Picotesla magnetometry of microwave fields with diamond sensors.

Developing robust microwave-field sensors is both fundamentally and practically important with a wide range of applications from astronomy to communication engineering. The nitrogen vacancy (NV) center in diamond is an attractive candidate for such purpose because of its magnetometric sensitivity, stability, and compatibility with ambient conditions. However, the existing NV center-based magnetometers have limited sensitivity in the microwave band. Here, we present a continuous heterodyne detection scheme that can enhance the sensor's response to weak microwaves, even in the absence of spin controls. Experimentally, we achieve a sensitivity of 8.9 pT Hz-1/2 for microwaves of 2.9 GHz by simultaneously using an ensemble of nNV ~ 2.8 × 1013 NV centers within a sensor volume of 4 × 10-2 mm3. Besides, we also achieve 1/t scaling of frequency resolution up to measurement time t of 10,000 s. Our scheme removes control pulses and thus will greatly benefit practical applications of diamond-based microwave sensors.

INTRODUCTION

Improving the sensitivity of microwave-field detection could directly advance many modern applications, such as wireless communication (), electron paramagnetic resonance (), high-field nuclear magnetic resonance (), and even astronomical observations (). Instead of conventional inductive detection, various quantum sensors have been developed in the past decades with enhanced capabilities. For instance, Rydberg atoms (), atomic magnetometers (), superconducting quantum interference devices (), and nitrogen vacancy (NV) centers in diamond (–) are highly sensitive to either the electric or magnetic field of microwaves. Among them, the NV center is distinguished by its unique properties including solid-state aspect and room temperature compatibility, which is essential for on-chip detection (), but suffers from relatively low sensitivity.

By using NV ensembles, the sensitivity of diamond magnetometer can be substantially improved with scaling of (), where nNV is the number of NV centers. It has been demonstrated for sensing of DC or low-frequency (<1-MHz) fields with an achieved sensitivity of several pT Hz−1/2 (–). However, for sensing of higher-frequency fields, higher spin-drive power is required (, ), making it available only for single or few NV centers (, ). Although this problem has recently been solved by pulsed Mollow absorption (, ) and concatenated continuous dynamical decoupling (), the uniform requirement of these sophisticated control pulses still limits the available nNV. To remove the barrier, one can directly observe the resonant absorption of microwave by NV ensembles and the subsequent spin transitions via measuring either optically detected magnetic resonance (ODMR) spectra (, ) or Rabi oscillations (, ). However, the absorption becomes inefficient and even loses the first-order response to the microwave field b1 when the corresponding Rabi frequency is smaller than the inhomogeneous transition linewidth , degrading the diamond sensor from field detection (∝ b1) to energy detection ().

Here, we propose a continuous heterodyne detection scheme to enhance the sensor’s response to weak microwave fields by introducing a moderate (still much weaker than ) and slightly detuned auxiliary microwave B1. The total energy after interference will have a slowly oscillating component with amplitude ∝B1b1. It means that the response to b1 is amplified by a factor of ∼B1/b1, which turns the diamond sensor back to field detection. We perform the demonstration on an ensemble of nNV ∼ 2.8 × 1013 NV centers within an effective sensor volume of 4 × 10−2 mm3. The diamond sensor maintains linear response to weak microwave fields down to subpicotesla. Within a total measurement time t of 1000 s, a microwave field of 0.28 pT is detectable, corresponding to a sensitivity of 8.9 pT Hz−1/2. Besides, the frequency resolution scales as 1/t down to 0.1 mHz for t = 10,000 s. We note that the strength of the auxiliary microwave required is just 220 nT, corresponding to a Rabi frequency of 4.4 kHz, which is nearly three orders lower than the typical control fields (, , ). The removal of sophisticated control pulses makes our scheme applicable to larger diamond sensors with further improved sensitivity. It also greatly benefits practical applications.

RESULTS

Continuous heterodyne detection scheme

The NV electron spin has a triplet ground state consisting of a bright state ∣mS = 0〉 and two degenerate dark states ∣mS = ±1〉 (denoting as ∣0〉 and ∣±1〉 hereinafter) with a zero-field splitting of D = 2.87 GHz. The degeneracy can be lifted by an external magnetic field B. Without loss of generality, we assume that the microwave of the form b1 cos ωt is resonant with ∣0〉 ↔ ∣1〉 transition (Fig. 1A) and then treat the NV center as a two-level system. If the microwave field b1 is strong enough, then one can observe a Rabi oscillation between ∣0〉 and ∣1〉 with frequency , where γNV = − 28.03 GHz/T (hertz refers to circular frequency hereinafter) is the gyromagnetic ratio of the NV electron spin. As shown in Fig. 1B, the oscillation slows down with reducing b1 and lastly degrades to an exponential decay. In addition to the intrinsic longitudinal relaxation Γ1 = 1/T1, the weak microwave opens an extra relaxation channel between ∣0〉 and ∣1〉 with a rate of ()where Γ2 = 1/T2 is the transitional relaxation rate and Δ is the detuning between ω and the NV transition frequency. In the presence of continuous laser, ∣1〉 will be polarized to ∣0〉 with a rate of Γp, which competes with the relaxation. A simple rate equation can describe the evolution of NV states (see Materials and Methods). In short, the competition leads to an equilibrium state, where the population of ∣0〉 is

Fig. 1.

Basic principle of continuous heterodyne detection.

(A) Simplified energy levels of NV centers. The ∣±1〉 states can be polarized to ∣0〉 state with a rate of Γp. A resonant microwave addresses the ∣0〉 ↔ ∣1〉 spin transition. (B) Evolution of the NV center driven by microwaves of different magnitudes. For a strong microwave, the spin state shows a Rabi oscillation between ∣0〉 and ∣1〉 with frequency Ω proportional to the microwave magnitude. For a weak microwave, the oscillation degrades to an exponential decay with a rate proportional to the square of microwave magnitude. (C and D) Comparison of direct and heterodyne detection. The competition between laser-induced polarization and microwave-induced relaxation leads to an equilibrium spin state. For direct detection (C), constant microwave magnitude results in DC fluorescence signal. For heterodyne detection (D), the microwave interference results in a time-varying magnitude and thus an AC fluorescence signal.

The extra relaxation Γ will result in decreased fluorescence (Fig. 1C). In the weak-field limit, i.e., Γ ≪ Γ1, this decrement iswhich is . It means that the NV center only preserves second-order response to the microwave field.

If we apply an auxiliary microwave of the form B1 cos [(ω + δ)t + ϕ] simultaneously, microwave interference will happen, resulting in modulation of the amplitude with beat frequency δ, as shown in Fig. 1D. In the situation of b1 ≪ B1, the amplitude can be simplified as . Here, we assume that both microwaves are resonant with the NV center, i.e., δ ≪ Γ2, and thus, the two microwaves can be treated as a single one, leading to a time-varying relaxation rate of , where . The constant term Γaux induces a constant decrement of fluorescence similar to Eq. 3, while the oscillating term induces an oscillation of fluorescence. The latter is predicted by the solution of rate equation (see Materials and Methods)

The oscillation frequency is just the heterodyne frequency δ, and the amplitude is . Now, the NV center has a linear response to the microwave field. We note that the relaxation between ∣0〉 and ∣−1〉 will no longer be negligible when Γp is comparable or smaller than Γ1 (fig. S1). Nevertheless, the linear response does not change but has a slightly different coefficient (section S1).

Experimental demonstration on NV ensembles

Benefiting from the removal of complicated control pulses, we can perform the experiment on a simple setup (Fig. 2A). We use an optical compound parabolic concentrator (CPC) to enhance the fluorescence collection efficiency (). As a proof-of-principle demonstration, both the signal and auxiliary microwaves are radiated from a 5-mm-diameter loop antenna. We apply an external magnetic field (∼12.5 G) perpendicular to the diamond surface, so that all the NV centers have the same Zeeman splittings. A common ODMR spectrum (Fig. 2B) can be obtained by sweeping microwave frequency. The triplet feature arises from the hyperfine coupling between the NV electron spin and the 14N nuclear spin. The ODMR linewidth of 482 kHz is defined by the full width at half maximum (FWHM). Here, the laser power is roughly 2 × 10−4 lower than the saturation power, so the optical power broadening is negligible (). The linewidth seems to be Γ2 according to Eqs. 1 and 3. However, for NV ensembles, the line broadening is usually dominated by other dephasing mechanisms, such as inhomogeneous strain fields, magnetic field gradients, and temperature fluctuations (). These mechanisms can be described by a static or quasi-static detuning Δ, which varies for different NV centers (see section S2 and fig. S2). Similarly, the Γ2 can also vary from NV to NV. Therefore, the ODMR spectrum after ensemble average can be calculated aswhere P(X) is the probability density function (PDF) of X (see Materials and Methods).

Fig. 2.

Proof-of-principle measurements on NV ensembles.

(A) Schematic of the setup. Both microwaves are radiated by a loop antenna with a diameter of 5 mm, which is parallel with the diamond surface. (B) ODMR spectrum of the NV centers. The blue line is the experimental result, while the red line is a three-peak Lorentz fit. The fitted FWHM linewidth is 482 kHz. The microwave (MW) field is 365 nT. The arrow marks the resonant frequency used in the following experiments. (C) Time traces of the photovoltage V with continuous laser and pulsed microwaves. For a single microwave (left), the photovoltage decreases to a constant value when the microwave is turned on and revives to the initial value after the microwave is turned off. The voltage difference marked by the blue arrow is the signal of direct detection. For dual microwaves (right), the phenomenon is similar but shows an additional oscillation when the microwave is turned on. The oscillation peak-to-peak amplitude marked by the yellow arrow is twice the signal of heterodyne detection. (D) Dynamical range of the diamond sensor. Blue circles indicate the means of measured voltage differences, where error bars indicate SEM. Yellow squares are extracted from the Fourier transform spectra, where error bars indicate the root mean square (RMS) of the baseline around δ with a span of 0.1 Hz. Here, B1 = 220 nT, and δ = 480 Hz. The lines are linear and parabolic fits for direct and heterodyne detection, respectively. Blue and yellow areas indicate the DC and AC noise floor for a total measurement time of 1000 s.

We first apply a single-channel resonant microwave. As shown in Fig. 2C, the photovoltage begins to decrease from V0 when the microwave is turned on and approaches to a saturation value V∞. When the microwave is turned off, the photovoltage returns to V0 with a slower revival rate of Γ1 + Γp. Measurements of the dependence of the revival rate on the laser power PL (watts) give Γ1 = 102 Hz and Γp = 250 × PL Hz (fig. S3), where PL ≤ 1.2 W is much lower than the saturation power (∼3.2 kW). The voltage difference gives the signal of direct measurement , specificallywhere Rd is the responsivity of direct measurement. As shown in Fig. 2D, the measured Sd indeed shows squared dependence on the microwave field b1 and is quickly lost in the noise, where the measured responsivity is 27.3 ± 0.3 μV/nT2, in agreement with the calculations (see Materials and Methods). Here, b1 is calibrated according to the relation , where PMW is the microwave power (fig. S4). Within a total measurement time of 1000 s, the minimum detectable b1 is just 4.9 nT. It means that this direct measurement is inefficient to sensing weak fields.

We then apply an additional auxiliary microwave. The time trace of photovoltage shows a similar behavior of decrease and revival with the microwave turning on and off. Besides, it also shows an oscillation (Fig. 2C), where the frequency and amplitude can be directly extracted from the Fourier transform spectrum (not shown in the figure). The extracted frequency gives the difference frequency of the two microwaves, while the extracted amplitude gives the signal of heterodyne measurementwhere Rh is the responsivity of heterodyne measurement. As shown in Fig. 2D, the measured Sh now shows linear dependence on b1, where the measured responsivity is 6.03 ± 0.01 mV/nT, also in agreement with the calculations (see Materials and Methods). Benefitting from measuring an AC signal, the noise is substantially reduced, and thus, this linear response can preserve over five orders in amplitude. Therefore, we have constructed a well field-to-voltage sensor with a dynamical range from 1 pT to 100 nT.

Optimization of sensitivity

To optimize the sensor’s performance, we need to improve the signal-to-noise ratio (SNR). A preferred option is to improve the responsivity Rh, which depends on Γp, B1, and δ. We first focus on the auxiliary microwave field B1, which is the key to enhance the response of NV centers to weak fields, serving as an amplifier. As shown in Fig. 3A, Rh first grows linearly with increasing B1, then tends to saturate, and eventually goes down. This trend consists of the theoretical expectation. According to Eq. 4, the peak should appear when , where 1/3 ≤ k < 1 positively depends on δ/(Γ1 + Γp), for example, k ≈ 0.51 for Γp = 200 Hz and δ = 480 Hz. After ensemble average over P(Δ) and P(Γ2), the optimal B1 does not have an analytic solution. Nevertheless, numerical calculations (see Materials and Methods) can well explain the experimental results (Fig. 3A). The optimal B1 is 220 nT, corresponding to a Rabi frequency of 4.4 kHz, well below the ODMR linewidth. We note that the removal of strong control fields is the most important advantage of our scheme.

Fig. 3.

Optimal sensitivity.

(A) Dependence of responsivity on auxiliary microwave field. Points are experimental results, where error bars indicate the RMS of baseline in Fourier transform spectra around δ = 480 Hz with a span of 0.1 Hz. The solid line is the theoretical calculation according to Eq. 16 in Materials and Methods. (B) Dependence of sensitivity on heterodyne frequency δ. The sensitivity is normalized according to the detection bandwidth. The red area indicates the optimal frequency window around 480 Hz. The blue area indicates the estimated shot noise–limited sensitivity. (C) Benchmark of sensitivity. The Fourier transform spectrum corresponds to a signal microwave field of 6.81 pT. The total measurement time is 1000 s. The measured SNR of 24.2 corresponds to a sensitivity of 8.9 pT Hz−1/2. Here, the auxiliary microwave field is 220 nT with δ = 480 Hz.

Another adjustable parameter is the laser power PL, which determines the spin polarization rate Γp ∝ PL. In principle, larger PL is preferred to maximal SNR if the noise is dominated by photon shot noise (see Materials and Methods). However, the increase in PL will introduce more experimental imperfections, such as sample heating and laser-induced photon noise. Therefore, we choose a moderate laser power PL = 0.8 W (see section S3 and fig. S5), corresponding to Γp = 200 Hz. At this point, the temperature of the diamond increases by 31.4 K (fig. S6). The measurement noise consists of the laser-induced noise (∝PL), the photon shot noise (), and the electric noise of the detection system (). Analysis of the dependence of noise on PL can extract the proportion of each noise (see section S4 and fig. S7). For the experimental condition here (PL = 0.8 W), the laser-induced noise dominates.

Because the laser has strong noise in a low-frequency (<400-Hz) band (fig. S7), we can increase δ to avoid it by tuning the frequency of the auxiliary microwave. However, it will also reduce the signal because of limited detection bandwidth, as predicted by Eq. 4, which we will discuss below. By dividing the measured noise spectrum by the calculated responsivity Rh(δ) (see Materials and Methods), we get the bandwidth-normalized sensitivity spectrum (Fig. 3B), where the optimal δ ∼ 480 Hz. This frequency window is currently determined by the laser-induced noise. To intuitively benchmark the sensitivity of our sensor, we perform the measurement on a weak microwave with a frequency of 2903.9 MHz and a field strength of 6.81 pT. Within a total measurement of 1000 s, the measured SNR is 24.2 (Fig. 3C), corresponding to a sensitivity of 8.9 pT Hz−1/2.

Frequency resolution and detection bandwidth

In addition to high sensitivity, another highlight of our scheme is unlimited frequency resolution, similar to previous heterodyne measurements (, , ). The frequency resolution can be promoted to 0.1 mHz by extending the total measurement time t to 10,000 s. As shown in Fig. 4A, the frequency resolution does not show obvious deviations from the 1/t scaling up to 10,000 s. Hence, further improvement is expectable by extending t, which is currently limited by the memory depth of the detector.

Fig. 4.

Linewidth and bandwidth.

(A) Dependence of linewidth on the total measurement time. The blue points are experimental results extracted from the Lorentz fits of Fourier transform spectra. The red line indicates the 1/t scaling. (B) Intuitive concept of bandwidth extension. The diamond “mixer” has a narrow-band response to the input microwave, where the band is centered at the frequency of the auxiliary microwave. If we cascade multiple mixers with different auxiliary microwaves, the band will be extended accordingly. (C) Measurements of bandwidth. All groups of measurements are normalized for better comparison of the bandwidth. The extended bandwidth consists of the ODMR linewidth. a.u., arbitrary units.

Another important figure of merit associated with frequency resolution is the detection bandwidth. According to Eq. 4, the −3-dB bandwidth should be . Although the actual bandwidth no longer has an analytic expression after ensemble average (see Materials and Methods), this formula is still a good estimation because Γaux is of the same order (∼102 Hz) as Γp + Γ1. An intuitive picture to understand this bandwidth is that the oscillation comes from the interference between the signal and auxiliary microwave, but the fluorescence has limited-speed response to variations of the microwave field (Fig. 2C). The limited-speed response in time domain corresponds to the limited bandwidth in frequency domain. The diamond sensor thus serves like a narrow-band mixer. To extend bandwidth, we can cascade multiple “mixers,” as shown in Fig. 4B. By simultaneously applying 240 channels of auxiliary microwaves with a frequency interval of 2 kHz, the “bandwidth” extends to 190 kHz, which is limited by the ODMR linewidth (Fig. 4C). Although the sensor has a response to all microwaves within the bandwidth, the measured frequency is the frequency difference between the signal microwave and the nearest auxiliary microwave. It means that frequency aliasing will happen, and thus, we need to repeat the measurement with a different frequency interval of the auxiliary microwaves to extract the actual frequency. We note that the extension of bandwidth comes at the expense of sensitivity. When the signal microwave is interfering with one of the auxiliary microwaves, the others serve like noise sources inducing additional longitudinal relaxation, corresponding to larger Γ1, and thus poorer sensitivity. Specifically, for m channels of auxiliary microwaves, the bandwidth scales as m, while the sensitivity scales as (see Materials and Methods). Another straightforward way is applying these auxiliary microwaves successively, which corresponds to sweeping the sensitive frequency band. The time consumption is directly proportional to bandwidth, so the sensitivity is proportional to .

DISCUSSION

We have shown that NV centers can be used as highly sensitive sensors for microwave magnetometry even in the absence of spin controls. Our scheme is based on resonant absorption of microwaves by NV centers. We have significantly improved the response to weak fields by introducing a moderate auxiliary microwave. Furthermore, we have applied the scheme to a diamond hosting NV ensembles, achieving a minimum detectable microwave field of 0.28 pT and a frequency resolution of 0.1 mHz.

Benefiting from the simplicity of our scheme, the measurements can be directly reproduced on larger sensors, resulting in further improved sensitivity. For example, if the diamond has a similar size to the photodiode (∼10 mm by 10 mm by 1 mm), then the sensitivity can be directly promoted to femtotesla level. Even so, the sensor is still far smaller than microwave wavelengths. An increase in NV density will also bring an improvement in sensitivity but needs to balance the increase in relaxation rate and the laser-heating problem. Although the sensor has a narrow bandwidth, the sensitive band is easily tunable by tuning the auxiliary microwave. The extended submegahertz detection bandwidth is limited by the ODMR linewidth. Improvement to gigahertz is possible by introducing a magnetic field gradient (). On the other hand, the sensitive frequency band is determined by the spin transition, which is magnetically adjustable up to hundreds of gigahertz () and even to the terahertz range in the future. Another small-range tuning method is using the Floquet dressed states that are independent of transition frequency ().

Our work paves the way for practical applications of diamond sensors, for example, microwave receivers in radars (), wireless communications (), and even radio telescopes (). This diamond device can work under extreme conditions, such as high temperature () or high pressure (–). The removal of spin controls will also efficiently reduce the complexity of contracting an on-chip diamond magnetometer ().

MATERIALS AND METHODS

Experimental setup and diamond samples

The core of our experimental setup is a sandwich of diamond, CPC (Edmund, 45DEG2.5MM), and optical filter (Semrock, LP02-638RU-25), which is mounted on a photodiode (Thorlabs, PDAPC2) for fluorescence detection. A high-power laser (Lighthouse, Sprout-D-5W) is used for illumination. The microwaves are generated from two radio frequency (RF) signal generators (Stanford, SG386) and/or an arbitrary waveform generator (Keysight, M8190a), amplified by a microwave amplifier (Mini-Circuits, ZHL-25W-63+) or attenuated by several attenuators, and radiated by a home-built loop antenna with a diameter of 5 mm. Another photodiode (Thorlabs, PDA36A2) is used for laser detection. The photovoltage is detected by an oscilloscope (NI, PXIe-5122) after a differential amplifier (Stanford, SR560). For the 10,000-s measurement (Fig. 4A), another oscilloscope (Keysight, MSOS254A) with a larger memory depth is used. All clocks are synchronized by one of the RF signal generators. The diamond is 100-oriented with an extra high-doping layer with ∼10-μm-thick growth with 99.99% 12C isotopic purity. The effective sensor volume is ∼4 × 10−2 mm3 with an estimated NV density of ∼4 parts per million. Hence, the number of NV centers is nNV ∼ 2.8 × 1013.

Rate equations

Considering a two-level system consisting of ∣0〉 and ∣1〉, there exists a relaxation between them of the rate Γ1 + Γ and a polarization from ∣1〉 to ∣0〉 of the rate Γp. Hence, the rate equations arewhere P is the population of ∣j〉 and satisfies ∑P ≡ 1. The solution iswhere is the equilibrium solution () given by Eq. 3. If we apply two microwaves, the rate equations become

Considering that Γ ≪ Γaux, the time-varying term is only a perturbation of the relaxation rate, so the solution should also be a perturbation of the equilibrium solution . Substituting a trail solution into Eq. 10 yields f = δ, φ = ϕ + arctan [δ/(Γp + Γ1 + Γaux)] + π, and

Calculations of responsivity

For a large number of NV centers, the local detuning Δ obeys normal distribution, so

Because the diamond is isotopically purified, the decoherence of NV centers is mainly induced by the surrounding electron spin bath. The bath spins generate a fast, randomly fluctuating field on the NV site, which is modeled by an Ornstein-Uhlenbeck process with variance of b2 and correlation time of τc (). Here, b (in unit of hertz) depends on the local distribution of bath spins, and its PDF is ()

Considering that an NV center is surrounded by many bath spins, the averaged correlation time of bath spins should have much smaller variance than a single one, so we assume that τc is constant. The decoherence rate Γ2 = b2τc and thuswhere σ2 = σ2τc/2. Here, σ1 ∼ 200 kHz and σ2 ∼ 8 kHz are roughly estimated from the measurement of ensemble T2 and ODMR spectrum (fig. S2).

The responsivity of direct measurement iswhere C is the fluorescence contrast, G = 375 kV/A is the total transimpedance gain, QEλ ≈ 89% is the quantum efficiency of the photodiode at λ = 700 nm, e = 1.602 × 10−19 C is the electron charge, η = 1% is the collection efficiency (corresponding to a saturation count rate of 0.8 Mcps for a single NV center), and α ≈ 15.4 is the ratio of the lifetime of the excited state versus the metastable state (). The responsivity of heterodyne measurement iswhere G = 750 kV/A is different from the direct measurement and A is from Eq. 11. Here, we take C as a free parameter to fit the experiment, because the contrast has slow variations on the time scale of days (see section S5 and fig. S8), which is possibly induced by the slow dynamics of the NV charge states (). For the direct measurements, the fitted contrast in Fig. 2D is 4.7%. For heterodyne measurements, the fitted contrasts in Figs. 2D and 3A are 5.8 and 6.8%, respectively. The contrast here corresponds to one of the nitrogen nuclear spin states, which is one-third of the full contrast.

Shot noise–limited sensitivity

If the laser power is much lower than the saturation power, then the photon emission rate of NV centers is proportional to the polarization rate Γp. In a sample period Ts, the detected photons are nNVηαΓpTs. Therefore, within a total measurement time of t, the SNR of Fourier transform spectra is

The noise is defined by the root mean square of the baseline in Fourier transform spectra, so there exists a coefficient 2 in the denominator. Here, one can see that larger Γp and Γaux are preferred, and the SNR quickly saturates to when Γp = 3Γaux > Γ1. Here, depends on the noise model. For simple estimation, we phenomenologically assume , where is the dephasing time of NV ensembles. Therefore, the shot noise–limited sensitivity iswhere β ∼ 2 is a modification factor to the phenomenological model (fig. S2). For the parameters in our experiment μs and C ∼ 6%, the best shot noise–limited sensitivity should be ∼1 pT Hz−1/2, which is smaller than the value of 2.2 pT Hz−1/2 in Fig. 3B. The deviation comes from the times more shot noise due to the balance detection and the unsaturated SNR due to insufficient laser power. For m channels of auxiliary microwaves, only one of them interferes with a specific single microwave, and others induce additional longitudinal relaxation. Hence, Eq. 17 becomes

It saturates to when Γp = 3mΓaux > Γ1.