# Anisotropic electron-nuclear interactions in a rotating quantum spin bath

A. A. Wood<EMAIL_ADDRESS>School of Physics, University of
Melbourne, Parkville Victoria 3010, Australia  R. M. Goldblatt  School of
Physics, University of Melbourne, Parkville Victoria 3010, Australia  R. P.
Anderson  La Trobe Institute of Molecular Science, La Trobe University,
Bendigo, Victoria 3550, Australia  L. C. L. Hollenberg  School of Physics,
University of Melbourne, Parkville Victoria 3010, Australia  Centre for
Quantum Computation and Communication Technology, University of Melbourne,
Victoria 3010, Australia  R. E. Scholten  School of Physics, University of
Melbourne, Parkville Victoria 3010, Australia  A. M. Martin  School of
Physics, University of Melbourne, Parkville Victoria 3010, Australia

###### Abstract

The interaction between a central qubit spin and a surrounding bath of spins
is critical to spin-based solid state quantum sensing and quantum information
processing. Spin-bath interactions are typically strongly anisotropic, and
rapid physical rotation has long been used in solid-state nuclear magnetic
resonance to simulate motional averaging of anisotropic interactions, such as
dipolar coupling between nuclear spins. Here, we show that the interaction
between electron spins of nitrogen-vacancy centers and a bath of 13C nuclear
spins in a diamond rotated at up to 300,000 rpm introduces decoherence into
the system via frequency-modulation of the nuclear spin Larmor precession. The
presence of an off-axis magnetic field necessary for averaging of the dipolar
coupling leads to a rotational dependence of the electron-nuclear hyperfine
interaction, which cannot be averaged out with experimentally achievable
rotation speeds. Our findings offer new insights into the use of physical
rotation for quantum control with implications for quantum systems having
motional and rotational degrees of freedom that are not fixed.

An electron spin qubit and a surrounding bath of nuclear spins in diamond is a
highly topical subject in solid-state quantum sensing and quantum information
processing. The spin bath is both a deleterious source of decoherence, and a
resource for storing and entangling quantum states. Understanding how the
qubit-bath interaction behaves and can be controlled in a range of settings is
an important consideration for future quantum technologies. For the nitrogen-
vacancy (NV) centre in natural abundance diamond, the central electron spin is
usually surrounded by a spin bath composed of spin-1/2 13C nuclei. The nuclear
spin bath is equally problematic for diamond-based quantum sensing [1, 2, 3]
and quantum information processing [4, 5], since it leads to a noisy
background magnetic field that limits the coherence time of the qubit and
consequently the precision of quantum measurements [6, 7, 8, 9]. Much effort
has been devoted to dynamical-decoupling schemes to protect coherence and
controllably isolate the central spin from the bath [10, 11].

The nuclear spins near optically-addressable electron spins like NV centers
are also a resource for storing and retrieving quantum information, and form
the basis of hybrid-spin quantum information processors [12, 13, 14, 15, 16,
17]. Nuclear spins are also useful ancillae for quantum sensing, where the
qubit-nuclear spin interaction is harnessed for more precise sensing [18, 19,
20, 21, 22] or accessing new sensing modalities [23, 24, 25, 26]. Novel means
of spin bath control are thus sought to tailor the environment around the
qubit as needed, decoupling spins in the bath from each other and also from
the central electron spin.

Physical rotation has long been employed in nuclear magnetic resonance (NMR)
experiments as a means of suppressing deleterious spin interactions that
broaden NMR transitions. Physical rotation of solid state NMR samples with the
magnetic field oriented at the magic angle ($\theta_{m}=54.74^{\circ}$) to the
rotation axis simulates motional averaging of anisotropic interactions, a
ubiquitous technique known as magic-angle spinning (MAS) [27, 28]. The NV-
nuclear spin hyperfine interaction is an archetypal case of a spin subject to
an anisotropic interaction Hamiltonian, much like the chemical shift
anisotropy or nuclear dipolar coupling in NMR [29] or anisotropic Zeeman
interactions encountered in electron paramagnetic resonance (EPR) [30].

In the NV system, the hallmark of the NV-13C hyperfine interaction is periodic
collapse and coherent revival of the NV spin-echo signal [31]. Homonuclear
dipole-dipole interaction between nuclear spins is much weaker ($\sim$kHz) and
drives spin-conserving flip-flop interactions, creating a noisy magnetic
environment for the NV and a reduced spin coherence time [32, 8] and
attenuates the coherent revivals. In principle, the latter can be eliminated
by physical rotation at kHz frequencies in the presence of a magnetic field at
the magic angle to the rotation axis, though just how this affects the NV-13C
hyperfine interaction is less clear.

In this work, we investigate the qubit-bath interaction in a diamond subjected
to physical rotation at rates comparable to the electron spin coherence time,
$T_{2}$, with an off-axis magnetic field to introduce anisotropic averaging.
Recent experimental advances have enabled quantum measurement and control of
rapidly-rotating NV centres [26, 33], making rotational averaging of
deleterious spin interactions in a quantum setting a feasible new avenue of
study for the central spin problem in diamond. We find that the NV - nuclear
spin hyperfine interaction is strongly affected by the rotation of the
diamond, inducing decoherence due to rotation-dependent frequency modulation
of the nuclear spin Larmor precession. While the coherence of the NV ensemble
in the presence of a 13C bath for varying magnetic field angles has been
previously examined in Ref. [34], our work shows that physical rotation of the
diamond modulates the NV-13C interaction in a way that leads to much faster
dephasing than in the stationary case. We show theoretically that small tilts
of the magnetic field from the rotation axis are expected to induce coherent
modulation of the spin-echo signal at the rotation period, but for larger tilt
angles the spin-echo signal is washed out by incoherent averaging of
hyperfine-augmented nuclear precession of 13C nuclei close to the NV. Any
effects of magic-angle spinning on the nuclear spin bath are therefore
obscured, and we conclude new strategies to eliminate the electron-nuclear
hyperfine coupling, or much faster rotation speeds, are required to realise
rotational decoupling in the NV system.

Our study is motivated by the prospects of eliminating nuclear homonuclear
couplings, which could potentially yield decoherence times comparable to
isotopically-purified 12C diamonds [7] and form a new mechanism for tuning the
qubit-bath interaction. In previous work, it was found that physical rotation
at a rate $\bm{\Omega}$ imparts an effective magnetic pseudo-field
$\bm{B}_{\Omega}=\bm{\Omega}/\gamma_{n}$ to the 13C spin bath, $\gamma_{n}$
the nuclear gyromagnetic ratio [26]. For the low magnetic fields that can be
studied with NV experiments (compared to NMR fields exceeding $1\,$T), pseudo-
fields can be made comparable to the bias magnetic field in the system, even
to the point of eliminating altogether Larmor precession in the frame of the
rotating NVs without any affect on the NV spin. The presence of the magnetic
pseudo-field from physical rotation also has a significant effect in MAS,
since the rotation itself generates a nontrivial field parallel to the
rotation axis: for $f_{\text{rot}}=5\,$kHz, $B_{\Omega}\sim 5\,$G.
Consequently, the NV-13C system represents a uniquely interesting system in
which to study the low-field dynamics of motional averaging.

This paper is structured as follows. We review in Section I the relevant
theory concerning the NV-13C interaction pertinent to our studies. Section II
describes the setup and our experimental examination of the loss of NV spin
coherence for varying magnetic field angles and rotation speeds. We use a
theoretical model of the NV coupled to a bath of interacting spins in Section
III to confirm our findings and examine situations too difficult to realise
experimentally. In Section IV we discuss our key findings in detail and give
an outlook on rotation as a means of controlling spin coherence in future
work.

##  I NV-nuclear spin interactions in the rotating frame

A simplified schematic of our experimental configuration, important
geometrical quantities and the NV energy level scheme is depicted in Fig.
1(a). The negatively-charged NV centre in diamond [35, 36] features a spin-1
ground state with an $m_{S}=0$ and degenerate $m_{S}=\pm 1$ spin sublevels
split by $D_{\text{zfs}}=2870\,$MHz. Application of a magnetic field $\bm{B}$
breaks the degeneracy of the magnetic sublevels, leading to a splitting of
$\omega_{m}\approx D_{\text{zfs}}+m\gamma_{e}|\bm{B}|\cos\theta_{B}$, where
$\gamma_{e}/2\pi=2.8\,$MHz/G is the electron gyromagnetic ratio and
$\theta_{B}$ the angle between the NV axis and the magnetic field, an
approximation valid for low magnetic fields ($<50$G) and $\theta_{B}\ll
90^{\circ}$. The diamond sample we consider in this work is natural abundance
13C CVD material ($98.9\%^{12}$C) and contains an ensemble density of NV
centres, with a typical nitrogen density of $0.1\,$ppm. We also work at
magnetic fields far from any cross-relaxation resonances between the NV and
its intrinsic nuclear spin [37], nearby P1 centers, and 13C spins [38, 39].
The Hamiltonian for the NV-13C system in the stationary laboratory frame is
[8]

$H=H_{\text{NV}}+H_{\text{Z}}+H_{\text{hfs}}+H_{\text{d-d}},$ (1)

where the NV Hamiltonian is $H_{\text{NV}}=D_{\text{zfs}}S_{z}^{2}$,
$H_{\text{Z}}=\gamma_{e}\bm{B}\cdot\bm{S}+\gamma_{n}\bm{B}\cdot\bm{I}$ is the
Zeeman Hamiltonian with $\gamma_{n}/2\pi=1071.5$Hz/G the 13C gyromagnetic
ratio, $H_{\text{hfs}}=\bm{S}\cdot\sum_{i}\bm{A}_{i}\cdot\bm{I}_{i}$ is the
hyperfine interaction between the NV spin $\bm{S}$ and the $i$th nuclear spin
$\bm{I}_{i}$, $\bm{A}_{i}$ the hyperfine tensor and $H_{\text{d-d}}$
represents the dipole-dipole interactions between 13C spins.

The focus of our analysis is the hyperfine interaction tensor between the NV
and nuclear spin, $\bm{A}_{i}$, which contains a contact term due to the
finite extent of the electron wavefunction, and a dipole-dipole interaction
term. The Fermi contact term scales exponentially with distance and thus
dominates for nuclear spins within a few lattice sites of the NV. The dipolar
term decreases with NV-nuclear spin separation as $r_{i}^{-3}$, and is
responsible for most dynamics observed in a $\tau\sim T_{2}$ duration spin-
echo experiment, such as coherent revivals at half the Larmor period [31]. The
dipolar term is also responsible for the behaviour we observe as the magnetic
field is tilted away from the rotation axis [34].

Figure 1: a) A diamond is mounted on an electric motor such that the $\langle
111\rangle$ crystallographic direction is approximately parallel to the
rotation axis. An ensemble of NV centers are optically prepared and readout by
a confocal microscope, and magnetic fields are applied to tilt the total field
$\bm{B}$ away from the rotation axis by an angle $\theta_{B}$ (inset). b) NV
electron spin (red) interacting with a bath of 20 13C nuclear spins within a
diamond physically rotating around the $z$-axis at $\omega_{\text{rot}}$. The
dashed circles depict radii in units of lattice constant $a_{0}=0.154$ nm. c)
Nuclear spin precession frequencies at $t=0$ for $|\bm{B}|=30\,$G and varying
tilt angles for the NV electron spin in the $m_{S}=0$ state (top) and
$m_{S}=-1$ state (bottom). Note the shift from the bare precession frequency
of $32.1\,$kHz due to the rotation at $3.33\,$kHz. The curve color corresponds
to the coloured vector in (b). d) Nuclear precession frequencies of the spin
ensemble for $m_{S}=0$, $\omega_{\text{rot}}=3.33\,$kHz as a function of time
for different tilt angles $\theta_{B}$. Rotation of the diamond modulates the
precession frequency of the nuclear spins due to anisotropic, non-secular
terms in the NV-13C hyperfine interaction.

Approximating the NV and 13C as point dipoles, the hyperfine interaction
tensor is given by [8]

$\bm{A}_{i}=\frac{\mu_{0}\gamma_{e}\gamma_{n}\hbar}{4\pi
r_{i}^{3}}\left(1-3\bm{\hat{r}}_{i}\bm{\hat{r}}_{i}\right),$ (2)

where $\mu_{0}$ is the vacuum permittivity and $\bm{\hat{r}}_{i}$ a unit
vector along the axis connecting the NV to the $i$th nuclear spin. The
presence of magnetic field components that are not parallel to the NV axis
(non-secular terms) has an important effect on the hyperfine interaction.
First reported in Ref. [40] for single NV centers and later by Ref. [34] for
NV ensembles, the off-axis magnetic field mixes electron spin states, encoding
each nuclear spin with an $r_{i}$-dependent precession frequency that
asymptotes to the bare precession frequency $\gamma_{n}B$ as $r_{i}$
increases. The increased spread of nuclear spin precession frequencies results
in faster decoherence in a spin-echo experiment. As we shall see, the encoding
of nuclear spin precession rates is not rotationally symmetric, meaning that
azimuthal rotation of the diamond (or magnetic field) results in frequency
modulation of the nuclear spins.

Figure 2: Measuring the coherence of a rapidly rotating NV - nuclear spin
ensemble. (a) Rotating the diamond with the NV axis tilted slightly away from
the rotation axis up-converts magnetic field components orthogonal to the
rotation axis to fields oscillating at the rotation frequency. This modulated
field can then be measured by a spin-echo pulse sequence applied synchronous
with the rotation of the diamond. Tilting the magnetic field further from the
rotation axis increases the amplitude of the up-converted field, and results
in fringes in the spin-echo signal. (b) Illustrative time-domain spin-echo
signals for $B=20\,$G (top) and $B=40\,$G (bottom) for $\theta_{B}=0$ showing
rotationally-induced shifting of the 13C contrast revivals. Adding a
transverse field to change $\theta_{B}$ increases the total field, which also
shifts the revival. We adjust the interpulse time $\tau_{R}$ so that the final
$\pi/2$-pulse of the spin echo sequence occurs at the same time as first (20
G) or second (40 G) contrast revival maximum for a given magnetic field
configuration. (c) Spin-echo signal at fixed $\tau_{R}$ vs. $\theta_{B}$ for
different rotation speeds. An exponentially-damped sinusoid (solid line) is
fitted to the experimental data (points), and a fully constrained theoretical
model of the envelope is overlaid (dashed line). We observe a rapid reduction
of contrast with $\theta_{B}$ as the rotation speed is increased that matches
well with theory; the discrepancy is attributed primarily due to imperfect
magnetic field calibration. Our observations are consistent with dephasing
introduced by NV-13C hyperfine modulation arising from rotation in the
presence of an off-axis magnetic field.

The Hamiltonian Eq.1 consists of terms intrinsic to the diamond,
$H_{\text{in}}=H_{\text{NV}}+H_{\text{hfs}}+H_{\text{d-d}}$, and extrinsic
lab-frame terms, $H_{\text{ex}}=H_{Z}$. Physical rotation of the diamond about
the $z$-axis in space corresponds to application of a unitary transformation
$\mathcal{R}(t)=\exp\left(-iJ_{z}\omega_{\text{rot}}t\right)$, with
$J_{z}=I_{z}+S_{z}$, to either the intrinsic or extrinsic terms. Since the NV
axis is very nearly parallel to the rotation axis in our experiments, there is
no substantive difference between the two transformations, although it is
marginally simpler to rotate the extrinsic Hamiltonian and consider the
effects in the NV frame. The transformed Hamiltonian becomes

$\displaystyle H_{\text{rot}}$
$\displaystyle=H_{\text{NV}}+H_{\text{hfs}}+H_{\text{d-d}}$
$\displaystyle+\mathcal{R}(t)H_{Z}\mathcal{R}^{\dagger}(t)+i\frac{d}{dt}\mathcal{R}(t)\mathcal{R}^{\dagger}(t).$
(3)

The final term in Eq. 3,
$i\frac{d}{dt}\mathcal{R}(t)\mathcal{R}^{\dagger}=\omega_{\text{rot}}J_{z}$,
is responsible for the rotationally-induced magnetic pseudo-field shift [26]
of the 13C nuclear spin precession frequency. Diagonalizing the Hamiltonian
Eq. 3 at time $t$ yields the eigenenergies $E^{i}_{m_{S},m_{I}}$ for the $i$th
nuclear spin, and computing

$\omega^{i}_{L}(m_{S})=\frac{1}{\gamma_{n}}\left|E^{i}_{m_{S},\uparrow}-E^{i}_{m_{S},\downarrow}\right|$
(4)

yields the $m_{S}$-dependent Larmor frequency for the $i$-th nuclear spin.
Figure 1(b,c) shows the nuclear spin Larmor precession frequencies for $N$
nuclear spins surrounding the NV centre for varying magnetic field tilt angle
$\theta_{B}$, and Fig. 1(d) depicts the time-dependence of $\omega_{L}(m_{S})$
for several tilt angles. The rotating magnetic field (in the frame of the NV)
leads to frequency modulation of the nuclear spin precession in a manner
dependent on the position of the nuclear spin. This modulation in turn
introduces a form of ensemble dephasing, since the spread of nuclear spin
precession frequencies will strongly suppress the observation of coherent
behaviour, such as 13C spin-echo revivals.

The non-secular terms in the NV-13C hyperfine interaction are responsible for
the modulation of the nuclear spin precession. A useful representation that
offers insight into the role of non-secular terms is considering an
$m_{S}$-dependent effective magnetic field acting on the nuclear spin [31, 41,
42, 43], which also depends on the angle the magnetic field makes to the NV
axis [32, 34]. Since the electron spin interacts much more strongly with
magnetic fields than the nuclear spins, there is significant back-action on
the bath [44] and as a result the electron spin state determines the evolution
of the coupled nuclear spins. The state-dependent field for the $i$th nuclear
spin is given by

$\bm{B}_{\text{eff},i}(m_{S})=m_{S}\bm{B}_{\text{dip}}+\bm{B}\cdot\bm{g}_{i}$
(5)

where $\bm{B}_{\text{dip}}$ is the electron spin dipole field and $\bm{g}_{i}$
is an effective nuclear spin $g$-tensor that expresses the NV-13C hyperfine-
mediated modification to the nuclear spin precession frequency

$\bm{g}_{i}=\mathbb{1}-\frac{\gamma_{e}}{\gamma_{n}D_{\text{zfs}}}(2-3|m_{S}|)\bm{A}_{i}.$
(6)

Equation 5 concisely expresses that the 13C spin experiences a magnetic field
depending on the NV electron spin state. The resulting Larmor precession rate
of the nuclear spins can therefore be written as

$\omega^{i}_{L}(m_{S})=\gamma_{n}\left|\bm{B}_{\text{eff},i}(m_{S})\right|.$
(7)

When either the diamond or the magnetic field is rotated, the
$\bm{B}\cdot\bm{g}_{i}$ term in Eq. 5 leads to time dependence of
$\omega^{i}_{L}$. In the following sections of this paper we will discuss
experimental detection of this frequency modulation effect.

##  II Experiment

Our experimental setup is essentially the same as that described previously
[26]. An optical-grade CVD diamond is mounted on the spindle of an electric
motor (Fig. 1(a)) that can spin at up to $350,000$ rpm. The diamond is a
$(111)$-cut sample with a $\sim 4^{\circ}$ polish (miscut) angle, and the
motor tip is machined so that one orientation class of NVs makes an angle of
$<0.2^{\circ}$ to the rotation axis. A 6 mm-working-distance microscope
objective focuses $500\,\upmu$W of green light close to the rotation center of
the spinning diamond, and collects emitted red fluorescence which is directed
to an avalanche photodiode, forming a confocal microscope. Three pairs of
current carrying coils are used to generate magnetic fields of up to $\approx
40\,$G with a maximum tilt angle from the rotation axis of
$\theta_{B}=40-50^{\circ}$. We used vector magnetometry to calibrate the field
produced from the coils, allowing us to estimate the tilt angle used in
experiments. A 20 $\upmu$m copper wire, 300$\upmu$m above the surface of the
diamond, is used to apply microwave fields with typical Rabi frequencies of
$5\,$MHz.

We use spin-echo interferometry [45] to measure the coherence of the rotating
ensemble of NV centers. A $3\,\upmu$s laser pulse is first applied, followed
by microwave pulses synchronous with the rotation [33]. Although the NV
orientation class we target is nearly parallel to the rotation axis, the small
misalignment $\delta\theta$ between the NV axis and the $z$ rotation axis
means that static magnetic field components $B_{\perp}$ transverse to $z$ are
effectively up-converted to alternating fields at the motor rotation
frequency, $B_{\text{ac}}\approx
B_{\perp}\delta\theta\cos(\omega_{\text{rot}}t-\phi_{0})$ [46], with
$\phi_{0}$ an inconsequential phase set by the orientation of the diamond on
the motor. Fluorescence detected from a spin-echo sequence with fixed total
time $\tau$ will therefore exhibit fringes as the magnetic field tilt angle
$\theta_{B}$ is increased. We use the amplitude of these fringes to infer the
coherence of the NV electron spin as a function of magnetic field tilt angle
and rotation speed of the diamond. The spin-echo time $\tau_{R}$ must be set
to an integer multiple of twice the 13C nuclear spin precession frequency,
where coherence revivals occur. These revivals shift significantly in time due
to the increasing rotation speed [26] and also slightly when a transverse
field $B_{\perp}$ is added to vary the tilt angle. The value of $\tau_{R}$ is
set to $2n(\gamma_{n}B_{\text{tot}}/2\pi+f_{\text{rot}})^{-1}$, for integer
$n$ and $B_{\text{tot}}=(B_{\perp}^{2}+B_{z}^{2})^{1/2}$ to ensure overlap
with the coherence revival, as depicted in Fig. 2(a,b).

Figure 2(c) shows spin-echo fringes for increasing magnetic field tilt angles
for two different magnetic field strengths, $20\,$G and $40\,$G, and rotation
speeds ranging from stationary to $5.17\,$kHz. We set the spin-echo time to
the first coherence revival for $20\,$G (at $\tau_{R}=75-93\,\mu$s) and the
second revival for $40\,$G (at $\tau_{R}=83-93\,\mu$s) and change the tilt
angle by applying a transverse magnetic field $B_{\perp}$ along the $x$-axis.
The spin-echo sequence is synchronised with an arbitrary phase to the
rotation, and hence up-converted magnetic field. This phase in turn sets the
period of the resulting fringes, and is scaled with rotation speed to remain
approximately constant. As the rotation speed increases, we observe a rapid
damping of contrast for increasing magnetic field tilt angle.

##  III Theoretical investigation

Figure 3: Simulated effective electron spin coherence time
$T_{2}^{\text{eff}}$as a function of magnetic field tilt angle $\theta_{B}$
and rotation speed $\omega_{\text{rot}}$ for $B=20\,$G (left) and 40 G
(right). Each value of $T_{2}$ derives from an exponential fit to the spin-
echo signal envelope, evaluated at the rotationally-shifted 13C revival
positions and averaged over 10 distinct bath configurations, each with
approximately 125 13C nuclei at random lattice sites. A decay envelope with
time constant $T_{2}^{\text{phenom}}$ limits the maximum coherence time to
$150\,\upmu$s to simulate the effects of other defects within the diamond
sample used in this work.

We compared our results to a numerical simulation of the NV spin-echo signal
generated by adapting the disjoint-cluster method of Ref. [32] to the time-
dependent, non-secular case. The large ensemble of $N$ interacting spins is
decomposed into $n$ groups $\left\\{G_{1},G_{2},...,G_{n}\right\\}$ of up to
$g$ spins, with strong interactions between elements of each group and
negligible interactions between elements in different groups. Considering a
single NV interacting with a group of up to $g$ 13C nuclei, the density matrix
of the system within the group is given by $\rho=\rho_{e}\otimes\rho_{B}$,
with $\rho_{e}$ the density matrix of the electron spin and
$\rho_{B}=1/2^{g}\mathbb{1}^{g}$ the density matrix for the thermally-
populated nuclear spin bath. The time-evolution of $\rho$ is then

$\rho(t)=U_{\tau}\rho(0)U^{\dagger}_{\tau},$ (8)

with $U_{\tau}=U_{\pi/2}U(\tau/2)U_{\pi}U(\tau/2)U_{\pi/2}$ a unitary operator
representing a spin-echo sequence applied to the electron spin with free-
evolution time $\tau$, $U_{\pi/2},U_{\pi}$ operators for $\pi/2$ and $\pi$
pulses, $U(t)$ the free evolution operator and $\rho(0)=|0\rangle\langle
0|\otimes\rho_{B}$. The normalised spin-echo signal for this particular group
of nuclear spins $G$ is then given by

$S_{G}(\tau)=2\text{Tr}\left[\mathcal{P}_{0}\rho(\tau)\right]-1,$ (9)

with $\mathcal{P}_{0}$ the projection operator onto the $|m_{S}=0\rangle$
electron spin state. The spin-echo signal from all groups is then
$S(\tau)=\prod_{G}S_{G}(\tau)$. To simulate ensemble averaging due to multiple
NV centres, we then compute the average signal
$S_{\text{ave}}=1/N_{\text{ave}}\sum S(\tau)$ for $N_{\text{ave}}$ different
configurations of 13C baths. The simulation results depicted in Fig. 2(c) are
for $N_{\text{ave}}=20$ different configurations of up to $N=310$ 13C spins
with a maximum group size of $g=3$. Our selection of $g=3$ includes spin
correlations up to third order, but to include any effects of potential
homonuclear dipolar decoupling (a two-body interaction) we need only use
$g=2$. Short experimentally realisable spin coherence times also prevent any
higher order effects from manifesting (see below).

Figure 4: Simulating the onset of spin-echo envelope modulation due to rapid
rotation with a tilted magnetic field. a) for small $\theta_{B}$ the spin-echo
signal collapses as the rotation speed is increased, before contrast revivals
at twice the rotation period become evident. The dashed traces depict the 13C
dipole-dipole limited coherence envelope for the relevant rotation speed and
$\theta_{B}=15^{\circ}$. b) for fixed rotation speeds of 200 krpm (3.33 kHz,
top) and 500 krpm (8.33 kHz, bottom), increasing the tilt angle causes the
rotational revivals to decay. At larger angles, small, intermittent revivals
reappear at integer multiples of the rotation period. Each simulated spin-echo
trace derives from the average of at least 20 different bath configurations of
up to 300 13C nuclear spins.

The theory simulations show concordance with the experimental results and
reproduce closely the dephasing induced by rapid rotation in the presence of
an off-axis magnetic field. Since this effect is entirely due to NV-nuclear
spin hyperfine coupling (i.e. $g=1$), retaining the homonuclear coupling
terms, or for that matter the grouping schema based on the homonuclear
couplings, is redundant. We include these terms, and correlations up to $g=3$
for the purposes of extending the model to situations where longer coherence
times are possible, as detailed later in this work.

It is useful to consider a figure of merit which parametrizes the coherence of
the system, such as an effective spin coherence time. We define the effective
spin-coherence time $T_{2}^{\text{eff}}(\theta_{B},\omega_{\text{rot}})$ as
the $1/e$-damping time of the spin-echo contrast for a given tilt angle and
rotation speed. We calculate the spin-echo signal at integer multiples of the
rotationally-shifted 13C contrast revival time
$\tau_{R}=4\pi/(\gamma_{13}B+\omega_{\text{rot}})$, and, assuming the spin-
echo signal damps as
$S(\tau)=S(0)e^{-(\tau/T^{\text{eff}}_{2}(\theta_{B},\omega_{\text{rot}}))^{n}}$,
fit a stretched exponential decay model to extract
$T^{\text{eff}}_{2}(\theta_{B},\omega_{\text{rot}})$. We include also a
phenomenological damping term to reproduce the stationary, $\theta_{B}=0$
coherence time $T_{2}^{\text{phenom}}=150\,\upmu$s, due to other defects in
the diamond (principally paramagnetic nitrogen P1 centers [47]). In Fig. 3, we
evaluate $T_{2}^{\text{eff}}$ for variable $\theta_{B}$ and
$\omega_{\text{rot}}$ for the magnetic field strengths examined in this work,
$20$ and $40$ G. As expected, we observe a rapid decay of spin coherence time
with increasing rotation speed and tilt angle that becomes sharper as the
magnetic field strength increases. Figure 3 broadly summarises the range of
potential coherence times possible with our current experimental
configuration.

With our experimental results well explained by the theoretical model, we can
confidently examine NV-13C dynamics outside the domain of measurement times,
magnetic fields, tilt angles and rotation speeds accessible with our current
experimental setup. When we remove the phenomenological damping term, the
simulations reveal an interesting effect at high rotation frequencies and
measurement times well beyond the $T_{2}$ of the diamond sample used in our
experiments. For rotation speeds $<200\,000\,$rpm, we observe the familiar
rapid damping of contrast at non-zero $\theta_{B}$ as the rotation speed
increases. However, as we exceed rotation speeds possible in our current
experiments, a revival of spin-echo contrast at twice-integer periods of the
rotation frequency appears for small tilt angles. This behavior persists even
after averaging over multiple baths to simulate measurement of an ensemble of
NV centers. Figure 4 (a) shows the spin-echo signal as a function of time,
evaluated at integer multiples of $4\pi/\omega_{13}$ (_i.e._ at each 13C
revival), for various rotation speeds at $\theta_{B}=15^{\circ}$. In Fig. 4(b)
we examine the onset of rotational modulation as the magnetic field tilt angle
is increased at rotation speeds of 200,000 rpm (3.333 kHz) and 500,000 rpm
(8.33 kHz).

The underlying cause of the initial spin-echo modulation at low magnetic field
angles can be easily understood as arising from modulation of the NV
$m_{S}=0\rightarrow-1$ transition energy due to the angular dependence of the
hyperfine interaction. This modulation is a small perturbation to the NV and
nuclear spin eigenstates at the rotation frequency $\omega_{\text{rot}}$. When
measured phase-incoherently due to the random positions of 13C nuclei, the
modulation leads to revivals in the spin-echo signal, in direct analogy and on
top of the original revivals at the 13C Larmor period. As the tilt angle is
increased, the degree of state mixing between electron spin levels and the
consequent electron-spin augmentation of the nuclear spin precession begins to
become significant. This process suppresses coherent revivals of the spin echo
signal due to the strongly perturbed nuclear spin precession rate.

Beyond the initial collapse of the spin-echo contrast with tilt angle
(vertical axis of Fig. 4), the simulation exhibits several low-amplitude
contrast revivals (particularly evident for the faster rotation speed in Fig.
4(b)). We attribute these effects to insufficient averaging over different
spin bath configurations in our simulations, rather than any possible coherent
effect induced by rotational averaging. We note that when the dipolar coupling
term in our simulations is set to zero, these features remain, suggesting they
originate from particular spin baths (_i.e._ no strongly coupled nuclear
spins). We would not expect to detect these in any experiment with an ensemble
of NV centers, but leave open the possibility that single NVs with a few
nearby 13C spins may exhibit similar effects.

##  IV Discussion

The spatial variation of the NV-13C hyperfine interaction in the presence of
an off-axis field can be best described as each nuclear spin precessing at a
rate determined by its position relative to the NV. When rotated, the nuclear
spins precess around an axis determined by the external magnetic field and the
local NV-dipole field. Physical rotation imparts a relative rotation between
axes within and external to the diamond, in this case the NV axis and the
nuclear precession axis. Therefore, the nuclear precession frequency becomes a
function of time. This can be intuitively understood as frequency-modulation
of the nuclear spin precession with a modulation depth determined by the
position of the 13C nucleus.

For practical purposes, any effects of rotation on the weak intra-bath
homonuclear couplings with the magnetic field at or near the magic angle is
totally obscured by the effects of the modulated NV-13C hyperfine interaction.
The theoretical model presented in the previous section corroborates this
reasoning and predicts that at small tilt angles, the NV spin echo signal is
modulated with contrast revivals at integer multiples of twice the rotation
period. Experimental limitations prevent us from observing this directly, but
no evidence of significant NV-13C decoupling is expected given the small
angles where contrast remains visible. To progress towards homonuclear
decoupling with rotation, strategies to mitigate the NV-nuclear spin
interaction must first be investigated. We discuss one such strategy, based on
slow rotation, below.

While sample rotation is a ubiquitous tool in NMR, the considerably stronger
electron-nuclear spin couplings in EPR systems mean that mechanical rotation
at rates comparable to the anisotropic interaction strength is impractical.
Our example system offers a particularly stark example, as the peak
homonuclear 13C-13C dipolar interactions $(10^{2}-10^{3}$ Hz) are easily
reached in our rotating experiments, in comparison to the typical NV-13C
coupling ($10^{5}$ Hz, 6 million rpm). However, it is not strictly necessary
to rotate faster than the anisotropic interaction strength in order to
eliminate it. If the system is subjected to phase evolution under three
equally spaced azimuthal angles (_i.e._ $120^{\circ}$) of the magnetic field
tilted by $54.7^{\circ}$ from the rotation axis, the anisotropic frequency
shifts sum to zero. The phases accumulated at each angular position therefore
sum in such a way as to yield only the isotropic component, a technique known
as ‘magic angle hopping’ [48, 49].

Practically, the magnetic field reorientation is done by either ‘hopping’ the
sample rotor to each new position and performing a phase measurement of the
spins, or rotating sufficiently slowly that the evolution is effectively
static during the phase measurement. In either case, the result of the phase
measurement is stored as a longitudinal spin state during the reorientation
step, making the relevant time scale for rotation $\sim T_{1}$, which for many
EPR systems, including the NV, can be on the order of rotation periods
achievable with gas-powered magic-angle spinners [50, 51], or electric motors,
as in our case. The NV-13C system, with $T_{1}$ typically $>1\,$ms, presents
an intriguing system in which magic-angle hopping could be investigated. At
rotation speeds of just a few kHz, three appropriate quantum measurements
could then be performed in sequence, with the resulting phase devoid of any
contribution from the NV-13C interaction. Remarkably, these speeds should also
be sufficient to induce homonuclear decoupling within the bath itself.

Further work is needed to refine the magic-angle hopping measurement procedure
for the case of the NV before an experimental demonstration is possible. For
example, spin-echo is no longer a suitable quantum measurement, since it
measures phase differences across the interrogation times, instead,
$T_{2}^{\ast}$-limited Ramsey interferometry is more appropriate. More
pressingly, the large zero-field splitting of the spin-1 NV effectively
freezes its orientation in the diamond lattice, and so it will not align along
the magnetic field at each rotation angle like a nuclear spin (or unbound
spin-$1/2$ electron) would as the magnetic field is rotated. This may require
a large magnetic field or a tilt angle exceeding the magic angle to observe
cancellation of the anisotropic interaction, though neither are available in
our current experimental configuration. In any case, some evidence of
rotational averaging should be detectable even with imperfect cancellation,
and this will be the subject of future work.

##  V Conclusions

We have examined the prospects of using physical rotation of a quantum system
for decoupling of deleterious spin bath interactions. We found that NV spin
coherence is rapidly lost as the magnetic field is tilted away from the
rotation axis, an observation we attribute to angular variation of the NV-13C
dipolar interaction. We have used theoretical models to support our
experimental findings and predict new phenomenology that could be explored in
an optimized experiment and different diamond sample. Our key finding is that
at the rotation speeds possible in our experiment, the NV-13C hyperfine
coupling is made time dependent, and induces dephasing that swamps any
observation of weaker homonuclear decoupling of the spin bath. Magic-angle
hopping may be a feasible alternative to our experiments, allowing the
electron-nuclear hyperfine interaction itself to be rotationally averaged.

Unlike many other quantum sensing and QIP platforms, the NV center in diamond
naturally lends itself to applications where motional and rotational degrees
of freedom are not fixed [52, 53, 54]. In these situations, rotation of the
diamond can be essential or detrimental to the intended application. The
results presented in this paper form a crucial first step of characterisation,
and lay the foundations for employing rotational averaging to decouple
interactions between the central spin and the nuclear spin bath in solid state
quantum sensing. Anisotropic magnetic dipolar interactions are becoming
ubiquitous in emerging applications of quantum systems. Novel characterisation
and control schemes based on rotation could allow for interesting new avenues
of investigation in quantum sensing and information processing, as well as
more fundamental directions such as many-body physics with strongly
interacting solid state systems [55, 56].

## Acknowledgements

We thank L. T. Hall for fruitful discussions and comments on the manuscript.
This work was supported by the Australian Research Council Discovery Scheme
(DP190100949).

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