# Tunable Gyromagnetic Augmentation of Nuclear Spins in Diamond

R. M. Goldblatt  A. M. Martin  A. A. Wood<EMAIL_ADDRESS>School of Physics, University of Melbourne, Parkville Victoria 3010, Australia

(August 30, 2024)

###### Abstract

Nuclear spins in solids exhibit long coherence times due to the small nuclear
gyromagnetic ratio. This weak environmental coupling comes at the expense of
slow quantum gate operations, which should be as fast as possible for many
applications in quantum information processing and sensing. In this work, we
use nitrogen-vacancy (NV) centers in diamond to probe the nuclear spins within
dark paramagnetic nitrogen defects (P1 centers) in the diamond lattice. The
gyromagnetic ratio of the P1 nuclear spin is augmented by hyperfine coupling
to the electron spin, resulting in greatly enhanced coupling to radiofrequency
control fields. We then demonstrate that this effect can be tuned by variation
of an external magnetic field. Our work identifies regimes in which we are
able to implement fast quantum control of dark nuclear spins, and lays the
foundations for further inquiry into rapid control of long-lived spin qubits
at room temperature.

Nuclear spins in solid-state substrates have attracted considerable interest
for applications in quantum information processing [1, 2, 3, 4, 5] and quantum
sensing [6, 7, 8, 9, 10, 11], as they exhibit long coherence times at room
temperature [12, 13, 14, 15, 16]. However, reduced coupling to magnetic noise
comes with concomitantly weaker coupling to control fields, meaning gate
operations are slow and error-prone.

When coupled to optically active electron spins, such as the NV center in
diamond [17, 18, 19], nuclear spins can be dynamically polarized [20],
controlled and measured, even down to the single-spin level [21], via
hyperfine coupling to the electron spin. Nuclear spins coupled to an electron
spin via a spin-mixing hyperfine interaction exhibit an enhanced coupling to
resonant driving fields [22, 23, 24, 25, 26, 27]. This gyromagnetic
augmentation has been demonstrated for nearby 13C nuclei [28] and the
intrinsic 14N nuclear spin hosted within the NV defect, enabling fast
manipulation of nuclear spins [25, 23] while still preserving the longer
coherence times relative to the electron spin [29, 30]. Gyromagnetic
augmentation offers a means of tuning a qubit between regimes of strong and
weak interaction, while still retaining long coherence times.

More recently, substitutional nitrogen atoms in diamond (P1 centers), composed
of a nitrogen nucleus coupled to an electron spin, have been identified as a
promising platform for quantum information processing [31, 32, 33]. The
ability to control and entangle individual electron and nuclear spins of P1
centers has been demonstrated at cryogenic temperatures [24]. In NV quantum
sensing, magnetic noise from the P1 spin bath limits the coherence of the NV
itself, particularly in diamonds with high nitrogen density [34, 35, 36].
Driving the P1 bath using resonant radiofrequency (rf) fields [37, 34, 35, 38,
39] has been demonstrated to mitigate P1 dephasing and preserve NV coherence.
Such continuous dynamical decoupling also requires detailed knowledge of the
spin bath frequency spectrum, which changes significantly with magnetic field.

In this work, we use an ensemble of NV centers in diamond to experimentally
characterize the electronic augmentation of optically-dark P1 nuclear spins in
diamond, and demonstrate rapid control of the nuclear spin state at room-
temperature. We show that the augmentation of the nuclear gyromagnetic ratio
is dependent on the external magnetic field strength, enabling fast quantum
gate operations on the P1 nuclear spins at low magnetic field strength where
spin mixing is significant, and explaining the vanishing of P1 nuclear spin
effects at higher field. Detailed characterization of the P1 bath has
implications for schemes using P1 qubits in quantum information processing and
harnessing continuous dynamical decoupling to eliminate P1-induced NV
dephasing.

A schematic of our experiment is depicted in Figure 1 and is described in
greater detail in refs. [40] and 111See Supplementary Information.. A type 1b,
$\langle 111\rangle$-cut diamond sample with a $1.1\%$ natural abundance of
13C and 1 ppm N concentration is mounted on the spindle of an electric motor,
which acts in this case as a precision rotation stage. Current-carrying coils
are used to generate a variable magnetic field of up to $100$ G aligned along
the $\hat{z}$-axis. Microwaves for NV driving and rf for P1 control are
produced by wires with diameters $20\,\upmu$m and $50\upmu$m respectively,
arranged in a cross as shown in Fig. 1(a). The required rf pulses are produced
by an I/Q modulated vector signal generator 222See Supplementary information.
We use sample where coherence times are dominated by 13C spins, instead of a
nitrogen dominated sample, so that the coherence time of the NV ensemble is
long enough to allow observation of slow (several microsecond) Rabi
oscillations.

The P1 center features an electron spin ($S$ = 1/2) associated with an
unpaired electron, coupled to an $I$ = 1 14N nucleus (99.6% nat. abundance).
The P1 center exhibits a static Jahn-Teller (JT) distortion [43, 44, 45],
which results in four possible crystallographic orientations for the P1
quantization axis due to an elongation of one of the N-C bonds, as shown in
Fig. 1(c). The interaction Hamiltonian for a single P1 center is given by:

$\displaystyle\begin{split}H_{P1}={}&-\gamma_{e}\vec{\boldsymbol{B}}\cdot\vec{\boldsymbol{S}}\,-\,\gamma_{N}\vec{\boldsymbol{B}}\cdot\vec{\boldsymbol{I}}+\,A_{\parallel}S_{z}I_{z}\\\
&+A_{\perp}(S_{x}I_{x}+S_{y}I_{y})\,+\,QI_{z}^{2}\,,\end{split}$ (1)

where $\gamma_{e}/2\pi=-2.8$ MHz/G and $\gamma_{N}/2\pi=307.7$ Hz/G are the
gyromagnetic ratios of the electron and of the 14N nuclear spin, respectively.
In Eq. 1, the hyperfine interaction between the electron spin and the 14N
nucleus has been separated into an axial coupling term,
$A_{\parallel}/2\pi=114$ MHz, and a transverse component,
$A_{\perp}/2\pi=81.34$ MHz. The nuclear quadrupole coupling term, with
$Q/2\pi=-4.2$ MHz, defines the zero-field splitting between the $m_{I}=\pm 1$
and $m_{I}=0$ states. Also, $\vec{\boldsymbol{S}}=(S_{x},S_{y},S_{z})$ and
$\vec{\boldsymbol{I}}=(I_{x},I_{y},I_{z})$ are the electron and nuclear spin
operators of the P1 center, and $\vec{\boldsymbol{B}}$ is an applied magnetic
field.

We assume the magnetic field is aligned approximately along a particular NV
crystallographic orientation class (and, therefore, one of the P1 orientation
axes), which we define as the $\hat{z}$-axis. The magnetic field has therefore
a polar angle $\theta$ to the $\hat{z}$-axis, such that $\theta=0^{\circ}$ for
the P1 orientation class parallel to the magnetic field, which will be
referred to as ‘on-axis’ and $\theta=109.5^{\circ}$, for the other three
degenerate P1 orientations, denoted ‘off-axis’. The energies for the six P1
spin states are calculated from the Hamiltonian in Eq. 1 and plotted as a
function of magnetic field strength in Fig. 2(a) for both the on-axis and off-
axis orientation classes. In the high magnetic field limit, $m_{S}=-1/2,+1/2$
and $m_{I}=-1,0,+1$ are good quantum numbers, and we label the eigenstates of
$H_{P1}$ as $\ket{m_{S},m_{I}}$. At lower magnetic fields, the eigenstates of
the Hamiltonian are superpositions of electronic and nuclear spin states. To
account for this state-mixing at low fields, we identify the asymptotic
eigenstates in the high field limit and label the states $\ket{a}-\ket{f}$ in
descending order as shown in Fig. 2(a).

Figure 1:  Experimental configuration and system geometry. a) Microwave and
radiofrequency fields are generated by crossed wires to control the NV and P1
centers. The ensemble of NVs are optically prepared and read out by a confocal
microscope using a 532nm laser. b) The NV centers interact with surrounding
P1s, which exhibit a Jahn-Teller distortion that gives rise to four
orientation classes (inset). c) The energy levels of the NV center and the
orientations of the P1 centers with respect to the external magnetic field,
which is aligned to the $\langle 111\rangle$ NV axis.

We use double electron-electron resonance (DEER) spectroscopy [46] to
characterize the dark spin bath surrounding the NV centers, as depicted in
Fig. 2 (b). A spin-echo pulse sequence decouples the NV ensemble from its
quasistatic environment, with the free evolution time of the sequence fixed at
a 13C-induced revival time in order to ensure signal visibility [28]. An rf
$\pi$-pulse recouples resonant spins in the environment to the NV centers as
we sweep the frequency of the rf field. The results from a DEER experiment at
$B=$100 G are shown at the bottom of Fig. 2 (b). The expected frequencies for
all possible P1 spin transitions, calculated from Eq. (1), are also shown. We
observe six spectral features, which are consistent with the electron spin
transitions in the theoretical spectrum (i.e. $\Delta m_{S}=\pm 1$ and $\Delta
m_{I}=0$). For both theory and experiment, the deeper amplitudes of the
resonant peaks correspond to the off-axis P1s, due to increased contrast
resulting from the three degenerate orientations.

The DEER measurement was then repeated at a magnetic field of 35 G, as shown
at the top of Fig. 2 (b), in which we see the same electron spin transitions.
We also see additional spectral features, with comparable amplitudes,
corresponding to nuclear spin transitions in which the nuclear spin projection
changes ($\Delta m_{I}=\pm 1$) while the electron spin is conserved ($\Delta
m_{S}=0$). At a magnetic field of 100 G, the nuclear spin transition features
are indistinguishable from the background of the spectrum. Nuclear spectral
features were also absent from previous DEER measurements at comparable
magnetic fields [34, 33, 39, 47], which suggests that the coupling of the P1
nuclear spin to the rf field depends on the magnitude of the external magnetic
field.

Figure 2:  Energy levels and DEER measurements for the P1 spin bath. (a)
Calculated energy levels for the coupled P1 electron spin and 14N nuclear spin
as a function of magnetic field. The solid (dashed) lines represent the energy
levels for the on-axis (off-axis) P1 orientation classes. The six energy
levels for each orientation class are labeled from $\ket{a}$ to $\ket{f}$,
according to their electron and nuclear spin state in the limit
$B\rightarrow\infty$. (b) DEER measurements and pulse sequence (inset). A
spin-echo sequence is applied to the NV centers, while an rf $\pi$-pulse with
swept frequency is applied at a constant time in the sequence. DEER
measurements at 35 G (top) and 100 G (bottom). The calculated resonant
frequencies for particular transitions are displayed over the experimental
signal, color-coded by transition type: blue for single electron spin
transitions, yellow for double quantum transitions and red for nuclear spin
transitions.

To better understand the variation in rf coupling, we consider how the nuclear
spin is coupled to the P1 electron spin. The augmentation factor,
$\alpha\equiv\gamma_{\text{N,eff}}/\gamma_{\text{N,bare}}$, has been
previously measured for a 14N nuclear spin coupled to the NV electron spin in
the presence of a magnetic field that is parallel to the N-V axis [25, 26] and
recently for off-axis field in the same system [30]. Here, the derivation of
the augmentation factor is extended to the coupled electron and nuclear spin
of the P1 center, and includes the case in which the magnetic field is not
aligned with the electron spin quantization axis in order to account for the
different orientation classes of the P1 ensemble.

The contribution of the transverse hyperfine coupling, parametrized by
$A_{\perp}$, dominates the P1 Hamiltonian at low magnetic fields. Hence, the
effective nuclear gyromagnetic ratio is expected to increase towards lower
field, where state mixing is most apparent. The amplitude of the DEER signal
is indicative of the strength of the dipolar coupling between the NV and the
P1 spin. As the effective nuclear gyromagnetic ratio reduces with increasing
field, so does the amplitude of the DEER signal, as we observe in the
difference between DEER measurements at 35 G and 100 G, presented in Fig. 2.
Following Ref. [25], the Hamiltonian in Eq. 1 can be diagonalized using
$E=P^{-1}\hat{H}P$, where $E$ is a diagonal matrix of energy eigenvalues and
$P$ is the matrix of the eigenvectors of the Hamiltonian. We compute the
matrix elements describing the coupling between states in the basis where
$\hat{H}$ is diagonal by transforming the rf Hamiltonian,
$\hat{H}_{\text{rf}}=B_{\text{rf}}(t)(\gamma_{E}S_{x}+\gamma_{N}I_{x})$, into
the diagonal basis. The time-dependence of $\hat{H}_{\text{rf}}$ is eliminated
using the rotating-wave approximation. We can then compute the augmentation
factor for a given nuclear spin transition, that is:

$\alpha_{a,b}=\frac{1}{\gamma_{N}B_{\text{rf}}}\bra{a}\tilde{H}_{\text{rf}}\ket{b}$
(2)

for transition $\ket{a}\rightarrow\ket{b}$, where $B_{\text{rf}}$ is the
amplitude of the oscillating field. The $\tilde{H}_{\text{rf}}$ term denotes
the interaction Hamiltonian in the diagonal basis defined by $P$, which
implicitly contains within it the field-dependence of this expression.

Figure 3:  Nuclear spin Rabi oscillations and the fitted Rabi frequencies and
amplitudes as a function of magnetic field. (a) Nuclear Rabi oscillations for
the off-axis transition $de$. An exponentially-damped sinusoid (solid line) is
fitted to the experimental data (points). (b) Nuclear Rabi oscillations for
transition $de$ for the off-axis orientations (top) and on-axis (bottom) for
increasing magnetic field strength. The fitted normalized Rabi frequencies and
amplitudes (inset) for these oscillations are shown in (c) together with the
calculated augmentation factor for these transitions. (d) The normalized Rabi
frequencies and amplitudes (inset) for transition $ab$ with the corresponding
theoretical augmentation curve. The error bars in the data are the standard
error in the fitted damped sinusoidal frequencies and amplitudes.

We then characterized the coupling between the P1 nuclear spin and the
external rf field by measuring the P1 nuclear spin Rabi frequency as a
function of magnetic field strength. Varying the length of the resonant rf
pulse reveals Rabi oscillations of the nuclear spin (Fig. 3(a)). The spin-echo
time was fixed at 13C revivals occurring at 45-65$\upmu$s to maintain
consistent sensitivity between measurements as the magnetc field strength is
changed. Using the labeling convention from Fig. 2(a), we identify two nuclear
spin transitions, $\ket{a}\leftrightarrow\ket{b}$ and
$\ket{d}\leftrightarrow\ket{e}$, which we denote as $ab$ and $de$. These
particular transitions are well isolated from other resonant frequencies in
the system across a range of magnetic fields and can therefore be individually
addressed.

Figure 3 (b) shows the measured Rabi oscillations for the $ab$ and $de$
transitions for magnetic field strengths between 10 G and 100 G. The contrast
obtained for the off-axis measurements is approximately three times greater
than for on-axis, due to the greater population of off-axis classes. For both
the off-axis and on-axis orientations, the Rabi frequency and amplitude of the
oscillations can be seen to decrease with increasing field strength. At each
magnetic field, the resonant frequency to drive each transition is extracted
from precise frequency-domain DEER measurements using low rf power, ensuring
that the observed changes in the amplitude and frequency are not caused by
off-resonant driving. The orientation of the rf field with respect to the P1
axes has only a minor effect on the gyromagnetic augmentation 333See
Supplemental information.

We fit a damped sinusoidal function of the form
$S(t)=S(0)e^{(-t/T_{D})^{n}}\cos{(\Omega t/2)}^{2}$ to the nuclear Rabi
oscillation data, where $T_{D}$ is the effective decay time, to extract the
Rabi frequency and oscillation amplitude for a given transition and magnetic
field. The Rabi frequency is $\Omega=\alpha\,\gamma_{N}B_{\text{rf}}$, so for
a constant rf field, the Rabi frequency scales with the effective gyromagnetic
ratio of the nuclear spin as a function of the external magnetic field. The
amplitude of the oscillation is determined by the coupling between the NV
ensemble and the varying magnetic field produced by the nuclear spins. The
magnetic moment of the nuclear spins is proportional to the effective
gyromagnetic ratio and, therefore, the oscillation amplitude also scales with
the augmentation factor.

The fitted frequencies and amplitudes of the Rabi oscillations for transition
$de$ are presented in Fig. 3(c,e), together with the normalized augmentation
factor calculated from Eq. 2. In order to effectively compare the results
across different P1 transitions, both the Rabi frequencies and amplitudes were
normalized with respect to the maximum values, with $\Omega_{\text{max}}=1.82$
MHz and $S(0)_{\text{max}}=0.012$ observed at 20 G. The off-axis and on-axis
data were then scaled according to the fitted frequencies and amplitudes
measured for each transition at 35 G. There is excellent agreement between the
measured Rabi frequencies and the theoretical values for the augmentation
factor of the nuclear spin gyromagnetic ratio. This agreement demonstrates
that the augmented gyromagnetic ratio of the 14N nuclear spins is well
explained by the hyperfine mixing with the P1 electron spin, which decreases
with increasing magnetic field.

The analysis of the nuclear Rabi oscillations for transition $ab$ (Fig.
3(d,f)) also shows an overall decrease in the frequency and amplitude with
increasing magnetic field, which is well modeled by the augmentation curve for
this transition. Given the low amplitude for the off-axis measurements for
transition $ab$, it was not possible to obtain statistically significant
measurements for the on-axis orientation for this particular transition.
Furthermore, certain magnetic field values were excluded from the analysis for
each transition where the resonant frequency at that field strength was too
close to other transition frequencies to individually resolve. The results
from our previous DEER measurements (Fig. 2) indicated that the nuclear spin
features disappear from the observed signal as the magnetic field is
increased. This observation is confirmed by the measured Rabi oscillation
amplitude, which decrease markedly as the magnetic field is increased from 10
G to 100 G.

We have shown that the gyromagnetic ratio of the P1 nuclear spin is augmented
by hyperfine coupling to the electron spin, and that the augmentation of the
nuclear gyromagnetic ratio can be tuned by the external magnetic field. At low
magnetic fields; that is, $<100$G, the augmentation factor increases sharply
and we are able to perform rapid quantum control of the nuclear spin state, at
Rabi frequencies comparable to the P1 or NV electron spin.

A remaining open question, the subject of further work, concerns the
consequences of gyromagnetic augmentation on the coherence of the P1 nuclear
spins. As more electron spin character is mixed into the nuclear spin, the
coupling to not only rf fields increases (as studied in this work) but also to
magnetic field noise which suppresses spin coherence. However, there is reason
to expect the augmented nuclear spins to still possess coherence times
significantly exceeding that of the P1 or NV electron spin if not the bare
nuclear spin coherence time. The NV nuclear spin augmentation factor is around
$\alpha=20$ at 500 G, but the response to static magnetic fields is still set
by the bare value of $\gamma_{N}$. As recently reported in Ref [24], augmented
nuclear spin transitions at low field possess coherence times a factor of 4
greater than that of the P1 electron spin. Furthermore, rapid state control
and long spin storage can potentially be combined with high speed magnetic
shuttling to dynamically tune the nuclear spin interaction with control
fields. In this work we examined a thermally distributed ensemble of P1 spins,
but with augmented driving, rapid hyperpolarisation of the P1 nuclear spins
may allow substantial enhancement, which is a possible avenue of future study.

Our work also shows explicitly that the NV center is sensitive to the P1
nuclear spin state at weak magnetic fields, adding further channels for
decoherence in NV quantum sensing. The detection of nuclear Rabi oscillations
in the NV spin-echo signal implies that the P1 nuclear spins are a significant
component in the interacting environment of the NV and will contribute to the
dephasing of NV spins at low magnetic fields. Suppression of NV dephasing
through spin-bath control and dynamical decoupling schemes [34, 39] could,
therefore, be enhanced at lower magnetic fields by addressing the nuclear
spins in addition to the P1 electron spins. At higher densities, the dipolar
coupling between the NV and P1 electrons may be used to indirectly couple the
associated nuclear spins, suggesting an interesting platform for quantum
information processing [49].

In conclusion, we have characterized the magnetic-field induced augmentation
of the gyromagnetic ratio for the P1 nuclear spin. We have found that at lower
magnetic field magnitudes, we are able to enact fast quantum control of the
nuclear spins within the defect. Tunable magnetic augmentation establishes a
potential path forward for using the P1 nuclear spin as a qubit in quantum
applications, by addressing the challenge of accessing and controlling nuclear
spins that are typically well isolated from external fields.

## Acknowledgements

We thank R. E. Scholten for a careful reading of the manuscript and
constructive discussions. This work was supported by the Australian Research
Council Discovery Scheme (DP190100949). R. M. G. was supported by a Research
Training Program Scholarship.

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