# Electron-to-nuclear spectral mapping via “Galton board” dynamic nuclear
polarization

Arjun Pillai  Department of Chemistry, University of California, Berkeley,
Berkeley, CA 94720, USA.  Moniish Elanchezhian  Department of Chemistry,
University of California, Berkeley, Berkeley, CA 94720, USA.  Teemu Virtanen
Department of Chemistry, University of California, Berkeley, Berkeley, CA
94720, USA.  Sophie Conti  Department of Chemistry, University of California,
Berkeley, Berkeley, CA 94720, USA.  Ashok Ajoy<EMAIL_ADDRESS>Department of Chemistry, University of California, Berkeley, Berkeley, CA
94720, USA.

###### Abstract

We report on a strategy to indirectly readout the spectrum of an electronic
spin via polarization transfer to nuclear spins in its local environment. The
nuclear spins are far more abundant and have longer lifetimes, allowing
repeated polarization accumulation in them. Subsequent nuclear interrogation
can reveal information about the electronic spectral density of states. We
experimentally demonstrate the method for reading out the ESR spectrum of
Nitrogen Vacancy center electrons in diamond via readout of lattice
${}^{13}\textrm{C}$ nuclei. Spin-lock control on the ${}^{13}\textrm{C}$
nuclei yields significantly enhanced signal-to-noise for the nuclear readout.
Spectrally mapped readout presents operational advantages in being background-
free and immune to crystal orientation and optical scattering. We harness
these advantages to demonstrate applications in underwater magnetometry. The
physical basis for the “one-to-many” spectral map is itself intriguing. To
uncover its origin, we develop a theoretical model that maps the system
dynamics, involving traversal of a cascaded structure of Landau-Zener anti-
crossings, to the operation of a tilted “Galton board”. This work points to
new opportunities for “ESR-via-NMR” in dilute electronic systems, and in
hybrid electron-nuclear quantum memories and sensors.

Introduction – The control and readout of electronic spins forms the basis for
quantum sensing [1]. Magnetometry with diamond Nitrogen Vacancy (NV) centers
leverages the ability of the NV electronic spin to be optically initialized
and read out at room temperature [2, 3]. Let $\mathcal{g}(B_{0},f)$ denote the
NV electronic spectrum at bias field $B_{0}$ and frequency $f$. Optical
readout (ODMR) [4] maps $\mathcal{g}$ to fluorescence contrast
$S[\mathcal{g}]$ by exploiting the differential optical contrast between the
triplet ground state levels $m_{s}{=}\\{0,\pm 1\\}$ originating from state-
selective branching in the excited state [5, 6]. Ensemble DC magnetometry then
entails measuring a fluorescence change upon a shift in the electronic
spectrum, $\delta S{=}S[\mathcal{g}(B_{0}+\delta
B,f)]-S[\mathcal{g}(B_{0},f)]$ when subject to a probe magnetic field $\delta
B$ [7, 8].

Spin-fluorescence mapping, while attractive in many contexts, suffers from
technical limitations that can reduce sensitivity in some operational
scenarios. A typical example is when the N-V axis is misoriented with respect
to $\vec{B}_{0}$ [9]. State mixing in the excited state then leads to a steep
drop in fluorescence and ODMR contrast [9]. Even when perfectly aligned,
typical fluorescence contrast levels are ${\lesssim}5\%$ and susceptible to
optical backgrounds and readout laser amplitude noise [10, 11, 8]. In
scenarios requiring quantum sensing in fluidic (or optically dense) media,
optical scattering can further reduce contrast. There has been broad interest
in alternate strategies that map the NV electronic populations into other
parameters (e.g. charge [12, 13]) that can be efficiently readout.

Fig. 1: Spin mapping protocol. (A) System. NV center electrons ($e$)
surrounded by a network of ${}^{13}\textrm{C}$ nuclei ($n$), which are
employed to report the electronic spectrum $\mathcal{g}$. (B) Principle.
NV${\rightarrow}{}^{13}\textrm{C}$ spectral maps rely on mapping the NV
spectral DOS $\delta\mathcal{g}$ in a narrow $\Delta f$ window (purple) to
${}^{13}\textrm{C}$ polarization. Panel shows the network of energy levels
corresponding to an NV center coupled to five ${}^{13}\textrm{C}$ nuclei.
Solid red line denotes the resulting NV electronic spectrum $\mathcal{g}$.
Starting with optically initialized populations at $m_{s}{=}0$ (black balls),
chirped microwaves (MWs) [14, 15] (curly purple line) drive transitions to the
$m_{s}{=}$+1 manifold in the $\Delta f$ window (purple). (C) Experimental
protocol consists of DNP applied in narrow $\Delta f$ windows (left) and
interrogating the ${}^{13}\textrm{C}$ nuclei by spin-locked readout [16]
(right), here at 7T, employing ${\sim}$100k pulses. (D) Single shot NMR
spectrum from a representative $\Delta f{=}$10MHz window. Inset (i): time-
domain ${}^{13}\textrm{C}$ signal under spin-lock control (red) and free
induction decay (blue). For the former, we estimate a rotating frame lifetime
$T_{2}^{\prime}{\approx}2$s [16]. Inset (ii): noise measured in the NMR
spectral wing; we estimate a single-shot SNR ${\approx}1.25{\times}10^{4}$.
Fig. 2: NV center electronic spectral mapping via ${}^{13}\textrm{C}$ nuclear
readout. (A) NV electronic spectra mapped to ${}^{13}\textrm{C}$ populations
via DNP performed for 20s at $B_{0}{\approx}$33.6mT in $\Delta f{=}$10MHz
windows (purple) whose edge is denoted by $f_{0}$ (see Fig. 1B). Red (blue)
data corresponds to 1% (3%) ${}^{13}\textrm{C}$-enriched crystals. The four NV
families here are aligned at $54^{\circ}$ to $\vec{B}_{0}$ and overlap in
frequency. Solid lines are Gaussian fits. Linewidth $\ell$ increases with
enrichment. Visible are satellites corresponding to NV centers with first-
shell ${}^{13}\textrm{C}$ nuclei. Inset (i): Zoomed view of satellites, offset
$-$99.7MHz from the main peak. Normalized spectra show ${\approx}2.75$ greater
satellite intensity for the 3% enriched sample. (B) Linewidth $\ell$ of
${}^{13}\textrm{C}$-mapped NV spectral peak for a single NV center family for
different $\Delta f$ window widths. Error bars are calculated from Gaussian
fits. Solid line is a spline-fit guide to the eye. Linewidths $\ell$ grow with
window size $\Delta f$, and saturate at $\ell{\approx}2$MHz for $\Delta
f{<}$1MHz. (C) Log scale representation of data for 1% enriched sample from
(A). Solid lines are Gaussian fits. Satellites are ${\sim}$0.02 the intensity
of the main peak. (D) DNP buildup curves showing ${}^{13}\textrm{C}$
polarization buildup for the different points (i)-(vi) labeled in (F). Solid
lines are biexponential fits. (E) DNP buildup in the small-time limit is
approximately linear (solid lines). (F) Extracted buildup rates (colored
points (i)-(vi)) from the linear fits in (E) with error bars. These rates
closely overlay the NV spectrum (lighter purple points), obtained similar to
(A). (G) Measured ${}^{13}\textrm{C}$ relaxation times for the different
spectral locations $f_{j}$ (i)-(vi) in (D-F) demonstrating that relaxation
rates are constant (${\approx}30$s-1). (H) ${}^{13}\textrm{C}$-mapped NV
spectra obtained under alternate MW sweep directions with $\Delta f$=3MHz.
Data here is on a single NV center family. Signal intensities are sweep-sign
dependent, but otherwise identical for the entire spectrum.

Here, we demonstrate an alternate route to reading out the NV electronic
spectrum via ${}^{13}\textrm{C}$ nuclear spins in the surrounding environment
(Fig. 1A). Our strategy relies on mapping the NV spectral density of states
(DOS) at frequency $f_{0}$, $\delta\mathcal{g}(B_{0},f_{0})$, into
${}^{13}\textrm{C}$ nuclear polarization $P$ (see Fig. 1B),

$\delta\mathcal{g}(B_{0},f_{0})=\int_{f_{0}}^{f_{0}+\Delta
f}\mathcal{g}(B_{0},f)df\xrightarrow{\textrm{map}}P,$ (1)

where $\xrightarrow{\textrm{map}}$ refers to an approximate map in the limit
$\Delta f{\rightarrow}0$. The ${}^{13}\textrm{C}$ nuclei are subsequently
inductively readout via RF techniques (NMR) providing certain natural
advantages over optical readout [17]: it is crystal orientation independent,
background-free, and immune to fluorescence fluctuations due to scattering.

The central focus of this paper is to unravel the mechanism of the
NV${\rightarrow}{}^{13}\textrm{C}$ spectral-map in Eq. (1). Such a one-to-many
spectral map appears challenging at first (Fig. 1A-B): every NV center is
surrounded by multiple ${}^{13}\textrm{C}$ nuclei, which are in turn dipole-
coupled to each other, and selective ${}^{13}\textrm{C}$ control is difficult.
Our approach employs dynamic nuclear polarization (DNP) performed within
narrow $\Delta f$ spectral windows [14, 15] (Fig. 1C). We demonstrate that
this permits globally mapping the NV DOS $\delta\mathcal{g}$ into
${}^{13}\textrm{C}$ populations (Fig. 1B). A key contribution here is
uncovering the physical basis behind this process, and hence Eq. (1); we
accomplish this by viewing the system dynamics in Fig. 1A, manifesting as
traversals through a cascade of Landau-Zener (LZ) anti-crossings, to the
analogous operation of a “Galton board” which ultimately yields analytic
solutions.

Fig. 3: “Galton board” nuclear hyperpolarization and DOS mapping. (A) LZ anti-
crossings (LZ-LACs). For $N{=}3$, lab frame energy levels of Fig. 1B manifest
in the rotating frame as a cascade of LZ anti-crossings (solid lines) between
nuclear states of alternate electronic manifolds. Dashed lines show
corresponding crossings and dots highlight LZ-LAC points. (B) Equivalent
Galton board. Isolating LAC points allows us to abstract the system in terms
of a Galton board where the crossing points (red) arrange in the tilted
checkerboard $\mathcal{I}_{k,\ell}$ in a 2D plot of energy and frequency
(marked $k$ and $\ell$ respectively). The two electronic manifolds are
highlighted for clarity. (C) Classical Galton board consisting of pegs
arranged in a pyramidal pattern. Balls striking pegs bounce left or right with
probabilities $p$ or $q$ respectively. Driving force is provided by gravity
(marked in box). In the analogous LZ Galton board (B), “balls” are nuclear
populations, the “pegs” are LACs at which populations bifurcate, and MWs
provide the driving force. (D) Transfer matrix description of traversals.
Panel considers a representative section of a larger LAC cascade as in (A),
but focused here on only one LAC. Two-component column vectors
$\textbf{p}_{k,\ell}^{\mp}$ denote nuclear populations entering (or leaving)
the node $(k,\ell)$. Bifurcation (arrows) of the populations is described by
the transfer matrix $\textbf{T}_{k,\ell}$. (E) Analytical evaluation of
population evolution. Panels show populations $\mathcal{P}_{n}$ in the $2^{N}$
numbered nuclear states in the $m_{s}{=}0$ and $m_{s}{=}$+1 manifolds (blue
and red points respectively) upon a full MW sweep following Eq. (6) for
$N{=}4$ and $N{=}8$ (left and right panels), with $p{=}q{=}0.5$ in (B).
Representative nuclear states are marked. Dashed lines represent the
hyperpolarization level. (F) Numerical simulation of DOS mapping. Panels show
hyperpolarization generated from different sized $\Delta f$ windows, assuming
a Gaussian distribution of energy levels in the $m_{s}{=}+1$ manifold (see
Fig. 1B). Curves track the underlying electronic DOS and become wider with
increasing $\Delta f$ (as in Fig. 2B). Forward and reverse sweeps yield
positive and negative polarization respectively (as in Fig. 2H).

System – Experiments here are carried out on a single-crystal sample with an
ensemble of ${\sim}$1ppm NV centers (Fig. 1A). The NV spectrum
$\mathcal{g}(B_{0},f)$ (red line in Fig. 1B) is constructed out of both
homogeneous and inhomogeneous contributions: the former primarily due to
interactions with lattice ${}^{13}\textrm{C}$ nuclei, and the latter on
account of strain and orientational effects [18]. Fig. 1B, for instance,
describes the spectral DOS $\delta\mathcal{g}$ in window $\Delta f$ (purple)
arising from an NV center hierarchically coupling to multiple
${}^{13}\textrm{C}$ nuclei [19].

Our strategy to map the electronic spectrum $\mathcal{g}$ is outlined in Fig.
1C. DNP is applied via chirped MWs [14, 15] in a $\Delta f$ window (Fig. 1B)
over $\mathcal{g}(B_{0},f_{0})$ to transfer the corresponding
$\delta\mathcal{g}$ information to ${}^{13}\textrm{C}$ population differences.
This typically involves ${\sim}10^{4}$ MW sweeps (left panel of Fig. 1C),
during which polarization is accumulated in the ${}^{13}\textrm{C}$ nuclei.
Fig. 2 demonstrates that the corresponding ${}^{13}\textrm{C}$ polarization
closely tracks the electronic DOS $\delta g$ in the $\Delta f$ window swept
(see Fig. 1B). A novelty in this work with respect to previous reports [20,
21] is the enhanced signal-to-noise (SNR) readout of the ${}^{13}\textrm{C}$
nuclei. We employ a pulsed spin-locking protocol [22, 23] (right panel of Fig.
1C) that allows the ${}^{13}\textrm{C}$ populations to be continuously
interrogated for long periods up to the limit set by the rotating-frame
lifetimes, typically $T_{2}^{\prime}{>}2$s [16]. It consists of a train of
$\theta$-pulses spin-locked with the initial ${}^{13}\textrm{C}$ spin state
(Fig. 1C), simultaneously inhibiting dipolar interactions and e-induced
dephasing. Inductive readout (with bandwidth ${\approx}$10kHz) occurs after
every pulse (red line in Fig. 1C); typical experiments involve ${>}$100k
pulses. This yields a ${}^{13}\textrm{C}$ readout SNR that is boosted by
${\gtrsim}10-10^{3}$ times [16] with respect to traditional free induction
decay (FID) measurements. Fig. 1D shows a representative single-shot
${}^{13}\textrm{C}$ signal obtained using a $\Delta f{=}10$MHz window. Fig.
1D(i) shows the corresponding time-domain signal, and for comparison, the
fast-decaying FID readout. We estimate here a single-shot SNR
${\approx}1.25{\times}10^{4}$ by comparing to the noise level at the spectral
wing (Fig. 1D(ii)).

NV${\rightarrow}{}^{13}\textrm{C}$ spectral map – Fig. 2 elucidates results of
spectral mapping. In Fig. 2A, we consider two single crystal samples with
${}^{13}\textrm{C}$ enrichment levels at 1% (red) and 3% (blue) respectively.
Points here denote the single-shot inductively detected ${}^{13}\textrm{C}$
signal intensity at 7T following Fig. 1C with $\Delta f{=}$10MHz (purple
window). The high SNR data traces the underlying NV electronic DOS
$\delta\mathcal{g}(B_{0},f)$ with excellent fidelity (solid lines are Gaussian
fits), reflecting that the map in Eq. (1) is indeed viable. For instance,
there is no measurable ${}^{13}\textrm{C}$ signal (to within noise) for
$f_{0}$ where the NV spectral DOS is zero. Intriguingly, however, for a
$\Delta f$ window even in the wing of the spectrum (purple window in Fig. 1B),
the ${}^{13}\textrm{C}$ signal reflects $\delta\mathcal{g}$.

We observe two satellites, offset at ${\pm}99.7$MHz from the spectral center
(zoomed in Fig. 2A(i)). We identify them as arising from NV centers possessing
a ${}^{13}\textrm{C}$ nucleus in the first shell [24]; the high SNR
${}^{13}\textrm{C}$ readout here allows them to be easily discerned despite
their low concentration (${\approx}3\%$ of all NV centers in the 1% enriched
sample). Fig. 2A(i) shows that while we probe only bulk ${}^{13}\textrm{C}$
nuclei (hyperfine shifted by ${<}$10kHz), one can unravel $\delta\mathcal{g}$
information via polarization funneled into them from NV-proximal nuclei. Fig.
2C shows data for the 1% enriched sample in a logarithmic scale; the ratio of
the satellite and main peak intensities (${\approx}$0.02) approximately
follows the expected 3.3% concentration. Returning to Fig. 2A, it is evident
that increasing ${}^{13}\textrm{C}$ enrichment yields a larger electronic
spectral broadening [25] and increased (relative) satellite intensity, both as
theoretically predicted. Fig. 2B describes the measured spectral linewidth
$\ell$ (arrows in Fig. 2A), as a function of the sweep window $\Delta f$
employed. It increases approximately linearly with $\Delta f$, as expected
from Eq. (1), and saturates at $\ell{\approx}2$MHz, which correlates with the
linewidth due to strain-broadening [26]. Error bars here are obtained from the
Gaussian fits.

Some key features of the spectra in Fig. 2A are worth mentioning. Here, the NV
center families have their NV axes at 54∘ to the bias field $\vec{B}_{0}$: a
regime of strong misorientation where ODMR suffers significant
(${\gtrsim}70\%$) contrast loss [10]. RF readout here is background free, and
spin-locking provides a significantly higher SNR compared to previous reports
[27, 28, 20, 21]. As opposed to solid-effect DNP [29] or that obtained at the
NV ESLAC [30, 28], the sign of the ${}^{13}\textrm{C}$ NMR signals obtained
via Fig. 1C is identical for the entire spectrum in Fig. 2A [31]. This makes
unraveling the electronic spectra feasible even when they are inhomogeneously
broadened (see Fig. 4).

What is the physical origin of a higher ${}^{13}\textrm{C}$ polarization for a
spectral point $f$ at the peak in Fig. 2A versus at the wing? Insight is
provided by Fig. 2D-E which consider how the growth of nuclear polarization
relates to the electronic DOS. We study polarization buildup curves for
representative points $f_{j}$ on the spectrum $\mathcal{g}(B_{0},f_{j})$
(marked (i)-(vi) in Fig. 2F) for the natural abundance (1%) sample. Indeed,
the points in Fig. 2A reflect a slice of this data at $t{=}20$s. Polarization
buildup reflects an interplay between spin injection into directly hyperfine
coupled ${}^{13}\textrm{C}$ nuclei, spin diffusion, and ${}^{13}\textrm{C}$
relaxation. To a good approximation, ${}^{13}\textrm{C}$ relaxation rates are
determined by interactions with paramagnetic impurities (P1 centers) [32, 25],
and are independent of the spectral location $f_{j}$ being probed. This is
shown in Fig. 2G, where we find $T_{1n}{{\approx}}30$s (dashed line)
independent of $f_{j}$ (see [33]). Similarly, spin diffusion rates, driven by
internuclear interactions, can be considered to be independent of $f_{j}$
[34]. One expects then that data in Fig. 2A arises from differences in
polarization injection rates conditioned on $f_{j}$, this in turn produces the
frequency-dependent polarization levels and yields Eq. (1). Experiments in
Fig. 2D-F confirm this intuition. We restrict attention to short-time
(${<}$1s) buildup (zoomed in Fig. 2E) and linearize the corresponding
polarization growth curves (solid lines) to extract the injection rate.
Normalizing and overlaying these values (colored points) on the normalized
${}^{13}\textrm{C}$-interrogated NV electronic spectra in Fig. 2F (lighter
purple points) shows that the polarization injection rates produce the signal
differences that tracks $\delta\mathcal{g}(B_{0},f_{j})$.

Finally, Fig. 2H illustrates the result of an experiment similar to Fig. 2A
(with $\Delta f{=}$3MHz), but focused on a single NV center family and
employing alternate MW sweeps conditions [14, 31] (low-to-high or high-to-low
frequency). Both cases in Fig. 2H reflect the electronic spectrum, but with an
opposite sign. There is a small observable shift ($\approx$2.3MHz) from the
spectral center.

“Galton board” hyperpolarization – Experiments in Fig. 2 give rise to the
intriguing question: why is the map in Eq. (1), even if just approximate,
possible at all? This can be recast to asking why DNP in a $\Delta f$ window
in the wing of the spectrum (as in the purple window in Fig. 2A) should
produce a signal tracking $\delta\mathcal{g}$. Modeling polarization transfer
at the spectral wing requires the ability to solve dynamics for an electron
coupled to multiple ($N$) nuclei (as in Fig. 1A-B) under a MW sweep. This
requires extending analytic descriptions of DNP mechanisms beyond commonly
considered $e$-$n$ or $e$-$e$-$n$ systems [29, 35, 31, 36]. Here we develop an
approach to make such solutions tractable at large $N$. A companion paper [37]
elucidates this theory in greater detail; here we present its salient aspects
and connections to experiments.

The instantaneous Hamiltonian of the e-n system at a particular frequency
$f_{0}$ along the MW sweep is,

$\mathcal{H}(f_{0})=(\Delta-
f_{0})S_{z}^{2}+\gamma_{e}B_{0}S_{z}+\Omega_{e}S_{x}+\sum_{j=1}^{N}\left[\omega_{j}^{(0)}\mathcal{P}_{0}I_{zj}+\omega_{j}^{(1)}\mathcal{P}_{1}I_{z^{\prime}j}\right]$
(2)

where we restrict attention to the $m_{s}{=}\\{0,+1\\}$ manifold, ignore
internuclear interactions, and where $\Delta$=2.87GHz is the NV center zero
field splitting, $\Omega_{e}$ is the electronic Rabi frequency, and
$\mathcal{P}_{0}{=}\left(\begin{array}[]{cc}{1}&{0}\\\
{0}&{0}\end{array}\right)\otimes\textbf{1}_{N}\>,\>\mathcal{P}_{1}{=}\left(\begin{array}[]{cc}{0}&{0}\\\
{0}&{1}\end{array}\right)\otimes\textbf{1}_{N}$ are projection operators. The
eigenenergies of $\mathcal{H}(f_{0})$ manifest as a cascade of LZ-LACs, the
number of which scale exponentially with $N$. Fig. 3A shows this for a
representative example of $N{=}3$ and positive hyperfine couplings
$A_{j}^{\parallel}{>0}$.

Modeling experiments in Fig. 2A requires a machinery to track the nuclear
populations as this LZ cascade is traversed. We first extract the $2^{2N}$ LAC
points to a 2D checkerboard $\mathcal{I}_{k,l}$ (Fig. 3B) of energy and
frequency (axes labeled $k$ and $\ell$), and identify their corresponding
energy gaps as $\varepsilon_{k,\ell}$. Upon optical pumping, the initial
nuclear populations are equally distributed among nuclear states in the
$m_{s}{=}0$ manifold. Then under the MW sweep, the nuclear populations
traverse through $\mathcal{I}_{k,l}$ and redistribute down or right at every
LAC (Fig. 3B). These traversals can be viewed as analogous to the operation of
a classical Galton board [38, 39] (Fig. 3C) in which balls fall through a
system of pegs under gravity, encountering which they bounce left or right. In
a similar manner, the LACs in Fig. 3B form the “pegs”, the “balls” are the
nuclear populations, and the swept MWs provide the driving force. The
probability of population bifurcation “right” at each LAC is conditioned on
the size of the energy gap $\varepsilon_{k,\ell}$, and given by the tunneling
probability $\eta_{k,\ell}{=}\exp(-\varepsilon_{k,\ell}^{2}/\dot{f_{0}})$,
where $\dot{f_{0}}$ is the MW sweep rate. Downward bifurcation then has
probability $1-\eta_{k,\ell}$. The energy gaps
$\varepsilon_{k,2^{N}-k+1}{\approx}\Omega_{e}$ are the largest on the
checkerboard, making traversals through them approximately adiabatic.

We assume that the LACs in Fig. 3B are hit sequentially due to the tilted
nature of the LZ Galton board, and that despite continuous action of the
laser, electronic repolarization happens far away from the LACs. If
$\textbf{p}_{k,\ell}^{-}$ and $\textbf{p}_{k,\ell}^{+}$ are two-element column
vectors denoting the nuclear populations entering or leaving the $(k,\ell)$
LAC (Fig. 3D), then
$\textbf{p}_{k,\ell}^{+}{=}\textbf{T}_{k,\ell}\>\textbf{p}_{k,\ell}^{-},$
where
$\textbf{T}_{k,\ell}=\left(\begin{array}[]{cc}{\eta_{k,\ell}}&{\left(1-\eta_{k,\ell}\right)}\\\
{\left(1-\eta_{k,\ell}\right)}&{\eta_{k,\ell}}\end{array}\right)$ is a
transfer matrix [40, 41] describing the bifurcation of populations analogous
to a Galton board. Traversals through LACs as in Fig. 3A can then be
recursively solved as,
$\textbf{p}_{k,\ell}^{+}{=}\textbf{T}_{k,\ell}\textbf{M}_{r}\textbf{p}_{k-1,\ell}^{+}+\textbf{T}_{k,\ell}\textbf{M}_{d}\textbf{p}_{k,\ell-1}^{+},$
where the operators, $\textbf{M}_{d}{=}\left(\begin{array}[]{cc}{0}&{0}\\\
{0}&{1}\end{array}\right)\ \textrm{and}\
\textbf{M}_{r}{=}\left(\begin{array}[]{cc}{1}&{0}\\\
{0}&{0}\end{array}\right)\>,$ describe “walks” down and right through the
$\mathcal{I}_{k,\ell}$ board in Fig. 3B. Ultimately, one can quantify the
traversal probability between two points with coordinates $(k_{i},\ell_{i})$
and $(k_{f},\ell_{f})$ in $\mathcal{I}_{k,\ell}$ as [37],

$\mathcal{P}[(k_{i},\ell_{i}){\rightarrow}(k_{f},\ell_{f})]=\sum_{\\{v\\}\in
L_{p}}\left(\prod_{j=1}^{n}\mathcal{C}_{v_{j}}\right),$ (3)

where $L_{p}{=}\binom{L}{k_{f}-k_{i}}$ represents the total number of paths v
involved, defined by the nearest neighbor vertices such that
$\textbf{v}{=}\\{v_{j}\\}{=}\\{(k_{1},\ell_{1}),(k_{2},\ell_{2}),\cdots,(k_{n},\ell_{n})\\}$,
and the coefficients,

$\mathcal{C}_{v_{j}}=\left\\{\begin{array}[]{ll}\eta_{k_{j},\ell_{j}}&\text{if
$k_{j-1}=k_{j+1}$ or $\ell_{j-1}=\ell_{j+1}$}\\\
(1-\eta_{k_{j},\ell_{j}})&\text{if $k_{j-1}\neq k_{j+1}$ and
$\ell_{j-1}\neq\ell_{j+1}$}\\\ \end{array}\right..$ (4)

After the action of the laser that resets the NV to $m_{s}{=}0$, the
population of any nuclear state $\mathcal{S}$ is,
$\mathcal{P}_{n}{=}{p}_{2^{N},\ell_{n}}^{+}(1)+{p}_{k_{n},2^{N}}^{+}(2),$
where $n\in 1{\cdots}2^{N}$ indexes $\mathcal{S}$ in a Hamming ordering.
Consequently, the nuclear hyperpolarization $P$ in Eq. (1) and measured in
experiments in Fig. 2A takes the form,

$P=\sum_{n=1}^{2^{N-1}}{\mathcal{P}}_{n}-\sum_{n=2^{N-1}+1}^{2^{N}}{\mathcal{P}}_{n},$
(5)

for $A^{\parallel}{>}0$. The alternate case of $A^{\parallel}{<}0$ is shown to
produce no hyperpolarization in the $m_{s}{=}+1$ manifold [37].

Drawing the analogy to the classical Galton board to yield the solution
procedure in Eq. (3)-Eq. (5) for the coupled e-$(n)^{N}$ system in Fig. 1A is
a key result of this work. It permits an analytical (or numerical) means to
evaluate traversals through the full $\mathcal{I}_{k,\ell}$ checkerboard in
Fig. 3B, or in narrow $\Delta f$ window through it (as in Fig. 2A). Consider
first the solution of the full sweep through $\mathcal{I}_{k,\ell}$. Assuming
the probability of redistribution down and right to be $p$ and $q$
respectively at every LAC and that the large energy gaps yield adiabatic
traversals ($\eta_{k,2^{N}-k+1}{=}0$), assumptions that do not alter the
essential physics of the problem, the probability of ending in the nuclear
state $n{\in}\mathcal{S}$ after a full MW sweep can be analytically written as
[37]:

$\displaystyle\mathcal{P}_{n}=\frac{1}{2^{N}}\Bigg{(}\sum_{\ell=1}^{2^{N}-1}\Bigg{[}{(n{-}2){+}(2^{N}{-}\ell{-}1)\choose
n{-}2}p^{2^{N}{-}\ell}q^{n{-}2}{+}{(2^{N}{-}n{-}\ell)+(2^{N}{-}2)\choose
2^{N}{-}n{-}\ell}p^{2^{N}{-}n{-}\ell}q^{2^{N}{-}1}\Bigg{]}+c(n)\Bigg{)},$ (6)

where $\cdot\choose{\cdot}$ represents the combinatoric operator and $c(n)$ is
a constant equal to 1 when $n{=}1$ and 0 otherwise. Eq. (6) alludes to the
binomial nature of Galton board traversals [42] and provides insight into the
physics of the polarization transfer process. Solutions to $N{=}4$ and $N{=}8$
are displayed in Fig. 3E, showing populations of the numbered nuclear states
in the $m_{s}{=}0$ and $m_{s}{=}+1$ manifolds (blue and red points
respectively). Representative nuclear states are marked. As is evident,
traversal through the Galton board in Fig. 3B is “biased”, resulting in the
hyperpolarization $P$ denoted by the horizontal dashed lines in Fig. 3E.

In a similar manner, one can carry out calculations of traversal through a
narrow $\Delta f$ window on the LZ Galton board (Fig. 2A). While an analytic
solution is tractable, it is unwieldy and here we resort to a numerical
solution assuming an exemplary Gaussian distribution of energy levels
$\mathcal{g}$ (with width 13.5MHz) in the $m_{s}{=}+1$ manifold as in Fig. 1B.
The edge of the window ($f_{0}$) is swept in an analogous fashion to the
experiments in Fig. 2A, and nuclear populations are calculated following Eq.
(3)-Eq. (5). Fig. 3F demonstrates the result for varying $\Delta f$ and under
alternate sweep conditions. We observe that the polarization levels $P$
approximately track the Gaussian DOS $\delta\mathcal{g}$, qualitatively
matching the experimental results in Fig. 2. Intuitively, this is because the
Galton board operation “moves” nuclear populations following the underlying
distribution $\delta\mathcal{g}$. Larger $\Delta f$ windows (panels left-to-
right in Fig. 3F) yield a broader spectrum, and opposite sweeps produce an
alternate hyperpolarization sign (Fig. 3F). These match the experimental
observations in Fig. 2B and Fig. 2H.

Fig. 4: Underwater DC magnetometry via ${}^{13}\textrm{C}$ readout performed
with the sample submerged in ${\approx}$4.9mL water. (A-B) NV spectral maps of
misoriented crystals for when the NV centers are at (A) $\theta{=}54^{\circ}$
and (B) arbitrarily oriented to the bias field $B_{0}{=}33.6$mT (shown in
insets (i) and (ii) respectively). (C) DC magnetometry. Nuclear mapped NV
spectra for A(i) with different applied magnetic fields $\delta B$ collinear
with $B_{0}$. (D) Scaling of spectral center frequency with field.

Outlook and conclusions — This work opens many interesting future directions.
Fig. 2A suggests the possibility to perform “ESR-via-NMR”, portending avenues
to probe electronic spins via nuclei coupled to them. We envision applications
in dilute electronic systems or photopolarizable radicals [43], where ESR
spectra may be inaccessible, but where information can be instead relayed to
more abundant, longer-lived nuclear spins to be readout. Such “one-to-many”
spectral maps also suggest applications in quantum memories [44].

From a technological perspective, we envision applications in RF
(${}^{13}\textrm{C}$) interrogated NV-center magnetometry, without the use of
a MW cavity [45]. As opposed to optical NV sensors, this can permit DC
magnetometers that function in turbid or scattering media and with arbitrarily
oriented crystals. Fig. 4 demonstates a proof-of-concept, employing
NV${\rightarrow}{}^{13}\textrm{C}$ maps for bulk DC magnetometry underwater, a
regime with several applications (e.g. undersea magnetic anomaly detection
[46]), but where current quantum sensor technologies are not viable. Fig. 4A-B
shows NV spectra for samples placed under ${\approx}4.9$mL of water,
corresponding to about 2000-fold the volume of the sample. Here the four NV
axes are identically aligned at 54∘ to the bias field (Fig. 4A) and
arbitrarily oriented (Fig. 4B). We clarify that only the sample (and not the
excitation and detection apparatus) is submerged, but these experiments
suggest that fully underwater quantum sensor magnetometers are feasible.

DC magnetometry can be carried out by monitoring the NV spectral shift under
an applied probe field $\delta B$. Fig. 4C-D shows the resulting individual
${}^{13}\textrm{C}$-interrogated spectra, from where we estimate an underwater
DC sensitivity ${\approx}363\textrm{nT}/\sqrt{\textrm{Hz}}$. Sensitivity is
governed by the NMR readout SNR, which is still limited here due to the
relatively low NMR coil filling-factor (${\approx}$0.004), ${}^{13}\textrm{C}$
hyperpolarization level ($\approx$0.2%) and small readout time (0.3s) with
respect to the dead-time. These factors can be considerably improved with
advancements to the experimental apparatus (see Ref. [16]); we estimate a
sensitivity improvement by four orders of magnitude is feasible. Fig. 4B also
suggests underwater vector magnetometry exploiting the four NV center families
simultaneously [47].

In conclusion, we have demonstrated a technique to readout electronic spectra
by mapping them to spin population differences in surrounding nuclear spins
via DNP. We applied this to mapping the NV center spectrum $\mathcal{g}$ into
the ${}^{13}\textrm{C}$ spin polarization levels. We showed that this
physically originates from the DNP process stemming from traversals of
cascaded structure of LZ anti-crossings. In an analogy to the operation of a
“Galton board”, we showed that these traversals can be analytically solved,
yielding a means to extract the resulting hyperpolarization levels in the $N$
nuclear spins coupled to the NV center. Finally, we demonstrated practical
applications of the method for underwater bulk-diamond DC magnetometry. We
envision applications of the spectral mapping technique introduced here to a
broader class of hybrid quantum systems.

We acknowledge discussions with J. Reimer, S. Bhave, P. Zangara, C. Meriles
and D. Suter. This work is funded by ONR under contract N00014-20-1-2806.

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Supplementary Information
Electron-to-nuclear spectral mapping via “Galton board” dynamic nuclear
polarization
Arjun Pillai, Moniish Elanchezhian, Teemu Virtanen, Sophie Conti and Ashok
Ajoy

_Department of Chemistry, University of California, Berkeley, Berkeley, CA
94720, USA._

Fig. S1: NV spectral mapping via ${}^{13}\textrm{C}$ readout.  Data here is
for four NV center families, extending Fig. 4B of the main paper. Red (blue)
curves here represent ${}^{13}\textrm{C}$ hyperpolarization signals obtained
under low-to-high (high-to-low) frequency sweeps. Fig. S2: Polarization
injection and relaxation dynamics.  Raw data corresponding to Fig. 2G of the
main paper. (A) Here ${}^{13}\textrm{C}$ relaxation rates measured for
different $\Delta f{=}20$MHz windows are measured by allowing the sample to
relax in the polarizing field subsequent to hyperpolarization for $t{=}20$s.
Both the polarization buildup curves (also shown in Fig. 2D-E) of the main
paper and the corresponding relaxation decays are shown. (B) Overlaying the
normalized relaxation curves in (A) shows close overlap, indicating that the
relaxation rates are independent of the $f_{j}$ spectral location chosen.

##  I Extended Data

Extended data for Fig. 2 and Fig. 4 of the main paper are shown in Fig. S1 and
Fig. S2. First, Fig. S1 considers data corresponding to Fig. 4B of the main
paper for an arbitrarily oriented diamond crystal showing the four NV center
families. We note that for each family, the ${}^{13}\textrm{C}$-obtained NV
ESR spectrum is of the same sign, which is opposite for the two MW sweep
directions. This highlights the main property that makes it easy to uncover
the electronic spectrum in spite of the spectra having inhomogenously
broadened components.

Fig. S2A shows the raw data corresponding to Fig. 2G of the main paper. Here,
for different parts of the ESR spectrum (shown in Fig. 2F) of the paper, we
determine the polarization buildup characteristics at a bias field $B_{0}$
(also shown in Fig. 2D). After 20s of polarization injection (dashed line),
the ${}^{13}\textrm{C}$ spins are then allowed to relax at the same bias field
$B_{0}$. This relaxation data is normalized and shown in Fig. S2B, from where
it is evident that the relaxation characteristics are independent of the NV
ESR spectral location being originally polarized. The extracted time constants
of this decay were plotted in Fig. 2G of the main paper.