measured using a vector network analyzer (VNA; Keysight, N5230C). The data presented in Figs. 2C and 3A, B, and D in the main text were measured at a power of $-110\,\mathrm{dBm}$, corresponding to a mean intra-resonator photon number of approximately $10^{6}$, which is sufficiently lower than the number of inverted spins, $\Delta N_{P1+}\approx 4\times 10^{14}$, ensuring minimal impact on the spin saturation. Although this power level almost reaches the typical saturation power of traveling-wave parametric amplifiers [2], it remains well below the saturation power of our maser amplifier as discussed in Fig. 4C in the main text.

Noise power spectra were measured by a spectrum analyzer (SPA; Keysight Technologies; N9010A). The output and input ports of the VNA are connected to the Probe and Output ports of the fridge via directional couplers (input: Pulsar Microwave, CS20-09-436/9; output: Marki, C10-0226). A signal generator (SG, Keysight Technologies, E8247C) generates a microwave probe tone during the signal-to-noise ratio (SNR) measurements, as presented in Fig. 4A.

Figure S2B displays the wiring configuration on the mixing chamber plate used to measure the $1\,\text{dB}$ compression points, as presented in Fig. 4C in the main text. To ensure the HEMT and subsequent components were not saturated, we bypassed the HEMT using a microwave switch.

In the pulse ESR measurement detailed in the Supplementary Text, we use an FPGA-based integrated measurement system (Quantum Machines, OPX+) to generate pulse envelopes, control microwave switches, and detect spin echo signals. Gaussian-shaped pulse envelope signals, modulated at $100\,\mathrm{MHz}$, are generated by the OPX arbitrary waveform generator (AWG). We use the single sideband modulation technique, where a baseband microwave tone (“LO” in Fig. S3B) is mixed with the pulse envelopes with an internal IQ mixer of the PSG. These pulses are subsequently amplified by a high-power amplifier (Cree, CMPA601C025F) and fed to the resonator through the “Pump” line. To minimize microwave leakage noise from the amplifier, a semiconductor-based microwave switch (Analog Devices Inc., ADRF5019-EVALZ) is installed immediately after the amplifier’s output. Similarly, another semiconductor-based microwave switch (Analog Devices Inc., HMC547ALP3E) is installed to protect the HEMT amplifier from the pulses.

For echo signal detection, the signals from the output port of the refrigerator are amplified by a series of room-temperature amplifiers (B&Z Technologies, BZ-02000800-090826-182020, and Mini-Circuits, ZVA-183+) and then down-converted using an IQ mixer (Marki, IQ-0307L). The local oscillator (LO) tones are extracted from a portion of the PSG’s locking signal (option HCC). After demodulation, the signals are passed through band-pass filters (Mini-Circuits, SBP-101+), and then amplified by a preamplifier (SRS, SR445A). Finally, they are detected by the OPX’s digitizer, which processes the signals digitally for heterodyne detection.

We characterized the noise temperature of the maser amplifier by using a procedure based on the Y-factor method with a calibrated noise source [18]. The noise source is weakly thermalized to the cold plate at about $100\,\mathrm{mK}$, which allows for quasi-independent temperature control using an attached heater and thermometer. This setup enables precise control over the input noise temperature $T_{\text{in}}$ into the maser amplifier device.

Thermal noise power is given by $P_{\text{noise}}=k_{\text{B}}T_{\text{noise}}\delta f_{\mathrm{BW}}$, where $k_{\text{B}}$, $T_{\text{noise}}$, and $\delta f_{\mathrm{BW}}$ are the Boltzmann constant, the noise temperature, and the measurement bandwidth, respectively. When a signal with a noise temperature of $T_{\text{in}}$ is fed into an amplifier, the output power $P_{\text{out}}$ is given by

where $G$ and $T_{\mathrm{amp}}$ are the power gain and noise temperature of the amplifier, respectively. In our experiment, without the maser amplifier, $P_{\text{out}}$ can thus be expressed as:

where $G_{\text{HEMT}}$ and $T_{\text{HEMT}}$ represent the gain and noise temperature of the HEMT amplifier, respectively. The additional noise contribution from the final room temperature amplifier, including cable attenuation from the HEMT to the room temperature setup, is denoted as $T_{\text{bkg}}$. Equation (S2) can be approximated to Eq. (1) in the main text, considering that $G_{\text{HEMT}}>10^{4}$ and $T_{\text{bkg}}\approx 300\,\text{K}$.

Similarly, the noise power with the maser amplifier on can be expressed as:

where $G_{\text{maser}}$ and $T_{\text{maser}}$ represent the gain and noise temperature of the maser amplifier, respectively. Therefore, Eq. (S3) can be approximated to Eq. (2) in the main text under the same considerations.

In our setup, a directional coupler and a circulator, as well as lossy non-superconducting cables, are placed between the noise source and the maser amplifier. Their insertion losses and attenuations must, therefore, be taken into account in the evaluation of the input noise temperature $T_{\text{in}}$, as they attenuate the noise photons from the noise source. To model this effect, we employ the beam-splitter model [31]. If the $j$-th component in the transmission line has an insertion loss $\beta_{j}$, the number of noise photons $n_{j+1}$ can be expressed as

where $n_{j}$ is the number of input noise photons of the $j$-th component, and $n_{\mathrm{th}}$ is the thermal noise photons corresponding to the physical temperature of the component. All the presented values of $T_{\text{in}}$ are corrected using Eq. (S4) based on the measurements detailed in the Supplementary Texts.

The single-spin coupling constant $g_{0}$, which represents the interaction strength between a single P1 center and a single microwave photon in a resonator, is expressed as [32]

where $\gamma_{e}$ and $\delta B_{0}$ denotes the gyromagnetic ratio of the electron spin and the magnetic vacuum fluctuation of the resonator, respectively, and $\mathbf{S}$ is the spin vector operator with $|{-1/2}\rangle$ and $|{+1/2}\rangle$ are the P1 center’s electron spin eigenstates. The magnetic vacuum fluctuation is given by

where $\mu_{0}$, $\omega_{r}$, and $\mathcal{V}_{\mathrm{eff}}$ are the vacuum permeability, the resonator frequency, and the effective mode volume of the resonator, respectively. $\mathcal{V}_{\mathrm{eff}}$ is defined as the ratio of the integral of the magnetic field energy density over the whole resonator mode volume $V$ to the maximum magnetic energy density [33]

where $\mathbf{B}(\mathbf{r})$ is the magnetic field in the resonator, maximized at the position $\mathbf{r}_{0}$. Using a finite element simulation software, COMSOL Multiphysics, the effective resonator mode volume $\mathcal{V}_{\mathrm{eff}}$ of our loop-gap resonator is estimated to be approximately $\mathcal{V}_{\mathrm{eff}}\approx 2.5\times 10^{-8}\,\mathrm{m^{3}}$. Therefore, the single-spin coupling constant of the P1 center with $S=1/2$ is obtained to be $g_{0}\approx 2\pi\times 0.15\,\mathrm{Hz}$.

To determine the number of inverted or non-inverted spins through the resonator reflection $r(\omega)$, we employed the input-output theory [34] for a coupled resonator-spin ensemble system with a coherent probe tone $\alpha_{\text{in}}=\alpha_{0}e^{-i\omega t}$ [17, 10]. The Hamiltonian can be expressed using the Tavis-Cummings model with Holstein-Primakoff approximation [35, 36, 37] as follows:

where $a$ ($s_{j}$) and $a^{\dagger}$ ($s_{j}^{\dagger}$) denote the annihilation and creation operators for the resonator mode ($j$-th spin), respectively. In addition, $N$ represents the number of spins, and $g_{j}$ is the single spin coupling constant between the resonator and the $j$-th spin. $\kappa_{\mathrm{ext}}=\omega_{r}/Q_{\mathrm{ext}}$ and $\kappa_{\mathrm{int}}={\omega_{r}}/{Q_{\mathrm{int}}}$ are the external and internal loss rates of the resonator, with $Q_{\mathrm{ext}}$ ($Q_{\mathrm{int}}$) representing the external (internal) quality factor.

Following the standard method for deriving the reflection coefficient [34], we obtain

where $K(\omega)$, called the “spin function”, includes all information about the spin ensemble. As will be discussed in the following section, $K(\omega)$ is linked to the spin susceptibility [17] and is defined as shown in Refs. [35, 36, 37],

with $g_{\text{ens}}=\sum_{j}g_{j}=g_{0}\sqrt{N}$ representing the ensemble coupling constant between the resonator and the spin ensemble. The decay rate of individual spins is denoted by $\gamma_{j}$, while $\omega_{s}$ and $\Gamma/2$ represent the center frequency and the half-width at half-maximum (HWHM) of the spin ensemble, respectively, assumed to follow a Lorentzian distribution here. The sign in front of $K(\omega)$ in Eq. (S9) represents the state of the spin ensemble, i.e., a positive sign ($+$) corresponds to an inverted spin ensemble, whereas a negative sign ($-$) represents a non-inverted spin ensemble.

We modify $K(\omega)$ to incorporate the spin polarization $\bar{\rho}=\Delta N/N_{\mathrm{tot}}$, where $\Delta N=N_{l}-N_{u}$ represents the population difference between the lower and upper levels of the $\text{P1}_{+}$ state, and $N_{\mathrm{tot}}=N_{l}+N_{u}$ is the total number of spins in the $\text{P1}_{+}$ transition. In a system with a population difference $\Delta N$, the ensemble coupling constant $g_{\text{ens}}$ is replaced by the effective coupling constant $g_{\text{ens,eff}}$, defined as $g_{\text{ens,eff}}^{2}=g_{0}^{2}\Delta N$. Consequently, $K(\omega)$ is reformulated in terms of $\bar{\rho}$ as

where $g_{\mathrm{ens,total}}^{2}=g_{0}^{2}N_{\mathrm{total}}$ represents the “conventional” collective coupling constant, assuming all spins are polarized into the lower energy state. The sign of $K(\omega)$ reflects the sign of $\Delta N$; it becomes negative in the case of population inversion and positive in a non-inverted state. To ensure consistency with Eq. (S9), we rewrite the reflection coefficient $r(\omega)$ as follows

The function $K(\omega)$ can be expressed in terms of the reflection coefficient $r(\omega)$ by rearranging Eq. (S12):

which allows $K(\omega)$ to be directly determined from the experimentally obtained $r(\omega)$ data. Fig. S4 shows such examples converted from experimentally obtained data and fits using the imaginary parts of Eq. (S11).

In an ideal spin ensemble, the imaginary part of $K(\omega)$ exhibits a Lorentzian profile, enabling straightforward determination of the population difference $\Delta N$ by fitting the measured spectrum using the function:

From this function and the imaginary part of Eq. (S11), the population difference $\Delta N$ can be extracted as $\Delta N=A/g_{0}^{2}B$.

However, significant deviations from the ideal Lorentzian profile are observed under certain experimental conditions. Immediately after initiating the pump, and before reaching the steady state, the spectrum becomes distorted due to the rapid inversion of the spins near the center of the distribution, while those farther from the center remain non-inverted. Furthermore, even in the steady state, intermediate pump powers lead to partial inversion of the spin ensemble, resulting in non-uniform inversion and persistent spectral distortion. This behavior is illustrated in Fig. S4. Panel A shows spectra at selected time points extracted from the time evolution in Fig. 3A of the main text. Among these, the spectra at 146.6 and 326.7 seconds clearly exhibit transient distortions following pump initiation. Panel B shows spectra at various pump powers extracted from the gain profiles in Fig. 3B. Notably, the spectrum at a pump power of $-59\,\mathrm{dBm}$ exhibits pronounced steady-state distortion due to partial inversion.

To accurately account for these spectral conditions, we employ a two-Lorentzian fitting model in which the spectrum is represented as a superposition of two separate Lorentzian contributions; one representing the inverted upper-level spins and the other the lower-level spins:

The results of fits using this model are illustrated in Fig. S4. From this fitting model, the population difference $\Delta N$ can be described as

The peak maser gain is defined by the absolute square of the reflection coefficient $r(\omega)$ at the resonance, where $\omega=\omega_{r}=\omega_{s}$. According to Eq. (S12), this relationship can be expressed as

where we defined a new parameter, $\kappa_{s}=4g_{\mathrm{ens}}^{2}/\Gamma=\mathrm{Im}[2K(\omega_{r}=\omega_{s})]$ [10, 37], as the “spin transition rate”. This parameter quantifies the effective transition rate of the spin ensemble to the resonator mode, i.e., the rate at which spins undergo upwards (downwards) transitions per unit time through the stimulated emission (absorption) with a rate $\Gamma_{\text{s}}=4g_{0}^{2}/\Gamma$. Moreover, $\kappa_{\text{s}}$ is directly correlated with the “magnetic quality factor” $Q_{\text{mag}}$ [8, 12], which characterizes a spin ensemble system as a maser gain medium:

where $\eta$ is the magnetic field filling factor and $\chi^{\prime\prime}$ is the imaginary part of the spin susceptibility at resonance, defined as $\chi(\omega)=-2K(\omega)/(\omega_{r}\eta)$ [17].

Using Eq. (S17), we calculated the gains $G(\omega_{r}=\omega_{s})$ for the two external quality factors, $Q_{\text{ext}}\approx 250$ and $730$, used in the experiments. The results are shown as dashed curves in the inset of Fig. 3C.

We modeled the dynamics of our diamond maser amplifier system as detailed below and solved them using semi-classical rate equations [19]. Although quantum Langevin equations would yield equivalent results [9, 10], our lack of expertise with that numerical treatment led us to opt for a semi-classical approach. The system under consideration, as depicted in Fig. S5A, consists of eight energy levels, which include two levels of the “fast-relaxing spins” and the three two-level systems associated with the P1 centers nuclear spin states, P1₊, P1₀, and P1_-. This configuration leads to eight rate equations, in addition to a photon number rate equation, to describe the population dynamics and photon numbers within the system. However, due to the computational complexity of solving nine equations, we simplified the model to seven equations by combining the two levels of $\text{P}1_{-}$ with the fast-relaxing spins. The resulting set of seven rate equations, including the photon number rate equation, are represented as follows [19, 38]:

where $N_{i}$ represents the number of spins of each level, $\Gamma_{41}$, $\Gamma_{52}$, and $\Gamma_{63}$ are the relaxation rates of each transition, $\Gamma_{\mathrm{cr}}$ is the four-spin cross-relaxation rate, and $\Gamma_{s}=4g_{0}^{2}/\Gamma$ is the stimulated emission rate. The rate equation for the intra-resonator photon number $n$ is:

The cross-relaxation rate $\Gamma_{\mathrm{cr}}$ in our system cannot be directly measured due to the presence of the “fast-relaxing spins”, contrasting with the previous reports [19, 20, 21]. To estimate this rate indirectly, we compared computational results from the rate equations with experimental results obtained under various pump powers, as depicted in Fig. S5B and explained in the next subsection. We determined $\Gamma_{\mathrm{cr}}$ to be within the range of $\sim 1$ to $10\,\text{mHz}$, with the parameters listed in Tables S1 and S2. While previously reported cross-relaxation rates for highly nitrogen-doped diamonds typically range from $\sim 10\,\text{Hz}$ to $\sim 100\,\text{Hz}$ [19, 20], theoretical models of the four-spin interactions imply that $\Gamma_{\mathrm{cr}}\propto d^{6}$ [20], where $d$ denotes the spin density. This supports our findings of $\Gamma_{\mathrm{cr}}\sim\text{mHz}$, as the P1 center density in our diamond sample of $\approx 16\,\text{ppm}$ is considerably lower than the $>100\,\text{ppm}$ in those previous studies.

Spins absorb incoming microwave pump tones and are excited to the upper state, a process governed by the same underlying physics as stimulated emission. Therefore, the photon absorption rate, i.e., the pump rate $\Gamma_{\text{p}}$, is calculated as a product of the stimulated emission rate and the number of incoming photons, as described in Eqs. (7.7) and (7.8) on p.260 in Ref. [38]:

where $\omega_{p}$ is the pump frequency, $\omega_{s}$ is the average spin resonance frequency of P1₀, and $n_{\mathrm{in}}$ is the number of input photons. The expression ${g_{0}^{2}\Gamma}/\left[{(\omega_{p}-\omega_{s})^{2}+\Gamma^{2}/4}\right]$ on the right-hand side is the rigorous stimulated emission rate with detuning between $\omega_{p}$ and $\omega_{s}$.

From the input-output theory, $n_{\mathrm{in}}$ with a coherent pump tone $\beta_{\mathrm{in}}=\beta_{0}e^{-i\omega t}$ applied to a single-port resonator is expressed as

where $P_{\mathrm{in}}=\hbar\omega|\beta_{0}|^{2}$ denotes the input pump power, and $\kappa_{\mathrm{tot}}=\kappa_{\mathrm{ext}}+\kappa_{\mathrm{int}}$ is the total loss rate of the resonator. Therefore, the pump rate $\Gamma_{\text{p}}$ can be expressed as a function of $P_{\mathrm{in}}$ as follows;

During maser operation, the pump frequency $\omega_{p}$ is tuned to match the resonance frequency of the P1₀ spins, i.e., $\omega_{p}=\omega_{s}=\omega_{0}$, but it is off-resonance with the resonator, thus $\omega_{p}\neq\omega_{r}$. Figure S5B displays the population difference in the P1+ transition, $\Delta N_{\text{P1+}}$, plotted against the calculated pump rates $\Gamma_{\text{p}}$. This population difference is extracted using the spin function $K(\omega)$, as discussed in the above section. The plot also includes overlayed simulation results from the rate equations Eqs. (S19) and (S20).

If the photons produced by stimulated emission exceed the resonator losses, the system enters the self-oscillating or free-running maser regime. In this regime, the maser is capable of emitting microwave photons into the resonator without the need for an externally applied probe tone. The threshold population difference $\Delta N_{\mathrm{thr}}$ can be calculated from the photon number rate equation given in Eq. (S20) [38]. At the threshold, the two terms in the equation are balanced, hence,

where $N_{4}^{\mathrm{thr}}$ is the number of spins in the state 4 at the threshold and $n_{\mathrm{ss}}$ is the photon number in the steady state in the free-running maser regime. Rearranging this equation gives

The second term can often be neglected since typically $n_{\mathrm{ss}}\gg 1$. Therefore,

In the case of $Q_{\mathrm{ext}}\approx 250$, $|\Delta N_{\mathrm{thr}}|$ is approximately $4.81\times 10^{14}$, while for $Q_{\mathrm{ext}}\approx 730$, it is approximately $2.42\times 10^{14}$. These thresholds are shown as the red dot-dashed lines in the insets of Fig. 3A and C in the main text.