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# “Galton board” nuclear hyperpolarization

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

###### Abstract

We consider the problem of determining the spectrum of an electronic spin via
polarization transfer to coupled nuclear spins and their subsequent readout.
This suggests applications for employing dynamic nuclear polarization (DNP)
for “ESR-via-NMR”. In this paper, we describe the theoretical basis for this
process by developing a model for the evolution dynamics of the coupled
electron-nuclear system through a cascade of Landau-Zener anti-crossings (LZ-
LACs). We develop a method to map these traversals to the operation of an
equivalent “Galton board”. Here, LZ-LAC points serve as analogues to Galton
board “pegs”, upon interacting with which the nuclear populations
redistribute. The developed hyperpolarization then tracks the local electronic
density of states. We show that this approach yields an intuitive and
analytically tractable solution of the polarization transfer dynamics,
including when DNP is carried out at the wing of a homogeneously broadened
electronic spectral line. We apply this approach to a model system comprised
of a Nitrogen Vacancy (NV) center electron in diamond, hyperfine coupled to
$N$ neighboring ${}^{13}\textrm{C}$ nuclear spins, and discuss applications
for nuclear-spin interrogated NV center magnetometry. More broadly, the
methodology of “one-to-many” electron-to-nuclear spectral mapping developed
here suggests interesting applications in quantum memories and sensing, as
well as wider applications in modeling DNP processes in the multiple nuclear
spin limit.

##  I Introduction

The central spin model is the physical description underlying a wide class of
quantum mechanical systems relevant in quantum information processing [1, 2],
sensing [3, 4] and DNP [5, 6]. This includes solid-state qubits or sensors
constructed out of electronic defects and donors in materials such as silicon
[7, 8] and diamond [9, 10], and in the context of DNP, radical species
embedded within nuclear spins of analytes of interest [11, 12]. These systems
are typically comprised of dilute electronic spin centers interacting with a
several-fold larger nuclear spin bath (schematically shown in Fig. 1A).
Assuming weak inter-electron couplings, the electronic spectrum is dominated
by hyperfine couplings to the nuclei (Fig. 1B). In this paper, we examine
whether this homogeneously broadened spectrum can be extracted via
interrogation of the nuclear spin bath alone. We discuss how this might allow
the possibility for “ESR-via-NMR” in some restricted settings. Such a method
would form a complement to other techniques such as dynamical decoupling
spectroscopy [13, 14].

Consider the scenario in Fig. 1A describing $N$ nuclear spins ($n$) coupled to
an electronic spin ($e$) in a magnetic field $B_{0}$, while ignoring
internuclear interactions. Fig. 1B depicts the resulting energy levels in two
electronic manifolds ($m_{s}{=}\\{0,\textrm{+}1\\}$), assuming a spin-1
electron and $N{=}3$ spin-1/2 nuclei. We refer to the electronic spectral
density of states (DOS) as the number of electronic levels per unit frequency
in a particular manifold (see Fig. 2B). Our approach utilizes a DNP protocol
[15, 16, 17] (Fig. 2A) for selective hyperpolarization injection from the
central spin to the nuclear bath. We show that the extent of polarization
transferred reflects the underlying electronic DOS (see Fig. 2B-C), and
therefore reports on the electronic spectrum (see accompanying paper [18]).
This exploits special features of polarization transfer when employed at the
wing of the electronic spectral line. Assume that the electron is polarized in
one of the manifolds, here $m_{s}{=}0$, and the nuclear spins start in a
completely mixed state $\textbf{1}_{N}$. DNP is excited by the application of
frequency-swept irradiation (Fig. 2A) across a narrow window $\mathcal{B}$
whose size is much narrower than the spectral linewidth (Fig. 2B). The
magnitude of hyperpolarization is then found to track the DOS being swept
over, as demonstrated by the companion paper to this article [18] (see Fig.
2C).

Fig. 1: Principle.  (A) Central spin model consisting of an electron ($e$)
coupled to $N$ nuclear spins ($n$) in a magnetic field $B_{0}$, ignoring
internuclear couplings. We focus here on an NV center electron coupled to
${}^{13}\textrm{C}$ nuclei. The NV electron is polarized by optical pumping
(green), and polarization is transferred (blue arrows) to ${}^{13}\textrm{C}$
nuclei by means of microwave (MW) irradiation (see Fig. 2). (B) Lab frame
energy levels of the $e$-$n$ system for $N{=}3$, showing a wide separation of
nuclear levels arranged by electronic manifold. (C) LZ anti-crossings. In the
rotating frame of the applied MWs, the levels in (B) transform to a cascade of
LZ anti-crossings (solid lines) between nuclear states in alternate electronic
manifolds. Dashed lines show crossings corresponding to the diagonal
Hamiltonian (see Fig. 5). Dots refer to the crossing points. (D) Equivalent
Galton Board. Isolating the crossing points allows us to abstract the system
in terms of a Galton board (see Fig. 3) where the crossing points (red)
arrange in the checkerboard $\mathcal{I}_{k,\ell}$ in a 2D plot of energy and
frequency (labeled $k$ and $\ell$ respectively). The two electronic manifolds
are highlighted for clarity.

A primary focus of this manuscript is to model this surprising result
(detailed in Ref. [18]) via the corresponding polarization transfer dynamics
to $N$ nuclei. Fig. 1C-D intuitively capture the physics of the DNP process;
lab frame energy levels in Fig. 1B are transformed to a series of $2^{2N}$
cascaded LZ level anti-crossings (“LZ-LACs”) in the rotating frame (see Fig.
1C), wherein ground state electronic levels successively anti-cross with their
excited state counterparts. Here, we describe a mechanism to model the
dynamics of traversals through such an “LZ-cascade”. Building an analogy, we
show that the process can be mapped onto the operation of a “Galton board”
[19, 20, 21] (Fig. 3). Here, we isolate the position of the LZ-LACs (points in
Fig. 1C) and lay them out in a 2D space of energy and frequency (axes labeled
$k$ and $\ell$) as in Fig. 1D, where the horizontal and vertical lines refer
to the two electronic manifolds (highlighted). The resulting checkerboard
pattern in Fig. 1D, which we refer to as $\mathcal{I}_{k\ell}$, makes concrete
the analogy to the Galton board (Fig. 3) — the LZ points form “pegs”, upon
interacting with which, under frequency swept drive, the nuclear populations
redistribute amongst the various levels within an electronic manifold. We show
that this provides a simple way to track the population changes at every time
instant, ultimately capturing the essential physics of the experiments.

Fig. 2: Experimental motivation. (A) DNP protocol.
NV${\rightarrow}{}^{13}\textrm{C}$ polarization transfer is driven by chirped
MWs performed in narrow frequency windows (centered at $f_{0}$ with bandwidth
$\mathcal{B}$) under continuous laser illumination [15, 16] for period $T$.
(B) Schematic of hyperpolarization as it occurs at the wing of the electronic
spectral line. Representative levels showing eigenstates of the diagonal
Hamiltonian in the upper electronic manifold with increasing numbers of
nuclear spins. Starting with polarization at $m_{s}{=}0$ (black balls),
chirped MWs (curly purple line) drive transitions to the $m_{s}{=}$+1 manifold
in the small $\mathcal{B}$ window (purple box) as shown. Solid red line
denotes the electronic DOS in the upper manifold. (C) Experimental data on a
single-crystal sample with natural abundance ${}^{13}\textrm{C}$ and
$\approx$1ppm NV concentration. DNP here is performed at
$B_{0}{\approx}$33.6mT in $\mathcal{B}$=10MHz windows (purple box) whose
center is scanned in frequency. ${}^{13}\textrm{C}$ NMR signal here is
measured at 7T. ${}^{13}\textrm{C}$ hyperpolarization signal tracks the local
electronic DOS (see also companion paper [18]). Solid line is a Gaussian fit.
Inset (i): Zoom in to satellites, hyperfine shifted by $\approx$99.7MHz
corresponding to NV centers with first-shell ${}^{13}\textrm{C}$ nuclei.  Fig.
3: Galton Board Schematic. (A) Classical “Galton Board” consisting of pegs
placed in a pyramidal fashion. Balls striking pegs bifurcate left or right
with probabilities $p$ or $q$ respectively. Driving force is provided by
gravity (marked in box). (B) LZ Galton Board. Analogous Galton board (see Fig.
1D) formed in our system where pegs are the LZ-LAC points. Shown is the case
when $N{=}3$ (Fig. 1C). Balls are nuclear populations, which bifurcate down or
right. Driving force is provided by the applied MW sweep (see Fig. 2).

###  I.1 Experimental motivation for this work

The motivation for this work is the experiment shown in Fig. 2C and in the
companion paper [18], performed in a model system in diamond. Herein, the
electron in Fig. 1A is a spin-1 NV defect center, surrounded by multiple
${}^{13}\textrm{C}$ nuclear spins. Attractive spin-optical properties allow NV
electronic polarization to the $m_{s}{=}0$ state via optical pumping at room
temperature. Simultaneously, long ${}^{13}\textrm{C}$ lifetimes permit the
ability to interrogate them for minute long periods [22], providing high
signal-to-noise means to read them out.

DNP is applied at low field ($B_{0}{\approx}$1-70mT) using a MW frequency
chirp over a narrow window $\mathcal{B}$ centered at frequency $f_{0}$ and at
a repetition rate $\omega_{r}$ (Fig. 2A) [15, 17]. Fig. 2B schematically
represents this (purple window) on the NV spectrum, hyperfine-broadened by
coupling to a few ${}^{13}\textrm{C}$ nuclei. The spectral DOS is also
schematically drawn in on the right side of Fig. 2B (red line). The resulting
${}^{13}\textrm{C}$ polarization is measured at high field (7T).
Interestingly, we observe (see Fig. 2C) that as $f_{0}$ is scanned across in
frequency, the ${}^{13}\textrm{C}$ NMR signals (points) closely track the
underlying NV ESR spectral DOS. The satellites in Fig. 2C, shifted
${\approx}$99.7MHz with respect to the spectral center, correspond to NV
centers with a ${}^{13}\textrm{C}$ nucleus in their first shell [23]. A more
detailed exposition of these experiments is presented in the accompanying
paper [18]. Here we also study the scaling of the obtained linewidth with the
size of the sweep window $\mathcal{B}$ and the influence of the MW sweep
direction. We find that the width of the DNP derived spectrum (as in Fig. 2C)
saturates at ${\approx}$2MHz on the lower end, closely resembling the
intrinsic ESR linewidth. These experiments suggest that “ESR-via-NMR” may be
feasible in such systems [18].

While some aspects of the physics here are easy to intuit — sweeping over a
part of the spectrum in Fig. 2C where there is negligible electronic DOS
intensity leads to no hyperpolarization — it is still mysterious that narrow
sweeps applied to the wing of a spectral line should produce a
${}^{13}\textrm{C}$ polarization that seemingly tracks the local spectral DOS.
We note, however, that the fact of DNP enhancement amplitudes reflecting the
underlying electronic spectrum, at least qualitatively, is common knowledge
[24, 25, 26, 27]. It is widely employed in experiments optimizing DNP levels
and to characterize the effectiveness of a radical species in yielding
hyperpolarization [28]. However, the difference in the present work is that we
seek to make this connection more quantitative, exploiting the DNP levels here
to “sense” the ESR spectrum. Ultimately, this also portends applications in
magnetometry, as explored in Ref. [18] exploiting the ability to readout the
nuclear spins with high signal-to-noise employing spin-locking techniques
[22].

We benefit in this endeavor from special features of the employed DNP
mechanism [15, 17] (Fig. 2A). Every part of the spectrum produces
hyperpolarization, but of an identical sign, as evidenced by Fig. 2C. This is
in contrast to common DNP techniques (e.g. solid effect [5]) where the DNP
sign is inverted at the red or blue shifted side of the spectral center. The
latter can cause cancellation in the DNP enhancements between different parts
of the spectrum, especially at low fields or for inhomogeneously broadened
spectra, making determining the underlying spectrum non-trivial [29].

It is evident that a full description of the DNP mechanism yielding the
results at the wing of the spectrum in Fig. 2C should include the physics of
polarization dynamics when the electron is coupled to $N$ nuclear spins. In
that respect, our work contributes to strategies for the analytic descriptions
of DNP in the large $N$ limit, extending common approaches that focus
primarily on $e$-$n$ or $e$-$e$-$n$ systems (e.g. solid and cross effects [5,
30, 6]), and where larger nuclear networks are evaluated fully numerically [5,
31, 32]. Indeed, previous calculations of the DNP technique [15] employed here
were also carried out only in the single $e$-$n$ limit [33]. Instead in this
paper, relying on a set of simplifying assumptions and a mapping to a Galton
board (Fig. 3), we show that a mechanistic description with $N$ coupled nuclei
as in Fig. 1A is tractable, and can lead to experimentally verifiable
predictions.

###  I.2 Summary of approach

Fig. 1C-D and Fig. 4 summarize the basic theoretical approach. The problem of
solving the hyperpolarization dynamics becomes that of tracking the
populations in a system as they encounter a cascade of LZ-LACs under a MW
sweep. The exponentially scaling number of anti-crossings (${\propto}2^{2N}$)
makes this problem challenging; however with some approximations (detailed
below), the dynamics can be restricted to that of populations at the LZ points
alone (Fig. 1D) .

Our solution entails building an analogy to the operation of a Galton board.
Consider the LZ checkerboard in Fig. 4. At $t{=}0$, the nuclear populations
are equally distributed along the levels of the $m_{s}{=}0$ manifold. Upon
encountering an LZ point, however, they are redistributed with a finite
probability either “right” or “down” on the checkerboard
$\mathcal{I}_{k,\ell}$ with the probability defined by the size of the local
LZ energy gap. Tracking the population dynamics can now be handled with a
transfer matrix formalism for any specific trajectory traversed through the LZ
checkerboard. If v defines a path through ${n}$ points in the 2D checkerboard
(black arrow in Fig. 4) with coordinates
$v{=}\\{v_{j}\\}{=}\\{(k_{1},\ell_{1}),(k_{2},\ell_{2}),\cdots,(k_{j},\ell_{j})\cdots(k_{n},\ell_{n})\\}$
(see Fig. 4B), and $\textbf{p}_{i}$ ($\textbf{p}_{f}$) are column vectors
denoting nuclear populations at the initial (final) LZ point, traversals can
be evaluated analogous to the operation of a Galton machine,

$\textbf{p}_{f}{=}\textbf{T}_{v_{n}}\textbf{M}_{(v_{n},v_{n-1})}\cdots\textbf{T}_{v_{j}}\textbf{M}_{(v_{j},v_{j-1})}\textbf{T}_{v_{2}}\textbf{M}_{(v_{2},v_{1})}\textbf{T}_{v_{1}}\textbf{M}_{r}\cdot\textbf{p}_{i}$
(1)

where $\textbf{T}_{v_{j}}$ is a “redistribution” operator at the LZ point
$v_{j}$, and $\textbf{M}_{(v_{n},v_{n-1})}$ and $\textbf{M}_{r}$ are “walk”
operators connecting the nearest neighbor points that define the path v.
Through this simple map, we demonstrate the origin of bias in the traversals
which is responsible for nuclear hyperpolarization. We then show that DNP
carried out in small windows $\mathcal{B}$ of the electronic spectrum occurs
such that the hyperpolarization follows the electronic DOS, a reflection of
the experimental data in Fig. 2C.

Fig. 4: Galton board of Landau-Zener LACs. (A) Panel shows the Landau-Zener
LACs for a representative case of an NV center coupled to $N{=}4$
${}^{13}\textrm{C}$ nuclei and positive $A_{j}^{\parallel}$. Representative
nuclear states are marked. The LZ-LACs (dots) form the $\mathcal{I}_{k,\ell}$
checkerboard as in Fig. 1D, where coordinates mark their individual positions.
Axes $\\{k,\ell\\}$ refer to energy and frequency respectively. Center of the
checkerboard is marked $\mathcal{O}$ and the conjugate diagonal (dashed line)
refers to positions $\mathcal{I}_{k,2^{N}-k+1}$. Eigenstates corresponding to
the diagonal Hamiltonian $\mathcal{H}_{\textrm{diag}}$ are labeled (see Fig.
5). At $t{=}0$, populations start equally distributed amongst the nuclear
states restricted to the $m_{s}{=}0$ manifold. A MW sweep causes the system to
encounter the LZ-LACs in a sequential manner. The black arrow shows an
exemplary trajectory v through the checkerboard. (B) Zoom-in to a section
(blue dashed box) of the checkerboard showing specific vertices $v_{j}$
forming the trajectory v. Nuclear populations are redistributed at every LZ-
LAC similar to the operation of a Galton board (Fig. 3).  Fig. 5: Emergence of
cascaded Landau-Zener LACs described in a systematic manner for an NV center
coupled to $N{=}3$ ${}^{13}\textrm{C}$ nuclei in a magnetic field $B_{0}$ (see
Fig. 1A) using the model in Eq. (10). We focus on the
$m_{s}{=}\\{0,\textrm{+}1\\}$ electronic manifolds, and introduce Hamiltonian
terms hierarchically to illustrate the physical origin of the LZ-LACs (shaded
insets). (A) Considering only the diagonal Hamiltonian
$\mathcal{H}_{\textrm{diag}}$ (Eq. (6)) with $A_{j}^{\parallel}{=}0$, and (B)
with finite $A_{j}^{\parallel}$. There are $2^{2N}$ crossing points between
nuclear states in the two electronic manifolds, labeled here by the
corresponding eigenstates (shown are representative states). (C) Including now
the Rabi frequency term $\Omega_{e}$ from $\mathcal{H}_{\textrm{non-diag}}$
(Eq. (7)) leads to LZ-LACs between states on the conjugate diagonal (see Fig.
4) involving a direct electronic spin flip,
$\left|{0,\mathcal{S}}\right\rangle{\leftrightarrow}\left|{\textrm{+}1,\mathcal{S}}\right\rangle$
for any nuclear state $\mathcal{S}$. (D) Finally, considering the full
Hamiltonian $\mathcal{H}_{\textrm{tot}}$ (Eq. (5)), LZ-LACs appear at each
crossing point. The corresponding energy gaps display the native hierarchy
based on the total number of spin flips they involve (see App. A).

##  II System of Cascaded Landau-Zener LACs

###  II.1 Central spin system and hyperpolarization protocol

The NV-${}^{13}\textrm{C}$ system is considered here as a model of a hybrid
$e$-$n$ system that captures two commonly encountered features reflected in
Fig. 1A:

  1. (i)

the electron concentration is orders of magnitude more dilute than that of the
nuclear spins, and thus inter-electron couplings can be neglected, and,

  2. (ii)

nuclear $T_{1n}$ far exceeds that of the electron, and polarization can be
made to accumulate in the nuclei through repeated polarization transfer from
the electron.

In our samples (e.g as employed in Fig. 2C), the NV centers are surrounded by
an average of ${\sim}10^{4}$ ${}^{13}\textrm{C}$ nuclei, and inter-electron
spacings are $\left\langle r_{e}\right\rangle{\approx}$12nm [34]. If $w_{L}$
defines the rate of NV electronic polarization buildup, the electron
polarization follows as $P_{e}(\tau)\approx 1-\exp(-\tau w_{L})$. At
sufficiently high optical powers, $w_{L}$ is set by the intensity of the laser
illumination applied. At low powers however, $w_{L}{\approx}T_{1e}^{-1}$ [17];
the electron polarization rate is then set by just the thermal regeneration
rate.

###  II.2 NV-${}^{13}\textrm{C}$ System Hamiltonian

The cascade of LZ-LACs as in Fig. 1C and Fig. 4 originates from diagonalizing
the instantaneous system Hamiltonian,

$\mathcal{H}_{\textrm{tot}}(t)=\mathcal{H}+\mathcal{H}_{\textrm{MW}}(t),$ (2)

at every step of the applied frequency sweep. The two terms here refer to the
spin Hamiltonian and applied chirped MW control respectively. The latter has
the form,

$\mathcal{H}_{\textrm{MW}}(t)=\Omega_{e}S_{x}\cos(\omega_{\textrm{MW}}t)$ (3)

where $S_{x}$ is a spin-1 Pauli operator and $\Omega_{e}$ is the electronic
Rabi frequency. The instantaneous MW frequency, $\omega_{\textrm{MW}}(t)$, is
of the form,

$\omega_{\textrm{MW}}(t)=\mathcal{B}\omega_{r}t+{f_{0}}-\mathcal{B}/2$ (4)

where $\omega_{r}$ is the sweep repetition rate. We will focus here on a
scenario as in Fig. 1A, considering a system of $N$ ${}^{13}\textrm{C}$
nuclear spins directly hyperfine-coupled to an NV center with hyperfine
strength $\|A_{j}\|$, where $j=1,{\cdots},N$. We ignore dipolar couplings
between the ${}^{13}\textrm{C}$ nuclei, focusing instead on a direct
NV${\rightarrow}{}^{13}\textrm{C}$ polarization transfer process. In the low-
field DNP regime, the bare ${}^{13}\textrm{C}$ Larmor frequency is smaller
than or comparable to the strength of the hyperfine couplings,
$\omega_{L}{=}\gamma_{n}B_{0}{\lesssim}\|A_{j}\|$ [15], where
$\gamma_{n}{=}1.07$kHz/G is the nuclear magnetogyric ratio.

Now considering the spin Hamiltonian $\mathcal{H}$ for a system of $N$
${}^{13}\textrm{C}$ nuclear spins hyperfine-coupled to the NV center as in
Fig. 1A, it is convenient to consider the nuclear Hamiltonian selectively in
each NV manifold. We focus here just on the two-level system formed by the
$m_{s}{=}\\{0,\textrm{+}1\\}$ manifolds (see Fig. 1B). We assume for
simplicity that $\textbf{B}_{0}$ is aligned with the NV axis, a direction we
label $\hat{z}$. We separate the hyperfine field into its longitudinal and
transverse components,
$\textbf{A}_{j}{=}A_{j}^{\parallel}\hat{z}+A_{j}^{\perp}\hat{\perp}$, where
$\hat{\perp}$ is a unit-vector in the $xy$ plane. Some comments are worth
noting:

  1. (i)

Experiments (e.g. Fig. 2C) are performed in the ensemble limit and thus, to a
good approximation, there is an equal probability of spins with positive or
negative $A^{\parallel}$ couplings.

  2. (ii)

The $m_{s}{=}0$ NV state is “non-magnetic”, and the nuclear spins are
predominantly quantized here in the Zeeman field $\textbf{B}_{0}$ with
frequency $\omega_{L}{=}\gamma_{n}B_{0}$. In reality, there is a weak
additional second-order coupling mediated by the hyperfine term, leading to an
effective nuclear frequency in the $m_{s}{=}0$ manifold,
$\omega^{(0)}_{j}{\approx}\omega_{L}{+}\frac{\gamma_{e}B_{0}A_{j}^{\perp}}{\Delta}$,
where $\Delta$=2.87GHz is the NV center zero field splitting. The nuclear
Hamiltonian in the $m_{s}{=}0$ manifold is then of the form,
$\mathcal{H}_{n}^{(0)}{=}\sum_{j=1}^{N}\omega_{j}^{(0)}I_{zj}$,where $I$ are
spin-1/2 Pauli operators.

  3. (iii)

In the $m_{s}{=}$+1 manifold, on the other hand, there is a combined action of
the Zeeman and hyperfine fields,
$\omega^{(1)}_{j}{=}\sqrt{(\omega_{L}+A_{j}^{\parallel})^{2}+(A_{j}^{\perp})^{2}}$.
Indeed, the quantization axis $\hat{z^{\prime}_{j}}$ here need not be
collinear with $\textbf{B}_{0}$; in general,
$\hat{z’_{j}}{=}\hat{z}\cos\phi_{j}{+}\hat{x}\sin\phi_{j}$, where the angle
$\phi_{j}{=}\tan^{-1}\left[A_{j}^{\perp}/(\omega_{L}+A_{j}^{\parallel})\right]$.
The $m_{s}{=}$+1 nuclear Hamiltonian is then of the form,
$\mathcal{H}_{n}^{(1)}{=}\sum_{j=1}^{N}\omega_{j}^{(1)}I_{z^{\prime}j}$, where
$I_{z^{\prime}j}{=}I_{zj}\cos\phi_{j}+I_{xj}\sin\phi_{j}$.

Going into a rotating frame with respect to $S_{z}^{2}$ at frequency
$\omega_{\textrm{MW}}$, gives the net system Hamiltonian in Eq. (2),

$\mathcal{H}_{\textrm{tot}}(\omega_{\textrm{MW}})=(\Delta-\omega_{\textrm{MW}})S_{z}^{2}+\gamma_{e}B_{0}S_{z}+\Omega_{e}S_{x}+\sum_{j=1}^{N}\left[\omega_{j}^{(0)}\mathcal{P}_{0}I_{zj}+\omega_{j}^{(1)}\mathcal{P}_{1}I_{z^{\prime}j}\right]\>,$
(5)

where $\mathcal{P}_{0}=\left(\begin{array}[]{cc}{1}&{0}\\\
{0}&{0}\end{array}\right)\otimes\textbf{1}_{N}\>;\>\mathcal{P}_{1}=\left(\begin{array}[]{cc}{0}&{0}\\\
{0}&{1}\end{array}\right)\otimes\textbf{1}_{N}$, are the respective projection
operators to the NV manifolds, and $\textbf{1}_{N}$ is a unit operator on the
nuclear spins.

###  II.3 Eigenstates and eigenenergies

Plotted with respect to the instantaneous frequency $\omega_{\textrm{MW}}$,
the eigenvalues of $\mathcal{H}_{\textrm{tot}}$ in Eq. (5) undergo a sequence
of cascaded LZ-LACs (as in Fig. 1C and Fig. 4). To determine their positions,
our strategy will be to first calculate the crossing points, focusing only on
the diagonal Hamiltonian, and subsequently promoting them to anti-crossings by
including off-diagonal terms (see Fig. 5). To this end, consider first
separating $\mathcal{H}_{\textrm{tot}}$ into its diagonal and off-diagonal
parts,
$\mathcal{H}_{\textrm{tot}}(\omega_{\textrm{MW}})=\mathcal{H}_{\textrm{diag}}(\omega_{\textrm{MW}})+\mathcal{H}_{\textrm{non-
diag}}$, where,

$\mathcal{H}_{\textrm{diag}}(\omega_{\textrm{MW}})=(\Delta-\omega_{\textrm{MW}})S_{z}^{2}+\gamma_{e}B_{0}S_{z}+\sum_{j=1}^{N}\left[\omega_{j}^{(0)}\mathcal{P}_{0}I_{zj}+\omega_{j}^{(1)}\mathcal{P}_{1}I_{zj}\cos\phi_{j}\right]\>\\\
$ (6) $\mathcal{H}_{\textrm{non-
diag}}=\Omega_{e}S_{x}+\sum_{j=1}^{N}\omega_{j}^{(1)}\mathcal{P}_{1}I_{xj}\sin\phi_{j}\>$
(7)

The latter encapsulates contributions from $\\{\Omega_{e},A_{j}^{\perp}\\}$
terms. Considering only $\mathcal{H}_{\textrm{diag}}$ allows us to label the
eigenstates (Fig. 1D), and identify points at which LZ-LACs will ultimately
arise (see also Fig. 4 and Fig. 5). Specifically, in Fig. 5A-D, we show the
systematic emergence of the LZ-LACs in a series of steps highlighted in the
insets —

  1. (i)

considering $\mathcal{H}_{\textrm{diag}}$, but with only the Zeeman terms of
the electron and nuclear spins (Fig. 5A);

  2. (ii)

considering $\mathcal{H}_{\textrm{diag}}$, but including the hyperfine terms
(Fig. 5B) results in the crossing points that closely define the checkerboard
$\mathcal{I}_{k,\ell}$ as in Fig. 1D;

  3. (iii)

including the Rabi term from $\mathcal{H}_{\textrm{non-diag}}$ (see Fig. 5C)
results in a set of anti-crossings at many former crossing points, here
corresponding to levels where the electronic, but not the nuclear, states
change;

  4. (iv)

and finally, including the full extent of off-diagonal terms in
$\mathcal{H}_{\textrm{non-diag}}$ (Fig. 5D), gives rise to anti-crossings at
each of the former crossing points, although with differing energy gaps
(described in Sec. IIE and App. A).

To label eigenstates in Eq. (6), we assume that $\mathcal{S}$ labels the
$2^{N}$ possible nuclear spin states in each electronic manifold consisting of
all combinations of spin up or down. For instance, for $N{=}2$, we have
$\mathcal{S}{=}\\{\left|{\downarrow\downarrow}\right\rangle,\left|{\downarrow\uparrow}\right\rangle,\left|{\uparrow\downarrow}\right\rangle,\left|{\uparrow\uparrow}\right\rangle\\}$.
These also correspond to the eigenstates of the diagonal Hamiltonian
$\mathcal{H}_{\textrm{diag}}$. Using Hamming binary notation with ${h_{j}}{=}$
0 or 1 representing $\left|{\uparrow}\right\rangle$ or
$\left|{\downarrow}\right\rangle$ respectively, the eigenenergies of the
diagonal Hamiltonian in the $m_{s}{=}\\{0,\textrm{+}1\\}$ manifolds take the
forms,

$E_{J}^{(0)}(\omega_{\textrm{MW}}){=}\sum_{j=1}^{N}(-1)^{h_{j}}\omega_{j}^{(0)}\\\
$ (8)
$E_{J}^{(1)}(\omega_{\textrm{MW}}){=}(\Delta+\gamma_{e}B_{0}-\omega_{\textrm{MW}})+\sum_{j=1}^{N}(-1)^{h_{j}}\omega_{j}^{(1)}\>.$
(9)

This allows us to constrain the locations of the LZ-LACs in the checkboard.
The index $J$ here runs over $2^{N}$ and we note from Eq. (8) that the
eigenvalues are independent of $\omega_{\textrm{MW}}$ in the $m_{s}{=}0$
manifold, and scale linearly with it in the $m_{s}{=}$+1 case (Eq. (9)). While
the discussion here is applicable for an arbitrary hyperfine network
$A_{j}^{\parallel}$, in this paper we consider an exemplary scenario where
hyperfine couplings are in a non-degenerate configuration such that,

$\omega_{j}^{(0)}=j^{p}\>;\>\omega_{j}^{(1)}={\alpha}j^{p}.$ (10)

For example, in Fig. 5, we employ this model with ${\alpha}{=}5$ and the
exponent $p{=}1.1$. The lack of degeneracies in this model illuminate the
physics of LZ traversals more clearly, while the same general principles
follow in the degenerate case as well.

###  II.4 Checkerboard of anti-crossings $\mathcal{I}_{k,\ell}$

Let us now return to the situation in Fig. 5B and determine the exact
positions of the LZ-LACs. Eigenenergies in each of the
$m_{s}{=}\\{0,\textrm{+}1\\}$ manifolds form separate sets of parallel lines,
and lines corresponding to different manifolds intersect. For $N$ nuclear
spins connected to the central NV center, there are in general $2^{2N}$
crossing points. Together, these crossing points form the checkerboard pattern
$\mathcal{I}_{k,\ell}$ (as in Fig. 1D and Fig. 4) in the 2D plot of energy and
frequency.

Let $\mathcal{I}$ refer to the matrix of coordinates of these crossing points
in a diagram as in Fig. 4, where the abscissa $\mathcal{I}_{k,\ell}(1)$ and
ordinate $\mathcal{I}_{k,\ell}(2)$ refer to frequency and energy respectively.
The label indices $k,\ell\in 1{\cdots}2^{N}$ refer to levels in the
$m_{s}{=}{\textrm{+}1}$ and $m_{s}{=}0$ manifolds respectively. We arrange the
crossings from left-to-right and top-to-bottom (i.e. increasing in frequency
and descending in energy). Referring to Fig. 4, the element
$\mathcal{I}_{1,1}$, for instance, refers to the crossing between levels
$\left|{0,\uparrow\uparrow\uparrow\uparrow}\right\rangle$ and
$\left|{\textrm{+}1,\downarrow\downarrow\downarrow\downarrow}\right\rangle$.
Following this convention, the elements can be written as,

$\mathcal{I}_{k,\ell}=\left(E_{k}^{(1)}(0)-E_{\ell}^{(0)},E_{\ell}^{(0)}\right)$
(11)

given that from Eq. (8),

$E_{k}^{(1)}(\omega_{\textrm{MW}})=E_{k}^{(1)}(0)-\omega_{\textrm{MW}}$ (12)

and thus, at the LZ-LACs,

$\omega_{\textrm{MW}}=E_{k}^{(1)}(0)-E_{\ell}^{(0)}.$ (13)

For instance, the LZ-LAC
$\mathcal{I}_{2,1}{=}\left(E_{2}^{(1)}(0)-E_{1}^{(0)},E_{1}^{(0)}\right)$ is
formed from the intersection of two levels in the
$m_{s}{=}\\{0,\textrm{+}1\\}$ manifolds. The $\mathcal{I}_{k,\ell}$
checkerboard pattern is displayed in Fig. 1D and Fig. 4, and is tilted since
the crossings corresponding to lower nuclear states occur at higher frequency,
$\mathcal{I}_{k,{\ell}+1}(1){>}\mathcal{I}_{k,\ell}(1)$. A key consequence is
that the system, upon a MW sweep, encounters LZ-LACs in a sequential manner
(Fig. 1D) — important for ultimately causing “bias” in the nuclear
polarization buildup.

###  II.5 LACs: Energy gaps, symmetries and hierarchies

Continuing our discussion onto Fig. 5C-D, we now elucidate the emergence of
the LZ-LACs and their associated energy gaps. We will refer to the
checkerboard energy gaps at positions (${k,\ell}$) as $\varepsilon_{k,\ell}$,
using the same notation convention as above. The gaps arise approximately at
the abscissa locations marked by $\mathcal{I}_{k,\ell}(1)$. Numerically
extracted gap center locations are plotted as points in Fig. 1D and Fig. 4,
while the dashed lines refer to the checkerboard crossing points obtained
using an approach similar to Fig. 5C. For simplicity, we refer to the
$\mathcal{I}_{k,k}$ positions on the checkerboard as those on the diagonal,
while those at the positions $\mathcal{I}_{k,2^{N}-k+1}$ belong to the
“conjugate diagonal” (see Fig. 4).

To intuitively unravel the formation of the LZ-LACs, consider first the action
of the Rabi frequency term $\Omega_{e}$, as in Fig. 5C. Energy gaps open up
only at the conjugate diagonal (for positive $A_{j}^{\parallel}$),
corresponding to the anti-crossings between states
$\left|{0,\mathcal{S}}\right\rangle{\rightarrow}\left|{\textrm{+}1,\mathcal{S}}\right\rangle$,
where $\mathcal{S}$ is any state of the nuclear spins. The energy gaps here
$\varepsilon_{k,2^{N}-k+1}{=}\Omega_{e}$, since this transition just
corresponds to a single electron flip — akin to the situation encountered in a
rapid adiabatic passage over the electrons. Including the perpendicular
hyperfine terms $A_{j}^{\perp}$ in Eq. (7) provides additional LZ-LACs at each
of the $\mathcal{I}_{k,\ell}$ crossing points in Fig. 5D.

App. A shows a detailed evaluation of the corresponding gaps, but we comment
here that they satisfy two mirror symmetry conditions (evident in Fig. 4),

$\displaystyle|\mathcal{O}-\mathcal{I}_{k,\ell}|$ $\displaystyle=$
$\displaystyle|\mathcal{O}-\mathcal{I}_{2^{N}-k+1,2^{N}-\ell+1}|$ (14)
$\displaystyle\varepsilon_{k,\ell}$ $\displaystyle=$
$\displaystyle\varepsilon_{2^{N}-k+1,2^{N}-\ell+1}$ (15)

  1. (i)

The first condition (Eq. (14)) describes a mirror symmetry of the checkerboard
diagram with respect to the center $\mathcal{O}=(\Delta+\gamma_{e}B,0)$ (see
Fig. 4), a consequence of the spin-1/2 ${}^{13}\textrm{C}$ nuclei and the fact
that all $A_{j}^{\parallel}$ are positive in the chosen model in Eq. (10).

  2. (ii)

Similarly, Eq. (15) elucidates a symmetry in the energy gaps about the
conjugate diagonal on the checkerboard (see Fig. 4). For instance, the gaps
corresponding to LZ-LACs
$\left|{0,\uparrow\uparrow\uparrow}\right\rangle{\leftrightarrow}\left|{\textrm{+}1,\downarrow\downarrow\uparrow}\right\rangle$
and
$\left|{0,\downarrow\downarrow\uparrow}\right\rangle{\leftrightarrow}\left|{\textrm{+}1,\uparrow\uparrow\uparrow}\right\rangle$
should be identical because they just differ in an exchange between nuclear
labels undergoing the crossing.

Fig. 6: Transfer matrix formalism.  Panel shows a schematic of traversal
through a Galton system of anti-crossings. We consider a representative
section of a larger $\mathcal{I}_{k,\ell}$ checkerboard as in Fig. 4, but
focus here on only four LZ points (dots). Populations bifurcate at the LZ
points similar to a Galton board (Fig. 3) upon a driven low-high frequency MW
sweep. Components of $\textbf{p}_{k,\ell}^{\mp}$, two-component column vectors
denoting nuclear populations entering (or leaving) the node $(k,\ell)$, are
shown. Bifurcation of the populations is described by the transfer matrix
$\textbf{T}_{k,\ell}$ (Eq. (20)). We assume there is no system evolution
between successive LZ points; it follows that
$p_{k,\ell}^{+}(1){=}p_{k+1,\ell}^{-}(1)$ and
$p_{k,\ell}^{+}(2){=}p_{k,\ell+1}^{-}(2)$, and yields a recursion relationship
connecting nearest-neighbor populations (Eq. (22)). This permits a simple
evaluation of system evolution in terms of nuclear population redistribution
probabilities upon traversal through the Galton board.

##  III Transfer matrix formalism

###  III.1 Traversals through cascaded Landau-Zener LACs

We now develop the analogy with the Galton board (see Fig. 3) to evaluate the
evolution of nuclear populations through the checkerboard
$\mathcal{I}_{k,\ell}$, and show how this evolution yields nuclear
hyperpolarization. We make two operational assumptions:

  1. (i)

The $\mathcal{I}_{k,\ell}$ LZ-LAC points are assumed to be hit sequentially
and their effects are evaluated individually. This is reasonable if the energy
gaps $\varepsilon_{k,\ell}$ are small compared to their frequency separation.

  2. (ii)

NV (re)polarization is assumed to happen far away from the exact LZ-LAC points
$\mathcal{I}_{k,\ell}$, a good approximation when
$\varepsilon_{k,\ell}{<}\mathcal{B}$. This allows us to consider the LZ system
traversals independent of optical pumping.

At each checkerboard point, let $\textbf{p}_{k,\ell}^{-}$ and
$\textbf{p}_{k,\ell}^{+}$ denote the nuclear populations of the two states
undergoing the LZ-LAC, before and after the anti-crossing is encountered. This
is shown in Fig. 6 for four LZ crossings in a 2x2 portion of the
$\mathcal{I}_{k,\ell}$ checkerboard. We denote the populations as column
vectors, with the first (second) element referring to nuclear populations in
the $m_{s}{=}0$ ($\textrm{+}1$) manifolds respectively,

$\textbf{p}_{k,\ell}^{-}=\left[\begin{array}[]{c}p_{k,\ell}^{-}(1)\\\
p_{k,\ell}^{-}(2)\\\
\end{array}\right]\>;\>\textbf{p}_{k,\ell}^{+}=\left[\begin{array}[]{c}p_{k,\ell}^{+}(1)\\\
p_{k,\ell}^{+}(2)\\\ \end{array}\right].$ (16)

The effect of any chosen trajectory through (as in Fig. 4) the LZ cascade can
then be built out of simple 2x2 matrices. Evolution starts with the population
restricted to the $m_{s}{=}0$ manifold, and with the ${}^{13}\textrm{C}$
nuclear spins in a mixed state $\textbf{1}_{N}$. This means that each of the
nuclear states in the $m_{s}{=}0$ manifold have an equal initial starting
probability. Traversal can then be calculated by considering populations
starting with the initial states,
$\textbf{p}_{1,\ell}^{-}=\left[\begin{array}[]{c}1\\\ 0\\\ \end{array}\right]$
and averaging the result over the $2^{N}$ $\ell$ states.

###  III.2 Transfer matrix formalism

Consider first the action of a single LZ-LAC point $\mathcal{I}_{k,\ell}$ with
energy gap $\varepsilon_{k,\ell}$. Encountering it leads to a rearrangement of
nuclear populations,

$\displaystyle p_{k,\ell}^{+}(1)$ $\displaystyle=$
$\displaystyle\eta_{k,\ell}p_{k,\ell}^{-}(1)+\left(1-\eta_{k,\ell}\right)p_{k,\ell}^{-}(2)$
$\displaystyle p_{k,\ell}^{+}(2)$ $\displaystyle=$
$\displaystyle\left(1-\eta_{k,\ell}\right)p_{k,\ell}^{-}(1)+\eta_{k,\ell}p_{k,\ell}^{-}(2)$
(17)

where $\eta_{k,\ell}$ refers to the tunneling probability through the anti-
crossing and has the form,

$\eta_{k,\ell}{=}\exp(-\varepsilon_{k,\ell}^{2}/\dot{\omega}_{\textrm{MW}}).$
(18)

Here $\dot{\omega}_{\textrm{MW}}$ is the MW sweep rate, and
$\dot{\omega}_{\textrm{MW}}{=}\mathcal{B}{\omega}_{r}$ for the linear chirp
employed (see Fig. 2A). Eq. (17) can then be recast in terms of a transfer
matrix $\textbf{T}_{k,\ell}$,

$\textbf{p}_{k,\ell}^{+}=\textbf{T}_{k,\ell}\>\textbf{p}_{k,\ell}^{-}\>,$ (19)

where,

$\textbf{T}_{k,\ell}=\left(\begin{array}[]{cc}{\eta_{k,\ell}}&{\left(1-\eta_{k,\ell}\right)}\\\
{\left(1-\eta_{k,\ell}\right)}&{\eta_{k,\ell}}\end{array}\right)\>.$ (20)

$\textbf{T}_{k,\ell}$ plays the role of a “redistribution” operator since it
causes the rearrangement of populations at the $\mathcal{I}_{k,\ell}$ anti-
crossing points (Fig. 6). Traversal is adiabatic if
$\dot{\omega}_{\textrm{MW}}\varepsilon_{k,\ell}^{-2}{\ll}1$ and diabatic if
$\dot{\omega}_{\textrm{MW}}\varepsilon_{k,\ell}^{-2}{\gtrsim}1$. In the
limiting case when the gap is large ($\eta_{k,\ell}{\rightarrow}0$) and there
is a complete transfer of population between the states at the LZ-LAC,
$\textbf{T}_{k,\ell}=\left(\begin{array}[]{cc}{0}&{1}\\\
{1}&{0}\end{array}\right)$. On the other hand, for a purely diabatic transfer,
$\textbf{T}_{k,\ell}=\textbf{1}_{2}=\left(\begin{array}[]{cc}{1}&{0}\\\
{0}&{1}\end{array}\right)$, and the populations are preserved. In a more
generic, yet diabatic case, there is a bifurcation of populations at the
$\mathcal{I}_{k,\ell}$ crossing point (Fig. 6). Note that the matrix
formulation above only considers the redistribution of nuclear populations at
each LZ-LAC and ignores coherences. In reality, there is a phase picked up by
the quantum state as it traverses through an LZ-LAC. However, in our
experiments (to a good approximation), this phase can be ignored because:

  1. (i)

In the high defect density sample we consider in Fig. 2, the electronic
$T_{2e}^{*}{\sim}50{\textrm{ns}}{\ll}\omega_{r}^{-1}$ is short [35] \- and
much smaller than the single sweep time $\omega_{r}^{-1}$ as well as the
traversal time between successive $\mathcal{I}_{k,\ell}$ points.

  2. (ii)

The bare nuclear coherence time over which electrons are repolarized —
$T_{2n}^{*}{\sim}$1ms — is of the order of the sweep period, but much smaller
than the total polarization time (which involves ${>}10^{3}$ sweeps) (Fig.
2A). To a good approximation then, any nuclear coherences produced as a result
of the traversal rapidly die away due to internuclear interactions.

  3. (iii)

In experiments (Fig. 2C), we perform an ensemble average over a large
collection of $e$-$n$ systems like in Fig. 1A. In this case, the coherence of
the electronic state can be considered randomized [36] and one can consider
only the redistributions of state populations at each LZ-LAC.

Eq. (19)-Eq. (20) make more concrete the analogy to the Galton board (Fig. 3).
A classical Galton board operates through balls falling through a system of
pegs under gravity (see Fig. 3A). At each peg, the balls can bounce left or
right (usually with 50% probability each). In a similar manner, the LZ-LACs
here form the “pegs”, the “balls” are the nuclear populations, and the swept
MWs provide the driving force analogous to gravity (Fig. 3B). The LZ Galton
board here is tilted, and the nuclear populations bounce at each LZ-LAC either
right or down, the probability being conditioned on the size of the
corresponding energy gap $\varepsilon_{k,\ell}$.

Building on this analogy, let us now evaluate the action of sequentially
encountering LZ anti-crossings under a low-to-high frequency MW sweep.
Extending Eq. (19), the net traversal can then be written as a product of
transfer matrices corresponding to a trajectory through the cascaded LZ-LACs.
Since there is no population evolution between successive LZ-LACs,
$p_{k,\ell}^{-}(1){=}p_{k-1,\ell}^{+}(1)$ and
$p_{k,\ell}^{-}(2){=}p_{k,\ell-1}^{+}(2).$ Therefore Eq. (19) can be rewritten
as,

$\textbf{p}_{k,\ell}^{+}=\textbf{T}_{k,\ell}\>\textbf{p}_{k,\ell}^{-}=\textbf{T}_{k,\ell}\left[\begin{array}[]{c}p_{k-1,\ell}^{+}(1)\\\
p_{k,\ell-1}^{+}(2)\\\ \end{array}\right]$ (21)

or equivalently,

$\textbf{p}_{k,\ell}^{+}=\textbf{T}_{k,\ell}\textbf{M}_{r}\textbf{p}_{k-1,\ell}^{+}+\textbf{T}_{k,\ell}\textbf{M}_{d}\textbf{p}_{k,\ell-1}^{+}$
(22)

where we define the “walk” operators,

$\textbf{M}_{d}=\left(\begin{array}[]{cc}{0}&{0}\\\
{0}&{1}\end{array}\right)\>;\>\textbf{M}_{r}=\left(\begin{array}[]{cc}{1}&{0}\\\
{0}&{0}\end{array}\right).$ (23)

$\textbf{M}_{d}$ and $\textbf{M}_{r}$ refer to walks over the checkerboard by
a single-step “down” or “right” respectively. Eq. (22) is interesting because
it connects populations at LZ-LAC sites $(k,\ell)$ with prior nearest neighbor
sites on the checkerboard, schematically described in Fig. 6. Combining Eq.
(19) and Eq. (22) provides the recursive relationship,

$\displaystyle\textbf{p}_{k,\ell}^{+}$ $\displaystyle=$
$\displaystyle\textbf{T}_{k,\ell}\textbf{M}_{r}\textbf{T}_{k-1,\ell}\textbf{M}_{r}\textbf{p}_{k-2,\ell}^{+}+\textbf{T}_{k,\ell}\textbf{M}_{r}\textbf{T}_{k-1,\ell}\textbf{M}_{d}\textbf{p}_{k-1,\ell-1}^{+}$
$\displaystyle+$
$\displaystyle\textbf{T}_{k,\ell}\textbf{M}_{d}\textbf{T}_{k,\ell-1}\textbf{M}_{r}\textbf{p}_{k-1,\ell-1}^{+}+\textbf{T}_{k,\ell}\textbf{M}_{d}\textbf{T}_{k,\ell-1}\textbf{M}_{d}\textbf{p}_{k-1,\ell-2}^{+},$

which connects an $\mathcal{I}_{k,\ell}$ checkerboard point to its nearest and
next-nearest neighbors (see Fig. 6). This process can be continued to
encompass every path of evolution through the cascaded LZ system. Consider for
instance the portion of the trajectory v (black line) in Fig. 4B. The final
nuclear population through it is then,
$\textbf{p}_{4,3}^{+}=\textbf{T}_{\textrm{eff}}\textbf{p}_{1,1}^{-}$, where
$\textbf{p}_{1,1}^{-}=\left[\begin{array}[]{c}1\\\ 0\\\ \end{array}\right]$
where,

$\textbf{T}_{\textrm{eff}}=\textbf{M}_{r}\textbf{T}_{43}\textbf{M}_{r}\textbf{T}_{33}\textbf{M}_{r}\textbf{T}_{23}\textbf{M}_{d}\textbf{T}_{22}\textbf{M}_{d}\textbf{T}_{21}\textbf{M}_{r}\textbf{T}_{11}\textbf{M}_{r}.$
(24)

In general, consider a path consisting of a sequence of nearest neighbor
points through $\mathcal{I}_{k,\ell}$:
$v{=}\\{v_{j}\\}{=}\\{(k_{1},\ell_{1}),\cdots,(k_{j},\ell_{j})\cdots(k_{n},\ell_{n})\\}$.
If $\textbf{p}_{i}^{-}$ and $\textbf{p}_{f}^{+}$ are the populations at the
initial and final coordinates respectively, this recursive condition can be
generalized,

$\displaystyle\textbf{p}_{f}^{+}$ $\displaystyle=$
$\displaystyle\textbf{T}_{v_{n}}\textbf{M}_{(v_{n},v_{n-1})}\cdots\textbf{T}_{v_{3}}\textbf{M}_{(v_{3},v_{2})}\textbf{T}_{v_{2}}\textbf{M}_{(v_{2},v_{1})}\textbf{T}_{v_{1}}\textbf{M}_{r}\cdot\textbf{p}_{i}^{-}$
(25) $\displaystyle=$
$\displaystyle\prod_{j=1}^{n}\textbf{T}_{v_{j}}\textbf{M}_{(v_{j+1},v_{j})}\cdot\textbf{p}_{i}^{-}$

where $\textbf{T}_{v_{j}}$ is the corresponding bounce operator at coordinate
$v_{j}{=}(k_{j},\ell_{j})$, and the operator M refers to the walk through the
checkerboard,

$\textbf{M}_{(v_{j+1},v_{j})}=\left\\{\begin{array}[]{ll}\textbf{M}_{r}&\text{for
$k_{j+1}=k_{j}+1$}\\\ \textbf{M}_{d}&\text{for $\ell_{j+1}=\ell_{j}+1$}\\\
\end{array}\right.$ (26)

based on whether the subsequent trajectory entails a movement right or down in
the board. From Eq. (22) and Eq. (23), and since
$\textbf{p}_{i}=\left[\begin{array}[]{c}1\\\ 0\\\ \end{array}\right]$, it is
now possible to evaluate the final probability of the path defined by the
coordinates $v=\\{v_{j}\\}$ as,

$\mathcal{P}=\prod_{j=1}^{n}\mathcal{C}_{v_{j}},$ (27)

with 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..$ (28)

These two factors entering Eq. (28) can be intuitively understood. If at a
specific LZ-LAC the trajectory v involves a path that continues straight and
does not take a “bend”, then the term that enters the product is the direct
tunneling probability $\eta_{k,\ell}$. Alternatively, a bend appears with the
factor $(1-\eta_{k,\ell})$. This allows for the final probability
$\mathcal{P}$ (and thus final population) for any path to be easily found. For
simplicity, we decompose $\mathcal{P}$ at any LZ-LAC point into two
components, $\mathcal{P}=[\mathcal{P}^{(1)},\mathcal{P}^{(2)}]^{T}$, with
$\mathcal{P}^{(1)}$ representing final $m_{s}{=}$0 populations and
$\mathcal{P}^{(2)}$ representing final $m_{s}{=}$+1 populations (see Fig. 7).
For instance, when applied to the full trajectory v in Fig. 4 (exiting in the
$m_{s}{=}$+1 manifold), one can just write down the trajectory,

$\displaystyle\mathcal{P}^{(2)}{=}\eta_{1,1}(1-\eta_{2,1})\eta_{2,2}(1-\eta_{2,3})\eta_{3,3}(1-\eta_{4,3})\eta_{4,4}\eta_{4,5}\eta_{4,6}\eta_{4,7}$
$\displaystyle\eta_{4,8}\eta_{4,9}\eta_{4,10}(1-\eta_{4,11})\eta_{5,11}(1-\eta_{6,11})\eta_{6,12}\eta_{6,13}\eta_{6,14}\eta_{6,15}\eta_{6,16}.$

Taking identical $\eta_{k,\ell}$ values for every energy gap
$\varepsilon_{k,\ell}$ gives the simple expression,

$\displaystyle\mathcal{P}^{(2)}{=}{\eta}^{16}(1-\eta)^{5}.$

This tractable method of tracking nuclear spin populations is the key result
of this paper.

Fig. 7: Calculation schematic for transfer probability $\mathcal{P}$.
Representative LZ Galton board showing a schematic of population evolution for
$N{=}2$ nuclei. Energy levels are marked. Panel shows the probability
$\mathcal{P}$ of arrival at an LZ-LAC (boxed), and probabilities
$\mathcal{P}^{(1)}$ and $\mathcal{P}^{(2)}$ of bifurcation right or down
(traveling towards the $m_{s}{=}0$ and $m_{s}{=}$+1 manifolds respectively).
Colored arrows represent contributions to the same nuclear state $\mathcal{S}$
from different electronic manifolds, which are eventually combined due to the
action of the laser (Eq. (30)). For example, the probability of ending in the
$\left|{\downarrow\downarrow}\right\rangle$ state (assuming unit value
starting population) is
$\mathcal{P}_{\downarrow\downarrow}=p^{+}_{4,4}(1)+p^{+}_{4,4}(2)$.
Contributing populations from both electronic manifolds, $p^{+}_{4,4}(1)$ and
$p^{+}_{1,4}(2)$, are highlighted by red arrows in this case.

###  III.3 General traversal through the $\mathcal{I}_{k,\ell}$ checkerboard

Following from Eq. (25)-Eq. (28), given a final coordinate in the
$\mathcal{I}_{k,\ell}$ checkerboard, it is possible to retrospectively
determine all paths that lead to it, and hence the transfer probability as the
sum over all the paths the system can traverse. Considering traversal from an
initial coordinate $(k_{i},\ell_{i})$ to a final coordinate
$(k_{f},\ell_{f})$, there are in general $(\ell_{f}{-}\ell_{i})$ vertical
steps and $(k_{f}{-}k_{i})$ horizontal steps required. This gives rise to
$L=(k_{f}+\ell_{f}-k_{i}-\ell_{i})$ total traversal steps, and there are hence
$L_{p}{=}\binom{L}{k_{f}-k_{i}}$ total paths of the traversal, where
$\left(\cdot\right)$ is the binomial operator. Following Eq. (27), we can
write the traversal probability from $(k_{i},\ell_{i})$ to $(k_{f},\ell_{f})$
as,

$\mathcal{P}[(k_{i},\ell_{i}){\rightarrow}(k_{f},\ell_{f})]{=}\sum_{\\{v\\}\in
L_{p}}\left(\prod_{j=1}^{n}\mathcal{C}_{v_{j}}\right).$ (29)

Eq. (29) generalizes finding traversal probabilities for any trajectory over
vertices v, where the index $j$ represents the position on the path.

###  III.4 Electron repolarization: Action of laser

Key to the hyperpolarization process is the ability to regenerate the
populations of the NV electrons by means of optical pumping. This permits the
ability to repeatedly accumulate polarization in the nuclear spins, using them
to indirectly report on the ESR spectrum of the electrons. We assume the
action of the laser instantaneously resets the population of the
$m_{s}{=\textrm{+}1}$ NV state to $m_{s}{=}0$, while keeping nuclear
populations unaffected (e.g.
$\left|{\textrm{+}1,\mathcal{S}}\right\rangle{\rightarrow}\left|{0,\mathcal{S}}\right\rangle$).
If $\textbf{p}_{k,\ell}^{{}^{\prime}}$ denotes nuclear populations after the
action of the laser, then for all points $(k,\ell)$ on the LZ-LAC
checkerboard,
$\textbf{p}_{k,\ell}^{{}^{\prime}\pm}{=}\textbf{L}\cdot\textbf{p}_{k,\ell}^{\pm}$,
with laser action described by the operator,

$\textbf{L}{=}\left(\begin{array}[]{cc}{1}&{1}\\\
{0}&{0}\end{array}\right)\>.$ (30)

We are making the assumption here that the extent of each of the LZ anti-
crossing regions (defined by the energy gaps $\varepsilon_{k,\ell}$) on the
checkerboard is small in comparison to the region between them. Note also that
in the DNP protocol in Fig. 2A, the laser is applied continuously with the MW
sweep. In principle, therefore, the action of the laser can occur at any point
during the traversal through the LZ checkerboard. The essential physics are,
however, unaffected if we consider the laser reset as just occurring at the
end (or beginning) of the traversal through the entire LZ checkerboard. This
physically means that finding the net populations after each cycle requires us
to evaluate the sum over corresponding populations in both electronic
manifolds. This is schematically shown in Fig. 7, where for instance, the
population of nuclei reaching the state
$\left|{\downarrow\downarrow}\right\rangle$ is calculated by the sum,
$\mathcal{P}_{\downarrow\downarrow}=p^{+}_{4,4}(1)+p^{+}_{4,4}(2)$.

##  IV Galton board hyperpolarization

###  IV.1 Calculating the nuclear polarization $P$

Elucidating the magnitude of polarization is essential to understanding the
origin of nuclear polarization “bias”, and is key to this is calculating
traversal probability through the LZ Galton board. Using Eq. (29), we can find
the population at any final point $(k_{f},\ell_{f})$, which then permits
finding the population of any nuclear state $\mathcal{S}$ after traversal
through an LZ cascade. Since each nuclear state appears once for each
electronic manifold, the imbalance upon system traversal can be accounted for
by evaluating the total population in each state at any time instant through
the diagram. The population in the $m_{s}{=}0$ state then takes the form (see
Eq. (17)),

${p}_{2^{N},\ell_{f}}^{+}(1){=}\sum_{n=1}^{2^{N}}\mathcal{P}^{(1)}[(1,n){\rightarrow}(2^{N},\ell_{f})],$
(31)

where the sum indicates the total probability starting from the left column of
the diagram. Similarly, the population in the $m_{s}{=}$+1 state, assuming
$A^{\parallel}{>}0$, takes the form,

${p}_{k_{f},2^{N}}^{+}(2){=}\sum_{n=1}^{2^{N}}\mathcal{P}^{(2)}[(1,n){\rightarrow}(k_{f},2^{N})].$
(32)

Due to the action of the laser (Eq. (30)), these populations are effectively
merged and the population of any nuclear state $\mathcal{S}$ is,

$\mathcal{P}_{n}{=}{p}_{2^{N},\ell_{n}}^{+}(1)+{p}_{k_{n},2^{N}}^{+}(2)$ (33)

where the subscript $n\in 1{\cdots}2^{N}$ here indexes the state
$\mathcal{S}$, and the individual terms can be calculated following Eq. (29).
For instance, this is schematically shown for the state
$\mathcal{P}_{\downarrow\downarrow}$ in Fig. 7. Finally, the nuclear
polarization $P$ is defined as the net excess of population in the nuclear
down states compared to the up states, and for the case of $A^{\parallel}{>}0$
that we consider,

$P{=}\sum_{n=1}^{2^{N-1}}{\mathcal{P}}_{n}-\sum_{n=2^{N-1}+1}^{2^{N}}{\mathcal{P}}_{n}.$
(34)

###  IV.2 Special case: Single-spin ratchet

As a simple application, consider an NV center coupled to a single
${}^{13}\textrm{C}$ nuclear spin. The mechanism here was referred to as a
“spin ratchet” in Ref. [15]. We note that while this case ($N{=}1$) was
considered before in Ref. [33], the Galton board analogies in Fig. 8 provide a
simple means to analyze the bifurcation and recombinations of the populations
in a intuitive and graphical manner. The analogy also provides a means to
generalize the mechanism of operation to larger $N$, an apriori non-trivial
problem that has not been previously considered elsewhere.

In Fig. 8, we consider both cases of hyperfine coupling $A^{\parallel}{>}0$
(Fig. 8A) and $A^{\parallel}{<}0$ (Fig. 8B), with Fig. 8C-D showing the
corresponding $\mathcal{I}_{k,\ell}$ checkerboards. Apart from the reversed
order of levels in the $m_{s}{=}$+1 manifold, the structures in these diagrams
are similar. This inversion arises from the difference in order in which the
LZ-LACs corresponding to the “small” and “large” energy gaps are encountered
during a MW sweep.

Fig. 8: “Spin ratchet” hyperpolarization corresponding to the case of an NV
center coupled to a single ${}^{13}\textrm{C}$ nuclear spin $(N{=}1)$. We
consider two cases, (A) $A^{\parallel}{>}0$ and (B) $A^{\parallel}{<}0$.
Panels show LZ anti-crossings similar to Fig. 1D, from which we abstract the
checkerboards in the lower panels. Nuclear states are shown in either case.
Large energy gaps corresponding to $\Omega_{e}$ are highlighted. (C-D) LZ
checkerboards corresponding to panels (A) and (B) respectively which elucidate
the change in the order of energy levels in the $m_{s}{=}$+1 manifold, and the
resulting change in the order in which the small and large energy gaps are
encountered. We assume here that large energy gaps have $\eta{=}0$ and thus do
not allow direct tunneling. The possible trajectories of populations starting
in the $\left|{0,\uparrow}\right\rangle$ and
$\left|{0,\downarrow}\right\rangle$ states are shown (purple and blue arrows
respectively). Polarization buildup occurs only in the former case
($A^{\parallel}{>}0$), while interference of the traversals through the Galton
system leads to a destruction of nuclear polarization buildup in the latter
case ($A^{\parallel}{<}0$).

(i) Case of $A^{\parallel}{>}0$: Consider a single traversal from left-to-
right starting from populations restricted to the $m_{s}{=}0$ manifold and
ending with NV repolarization. Using Eq. (33), and following from Eq. (29),
the probability of the nuclear state $\left|{\downarrow}\right\rangle$
remaining in $\left|{\downarrow}\right\rangle$ can be written as,

$\mathcal{P}(\downarrow\>\rightarrow\>\downarrow)={p}_{2,2}^{+}(1)+{p}_{1,2}^{+}(2),$
(35)

taking,

${p}_{2,2}^{+}(1)=\mathcal{P}^{(1)}[(1,2){\rightarrow}(2,2)],\>\textrm{and}\>,{p}_{1,2}^{+}(2)=\mathcal{P}^{(2)}[(1,2)],$
(36)

in the Galton board in Fig. 8C. This provides us a method to calculate the
individual probabilities as,

$\displaystyle\mathcal{P}(\downarrow\>\rightarrow\>\downarrow)$
$\displaystyle=$ $\displaystyle\left(1-\eta_{1,2}\right)+\eta_{1,2}\eta_{1,1}$
$\displaystyle\mathcal{P}(\downarrow\>\rightarrow\>\uparrow)$ $\displaystyle=$
$\displaystyle\eta_{1,2}\left(1-\eta_{1,1}\right)$
$\displaystyle\mathcal{P}(\uparrow\>\rightarrow\>\downarrow)$ $\displaystyle=$
$\displaystyle\eta_{1,2}\left(1-\eta_{1,1}\right)+2\eta_{1,1}\left(1-\eta_{1,1}\right)\left(1-\eta_{1,2}\right)$
$\displaystyle\mathcal{P}(\uparrow\>\rightarrow\>\uparrow)$ $\displaystyle=$
$\displaystyle\eta_{1,1}\eta_{1,2}+\eta_{1,1}^{2}\left(1-\eta_{1,2}\right)+\left(1-\eta_{1,1}\right)^{2}\left(1-\eta_{1,2}\right),$

where the tunneling probabilities follow Eq. (18) and we have exploited
symmetries in energy gaps (described in Sec. IIE). Each term in these
expressions corresponds to a different trajectory through
$\mathcal{I}_{k,\ell}$. For instance, there are two paths that constitute the
term, $\mathcal{P}(\downarrow\>\rightarrow\>\downarrow)$, corresponding to the
probabilities of
$\left|{0,\downarrow}\right\rangle{\rightarrow}\left|{0,\downarrow}\right\rangle$
and
$\left|{0,\downarrow}\right\rangle{\rightarrow}\left|{\textrm{+}1,\downarrow}\right\rangle$.
This permits a means to determine the nuclear hyperpolarization, where we
evaluate the difference in populations between the nuclear states at the end
of the sweep as,

$\displaystyle P$ $\displaystyle=$
$\displaystyle\left[\mathcal{P}(\downarrow\>\rightarrow\>\downarrow)+\mathcal{P}(\uparrow\>\rightarrow\>\downarrow)\right]-\left[\mathcal{P}(\downarrow\>\rightarrow\>\uparrow)+\mathcal{P}(\uparrow\>\rightarrow\>\uparrow)\right]$
(37) $\displaystyle=$
$\displaystyle\left(1-\eta_{1,2}\right)\left[1-\left(2\eta_{1,1}-1\right)^{2}\right].$

Ultimately, the expression in Eq. (37) demonstrates that the polarization
levels are obtained as the difference of the probabilities of transitions to
spin down states and transitions to spin up states.

It is simplest to evaluate Eq. (37) for the situation when the Rabi frequency
$\Omega_{e}$ is large so that the traversals through the energy gaps
$\varepsilon_{1,2}$ and $\varepsilon_{2,1}$ are adiabatic, but those through
the others are diabatic. We then have $P{=}1-\left(2\eta_{1,1}-1\right)^{2}$;
a non-zero hyperpolarization develops through an accumulation of nuclear spins
in the $\left|{\downarrow}\right\rangle$ state with a single MW sweep.
Building on Eq. (37) then, the net hyperpolarization developed in a total time
$T$ (Fig. 2A) is,

$P_{\textrm{net}}=\left[1-\exp\left(\frac{-w_{L}}{\omega_{r}}\right)\right]\cdot
T\omega_{r}\cdot\left(1-\eta_{1,2}\right)\left[1-\left(2\eta_{1,1}-1\right)^{2}\right],$
(38)

where the first term encapsulates the starting electron polarization, the
second term ($\propto T\omega_{r}$) gives the total sweeps in time $T$, and
the last term is Eq. (37). The mechanism can therefore be thought of as a
“ratchet” — every MW sweep develops a finite amount of polarization.

(ii) Case of $A^{\parallel}{<}0$: A similar analysis can be carried out for
the situation with $A^{\parallel}{<}0$ (see Fig. 8B), yielding,

$\displaystyle\mathcal{P}(\downarrow\>\rightarrow\>\downarrow)$
$\displaystyle=$ $\displaystyle\eta_{1,1}\eta_{1,2}+\eta_{1,2}(1-\eta_{1,1})$
$\displaystyle\mathcal{P}(\downarrow\>\rightarrow\>\uparrow)$ $\displaystyle=$
$\displaystyle\left(1-\eta_{1,2}\right)$
$\displaystyle\mathcal{P}(\uparrow\>\rightarrow\>\downarrow)$ $\displaystyle=$
$\displaystyle\left(1-\eta_{1,1}\right)^{2}(1-\eta_{1,2})+{\eta_{1,1}}^{2}(1-\eta_{1,2})$
$\displaystyle+$ $\displaystyle 2\eta_{1,1}(1-\eta_{1,1})(1-\eta_{1,2})$
$\displaystyle\mathcal{P}(\uparrow\>\rightarrow\>\uparrow)$ $\displaystyle=$
$\displaystyle\eta_{1,1}\eta_{1,2}+\eta_{1,2}(1-\eta_{1,1}),$

once again exploiting symmetrical energy gaps. The polarization developed upon
one MW sweep (similar to Eq. (37)) now gives,

$\displaystyle P$ $\displaystyle=$
$\displaystyle\left[\mathcal{P}(\downarrow\>\rightarrow\>\downarrow)+\mathcal{P}(\uparrow\>\rightarrow\>\downarrow)\right]-\left[\mathcal{P}(\downarrow\>\rightarrow\>\uparrow)+\mathcal{P}(\uparrow\>\rightarrow\>\uparrow)\right]$
(39) $\displaystyle=$ $\displaystyle 0.$

There is therefore no net buildup of polarization in this case, a consequence
of a fine balance between the population traversals in the two arms of the
Galton board in Fig. 8B.

(iii) Sweep direction dependence: The discussion above is centered on low-to-
high frequency sweeps for the two cases of positive and negative hyperfine
coupling. Consider now the opposite scenario to Fig. 8A-B - where sweeps are
applied from high-to-low frequency. The order of the two energy gaps in (i)
and (ii) above are now reversed. This results in (i) no effective
hyperpolarization in the $A^{\parallel}{>}0$ case, and (ii) hyperpolarization
in the $\left|{\uparrow}\right\rangle$ state in the $A^{\parallel}{<}0$ case.
The expressions are identical to that in Eq. (37). We note that this results
in two consequences that are borne out by experiments (see companion paper
[18]):

  1. (i)

Due to equal proportions of $A^{\parallel}{<}0$ and $A^{\parallel}{>}0$
present in experiments, there is an inversion in the hyperpolarization sign
upon a reversal in the direction of the MW sweep. This is in contrast with
other DNP methods (e.g solid effect) where the DNP sign is different for the
two halves of the spectrum.

  2. (ii)

A consequence is that all parts of the spectrum yield the same sign in the
${}^{13}\textrm{C}$ hyperpolarization signal, making unravelling the
underlying electronic spectrum a simpler experimental undertaking.

  3. (iii)

Finally, in the case of large $N$ systems, one would expect a slight shift in
the frequency center of the DNP derived spectrum in the case of one sweep
direction with respect to the other.

Fig. 9: Evolution of nuclear populations through the LZ Galton board. Panels
show the emergence of nuclear hyperpolarization upon a low-to-high frequency
MW sweep (assuming $\eta_{k,2^{N}-k+1}{=}0$ along the conjugate diagonal). (A)
LZ cascade for $N{=}3$ nuclei (shown in inset) and $A^{\parallel}{>}0$ as in
Fig. 5D. (B) Traversal through LZ cascade. We extract the LZ-LAC points (red)
in (A) to define the LZ Galton board (dashed lines). Representative nuclear
states are marked. Panels show evolution of nuclear populations (colored
points), starting from different nuclear states $\mathcal{S}$ (indexed by $n$
in a Hamming notation) in the $m_{s}{=}0$ manifold, under a MW sweep. Opacity
of dots represents relative population. There is a clear bias towards nuclear
down states. (C) Population bias and emergence of hyperpolarization. Final
populations in different nuclear states (numbered in a Hamming ordering) in
the $m_{s}{=}0$ and $m_{s}{=}$+1 manifolds (blue and red points respectively)
after the traversals through the LZ Galton board shown in (B). Dashed line
represents the hyperpolarization level. (D) Panel shows populations similar to
(C), but for $N{=}4$. (E) Panel shows populations similar to (D), but now for
$A^{\parallel}{<}0$. Due to the change in the order of energy levels and
energy gaps, there is no net polarization developed (see also Fig. 8D).

###  IV.3 Origin of nuclear polarization bias

The origin of hyperpolarization can be traced to the differential traversal
through the $\mathcal{I}_{k,\ell}$ checkerboard in a manner that “biases” the
system to increase population in particular nuclear states at the cost of
others. Considering the example of an NV coupled to a single nuclear spin, Eq.
(37) shows a biased traversal that follows,

$p_{f}(\left|{\downarrow}\right\rangle)\approx
p_{i}(\left|{\downarrow}\right\rangle)+\beta
p_{i}(\left|{\uparrow}\right\rangle)\>;\>p_{f}(\left|{\uparrow}\right\rangle)\approx(1-\beta)p_{i}(\left|{\uparrow}\right\rangle),$
(40)

where $\beta$ is a finite probability and $p_{i}(\cdot)$ and $p_{f}(\cdot)$
refer to nuclear populations before and after traversal through the LZ Galton
board, and have the form,

$\displaystyle p_{i}(\left|{\downarrow}\right\rangle)$ $\displaystyle=$
$\displaystyle
p_{1,2}^{-}(1);p_{i}(\left|{\uparrow}\right\rangle)=p_{1,1}^{-}(1);$
$\displaystyle p_{f}(\left|{\downarrow}\right\rangle)$ $\displaystyle=$
$\displaystyle
p_{2,2}^{+}(1)+p_{1,2}^{+}(2);p_{f}(\left|{\uparrow}\right\rangle)=p_{2,1}^{+}(1)+p_{2,2}^{+}(2),$

Effectively, one of the nuclear states (here
$\left|{\downarrow}\right\rangle$) is unaffected, while the other (here
$\left|{\uparrow}\right\rangle$) is flipped with the probability $\beta$.
Successive sweeps through the LZ system then set up a “ratchet” that builds
nuclear hyperpolarization in the $\left|{\downarrow}\right\rangle$ state.

To understand the origin of this bias for larger $N$, consider traversal
through the full checkerboard in Fig. 4. As we demonstrate in Fig. 9, a
combined consequence of (i) the large energy gaps along the conjugate
diagonal, (ii) the order that the energy gaps are encountered in, and (iii)
the tilted nature of the LZ Galton board, there is a natural bias in evolution
towards the left and bottom of the diagram (Fig. 9). This net excess of
population in the nuclear down states over the nuclear up states (Eq. (34))
ultimately results in the hyperpolarization that we measure in experiments
(e.g. in Fig. 2C and Ref. [18]). In Sec. IVE, we illustrate the development of
this bias via numerical simulations of LZ-LAC traversals (Fig. 9).

###  IV.4 Analogy to Mach-Zender interferometry

We note finally that the Galton board illustrated in the panels in Fig. 8C-D
bear resemblance to the construction of a Mach-Zender (MZ) interferometer
[37], where the LZ-LACs serve analogous to beam splitters. One might then
consider the fine cancellation occurring in Fig. 8D as an interference effect
between the two arms of the MZ interferometer. A more detailed discussion of
this connection is beyond the scope of this manuscript; however, we note that
such connections could elevate the LZ-LAC checkerboard, naturally occurring in
spin systems as in Fig. 1 to applications exploiting the power of parallelized
interferences in cascaded MZ interferometers (e.g. BosonSampling [38]).

###  IV.5 Numerical simulations

The formalism developed via Eq. (27), Eq. (29), Eq. (33), and Eq. (34) permits
a tractable solution for the traversals through a LZ-LAC Galton board. Before
an analytical solution, we first consider a numerical evaluation in Fig. 9,
considering the case of $N{=}3$ and $A^{\parallel}{>}0$ \- and assuming for
simplicity $\theta_{j}{=}\pi/2$ \- with the LZ cascade structure shown in Fig.
9A. We consider evolution under a low-to-high frequency MW sweep, and solve
the traversal following Eq. (33). We assume that, as in experiments, the Rabi
frequency $\Omega_{e}$ is large, such that the energy gaps at the conjugate
diagonals are large and traversals through them are adiabatic. The individual
panels in Fig. 9B then show the populations of nuclear states at each point on
the $\mathcal{I}_{k,\ell}$ checkerboard, starting with populations confined to
starting $m_{s}{=}0$ nuclear states $\mathcal{S}$ (indexed by $n$). We track
the populations (dots) after each of the $\mathcal{I}_{k,\ell}$ LZ-LACs (red
points), where the opacity of the dots denotes the normalized population
level. The analogy to the classical Galton board is evident here (Fig. 3A);
there is a “sieving” effect as the populations bifurcate at each LZ-LAC,
yielding progressively smaller populations as one traverses deeper into the
checkerboard. There is an intrinsic bias towards the left and bottom of the
Galton board (evident in Fig. 9B); this ultimately yields hyperpolarization.

Fig. 9C makes this more clear by plotting the resulting nuclear state
populations in the $m_{s}{=}0$ and $m_{s}{=}$+1 manifolds (blue and red points
respectively) following a sweep over the full Galton board. Fig. 9D shows an
analogous calculation for $N{=}4$. Representative nuclear states are marked
and state $n$ follows a Hamming ordering. In both panels, a decrease in
population is evident with increasing state number $n$ in each manifold, while
the population of a particular nuclear state is higher in the $m_{s}{=}$+1
(red) manifold than in the $m_{s}{=}0$ (blue) one. Both these observations
reflect the bias in the Galton board traversal. Ultimately, upon application
of the laser to repolarize all the electronic polarization to $m_{s}{=}0$, one
obtains nuclear hyperpolarization, the value of which is denoted by the black
dashed lines in Fig. 9C-D.

Finally, Fig. 9F considers the inverted scenario where $A^{\parallel}{<}0$
(for an identical Galton board) and a full MW sweep is carried out. In this
case, while the LZ cascade has a similar structure to the diagram in Fig. 9A,
the order of the nuclear states in the $m_{s}{=}$+1 manifold are reversed, and
the large energy gaps appear on the diagonal instead of the conjugate
diagonal. As a result, the development of polarization bias is impeded through
a fine counterbalance of the nuclear populations as they traverse the Galton
board. A type of destructive interference then ensues, resulting in no
hyperpolarization in this case. This is shown in Fig. 9E, where we consider
the populations in each of the nuclear states similar to Fig. 9C-D, assuming
the diagonal energy gaps are large so that traversals through them are
adiabatic. All of the nuclear states in each manifold then have the same
population, and there is no net development of hyperpolarization. We note,
however, that the situations for $A^{\parallel}{>}0$ and $A^{\parallel}{<}0$
are exactly reversed when the MW sweep is applied from high-to-low frequency.

Fig. 10: Analytic solution to LZ Galton board traversal. Panels show the
evaluation of the analytic expression Eq. (42) for the solution of total
nuclear populations following a full MW sweep through a (A) $N{=}4$ and (B)
$N{=}8$ LZ Galton board. Similar to Fig. 9, the populations
($2^{N}\mathcal{P}_{n}$) of nuclear states in the $m_{s}{=}0$ and $m_{s}{=}$+1
manifolds (blue and red points respectively) are shown, with $\mathcal{P}_{n}$
following Eq. (42). We have assumed here $p{=}q{=}0.5$ (see Fig. 3B). Dashed
lines represent the polarization level. Panels demonstrate close resemblance
to the numerical simulations in Fig. 9.

###  IV.6 Theoretical evaluation of Galton board traversal

The numerical solution to traversals through an LZ Galton board in Fig. 9,
while revealing the essential physics of the biased traversal and buildup of
hyperpolarization, is still unwieldy to solve for large $N$. This is because
of the exponentially scaling number of LZ-LACs the system entails ($\propto
2^{2N})$. That said however, the analogy to the classical Galton board
developed above (see Fig. 3), makes feasible a full analytic solution under
certain simplifying assumptions. Consider first that the initial nuclear
populations are equally distributed among states in the $m_{s}{=}0$ electronic
manifold before entering the LZ Galton board at points $(1,\ell)$. Nuclear
populations travel along horizontal lines $\ell$ and vertical lines $k$ in the
LZ Galton board and redistribute at every LZ-LAC on $\mathcal{I}_{k,\ell}$,
where $k,\ell\in 1{\cdots}2^{N}$. Similar to the assumptions made in a
classical Galton board, we will define the probability of redistribution down
and right to be $p$ and $q$ respectively at every LZ-LAC, with $p{+}q{=}1$. In
addition, for points along the conjugate diagonal, we assume that the energy
gaps are large enough that $\eta_{k,2^{N}-k+1}{=}0$, and thus traversals are
adiabatic. We demonstrate below that these assumptions permit us to capture
the physics of the cascaded LZ-LAC traversal while permitting an analytical
solution (Fig. 10).

Towards this end, we evaluate the Galton board traversal to determine the
probability of the total nuclear population that ends up in each specific spin
state $\mathcal{S}$, indexed by $n$ (where $n\in 1{\cdots}2^{N}$). Due to the
action of the laser (Eq. (30)), the nuclear population in any $n$ state is
then just the sum of the population that exits point $(n,2^{N})$ downward and
point $(2^{N},2^{N}{-}n{+}1)$ to the right (as described in Eq. (33)).

We first consider the traversal of a nuclear population starting at a point
(1,$\ell$) before cascading through the board. The population entering at
point $(1,2^{N})$ will necessarily travel down, due to the large energy gap it
immediately encounters. We can express this with a function $f(n)$ which is
equal to 1 when $n{=}1$ and is equal to 0 otherwise.

For nuclear populations starting at any other $\ell\in 1{\cdots}2^{N}{-}1$, we
can now employ logic similar to that in a classical Galton board to evaluate
the probabilities of each spin state. For states ending in the $m_{s}{=}$+1
manifold, any nuclear population must travel $2^{N}{-}\ell{-}1$ steps down and
$n{-}2$ steps right (disregarding movement through the conjugate diagonal).
Because the nuclear population can take a variety of trajectories (e.g. Fig.
4) to reach the same spin state, we can use a combinatoric to find the number
of ways to choose $n{-}2$ right steps from the total steps. Thus, we can write
the probability as the exact form, ${(n-2)+(2^{N}-\ell-1)\choose
n-2}p^{2^{N}-\ell}q^{n-2}$, where $\cdot\choose{\cdot}$ is a combinatoric
operator, revealing a binomial form similar to a classical Galton board.
Likewise, we can compute the probability for reaching a spin state $n$ on the
right (in the $m_{s}{=}0$ manifold), which requires $2^{N}-n-\ell$ down steps
and $2^{N}-2$ right steps. This probability is written as
${(2^{N}-n-\ell)+(2^{N}-2)\choose 2^{N}-n-\ell}p^{2^{N}-n-\ell}q^{2^{N}-1}$.
Combining both these results, we can write the total probability that a
nuclear population from a point $(1,\ell)$, where $\ell\in
1{\cdots}2^{N}{-}1$, will end up in a specific spin state $n$ as,

$\displaystyle\mathcal{P}_{n,\ell}={(n{-}2){+}(2^{N}{-}\ell{-}1)\choose
n{-}2}p^{2^{N}{-}\ell}q^{n{-}2}$
$\displaystyle{+}{(2^{N}{-}n{-}\ell)+(2^{N}{-}2)\choose
2^{N}{-}n{-}\ell}p^{2^{N}{-}n{-}\ell}q^{2^{N}{-}1},$ (41)

where
$\mathcal{P}_{n,\ell}{=}\mathcal{P}^{(1)}[(1,\ell){\rightarrow}(2^{N},n)]+\mathcal{P}^{(2)}[(1,\ell){\rightarrow}(n,2^{N})]$
as in Eq. (41). The total proportion of nuclear population that ends up in a
specific state $n$ is then just the average of the probabilities from the
above formula for all possible entrance points, and takes the form,

$\displaystyle\mathcal{P}_{n}=\frac{1}{2^{N}}\Bigg{(}\sum_{\ell=1}^{2^{N}-1}\Bigg{[}{(n{-}2){+}(2^{N}{-}\ell{-}1)\choose
n{-}2}p^{2^{N}{-}\ell}q^{n{-}2}{+}{(2^{N}{-}n{-}\ell)+(2^{N}{-}2)\choose
2^{N}{-}n{-}\ell}p^{2^{N}{-}n{-}\ell}q^{2^{N}{-}1}\Bigg{]}+f(n)\Bigg{)},$ (42)

where $\mathcal{P}_{n}$ is the same probability as in Eq. (33). Fig. 10
displays the result of this analytic expression (Eq. (42)) for a LZ Galton
board of size $N{=}4$ and $N{=}8$. We emphasize the strong resemblance to the
numerical simulations in Fig. 9C-D. This also illustrates that the essential
physics of the LZ Galton traversal and the development of hyperpolarization is
less affected by the exact energy gaps at the individual LZ-LACs.

Interesting to note is the case of the reverse sweep (high-to-low frequency),
in which the population starts from the right side of the Galton board and
traverses through the board by moving either up or left at each LZ-LAC with
probabilities $p$ and $q$. Because this is the same board and all other above
assumptions apply, it becomes clear that there is symmetry between the two
cases. Specifically, the reverse sweep is identical in behavior to the
original sweep when the board is flipped across the conjugate diagonal. This
means that we can use the same analytical solutions we have acquired for the
original sweep to the reverse sweep, with a relabeling of the states as $s$
according to the symmetrical flip (where $s{=}2^{N}{-}n$). This takes the
form,

$\displaystyle
P_{s}{=}\frac{1}{2^{N}}\Bigg{(}\sum_{\ell=1}^{2^{N}-1}\Bigg{[}{(2^{N}{-}s{-}2){+}(2^{N}{-}\ell{-}1)\choose
2^{N}{-}s{-}2}p^{2^{N}{-}\ell}q^{2^{N}{-}s{-}2}{+}{(s{-}\ell)+(2^{N}{-}2)\choose
s{-}\ell}p^{s{-}\ell}q^{2^{N}{-}1}\Bigg{]}+f(2^{N}{-}s)\Bigg{)}.$ (43)

An evaluation of Eq. (43) then shows, by symmetry, the development of opposite
polarization in this case, matching experimental results [18].

Fig. 11: Numerical simulation showing “ESR-via-NMR” spectral mapping. (A)
Lower panel shows an exemplary LZ Galton board with $N{=}8$ nuclei employed in
simulations. Energy levels in the $m_{s}{=}$+1 manifold have a Gaussian DOS of
width ${\approx}$13.5MHz (upper panel) similar to Fig. 2B. Units chosen here
for the Galton board extent are arbitrary and just used for illustration.
Since numerical evaluation is performed column wise, the tilt of the Galton
board is not shown here. (B) Numerical result showing the developed
${}^{13}\textrm{C}$ hyperpolarization using sweep windows of size
$\mathcal{B}$ of increasing size through the Galton board in (A), similar to
the experiments in Fig. 2C. Forward (dark) and reverse (light) sweeps result
in positive and negative polarization respectively. Increasing $\mathcal{B}$
yields a greater hyperpolarization signal and a broader spectral profile,
matching experimental observations (Fig. 2C and Ref. [18]). For narrow
$\mathcal{B}$ (panel (i)) we find three distinct regimes in the
hyperpolarization profiles (circled), where there is a sharp increase, a rapid
linear decrease and no net polarization respectively. With increasing
$\mathcal{B}$, the profiles begin to approximate the DOS profile of the
underlying Galton board being swept over. (C) Unraveling the origin of the
hyperpolarization profile in panel B(i). Representative nuclear states are
marked. For each of the three regimes, we plot the resulting nuclear
populations in the $2^{N}{=}$256 nuclear states, numbered $n$ following a
Hamming convention. Hyperpolarization is evaluated as the difference in
populations between two halves of this plot (indicated by the dashed line),
following Eq. (34). In regime 1 (i), there is an emergence of
hyperpolarization because the population imbalances constructively add. In
regime 3 (iii), however, the population imbalances are in the same half of the
plot, yielding no net polarization.

###  IV.7 Evaluation of a window sweep on the Galton board

While the previous section explores the dynamics of a full sweep of the Galton
board, we now turn our focus to the case when a small window $\mathcal{B}$ is
swept (corresponding to experiments in Fig. 2C). Analytic solutions similar to
Eq. (29) are still tractable, but due to expressions being considerably more
complex than the full sweep behavior, here we instead carry out a numerical
solution. We employ the same assumptions above that each LZ-LAC is encountered
sequentially and that the energy gaps along the conjugate diagonal are large.
First, motivated by experiment, we assume a Gaussian distribution for energy
levels (DOS) in the $m_{s}{=}$+1 manifold, as seen in Fig. 11A. This modulates
the spacing of the $\mathcal{I}_{k,\ell}$ columns and impacts the number of
$m_{s}{=}$+1 energy levels that are swept in any specific frequency window
$\mathcal{B}$.

We employ a physically motivated method of evaluating traversals through the
board. First note that the ability to uncover spectral DOS mapping dynamics
necessarily requires simulations of large $N$ Galton boards to prevent
artifacts stemming from edge effects in small $N$ systems. For large Galton
boards as in Fig. 11A (which is for $N{=}8$ nuclei, thus having 256 nuclear
states) however, the full simulation approach employed in Fig. 9 proves
expensive and unwieldy due to the exponential scaling system size. Instead
here, for simplicity, we choose a specific order in which the LZ-LACs are
evaluated — solving the result of traversals through each
$\mathcal{I}_{k,\ell}$ column from the top to the bottom, working through the
board from left to right. This is an acceptable assumption because the
bifurcation action at any LZ-LAC is only dependent on previously encountered
LZ-LACs above or to the left of it. With this additional assumption in place,
we can evaluate the traversals of populations, starting and ending within a
specified window $\mathcal{B}$ defined by its starting column and width. We
choose a large Galton board (N=8) in Fig. 11A, and since evaluations are
column wise here, we show the board without its tilt.

Following this approach, Fig. 11B shows the numerical evaluations of the
traversal of the board in Fig. 11A with varying $\mathcal{B}$ and both forward
(dark) and reverse (light) sweeps. These plots are produced by taking repeated
window sweeps of a set size $\mathcal{B}$ while gradually moving the starting
point (corresponding to the starting point $f_{0}$ in Fig. 2) through the
board and calculating the polarization after the population in each state has
redistributed following Eq. (29). With increasing window size (e.g. in Fig.
11B(ii)-(iii)), we observe that the obtained hyperpolarization profiles
approximately trace the underlying electronic DOS of the underlying LZ Galton
board, in this case the Gaussian function chosen. There is also an increase in
overall polarization as the window size $\mathcal{B}$ increases, and an
increasing width in the hyperpolarization profile with increasing
$\mathcal{B}$ (see Fig. 11B). Both these features match experimental
observations (Fig. 2C and Ref. [18]).

Why does the observed hyperpolarization signal approximately track the
electronic spectral DOS? Insight into this is revealed in Fig. 11B(i) and Fig.
11C. For narrow window sizes $\mathcal{B}$, the simulations in Fig. 11B(i)
reveal that one half of the Galton board produces positive nuclear
polarization, while the other half produces no net polarization. Moving from
left-to-right in Fig. 11B(i) for instance, we identify three regimes in the
obtained hyperpolarization profiles: a regime 1, in which there is a increase
in polarization; regime 2, in which there is a linear decrease; and regime 3
with zero net polarization. As the window size $\mathcal{B}$ increases, the
differences in the profiles between the three regimes become less prominent,
and ultimately in Fig. 11C(iii) the profile appears Gaussian-like. At this
point, the profiles approximate the underlying spectral DOS in the starting LZ
Galton board.

To unravel the internal dynamics of the Galton traversals that yield Fig. 11B,
let us start by considering a representative sweep in regime 1 of Fig. 11B(i).
The result is shown in Fig. 11C(i), where we plot the nuclear populations in
each of the $2^{N}$ levels (labeled $n$) after the sweep over $\mathcal{B}$ is
completed. In this regime, it is predominantly the net nuclear-down states
that are being swept over. A (positive) increase in the nuclear populations is
then evident in nuclear states that are being swept over; this arises from the
populations effectively moving towards the bottom of the LZ-Galton board.
Similarly, there is a decrease of populations in high-$n$ nuclear states,
corresponding to predominantly nuclear up states. Since the final polarization
plotted in Fig. 11B(i) is evaluated (following Eq. (34)) as the difference in
populations between the nuclear states on either half of the plot in Fig.
11C(i) (denoted by the dashed line), there is a net development of
hyperpolarization in this case. Indeed, Fig. 11C(i) suggests that the
polarization produced is approximately proportional to the number of levels
swept over, since the “positive” portion of this population imbalance has a
width that scales with $\mathcal{B}$.

In a similar manner, Fig. 11C(ii) shows a sweep over a window $\mathcal{B}$
that contains a mix of net nuclear-down and up states (regime 2). In this
case, the positive and negative population regions move closer together (see
Fig. 11C(ii)). A difference (across the dashed line) then yields the
approximately linear decrease in polarization observed in regime 2 as the
sweep window passes into the net nuclear-up states. Finally, an illuminating
view is provided by a sweep carried out in regime 3), shown in Fig. 11C(iii).
Here, both population imbalances occur in the same half of the plot; hence
there is no hyperpolarization buildup. Ultimately a combination of these
features yields the profile in Fig. 11B(i). It is illustrative that the
reverse sweep (light line in Fig. 11B(i)) has a mirror-inverted profile about
the Galton board center. Finally, upon increasing $\mathcal{B}$ (Fig.
11B(ii)-(iii)), the sharp differences between the three regimes in Fig. 11B(i)
become less prominent; since the profile becomes a convolution with the
(larger) sweep window. In this case, both positive and negative sweeps yield
profiles that approximately map the spectral DOS.

##  V Outlook

Our work opens many interesting directions for future research. First, the
formalism developed here allows one to model the effect of a driven $N$-spin
quantum system undergoing evolution through a cascaded series of level anti-
crossings. Given that the number of anti-crossings scales exponentially with
system size $\propto 2^{2N}$, this problem appears intractable at first.
However, the description in terms of traversals of an analogous “Galton board”
permits both analytic and numeric solutions, allowing one to extract
physically relevant information in the large $N$ limit. The cascaded LZ-LAC
structure studied here (Fig. 1) is a motif that occurs commonly in several
contexts, such as in quantum walks in Hilbert spaces [39, 40, 41, 42] and
operator scrambling [43, 44], as well as in applications such as BosonSampling
[45] that involve multiple cascaded Mach-Zender interferometer traversals [46,
47, 48] (see Fig. 8). We therefore envision that the approach developed here
might contribute means to employ engineered central spin systems, such as in
Fig. 1, to study these phenomena.

More direct applications of this work lie in mechanistic descriptions of
dynamic nuclear polarization (DNP) in the large nuclear spin limit: the
predominant experimental regime of interest. Our work shows a systematic
approach to extend DNP models beyond standard descriptions of $e$-$n$ or
$e$-$e$-$n$ systems and can inform physics at large $N$ [36]. Apart from the
MW driven DNP problem considered here, we envision applications to
hyperpolarization mechanisms employing field sweeps [49], or with magic angle
spinning [32, 30, 50], where a similar cascaded structure of LACs is
encountered [51, 52]. Similar descriptions to DNP arise in the context of
quantum state “cooling” in a variety of systems [53, 54, 55]; we therefore
envision wider applications of this methodology developed here in these
systems.

An application we considered was in performing “ESR-via-NMR” [18], wherein the
electronic spectrum is effectively coaxed out by only reading out the nuclear
populations. This relies on mapping the electronic spectral density of states
(DOS) to nuclear population imbalances via hyperpolarization transfer (similar
to Fig. 2C). The utility of this approach may be particularly relevant in
systems where ESR spectra may be inaccessible, e.g when the electrons are so
dilute that ESR signals from them are too weak, or their short $T_{2e}$ makes
probing them directly more difficult. Instead, one could leverage the more
abundant, longer-lived (high $T_{1n}$) nuclear spins in their immediate
surroundings as a means to indirectly probe them by accumulating electronic
DOS information in nuclear polarization prior to readout. We envision
particularly compelling applications in photo-polarizable electronic systems
in molecular complexes [56].

From a technological perspective, the approach developed here portends
applications in quantum memories [57] constructed out of central spin systems
(such as Fig. 1A), wherein information from the electronic spins is mapped in
a “one-to-many” fashion to long-lived nuclear registers. Similarly, there are
interesting applications in quantum sensing, including modalities where the
electronic spin sensors are interrogated via polarization transfer of
surrounding nuclear spins. In the context of NV centers, this opens
applications in RF interrogated magnetometry, wherein the NV center
populations are readout by ${}^{13}\textrm{C}$ nuclei without a MW cavity (see
accompanying manuscript [18]). As opposed to conventional optically
interrogated NV magnetometers, this can permit DC magnetometers that function
in turbid or optically dense media, harnessing the immunity of RF readout to
optical scattering, permitting the arbitrary orientation of the diamond
crystals. This will open avenues for a networked quantum sensing “underwater”
[18], with a variety of applications (e.g. undersea magnetic anomaly detection
[58]).

##  VI Conclusion

In conclusion, we theoretically considered the problem of polarization
transfer in a system of an electronic spin coupled to $N$ surrounding nuclei
via frequency swept microwaves. We described how, in an appropriate rotating
frame, the system can be described by traversals through a cascade of Landau-
Zener anti-crossings (LZ-LACs). We showed that the traversals can be mapped
into an equivalent evolution through a “Galton board”, where individual LZ-
LACs map onto “pegs”, upon encountering which the nuclear populations
bifurcate; this makes the solution of the complex dynamics through a coupled
$e$-$n$ system consisting of $2^{2N}$ anti-crossings analytically tractable.
Our formal solution procedure relied on elucidating trajectories of evolution
through the LZ Galton board via transfer matrices, allowing one to extract
insights into the development of “bias” in the traversals that then leads to
hyperpolarization. An implication is that spin bath polarization levels track
the electronic spectral DOS (Fig. 2C), allowing the nuclear spins to
indirectly report on the underlying electronic spectrum, in effect achieving
“ESR-via-NMR” [18]. The Galton board mapping developed here suggests that a
similar approach can be extended to problems involving traversals through a
series of cascaded energy crossings, with applications in DNP [51], spin qubit
or resonator “cooling” [53, 55] and in quantum sensing and information
processing with central-spin systems.

### Acknowledgments

We gratefully acknowledge discussions with J. Reimer, C. Meriles and P.
Zangara. This work was funded by ONR under N00014-20-1-2806.

Fig. 12: Energy of the LZ anti-crossings $\varepsilon_{k,\ell}$ here plotted
with respect to their positions on the $\mathcal{I}_{k,\ell}$ checkerboard for
an $N{=}3$ system. The conjugate diagonal is illustrated by the dashed line
and representative nuclear states are marked. We focus here on the scaling of
the energy gaps $\varepsilon_{k,\ell}$ based on the number of nuclear spin
flips required, hence we fix $A_{j}^{\perp}$ equal for all three spins in this
calculation. Along any column or row, a hierarchy in the energies is evident.
Also note the intrinsic symmetry of energies with respect to the checkerboard
conjugate diagonal (see Fig. 4).

##  Appendix A Hierarchy in energy gaps

In this Appendix, we present details of the diagonalization of the Hamiltonian
resulting in calculations of the energy gaps and their native hierarchy in the
checkerboard $\mathcal{I}_{k,\ell}$ (see Sec. IID). The procedure for the
calculation of the $\varepsilon_{k,\ell}$ energy gaps is most simply
illustrated considering first an $N{=}1$ system and evaluating the anti-
crossing between states $\left|{0,\uparrow}\right\rangle$ and
$\left|{\textrm{+}1,\downarrow}\right\rangle$ corresponding to a “joint”
$e$-$n$ flip at checkerboard point $\mathcal{I}_{1,1}$. There is no direct
term in the Hamiltonian that can accomplish a dual spin-flip, but as we
described in Sec. IID, it can be constructed as a combined action of the
$\\{\Omega_{e},A_{1}^{\perp}\\}$ terms. Indeed, the Hamiltonian $\mathcal{H}$
in Eq. (5) in the basis indicated (blue terms) has the form,

$\mathcal{H}-\mathcal{I}_{1,1}(1)\textbf{1}=\bNiceMatrix[first-row,,first-
col,last-
col]&\left|{0,\uparrow}\right\rangle\left|{+1,\downarrow}\right\rangle\left|{0,\downarrow}\right\rangle\left|{+1,\uparrow}\right\rangle\\\
0\Omega_{e}\left|{0,\uparrow}\right\rangle\\\
0\Omega_{e}A_{1}^{\perp}\left|{+1,\downarrow}\right\rangle\\\
\Omega_{e}-\omega_{1}^{(0)}\left|{0,\downarrow}\right\rangle\\\
\Omega_{e}A_{1}^{\perp}\omega_{1}^{(1)}\left|{+1,\uparrow}\right\rangle\\\ $
(44)

where we employ a $\omega_{\textrm{MW}}$ value in Eq. (5) to be resonant with
the checkerboard point $\mathcal{I}_{1,1}(1)$. Clearly the states
$\left|{0,{\uparrow}}\right\rangle$ and
$\left|{\textrm{+}1,{\downarrow}}\right\rangle$ are then degenerate, and while
no direct coupling (off-diagonal term) exists between them, it can be
constructed by first diagonalizing with respect to the $A_{1}^{\perp}$ term.
The new eigenstates then can be defined as,

$\displaystyle\left|{\textrm{+}1,{\downarrow}}\right\rangle_{A}$
$\displaystyle=$
$\displaystyle\left|{\textrm{+}1,{\downarrow}}\right\rangle\cos\theta_{A}^{(1)}+\left|{\textrm{+}1,{\uparrow}}\right\rangle\sin\theta_{A}^{(1)}$
$\displaystyle\left|{\textrm{+}1,{\uparrow}}\right\rangle_{A}$
$\displaystyle=$
$\displaystyle-\left|{\textrm{+}1,{\downarrow}}\right\rangle\sin\theta_{A}^{(1)}+\left|{\textrm{+}1,{\uparrow}}\right\rangle\cos\theta_{A}^{(1)}$
$\displaystyle\left|{0,{\downarrow}}\right\rangle_{A}$ $\displaystyle=$
$\displaystyle\left|{0,{\downarrow}}\right\rangle\>;\>\left|{0,{\uparrow}}\right\rangle_{A}=\left|{0,{\uparrow}}\right\rangle$
(45)

where $\tan\theta_{A}^{(1)}{=}A_{1}^{\perp}/\omega_{1}^{(1)}$, and we have
used the fact that the hyperfine coupling is predominantly operational only in
the $m_{s}{=}$+1 manifold. Restricting our attention to the degenerate two-
level subspace in Eq. (44) gives,

$\mathcal{H}-\mathcal{I}_{1,1}(1)\textbf{1}\approx\bNiceMatrix[first-
row,,first-col,last-
col]&\left|{0,\uparrow}\right\rangle_{A}\left|{1,\downarrow}\right\rangle_{A}\\\
0\Omega_{e}\sin\theta_{A}^{(1)}\left|{0,\uparrow}\right\rangle_{A}\\\
\Omega_{e}\sin\theta_{A}^{(1)}\frac{1}{2}(A_{1}^{\perp})^{2}/\omega_{1}^{(1)}\left|{1,\downarrow}\right\rangle_{A}\\\
$ (46)

allowing us to read off the energy gap,

$\varepsilon_{1,1}=\left[\left(\frac{1}{2}(A_{1}^{\perp})^{2}/\omega_{1}^{(1)}\right)^{2}+(2\Omega_{e}\sin\theta_{A}^{(1)})^{2}\right]^{1/2}$
(47)

In the simple case that $A_{1}^{\perp}{\ll}\omega_{1}^{(1)}$, this gives,
$\varepsilon_{1,1}\approx\frac{2\Omega_{e}A_{1}^{\perp}}{\omega_{L}+A_{1}^{\parallel}}$.
Indeed, this result could also have been predicted out of simple first-order
perturbation theory arguments. A similar calculation for the energy gap
$\varepsilon_{2,1}$ shows that a hierarchy in the $N{=}1$ checkerboard is
operational, $\varepsilon_{1,1}{<}\varepsilon_{2,1}{=}\Omega_{e}$. This is
illustrated for instance in Fig. 8.

Similarly, now consider the diagonalization of the Hamiltonian for the $N{=}2$
case. We begin the discussion by noting that in the basis indicated, the
Hamiltonian has the form,

$\mathcal{H}=\bNiceMatrix[first-row,,first-col,last-
col]&\left|{0,\uparrow\uparrow}\right\rangle\left|{\textrm{+}1,\downarrow\downarrow}\right\rangle\left|{\textrm{+}1,\uparrow\downarrow}\right\rangle\left|{\textrm{+}1,\downarrow\uparrow}\right\rangle\left|{\textrm{+}1,\uparrow\uparrow}\right\rangle\\\
0\Omega_{e}\left|{0,\uparrow\uparrow}\right\rangle\\\
0A_{1}^{\perp}A_{2}^{\perp}\left|{\textrm{+}1,\downarrow\downarrow}\right\rangle\\\
A_{1}^{\perp}\omega_{1}^{(1)}\left|{\textrm{+}1,\uparrow\downarrow}\right\rangle\\\
A_{2}^{\perp}\omega_{1}^{(2)}\left|{\textrm{+}1,\downarrow\uparrow}\right\rangle\\\
\Omega_{e}-(\omega_{1}^{(1)}+\omega_{1}^{(2)})\left|{\textrm{+}1,\uparrow\uparrow}\right\rangle\\\
$ (48)

where there is an energy degeneracy between states
$\left|{0,\uparrow\uparrow}\right\rangle$ and
$\left|{\textrm{+}1,\downarrow\downarrow}\right\rangle$, but no direct
coupling between them. Once again, however, we can carry out a diagonalization
with respect to the $\\{A_{1}^{\perp},A_{2}^{\perp}\\}$ terms, identifying a
rotated basis similar to Eq. (45),

$\displaystyle\left|{\textrm{+}1,{\downarrow\downarrow}}\right\rangle_{A}$
$\displaystyle=$
$\displaystyle\left|{\textrm{+}1,{\downarrow\downarrow}}\right\rangle\cos\theta_{A}^{(1)}+\left|{\textrm{+}1,{\uparrow\downarrow}}\right\rangle\sin\theta_{A}^{(1)}$
$\displaystyle\left|{\textrm{+}1,{\uparrow\uparrow}}\right\rangle_{A}$
$\displaystyle=$
$\displaystyle-\left|{\textrm{+}1,{\uparrow\downarrow}}\right\rangle\sin\theta_{A}^{(2)}+\left|{\textrm{+}1,{\uparrow\uparrow}}\right\rangle\cos\theta_{A}^{(2)}$
(49)

allowing one to write the Hamiltonian matrix in the degenerate subspace,

$\mathcal{H}\approx\bNiceMatrix[first-row,,first-col,last-
col]&\left|{0,\uparrow\uparrow}\right\rangle_{A}\left|{1,\downarrow\downarrow}\right\rangle_{A}\\\
02\Omega_{e}\sin\theta_{A}^{(1)}\sin\theta_{A}^{(2)}\left|{0,\uparrow\uparrow}\right\rangle_{A}\\\
2\Omega_{e}\sin\theta_{A}^{(1)}\sin\theta_{A}^{(2)}\frac{1}{2}(A_{1}^{\perp})^{2}/\omega_{1}^{(1)}+\frac{1}{2}(A_{2}^{\perp})^{2}/\omega_{2}^{(1)}\left|{1,\downarrow\downarrow}\right\rangle_{A}\\\
$ (50)

ultimately resulting in an energy gap that can approximately be written as,

$\varepsilon_{1,1}\approx
4\Omega_{e}\sin\theta_{A}^{(1)}\sin\theta_{A}^{(2)}=4\Omega_{e}\left(\frac{A_{1}^{\perp}}{\omega_{L}+A_{1}^{\parallel}}\right)\left(\frac{A_{2}^{\perp}}{\omega_{L}+A_{2}^{\parallel}}\right)\>,$
(51)

Clearly a hierarchy in the energy gaps is then evident, since
$\varepsilon_{2,1}{\approx}2\Omega_{e}\sin\theta_{A}^{(1)}$ and
$\varepsilon_{3,1}{\approx}\Omega_{e}$, and if the transverse hyperfine terms
are approximately the same order of magnitude,
$\varepsilon_{1,1}{<}\varepsilon_{2,1}{<}\varepsilon_{3,1}$, generally
decreasing with the number of successive nuclear flips required between the
two states at each anti-crossing. Fig. 12 displays the energy gaps
$\varepsilon_{k,\ell}$ for an $N{=}3$ system — the heights of the bars here
are the size of the energy gaps, while their positions correspond to those on
the checkerboard. We have chosen $A_{j}^{(1)}$ equal here to highlight the
hierarchy just coming from the position of the anti-crossing on the
checkerboard.

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