Latched detection of zeptojoule spin echoes with a kinetic inductance parametric oscillator

When strongly pumped at twice their resonant frequency, nonlinear resonators develop a high-amplitude intracavity field, a phenomenon known as parametric self-oscillations. The boundary over which this instability occurs can be extremely sharp and thereby presents an opportunity for realizing a detector. Here, we operate such a device based on a superconducting microwave resonator whose nonlinearity is engineered from kinetic inductance. The device indicates the absorption of low-power microwave wavepackets by transitioning to a self-oscillating state. Using calibrated pulses, we measure the detection efficiency to zeptojoule energy wavepackets. We then apply it to measurements of electron spin resonance, using an ensemble of 209Bi donors in silicon that are inductively coupled to the resonator. We achieve a latched readout of the spin signal with an amplitude that is five hundred times greater than the underlying spin echoes.


INTRODUCTION
Over the past decade, quantum-limited parametric amplifiers operating at microwave frequencies have progressed from proof-of-principle to ubiquity within circuit quantum electrodynamics (cQED) experiments.Typically, these devices are operated in a linear regime, but several types of parametric amplifiers are explicitly nonlinear, such as the Josephson bifurcation amplifier (JBA) [1] and the Josephson parametric oscillator (JPO) [2,3].These devices essentially act as a microwave "click"detector, in that threshold detection is employed to discriminate the presence or absence of a signal.Central to the design of JBAs and JPOs is the use of Josephson junctions, which provide the non-linearity required for signal mixing.An alternative source of non-linearity is the kinetic inductance intrinsic to thin films of disordered superconductors [4].In contrast to Josephson junctionbased devices, superconducting microwave resonators engineered from high kinetic inductance materials retain high quality-factors when operated in tesla-strength magnetic fields [5,6] and at elevated temperatures.This has recently inspired the development of magnetic fieldcompatible resonant parametric amplifiers that operate close to the quantum noise limit [7][8][9][10], which have a range of applications including axion detection [11] and quantum computation with spin qubits [12].
Another application of these devices is the measurement of electron spin resonance (ESR).JPAs have already been used to push noise in ESR experiments to the quantum limit, where vacuum fluctuations of the electromagnetic field dictate the spin detection sensitivity [13,14].Several other recent works have applied nonlinear microwave amplifiers [15] and qubit-based sensors [16][17][18][19] employing Josephson junctions to measurements of ESR in order to push detection sensitivities to record levels.Kinetic inductance parametric amplifiers (KIPAs) have also recently been used for ESR, where they have been demonstrated to have several advantages over JPAs [10].Due to their compatibility with moderate magnetic fields, KIPAs can serve as both the resonator for inductive detection of spin echo signals as well as the firststage amplifier.This not only simplifies the measurement setup by obviating a separate quantum-limited amplifier, it also eliminates any insertion loss between the resonator and the first cryogenic amplifier.
Here, we extend previous works with KIPAs by operating one as a "click"-detector, rather than as a linear amplifier.By biasing the device near the threshold where its behaviour transitions from a linear amplifier to a parametric oscillator, the onset of parametric self-oscillations (PSO) serves as an indicator for the absorption of microwave wavepackets.To distinguish this operating regime, which has not been previously demonstrated, we refer to the device here as a kinetic inductance parametric oscillator (KIPO), in analogy to the JPO which operates under a similar principle.In the following we describe the concept of the detector, calibrate its sensitivity, and demonstrate its application in ESR measurements of an ensemble of bismuth ( 209 Bi) donors in silicon (Si) that are directly coupled to the device.

RESULTS
Device Design Our device is similar to the KIPAs described in previous works [7,10].It is patterned in a single lithographic step from a 50 nm thick film of nio- The device is represented by a parallel RLC resonator with frequency ω0 and a band-stop stepped impedance filter (BS-SIF).The resonator has a geometric inductance (Lg) and a kinetic inductance (L k ).We operate the device as a "click"-detector by biasing it with a strong pump with frequency ωp and a DC current IDC.In this work we detect two types of signals: weak classical signals with frequency near ω0 generated by a microwave source (1.), which we use to calibrate the detector's sensitivity, and spin echoes from an ensemble of 209 Bi donor spins (2.) that are resonantly coupled to the resonator via Lg.Detailed schematics of the measurement setups are presented in the Supplementary Material.(b) An artist's depiction of the resonator, which is formed from a λ/4 section of transmission line with a dense interdigitated capacitor. 209Bi spins are implanted into the silicon substrate.(c) The maximum power measured with a spectrum analyzer centered at ω0 as a function of ∆p = ωp − 2ω0 and the pump power Pp.The dark blue region corresponds to the parameter space where the device self-oscillates.The powers are referred to the output of the device and Pmax is truncated at -100 dBm to enhance clarity.The red lines correspond to the PSO threshold (P th ) predicted from a model of the device using three different values of Qi.The stationary and swinging pendulums are used to depict the quiet and self-oscillating states of the device, respectively.(d) Qi extracted from measurements of S11 performed with a VNA as the signal power is varied.Measurements were taken at T = 10 mK.
bium titanium nitride (NbTiN) with a kinetic inductance of L k = 3.45 pH/□.The NbTiN is deposited on a Si substrate enriched in the isotope 28 Si (750 ppm residual 29 Si) that was implanted with 209 Bi donors at a concentration of 10 17 cm −3 over a depth of 1.35 µm.The device has a single port and consists of a quarter-wavelength (λ/4) coplanar waveguide resonator that is shorted to ground at one end and galvanically connected to a band-stop stepped impedance filter (BS-SIF) [20] at the other end (Figs.1a,b).The resonator features a dense interdigitated capacitor with 1 µm wide fingers and a 1.5 µm gap to ground (see Supplementary Material), which compensates for the strong kinetic inductance and reduces the impedance of the mode to Z 0 ≈ 33 Ω.The BS-SIF serves to confine the resonant mode of the device while simultaneously allowing the application of a DC bias current (I DC ).An I DC can be used to tune the res-onance frequency from ω 0 (I DC = 0)/2π = 7.776 GHz to ω 0 (I DC = 4.89 mA)/2π = 7.530 GHz via the quadratic dependence of L k on the total current [21].An I DC also enables three-wave mixing so that a pump with frequency ω p ≈ 2ω 0 can be used to amplify signals with frequencies about ω 0 [7,22].The device is connected to the cold finger of a dilution refrigerator with a base temperature of 10 mK or a pumped 3 He cryostat with a base temperature of 400 mK, depending on the experiment.The DC current and microwave tones are combined at base temperature using a bias tee and diplexer (Fig. 1a).The device is measured in reflection with the signal being routed through a cryogenic high electron mobility transistor (HEMT) amplifier at 4 K.Further details of the device design and measurement setup are provided in the Supplementary Material.

Detector Concept
The defining characteristic of PSO is the formation of a large intracavity field at ω 0 whenever the three-wave mixing strength exceeds the average rate at which resonant photons can escape the cavity.Experimentally, this manifests as the sudden generation of a large power at ω 0 that is emitted from the device whenever the pump power P p is raised beyond a sharp threshold P th .In Fig. 1c we report the maximum power recorded by a spectrum analyzer (P max ) centered at ω 0 as a function of P p and a detuning of the pump frequency ∆ p = ω p − 2ω 0 .The dark blue region corresponds to the parameter space where the device undergoes PSO, and the boundary of this region provides a direct measurement of P th .
Using cavity input-output theory and the Hamiltonian for the KIPA [7], we model our device as a parametrically-driven Duffing oscillator (see Supplementary Material), as has been done previously for JPOs [2,23,24].Using parameters extracted from measurements of the device we directly compare the model and our experimental measurement of the threshold power P th (red lines in Fig. 1c).Crucially, the model predicts that when , where Q i and Q c are the internal and coupling quality factors of the resonant mode, respectively.In Fig. 1c we compare three models of the device which are equivalent except for Q i .For this particular measurement we find there is reasonable agreement between the data and model for Q i = 18×10 3 .We also highlight that P th shifts to smaller values as Q i is increased.
The quality factors Q i and Q c can be extracted from measurements of the device in reflection (S 11 ) using a vector network analyzer (VNA).As is common for superconducting microwave resonators, we observe that Q i is non-linear with the applied signal power, or equivalently the average number of intracavity photons n (Fig. 1d).This indicates that two level systems (TLSs) interact with the resonant mode [25] and limit Q i at low signal power.We observe Q i to vary between 4.7 × 10 3 and 29 × 10 3 for 10 −2 < n < 3 × 10 5 , which in all cases is much smaller than Q c ≈ 200 × 10 3 (see Supplementary Material).An important consequence of this is that the resonant mode's linewidth ω 0 (Q −1 i +Q −1 c ), and hence P th , are sensitive to the signal power inside the resonator.
The experiments shown in Fig. 2 demonstrate how the dependence of P th on n can be used to create a device that detects the absorption of low-power microwave wavepackets.First, the device is biased with a DC current I DC = 2.55 mA and a pump tone with frequency ω p = 2ω 0 .Following a delay τ 0 = 1 ms, a stimulus pulse with duration τ 1 = 10 µs, frequency ω 0 , and power P 0 = −116.4dBm is delivered to the input of the device (Fig. 2a).Throughout the experiment we monitor the device by performing a homodyne measurement (i.e.demodulating the signal using a local oscillator with frequency ω LO = ω 0 ) and recording the amplitude of the signal quadrature components X and Y .For the lowest a b i ii iii FIG. 2. Detector concept.(a) A depiction of the two-tone pulse sequence.The device is biased with IDC and pumped at ωp, before a short stimulus pulse with frequency ω0 is supplied to the device.A threshold applied to the amplitude of the demodulated signal is used to determine if the device is self-oscillating.(b) Single shots of the pulse sequence for decreasing Pp (i to iii).For certain Pp, the stimulus pulse triggers the onset of PSO (ii).For these experiments, τ0 = τ2 = 1 ms, τ1 = 10 µs, IDC = 2.55 mA, P0 = −116.4dBm and T = 400 mK.The time is measured from the leading edge of the pump pulse.The stimulus pulse is seen directly in (iii) at t = 1000 µs.pump power P p (Fig. 2b, iii), we observe no PSO.For the largest P p (Fig. 2b, i), we observe PSO, but at a time that is uncorrelated with the stimulus pulse.But for an appropriately chosen P p ≈ P th (Fig. 2b, ii) we observe the PSO onset at a time that is correlated with the stimulus pulse (the quiet and self-oscillating states are represented schematically with the pendulum).We hypothesize that absorption of the stimulus pulse triggers the onset of PSO due to the partial saturation of the TLSs; the increased Q i associated with this results in P th being dynamically reduced below the P p setpoint, thereby triggering PSO.
From Fig. 2b it is clear that the self-oscillating state has a large amplitude (α), relative to the stimulus pulse.Indeed, Fig. 1c shows that the peak power of PSO can exceed -75 dBm.This can be understood by noting that α ∝ 1/K for a Duffing oscillator, where K is the self-Kerr strength (see Supplementary Material).The Kerr effect for these devices is known to be negligible relative to Josephson junction-based devices, due to the weak and distributed nature of the kinetic inductance non-linearity [7].By comparing the power of PSO measured with a spectrum analyzer to the average number of intracavity photons n calculated from VNA measurements of S 11 with a known power, we estimate that n > 10 5 during PSO.For this device, α is large enough to enable subsequent four wave mixing processes, which results in the generation of a frequency comb about ω 0 whenever the device self-oscillates (see Supplementary Material).This results in oscillations in the amplitude of the demodulated signal when the device is self-oscillating (Fig. 2b).Detection Efficiency and Sensitivity To calibrate the sensitivity of the detector, we measure its response using two pulse sequences.The first is depicted in Fig. 3a and is identical to that used in the previous section (Fig. 2a) but with different timings τ 0 , τ 1 and τ 2 .The second sequence is similar, differing only in that the stimulus pulse is omitted (Fig. 3b).Using threshold detection, this allows us to measure the efficiency of the sensor, E = P (T |S) − P (T | S), where T indicates that the device self-oscillates and S ( S) indicates that the device receives (does not receive) a stimulus pulse.P (T |S) is therefore the conditional probability describing the successful detection of the stimulus and P (T | S) is the probability of observing a dark count.
In Fig. 3c, we measure the detection efficiency E as a function of the pump power P p with a DC current I DC = 2.0 mA, stimulus pulse duration τ 1 = 10 µs, and stimulus pulse powers P 0 in the range [−137, −111] dBm.For these experiments we also phase modulate the pump microwave source at a rate of 15 kHz because the gain of a linear parametric amplifier (P p < P th ) operated in degenerate mode (ω p = 2ω 0 ) is dependent upon the relative phase between the pump and signal (ϕ p ) [7].While the modulation rate is slower than 1/τ 1 = 100 kHz, for each data point we measure 10 4 shots of the pulse sequence to ensure the unbiased sampling of all ϕ p , thereby mitigating its effect.E is found to grow monotonically with P 0 and is non-zero over a 7 dB range in P p .For the largest P 0 measured (P 0 = −111 dBm), E reaches a maximum of 0.98, which indicates that the device functions as a near perfect detector.As P 0 is reduced, the P p at which E is maximized grows slightly (from P p = −51.2dBm for P 0 = −111 dBm to P p = −49.4dBm for P 0 = −137 dBm).This reflects that as P 0 is reduced, the device needs to be biased increasingly closer to P th for successful detection, which results in a corresponding increase to the number of dark counts.In Fig. 3d we show that the probability of dark counts P (T | S) grows from < 0.01 to > 0.99 over a 3.1 dB range in P p .This corresponds to a dark count rate that is < 0.8 Hz for P p < −51.4 dBm and > 23 kHz for P p = −48.3dBm (see Supplementary Material).We note that while we chose to focus on the setpoint I DC = 2.0 mA in the main text, the device achieves high E over a tunable range of ∼ 100 MHz (see Supplementary Material).
Next we measure the detection efficiency E as a function of the stimulus pulse duration τ 1 (Fig. 3e).We fix the pump power P p to the value where E was maximized in the previous experiment (−51.2 dBm) and continue to phase modulate the pump.In this experi-ment the dark count probability P (T | S) is at maximum 1.5×10 −2 , which ensures that E mainly reflects the probability of true detection P (T |S).E reaches a maximum of E = 0.995 and is found to grow monotonically with both P 0 and τ 1 .This suggests that E is strongly correlated with the energy of the wavepacket J = P 0 τ 1 .We confirm this by plotting E(J) (Fig. 3f), where we see that for all P 0 , E(J) resembles an activation curve.Fitting the entire dataset with a sigmoid we infer that E = 0.5 for J = 5.0 6.3  4.0 zJ, where the upper and lower bounds correspond to the uncertainty in the calibration of P 0 .We note that the E achieved for a given J does show some dependence on P 0 , with the two largest P 0 measured having the lowest E. For these two measurements the cavity ringdown time is as long as 2ω 0 (Q 1 µs, which reduces the detection sensitivity to pulses with short τ 1 at such high powers.This may explain why their activation curves (Fig. 3f) do not align with those taken at lower P 0 , where Q i (and therefore the ring-down time) is reduced.Excluding the data with P 0 ≥ −116 dBm from the fit (red shaded area in Fig. 3f) we find the lower bound for the sensitivity to be J = 3.9 4.9 3.1 zJ for E = 0.5.This corresponds to wavepackets containing 756 952 601 microwave photons, measured at the input of the device.A complementary method for determining the sensitivity of a detector is plotting the receiver operating characteristic curve (see Supplementary Material), where we find that the detector performs better than a random binary classifier for pulse energies above 0.21 0.27 0.17 zJ (42 52 33 photons).Finally, we examine the influence of the pump phase ϕ p on the detection efficiency E by turning off phase modulation on the pump microwave source and instead controlling for and stepping ϕ p throughout the experiment (Fig. 3g).For each pump power P p , we found that E could be both enhanced and suppressed by controlling ϕ p , relative to a control experiment where the pump microwave source was phase modulated.We also found that E averaged across all ϕ p agreed closely with the efficiency obtained with phase modulation.This confirms for the experiments in Figs.3c-f that even though 1/τ 1 is faster than the phase modulation rate, the large number of shots taken ensures that E is independent of ϕ p when phase modulation is enabled.For P 0 = −127 dBm (the weakest power measured in this experiment), E could be suppressed to zero or made as large as 0.68, while with phase modulation E = 0.21.The phase-dependence of E highlights an essential aspect underlying the operation of this detector: the microwaves absorbed into the resonator are first parametrically amplified.The amplified signal saturates the TLSs, thereby triggerring PSO.Latched Readout of a Spin Ensemble To measure ESR we apply an in-plane magnetic field of strength B 0 = 13.71mT to bring an ESR transition of the 209 Bi donors, which were implanted into the Si substrate prior to fabrication, into resonance with the device (see Methods for a description of the spin transition measured).Resonant pulses delivered to the device are then used to control the sub-ensemble of spins with Larmor frequencies within the bandwidth of the resonator.The spins are measured using a Hahn echo pulse sequence, which is depicted in Fig. 4a.The first pulse in the sequence (x π/2 ) is a π/2 pulse which causes the spins to precess about B 0 and dephase due to interactions with their environment.After a time delay τ 1 , a phase-shifted π pulse (y π ) partially reverses the dephasing and causes the spins to refocus and emit an echo, temporarily populating the resonator with photons [26].For the Hahn echo pulse sequence this refocusing procedure is performed only once, which we designate N = 1 (see Fig. 4a).
To detect the spins via PSO, we modify the standard Hahn echo pulse sequence by adding in a strong parametric pump following the refocusing pulse.As in the previous experiments, we set the pump power P p close to but below the threshold power P th .PSO may then be triggered when the echo populates the resonator.Indi-vidual shots of the pulse sequence are depicted in Fig. 4b.When no pump tone is supplied (blue trace) the device functions simply as a resonator.The first amplifier in the measurement chain is a HEMT at 4 K.As a result, the signal-to-noise ratio (SNR) is poor and to reliably resolve the echo, the sequence must be repeated so that the signal can be averaged (inset of Fig. 4b).When the pump is on, the echo may trigger PSO, resulting in a detection signal with an amplitude that is a factor ∼ 500 greater than that of the echo (red trace).Moreover, while the echo itself is ≈ 10 µs long, corresponding to the duration of the resonant pulses used for spin control, the PSO persists until the pump is turned off.This latched ESR readout of a spin ensemble constitutes a novel technique for detecting spin resonance.
A common technique used to enhance the SNR of pulsed ESR measurements, and thereby reduce the measurement time, is to average multiple spin echoes collected with a Carr-Purcell-Meiboom-Gill (CPMG) se- quence.The CPMG pulse sequence is an extension of the Hahn-echo pulse sequence that refocuses the spins with y π pulses a total of N times (Fig. 4a).In Fig. 4c we show that this technique can similarly be applied to boost the detection effieciency E of our detector.For each P p we perform 1040 shots of the pulse sequence with N = 20 refocusing pulses.We calculate E(N ) by determining whether PSO were triggered for any of the N repetitions within each shot of the pulse sequence.We find that by repeatedly refocusing the spin echo, E can be increased by up to a factor of 25 for P p = −38.9dBm, which in absolute terms corresponds to an improvement in E by 0.28.For the three larger values of P p measured, the absolute improvement to E is even greater, with all being enhanced by more than 0.5 for a modest number of repetitions N .
The experimental data in Fig. 4c is well-fit by the simple equation which is a good approximation provided the amplitude of the spin echo does not decay appreciably within the time required to complete the N refocusing pulses.In Fig. 4d we compare the values of the true detection probability P (T |S) and the dark count probability P (T | S) fit to the data with those measured using a Hahn echo sequence (N = 1), which shows good agreement between the two sets of values.This suggests that for the present experiment, detection of the spin echoes can be modelled as a series of statistically independent events with constant probabilities for spin echo detection and dark counts.This has important implications for optimizing the detection of spin echoes with the KIPO, namely that for finite P (T | S), the number of refocusing pulses N required to maximize E scales as O(ln[P (T |S)/P (T | S)]) (see Supplementary Material).Indeed, we see in Fig. 4c that for P p = −38.4dBm where P (T |S)/P (T | S) = 5.3, E could be maximized with only N = 13 refocusing pulses.This rapid convergence on the optimal efficiency means that the CPMG detection scheme should also be effective on spin systems with shorter coherence times, as are commonly found in conventional ESR spectroscopy experiments [27], where the number of refocusing pulses that can be applied is limited.Measuring spins with lower coherence would likely require an increase in both the excitation and detection bandwidth relative to the present device.

DISCUSSION
The sensitivity of the detector could be improved beyond what is demonstrated here by designing a device that is critically coupled (Q i = Q c ), in which case the wavepacket would be more efficiently absorbed by the device.For the current detector, Q c /Q i ≈ 24 for n ≈ 1, in which case the fraction of power that is absorbed is only 1 − |S 11 | 2 ≈ 0.15.Reducing Q c to achieve critical coupling, e.g. by modifying the BS-SIF, could therefore increase its sensitivity by nearly an order of magnitude, while simultaneously increasing the bandwidth.We also note that the peak sensitivity was measured in an experiment utilizing phase modulation of the pump microwave source, while in Fig. 3g we have already demonstrated that E can be increased by more than a factor of three, provided the pump and signal are phase-matched.
The sensitivity of future devices could be further enhanced by increasing the strength with which the device couples to TLSs.While this approach is counter to most cQED experiments, in the present case, it will further increase the sensitivity of Q i to the average number of intracavity photons n, which is central to signal detection.This could be achieved by reducing the widths of the coplanar waveguide gap (g) and conductor (w) below the dimensions of the present device (g = 1.5 µm, w = 1 µm).Because TLSs reside at the dielectric interface, this will have the effect of concentrating the resonant mode volume in the material hosting the TLSs [25,28].In addition, dielectrics with higher concentrations of TLSs [29][30][31] could be intentionally deposited to improve both the detection sensitivity and bandwidth.
The experiments presented in Fig. 4 are a proof-ofconcept that demonstrate how the detector might be used in ESR experiments.While the experiments in Fig. 4 are taken at a single magnetic field B 0 , we also compare the conventional ESR and KIPO methods over a range of B 0 fields, simulating how the KIPO might be used as a spectroscopic probe (Supplementary Material).This experiment further highlights that while detection of the spin echoes with the KIPO occurs with probability < 1, the KIPO signal is greater than conventional ESR methods even when averaging over all shots and accounting for dark counts.This is directly related to the large power of the PSO signal, which we show in Fig. 1c can exceed −75 dBm at its peak, referred to the output of the device.This is ∼ 20 dB greater than the equivalent room-temperature Johnson-Nyquist noise measured over a bandwidth of 100 MHz, which means that it would be possible to measure "clicks" produced by the KIPO even without a cryogenic HEMT amplifier.
The experiments shown in Figs.4c,d show that ESR detection using the KIPO benefits from the extension of conventional ESR techniques.Further, it was recently demonstrated how "click"-detectors can be used to perform a full-suite of ESR techniques [17], which may be similarly adapted to the KIPO.Continuous-wave ESR detection might also be explored, as demonstrated using classical oscillator circuits that operate deep in the oscillator regime [32].In comparing the KIPO to works using Josephson junction-based devices, we emphasize the simplicity of our experiment.The KIPO does not require the operation of any qubits for detecting the microwave signals and can be operated at elevated temperatures and directly in a magnetic field (here 400 mK and 13.71 mT, which would preclude the use of aluminum Josephson junctions).Ultimately, this enables the KIPO to be coupled directly to the spin ensemble, which obviates a following quantum-limited amplifier or a cryogenic HEMT amplifier, as noted above.Aside from simplifying near quantum-noise-limited ESR setups, these experiments also suggest that the probability of detecting spins with the KIPO may scale exponentially with the number of measurements N .This might be exploited in some measurements to increase the speed of spin detection relative to conventional techniques that utilize linear quantum-noise-limited amplifiers, where the SNR grows as √ N .Future work will focus on comparing both detection techniques and further exploring the advantages of latched echo readout in ESR spectroscopy.

MATERIALS AND METHODS
Device Fabrication.The silicon substrate is enriched in the isotope 28 Si with a residual 29 Si concentration of 750 ppm. 209Bi ions with multiple energies were implanted to a concentration of 10 17 cm -3 between 0.75 µm and 1.75 µm depth and electrically activated with a 20 min anneal at 800 • C in a N 2 environment.A 50 nm film of NbTiN was then sputtered (STAR Cryoelectronics).The film was determined to have a kinetic inductance of 3.45 pH/□ by matching the resonant frequencies of capacitively coupled λ/4 resonators to microwave simulations (Sonnet).The device was patterned in a single step with electron beam lithography and subsequently dry-etched with a CF 4 /Ar plasma.The device was then mounted and wire bonded to a printed circuit board and enclosed in a 3D copper cavity.
Measurement Setups.The measurements are performed in two different cryogenic systems: a dry dilution refrigerator with a base temperature of 10 mK, and a pumped 3 He cryostat with a base temperature of 400 mK.In both setups DC and microwave signals are combined at the coldest stage with a bias-tee and diplexer and two cryogenic circulators are used to route the reflected microwave signals to cryogenic HEMT amplifiers.In some experiments additional amplification and filtering of the microwave and baseband signals is performed at room temperature.Time resolved measurements are performed via homodyne detection with ω LO = ω p /2 utilizing a homemade microwave bridge.The Supplementary Material includes detailed schematics of both setups.VNA Measurements.VNA measurements were fit to cavity input-output theory [33] in order to extract ω 0 , Q i and Q c .For the data in Fig. 1d, a baseline was measured and subtracted from each measurement, by setting I DC = 0 mA, which shifted ω 0 outside the measurement window.Measurement of E. The measurements of E presented in Fig. 3 were collected by repeating each pulse sequence 500 shots at a time, with a 33 Hz repetition rate (pump duty cycle ≈ 1/250).In each experiment, the parameters being swept (P 0 , P p , τ 1 , ϕ p , and the corresponding control experiments) were selected in pseudo-random order.For each datapoint a minimum of 10 4 shots of both the experimental and control sequences were run.A threshold applied to the amplitude of the demodulated signal √ X 2 + Y 2 was used to determine if the device self-oscillated.For the experiments in Figs.3a-d the pump was phase-modulated at a rate of 15 kHz to ensure E would not be dependent on ϕ p .In Fig. 3e phase modulation was disabled.Despite a 3 GHz clock used to sync the two microwave sources, a small phase drift was still evident in P (T |S) (the control experiments used for calculating P (T | S) showed no trend, as expected).To correct for the phase drift, P (T |S) for each experimental repetition (500 shots for each ϕ p ) was fit with a phenomenological function and aligned with the minimum E on ϕ p = 0.For all measurements of E, the uncertainties were found by adding in quadrature the 95% confidence intervals for the binomnial distributions of P (T |S) and P (T | S).Measurements of 209 Bi.All measurements of 209 Bi were performed at 400 mK.The device is mounted so that B 0 is aligned parallel to the long axis of the resonator.This orientation is chosen so that the magnetic field of the resonant mode B 1 is perpendicular to B 0 for spins located underneath the resonator.Numerically solving the 209 Bi spin Hamiltonian allows one to target specific 209 Bi ESR transitions by adjusting B 0 until the spins are resonant with the cavity.In this work we measure the ⟨−, 4| S x |+, 5⟩ transition.The small discrepancy between the B 0 at which we measure ESR (13.71 mT) and the numerically-predicted field (13.54 mT) might be attributed to strain caused by the different coefficients of thermal expansion of the Si substrate and NbTiN thin film [34] or a slight miscalibration of the superconducting solenoid used to generate B 0 .For the Hahn echo and CMPG sequences used for the experiments in Fig. 4, we use τ 0 = τ 2 = 10 µs, τ 1 = 150 µs, and τ 3 = 2τ 1 + 4τ 0 /π [35].The leading edge and trailing edges of the pump pulses are padded by 50 µs and 30 µs with respect to the π y pulses, to avoid their amplification.The sequences were measured with a repetition time of 7 s.For the experiment in Fig. 4c a total of 1040 shots of the full sequence with N = 20 were measured for both the control and experimental sequences at each P p , while alternating between the control and experimental sequences.We performed two independent control sequences: one where the x π/2 pulse was excluded and a second where the twenty y π pulses were excluded.The two values of P (T | S) were found to agree closely with one another, despite the different timing of the pulses relative to the onset of the pump.This indicates that the dark count rate was not influenced by residual fields of the x π/2 and y π pulses in this experiment.Histograms of the detector counts for the experimental and control sequences as a function of time and repetition N are provided in the Supplementary Material.

Calibration of Powers.
All powers mentioned throughout the text are referred to the input of the device enclosure.To calibrate these powers we performed three separate cool-downs of the cryostats with two additional high-frequency lines (L 1 and L 2 ).In the first two cool-downs, L 1 or L 2 were connected in place of the device, and S 21 measurements of each pair of lines were taken.In the third cool-down, an S 21 measurement was taken where L 1 and L 2 were connected to one another at the base temperature plate.The combination of S 21 measurements taken over the three cool-downs allowed for a full reconstruction of the gain (loss) of each line, which agreed closely with the designed amplification (attenuation).We estimate the powers to be accurate to within ±1 dB.performed the 209 Bi implantation.D.P. helped with the measurement electronics.A.M. and J.J.P. supervised the project.W.V. and J.J.P. wrote the manuscript with input from all authors.2) and (3) were combined with a bias-tee (Pasternack PE1615) and a diplexer (Marki DPX114) and fed into the device via a short semi-rigid cable connected to the common port of the diplexer.
To ensure that the noise reaching the device at ω 0 was limited by the equilibrium fluctuations at the lowest temperature stage (10 mK), 70 dB of fixed broadband attenuation was used on line (1) and a series of reflective low-pass filters were used on line (2).To facilitate the application of a strong pump, line (3) had only 16 dB of fixed attenuation, but high-pass filters at the lowest temperature stages ensured > 60 dB of attenuation for frequencies about ω 0 .
The reflected microwave signals were routed through two cryogenic HEMT amplifiers situated at 1 K and 4 K.To protect the amplifiers from damage and radiation from reaching the sample, a DC-block, two band-pass filters, and two isolators were used.

B. Measurements at 400 mK
Measurements at 400 mK were performed using a 3 He refrigerator.The copper sample enclosure was mounted to the base-plate of the insert and enclosed in a vacuum can that was dipped directly into liquid 4 He.A closed-cycle, single-shot 3 He system maintained the sample at ∼ 400 mK throughout the duration of the measurements.The attenuation and filtering of the four lines was configured similarly to that of the 10 mK measurements.

C. Microwave Bridge
Time-resolved experiments were performed using a homemade microwave spectrometer which is depicted in Fig. S3.The system has two main paths: one for generating phase-coherent pulses and one for performing homodyne demodulation of the signals coming from the device.Pulses are generated by IQ mixing a local oscillator (LO) with baseband signals from an arbitrary waveform generator (AWG).To extend the dynamic range of the system, a programmable attenuator is used to vary the output power of the system.To suppress leakage of signals from the box when no pulses are being sent, a fast microwave switch is placed before the output of the system and is actuated throughout the pulse sequences.Similar switches are placed at the input of the box and provide > 40 dB of attenuation when open to ensure that the high power pulses used for performing ESR do not reach the demodulator with a power exceeding its damage threshold.Power entering the box is further amplified before being homodyne demodulated.The resulting quadrature signals, X(t) and Y (t), are digitized by a data acquisition system (DAQ).For experiments where the pump is added to a pulse sequence, it is routed through the box and gated by a fast microwave switch.Triggering of the AWG, DAQ, and microwave switches is achieved with TTL logic supplied by a pulse generator (Spin-Core PulseBlaster ESR Pro).

II. DEVICE DESIGN
The design of the device is similar to that of several previously reported Kinetic Inductance Parametric Amplifiers (KIPAs) [1,2].It consists of two main components: a high-quality factor quarter-wavelength (λ/4) microwave resonator and a Band-Stop Stepped Impedance Filter (BS-SIF, Fig. S4).The former is constructed as a dense Interdigitated Capacitor (IDC) with forty fingers of width 1 µm extending from the center conductor, each of which has a spacing of 1.5 µm to ground (Fig. S4b).The fingers of the IDC were made to have two lengths (twenty with length 125 µm and twenty with length 135 µm) to produce a frequency detuning of the next harmonic at ω 1 .This ensures ω 1 ̸ = 3ω 0 , so that a strong pump tone at ω p = 2ω 0 does not couple the λ/4 and 3λ/4 modes.The BS-SIF consists of eight total segments of CPW with alternating low Z lo = 29.5 Ω and high Z hi = 123 Ω impedance.The electrical lengths of each segment are λ/4 at the resonant frequency ω 0 , which results in a deep stop-band centered at ω 0 , and pass-bands at DC and ω p .The BS-SIF therefore serves two purposes: it isolates the resonant mode from the measurement port (i.e.creates a large coupling quality factor Q c ) and it enables the resonator to be galvanically connected to the measurement port so that it can be biased with a DC current I DC .The design parameters of the device are summarized in Table S1.

III. MEASUREMENTS OF RESONATOR FREQUENCY AND Q-FACTOR
Measurements of S 11 were performed with a Vector Network Analyzer (VNA) (Keysight PNA-L N5231B).Due to the narrow bandwidth of the resonator relative to its frequency tunability, measurements of S 11 with the resonator far-detuned were subtracted to remove background ripple and correct for the line delay.For the lowest power measurements in Fig. 1d of the main text, the signal was digitally filtered with a Savitzky-Golay filter to improve the signal to noise ratio.
Fig. S5 shows ω 0 , Q i and Q c measured as a function of I DC , which are extracted from fits of S 11 to cavity input-output theory.The resonance frequency ω 0 /2π could be tuned by 246 MHz by applying I DC = 4.89 mA (Fig. S5a).The kinetic inductance is expected to vary with I DC according to where I * is a constant [3].This in turn results in the frequency of the resonator varying as [1,3] Fitting the measured ω 0 (I DC ) with Eq. 2 yields I * = 21.5 mA for this device.Notably, the device is extremely under-coupled over the entire range of operation, with Q i /Q c in the range [0.033, 0.064] (Fig. S5b,c).
Fig. 1d of the main text shows measurements of Q i taken as a function of the applied microwave power P 0 .This can be converted to an average number of intracavity photons n = 2Q 2 is the loaded quality factor [4].The general improvement of Q i with n is consistent with many other studies of high-Q superconducting microwave resonators, where the influence of Two Level Systems (TLSs) on Q i has been well documented [5].Notably, where TLSs limit Q i , one typically expects Q i to flatten below n = 1, because the microwave power is insufficient to depolarize the TLSs.This is not observed in the range of signal powers studied here.We note that some authors have reported that Q i reduces with signal power down to n = 10 −3 [6].

IV. MODEL OF A PARAMETRICALLY-PUMPED DUFFING OSCILLATOR
To understand the behaviour of the KIPA when strongly pumped, we seek to develop a theoretical model based on a parametrically-pumped Duffing oscillator, which has previously proven to be an excellent description for Josephson Parametric Oscillators [7][8][9].For a classical oscillator with position x, the Duffing equation is given by where λ is the linewidth of the oscillator, ω is the frequency of a periodic drive with strength F , and d is the Duffing constant.For a mass on a spring, the Duffing constant describes a softening or stiffening of the spring as it extends from its equilibrium position.We show below that for a superconducting microwave resonator it is related to the Kerr effect.
To relate the Duffing equation to the KIPA we take the approach of Lin, et al. [9] and begin with the master equation for the intracavity field, which is given by ȧ Here a (a † ) is the bosonic annihilation (creation) operator for the cavity mode, a in (a † in ) is the bosonic annihilation (creation) operator for the port mode, and b in (b † in ) is the bosonic annihilation (creation) operator for the bath mode.κ = ω 0 /Q c and γ = ω 0 /Q i are the rates at which photons couple from the cavity mode to the port mode and bath mode, respectively.
The Hamiltonian for the KIPA was previously derived in Reference [1].In the frame rotating at half the pump frequency it is given by where Here, ∆ is a detuning of half of the pump frequency from the point of degeneracy (i.e. the cavity frequency), ζ is the three-wave mixing strength, and K is the Kerr strength.∆ accounts for the shift of the cavity with an applied DC current (δ DC ) and pump current (δ p ) due to the nonlinear kinetic inductance in the KIPA.I * is a constant that sets the scale of the non-linearity of the KIPA and can be experimentally determined by measuring δ DC with a Vector Network Analyzer (VNA), as demonstrated in Fig. S5a.I DC is the applied DC current that enables three-wave mixing and I p is the pump current peak amplitude.L T is the total kinetic inductance and ϕ p is the phase of the pump tone.Substituting Eq. 5 into Eq. 4 yields Next we consider the simplification ⟨a in ⟩ = ⟨b in ⟩ = 0, which corresponds to the case where no resonant signals are sent to the cavity via the port or bath modes.In this case, Eq. 11 can be transformed into two real-valued coupled differential equations by substituting a = X − iY , where X and Y are the operators for the quadrature amplitudes of the intracavity field.Doing so yields where γ = (κ + γ)/2.This can be recast in the form where Previous authors have noted the similarity of Eq. 13 to Hamilton's equations [7,8].This has lead to Eq. 14 being described as a metapotential because the behaviour of the superconducting circuit is analogous to a particle traversing the potential surface g(X, Y ).
In Eq. 14, the terms ∆/(2γ), ζ/(2γ), and K/(2γ) correspond generalized versions of the detuning of the pump frequency, the pump strength, and the Duffing non-linearity, respectively.Their influence on the behaviour of the oscillator become clear by taking the steady state solution of Eq. 13.We also make use of a change of variables for X and Y so that they can be described in terms of a single amplitude α and phase θ by making the substitutions X = α cos(θ) and Y = α sin(θ).Doing so yields Equations 15 and 16 are π-periodic, and thereby imply the existence of two non-trivial solutions.Moreover, it is straightforward to solve for α by taking their product, which yields a solution that is independent of θ and is equal to Because α is an amplitude, we require Eq. 17 to yield real solutions.This is true only when so that we can interpret Eq. 18 as a boundary in parameter space.When the inequality is false, the device functions as a linear parametric amplifier.In this case, provided ⟨a in ⟩ = 0, α will remain equal to zero.When the inequality is true, however, the resonator will quickly develop a large intracavity field with amplitude α and phase θ = 0 or θ = 1π.

V. HYSTERETIC PARAMETRIC SELF-OSCILLATIONS
In Fig. 2 of the main text it is demonstrated that the detector signal latches.This is related to the fact that Q i is dependent upon n, which results in hysteresis of P th .This hysteresis can be directly observed when measuring |S 11 | with a VNA while sweeping P p .In Fig. S6a we increase P p and observe a sharp transition in the behaviour of the device as P p is raised beyond −25.5 dBm.In contrast, when P p is decreased, the behaviour changes at P p = −32.4dBm (Fig. S6b).
Line cuts taken from the two measurements show that the device functions as a simple resonator for P p < −32.3 dBm (Fig. S6e) and as a parametric oscillator for P p > −25.4 dBm (Fig. S6c).The behaviour observed at large P p can be clearly identified as parametric self-oscillations due to the large power generated at exactly half the pump frequency.For P p intermediate to these values, the device functions as a parametric amplifier prior to latching.Notably, the linear parametric gain achieved is much smaller than in previous studies using KIPAs [1,2], which is a consequence of the fact that Q i < Q c for this device.The power generated in the self-oscillating state is large enough to cause the amplification chain to compress, which results in the baseline shifting by -5.6 dB.For a parametric oscillator, one would also expect that the amplitude of the selfoscillations would be arrested by a non-linear shift in the resonance frequency due to the Duffing non-linearity.This is because for a fixed ω p and P p , shifting ω 0 will reduce the rate of downconversion, which in combination with a finite cavity bandwidth set by Q L , eventually leads the intracavity field to a high-amplitude equilibrium.Fig. S6b clearly reveals this behaviour, with the detuning (ω p /2 − ω 0 (P p ))/2π reaching a maximum of 3.2 MHz at the highest P p measured.
It is also notable that the depth of the resonance appears to increase when the device selfoscillates.Fitting the resonance in isolation from the other features confirms this and reveals that it is primarily the result of Q i , which increases from 13 × 10 3 to 50 × 10 3 when the device selfoscillates.We attribute this to be the result of the down-converted power partially saturating the TLSs.

VI. EVIDENCE OF ADDITIONAL MIXING PROCESSES
The measurements of S 11 shown in Fig. S6 reveal several features that are not expected from the basic theory of a parametric oscillator, namely several sharp peaks on the red sideband and fine structure near ω = ω p /2.To better understand these features, we measure the output signal with a spectrum analyser centered on ω 0 (as measured with a VNA with the pump off) when the device is supplied only a DC current and a pump with frequency ω p = 2ω 0 .Fig. S7 shows one spectrum obtained with I DC = 2.5 mA and P p = −50.3dBm.For this setpoint, P p is just above the level required to initiate parametric self-oscillations, but similar results are obtained for larger P p and for different I DC .The spectrum reveals a frequency comb that spans nearly the entire 3 MHz bandwidth of the measurement, with teeth that are equally spaced by 54 kHz.The tooth of the comb with the largest power is indeed centered at ω p /2, as expected for a parametric oscillator.The generation of frequency combs with superconducting microwave resonators has been demonstrated in several works including References [10][11][12][13][14] with several mechanisms underlying comb formation.We also note that the spectrum is obtained with phase modulation of the pump disabled, and direct measurements of the microwave sources do not reveal any sidebands that might otherwise explain the frequency comb generated by the KIPO.While a full explanation of the comb generated by the KIPO is beyond the scope of this work, it nevertheless has important implications to the experiments of this manuscript.First, when the KIPO is made to self-oscillate, the combs introduce a time-varying component to the amplitude of the signal demodulated with a local oscillator of frequency ω LO = ω p /2, as in Figs.2b and 4b of the main text.Second, the equal spacing of the teeth suggests they are likely generated via higher-order mixing processes (e.g.four wave mixing), which would thereby deplete power from the tooth at ω p /2.As the frequency comb is not captured in the model of a Duffing oscillator used to describe the KIPO, the amplitude of the self-oscillating state given by the model (Eq.17) will not match that of the experiments.Because the frequency comb is generated only upon the initiation of parametric self-oscillations, however, the Duffing oscillator model nevertheless remains applicable for estimating the parametric self-oscillation threshold P th (P p , ω p ).

VII. MAPPING THE SELF-OSCILLATION BOUNDARY
In this section we show how P th , which is written in terms of the KIPA Hamiltonian in Eq. 18, can be compared to experiments.
We begin by expanding the ∆ 2 term using Eq.6, which yields To simplify this we make note of the relative scales of each term.ω 0 /2π and ω p /2π are both of order GHz.For our experiments, δ DC /2π is tens of MHz.The expression for δ p is given by Eq. 8 where it is written in terms of I 2 p .To get a sense of its magnitude we can write it in terms of a microwave power P p by substituting I 2 p → 2P p /Z p , where Z p is the impedance of the device at frequency ω p .From simulations of the device (Sonnet) we expect Z p ≈ 33 Ω.For P p = −33 dBm, the largest power used for the experiment shown in Fig. 1c of the main text, this yields δ p /2π ≈ 64 kHz.From previous works we know that K/2π is of order Hz for these devices [1].We therefore conclude that ω 0 , ω p ≫ δ DC ≫ δ p ≫ K which allows us to approximate Eq. 19 as where we have omitted the terms which are lower in frequency.
Here we note that in the main text we define ∆ p = ω p − 2ω 0 (I DC ), where ω 0 (I DC ) is an experimentally measured value of the resonant frequency with an I DC applied.Written in terms of Eqs.7-9 this is where we neglect the small shift in the resonance frequency due to the microwave power of the tone used to measure S 11 .This is an important difference given that ∆ accounts for the shift δ p due to the pump power, whereas ∆ p does not.Using the approximation in Eq. 20 we find Here we see that Eq. 23 contains a term that is proportional to δ p that accounts for the shift of the resonant frequency with the pump power.Using Eqs.6-10 and Eq. 23, Eq. 18 can be written as 1 16 We then solve for I 2 p and convert it to a pump power as above.Without loss of generality, we assume ζ to be real and a positive value, i.e. we ignore the phase ϕ p , which is only relevant when a coherent signal is being amplified, whereas here we consider only vacuum noise.We also substitute Z p → α p Z p , where α p is a constant that accounts for ripple in the transmission of the pump power through the device.This yields For the models shown in Fig. 1c of the main text we set Q c = 220 × 10 3 and I * = 21.5 mA based on the measurements shown in Fig. S5.We set Z p = 33 Ω based on simulations of the device (Sonnet) and α p = 1.3 (1.13 dB) because it reproduces the experimental data well (the width of the self-oscillating region is primarily set by α p Z p ).With other KIPAs we have observed that α p can vary by as much as 10 dB over the operating frequency range (the ripple can can be inferred from measurements of the P p required to achieve a fixed gain while tuning ω 0 (I DC )).We also note that in Fig. 1c we apply an offset of ∆ p /2π = −500 kHz to the x-axis to center the parameter region where we observe parametric self-oscillations on ∆ p = 0; this likely indicates that the frequency of the device drifted between the times Fig. 1c and ω 0 (I DC ) were measured.
Overall, the model qualitatively reproduces the self-oscillation boundary that is measured.The main effect of increasing Q i is to shift the boundary to lower P p .Good agreement between the model and the data occurs when Q i = 18 × 10 3 .This is notable because Q i (n = 0.01) ≈ 5 × 10 3 is measured when the pump is off (Fig. 1d of the main text).This might indicate that our model is incomplete, or that Q i is not independent of P p , as we have assumed thus far.Supporting the latter hypothesis is the recent work by Qiu, et al. who found that it was essential to consider the P p -dependent saturation of the TLSs due to amplified vacuum noise to quantitatively explain the behaviour of their amplifier [15].
The main takeaway we highlight is that the model gives reasonable estimates for the observed P th .Moreover, because the device is undercoupled, its linewidth, and hence P th , is very sensitive to n.This is the basis of our detector: P th can be dynamically reduced below a fixed P p when the device absorbs resonant power, resulting in onset of parametric self-oscillations.
VIII.PHASE-COHERENT MEASUREMENTS RESOLVING THE QUIET, 0π, AND 1π STATES So far, our measurements have shown that when P p > P th the device generates a high-amplitude field at frequency ω p /2. Central to the model of a parametrically driven Duffing oscillator, however, is that there are two phase-coherent self-oscillating states, referred to as the 0π and 1π states.
To directly resolve these states, we down-convert the signal emitted from the device using a mixer driven by a local oscillator with frequency ω LO = ω p /2 and digitize the resulting quadrature amplitudes, X(t) and Y (t).In Fig. S8b we show a histogram of the signal in the XY -plane when the device is pumped with P p < P th .This is the quiet state of a parametric oscillator and corresponds to amplified vacuum noise; it is therefore a circle that rests at the origin of the XY -plane.In Fig. S8c we set P p > P th and observe the emergence of two high-amplitude phase-coherent states.These are the 0π and 1π states.Their phase with respect to one another is fixed and equal to π, but their specific orientation in the XY -plane is set by their phase with respect to the local oscillator.We therefore choose to align both states along the Y axis during post-processing.We note that to observe both the 0π and 1π states requires measuring the system many times.The histograms in Fig. S8b,c correspond to 250 ms of digitized data, which was acquired from 500 shots of the pulse sequence depicted in Fig. S8a.The pulse sequence provides 1 ms of dead time to ensure the device resets, followed by 1 ms where the pump is turned on, and then 500 µs where the pump remains on and the demodulated signal is digitized.The 1 ms period where the pump is turned on prior to digitizing the signal is intended to allow the device to reach its steady state.Nevertheless, when P p ≈ P th , occasionally the device is seen to transition from the quiet state to either the 0π or 1π states.These transitions are effectively irreversible while the pump remains on, due to the hysteresis observed in Fig. S6.These transitions can be seen clearly by renormalizing the colormap so that every bin is either zero (no counts) or one (one or more counts), which show that the trajectories taken from the quiet to the 0π and 1π states are coherent and reproducible (Fig. S8d).It is important to ensure no resonant power is supplied to the device during these measurements.It is well known that even a weak tone can cause the device to favour the 0π or 1π states, a phenomenon commonly referred to as injection locking [9].In the present experiments, the balanced weighting of the 0π and 1π states confirms injection locking is not occurring.
By acquiring a series of these histograms, we can map the time-averaged populations of the quiet, 0π, and 1π states as a function of experimental parameters.In Figs.S8e,f we show series of histograms measured as a function of P p and ∆ p , respectively.These measurements are equivalent to taking vertical and horizontal line-cuts across the self-oscillation boundary, which is mapped in Fig. 1c of the main text.For both sets of measurements, the self-oscillation boundary as predicted by the model described above is indicated by red arrows and shows good agreement with the measurements.The device parameters used in the model are the same as in Fig. 1c, with Q i = 18 × 10 3 .Note that as we did for Fig. 1c, we apply here an offset of ∆ p = −400 kHz to account for drift in the resonance frequency.
While the amplitude of the self-oscillating states increases with P p and decreases with ∆ p , as broadly predicted by the model in Eq. 17, we note that the model does not produce a quantitative match to this aspect of our experiments.As stated above, this is because the model does not account for the emergence of the frequency comb shown in Fig. S7 which deplete the power at ω p /2. Future refinements to the model, such as the inclusion of additional mixing processes and a P p -dependent Q i , may facilitate a quantitative match to all aspects of our experiments.

IX. DETECTOR DARK COUNT RATE
In Fig. 3d of the main text we show the probability of measuring dark counts P (T | S) as a function of the pump power P p .From this measurement we also determine the dark count rate (DCR).To do so, we make use of the fact that the detector latches in the self-oscillating state.We can then calculate the DCR as where τ tot = τ 0 + τ 1 + τ 2 = 120 µs is the total duration of the pump pulse in each shot of the control experiment (Fig. 3b of the main text), N = 10 4 is the total number of shots of the control pulse sequence (depicted in Fig. 3b of the main text), n is the number of dark counts, and t i is the time at which the detector "clicks" measured from the rising edge of the pump pulse.
In Fig. S9 we plot the DCR as a function of P p .We also directly compare it to the detector efficiency E measured during the same experiment with pulses of duration τ 1 = 10 µs and power P 0 = −111 dBm.The minimum dark count rate we can measure is limited by the total measurement time and is equal to 1/(N τ tot ) = 0.83 Hz.For P p < −51.4 dBm we detect no dark counts, whereas for P p > −50.4 dBm the DCR exceeds 1 kHz.
In Figs.3e-g of the main text we determine the detection sensitivity and demonstrate that E is phase-sensitive.For these experiments we set P p = −51.2dBm, where in Fig. S9 we measure a total of n = 13 dark counts in N = 10 4 shots, corresponding to a DCR of 10.8 Hz.

X. RECEIVER OPERATING CHARACTERISTIC CURVE
There are a wide variety of metrics that can be used to benchmark the performance of a detector.Throughout the main text we use the detector efficiency E = P (T |S) − P (T | S) as a single metric.One disadvantage of this approach, however, is that by itself E does not communicate the specific balance between the probability of successful detection P (T |S) and the probability of dark counts P (T | S), both of which may vary with different settings of the detector.A common approach to evaluating these trade-offs is to plot an receiver operating characteristic (ROC) curve, which compares P (T |S) directly to P (T | S).In Fig. S10 we show an ROC curve for the experiment depicted in Fig. 3c of the main text.The black diagonal line corresponds to the scenario where P (T |S) = P (T | S), in which case the detector behaves as a random binary classifier, i.e. using the detector is equivalent to guessing.For the pulses with power P 0 = −137 dBm, the detector outperforms the random binary classifier.For this experiment, the pulses had duration τ 1 = 10 µs, so that the total energy within the wavepacket was J = P 0 τ 1 = 0.21 0.27 0.17 zJ (42 52 33 photons), where the upper and lower values correspond to a 1 dB uncertainty in P 0 .

XI. DETECTOR EFFICIENCY WITH BIAS CURRENT
In Fig. S11 we compare measurements of E completed over a series of I DC setpoints, corresponding to a tunable frequency range of δ DC /2π ≈ 95 MHz.For each setpoint, the device achieves E near unity for pulses with duration τ 1 = 10 µs and P 0 = −111 dBm.The similar performance of the device at each setpoint indicates it is generally insensitive to I DC , and can therefore be used as a detector over much of its total tunable frequency range.

XII. MEASURING ESR AS A FUNCTION OF MAGNETIC FIELD
To simulate the use of the KIPO as a spectroscopic probe, we measure ESR as a function of the magnetic field strength B 0 for the ⟨−, 4| S x |+, 5⟩ transition of 209 Bi.In Fig. S12a we show a conventional ESR measurement using a CPMG sequence with N = 20 refocusing pulses.For each B 0 we execute the full pulse sequence a total of twenty times, and average the full record of 20 × 20 = 400 echoes.We show the signal measured on X, the quadrature onto which the spin echo signal was emitted, and find the spin signal to have a maximum at 13.7 mT.
We compare the above to a measurement that is otherwise equivalent, except that the KIPO is used as a "click"-detector by biasing it with a pump with power P p = −40.0dBm during the time period the echoes are refocused (Fig. S12b).The self-oscillation signal that is triggered by the spin echoes cannot be aligned onto a single quadrature, so we instead plot the amplitude of the full homodyne-demodulated signal √ X 2 + Y 2 .We show the average signal acquired over all 20 × 20 = 400 refocusing pulses which is justified for this measurement because no dark counts were recorded for the control experiment (the sequence omitting the x π/2 pulse was used).As for the conventional measurement, the spectrum shows a maximum at 13.7 mT, indicating that the detector can be used to determine the B 0 (frequency) of the ESR transition.Moreover, the amplitude of the detector signal with respect to the noise of the measurement is far greater than in the conventional case.This is despite the fact that we purposefully operate the detector at a point of low detection efficiency to avoid any dark counts.We refrain from quantitatively comparing the signal to noise ratio (SNR) of these measurements as has been done previously for experiments using linear amplifiers because the detection signals are fundamentally different; because the detector signal latches, it can be used to achieve an arbitrary SNR by extending the time the signal is integrated.

XIII. HISTOGRAMS OF DETECTOR "CLICKS" IN A CPMG ESR MEASUREMENT
In Figs.S13a,b we plot histograms showing the time within the CPMG pulse sequence at which the detector "clicks."The data is from the same experiment shown in Fig. 4c of the main text where N = 20 refocusing pulses were used, and corresponds to the minimum and maximum values of P p used in that experiment.For each shot of the pulse sequence, we consider only the first time the detector "clicks," because the amplitude of the self-oscillating state is sufficiently large to drive the spin system and scramble the echoes.For both P p , the "clicks" of the experimental pulse sequence (blue bars) are triggered within a short time window near t = 170 µs, corresponding to the time at which the spin echo refocuses (see the inset of Fig. 4b of the main text).In contrast, the dark counts for the two control experiments (red and orange bars, plotted as negative values) show no trend with t.These histograms give further confidence that the "clicks" triggered in the experimental pulse sequence are indeed caused by spin echoes and are not an artifact of the measurement sequence.
In Figs.S13c,d we compare the distribution of the "clicks" with the repetition N , i.e. on which of the refocusing y π pulses the detection is made.The data clearly reveals that for larger P p , the "clicks" occur more frequently at smaller N .This is intuitive, as it suggests that as P p approaches P th , the sensor is more likely to "click" on any given repetition, in accordance with the model described in Section XIV.
In Fig. S14 we plot Eq. 30 as a function of the ratio ϵ = P (T |S)/P (T | S) for several fixed values of P (T |S).The solid lines are linear fits of N as function of ln(ϵ), and capture the points well.This shows that the number of refocusing pulses N in a CPMG experiment required to maximize the detection efficiency E scales as O[ln(ϵ)].

FIG. 1 .
FIG. 1. Measurement schematic and device characterization.(a) The device is represented by a parallel RLC resonator with frequency ω0 and a band-stop stepped impedance filter (BS-SIF).The resonator has a geometric inductance (Lg) and a kinetic inductance (L k ).We operate the device as a "click"-detector by biasing it with a strong pump with frequency ωp and a DC current IDC.In this work we detect two types of signals: weak classical signals with frequency near ω0 generated by a microwave source (1.), which we use to calibrate the detector's sensitivity, and spin echoes from an ensemble of 209 Bi donor spins (2.) that are resonantly coupled to the resonator via Lg.Detailed schematics of the measurement setups are presented in the Supplementary Material.(b) An artist's depiction of the resonator, which is formed from a λ/4 section of transmission line with a dense interdigitated capacitor. 209Bi spins are implanted into the silicon substrate.(c) The maximum power measured with a spectrum analyzer centered at ω0 as a function of ∆p = ωp − 2ω0 and the pump power Pp.The dark blue region corresponds to the parameter space where the device self-oscillates.The powers are referred to the output of the device and Pmax is truncated at -100 dBm to enhance clarity.The red lines correspond to the PSO threshold (P th ) predicted from a model of the device using three different values of Qi.The stationary and swinging pendulums are used to depict the quiet and self-oscillating states of the device, respectively.(d) Qi extracted from measurements of S11 performed with a VNA as the signal power is varied.Measurements were taken at T = 10 mK.

FIG. 3 .
FIG. 3. Detection efficiency measured with calibrated pulses.(a,b) The experimental pulse sequences used to measure the efficiency of the detector E. The sequence in panel b is a control experiment that measures dark counts.(c) E measured as a function of the stimulus pulse power P0 and pump power Pp.(d) The probability of dark counts P (T | S) as a function of Pp.(e) E measured as a function of P0 and τ1 for Pp = −51.2dBm.(f) The data in panel (e) re-plotted as a function of total wavepacket energy J = P0τ1.We determine the lower bound of the detector's sensitivity by fitting the data with P0 < −116 dBm with a sigmoid function (red shaded region).(g) E measured as a function of the pump phase ϕp with τ1 = 10 µs.For all experiments IDC = 2.0 mA, τ0 = 10 µs, τ2 = 100 µs and T = 10 mK.For each data point, at least 10 4 shots of both the pulse sequences are measured.For all panels, the uncertainties in E are too small to be seen.

FIG. 4 .
FIG. 4. Latched readout of 209 Bi spin echoes.(a) A schematic of the Hahn echo (N = 1) and CPMG-N pulse sequences.For the pulses, x and y refer to the phase of the signal while π/2 and π refer to the tipping angle.P (T | S) was measured using sequences where either the x π/2 or yπ pulses were omitted, so that no spin echo was produced.Both control sequences were performed and resulted in a similar number of dark counts.(b) Single shots of the Hahn echo pulse sequence with the pump off (blue) and with with Pp = −38.4dBm (red), measured at a field of B0 = 13.71mT.Inset: an average of 100 Hahn echoes measured with the pump offnote the scaling of the y-axis.(c) E measured as a function of N .For each Pp, a total of 1040 shots of the pulse sequence with N = 20 were measured.The errorbars correspond to the 95% confidence interval.The solid lines are fits of the data to Eq. 1.(d) A comparison between the experimental values of P (T |S) and P (T | S) for N = 1 and those fit to the data using Eq. 1.All experiments were completed at T = 400 mK and the pump was phase modulated at a rate of 15 kHz.
FIG. S5.The resonance frequency and quality factors measured as a function of IDC.The values are extracted for measurements of S11 using a VNA with a power such that n ≈ 1.The data in (a) is fit with Eq. 2 to extract the constant I * .

FIG. S6 .
FIG. S6.Hysteresis of self-oscillations.(a) VNA measurements of |S11| as Pp is increased.(b) |S11| as Pp is decreased.The labels ω0(Pp) and NDG show the shifted resonance frequency and the idler associated with a non-degenerate gain process, respectively.(c-e) Line cuts from (a) and (b) at various powers.The DC current was set to IDC = 0.83 mA for these measurements.The sharp change in the baseline power is due to the saturation of the amplification chain when the device self-oscillates.The colourmap for (a) was truncated at +12 dB and also applies to panel (b).
FIG. S7.A spectrum of the self-oscillation signal.For this measurement IDC = 2.5 mA and the resonator frequency was measured to be ω0/2π = 7.7203 GHz.The only signals supplied to the device during the measurement are the DC current IDC and a pump with frequency ωp = 2ω0 and power Pp = −50.3dBm.All powers are referred to the device input.
FIG. S8.Histograms of demodulated self-oscillations.(a) The pulse sequence used to measure the histograms.It is repeated 500 times for each measurement.(b) A histogram where Pp < P th , showing only the quiet state.(c) A histogram where Pp > Pth, showing the 0π and 1π states.(d) A histogram where Pp ≈ P th , showing the transition from the quiet to the self-oscillating states.Note that for this panel the histogram values are binary so the trajectories between the states can be easily seen.(e) A series of histograms in the XY -plane that have been projected onto X, showing the population evolve from the quiet state to the 0π and 1π states as Pp is increased.The red arrows correspond to Pth predicted from our model.(f) The same as in (e), except ∆p is varied for a fixed Pp = −46.2dBm.The red arrows correspond to the ∆p at which the model predicts the onset of parametric self-oscillations.
FIG. S9.KIPO dark count rate.A comparison between the detection efficiency E and DCR.The data is from the same experiment as shown in Fig. 3c of the main text, where E was measured for pulses with duration τ1 = 10 µs and power P0 = −111 dBm.The vertical dashed line corresponds to Pp = −51.2dBm, which is the Pp used in Fig. 3e-g of the main text.
FIG. S10.Receiver operating characteristic curve.The data is the same as is shown in Fig. 3c of the main text.A black diagonal line is plotted along the diagonal P (T |S) = P (T | S) and corresponds to a random binary classifier.
FIG. S12.Spectroscopy with CPMG measurements.(a) The amplitude of a signal measured with the conventional CPMG measurement sequence.(b) The same as (a) but measured with the modified CPMG sequence, with Pp = −40.0dBm.In both measurements we perform twenty shots of the CPMG sequence with N = 20 refocusing pulses and average the entire record.We recorded no dark counts for the control sequence of the measurement in (b).The measurements in both panels were taken during the same field sweep, with IDC = 2.53 mA and a waiting period of 7 s between each shot of each pulse sequence.
FIG. S13.Histograms of detector "clicks" for a CPMG measurement.(a,b) The distribution of "clicks" with time t for two Pp.The counts associated with the two control experiments are summed and plotted as negative values, so that they can be easily compared with the counts from the experimental sequence.(c,d) The distribution of clicks with refocusing pulse repetition N .The data is from the experiment shown in Fig. 4c of the main text.