††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# All-Optical Nuclear Quantum Sensing using Nitrogen-Vacancy Centers in
Diamond

B. Bürgler  Department of Physics, University of Basel, Klingelbergstrasse 82,
Basel CH-4056, Switzerland  T. F. Sjolander  Department of Physics, University
of Basel, Klingelbergstrasse 82, Basel CH-4056, Switzerland  O. Brinza
Laboratoire des Sciences des Procédés et des Matériaux, LSPM, CNRS-UPR 3407,
Université Sorbonne Paris Nord, 99 Avenue JB Clément, Villetaneuse 93430,
France  A. Tallaire  Laboratoire des Sciences des Procédés et des Matériaux,
LSPM, CNRS-UPR 3407, Université Sorbonne Paris Nord, 99 Avenue JB Clément,
Villetaneuse 93430, France  Institut de Recherche de Chimie Paris, CNRS,
Chimie ParisTech, Université PSL, 11 rue Pierre et Marie Curie, 75005 Paris,
France  J. Achard  Laboratoire des Sciences des Procédés et des Matériaux,
LSPM, CNRS-UPR 3407, Université Sorbonne Paris Nord, 99 Avenue JB Clément,
Villetaneuse 93430, France  P. Maletinsky<EMAIL_ADDRESS>Department of Physics, University of Basel, Klingelbergstrasse 82, Basel
CH-4056, Switzerland

###### Abstract

Solid state spins have demonstrated significant potential in quantum sensing
with applications including fundamental science, medical diagnostics and
navigation. The quantum sensing schemes showing best performance under ambient
conditions all utilize microwave or radio-frequency driving, which poses a
significant limitation for miniaturization, energy-efficiency and non-
invasiveness of quantum sensors. We overcome this limitation by demonstrating
a purely optical approach to coherent quantum sensing. Our scheme involves the
15N nuclear spin of the Nitrogen-Vacancy (NV) center in diamond as a sensing
resource, and exploits NV spin dynamics in oblique magnetic fields near the
NV’s excited state level anti-crossing to optically pump the nuclear spin into
a quantum superposition state. We demonstrate all-optical free-induction decay
measurements - the key protocol for low-frequency quantum sensing - both on
single spins and spin ensembles. Our results pave the way for highly compact
quantum sensors to be employed for magnetometry or gyroscopy applications in
challenging environments.

##  I Introduction

Spin based quantum sensors can be employed to measure a wide range of relevant
physical quantities, including magnetic [1] or electric fields [2],
temperature [3], or rotary motion [4]. This abundance of potential
observables, combined with their high sensitivity at the nano-scale makes
quantum sensors highly interesting for many fields of application, such as
life sciences [5], geological sciences [6, 7], navigation [8] and material
sciences [9].

Nitrogen-Vacancy (NV) centers in diamond (Fig. 1a) are a particularly
promising platform for such spin based quantum sensing applications, because
they host a single electron spin [10] with long coherence times [11, 12] even
at room temperature [13]. Upon optical excitation with green light, the NV
center emits spin-dependent red photoluminescence (PL) [14], which enables
all-optical electron spin readout. At the same time, such optical excitation
pumps the NV electron spin [15, 16] into a specific spin eigenstate, enabling
all-optical spin initialization. Time-varying (AC) driving fields, mostly in
the microwave (MW) or radio-frequency (RF) domain can then be used to
coherently control the spin, and create superposition states for sensing. This
combination of optical initialization, readout, and coherent spin manipulation
by AC driving fields form the basis of almost all established spin based
approaches to sensing [17].

The NV electron spin is inherently coupled to the nuclear spin of its Nitrogen
atom – a spin which exhibits significantly longer coherence times compared to
the NV electron spin [18, 19] and therefore provides another interesting
resource for quantum technology applications. Specifically, nuclear spins have
been exploited as a quantum register for quantum communication [20] and
enhanced spin readout techniques [21], but they also offer interesting
opportunities for sensing, be it for magnetometry [18, 22, 23, 24] or for
gyroscopy [25, 26].

Many Nitrogen spin based quantum sensing schemes in diamond rely on the
resonant coupling of the NV spin and the nuclear Nitrogen spin at a magnetic
field of about $500~{}$G [27, 28], where spin flip-flop processes occur in the
NV’s orbital excited state at the excited state level anti-crossing (ESLAC).
It has been shown that optical pumping in the vicinity of the ESLAC results in
nuclear spin hyperpolarization [29, 30], and that – by virtue of the same
mechanism – the NV center shows a nuclear spin-state dependent rate of
transient PL [28]. As a result, optical pumping close to the ESLAC enables
both all-optical readout of the nuclear spin state and initialization into a
nuclear spin eigenstate, which, together with RF driving, forms the basis for
nuclear spin based sensing schemes [28].

However, the ubiquitous need for AC coherent driving in spin based quantum
sensing is a severe limitation for many applications. Specifically, such AC
driving fields can adversely affect investigated samples and for integrated or
portable sensing devices, their delivery increases power-consumption and
overall complexity, and thereby size and cost of the system. Recent
experiments have demonstrated microwave-free NV magnetometry schemes that are
based on sharp changes in NV PL at level anti-crossings, that occur at
specific magnetic fields [31, 32]. While avoiding the need for MW or RF
delivery, these approaches do not exploit quantum coherence (and are therefore
limited in sensitivity) and are furthermore highly vulnerable to background
drifts in NV PL.

Here, we present a novel method for coherent, microwave-free quantum sensing
using the 15N nuclear spin of the NV center in diamond. Our approach is based
on optical driving of the NV center near the ESLAC in the presence of a small,
static magnetic field transverse to the NV symmetry axis denoted by the unit
vector $\bm{e}_{z}$ (see Fig. 1a). As we will show, such a small transverse
magnetic field component has a striking effect, in that optical pumping
prepares the 15N nuclear spin in a coherent superposition state within the
NV’s ground state spin manifold. Following optical pumping, this
initialization leads to Larmor precession of the nuclear spin about an
effective magnetic field; a precession we directly monitor via nuclear spin-
state dependent PL [28]. Figure 1b shows an example of such an all-optical
nuclear free induction (FID) measurement, obtained using the pulse sequence
depicted in Fig. 1c. Data for this work were recorded on a home-built confocal
optical microscope (see methods) with magnetic field control; here at a
magnetic field of strength $|\bm{B}_{\rm ext}|=540~{}$G, tilted by
$\Phi=1^{\circ}$ away from $\bm{e}_{z}$.

The negatively charged NV center possesses an electron spin $S=1$ quantized
along the NV symmetry axis $\bm{e}_{z}$. For NV centers formed by 15N (denoted
as “15NV” in the following), the Nitrogen nucleus exhibits a spin $I=1/2$. The
Hamiltonian for the orbital ground (gs) and excited state (es) of such a 15NV
can be expressed as

$\frac{\hat{H}^{\rm gs,es}}{h}=D_{0}^{\rm
gs,es}\hat{S}_{z}^{2}+\hat{\bm{S}}\cdot\bm{A}^{\rm
gs,es}\cdot\hat{\bm{I}}+\gamma_{S}\bm{B}_{\rm
ext}\cdot\hat{\bm{S}}+\gamma_{I}\bm{B}_{\rm ext}\cdot\hat{\bm{I}}\,,$ (1)

where $\hat{\bm{S}}$ and $\hat{\bm{I}}$ are the NV electron and nuclear spin
operators, $\gamma_{S}=2.8~{}$MHz/G and $\gamma_{I}=431.7~{}$Hz/G are the
respective gyromagnetic ratios, $D_{0}^{\rm gs}=2.87~{}$GHz and $D_{0}^{\rm
es}=1.42~{}$GHz are the zero-field splittings, and $\bm{B}_{\rm ext}$ is the
applied magnetic field. The hyperfine coupling tensor $\bm{A}^{\rm gs,es}$ has
two independent components $A_{\parallel}^{\rm gs}=3.03~{}$MHz and
$A_{\perp}^{\rm gs}=3.65~{}$MHz for the ground state; and $A_{\parallel}^{\rm
es}=-57.8~{}$MHz, $A_{\perp}^{\rm es}=-39.2~{}$MHz for the excited state [33,
34]. This Hamiltonian is conveniently expressed in a basis of spin eigenstates
$\\{\ket{m_{S},m_{I}}\\}$, where $m_{S}$ and $m_{I}$ are the magnetic quantum
numbers associated with $\hat{S}_{z}$ and $\hat{I}_{z}$. The coherent FID
dynamics that are studied in this work (Fig. 1b) can be completely encompassed
by a reduced subspace spanned by $\\{\ket{0,-1/2},\ket{0,+1/2}\\}$.

##  II Results

Figure 1:  a Crystal structure of the Nitrogen-Vacancy center with
illustration of its associated spins and coordinate axes. b All-optical
nuclear spin precession of the 15N Nuclear spin observed at a magnetic field
of $|\bm{B}|=540~{}$G tilted away from the NV symmetry axis by
$\Phi=1^{\circ}$. Fitting of a harmonic function (black) yields a precession
frequency $251.18\pm 0.12~{}$kHz. c Pulse sequence employed for b, consisting
of a $3~{}$µs green laser pulse separated by a variable delay $\tau$. The
first $350~{}$ns of each green pulse are utilized for optical nuclear spin
readout, while the remainder of the pulse reinitializes the spin system. d
Quantitative Bloch-Sphere representation of the 15N spin in the
$\ket{m_{s}=0}$ ground state manifold. For a magnetic field $\bm{B}_{\rm ext}$
tilted from the NV symmetry axis by the angle $\Phi=1^{\circ}$, optical
pumping initializes the nuclear spin into $\hat{\rho}_{\rm init}^{\rm nuc}$.
The nuclear spin subsequently precesses around an effective magnetic field
$\bm{B}_{\rm eff}$. The measurement axis for all-optical readout of this
precession is given by $\bm{e}_{\rm init}$. e Experimentally observed
precession frequency (blue crosses) at different values of $|\bm{B}_{\rm
ext}|$ and fixed $\Phi=1^{\circ}$, together with analytic (solid orange line)
and numerical predictions (black dots).

Derivation of an effective Hamiltonian. In the following we explain the
nuclear precession data presented in Fig. 1b by discussing how the presence of
the transverse magnetic field component $B_{\perp}$ significantly affects the
system’s optical pumping and subsequent FID dynamics. We start by calculating
an effective Hamiltonian for the 15N spin in the $m_{S}=0$ ground state
subspace, $\\{\ket{0,-1/2},\ket{0,+1/2}\\}$, using Van Vleck perturbation
theory [35] (see the supplementary information, Section I, for a detailed
derivation). Without loss of generality, we set the transverse magnetic field
to point along the unit vector $\bm{e}_{x}$. We then obtain the effective
Hamiltonian

$\displaystyle\frac{H^{m_{S}=0}_{\rm{eff}}}{h}=\frac{1}{2}\begin{bmatrix}\gamma_{I}B_{z}+\nu_{z}&\hskip
5.69046pt\gamma_{I}B_{\perp}+\nu_{\perp}\vspace{0.2cm}\\\
\gamma_{I}B_{\perp}+\nu_{\perp}&\hskip
5.69046pt-\gamma_{I}B_{z}-\nu_{z}\end{bmatrix},$ (2)

where

$\nu_{z}=\frac{\gamma_{S}B_{z}(A_{\perp}^{\rm gs})^{2}}{(D_{0}^{\rm
gs})^{2}-(\gamma_{S}B_{z})^{2}}$ (3)

denotes the correction to the diagonal elements caused by mixing between
states of different $m_{S}$, and

$\nu_{\perp}=\frac{-2\,\gamma_{S}B_{\perp}A_{\perp}^{\rm gs}D_{0}^{\rm
gs}}{(D_{0}^{\rm gs})^{2}-(\gamma_{S}B_{z})^{2}}$ (4)

is the corresponding correction to the off-diagonal elements. We note that
such an effective hyperfine Hamiltonian for 15NV’s has been discussed earlier
as a perturbation in $B_{\perp}$ in the limit $B_{z}\ll D_{0}^{\rm gs}$ [36],
or as a perturbation in $A_{\perp}^{\rm gs}$ [28], but never as a perturbation
in both simultaneously as we present it here. Additionally, exact analytic
expressions for $\nu_{\perp}$ have previously been derived in [37].

Hamiltonian $H^{m_{S}=0}_{\rm{eff}}$ yields that the 15N nuclear spin in the
NV ground state is quantized along an effective magnetic field
$\bm{B}_{\rm{eff}}$. Diagonalization of $H^{m_{S}=0}_{\rm{eff}}$ yields

$\gamma_{I}|\bm{B}_{\rm{eff}}|=\sqrt{\left(\gamma_{I}B_{z}+\nu_{z}\right)^{2}+\left(\gamma_{I}B_{\perp}+\nu_{\perp}\right)^{2}}=:f_{\rm
nuc}\,,$ (5)

where $f_{\rm nuc}$ is the expected nuclear spin precession frequency.
Importantly, $\bm{B}_{\rm{eff}}$ is neither aligned with $\bm{B}_{\rm ext}$
nor with the NV symmetry axis. Instead, it lies in the plane spanned by
$\bm{B}_{\rm{eff}}$ and $\bm{B}_{\rm ext}$, and is tilted away from
$\bm{e}_{z}$ by an angle
$\vartheta=\tan^{-1}\left[(\gamma_{I}B_{\perp}+\nu_{\perp})/(\gamma_{I}B_{z}+\nu_{z})\right]$.
Interestingly, $\vartheta$ is significantly larger than the misalignment angle
$\Phi$ between $\bm{B}_{\rm ext}$ and $\bm{e}_{z}$, and $\vartheta$ has a sign
opposite to $\Phi$ due to the negative sign of $\gamma_{I}$ (c.f. Fig. 1d).
Finally, note that according Eq. (4), $\nu_{\perp}=0$ when $B_{\perp}=0$,
which in turns causes $\vartheta=0$. In this case, both $\bm{B}_{\rm eff}$ and
$\bm{B}_{\rm ext}$ are aligned with the NV symmetry axis.

Analysis of 15N spin dynamics. We now discuss how the presence of $B_{\perp}$
affects the 15N nuclear spin dynamics and enables all-optical initialization
into a nuclear spin superposition state. The use of 15N is key to this, since
it does not have a quadrupolar spin splitting which would prevent any nuclear
Larmor precession. It is clear from Eq. (4) that when $B_{\perp}\neq 0$, the
15N nuclear quantization axis depends sensitively on the hyperfine coupling
parameter $A_{\perp}$, and the splitting of the involved spin-levels.
Therefore, the ground and excited state nuclear spin quantization axes are in
general different, because the hyperfine parameters differ in both magnitude
and sign between the two cases. This difference in 15N quantization axes
results in optical pumping of the nuclear spin into a state that does not
correspond to an eigenstate of the effective ground state Hamiltonian
$H^{m_{S}=0}_{\rm{eff}}$. To be specific, optical pumping accumulates NV
excited state population in the eigenstate with the largest $m_{S}=0$
character (i.e. the state $\ket{\psi}$ for which
$\bra{\psi}\hat{\bm{S}}_{z}/\hbar\ket{\psi}$ is closest to zero), since this
state has the lowest probability of shelving into the NV’s singlet state – we
denote this state as $\ket{\tilde{0}_{\rm es}}$. By the same argument,
$\ket{\tilde{0}_{\rm es}}$ is also the “brightest” state in that it yields the
largest rate of emission of NV fluorescence photons. For state
$\ket{\tilde{0}_{\rm es}}$, the expectation value of the nuclear spin lies
along the vector $\bm{e}_{\rm init}=\bra{\tilde{0}_{\rm
es}}\hat{\bm{I}}/\hbar\ket{\tilde{0}_{\rm es}}$. Vector $\bm{e}_{\rm init}$
therefore defines both the direction along which the nuclear spin is
initialized under green illumination, as well as the measurement axis for
optical readout of the nuclear spin. For $B_{\perp}\neq 0$, $\bm{e}_{\rm
init}$ is not collinear with $\bm{B}_{\rm{eff}}$, and thus optical pumping
will initialize the 15N nuclear spin in a state that is not an eigenstate of
$H^{m_{S}=0}_{\rm{eff}}$. Disengaging green laser excitation after optical
pumping will therefore result in precession of the 15N nuclear spin around
$\bm{B}_{\rm{eff}}$. Finally, note that for NV centers formed with 14N, no
Larmor precession occurs because its quadrupolar splitting locks
$\bm{B}_{\rm{eff}}$ onto the NV axis.

Comparison with numerical results. To further verify this picture, we
developed a numerical model that simulates the dynamics of the 15NV system
during and after optical pumping. The model is based on classical rate
equations for the optical pumping process [16], coupled with master equations
describing the quantum-mechanical evolution of the system’s density matrix
within each relevant orbital manifold: the orbital ground and excited states
as well as the singlet state (see further details in the supplementary
information, Section II.1).

In Fig. 1d we summarize the numerical and theoretical results in a Bloch
sphere representation of the 15N nuclear spin dynamics for the same magnetic
field that was used to obtain the experimental results in Fig. 1b. The
effective field $\bm{B}_{\rm eff}$ is calculated numerically through exact
diagonalisation of the ground state Hamiltonian $\hat{H}^{\rm gs}$, and
$\bm{e}_{\rm init}$ is calculated via diagonalization of the excited state
Hamiltonian $\hat{H}^{\rm es}$. The nuclear spin density matrix
$\hat{\rho}_{\rm init}^{\rm nuc}$ following optical pumping is obtained by
propagating the system density matrix $\hat{\rho}$ for $3$ µs of laser
excitation, followed by a $50~{}$ns dark time (to let the system relax fully
to the ground state), and by subsequently taking the trace over the NV
electron spin degrees of freedom. The nuclear spin precession dynamics is then
described by propagating $\hat{\rho}$ under the influence of $\hat{H}^{\rm
gs}$.

We make two observations that underline the excellent agreement of this
numerical model with our analytical discussion. First, the orientation of
$\bm{B}_{\rm eff}$ obtained from numeric diagonalization of the full ground
state Hamiltonian $\hat{H}^{\rm gs}$ shows perfect agreement with the
analytical prediction from Eq. (2) (see further details in the supplementary
information, Section II.3). Second, the initial nuclear spin direction
Tr$(\hat{\bm{I}}\hat{\rho}_{\rm init}^{\rm nuc})$ obtained from our numerical
model is perfectly collinear with $\bm{e}_{\rm init}$, as long as we set the
intersystem crossing rate for the $m_{S}=0$ states to zero (see the
supplementary information, Section II.3). Both observations strongly support
the validity of our model and its applicability to quantitatively describe our
data.

All-optical nuclear 15N precession The presented theory framework allows us to
further analyze the data presented in Fig. 1b. The observed FID oscillation
frequency was determined by least-squares fitting to $f_{\rm nuc}=251.18\pm
0.12~{}$kHz, in good agreement with Eq. (5), which yields $f_{\rm
nuc}=252.71~{}$kHz for the experimental conditions $|\bm{B}_{\rm ext}|=540$ G
and $B_{\perp}=10.6$ G. The small remaining discrepancy can be assigned to
uncertainties in controlling the tilt angle $\Phi$, and determining the exact
field components $B_{\perp}$ and $B_{\parallel}$. To demonstrate that the
observed oscillations indeed originate from nuclear spin precession, we
repeated the same experiment at fixed angle $\Phi$, while varying
$|\bm{B}_{\rm ext}|$. Figure 1e shows the resulting, near-linear dependance of
$f_{\rm nuc}$ on $|\bm{B}_{\rm ext}|$ and the excellent agreement with the
predictions of both Eq. (5) and the numeric model. The enhancement of $f_{\rm
nuc}$ over the bare Larmor frequency results from the terms $\nu_{z}$ and
$\nu_{\perp}$ in Eq. (5).

Figure 2:  a Measured free induction decay (FID) contrast of a single NV as a
function of magnetic field orientation, for a fixed total field of
$|\bm{B}_{\rm ext}|=533~{}$G. b Same data as in a, but plotted against total
transverse magnetic field $B_{\perp}$. The black line is the prediction of our
numerical model, which is normalized to the mean of the data at $\Phi_{\rm
opt}\pm 0.03^{\circ}$. c Nuclear FID contrast as a function of magnetic tilt
angle $\Phi$ and total magnetic field $|\bm{B}_{\rm ext}|$. d Maximal observed
contrast $C_{\rm max}$ and corresponding tilt angle $\Phi_{\rm opt}$ for each
value of $|\bm{B}_{\rm ext}|$. The prediction of our numerical model is shown
in black, which is normalized to the maximal data point.

Angle ($\Phi$) and field ($|\bm{B}_{\rm ext}|$) dependance of the all-optical
15N FID signal. The requirement of applying a transverse magnetic field
$B_{\perp}$ to obtain an observable, all-optical 15N FID signal motivates the
question of how the FID readout contrast $C$ depends on both $\Phi$ and
$|\bm{B}_{\rm ext}|$. Figure 2a shows single NV data, where we determined $C$
as a function of $\Phi$ for a fixed field $|\bm{B}_{\rm ext}|=533~{}$G. We
determined $C$ from the Fourier space amplitude of individual FID curves, for
varying field misalignment angles $\Phi_{x}$ and $\Phi_{y}$, applied in the
$x$-$z$ and $y$-$z$-planes, respectively. For the small angles we
investigated, $\Phi\approx(\Phi_{x}^{2}+\Phi_{y}^{2})^{1/2}$. As expected,
when $\Phi=0$, no nuclear FID contrast is observed, because in this case
$\bm{B}_{\rm eff}$ and $\bm{e}_{\rm init}$ are both collinear with
$\bm{e}_{z}$ such that the nuclear spin is optically pumped into the non-
precessing eigenstate $\ket{0,+1/2}$ of $\hat{H}^{\rm gs}$. Upon increasing
$\Phi$, $\bm{B}_{\rm eff}$ and $\bm{e}_{\rm init}$ both tilt away from
$\bm{e}_{z}$ in different directions, resulting in nuclear precession of
increasing contrast $C$. At the same time, increasing $\Phi$ (i.e.
$B_{\perp}$) tends to reduce the nuclear hyperpolarization efficiency [29, 28]
and the NV optical spin readout contrast [16], both of which reduce $C$.
Overall, these counteracting effects imply that there is an optimal tilt angle
$\Phi_{\rm opt}$ which maximizes $C$ by balancing the magnitude of the nuclear
spin coherences with nuclear spin readout efficiency. We call this maximized
contrast $C_{\rm max}$.

To determine $\Phi_{\rm opt}$ and $C_{\rm max}$, we show in Fig. 2b the data
from Fig. 2a as a function of transverse field $B_{\perp}$, where for each
data point, we determined $B_{\perp}$ from the NV’s full optically detected
magnetic resonance spectrum. Figure 2b reveals a clear maximum in $C$ at
$B_{\perp}\approx 8.6~{}$G, which corresponds to $\Phi_{\rm opt}\approx
0.86^{\circ}$ (see further details in the supplementary information, Section
III). These results are in good agreement with the predictions of our
numerical model (black curve in in Fig. 2b). The quality of the simulation
depends sensitively on the NV intersystem crossing rates, all of which were
kept fixed to literature values [16] in our calculations (see further details
in the supplementary information, Section II.3). We assign remaining
discrepancies between data and simulations to uncertainties on optical
transition rates employed in the model.

Interestingly, we find that our all-optical 15N nuclear FID protocol is
relatively resilient to deviations of $\bm{B}_{\rm ext}$ away from ideal ESLAC
conditions. For this, we investigated the dependance of the contrast $C$ on
the applied magnetic field $|\bm{B}_{\rm ext}|$ and tilt angle $\Phi_{x}$,
where for each data point, we ensured that $\Phi_{y}=0$ is maintained to
within experimental accuracy. The resulting data show a nontrivial dependance
of $C$ on $|\bm{B}_{\rm ext}|$ and $\Phi_{x}$ (Fig. 2c), and in particular
reveal that both $\Phi_{\rm opt}$ and $C_{\rm max}$ change with $|\bm{B}_{\rm
ext}|$ (Fig. 2d). These dependencies are qualitatively captured by our
numerical model. We find a global maximum of $C_{\rm max}\approx 4.2~{}\%$ for
$|\bm{B}_{\rm ext}|=533~{}$G, and a drop of $C$ over a full-width at half
maximum (FWHM) range of $\sim 50$ G.

Figure 3:  a 15N nuclear precession frequency $f_{\rm nuc}$ corresponding to
the data shown in Fig. 2a. Only pixels for which $C>1\%$ are shown. b Same
data as in a, plotted against total transverse magnetic field $B_{\perp}$,
together with the numerical model prediction (black), the prediction from Eq.
(5) (orange), and the bare Larmor frequency $\gamma_{I}B_{z}$ (green).

Nuclear spin precession frequency. Further, we investigate the dependance of
$f_{\rm nuc}$ on the magnetic field tilt angles $\Phi_{x}$ and $\Phi_{y}$. For
this, we determine $f_{\rm nuc}$ by Fourier analysis of the FID data for each
data point sampled in Fig. 2a. The result is shown in Fig. 3a, along with the
corresponding plot of the same data as a function of $|B_{\perp}|$ in Fig. 3b.
The precession frequency $f_{\rm nuc}$ increases with $B_{\perp}$ in a way
that is excellently described by both Eq. (5) and our numerical model. We
again assign small discrepancies between measured and predicted values of
$f_{\rm nuc}$ to experimental uncertainties in determining $\bm{B}_{\rm ext}$.

Figure 4:  All-Optical nuclear spin precession of a, a single NV in a diamond
nanopillar, and b, an ensemble of NVs in bulk diamond, both measured at
$|\bm{B}_{\rm ext}|=533~{}$G with tilt angles $\Phi=0.65^{\circ}$ and
$\Phi=0.8^{\circ}$, for a and b respectively. Each data set is fitted with a
damped harmonic function to determine the nuclear spin coherence time
$T_{2}^{*}$, yielding $T_{2}^{*}=248.1\pm 12.4~{}$µs and $T_{2}^{*}=508.5\pm
17.4~{}$µs, for a and b respectively.

Nuclear coherence time. The ease of use of our all-optical 15N FID experiments
enables facile assessment of the 15N inhomogeneous nuclear spin coherence time
$T_{2}^{*}$. To determine $T_{2}^{*}$, we extend the measurement pulse
sequence shown in Fig. 1c to longer FID evolution times $\tau$, to resolve the
full decay of the signal (Fig. 4a). Fitting an exponentially decaying harmonic
function to the data yields $C=3.09\pm 0.11~{}$% and $T_{2}^{*}=248.1\pm
12.4~{}$µs for the single NV center under investigation. This decoherence time
is somewhat shorter than previously reported values [38], but consistent with
the rather short NV electron spin relaxation time of $T_{1}=315\pm 16~{}$µs
for our shallow NV – a timescale which is known to limit the NV’s nuclear spin
decoherence time [39, 40].

Scaling to NV ensembles An interesting question with particular relevance for
potential applications in quantum sensing is whether our all-optical scheme
also scales to ensembles of NV centers. To address this question, we repeated
our experiments on an ensemble of NV centers in an unstructured, CVD grown
diamond sample with a preferential orientation of NV centers along one of the
four possible crystal directions [41] (see methods). We maximize the contrast
$C$ in the same fashion as done in the single NV case, and determine
$T_{2}^{*}$ through the full FID trace shown in Fig. 4b. Using a least-squares
fit as before, we find $C=2.08\pm 0.04$%, and $T_{2}^{*}=508.5\pm 17.4~{}$µs
for this NV ensemble. While $C$ is comparable to the single NV case (with a
slight deterioration due to the minority of non-aligned NVs), $T_{2}^{*}$
almost doubles. This value of $T_{2}^{*}$, however, is still short of the best
reported values of $\approx 2.2~{}$ms for 15NV nuclear spin coherence times
[38, 26]. We exclude electron spin $T_{1}$ relaxation as a source for this
fast nuclear spin decoherence, as we measured $T_{1}=5.8\pm 0.5~{}$ms in this
sample. Possible other sources for nuclear spin dephasing in our ensemble
experiment include fluctuations in external magnetic field or temperature [40,
38].

##  III Discussion

Our all-optical 15N FID scheme lends itself to applications in quantum
sensing, e.g. in magnetometry and gyroscopy (rotational sensing). In the
following we discuss the predicted performance of such all-optical coherent
quantum sensing schemes.

The shot noise limited FID sensitivity for spin based, low-frequency
magnetometry is given by [42, 17, 43]

$\eta_{\rm mag}\approx\frac{1}{\gamma C\sqrt{NT_{2}^{*}}}\,.$ (6)

Here, $N$ is the average number of detected photons per readout pulse, $C$ is
the readout contrast, and $\gamma$ is the gyromagnetic ration of the spins
employed for sensing. Further sensitivity reductions due to overhead in
preparation and measurement of quantum states are not included in this
expression, but of little relevance to our conclusion, given the long
$T_{2}^{*}$ times at hand.

Evaluating Eq. (6) for our single NV data ($T_{2}^{*}=250~{}$µs, $C=4\%$,
$N=0.1$) using the effective nuclear gyromagnetic ratio $\gamma=1.2\gamma_{I}$
determined from the slope of the data in Fig. 1e (or from Eq. (5)), we obtain
a photon shot noise limited magnetometry sensitivity of $\eta_{\rm
mag}=154~{}\text{\textmu T}/\sqrt{\text{Hz}}$. Further, given that our
approach scales to NV ensembles, we make predictions on future ensemble NV
magnetometry sensitivity. For this, we assume a laser power of $100~{}$mW, a
$350~{}$ns readout window, and a conversion ratio of excitation photons to
detected PL photons of $3.4~{}$% [44], to obtain $N=3.2\cdot 10^{9}$. Together
with the measured ensemble values $T_{2}^{*}=500~{}$µs and $C=2~{}\%$, we
obtain $\eta_{\rm mag}^{\rm ensemble}=1.22~{}$nT$/\sqrt{\rm Hz}$.

For spin based gyroscopy, the sensitivity is determined in analogy to Eq. (6),
but with omission of the gyromagnetic ratio, i.e. $\eta_{\rm
gyro}=\gamma\cdot\eta_{\rm mag}$ [28]. Nuclear spins are therefore
particularly attractive for gyroscopy, since their long $T_{2}^{*}$ times
generally offer them better sensitivities compared to electron spins, while
they are less susceptible to magnetic fields and their fluctuations. Employing
the same procedure as before, we obtain a projected ensemble gyroscope
sensitivity of $\eta_{\rm gyro}^{\rm ensemble}\approx 135~{}^{\circ}/\sqrt{\rm
hour}$.

To place these estimates in context, we note that best reported magnetometry
sensitivities using electron spin ensembles in diamond were $\tilde{\eta}_{\rm
mag}^{\rm ensemble}<10$ pT$/\sqrt{\rm Hz}$ [45], while NV-based nuclear spin
gyroscopes have recently achieved sensitivities $\tilde{\eta}_{\rm gyro}^{\rm
ensemble}=280~{}^{\circ}/\sqrt{\rm hour}$ [26]. While for magnetometry, our
projected sensitivity alone is not competitive with the state-of-the-art, the
microwave-free modality we present still lends itself to specific
applications, e.g. remote sensing through optical fibres [46], or for cases
where the MW drive would critically affect the sample of interest. Conversely,
for gyroscopy, we project numbers competitive with previous approaches. The
added feature of all-optical NV gyroscopy is hereby a key asset, which may
enable future integrated and power-efficient NV gyroscopes.

Looking forward, we note that our all-optical nuclear spin sensing scheme is
also amenable to alternate high fidelity readout schemes to increase
measurement contrast, based on spin-to-charge conversion [47, 48]. Another
potential path to improving contrast $C$ is to dynamically pulse the field
misalignment angle between spin initialisation and readout, to separately
optimise the two processes.

In conclusion, we have presented an all-optical scheme for observing FID
dynamics of 15N nuclear spins in diamond NV centers. Our technique is based on
optical pumping of the 15N nuclear spin into a quantum superposition state – a
novel optical pumping process that occurs near the NV’s ESLAC in presence of a
small transverse magnetic field. These results may find applications in
various fields of quantum sensing, most notably all-optical magnetometry and
gyroscopy, for which we give benchmark comparisons which compare favourably
with the state-of-the-art. Our results also suggest possible extensions to a
range of other, relevant scenarios, including analogous dynamics near the NV’s
ground-state level anti-crossing, or all-optical addressing of nearby
${}^{13}C$ nuclear spins. The nuclear spin dynamics we discussed should
generally be observable in any color center exhibiting suitable level anti-
crossing dynamics and coupling to nuclear spins, and might as such offer
interesting opportunities for sensing with and characterization of novel color
centers in a variety of solid state hosts.

##  IV Methods

Single NV diamond sample The majority of our experimental results (Figs. 1-3
and Fig 4a) were obtained on a single NV center that was created in an
“electronic grade” diamond sample (Element Six) by ion implantation and
subsequent sample annealing [49]. For implantation, we employed singly charged
15N ions at a flux of $10^{11}~{}$cm-2 and an energy of $6~{}$keV,
corresponding to a nominal implantation depth of about $9~{}$nm [50]. To
increase PL collection efficiency, parabolic diamond pillars were fabricated
in the diamond surface [51] subsequent to NV creation – a single pillar
containing an individual NV center was studied in this work.

NV ensemble diamond sample The NV ensemble sample used to obtain data shown in
Fig. 4b was grown on a CVD diamond substrate along the $(113)$ crystal
orientation to facilitate Nitrogen incorporation and create NVs preferentially
oriented along the NV-axis lying closest to the growth plane [41]. A $15$ µm
thick layer containing NVs was obtained using 12C and 15N enriched gas
mixture, which led to an estimated NV density of $\sim 300~{}$ppb [52].

Experimental setup A home-built confocal microscope (Olympus LMPLFLN-$100$
objective, $\text{NA}=0.8$) was used to focus a green laser (Cobolt 06-MLD;
emission wavelength $515~{}$nm) on the sample and to simultaneously collect
the emitted red PL. All data shown in this paper were taken by optically
exciting the NV(s) near saturation, which for our setup corresponded to a
laser power of about $50~{}$µW for the single NV in a nano-pillar, and
$2.2~{}$mW for the ensemble of NVs in unstructured diamond. A static magnetic
field was applied using a permanent neodymium disk magnet (supermagnete,
S-60-05-N) mounted on a linear translation stage, to tune the magnetic field
strength at the NV location. For precise magnetic field alignment near the
ESLAC, the magnet is mounted on a goniometric stage (SmarAct SGO-60.5 and
SGO-77.5). Finally, the laser and photon-detectors were gated with pulses
which were created and synchronized using a high-frequency signal generator
(Zurich Instruments SHFSG), which also served as a source for microwave pulses
used for the characterization of the magnetic field via optically detected
magnetic resonance experiment.

## Acknowledgments

We gratefully acknowledge Vincent Jacques, David Broadway, Andrey Jarmola,
Sean Lourette and Dima Budker for fruitful discussions. We further acknowledge
financial support through the NCCR QSIT (Grant No.$185902$), the Swiss
Nanoscience Institute, and through the Swiss NSF Project Grant No.$188521$.

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##  V Supplementary Information

###  V.1 I. Effective Nuclear Hamiltonian

Here, we calculate an effective Hamiltonian for the 15N spin in the electronic
$m_{S}=0$ subspace. To that end, we employ Van Vleck perturbation theory,
which is applicable to Hamiltonians $\hat{H}$ that can be written in the form
$\hat{H}=\hat{H}_{0}+\hat{V}$, where $\hat{H}_{0}$ is block diagonal, made of
distinct subspaces, and $V$ is a perturbation that couples the initially
uncoupled different subspaces of $\hat{H}_{0}$. Following the notation in
[35], to second order, the effective Hamiltonian for each individual subspace
is given by

$\displaystyle\bra{i}\hat{H}^{\alpha}_{\rm{eff}}\ket{j}=$
$\displaystyle\bra{i,\alpha}H_{0}+V\ket{j,\alpha}$
$\displaystyle+\frac{1}{2}\sum_{k,\gamma\neq\alpha}\bra{i,\alpha}V\ket{k,\gamma}\bra{k,\gamma}V\ket{j,\alpha}$
$\displaystyle\quad\times\left[\frac{1}{E_{i,\alpha}-E_{k,\gamma}}+\frac{1}{E_{k,\alpha}-E_{k,\gamma}}\right].$
(7)

Here, latin indices denote states within a given subspace, and greek indicies
count over the subspaces. Equation (7) is valid if the energy difference
between states in different blocks is much larger than the coupling between
them, $|E_{i,\alpha}-E_{j,\beta}|\gg\bra{i,\alpha}V\ket{j,\beta}$.

Hamiltonian $\hat{H}^{\rm gs}$ from the main text is such a block diagonal
Hamiltonian. Written in the basis $\\{\ket{m_{S},m_{I}}\\}$, it reads

$\tiny\frac{\hat{H}^{\rm gs}}{h}=\left[\begin{matrix}D_{0}^{\rm
gs}+\gamma_{S}B_{z}+\frac{\gamma_{I}B_{z}+A_{\parallel}^{\rm
gs}}{2}&\frac{\gamma_{I}}{2}B_{x}&\frac{\gamma_{S}}{\sqrt{2}}B_{x}&0&0&0\\\
\frac{\gamma_{I}}{2}B_{x}&D_{0}^{\rm
gs}+\gamma_{S}B_{z}-\frac{\gamma_{I}B_{z}+A_{\parallel}^{\rm
gs}}{2}&\frac{1}{\sqrt{2}}A_{\perp}^{\rm
gs}&\frac{\gamma_{S}}{\sqrt{2}}B_{x}&0&0\\\
\frac{\gamma_{S}}{\sqrt{2}}B_{x}&\frac{1}{\sqrt{2}}A_{\perp}^{\rm
gs}&+\tfrac{1}{2}\gamma_{I}B_{z}&\frac{\gamma_{I}}{2}B_{x}&\frac{\gamma_{S}}{\sqrt{2}}B_{x}&0\\\
0&\frac{\gamma_{S}}{\sqrt{2}}B_{x}&\frac{\gamma_{I}}{2}B_{x}&-\tfrac{1}{2}\gamma_{I}B_{z}&\frac{1}{\sqrt{2}}A_{\perp}^{\rm
gs}&\frac{\gamma_{S}}{\sqrt{2}}B_{x}\\\
0&0&\frac{\gamma_{S}}{\sqrt{2}}B_{x}&\frac{1}{\sqrt{2}}A_{\perp}^{\rm
gs}&D_{0}^{\rm gs}-\gamma_{S}B_{z}+\frac{\gamma_{I}B_{z}-A_{\parallel}^{\rm
gs}}{2}&\frac{\gamma_{I}}{2}B_{x}\\\
0&0&0&\frac{\gamma_{S}}{\sqrt{2}}B_{x}&\frac{\gamma_{I}}{2}B_{x}&D_{0}^{\rm
gs}-\gamma_{S}B_{z}-\frac{\gamma_{I}B_{z}-A_{\parallel}^{\rm gs}}{2}\\\
\end{matrix}\right]\normalsize$ (8)

where the blocks are defined by states of equal values of $m_{S}$, and where,
without loss of generality, we define the direction of the transverse magnetic
field as the $x$-direction, e.g. that $B_{y}=0$ and $B_{x}=:B_{\perp}$.

We now evaluate Eq. (7) for $\hat{H}^{\rm gs}=\hat{H}_{0}+\hat{V}$ with
$\hat{H}_{0}=D_{0}^{\rm
gs}\hat{S}_{z}^{2}+\gamma_{S}b_{z}\hat{S}_{z}+\gamma_{I}b_{z}\hat{I}_{z}$ and
$\hat{V}=B_{\perp}(\gamma_{I}\hat{I}_{x}+\gamma_{S}\hat{S}_{x})+\hat{\bm{S}}\cdot\bm{A}\cdot\hat{\bm{I}}$,
and $\alpha$ corresponding to the $m_{S}=0$ subspace. This leads to the
following matrix elements,

$\displaystyle\small\frac{{\bra{1}\hat{H}_{\rm eff}^{m_{S}=0}}\ket{1}}{h}=\ $
$\displaystyle\frac{+\gamma_{I}B_{z}}{2}+\frac{(A_{\perp}^{\rm
gs})^{2}}{A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}+2\gamma_{I}B_{z}-2\gamma_{S}B_{z}}$
$\displaystyle+\frac{4\gamma_{S}^{2}B_{\perp}^{2}D_{0}^{\rm
gs}}{(A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}+2\gamma_{S}B_{z})(A_{\parallel}^{\rm gs}+2D_{0}^{\rm
gs}+2\gamma_{S}B_{z})}\ ,$ (9) $\displaystyle\frac{{\bra{2}\hat{H}_{\rm
eff}^{m_{S}=0}}\ket{2}}{h}=\ $
$\displaystyle\frac{-\gamma_{I}B_{z}}{2}+\frac{(A_{\perp}^{\rm
gs})^{2}}{A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}-2\gamma_{I}B_{z}+2\gamma_{S}B_{z}}$
$\displaystyle+\frac{4\gamma_{S}^{2}B_{\perp}^{2}D_{0}^{\rm
gs}}{(A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}-2\gamma_{S}B_{z})(A_{\parallel}^{\rm gs}+2D_{0}^{\rm
gs}-2\gamma_{S}B_{z})}\ ,$ (10) $\displaystyle\frac{{\bra{1}\hat{H}_{\rm
eff}^{m_{S}=0}}\ket{2}}{h}=\ $ $\displaystyle\frac{{\bra{2}\hat{H}_{\rm
eff}^{m_{S}=0}}\ket{1}}{h}=\frac{+\gamma_{I}B_{\perp}}{2}+\frac{\gamma_{S}B_{\perp}A_{\perp}^{\rm
gs}/2}{A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}+2\gamma_{S}B_{z}}+\frac{\gamma_{S}B_{\perp}A_{\perp}^{\rm
gs}/2}{A_{\parallel}^{\rm gs}-2D_{0}^{\rm gs}-2\gamma_{S}B_{z}}$
$\displaystyle+\frac{\gamma_{S}B_{\perp}A_{\perp}^{\rm
gs}/2}{A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}+2\gamma_{S}B_{z}-2\gamma_{I}B_{z}}+\frac{\gamma_{S}B_{\perp}A_{\perp}^{\rm
gs}/2}{A_{\parallel}^{\rm gs}-2D_{0}^{\rm
gs}-2\gamma_{S}B_{z}+2\gamma_{I}B_{z}}\ .$ (11)

Since $\gamma_{I}\ll\gamma_{S}$, we simplify in all denominators the terms
$(\gamma_{S}\pm\gamma_{I})\approx\gamma_{S}$. In similar fashion, we use the
fact that $A_{\parallel}^{\rm gs}\ll D_{0}^{\rm gs}$ and thus set
$(A_{\parallel}^{\rm gs}\pm 2D_{0}^{\rm gs})\approx\pm 2D_{0}^{\rm gs}$ in all
denominators. We obtain the following effective Hamiltonian for the 15N spin
in the $m_{S}=0$ manifold:

$\displaystyle\small\frac{\hat{H}^{m_{S}=0}_{\rm{eff}}}{h}$
$\displaystyle=\frac{1}{2}\begin{bmatrix}+\gamma_{I}B_{z}+\frac{+\gamma_{S}B_{z}(A_{\perp}^{\rm
gs})^{2}-D_{0}^{\rm gs}(A_{\perp}^{\rm gs})^{2}-2D_{0}^{\rm
gs}(\gamma_{S}B_{\perp})^{2}}{(D_{0}^{\rm
gs})^{2}-(\gamma_{S}B_{z})^{2}}&\gamma_{I}B_{\perp}-\frac{2\gamma_{S}B_{\perp}A_{\perp}^{\rm
gs}D_{0}^{\rm gs}}{(D_{0}^{\rm gs})^{2}-(\gamma_{S}B_{z})^{2}}\\\
\gamma_{I}B_{\perp}-\frac{2\gamma_{S}B_{\perp}A_{\perp}^{\rm gs}D_{0}^{\rm
gs}}{(D_{0}^{\rm
gs})^{2}-(\gamma_{S}B_{z})^{2}}&-\gamma_{I}B_{z}+\frac{-\gamma_{S}B_{z}(A_{\perp}^{\rm
gs})^{2}-D_{0}^{\rm gs}(A_{\perp}^{\rm gs})^{2}-2D_{0}^{\rm
gs}(\gamma_{S}B_{\perp})^{2}}{(D_{0}^{\rm
gs})^{2}-(\gamma_{S}B_{z})^{2}}\end{bmatrix}.$ (12)

Next, we add an energy-offset of $\left(D_{0}^{\rm gs}(A_{\perp}^{\rm
gs})^{2}-2D_{0}^{\rm gs}(\gamma_{S}B_{\perp})^{2}\right)/\left((D_{0}^{\rm
gs})^{2}-(\gamma_{S}B_{z})^{2}\right)$ to place the energy levels
symmetrically around zero, and thereby obtain the Hamiltonian given in the
main text,

$\displaystyle\frac{\hat{H}^{m_{S}=0}_{\rm{eff}}}{h}=\frac{1}{2}\,\begin{bmatrix}\gamma_{I}B_{z}+\nu_{z}&\hskip
5.69046pt\gamma_{I}B_{\perp}+\nu_{\perp}\vspace{0.2cm}\\\
\gamma_{I}B_{\perp}+\nu_{\perp}&\hskip
5.69046pt-\gamma_{I}B_{z}-\nu_{z}\end{bmatrix}.$ (13)

where

$\nu_{z}=\frac{\gamma_{S}B_{z}(A_{\perp}^{\rm gs})^{2}}{(D_{0}^{\rm
gs})^{2}-(\gamma_{S}B_{z})^{2}}$ (14)

denotes the correction to the diagonal elements caused by mixing between
states of different $m_{S}$, and

$\nu_{\perp}=\frac{-2\,\gamma_{S}B_{\perp}A_{\perp}^{\rm gs}D_{0}^{\rm
gs}}{(D_{0}^{\rm gs})^{2}-(\gamma_{S}B_{z})^{2}}$ (15)

is the corresponding correction to the off-diagonal elements. Note that these
expressions for $\nu_{z}$ and $\nu_{\perp}$ are diverging for $D_{0}^{\rm
gs}=\gamma_{S}B_{z}$, e.g. near the ground state level anti-crossing. However,
Van Vleck formalism is not applicable to that regime since the corresponding
electronic subspaces of $H_{0}$ are not sufficiently spaced in energy once
this condition is approached.

###  V.2 II. Numerical Model for Optical Pumping under ESLAC Conditions

In this section, we present in detail our numerical model for simulating the
spin dynamics of the NV center with and without green illumination for a given
magnetic field $\bm{B}$. To calculate the trajectory of both the electron and
nuclear spin under optical pumping it is necessary to consider not only
classical rate equations coupling the orbital states, but also to incorporate
the quantum mechanical evolution of the spins within each orbital state. Our
model follows an approach previously taken to model the effect of chemical
reaction kinetics on NMR spectra [53].

####  V.2.1 II.1 Mathematical Description of the Model

We model the room temperature NV- center as a system made up of three distinct
electronic states: The ground state (gs), the excited state (es), and a meta
stable singlet state (s). We neglect the distinct $E_{x}$ and $E_{y}$ orbital
branches in the excited state as they are efficiently averaged at room
temperature, as well as the existence of two singlet states - we assume that
there is only one such singlet state. We also neglect laser induced ionization
to the NV0 state.

First we define a separate spin density operator for each orbital state
(labelled with $\alpha$) which evolves coherently per the Liouville-von
Neumann equation of motion

$\frac{\text{d}}{\text{d}t}\hat{\rho}_{\alpha}=\hat{\hat{L}}_{\alpha}\hat{\rho}_{\alpha},$
(16)

where the carets denote operators, double carets denote superoperators, and
$\alpha$ indexes over the different orbital states. The commutation
superoperator $\hat{\hat{L}}_{\alpha}$ in Equation (16) can be calculated from
the corresponding Hamiltonian $\hat{H}_{\alpha}$ as

$\hat{\hat{L}}_{\alpha}=-i\left(\hat{H}_{\alpha}\otimes
E_{\alpha}-E_{\alpha}\otimes\hat{H}_{\alpha}^{T}\right)\,,$ (17)

where $\alpha\in\\{\text{es},\text{gs},\text{s}\\}$ denotes the orbital,
$E_{\alpha}$ is the identity matrix of the same dimensionality as $H_{\alpha}$
and $T$ denotes matrix transposition. The Hamiltonians for each orbital state
are given by:

$\displaystyle\hat{H}^{\rm gs,es}/h$ $\displaystyle=D_{0}^{\rm
gs,es}\hat{S}_{z}^{2}+\hat{\bm{S}}\cdot\bm{A}^{\rm
gs,es}\cdot\hat{\bm{I}}+\gamma_{S}\bm{B}\cdot\hat{\bm{S}}+\gamma_{I}\bm{B}\cdot\hat{\bm{I}}\,$
(18) $\displaystyle\hat{H}^{\rm s}/h$
$\displaystyle=\bm{B}\cdot\hat{\bm{\sigma}}\,.$ (19)

where $\hat{\bm{S}}$ and $\hat{\bm{I}}$ are angular momentum operators for the
electron and 15N nucleus acting on the joint electron/nuclear Hilbert space,
and $\hat{\bm{\sigma}}$ are the 2x2 spin-1/2 matrices. The constants and
coupling tensors are defined in table 1. Next, we couple these differential
equations with additional (real valued) superoperators corresponding to the
incoherent optical pumping process. These superoperators act to reduce or
increase the population of a given spin state and thus take the role of spin-
selective relaxation superoperators

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{gs}=\ $
$\displaystyle\hat{\hat{L}}_{gs}\hat{\rho}_{gs}-k_{\rm
green}\hat{\rho}_{gs}+k_{\rm
red}\hat{\rho}_{es}+k_{s1}(\hat{\hat{S}}^{\otimes}_{+1}+\hat{\hat{S}}^{\otimes}_{-1})\hat{\rho}_{s}+k_{s0}\hat{\hat{S}}^{\otimes}_{0}\hat{\rho}_{s}$
$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{s}=\ $
$\displaystyle\hat{\hat{L}}_{s}\hat{\rho}_{s}-(2k_{s1}+k_{s0})\hat{\rho}_{s}+k_{\rm
ISC}^{ms=|1|}\hat{\hat{T}}^{e}(\hat{\hat{P}}_{1}+\hat{\hat{P}}_{-1})\hat{\rho}_{es}+k_{\rm
ISC}^{ms=0}\hat{\hat{T}}^{e}\hat{\hat{P}}_{0}\hat{\rho}_{es}$
$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{es}=\ $
$\displaystyle\hat{\hat{L}}_{es}\hat{\rho}_{es}+k_{\rm
green}\hat{\rho}_{gs}-k_{\rm red}\hat{\rho}_{es}-k_{\rm
ISC}^{ms=|1|}(\hat{\hat{P}}_{+1}+\hat{\hat{P}}_{-1})\hat{\rho}_{es}-k_{\rm
ISC}^{ms=0}\hat{\hat{P}}_{0}\hat{\rho}_{es},$ (20)

where $\hat{\hat{P}}_{\pm 1,0}$ are projection superoperators that project
$\hat{\rho}_{\alpha}$ onto the NV-electron spin state with $m_{S}=\pm 1,0$
while leaving the dimensionality of $\hat{\rho}_{\alpha}$ unchanged. This
superoperator ensures that the rate of inter system crossing (ISC) out of the
excited state depends on the instantaneous spin state population of
$\hat{\rho}_{es}$. Next, $\hat{\hat{T}}^{e}$ is a partial trace superoperator
that acts on a 36-dimensional joint electron/nuclear density operator and
traces out the NV-electron degrees of freedom, leaving a 4-dimensional density
operator corresponding only to 15N. Finally, $\hat{\hat{S}}^{\otimes}_{0,\pm
1}$ is a direct product superoperator that acts on a 4-dimensional 15N density
operator and turns it into a joint electron/nuclear density operator with the
NV-electron in the state $ms=0,\pm 1$. Note that because their effect changes
dimensionality of $\hat{\rho_{\alpha}}$, the matrix representations of
$\hat{\hat{T}}^{e}$ and $\hat{\hat{S}}^{\otimes}_{0,\pm 1}$ are not square.
Further, since none of these superoperators act on the 15N degrees of freedom
this model assumes that the nuclear spin state is preserved throughout the
optical pumping process. Equation (20) is normalized such that the sum of the
traces of the spin density operators $\rho_{\alpha}$ is equal to 1, meaning
$\text{Trace}\\{{\hat{\rho}_{\alpha}}\\}$ is the fractional population of the
total system in orbital state $\alpha$. We will give procedures for generating
the corresponding matrix representations for the superoperators in Eq. (20)
later in this section.

Figure 5 shows a pictorial representation of the processes modeled by Eq.
(20). The ground and excited states are coupled via optical excitation with
rate $k_{\rm green}$ and radiative decay with rate $k_{\rm red}$ respectively.
The spin selective ISC causes non-radiative transitions from the excited state
to the singlet, described by the rates $k_{\rm ISC}^{ms=\pm 1,0}$. Relaxation
from the electronic singlet into the ground state is also electron spin
selective, with rates given by $k_{ms=1}$ and $k_{ms=0}$ respectively. The
precise values for the ISC and relaxation rates have been subject to
considerable debate. At the end of these section we present modeling results
for different sets of these parameters.

Figure 5:  Level Structure for the optical pumping model. Each of the three
orbital states is governed by its own Hamiltonian.

By concatenating the vector representations of $\hat{\rho}_{gs}$,
$\hat{\rho}_{s}$, and $\hat{\rho}_{es}$ into a single 76x1 spin density
operator for the entire system, $\hat{\rho}$, Eq. (20) can conveniently be
cast in matrix form:

$\displaystyle\tiny\frac{\rm{d}}{{\rm{d}}t}\begin{bmatrix}\hat{\rho}_{gs}&\\\
\hat{\rho}_{s}&\\\
\hat{\rho}_{es}\end{bmatrix}=\begin{bmatrix}\hat{\hat{L}}_{gs}-k_{\rm
green}\hat{\hat{E}}_{36}&+k_{s\pm
1}(\hat{\hat{S}}^{\otimes}_{1}+\hat{\hat{S}}^{\otimes}_{-1})+k_{s0}\hat{\hat{S}}^{\otimes}_{0}&+k_{\rm
red}\hat{\hat{E}}_{36}\\\ 0&\hat{\hat{L}}_{s}-(2k_{s\pm
1}+k_{s0})\hat{\hat{E}}_{4}&+k_{ISC}^{m_{S}=|1|}\hat{\hat{T}}^{e}(\hat{\hat{P}}_{1}+\hat{\hat{P}}_{-1})+k_{ISC}^{m_{S}=0}\hat{\hat{P}}_{0}\\\
k_{\rm
green}\hat{\hat{E}}_{36}&0&\hat{\hat{L}}_{es}-k_{f}\hat{\hat{E}}_{36}-k_{ISC}^{m_{S}=|1|}(\hat{\hat{P}}_{1}+\hat{\hat{P}}_{-1})-k_{ISC}^{m_{S}=0}\hat{\hat{P}}_{0}\end{bmatrix}\begin{bmatrix}\hat{\rho}_{gs}&\\\
\hat{\rho}_{s}&\\\ \hat{\rho}_{es}\end{bmatrix},$ (21)

where $E_{36(4)}$ is a 36x36(4x4) identity matrix. In the following, we will
call the resulting 76x76 dimensional matrix $\hat{\hat{A}}$. Note that because
of how we formed $\hat{\rho}$ by concatenation it cannot represent coherences
between different orbital states, however such coherences are unlikely to be
significant and are not necessary to explain the physics of interest in this
work. We also note that since we use identity superoperators to represent the
radiative processes, spin-spin coherences will be preserved under optical
excitation and radiative decay within this model. Since the matrix
$\hat{\hat{A}}$ commutes with itself for all values of time $t$, Eq. (21) can
easily be integrated and thus the time evolution of the system can be
calculated as

$\hat{\rho}(t)=\mathrm{e}^{2\pi\hat{\hat{A}}t}\hat{\rho}(0),$ (22)

for any time $t$. At any given time the first 36 entries of $\hat{\rho}(t)$
corresponds to $\hat{\rho}_{gs}(t)$ written in vector form, the next 4 entries
are $\hat{\rho}_{s}(t)$ and the last 36 are $\hat{\rho}_{es}(t)$. We take the
predicted instantaneous photoluminescence for $\hat{\rho}(t)$ as the
fractional population in the excited electronic state,

$\text{PL}(t)=\text{Trace}\\{\hat{\rho}_{es}(t)\\}\,.$ (23)

We note that in this work we use Eq. (22) to calculate the time evolution of
the system both in the presence and and absence of green illumination by
choosing different values for the parameter $k_{\rm green}$ and calculating
the propagation piecewise with different $\hat{\hat{A}}$ matrices.

####  V.2.2 II.2 Matrix Representations of Superoperators

We define the matrix representations of the 15N spin operators on the joint
electron/nuclear space as
$\hat{I}_{x/y/z}=\hat{\sigma}_{x/y/z}\otimes\hat{E}_{3}$ and the NV-electron
spin operators as $\hat{S}_{x/y/z}=\hat{E}_{2}\otimes\hat{\lambda}_{x/y/z}$,
where $\hat{\lambda}$ are the 3x3 spin-1 matrices. The action of the partial
trace superoperator is defined by
$\hat{\hat{T}}^{e}\hat{I}_{x/y/z}=\text{Trace}\\{\hat{E}_{3}\\}\hat{\sigma}_{x/y/z}$.
In the numerical implementation of our model we vectorize operators as 36x1 or
4x1 dimensional column vectors. The matrix representation of
$\hat{\hat{T}}^{e}$ is determined following the procedure layed out in [53]
and reproduced here for completeness:

$\\{\hat{\hat{T}}^{e}\\}_{\alpha\beta}=\begin{cases}\quad 1&\text{, if }m=n\\\
\quad 0&\text{, else}\\\ \end{cases}$ (24)

where

$\displaystyle\alpha$ $\displaystyle=(i-1)\cdot d_{I}+j$ $\displaystyle\beta$
$\displaystyle=(((i-1)\cdot d_{S}+m)-1)\cdot d_{S}\cdot d_{I}+(j-1)\cdot
d_{S}+n\,,$ (25)

where $i,n=1\,...d_{I}$ count through the degrees of freedom of the first
subspace (nuclear in our case), and $j,m=1\,...d_{S}$ count through the
degrees of freedom of the second subspace (NV-electron in our case). The
resulting $\hat{\hat{T}}^{e}$ matrix is 4x36 dimensional.

The matrix representation of the direct product superoperator
$\hat{\hat{S}}^{\otimes}_{0,\pm 1}\hat{E}_{2}=\hat{S}_{0,\pm 1}$ is

$\\{\hat{\hat{S}}^{\otimes}_{0,\pm 1}\\}_{\alpha\beta}=\\{\hat{\kappa}_{0,\pm
1}\\}_{mn}$ (26)

where

$\displaystyle\alpha$ $\displaystyle=(((i-1)\cdot d_{S}+m)-1)\cdot d_{S}\cdot
d_{I}+(j-1)\cdot d_{S}+n$ $\displaystyle\beta$ $\displaystyle=(i-1)\cdot
d_{I}+j\,,$ (27)

and

$\hat{\kappa}_{+1}=\left(\begin{matrix}1&0&0\\\ 0&0&0\\\ 0&0&0\\\
\end{matrix}\right)\qquad\hat{\kappa}_{0}=\left(\begin{matrix}0&0&0\\\
0&1&0\\\ 0&0&0\\\
\end{matrix}\right)\qquad\hat{\kappa}_{-1}=\left(\begin{matrix}0&0&0\\\
0&0&0\\\ 0&0&1\\\ \end{matrix}\right)\,.$ (28)

The numbers $i$, $j$, $n$, $m$, $d_{S}$ and $d_{I}$ are the same as defined
above. The resulting dimensionality of the matrix representation of
$\hat{\hat{S}}^{\otimes}_{0,\pm 1}$ is 36x4.

Finally, we define the matrix representation for the projection superoperator,

$\hat{\hat{P}}_{0,\pm 1}=\sum_{k=1}^{4}\rho_{0,\pm 1}^{k}\cdot(\rho_{0,\pm
1}^{k})^{T}$ (29)

where $\rho_{0,\pm 1}^{k}=\hat{\kappa}_{\rm nuc}^{k}\otimes\hat{\kappa}_{0,\pm
1}$ is the 36x1 column vector representation of the joint 6x6 density matrix
operator and

$\hat{\kappa}_{\rm nuc}^{1}=\left(\begin{matrix}1&0\\\ 0&0\\\
\end{matrix}\right)\qquad\hat{\kappa}_{\rm nuc}^{2}=\left(\begin{matrix}0&1\\\
0&0\\\ \end{matrix}\right)\qquad\hat{\kappa}_{\rm
nuc}^{3}=\left(\begin{matrix}0&0\\\ 1&0\\\
\end{matrix}\right)\qquad\hat{\kappa}_{\rm nuc}^{4}=\left(\begin{matrix}0&0\\\
0&1\\\ \end{matrix}\right).$ (30)

While $\hat{\hat{T}}^{e}$ and $\hat{\hat{S}}^{\otimes}_{0,\pm 1}$ both change
the dimensionality of the state they operate on, $\hat{\hat{P}}_{0,\pm 1}$ is
a square 36x36 matrix and thus preserves dimensionality. The sum in equation
(29) ensures that $\hat{\hat{P}}_{0,\pm 1}$ projects onto a particular
electronic spin state, while leaving the nuclear spin unchanged.

####  V.2.3 II.3 Details on Simulation Evaluation

In order to simulate the PL of the all-optical pulse sequence introduced in
the main text using the model described above, one needs to choose values for
the various NV parameters and transition rates $k$. Here, apart from
$D_{0}^{\rm gs}$ which we determine experimentally as described later in the
SI, we use literature values for these simulation parameters. The exact
numbers employed are listed in table 1 and originate from [33, 34]. For the
optical transition rates, we consider five different sets of numbers, labeled
parameter sets 1 to 5, as shown in the table 2.

$D_{0}^{\rm gs}$ | $\left[\text{MHz}\right]$ | $+2870.760402$
---|---|---
$A_{\parallel}^{\rm gs}$ | $\left[\text{MHz}\right]$ | $+3.03$
$A_{\perp}^{\rm gs}$ | $\left[\text{MHz}\right]$ | $+3.65$
$D_{0}^{\rm es}$ | $\left[\text{MHz}\right]$ | $+1420$
$A_{\parallel}^{\rm es}$ | $\left[\text{MHz}\right]$ | $-57.8$
$A_{\perp}^{\rm es}$ | $\left[\text{MHz}\right]$ | $-39.2$
$\gamma_{S}$ | $\left[\text{MHz/G}\right]$ | $0.000431744$
$\gamma_{I}$ | $\left[\text{MHz/G}\right]$ | $2.802494716$
Table 1: Numeric values for the employed NV parameters. |  | Parameter | Parameter | Parameter | Parameter | Parameter
---|---|---|---|---|---|---
|  | Set 1 | Set 2 | Set 3 | Set 4 | Set 5
$k_{\rm red}$ | $\left[\text{MHz}\right]$ | 66 | 77 | 63.70 | 63.2 | 67.4
$k_{\rm ISC}^{m_{S}=0}$ | $\left[\text{MHz}\right]$ | 0 | 0 | 12.97 | 10.8 | 9.9
$k_{\rm ISC}^{m_{S}=|1|}$ | $\left[\text{MHz}\right]$ | 57 | 30 | 80.00 | 60.7 | 96.6
$k_{s0}$ | $\left[\text{MHz}\right]$ | 1.0 | 3.3 | 3.45 | 0.8 | 4.83
$k_{s1}$ | $\left[\text{MHz}\right]$ | 0.7 | 0 | 1.08 | 0.4 | 1.055
Reference |  | [45] | [54] | [55] | [16] | [56]
Table 2: Numeric values for of the optical transition rates used in our model.
We consider five different parameter sets. In the main text, we use set 4.

Note that parameters sets 1 and 2 assume $k_{\rm ISC}^{m_{S}=0}=0$,
prohibiting inter system crossing for $m_{S}=0$ population in the excited
state. Using the values of either of these sets, we fully simulate the all-
optical sequence described in the main text, that consists of a green
initialization pulse, followed by a interval of free evolution in the dark and
subsequent PL readout during a short green laser pulse. First, we simulate
optical pumping of $\hat{\rho}(0)$ at NV saturation for $3000$ ns, e.g.
setting $k_{\rm green}=s\cdot k_{\rm red}$ with $s=35\%$ as determined
experimentally on the single NV sample. To that end, we choose $\hat{\rho}(0)$
such that all six states in the ground electronic orbital are occupied with
each $1/6$, and the excited orbital states and the single states are empty.
Afterwards, we run the model with $k_{\rm green}=0$, propagating the optically
pumped state in the dark for $290$ ns. This empties out the singlet into the
ground state. Subsequently, still with $k_{\rm green}=0$, we propagate the
system for a duration $\tau$, followed by a readout made up of $350$ ns with
$k_{\rm green}=s\cdot k_{\rm red}$. This readout pulse is simulated in steps
of $1$ ns, and the total PL($\tau$) is the sum of the PL of each step.
Plotting $\text{PL}(\tau)$ yields an oscillation whose frequency $f$ we
determine by taking the position of maximum of the corresponding Fourier
spectrum. The contrast $C$ of the oscillation in $\text{PL}(\tau)$ is also
determined via the Fourier spectrum: We compute it as $C=4V/V_{0}$ where $V$
is the amplitude of the detected Fourier space peak, and $V_{0}$ is the
Fourier space peak at zero.

Figure 6:  The data shown in Fig 2b and d in the main text, together with the
results of the numerical simulation, run with each of the five optical
transition rate parameter sets. Overall, set 4 has the best agreement with the
data, which is why model 4 is used for all numerical results shown in the main
text.

With this procedure, we simulate both the nuclear precession frequency $f$ as
well as the precession contrast $C$, for $\bm{B}$-field values corresponding
to every pixel shown in Fig 2b in the main text. The numeric result for the
precession contrast $C$ is shown in Fig. 6a, where for ease of comparison, the
experimental data has been normalized to 1 and the simulation to 0.75. The
plot demonstrates that the numeric result depends strongly on the chosen
optical transition rates. Specifially, depending on the employed optical set
of parameters, we find maximal contrast for $B_{\perp}$ between $7.9$ and
$10.2$ G.

In Fig 6b we show the results of running the model for $\bm{B}$-field values
that correspond to the measurement shown in Fig 2d in the main text. Again, it
is evident that the choice of optical transition rates has a significant
effect on the numeric result. All set of parameters predict a global maximum
in $C$ between 530 and 550 G. Interestingly, sets 1 and 2 which dictate
$k_{\rm ISC}^{m_{S}=0}=0$ differ drastically from the other parameter sets
with $k_{\rm ISC}^{m_{S}=0}\neq 0$ in that they show a second local maximum
for $C$ at about 470 G. This is not consistent with our data.

Overall, we find that parameter set 4 has the best agreement with our data,
and thus we use set 4 for all simulations shown in the main text’s figures.
Note that since $\text{PL}(\tau)$ is not true to scale, we normalize all
numeric results for $C$ shown in the main text to the corresponding data.

Finally, we note that the numeric model agrees very well with the analytic
predictions for the nuclear precession frequency. This excellent agreement is
demonstrated in the main text’s figure 3b. We want to emphasize, however, that
not only the precession frequency $f_{\rm nuc}=\gamma_{I}|\bm{B}_{\rm eff}|$
of numerical and analytical predictions agree well, but also the tilt angle
$\vartheta$ of $\bm{B}_{\rm eff}$, as shown in Fig. 7. Here, $\bm{B}_{\rm
eff}$ and its tilt angle $\vartheta$ are calculated numerically by
diagonalization of $\hat{H}_{\rm gs}$, or analytically via Eq. (2) in the main
text. Combined, this means that both approaches yield the same $\bm{B}_{\rm
eff}$.

Figure 7: Prediction of the tilt angle $\vartheta$ of $\bm{B}_{\rm eff}$ of
both our the numeric simulation and the analytic approach based on Van Vleck
perturbation theory. These predictions have excellent agreement.

###  V.3 III. Magnetic Field Map

In the main text’s Fig. 2b (and 3b), we have shown the same data as in Fig 1a
(and 3a), but plotted against the transverse magnetic field $B_{\perp}$. Here,
we elaborate on how this magnetic field is determined experimentally. Since we
generate the external magnetic field with a permanent magnet that is mounted
on a goniometer, and since this goniometer does not rotate about the NV
perfectly, it is in general not accurate to simply measure $|\bm{B}_{\rm
ext}|$ and set $B_{\perp}=\sin(\Phi)|\bm{B}_{\rm ext}|$. Instead, we measure
the magnetic field vector components $B_{\perp}$ and $B_{\parallel}$
individually for every other pixel of the main text’s Fig. 2a, using the same
experimental conditions, by taking an optically detected magnetic resonance
(ODMR) spectrum of each individual pixel. This way, we experimentally
determine all four transition frequencies for a given magnetic field; two
corresponding to the hyperfine transitions from the electron spin manifold
$\ket{0}$ to the electron spin manifold $\ket{+1}$, and two corresponding to
the hyperfine transitions from $\ket{0}$ to $\ket{-1}$. Then, we determine
$D_{0}^{\rm gs}$ by fitting the ground state Hamiltonian $\hat{H}_{\rm gs}$ as
defined in the main text to the four measured transition frequencies of the
center pixel, where we enforce $B_{\perp}=0$ since $\Phi_{x}=\Phi_{y}=0$. We
find $D_{0}^{\rm gs}=2870.760402$ MHz. Using this calibration of $D_{0}^{\rm
gs}$, we then determine the magnetic components
$B_{\parallel}(\Phi_{x},\Phi_{y})$ and $B_{\perp}(\Phi_{x},\Phi_{y})$ of all
other pixels by fitting $\hat{H}_{\rm gs}$ to the transition frequencies of
these pixels, using $D_{0}^{\rm gs}$ as determined before. In the end, we
interpolate both $B_{\parallel}(\Phi_{x},\Phi_{y})$ and
$B_{\perp}(\Phi_{x},\Phi_{y})$ to twice the density in $\Phi_{x}$ and
$\Phi_{y}$ to match the pixel density of Fig. 2a (respectively 3a). This
interpolated result is shown in Fig. 8a and b, where $|\bm{B}|=533$ G. The
simulations shown in the main text, as well as the data in Fig. 2b and 3b are
based on these experimentally determined magnetic field values.

Figure 8:  a Parallel magnetic field component $B_{\parallel}$ and b
transverse magnetic field component $B_{\perp}$ as a function of tilt angles
$\Phi_{x}$ and $\Phi_{y}$, measured via optically detected magnetic resonance
of all four microwave-driving 15N transitions. Each of the shown pixels
corresponds to one pixel in the main text’s Fig. 2a and 3a. The total magnetic
field is about $|\bm{B}|=533$ G. c Measured transverse magnetic field as a
function of total tilt angle $\Phi$, revealing a linear dependance with slope
$10.01\,\text{G}/\text{deg}$, allowing for a simple conversion of $B_{\perp}$
to corresponding $\Phi$ and vice versa.

Further, these magnetic field data can be plotted against total angle,
$\Phi=(\Phi_{x}^{2}+\Phi_{y}^{2})^{1/2}$, as shown in Fig. 8c, revealing a
near-linear dependance of $B_{\perp}$ on $\Phi$. This is to be expected, since
for $|\bm{B}|=533$ G and small angles $\Phi$ as investigated here,

$B_{\perp}=\sin(\Phi\cdot\pi/180^{\circ})|\bm{B}|\approx\Phi\cdot\pi/180^{\circ}|\bm{B}|=\Phi\cdot
9.3\,\text{G}/\text{deg}\,.$ (31)

The linear fit shown in Fig. 8c gives that in our experiment the slope is
$10.01\,\text{G}/\text{deg}$. This slightly larger value compared to above
prediction (and its not perfeclty linear shape) is explained by the fact that
the goniometer employed in our experiments is not rotating the permanent
magnet perfectly about the NV, resulting a slight change in total magnetic
field whenever the magnet is rotated.

Finally, note that we use the slope of $10.01\,\text{G}/\text{deg}$ to convert
$B_{\perp}$ to $\Phi$ – but only for data taken at $|\bm{B}|=533$ G because
this is the only field where we acquired ODMR data for the entire range of the
goniometer. Specifically, as discussed in the main text, at $|\bm{B}|=533$ G
the best contrast is achieved at $B_{\perp}=9.4$ G which thus corresponds to
$\Phi_{\rm opt}=0.94^{\circ}$.

Figure 9:  Same measurement as shown in the main text in Fig. 2a, c, d and
Fig. 3a, but here the data were taken on an ensemble of NVs rather than on a
single NV. We find the same qualitative and quantitative result, with the only
difference being a slightly lower contrast $C$.

###  V.4 IV. Ensemble Data

In the main text’s Fig. 2 and 3, we discuss the dependance of the of the
observed nuclear precession on the applied magnetic field in the context of a
single NV. However, all that data can also be reliably reproduced on a full
ensemble of NVs. In Fig. 9, we show the results of these ensemble measurements
which were taken on the NV ensemble described in the main text: an
unstructured, grown diamond with 15NVs that are preferentially aligned with a
specific crystal orientation. These ensemble measurements reproduce the
single-NV data in both qualitative and quantitative way, demonstrating that
the all-optical nuclear quantum sensing scheme proposed in this work does
scale well to NV ensembles. We note, however, that the ensemble shows a
contrast $C$ of about half the single-NV contrast, which we assign to the
minority of non-aligned NVs.


Gain Profile Characterization and Modelling
for an Accurate EDFA Abstraction and Control

Giacomo Borraccini(1),
Vittorio Gatto(1),
Andrea D'Amico(1),
Stefano Straullu(2),
Francesco Aquilino(2),

Stefano Piciaccia(3),
Alberto Tanzi(3),
Gabriele Galimberti(4) and
Vittorio Curri (1)

(1)Politecnico di Torino, Turin, Italy,
<EMAIL_ADDRESS>(2)LINKS Foundation, Turin, Italy; (3)Cisco Photonics, Vimercate, Italy; (4)Internet Engineering Task Force (IETF)

Relying on a two-measurement characterization phase, a gain profile model for dual-stage EDFAs is presented and validated in full spectral load condition.

It precisely reproduces the EDFA dynamics varying the target gain and tilts parameters as shown experimentally on two commercial items from different vendors. ©2023 The Author(s)

§ INTRODUCTION

Maximizing the capacity of optical infrastructures is one of the main objectives for operators and service providers, aiming at minimizing costs at the same time <cit.>.

For this purpose, quality-of-transmission estimation (QoT-E) represents a fundamental aspect for both the optical control and data planes, and predicting the expected behavior of the system with a reasonable margin <cit.>.

To this purpose, with an open networking perspective, the open source GNPy model <cit.> is an extensively validated vendor-agnostic solution.

As a result, optical amplifiers are key network elements that allow signal power levels to be restored at the cost of signal-to-noise ratio degradation.

The characterisation of optical amplifiers includes two main parameters that are critical in the search for the optimum operating point of the system: the gain profile, $g$, and the noise figure profile.

In general, an adequate description of the frequency-dependent physical layer parameters plays an important role in the QoT-E, especially when considering wideband transmission scenarios <cit.>.

Focusing on Erbium-doped fiber amplifiers (EDFAs), an ideal gain profile can be expressed in decibels as:

\begin{equation}\label{eq:e}
g(f; G, T) = G + \frac{T}{B}(f -f_c)
\end{equation}

where $f$ is the optical frequency,  $G$ and $T$  are the target gain and tilt parameters set in the amplifier, $f_c$ is the rotation pivot point of the tilt, and $B$ is the operative EDFA bandwidth.

Accurate application of the tilt is important to compensate for stimulated Raman scattering (SRS) due to the propagation through the fiber span <cit.>.

Eq. <ref> completely neglects the additional gain profile ripple due to the EDFA manufacturing and the physical behaviour.

The design of the EDFA gain profile can be achieved through the evaluation of the dynamic gain tilt (DGT) parameter, evaluated as the normalized difference, with respect to a specific frequency of two different gain profiles <cit.>.

Starting from the DGT definition and from the explicit interpretation of the gain profile, it is possible to define a constant parameter regardless of the specific configuration of tilt and gain parameters that is set in the amplifier.

Different machine learning (ML) strategies have already tackled the problem of accurately modelling the gain ripple profile <cit.>, requiring relatively large datasets.

ML solutions can also compensate for gain profile fluctuations induced by a varying spectral load, as shown in <cit.>.

In this work, a semi-analytical EDFA model is presented and validated, focusing on the accurate reproduction of the gain profile, including the gain ripple, in a full spectral load transmission scenario.

Extending Eq. <ref>, the proposed methodology provides a low computational cost procedure for the gain ripple profile evaluation.

The implemented abstraction is composed of two different steps: first, the EDFA gain profile is characterized considering only two gain/tilt settings in full spectral load condition, that is the most interesting solution by on case exploiting ASE shaped channels.
Then, the semi-analytical model enables an accurate estimation of the gain profile for any pair of gain/tilt settings.

This implementation is very promising for its precision and simplicity, fundamental features for efficient and agile QoT-E such as the one within GNPy <cit.>.

§ METHODOLOGY

Without any loss of generality, an accurate EDFA gain profile modeling is described as follows:

\begin{equation}\label{eq:eq1}
g(f; G, T) = G + \frac{T}{B}(f -f_c) + r_T(f)
\end{equation}

where the term $r_T(f)$ refers to the gain ripple profile, that can be defined as the deviation between the real and ideal amplifier gain profiles <cit.>.

In general, all the gain ripple profiles, $r_T(f)$, generated in a different tilt condition are dependent on the set target tilt parameter, $T$.

In particular, the gain ripple profile of a real EDFA evaluated when the tilt parameter is set to 0 dB is a manufacturing footprint of the EDFA design procedure and, indeed, does not depend on $T$:

\begin{equation}\label{eq:eq2}
g(f; G, T=0) = G + r_0(f)\,.
\end{equation}

Bearing in mind the DGT-based EDFA design that describes the physical implementation of the tilt, it is possible to define a tilt-independent parameter, $K(f)$, defined as:

\begin{equation}\label{eq:eq3}
K(f) = \frac{r_0(f) - r_T(f)}{T}\,,
\end{equation}

which is a feature of the EDFA figure of merit, and $r_T(f)$ is derived from the gain profile measurement setting the tilt parameter at $T$.

From Eq. <ref>, it is possible to evaluate a generic tilt-dependent gain ripple profile in function of the $r_0(f)$ and $K(f)$:

\begin{equation} \label{eq:eq6}
r_T(f) = r_0(f) - T\,K(f)
\end{equation}

Using the Eq. <ref>, it is possible to obtain an accurate evaluation of a generic gain profile for a specific EDFA:

\begin{equation}\label{eq:eq4}
g(f; G, T) = G + \frac{T}{B}(f-f_c) + r_0(f) - T\,K(f)
\end{equation}

Sketch of the experimental setup used to characterize a single EDFA item.

Gain profile comparison between the experimental measurements and the presented EDFA model for two different configurations: (a) EDFA 1; (b) EDFA 2.

§ MEASUREMENTS

Box plot resuming the error estimation between the all the performed experimental measurements, $g^m$, and the corresponding predictions using the presented model, $g^p$: (a) EDFA 1; (b) EDFA 2.

To validate the model, a full experimental characterization of two set of two distinct EDFA families (referred as EDFA 1 and EDFA 2 in the following) is performed.

For each item, an experimental dataset consisting of different gain profiles for a given combinations of total input power, target gain and tilt parameters, have been collected.

The configuration setup for the acquisition of the dataset is depicted in Fig. <ref>.

A commercial wavelength selective switch (WSS) is programmed in order to shape the ASE noise generating a wavelength division multiplexed (WDM) comb of either 40 or 48 channels, 100 GHz spaced, each modulated at 32 GBaud; this way, two different spectral loads are provided at the EDFAs' input, according to each device specifications: 4 THz for EDFA 1, 4.8 THz for EDFA 2, respectively.

Each amplifier can be controlled via secure shell protocol (SSH), which has been exploited to set the gain and the tilt parameters, whereas its optical input power is changed acting on the variable optical attenuator (VOA) placed in front of it.

The optical spectrum at both the EDFA's input and output is captured using an optical spectrum analyzer (OSA).

The dataset for EDFA 1 has been collected changing the input power from -10 dBm to +6 dBm, with steps of 2 dB, and setting EDFA's gain and tilt values in the range [14:20] dB and [-5:4] dB, respectively, each with steps of 1 dB.

Regarding EDFA 2, the gain is extend to a range of to [12:27] dB with a 1 dB step, consequently reducing the upper limit of the input power range to +4 dBm.

The presented model has been derived from both the EDFA items under investigation.

By fixing a target gain parameter value, the gain ripple imposing tilt equal to 0 dB, $r_0(f)$, has been derived simply by measuring the corresponding gain profile and removing the average gain.

Since the target tilt parameter is generally defined in wavelength, the $K(f)$ has been evaluated with the gain profile with tilt 0 dB and with the gain profile that has the larger negative target tilt parameter value.

The choice of the second gain profile is arbitrary, but the experimental evidence has shown that a larger tilt results in a smaller relative measurement error.

Furthermore, considering the non-zero target tilt parameter, the tilt pivot point, $f_c$, is evaluated as the linear regression of the measured gain profile assumes the value of the target gain parameter.

The EDFA operating band, $B$, is estimated by considering the same linear regression so that the value of the set target tilt parameter agrees with the estimate.

§ RESULTS

The Eq. <ref> has been used to evaluate the gain profile with a specific gain target and a specific tilt target.

In Fig. <ref>, it is possible to observe the measured gain profile and the evaluated gain profile for both the considered EDFAs setting the different configurations of gain and parameter values.

A single gain profile is provided using the proposed model for all the values of total input powers as it is independent with respect to the latter.

It is worth noting that the measured profiles are affected by measurement post-processing uncertainty, as all the gain profiles have been obtained as a differential calculation of the input/output power profiles, measured with the OSA, and re-scaled to the EDFA input/output total powers, which are affected by an inaccuracy of $\pm$0.1 dB.

Fig. <ref> provides a larger view of the model effectiveness, showing the estimated errors comparing the measured gain profiles and the gain profiles evaluated by the model for all the possible combinations of total input power, gain and tilt values.

In both cases, the error is centered in 0 dB, demonstrating an unbiased estimation, showing a worst case value of 0.1 dB for the EDFA 1, and roughly 0.3 dB for the EDFA 2, without considering the outliers.

Furthermore, it can be observe that the model error is slightly higher in the high-frequency amplification region.

This can be explained with the well-known spectral hole burning phenomenon <cit.>.

§ CONCLUSIONS

An accurate and simple gain profile modelling for dual-stage EDFAs working in full spectral load condition is proposed and experimentally verified on two classes of devices coming from different vendors.

In particular, by means of the characterization of two device-specific invariant profiles, $K(f)$ and $r_0(f)$, which can be evaluated with only two gain profile samples, all the gain profiles for all the combination of input power, gain and tilt values can be obtained with a semi-analytical expression, Eq. <ref>.

As the characterization procedure does not include any complexity, it can be performed both by the vendors and final users, and the device-specific characterization can be easily stored or shared.

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journaltitleJournal of Lightwave Technology
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titleMachine learning for ultrawide bandwidth amplifier configuration

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titleIterative supervised learning approach using transceiver bit-error-rate measurements for optical line system optimization

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titleSpectral hole burning in erbium-doped fiber amplifiers

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