# DC Quantum Magnetometry Below the Ramsey Limit

Alexander A. Wood  School of Physics, University of Melbourne, Victoria 3010,
Australia.  Alastair Stacey  School of Science, STEM College, RMIT University,
Melbourne, VIC 3001, Australia  Andy M. Martin  School of Physics, University
of Melbourne, Victoria 3010, Australia.

###### Abstract

We demonstrate quantum sensing of dc magnetic fields that exceeds the
sensitivity of conventional $T_{2}^{\ast}$-limited dc magnetometry by more
than an order of magnitude. We used nitrogen-vacancy centers in a diamond
rotating at periods comparable to the spin coherence time, and characterize
the dependence of magnetic sensitivity on measurement time and rotation speed.
Our method up-converts only the dc field of interest and preserves the quantum
coherence of the sensor. These results definitively improve the sensitivity of
a quantum magnetometer to dc fields, an important and useful addition to the
quantum sensing toolbox.

Magnetometers based on the nitrogen-vacancy (NV) center in diamond [1, 2]
provide $\upmu$T - nT sensitivity for single centers at ambient temperature
and mm-to-sub-$\upmu$m length scales, making them attractive resources for
studying biomagnetism [3], solid state systems [4, 5] and nanoscale NMR [6] in
challenging real-world sensing environments [7]. Since many magnetic phenomena
of importance in navigation and biomagnetism manifest as slowly varying or
static magnetic fields, intense effort has been devoted in particular to
improving dc sensitivity [8], focusing on the diamond material [9, 10], photon
collection efficiency [11], quantum control sequences to eliminate decoherence
[12, 13, 14] and more recently, the addition of ferrite flux-concentrators
[15, 16]. Many approaches to improving the measurement signal are frustrated
by a commensurate increase in noise, limiting the attainable sensitivity.

To date, Ramsey-type interferometry [17] is the optimum dc measurement
sequence [18, 8], and the sensitivity of Ramsey magnetometry is limited by the
ensemble dephasing time $T_{2}^{\ast}$, which reflects the magnitude of low
frequency noise in the system. In diamond, impurities such as 13C [19] or
paramagnetic nitrogen [20] are the dominant contributions to $T_{2}^{\ast}$,
resulting in $T_{2}^{\ast}<1\,\upmu$s for readily-available CVD diamond
samples. Additionally, $T_{2}^{\ast}$ varies considerably between diamond
samples, and often significantly _within_ a single sample due to gradients or
spatial variations of crystal strain or varied dopant density. This
decoherence can be largely eliminated by employing time-reversal dynamical
decoupling measurement schemes, such as Hahn spin-echo [21], but at the cost
of insensitivity to slowly varying or static dc fields [22]. Alternative NV-
magnetometry techniques [23, 24, 25] that eschew conventional quantum sensing
protocols have also been proposed and demonstrated, though to date yielded
comparable sensitivities to standard methods.

In this work, we use physical rotation to realize a significant improvement in
the sensitivity of a dc quantum magnetometer. Mechanically rotating a diamond
at rates comparable to the spin coherence time $T_{2}\sim 0.1-1$ ms up-
converts an external dc magnetic field to the rotation frequency, which is
then detected using NV spin-echo magnetometry [26]. The quantum sensing time
is increased from $T_{2}^{\ast}$ up to $T_{2}$, with a potential
$\sqrt{T_{2}/T_{2}^{\ast}}$ sensitivity gain, typically an order of magnitude
[22]. We demonstrate a 30-fold improvement in sensitivity over comparable
Ramsey magnetometry. Furthermore, our demonstrated sensitivity exceeds, by a
factor of 4.5, the theoretical shot-noise-limited, unity duty cycle
sensitivity of $T_{2}^{\ast}$-limited Ramsey magnetometry with this diamond.

Figure 1: Diamond rotation up-conversion magnetometry (DRUM) schematic and key
results. a) Schematic model of NV center, axis tilted from $z$ by
$\theta_{\text{NV}}=30.2^{\circ}$, rotating at $\omega_{\text{rot}}$ with
external magnetic field components $B_{z}$ and $B_{x}$. b) Rotation of the
diamond modulates the Zeeman shift in proportion to $B_{x}$, at a rate set by
the rotation frequency. c) Typical stationary spin-echo (blue circles) and
Ramsey (orange squares, detail inset) signals at $B_{z}=2.3\,$mT for the
diamond sample used in this work. d) Comparing magnetometry signals: varying
an applied dc $B_{x}$ field (DRUM) or effective $z$-field (Ramsey) at the
optimum sensing times traces out fringes. We used 10 s and 30 s measurement
times per point, for Ramsey and DRUM, respectively. (e, zoom-in of d) The DRUM
fringes are far faster, as considerably more phase accumulates within the
$500\times$ longer sensing time. f) Allan deviation of DRUM and Ramsey when
sensitive (blue) and insensitive (orange, microwaves detuned) to magnetic
fields. Both measurements follow $T^{-1/2}$ scaling for a few hundred seconds,
before drifts become dominant.

Our method, depicted in Fig. 1(a,b), is called diamond rotation up-conversion
magnetometry (“DRUM”) and was introduced in Ref. [26]. The NV Hamiltonian in
the presence of a magnetic field $\bm{B}$ is
$H=D_{\text{zfs}}S_{z^{\prime}}^{2}+\gamma_{e}\bm{B}\cdot\bm{S}$, with
$D_{\text{zfs}}/2\pi=2870\,$MHz at room temperature,
$\gamma_{e}/2\pi=2.8\,$kHz $\upmu$T-1 and
$\bm{S}=(S_{x^{\prime}},S_{y^{\prime}},S_{z^{\prime}})$ the vector of spin-1
Pauli matrices. We have ignored strain, electric fields and coupling to
temperature. The quantization axis $z^{\prime}$ is taken as the nitrogen-
vacancy axis, lying along one of the four $[111]$ crystallographic axes of the
crystal and therefore at an angle $\theta_{\text{NV}}$ to $z$. We use a
single-crystal diamond containing an ensemble of NV centers, with a surface
normal parallel to the rotation axis. A bias magnetic field is applied along
the rotation axis $z$ to spectrally select one NV class and isolate $m_{S}$
transitions. The energies of the NV $m_{S}=0,\pm 1$ spin states for weak
applied fields ($\hbar=1$) are $\omega(m_{S})\approx
D_{\text{zfs}}+m_{S}\gamma_{e}B\cos(\theta_{\text{NV}})$, with $B=|\bm{B}|$. A
weak magnetic field $B_{x}$ is applied along the lab-frame $x$-axis, making
the Zeeman shift time-dependent during rotation at an angular frequency
$\omega_{\text{rot}}$. Considering the
$|m_{S}=0\rangle-\leftrightarrow|m_{S}=-1\rangle$ transition, the time
dependent component is

$\omega_{-1,0}(t)\approx\gamma_{e}B_{x}\sin\theta_{\text{NV}}\cos(\omega_{\text{rot}}t-\phi_{0}),$
(1)

with $\phi_{0}$ set by the initial orientation of the diamond, and adjusted to
maximize sensitivity to either $x$ or $y$-oriented fields. The dc field in the
lab frame is now effectively an ac field in the NV frame, with amplitude
$\delta B_{x}\sin\theta_{\text{NV}}$ and frequency $\omega_{\text{rot}}$.

Our previous realization of DRUM used an NV tilt angle
$\theta_{\text{NV}}=4^{\circ}$, resulting in up-conversion of only a small
fraction of the dc field. In this work, we use a $\langle 110\rangle$-cut CVD-
grown type IIa diamond with a natural abundance of 13C and approximately [N] =
1 ppm, [NV] = 0.01 ppm, mounted on an electric motor that can spin at up to
5.83 kHz. We choose an NV orientation class making an angle of $30.2^{\circ}$
to the $z$-axis for our measurements, yielding $\sin\theta_{\text{NV}}=0.5$,
which is easily resolved and still sufficient to attain sensitivities
exceeding that of optimized Ramsey sensing. Typical stationary spin-echo and
Ramsey signals are shown in Fig. 1(c) for $B_{z}$ = 2.3 mT. The 13C spin bath
is the dominant source of decoherence [8], limiting $T_{2}^{\ast}$ to less
than half a microsecond, but the sample exhibits a much longer
$T_{2}=250(9)\,\upmu$s, with interferometric visibility restricted to revivals
spaced at twice the 13C Larmor period [19]. Such a wide gulf separating
$T_{2}$ and $T_{2}^{\ast}$ is not unusual, reflecting the much greater
sensitivity of $T_{2}^{\ast}$ to the presence of magnetic impurities and
sample-specific imperfections [14, 20].

The remaining experimental details are similar to that described in Refs. [27,
26, 28], and further details are provided in the Supplementary Material.
Briefly, a scanning confocal microscope optically polarizes (1 mW 532 nm) and
reads (600-800 nm, 9$\times 10^{6}$ cts/s) the NV fluorescence. Microwave
fields are applied along the $z$-axis with a coil antenna to ensure rotational
symmetry. Further coils supply the $z$-oriented bias field and create the
transverse dc test fields. A fast pulse generator controls the timing of laser
and microwave pulses, and is triggered synchronously with the rotation. The
laser is pulsed for $3\,\upmu$s to optically pump NVs to the $m_{S}=0$ state,
and the subsequent microwave spin-echo sequence is timed so that the
$\pi$-pulse is applied at the zero-crossing of the up-converted field (Fig.
1(b)), conferring maximum sensitivity. After the microwave pulses, optical
readout is performed after a shuttling time so that the whole sequence takes
one rotation period. A second sequence is applied back-to-back, with the final
$\pi/2$-pulse phase shifted by $180^{\circ}$. The detected photoluminescence
from each trace is then normalized and the difference computed to extract the
contrast $\mathcal{S}$, which constitutes the DRUM signal.

The concept of rotational dc up-conversion should be applicable to a wide
range of competing magnetometry architectures, including laser and absorption
based readout schemes. The point of this paper is to demonstrate the
sensitivity advantage of the DRUM technique over standard
$T_{2}^{\ast}$-limited Ramsey magnetometry. Our priorities are therefore not
to improve the ultimate sensitivity of the measurement, but rather ensure that
the two techniques are compared on an equal basis. Using isotopically-purified
diamonds with very high NV densities, wide-area collection optics and powerful
excitation beams has been shown to be the most effective means to achieve high
sensitivities [29], and these techniques should be compatible with diamond
rotation, with sufficient engineering application. The shorter coherence times
intrinsic to these NV-dense configurations place additional requirements on
rotation speed, and these will be discussed later in this work.

Due to strain and impurity inhomogeneity, $T_{2}^{\ast}$ exhibits considerable
spatial variation111See Supplementary Material. To assess the peak Ramsey
sensitivity, we locate a region with a comparatively high
$T_{2}^{\ast}=360\,$ns and determine the sensitivity, $\delta
B=\left(\frac{d\mathcal{S}}{dB}\right)^{-1}\sigma(\mathcal{S})\sqrt{T}$, with
$d\mathcal{S}/dB$ the mid-fringe signal slope, $T$ the total integration time,
and $\sigma(\mathcal{S})$ the standard deviation of the Ramsey signal. For
Ramsey, changing the microwave frequency is equivalent to varying a magnetic
field exactly parallel to the NV axis. Ramsey fringes as a function of
effective magnetic field are shown in Fig. 1(d). The sensitivity of Ramsey
magnetometry was found by detuning to the mid-fringe point and repeating the
same 10 s averaging interval 10 times. The best we achieved was a standard
deviation of this data yielding $\delta B=0.86\upmu$T $\text{Hz}^{-1/2}$.

Next, we rotated the diamond at 3.75 kHz and performed DRUM with a spin-echo
time of $\tau=180\,\upmu$s ($B_{z}=0.7\,$mT), yielding fringes as an applied
$x$-field is varied as shown in Fig.1(d,e), with each point averaged for 30 s.
We calculate the operational sensitivity to be
$28\,\text{nT}\,\text{Hz}^{-1/2}$, about 30 times better than that of Ramsey
in the same diamond sample. DRUM exhibits similar long-time averaging behavior
to Ramsey, but towards a much lower minimum detectable field. Figure 1(f)
shows the Allan deviation 222A comprehensive description of Allan deviation as
it applies to our work is provided in the Supplementary Material. of DRUM and
Ramsey as a function of averaging time. To assess the relative magnitudes of
magnetic drifts and intrinsic noise in each technique, we measure while
sensitive to magnetic fields, and then again with the microwaves detuned by 15
MHz, so that only intrinsic measurement noise is present. Ramsey, sensitive to
drifts in temperature, local strain as well as magnetic drifts (common also to
DRUM) exhibits a significantly higher Allan deviation compared to DRUM. No
amount of averaging time with Ramsey can exceed the performance of DRUM. This
demonstration constitutes a significant achievement, showing that
$T_{2}^{\ast}$ does not intrinsically limit dc magnetic sensitivity for a
spin-based quantum magnetometer. In what follows, we describe how the optimum
parameters were deduced, and how the ultimate sensitivity depends on
parameters we can control.

The shot-noise limited dc sensitivity of the DRUM measurement is given by
333See Supplementary Material

$\delta B=\frac{\pi
e^{\left(\frac{\tau}{T_{2}}\right)^{n}}}{4C\gamma_{e}\sin\theta_{\text{NV}}\sin\left(\frac{\pi\tau}{2t_{\text{rot}}}\right)}\frac{1}{\sqrt{t_{\text{rot}}}},$
(2)

with $n\approx 3$ reflecting sample-specific decoherence processes, $C$ the
readout efficiency [33] and $t_{\text{rot}}$ the rotation period. Extracting
the optimum performance for DRUM amounts to increasing the slope
$d\mathcal{S}/dB$, which depends on the rotation speed and spin-echo
measurement time, and increasing the measurement signal-to-noise, largely via
well-understood means [8] that benefit both Ramsey and DRUM.

The rotation speed can be chosen to optimize sensitivity. The intrinsic
amplitude of the up-converted field is set by the lab-frame dc field, but the
phase accumulated by the NV centers depends on the rotation speed. Faster
rotation speeds enable the same integration time to sample a greater fraction
of the modulated field, and more measurements are possible within a given
averaging time, increasing the number of photons collected. However, without
recourse to higher-order dynamical decoupling sequences, which supplant spin-
echo as the optimum measurement only when $T_{2}\gg t_{\text{rot}}$,
increasing the rotation speed eventually reduces the accumulated phase due to
the smaller integrated area under $\omega_{-1,0}(t)$.

Maximizing the slope also requires minimizing decoherence sources. For a
natural abundance diamond, the NV-13C bath interaction depends on the strength
[34, 35] and direction [36, 28] of the magnetic field. Consequently, the
$z$-bias field tunes the spin-echo time $\tau$, and must be set so that
$\tau=n_{c}/(B\times 10.71\,\text{kHz\,mT}^{-1}\pm\omega_{\text{rot}}/2\pi)$,
$n_{c}\in\\{1,2,...\\}$. Stronger $z$-fields result in faster decoherence due
to the anisotropic NV-${}^{\text{13}}$C hyperfine coupling [36, 28]. We
therefore minimize the requisite $z$-field by ensuring just the _first_ 13C
revival coincides with the desired measurement time, and choose the rotation
direction so that the induced magnetic pseudo-field [37] adds to the $z$-bias
field, _i.e._ $\omega_{\text{rot}}>0$.

The slope $d\mathcal{S}/dB$ is given by

$\frac{d\mathcal{S}}{dB}=\frac{4A}{\omega_{\text{rot}}}e^{-\left(\frac{\tau}{T_{2}}\right)^{n}}\gamma_{e}\sin\theta_{\text{NV}}\sin\left(\frac{\omega_{\text{rot}}\tau}{4}\right).$
(3)

For comparison with data, we leave $A$ and $T_{2}$ as free parameters,
possibly dependent on rotation speed, and fix $n=3$, which best describes our
observed relaxation and is consistent with other experiments [36, 35]. We
measured the mid-fringe slope of DRUM fringes as a function of $\tau$ and
$\omega_{\text{rot}}$. Fig. 2(a, inset) shows DRUM fringes as the spin-echo
time is varied for $\omega_{\text{rot}}/2\pi=3.75\,$kHz, and the slope
extracted from fits of Eq. 3 to such data is plotted versus $\tau$ for
rotation speeds from $1-6\,$kHz (Fig. 2(a)). We find that as the rotation
speed is increased, an optimal sensing time appears around 3 to 4 kHz.

Figure 2: Optimization of DRUM. (a) Peak slope $d\mathcal{S}/dB$ as a function
of r$\omega_{\text{rot}}$, deduced from fits of Eq. 3 to slope-vs-$\tau$ data
gathered for DRUM fringes at different rotation speeds (inset, 3.75 kHz) (b)
Measured $T_{2}$ (gray dashed line is linear fit) and spin-echo time
$\tau_{\text{opt}}$ where the peak slope occurs: decoherence results in
$\tau_{\text{opt}}<T_{2}$. Dashed lines in (a) and (b) correspond to
theoretical predictions from measured $T_{2}$ data. (c) Increasing photon
number per measurement bin (blue circles), and reducing measurement noise
(orange squares). (d) Sensitivity $\delta B$ as a function of rotation speed.
Shaded regions denote maximum and minimum ranges from 5 repetitions.

We also monitored the coherence time $T_{2}$ and time at which the peak slope
occurs $\tau_{\text{opt}}$ as a function of rotation speed (Fig. 2(b)). The
measured $T_{2}$ is higher than that in Fig. 1(c) due to the overall lower
magnetic field strengths, and drops slightly as a function of rotation speed,
which we believe is due to imperfectly canceled $y$-fields or the test field
itself inducing weak anisotropy in the NV-13C interaction 444See Supplementary
Material. Numerically maximizing Eq. 3 for the measured $T_{2}$ yields the
dashed theoretical predictions in Fig. 2(a, b), confirming the key role
$T_{2}$ plays. While the theory accurately reproduces the peak slope, the
optimum time $\tau_{\text{opt}}$ where the slope is maximized is lower than
predicted; we attribute this to the particular choice of $n=3$ in Eq. 3.

The variation in slope is tempered by increasing photon collection rates (and
hence reduction in shot noise) due to increased duty cycle at higher rotation
speeds, as shown in Fig. 2(c). This leads to an almost flat dependence on
sensitivity with rotation speed, as shown in Fig. 2(d). DRUM operates close to
the shot-noise predicted sensitivity limit, 30 times better than the Ramsey
sensitivity. This is actually greater than $\sqrt{T_{2}/T_{2}^{\ast}}=26$ due
to the $10\%$ duty cycle of Ramsey measurements (including laser preparation
and readout), and the trigonometric factor and integration for less than a
full period in DRUM. However, computing the idealized Ramsey sensitivity with
unity duty cycle yields $122$ nT Hz-1/2, still a factor of 4.5 worse than
experimentally demonstrated DRUM with $\tau<t_{\text{rot}}$ and
$\theta_{\text{NV}}=30^{\circ}$.

_Discussion._ We have shown in this work that dc up-conversion magnetometry
with a rotating diamond can significantly exceed the sensitivity of
conventional Ramsey magnetometry. Unlike previous proposals [39], our scheme
up-converts just the magnetic fields of interest and not the deleterious noise
that limits quantum coherence, thus definitively improving sensitivity to dc
fields. We also retain the vector sensitivity of the NV center, with a timing
adjustment to the synchronization allowing for exclusive $y$-field detection.
In principle, our scheme is equally applicable to any quantum system where the
coupling between qubit and parameter of interest can be modulated in time, and
this paper shows that sensitivity exceeding the $T_{2}^{\ast}$ limit is thus
possible.

For up-conversion to be worthwhile, $T_{2}\gg T_{2}^{\ast}$ and
$t_{\text{rot}}\sim T_{2}$. More practically, however, the
$\sqrt{\omega_{\text{rot}}}$ scaling of sensitivity in Eq. 2 requires a
commensurate increase in $C$, _i.e._ by increasing NV density, which in turn
reduces $T_{2}$ [20]. However, with the simplified assumption that
$\tau=t_{\text{rot}}=T_{2}$ and $\theta_{\text{NV}}=90^{\circ}$, DRUM can
confer a sensitivity gain over Ramsey in any diamond sample of $\delta
B_{\text{DRUM}}/\delta B_{R}=\frac{\pi}{4}\sqrt{T_{2}^{\ast}/T_{2}}$. For DRUM
to have a factor of 10 better sensitivity than unity-duty-cycle Ramsey,
$T_{2}=62\,T_{2}^{\ast}$.

Mechanical rotation can be challenging, though not impossible, to achieve for
the short $T_{2}\sim 5-10\,\upmu$s exhibited by NV-dense diamond samples.
Commercial NMR magic-angle spinning devices can achieve rotation rates of
mm3-scale samples now up to 150 kHz [40], and demonstrations of ultrafast
rotation of optically-trapped microscale structures [41, 42] yield GHz
rotation frequencies. Since rapid libration can be substituted for rotation,
alternative approaches could potentially leverage fast piezoelectric tip-tilt
transducers [43] for larger samples or micro-to-nano structures for smaller
length scales, for instance in optically or electrically-trapped [44] micro-
to-nano diamonds. Modulation can also be achieved by position displacement in
a spatially-varying field, and recent work has demonstrated up-conversion of
dc fields to ac using scanning single NV magnetometry by rapidly modulating
the distance between tip and sample [45]. Another option could be modulation,
_i.e._ via position displacement, of the sensitivity enhancement conferred by
ferrite flux concentrators [15].

In conclusion, we have demonstrated that $T_{2}$-limited dc magnetometry can
exceed the sensitivity of $T_{2}^{\ast}$-limited Ramsey magnetometry for
diamond-based quantum magnetometers. We anticipate our work will stimulate
other approaches, not necessarily based on sample rotation, to combine up-
conversion with existing schemes to improve magnetic sensitivity. Augmented
with fast rotation, improvements such as engineered diamonds with higher NV
densities and larger optical excitation and collection areas may be sufficient
to achieve the much sought-after fT Hz-1/2 regime of dc magnetic sensitivity
with room-temperature, microscale diamond sensors, where applications such as
magnetoencephalography in unshielded environments become possible [46].

This work was supported by the Australian Research Council (DE210101093,
DE190100336). We thank R. E Scholten for insightful discussions and a careful
review of the manuscript. The authors (AAW, AS, AMM) are inventors on a United
States Patent, App. 16/533,167 which is based on this work.

## References

  * Doherty  _et al._ [2013] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, Physics Reports 528, 1 (2013).
  * Schirhagl  _et al._ [2014] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Annual Review of Physical Chemistry 65, 83 (2014).
  * Gille  _et al._ [2021] R. W. d. Gille, J. M. McCoey, L. T. Hall, J.-P. Tetienne, E. P. Malkemper, D. A. Keays, L. C. L. Hollenberg, and D. A. Simpson, PNAS 118 e2112749118 (2021).
  * Tetienne  _et al._ [2017] J.-P. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, Science Advances 3, e1602429 (2017).
  * Casola  _et al._ [2018] F. Casola, T. van der Sar, and A. Yacoby, Nat Rev Mater 3, 1 (2018).
  * Mamin  _et al._ [2013] H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, Science 339, 557 (2013).
  * Fu  _et al._ [2020] K.-M. C. Fu, G. Z. Iwata, A. Wickenbrock, and D. Budker, AVS Quantum Sci. 2, 044702 (2020).
  * Barry  _et al._ [2020] J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A. Hart, L. M. Pham, and R. L. Walsworth, Rev. Mod. Phys. 92, 015004 (2020).
  * Balasubramanian  _et al._ [2009] G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Nat Mater 8, 383 (2009).
  * Herbschleb  _et al._ [2019] E. D. Herbschleb, H. Kato, Y. Maruyama, T. Danjo, T. Makino, S. Yamasaki, I. Ohki, K. Hayashi, H. Morishita, M. Fujiwara, and N. Mizuochi, Nat Commun 10, 3766 (2019).
  * Clevenson  _et al._ [2015] H. Clevenson, M. E. Trusheim, C. Teale, T. Schröder, D. Braje, and D. Englund, Nat. Phys. 11, 393 (2015).
  * Lange  _et al._ [2012] G. d. Lange, T. v. d. Sar, M. Blok, Z.-H. Wang, V. Dobrovitski, and R. Hanson, Sci. Rep. 2, 382 (2012).
  * Mamin  _et al._ [2014] H. Mamin, M. Sherwood, M. Kim, C. Rettner, K. Ohno, D. Awschalom, and D. Rugar, Phys. Rev. Lett. 113, 030803 (2014).
  * Bauch  _et al._ [2018] E. Bauch, C. A. Hart, J. M. Schloss, M. J. Turner, J. F. Barry, P. Kehayias, S. Singh, and R. L. Walsworth, Phys. Rev. X 8, 031025 (2018).
  * Fescenko  _et al._ [2020] I. Fescenko, A. Jarmola, I. Savukov, P. Kehayias, J. Smits, J. Damron, N. Ristoff, N. Mosavian, and V. M. Acosta, Phys. Rev. Research 2, 023394 (2020).
  * Zhang  _et al._ [2021] C. Zhang, F. Shagieva, M. Widmann, M. Kübler, V. Vorobyov, P. Kapitanova, E. Nenasheva, R. Corkill, O. Rhrle, K. Nakamura, H. Sumiya, S. Onoda, J. Isoya, and J. Wrachtrup, Phys. Rev. Applied 15, 064075 (2021).
  * Ramsey [1950] N. F. Ramsey, Phys. Rev. 78, 695 (1950).
  * Rondin  _et al._ [2014] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, Rep. Prog. Phys. 77, 056503 (2014).
  * Childress  _et al._ [2006] L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, Science 314, 281 (2006).
  * Bauch  _et al._ [2020] E. Bauch, S. Singh, J. Lee, C. A. Hart, J. M. Schloss, M. J. Turner, J. F. Barry, L. M. Pham, N. Bar-Gill, S. F. Yelin, and R. L. Walsworth, Phys. Rev. B 102, 134210 (2020).
  * Hahn [1950] E. L. Hahn, Phys. Rev. 80, 580 (1950),.
  * Taylor  _et al._ [2008] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, Nat Phys 4, 810 (2008).
  * Acosta  _et al._ [2010] V. M. Acosta, E. Bauch, A. Jarmola, L. J. Zipp, M. P. Ledbetter, and D. Budker, Applied Physics Letters 97, 174104 (2010).
  * Jeske  _et al._ [2016] J. Jeske, J. H. Cole, and A. D. Greentree, New J. Phys. 18, 013015 (2016).
  * Wickenbrock  _et al._ [2016] A. Wickenbrock, H. Zheng, L. Bougas, N. Leefer, S. Afach, A. Jarmola, V. M. Acosta, and D. Budker, Appl. Phys. Lett. 109, 053505 (2016).
  * Wood  _et al._ [2018a] A. A. Wood, A. G. Aeppli, E. Lilette, Y. Y. Fein, A. Stacey, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Phys. Rev. B 98, 174114 (2018a).
  * Wood  _et al._ [2018b] A. A. Wood, E. Lilette, Y. Y. Fein, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Science Advances 4, eaar7691 (2018b).
  * Wood  _et al._ [2021] A. A. Wood, R. M. Goldblatt, R. P. Anderson, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Phys. Rev. B 104 085419 (2021).
  * Wolf  _et al._ [2015] T. Wolf, P. Neumann, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, and J. Wrachtrup, Phys. Rev. X 5, 041001 (2015).
  * Note [1] See Supplementary Material.
  * Note [2] Derived in full in the Supplementary Material.
  * Note [3] A comprehensive description of Allan deviation as it applies to our work is provided in the Supplementary Material..
  * Degen  _et al._ [2017] C. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).
  * Zhao  _et al._ [2012] N. Zhao, S.-W. Ho, and R.-B. Liu, Phys. Rev. B 85, 115303 (2012).
  * Hall  _et al._ [2014] L. T. Hall, J. H. Cole, and L. C. L. Hollenberg, Phys. Rev. B 90, 075201 (2014).
  * Stanwix  _et al._ [2010] P. L. Stanwix, L. M. Pham, J. R. Maze, D. Le Sage, T. K. Yeung, P. Cappellaro, P. R. Hemmer, A. Yacoby, M. D. Lukin, and R. L. Walsworth, Phys. Rev. B 82, 201201 (2010).
  * Wood  _et al._ [2017] A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Nature Physics 13, nphys4221 (2017).
  * Note [4] Discussed in greater detail in the Supplementary Material.
  * Ajoy  _et al._ [2016] A. Ajoy, Y. X. Liu, and P. Cappellaro, arXiv:1611.04691 (2016).
  * Schledorn  _et al._ [2020] M. Schledorn, A. A. Malär, A. Torosyan, S. Penzel, D. Klose, A. Oss, M.-L. Org, S. Wang, L. Lecoq, R. Cadalbert, A. Samoson, A. Böckmann, and B. H. Meier, Chembiochem 21, 2540 (2020).
  * Reimann  _et al._ [2018] R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, Phys. Rev. Lett. 121, 033602 (2018).
  * Ahn  _et al._ [2020] J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, and T. Li, Nat. Nanotechnol. 15, 89 (2020).
  * Csencsics  _et al._ [2019] E. Csencsics, B. Sitz, and G. Schitter, IFAC-PapersOnLine 8th IFAC Symposium on Mechatronic Systems MECHATRONICS 2019, 52, 289 (2019).
  * Perdriat  _et al._ [2021] M. Perdriat, C. Pellet-Mary, P. Huillery, L. Rondin, and G. Hétet, Micromachines 12, 651 (2021).
  * Huxter  _et al._ [2022] W. S. Huxter, M. L. Palm, M. L. Davis, P. Welter, C.-H. Lambert, M. Trassin, and C. L. Degen, arXiv:2202.09130 (2022).
  * Boto  _et al._ [2018] E. Boto, N. Holmes, J. Leggett, G. Roberts, V. Shah, S. S. Meyer, L. D. Muñoz, K. J. Mullinger, T. M. Tierney, S. Bestmann, G. R. Barnes, R. Bowtell, and M. J. Brookes, Nature 555, 657 (2018).

DC Quantum Magnetometry Below the Ramsey Limit - SUPPLEMENTARY INFORMATION

##  S1 Experimental setup

The basic configuration of the experiment is the same as reported in Ref. [2],
with several differences and improvements. The experimental configuration is
depicted in Fig. S1 and a description of the key elements follows.

###  S1.1 Confocal Microscope

The optical illumination and photon collection is performed by a scanning
confocal microscope, mounted in a top-down configuration as shown in Fig. S1.
A 50x magnification, $NA=0.6$ Nikon SLWD objective with an $11\,$mm working
distance is mounted to a $xyz$-piezoelectric nanopositioner stage (Physik
Instrumente P-545.R7 series) with $200\,\upmu$m travel range in each
direction. About $1\,$mW of 532 nm laser light (Laser Quantum Opus) is input
to the back aperture of the objective. We collect $8-10\times 10^{6}\,$
photons/s on a single-photon counting module (SPCM, Excelitas SPCM-AQRH-14)
from the diamond sample after coupling into a $50\,\upmu$m-diameter multi-mode
optical fiber with a 250 mm achromatic doublet lens. Photon counts and arrival
times are registered on a fast counting module (FAST ComTech MCS6A) and
analyzed on a dedicated computer.

###  S1.2 Diamond Sample and motor

We used two similar diamond samples and two ideally identical motors in this
work to better assess the implementation variability of DRUM. Each diamond
sample is an optical-grade ($[N]<1\,$ppm) type IIa single-crystal diamond
plate from Delaware Diamond Knives (DDK), $3\times 1.2\times 0.2$ mm and
$\langle 110\rangle$-cut wide face with a $\sim 4^{\circ}$ polishing miscut.
Each sample has a reasonably uniform distribution of NV centers throughout
though with notable bright stripes of NV fluorescence, and we undertake no
pre-treatment other than sample cleaning. The diamond samples are bonded with
UV-curing epoxy (Norland Optical Adhesive Type 63) to the end of a high-speed
electric motor (Celeroton CM-2-500) driven by a three-phase converter
(Celeroton CC-75-500 with E08 synchronization extension board). The motors can
be ran at a maximum speed of 500,000 rpm in either direction, though for
mechanical longevity we restrict speeds below 350,000 rpm.

Individual motors exhibit some variation in stability for different rotation
speeds (occasionally, audibly so), and we observe a run-in effect whereby
sustained operation at a single speed (_i.e._ for months at a time) makes the
motor unstable at other speeds, larger or smaller. We experienced this effect
with one motor and replaced it with a different motor and diamond that had not
been run-in, and this data was presented in the paper. We also examined the
effects of different rotation directions on DRUM, as discussed later in
Section S7.

We determine the ‘rotation center’, _i.e._ the $r=0$ point about which the
diamond rotates by performing unsynchronized $x-y$ scanning confocal
microscopy and identifying the characteristic patterns of circular motion.
Inhomogeneities are blurred into circular patterns centered at $r=0$, and we
position the microscope objective at this point to reduce the effects of off-
axis rotation, as shown in Fig. S1. The motor is securely recessed into a
thick block of aluminium, and a commercial PID controller stabilizes the motor
temperature to better than 0.01 K. A 1.0 mm thick mu-metal shield covers the
exposed part of the motor and suppresses magnetic interference from the
rotating pole piece.

Figure S1: Block diagram of experimental setup and magnetic field geometry.
Inset image shows scanning confocal image near rotation center, crosshair
denotes point of symmetry.

###  S1.3 Microwave generation and control

We use a Windfreak Technologies SynthHD PLL-based microwave generator as a
local oscillator (LO) feeding an I/Q modulator (Texas Instruments TRF37T05EVM
IQ modulator evaluation board). We drive the $I,\bar{I},Q,\bar{Q}$ inputs of
the I/Q modulator using TTL pulses from a Swabian Instruments Pulsestreamer
8/2, which allows for fast, almost distortionless pulsing with phase control:
a single TTL pulse into one of the I/Q channels yields the microwave pulse
phases
$\left[I,\bar{I},Q,\bar{Q}\right]=\left[0,180^{\circ},90^{\circ},270^{\circ}\right]$,
and we observe LO suppression comparable to a commercial vector signal
generator. The phase control, while limited is still useful: we use
$180^{\circ}$ pulses, _i.e._ a digital high into the $\bar{I}$ channel to
normalize the spin-echo (or Ramsey) signals. The Pulsestreamer is hardware-
triggered by a digital signal from a delay generator synchronized to the
rotation of the motor.

Microwave fields are applied to the NV centers using a 2 mm diameter, 5-turn
coil fabricated from $150\,\upmu$m enamel-insulated copper wire located about
$200\,\upmu$m above the surface of the diamond. We drive the coil with a 50 W
microwave amplifier (Microwave Amps AM6-1-3.5-47-47). As a result of the coil
geometry and standoff distance, we achieve only modest Rabi frequencies,
ranging from $6.24\,$MHz at a frequency of $2754\,$MHz ($B_{z}=4.8\,$mT) to
2.45 MHz at 2845 MHz (1.0 mT). A coil-type geometry is required to ensure a
rotationally-symmetric mw Rabi frequency, and we carefully position the coil
so that the Rabi frequency varies as little as possible as the diamond is
stepped through fixed rotation angles (typically better than $50\,$kHz across
a full period).

###  S1.4 Magnetic fields

The $z$-oriented dc magnetic field for spectrally selecting NV orientation
classes and tuning the 13C revival time is provided by a single coil coaxial
with the laser, with a characteristic calibration of $5.3\,$mT/A. A
programmable switched-mode power supply (HMP2030) provides up to 1 A to drive
the coil. Two pairs of coils enclosing the system provide $x$\- and
$y$-oriented magnetic fields, which are used cancel non-axial components of
the $z$-applied field, and provide the test magnetic fields used in our
experiments. The test fields are defined as the added current above that
required to cancel the $z$-coil misalignment, _i.e._
$B_{x}=i_{x}B_{x}^{\prime}+B_{x0}(i_{z})$ ,
$B_{y}=i_{y}B_{y}^{\prime}+B_{y0}(i_{z})$, with $i_{x,y,z}$ the $x$, $y$ or
$z$ coil currents, $B_{x0},B_{y0}$ the ‘nulling’ fields, which depend on the
$z$-coil current $i_{z}$, $B_{x}^{\prime}$ = 0.38 mT/A and $B_{y}^{\prime}$ =
0.56 mT/A. We determined these calibration factors using NV vector
magnetometry and comparing against diagonalization of the NV Hamiltonian, the
process is described fully in the Supplemental Material of Ref. [2]. The $x$
and $y$-fields are driven by separate precision laser diode current drivers
(Newport Model 505) which ensures low-noise, highly stable test magnetic
fields. We drive the 20 mA/V analog voltage input of the current drivers with
a 16-bit DAC, yielding 15 $\upmu$A current resolution.

###  S1.5 Improvements over previous experiment

Most improvements over the previous demonstration are easy to quantify
directly, such as the increased NV tilt angle $\theta_{\text{NV}}=30^{\circ}$,
yielding a trigonometric factor $7\times$ the previous experiment. We also
work with higher laser powers, a slightly higher NV density, $\sim 2\times$
larger optical preparation and better photon collection, yielding $3$ times
higher photon counts than Ref. [2], enabling us to reduce the averaging time
for DRUM by a factor of 10. These same benefits (excepting the trigonometric
factor) also apply to Ramsey, and we achieved a $4\times$ improvement in
Ramsey over the previous measurement as well. The $T_{2}$ of the diamond we
used in this work enables longer measurement times, $\tau=180\,\upmu$s (and up
to $200\,\upmu$s) vs $124\,\upmu$s previously.

Other improvements are less easy to explicitly quantify, though definitively
beneficial. In this work, we were able to run two DRUM sequences back-to-back
in consecutive periods, alternating the readout $\pi/2$-pulse phase between 0
and $180^{\circ}$, which provides common-mode rejection of magnetic noise,
notably that arising from AC mains interference. Furthermore, we optimized the
pulse timing to eliminate trigger skipping, which randomly added dead time and
occurs when the measurement pulse sequences is very near to the rotation
period of the diamond. Neither of these issues applied to Ramsey magnetometry
in either this or the previous realisation. Implementing ultra-stable current
drivers for the shim and test fields is also highly beneficial, the resolution
alone of current drivers used in the current work being 25 times finer than
the previous case.

###  S1.6 General

We use the ‘Qudi’ experimental control software [3] for managing computer
control of the experiment. We note that the current version of the experiment
exhibits remarkable magnetic stability, as depicted in Fig. 1(f) of the main
text, and elaborated on further when we discuss Allan deviation in Sec. S3. We
are able to run the motor at very high speeds (5 kHz) for several days
continuously, for periods of several months without noticeable degradation of
the rotation. We note that such sustained operation at a fixed rotation speed
appears to adversely affect the motor’s stability at other speeds, the ‘run-
in’ effect mentioned previously. Given the relatively flat scaling of DRUM
sensitivity, run-in poses an issue only for the characterisation measurements
we performed. For general operation, we do not anticipate a case where varied
rotation speed is essential.

##  S2 Ramsey Variation

A characteristic feature of both samples we investigated is strong spatial
variation of $T_{2}^{\ast}$ over $\upmu$m-scale regions. We measured time-
domain Ramsey signals as a function of confocal $x-y$ position in a
(10$\,\upmu$m)2 region, as shown in Fig. S2(a). The $T_{2}^{\ast}$ extracted
from damped sinusoidal fits varies between $160\,$ns and $390\,$ns over a few
micrometers. This variation is attributed to local variation in crystal
strain, as reported in Ref. [4] and possibly nitrogen density: we note a
correlation between low values of $T_{2}^{\ast}$ and high NV fluorescence, as
shown in Fig. S2(c). Notably, this spatial pattern rotates when we rotate the
diamond by ‘parking’ the motor at different angles, as shown in Fig S2(b).
This final observation confirms the origin of the inhomogeneity as inside the
diamond. We also note (data not shown) that $T_{2}$ extracted from spin-echo
measurements shows essentially no spatial variation.

Figure S2: Spatial variation of Ramsey $T_{2}^{\ast}$. (a) We measured Ramsey
fringes (right) as a function of confocal $x,y$ position in a 10 $\upmu$m
square. Fitting to the fringes yields $T_{2}^{\ast}$, which varies
considerably over the region of interest (left). Black dot denotes approximate
position of rotation center. (b) Rotating the diamond clockwise by 45∘ and
repeating the measurement shows the spatial variation to vary with rotation
angle, confirming it is predominantly intrinsic to the diamond. (c) While
spatially varying crystal strain is the principal culprit for $T_{2}^{\ast}$
variation, we observe a rough correlation between NV PL (at $0^{\circ}$ motor
park angle, a proxy for nitrogen density) and the variability in Ramsey
coherence time in (a), note in particular the large bright patch in the top
left corner corresponds to an equally sized region of low $T_{2}^{\ast}$.

##  S3 Allan deviation

Fig. 1(f) of the main text depicts the long-term stability and averaging
characteristics of DRUM, encapsulated in the _Allan deviation_. The Allan
deviation is primarily used a measure of precision oscillator and inertial
sensor stability to date, has featured in several recent works in NV precision
sensing, eg. Refs [5, 6, 7]. While much of the existing body of literature
concerns application to precision time standards, the key features of Allan
deviation can be ported to magnetometers, though not without some
clarification to prevent misinterpretation.

A key difference is in the intended use of the measure. For a clock, the
variation of time is assumed to be well known, indeed immutable, and
variability in an oscillator’s measurement of time assumed to arise from noise
and imperfections. A magnetometer is fundamentally different, in that the
device itself is subject to imperfections and noise, but the quantity being
measured, _i.e._ the magnetic field, does _not_ follow a well-defined
variation. For example, the NV zero-field splitting can drift as a result of
temperature variations, a ‘device’ defect, but the magnetic environment may
also drift and fluctuate, an environmental issue unrelated to the performance
of the magnetometer itself. This being the case, a magnetometer with low Allan
deviation that follows a particular scaling is fairly meaningless in isolation
and requires comparison with some measure of either the device’s intrinsic
defects or that of the environment. Examples are operation in a carefully
controlled or magnetically shielded environment, or, operation without
sensitivity to magnetic fields. The former is ideal though difficult to
achieve, the latter is simple though somewhat less informative.

Allan deviation is fundamentally a measure of oscillator stability. The
oscillator in our case is the NV two-level splitting, and we measure the
relative phase between this and an applied microwave field in any
interferometry experiment. We assume that the microwave field is essentially
perfect on the timescale of our experiments, so that the ‘noisy’ oscillator is
the NV itself. In our experiment, we set DRUM to the mid-fringe sensing point
with an appropriate applied $x$-field. We then run the same measurement
continuously for several hours or more. The fast counting hardware accumulates
a photoluminescence trace $S$ continuously, and we compute the NV contrast
using only the photons counted in between polling times, typically 10 s,
_i.e._

$S_{n}=S(t_{n})-S(t_{n}+t_{\text{poll}}).$ (1)

The time series $\\{S_{n}\\}$ is then converted from NV fluorescent contrast
to frequency perturbations using

$\omega_{n}=\gamma\left(\frac{d\mathcal{S}}{dB}\right)^{-1}S_{n}.$ (2)

Assuming the deviation of the NV contrast is constrained to the linear region
of $\mathcal{S}$, which it is in our experiments, there is no need for phase
unwrapping, and the total phase $\phi_{n}$ of the NV as a function of time can
be determined by the cumulative sum of $\omega_{n}$ multiplied by the polling
time interval $\delta T$

$\phi_{n}=\sum_{k=1}^{n}\delta T\omega_{k}.$ (3)

This list of phase samples as a function of measurement time is what the Allan
deviation assesses the stability of. The Allan deviation at an averaging time
of $m\delta T$ for $N$ samples of total phase $\phi_{n}$ using the overlapped
$T$ estimator is

$\sigma_{A}^{2}(m\delta T)=\frac{1}{2(m\delta
T)^{2}(N-2m)}\sum_{i=0}^{N-2m-1}\left(\phi_{i+2m}-2\phi_{i+n}+\phi_{i}\right)^{2}.$
(4)

Figure S3 shows the raw NV contrast as a function of time (sampled every 10
s), the corresponding total phase as a function of time and finally the Allan
deviation. We confirm that our measurement stays ‘on the same fringe’ by
measuring DRUM fringes at the beginning and end of the Allan deviation data
run, as shown in Figure S3(d). The amplitude modulation we observe here is due
to the long microwave pulses used, typically $t_{\pi}=200\,$ns.

Figure S3: Allan deviation measurement of oscillator stability. (a) Raw NV
contrast at the mid-fringe point, logged every 10 s. (b) accumulated total
phase, according to Eq. 3. (c) Calculated overlapping Allan deviation as a
function of averaging time. The $1/\sqrt{T}$ scaling indicates white frequency
noise, which originates from photon shot noise. (d) DRUM fringes (traced here
as a function of bare $B_{x}$-coil current) before and after a 16 hr run,
showing minimal phase drift.

So, what does the Allan deviation _mean_ for a magnetometer? In the absence of
a power spectral density plot, the Allan deviation allows some information
about the dominant sources of noise to be deduced. The $1/\sqrt{T}$ scaling of
$\sigma_{A}$ we observe for averaging times up to 500 s is consistent with
white frequency noise, that is, random fluctuations in the measured oscillator
frequency consistent with errors originating from photon shot noise. Beyond
this time, the Allan deviation inflects and then increases as $\sqrt{T}$,
consistent with a random walk (brown noise) and slow drift.

Considering the data shown in the main text (Fig. 1(f)), the Allan deviation
when insensitive to magnetic fields (i.e. microwaves detuned) continues to
decrease as $1/\sqrt{T}$ for Ramsey, but shows more complicated behaviour
(when $B$-sensitive _and_ $B$-insensitive) for DRUM. That the same features
are present on the magnetometer signal when sensitive _and_ insensitive to
magnetic fields says that the noise affects the NV-spin state readout. We
attribute this primarily to drift of the confocal focus away from the rotation
center, which if severe ($>2\,\upmu$m, the confocal spot size) can affect
optical state preparation and readout due to timing errors and motor jitter.
This drift primarily occurs due to lab temperature variations of order
$1-2^{\circ}$C throughout long measurement runs. Drift of the confocal spot
also introduces slow magnetic drifts, since magnetic field gradients (which we
measure using DRUM by displacing from the rotation center) are present,
resulting in a phase shift in DRUM fringes as the confocal spot is displaced.

##  S4 Peak sensing time and vector sensitivity

We time the DRUM pulse sequence so that the $\pi$-pulse in the spin-echo
sequence occurs at the same time as the zero-crossing of the AC field in the
rotating frame. Practically, we determine this point by optimising the
frequency of the DRUM fringes. Additionally, DRUM can be made sensitive to
only $x$ or $y$ fields by adjusting the time between the motor synchronization
signal and application of the microwave pulses. We confirm this behaviour by
measuring DRUM fringes as a function of delay time $t_{\text{del}}$ and
determining the fringe frequency. Fig S4 shows DRUM fringes at a measurement
time of $\tau=70\,\upmu$s for $x$-applied fields and $y$-applied fields as a
function of $t_{\text{del}}$. The differing peak frequencies, expressed in
mA-1, reflect the different coil calibration constants. A degree of non-
orthogonality is evident in our coils, as the $x$ and $y$ fringe frequency
variation is $97^{\circ}$ out of phase, rather than exactly $90^{\circ}$.

Figure S4: Optimisation of DRUM fringe frequency and vector sensitivity. (a)
Schematic of procedure; we conduct a spin-echo measurement with
$\tau=70\,\upmu$s at $\omega_{\text{rot}}/2\pi=3.33$ kHz and vary the time
$t_{\text{del}}$ between trigger synchronization and when the microwave
sequence is applied, effectively changing the phase of the ac field sampled in
the DRUM measurement. (b) Measured DRUM fringes, with sinusoidal fits, as a
function of $B_{y}$ coil current for different delay times $t_{\text{del}}$.
(c) Fitted $x$-coil and $y$-coil fringe frequencies for varied
$t_{\text{del}}$. The almost $90^{\circ}$ phase shift between the traces is
evidence of vector selectivity tunable with $t_{\text{del}}$. Error bars are
standard error in fitted frequency.

From these results, the known rotation speed and the known spin-echo time
$\tau$, we can deduce at what delay time the zero-crossing occurs (in either
rotation direction) and time sequences accordingly to maximize the DRUM
sensitivity. While _exclusive_ $x$ or $y$-field sensitivity cannot be
demonstrated with the coil geometry used in this work, for applied transverse
fields exactly orthogonal to each other DRUM exhibits true vector sensitivity,
adjusting the delay time can then be used to deduce the bearing of an applied
magnetic test field.

##  S5 Calibration to absolute field

When DRUM is synchronized to the rotation of the diamond, increasing $B_{x}$
or $B_{y}$ traces out monotonic sinusoidal fringes essentially indefinitely.
When synchronization is absent, there is no well-defined relationship between
the unconverted field and the measurement sequence. As such, the fringes
phase-average to zero, except when the transverse fields are canceled to zero
via application of unique $B_{x0}$ and $B_{y0}$ ‘nulling fields’. In this
case, spin-echo contrast is maximized and sharply reduces as soon as the
applied fields deviate from the nulling fields. This allows us to determine
the currents corresponding to $B_{x0}$ and $B_{y0}$, and hence, calibrate DRUM
fringes to an absolute field strength, namely, $B_{x}=B_{y}=0$.

Figure S5: Calibration of DRUM to an absolute field. (a) when running
asynchronously with the rotation ($\omega_{\text{rot}}/2\pi=3.750\,$kHz,
$\tau=180\,\upmu$s), DRUM fringes rapidly damp, except when the transverse
field is zero. Here, we vary the applied $B_{x}$ and $B_{y}$ fields, and find
a sharp local maximum where the DRUM contrast peaks, which corresponds to
$B_{x0}=B_{y0}=0$, _i.e._ the magnetic field is parallel to the rotation axis.
(b) $B_{x}$ and (c) $B_{y}$ detailed optimisation, showing optimum contrast
point and damping DRUM fringes.

Figure S5 shows the results of this calibration, for
$\omega_{\text{rot}}/2\pi=3.750\,$kHz and $\tau=180\,\upmu$s, clearly
indicating the values of $B_{x0}$ and $B_{y0}$. Much of the off-axis fields
responsible for needing $B_{x0}$ and $B_{y0}$ in the first place is due to a
slight tilt in the $z$-coil orientation, making it not perfectly parallel to
the rotation axis. The required $B_{x0}$ and $B_{y0}$ therefore change with
applied $z$-current, and we adjust $B_{x0}$ and $B_{y0}$ accordingly when
changing $\tau$ (which requires variation of $B_{z}$), as discussed in the
main text. The test fields applied with the $x$-coils in our demonstration of
DRUM are therefore centered approximately at $B_{x}=0$. In practice, we
observe synchronized DRUM signals phase shifted from zero at $B_{x}=B_{x0}$,
possibly due to drifting nulling field calibrations or imprecise knowledge of
$B_{x0}$ and $B_{y0}$ in the first place: the fringe period when synchronised
is much faster than the observed ripples in Fig. S5, so small errors in
$B_{x0}$ and $B_{y0}$ translate to larger phase offsets when synchronized.

##  S6 Sensitivity derivation

We follow the example of Ref. [8] in our derivation of the shot-noise limited
DRUM sensitivity. We begin with the rotating NV, represented by the unit
vector

$\mathbf{n}(t)=\left(\sin\theta_{\text{NV}}\cos\left(\omega_{\text{rot}}t-\phi_{0}\right),\sin\theta_{\text{NV}}\sin\left(\omega_{\text{rot}}t-\phi_{0}\right),\cos\theta_{\text{NV}}\right),$
(5)

with $\theta_{\text{NV}}$ the polar angle between the NV and rotation axis
$z$, $\phi_{0}$ set by the initial orientation of the NV axis and
$\omega_{\text{rot}}$ the rotation angular frequency. We consider a static
magnetic field in the stationary (lab) frame,

$\mathbf{B}=\left(B_{x},0,B_{z}\right)$ (6)

where, without loss of generality we have assumed $B_{\perp}=B_{x}$. The
Zeeman shift experienced by the NV is well approximated for weak $B_{x}$ and
$\theta_{\text{NV}}\ll 90^{\circ}$ by ($\hbar=1$)

$\displaystyle E(m_{S})$ $\displaystyle=$ $\displaystyle
m_{S}\gamma_{e}\mathbf{B}\cdot\mathbf{n}(t)$ (7) $\displaystyle=$
$\displaystyle
m_{S}\gamma_{e}B_{z}\cos\theta_{\text{NV}}+m_{S}\gamma_{e}B_{x}\sin\theta_{\text{NV}}\cos\left(\omega_{\text{rot}}t-\phi_{0}\right),$

with $m_{S}$ the magnetic quantum number of the NV and
$\gamma_{e}/2\pi=28\,$kHz/$\upmu$T. We consider microwaves resonant only with
the $m_{S}=0\leftrightarrow m_{S}=-1$ transition, in which case the time-
dependent part of the transition energy is the second term in Eq. 7 with
$m_{S}=-1$. In a spin-echo experiment with inter-pulse time $\tau$, the NV
eigenstates will accumulate a relative phase

$\Phi=\int_{-\tau/2}^{0}E(t)dt-\int_{0}^{\tau/2}E(t)dt.$ (8)

Setting $\phi_{0}=\pi/2$ ensures symmetric integration for $\tau$ either side
of the zero crossing of the modulated energy, and this yields the optimum
phase accumulation as a function of applied magnetic field. Equation 8
evaluates to

$\Phi=\frac{4B_{x}\gamma_{e}}{\omega_{\text{rot}}}\sin\theta_{\text{NV}}\sin^{2}\left(\frac{\omega_{\text{rot}}\tau}{4}\right).$
(9)

Readout with a single $\pi/2$-pulse yields an NV fluorescence signal that
varies as $\mathcal{S}_{\pi/2}=\cos^{2}\left(\Phi/2\right)$, while computing
the (normalised) difference using a second sequence with a $180^{\circ}$-phase
shifted (_i.e._ -x) $\pi/2$-pulse yields
$\mathcal{S}=\mathcal{S}_{\pi/2}-\mathcal{S}_{-\pi/2}=\cos^{2}\left(\Phi/2\right)-\sin^{2}\left(\Phi/2\right)=\cos\Phi$.
Operating at the mid-fringe point of the DRUM signal amounts to adding a given
$B_{x}$ to induce a $\pi/2$ phase shift, _i.e._ so that
$\mathcal{S}=\cos(\Phi+\pi/2)=\sin\Phi\approx\Phi$. The transduction
parameter, or slope $d\mathcal{S}/dB_{x}$, which converts magnetic field to
quantum sensor readout is then

$\frac{d\mathcal{S}}{dB_{x}}=\frac{4\gamma_{e}}{\omega_{\text{rot}}}\sin\theta_{\text{NV}}\sin^{2}\left(\frac{\omega_{\text{rot}}\tau}{4}\right).$
(10)

The sensitivity per unit bandwidth is the smallest detectable field $\delta
B_{x}=\delta B$ that yields unity signal-to-noise in one second of averaging
time. Regardless of $\tau$, we can perform a maximum of $1/t_{\text{rot}}$
measurements in one second. Thus

$SNR=\delta
Be^{-\left(\frac{\tau}{T_{2}}\right)^{n}}\frac{d\mathcal{S}}{dB}\frac{2C}{\sqrt{t_{\text{rot}}}},$
(11)

with $T_{2}$ the spin coherence time and
$C=(1+4/(\epsilon^{2}\mathcal{N}t_{L}))^{-1}\approx 0.1$ the readout
efficiency; $t_{L}=500\,$ns the laser pulse integration region,
$\mathcal{N}=9\times 10^{6}$ photons/s the steady-state count rate and
$\epsilon=0.1$ the maximum readout contrast ($\tau=0$).

Setting $SNR=1$, substituting $\omega_{\text{rot}}=2\pi/t_{\text{rot}}$ and
rearranging for $\delta B$,

$\delta
B=\frac{\pi\,e^{\left(\frac{\tau}{T_{2}}\right)^{n}}}{4C\gamma_{e}\sin\theta_{\text{NV}}\sin^{2}\left(\frac{\pi\tau}{2t_{\text{rot}}}\right)\sqrt{t_{\text{rot}}}}.$
(12)

The sensitivity is reduced by a factor of $2$, on account of self-
normalisation to the tail of each PL measurement ($\sqrt{2}$) and halved duty
cycle for normalisation with the $-\pi/2$-pulse ($\sqrt{2}$). However, due to
$n=3$ in the decay exponent, and as we have found from experiment,
$\tau=T_{2}$ is not the optimum measurement time where the peak slope occurs
at. Instead, we find typically $\tau_{\text{opt}}\sim 0.7-0.8\,T_{2}$, meaning
the sensitivity reduces by a factor $e^{(0.7)^{3}}\approx e/2$ (at 3.750 kHz).
The shot-noise limited sensitivity is then

$\delta B=\frac{\pi
e}{4C\gamma_{e}\sin\theta_{\text{NV}}\sin^{2}\left(\frac{\pi\tau}{2t_{\text{rot}}}\right)\sqrt{t_{\text{rot}}}}.$
(13)

For our current experimental parameters at $3.750$ kHz, _i.e._
$\theta_{\text{NV}}=30^{\circ}$, $\tau=180\,\upmu$s,
$t_{\text{rot}}=267\,\upmu$s, this evaluates to $18\,$nT Hz-1.

The operating Ramsey sensitivity, which includes the same scaling due to self-
normalisation and normalisation with the $-\pi/2$-pulse (multiplication by 2)
is given by

$\delta
B_{\text{Ramsey}}=\frac{\exp\left(\tau/T_{2}^{\ast}\right)}{\gamma_{e}C}\frac{\sqrt{\tau+t_{D}}}{\tau},$
(14)

with $C$ the same as for DRUM, $T_{2}^{\ast}=360\,$ns, $\tau$ the Ramsey time
and $t_{D}$ the dead time, which includes a $3\,\upmu$s laser pulse,
$1.2\,\upmu$s of dark time to maximise optical pumping efficiency due to spin
state shelving and the combined durations of the two microwave pulses,
$2t_{\pi/2}=t_{\pi}=200\,$ns. The idealised, zero-dead time Ramsey sensitivity
with which we quote comparison with in the main text is given by Eq. 14 with
$\tau=T_{2}^{\ast}$, $t_{D}=0$ and is given by

$\delta B_{\text{Ramsey,
ideal}}=\frac{e}{2\gamma_{e}C}\frac{1}{\sqrt{T_{2}^{\ast}}}.$ (15)

The gain of DRUM over Ramsey is computed by taking the ratio of $\delta
B_{\text{DRUM}}/\delta B_{\text{Ramsey}}$, with $\theta=90^{\circ}$,
$\tau=T_{2}=t_{\text{rot}}$ in Eq. 13 and the idealised shot-noise limited
Ramsey sensitivity in Eq. 15. We assume $C$ to be the same for both
measurements.

##  S7 The effects of rotation direction

We also examined the effect of changing the direction of the motor’s rotation.
The data presented in the main text was using a rotation direction that
created an effective magnetic ‘pseudo-field’ [9]
$B_{\omega}=\frac{\omega_{\text{rot}}}{\gamma_{13}}>0$
($\gamma_{13}=10.71\,\text{Hz\,}\upmu\text{T}^{-1}$) that added to the
$z$-oriented magnetic bias field, _i.e._

$\mathbf{B}=B\,\hat{\bm{z}}=\left(B_{z}+B_{\omega}\right)\hat{\bm{z}}.$ (16)

Importantly, this has the consequence of shifting 13C contrast revivals to
_shorter_ times for an applied magnetic field. Therefore, to achieve a
particular spin-echo measurement time $\tau$ at the first carbon-13 revival, a
lower magnetic field is required

$\tau=\frac{4\pi}{\gamma_{13}(B_{z}+B_{\omega})}.$ (17)

We found that when the rotation direction was reversed, and larger $z$-fields
required for the same revival time, $T_{2}$ was lower. Figure S6 shows the
pertinent data, as well as the slope $d\mathcal{S}/dB$ as a function of
rotation speed. Different diamond samples were attached to Motor I and Motor
II, though both samples exhibited nearly identical count rates, NV
orientations and coherence times. In all cases, we observe $T_{2}$ to fall
with rotation speed, at a similar rate, though the starting values are quite
different, and these differences are reflected in the peak sensing slope data.
It was found also that Motor I exhibited some form of instability near
$5\,$kHz, leading to a marked drop in sensing slope (this observation led to
the implementation of Motor II). We suspect motor run-in at a fixed speed (in
this case, 3.333 kHz) as a potential culprit for this instability.

Without $\omega_{\text{rot}}>0$ data for Motor I allowing a conclusive
determination, we posit with some confidence that the different coherence
times for different rotation directions are due to the magnitude of $B_{z}$.
While rotationally-symmetric, the $z$-bias field still makes an angle of
$30^{\circ}$ to the NV axis. Larger $B$ fields tilted away from the NV axis
induce more electron-spin state mixing, and, due to the anisotropy of the
NV-13C nuclear spin hyperfine interaction, a spread in nuclear spin precession
frequencies [10]. Consequently, the amplitude of the spin-echo signal is
suppressed, and $T_{2}$ reduced. In other experiments (data not shown), we
kept the spin-echo time $\tau$ constant but changed the magnetic field
strength and order of the 13C revival we perform the DRUM measurement at,
_i.e._ for $\omega_{\text{rot}}/2\pi=3.750\,$kHz, $\tau=80\,\upmu$s and
$B=2.68\,$mT ($n=1$) to $B=9.0\,$mT ($n=4$), and observed that higher fields
indeed reduced the revival amplitude.

Figure S6: The effects of changing rotation direction, and instability in
motor rotation. (a) $T_{2}$ as a function of rotation speed for two motors,
Motor I and Motor II, each with a different diamond sample mounted. We observe
a markedly higher $T_{2}$ for rotation against the direction of Larmor
precession. (b) Peak sensing slopes for the same motor and diamond
configurations in (a). A sharp dip near 5 kHz in Motor I we believe is
evidence for unstable rotation.

## References

  * [1]
  * Wood  _et al._ [2018] A. A. Wood, A. G. Aeppli, E. Lilette, Y. Y. Fein, A. Stacey, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Phys. Rev. B 98, 174114 (2018).
  * Binder  _et al._ [2017] J. M. Binder, A. Stark, N. Tomek, J. Scheuer, F. Frank, K. D. Jahnke, C. Müller, S. Schmitt, M. H. Metsch, T. Unden, T. Gehring, A. Huck, U. L. Andersen, L. J. Rogers, and F. Jelezko, SoftwareX 6, 85 (2017).
  * Bauch  _et al._ [2018] E. Bauch, C. A. Hart, J. M. Schloss, M. J. Turner, J. F. Barry, P. Kehayias, S. Singh, and R. L. Walsworth, Phys. Rev. X 8, 031025 (2018).
  * Wolf  _et al._ [2015] T. Wolf, P. Neumann, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, and J. Wrachtrup, Phys. Rev. X 5, 041001 (2015).
  * Michl  _et al._ [2019] J. Michl, J. Steiner, A. Denisenko, A. Bülau, A. Zimmermann, K. Nakamura, H. Sumiya, S. Onoda, P. Neumann, J. Isoya, and J. Wrachtrup, Nano Lett. 19, 4904 (2019).
  * Fescenko  _et al._ [2020] I. Fescenko, A. Jarmola, I. Savukov, P. Kehayias, J. Smits, J. Damron, N. Ristoff, N. Mosavian, and V. M. Acosta, Phys. Rev. Research 2, 023394 (2020).
  * Degen  _et al._ [2017] C. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).
  * Wood  _et al._ [2017] A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Nature Physics 13, nphys4221 (2017).
  * Stanwix  _et al._ [2010] P. L. Stanwix, L. M. Pham, J. R. Maze, D. Le Sage, T. K. Yeung, P. Cappellaro, P. R. Hemmer, A. Yacoby, M. D. Lukin, and R. L. Walsworth, Phys. Rev. B 82, 201201 (2010).