††thanks: M.B.P and I.S. contributed equally to this work and its leadership.
Correspondence should be addressed to<EMAIL_ADDRESS>or
<EMAIL_ADDRESS>M.B.P and I.S. contributed equally to this
work and its leadership. Correspondence should be addressed to
<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
# Towards a unified picture of polarization transfer - equivalence of DNP and
PHIP

Martin C. Korzeczek  Institute of Theoretical Physics and IQST, Albert-
Einstein Allee 11, Ulm University, 89081 Ulm, Germany  Laurynas Dagys  NVision
Imaging Technologies GmbH, 89081 Ulm, Germany  Christoph Müller  NVision
Imaging Technologies GmbH, 89081 Ulm, Germany  Benedikt Tratzmiller  Institute
of Theoretical Physics and IQST, Albert-Einstein Allee 11, Ulm University,
89081 Ulm, Germany  Carl Zeiss MultiSEM GmbH, 73447 Oberkochen, Germany  Alon
Salhov  NVision Imaging Technologies GmbH, 89081 Ulm, Germany  Racah Institute
of Physics, The Hebrew University of Jerusalem, Jerusalem, 91904, Givat Ram,
Israel  Tim Eichhorn  NVision Imaging Technologies GmbH, 89081 Ulm, Germany
Jochen Scheuer  NVision Imaging Technologies GmbH, 89081 Ulm, Germany  Stephan
Knecht  NVision Imaging Technologies GmbH, 89081 Ulm, Germany  Martin B.
Plenio  Institute of Theoretical Physics and IQST, Albert-Einstein Allee 11,
Ulm University, 89081 Ulm, Germany  Ilai Schwartz  NVision Imaging
Technologies GmbH, 89081 Ulm, Germany

###### Abstract

Hyperpolarization, the elevation of nuclear spin polarization beyond its value
in thermal equilibrium, is a leading method for enhancing the sensitivity of
nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). In this
approach, a readily available source of higher spin order, either from
electron spins in dynamic nuclear polarization (DNP) or singlet states in
hydrogen for PHIP / SABRE, is placed in close contact with the nuclear spin of
interest to allow for efficient transfer of the spin order. However, the
physics of polarization transfer from electron spins and nuclear singlet
states appears to differ substantially due to the marked difference in energy
scale between electron and nuclear spins and due to the different physical
origin and strength of the interaction with the target nuclei. Nevertheless,
utilizing a pseudo-spin formalism we show that a critical regime of PHIP /
SABRE, namely the low-field polarization transfer regime, can be mapped to an
equivalent system. Here the heteronuclear target spin takes a role analogous
to that of the DNP-electron and the pseudospin of the parahydrogen takes the
role of the nuclear spin in DNP. Using this equivalence we provide a detailed
mapping between some of the most important polarization transfer sequences in
PHIP and DNP and we show how one can transfer sequences that have been
developed in one field to the other. Building on this mapping we foresee new
insights generated and potentially a strong cross-pollination between
polarization sequence developers in the two fields.

Hyperpolarization $|$ DNP $|$ PHIP $|$ SABRE $|$ Nuclear magnetic resonance

##  I Introduction

Nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI)
applications are severely limited by low detection sensitivities. These are
rooted in the weak thermal equilibrium nuclear spin polarization of the
detection targets at room temperature which, typically, amounts to a few parts
per million (ppm) per Tesla of applied magnetic field. Achievable magnetic
field strengths in magnetic resonance (MR) devices are approaching their
physical limits and significant future increases in the sensitivities of
detection coils appear challenging. This places stringent limits on the
minimally detectable quantities under MR and thus presents a major obstacle
towards the goal of metabolic imaging applications at physiologically relevant
metabolite concentrations, which, in turn, could have a major impact on cancer
treatment [1, 2].

Therefore, MR-based metabolic imaging requires an increase in the MR signal by
orders of magnitude and has motivated the search for methods that can deliver
such gains. The last decades have seen a wide variety of proposals to overcome
the MR detection challenge by increasing the nuclear spin polarization beyond
its thermal equilibrium value - thus achieving hyperpolarised nuclei.
Promising approaches include parahydrogen-induced polarization (PHIP) [3, 4,
5] by catalytic addition [6] or by reversible exchange [7], dynamic nuclear
polarization (DNP) at low temperatures [8] and by optically polarised
persistent [9] and non-persistent [10, 11] electron spins. These methods have
the demonstrated potential of achieving nuclear spin polarization in the
percent range which enables MR signal increases of around four orders of
magnitude.

Figure 1: Depiction of the spin system, energy levels and relevant Hamiltonian
terms, demonstrating the mapping between DNP and PHIP. (a) the system of two
hydrogen spins $I^{(1)}$, $I^{(2)}$ and one heteronuclear spin $S$. (b) The
same system with the hydrogen spins $I^{(1)}$, $I^{(2)}$ in the singlet-
triplet basis notation, as typically done in PHIP. (c) decomposing the
hydrogen energy states into the interacting ($I$) and non-interacting
($\tilde{I}$) pseudospins. (d) Hamiltonian of nuclear spin and electron spin
in typical DNP systems. From (c), (d) it becomes evident that the pseudospin
formalism for PHIP has significant similarities to a DNP system.

Despite significant differences in the underlying physical systems and
experimental detail these methods share important conceptual similarities.
Indeed, the common motif of these methods is the combination of three key
ingredients. First, a readily available source of polarization albeit in a
form that is not directly usable for imaging such as thermally or optically
polarised electron spins or singlet order in parahydrogen; secondly, the
ability to bring these spins into close proximity with the target nuclei, such
as 13C, that need to be hyperpolarised and, thirdly, control methods based on
externally applied fields that enable robust and efficient transfer of
polarization from source to target.

Each of the proposed hyperpolarization methodologies have developed their own
control protocols and languages in which to describe them. The different
energy scales between DNP (electron to nuclear spin transfer, three orders of
magnitude difference in energy levels) and PHIP (nuclear spin to nuclear spin
transfer) have resulted in seemingly distinct constraints for the control
methods, keeping development in these fields separated. The realization of
potential commonalities between the fields would benefit considerably from the
availability of "dictionaries" that facilitate the translation between control
protocols.

In this work we will expose the commonality between PHIP and DNP control
protocols for polarization transfer. We use the framework of [12] in the near-
equivalence regime for PHIP molecules, where we define a pseudospin for the
strongly coupled protons which has an energy separation equal to the J
coupling. In this framework, a heteronuclear spin such as 13C has an energy
splitting orders of magnitude larger than that of the pseudospin, leading to a
regime similar to that encountered in DNP systems (see Fig. 1 for a
schematics). We demonstrate the mathematical equivalence of the descriptions
of the two systems and exemplify at the hand of explicit examples that this
enables the direct translation of a large number of control protocols from one
field to the other.

We then proceed by concentrating on two practical examples: (1) the PulsePol
family of protocols [13] that has been developed theoretically and
demonstrated experimentally in the framework of DNP with colour centers in
diamond to facilitate polarization transfer that is highly robust to a broad
range of control errors that include detuning errors due to $B_{0}$ field
inhomogeneities and temporal and spatial $B_{1}$-field fluctuations of
external driving fields. We demonstrate by numerical simulation that the
beneficial features of robustness translate to the PHIP setting and proceed
with a experimental work to verify that the predicted efficiency and
robustness of this new control sequence in the realm of PHIP translates into
high levels of polarization of ${}^{13}\mathrm{C}$ in dimethyl maleate. (2)
The recently introduced adiabatic amplitude sweeps in the context of PHIP [14]
was shown to be robust and efficient for the regime of in the regime where the
coupling still allows for a sweep duration much shorter than the relevant
coherence times. We demonstrate the applicability of this protocol for DNP
with optically polarised electrons in pentacene molecules inside a naphthalene
crystal and the resulting efficiency advantage over NOVEL, a standard method
for polarization transfer in such systems.

##  II Theory

In this section we derive a mapping between the key physical degrees of
freedom underlying dynamical nuclear polarization and those of parahydrogen
induced polarization. We identify the effective qubit degrees of freedom and
show that the Hamiltonians in both settings can be identified by a suitable
basis transformation. Armed with this equivalence, we proceed to reveal and
describe the parallels between various polarization schemes commonly used in
PHIP and in DNP, thereby exemplifying general translation rules between DNP
and PHIP control protocols.

###  II.1 Polarization transfer in DNP

Dynamical nuclear polarization has the goal of mediating the transfer of
polarization from a source spin ($S$), typically of large energy splitting
such as an electron, to a target spin ($I$) of lower energy splitting,
typically a nuclear spin (see Fig. 1 where the level splittings of $I$ and $S$
are denoted by $\omega_{I}$ and $\omega_{S}$ respectively). These spins are
coupled via the Hamiltonian term $A_{\perp}\hat{I}_{x}\hat{S}_{z}$. In the
case of NV- centers in diamond, for example, $S$ is given by a two-level
subsystem of the electronic ground-state triplet [15]. Furthermore, the spin
$S$, with gyromagnetic ratio $\gamma_{S}$, may be driven by a time-dependent
magnetic field $B_{1}(t)$ whose Rabi frequency $\Omega=|\gamma_{S}B_{1}|$ may
reach or even exceed significantly $\omega_{I}$. Typically, the energy scales
are assumed to fulfill the conditions

$\displaystyle\omega_{S}\gg\ \gamma_{S}|B_{1}|,\omega_{I}\ \gg A_{\perp}.$ (1)

The corresponding Hamiltonian reads ($\hbar=1$)

$\displaystyle
H=\omega_{I}\hat{I}_{z}+\omega_{S}\hat{S}_{z}+A_{\perp}\hat{I}_{x}\hat{S}_{z}+\gamma_{S}B_{1}(t)\hat{S}_{x}.$
(2)

Under the assumptions of eq. 1 and a suitable choice of $B_{1}(t)$, pulsed or
continuous wave, an application of Average Hamiltonian Theory [16] then
results in the effective Hamiltonian

$\displaystyle
H^{\text{eff}}=\dfrac{A_{\ast}}{2}(\hat{I}_{x}\hat{S}_{x}+\hat{I}_{y}\hat{S}_{y}).$
(3)

in the frame corotating with the Larmor precessions. Here, the effective
coupling $A_{\ast}=A_{\ast}[B_{1}(t)]\leq A_{\perp}$, effects coherent
polarization transfer between the two spin degrees of freedom $I$ and $S$. In
order to achieve this type of Hamiltonian, for pulsed polarization schemes,
the waiting times between pulses typically need to scale with a single free
parameter $\tau$ which has to be matched to $\omega_{I}$ to reach resonance
and achieve a non-zero $A_{\ast}$. For continuous wave schemes it is the Rabi
frequency that typically has to match $\omega_{I}$.

Important criteria for good polarization schemes are the attained coupling
strength $A_{\ast}/A_{\perp}$, the scheme’s robustness to errors in the
driving field such as a resonance-offset ($\Delta$) due to
$B_{0}$-inhomogeneities, temporal and $B_{1}$-field fluctuations
($\Omega_{error}$), and also the simplicity of the scheme’s implementation in
practice.

###  II.2 Polarization transfer in PHIP

We proceed to establish the equivalence of the DNP setting of section A and a
typical setting for PHIP in the low-field regime and heteronuclear spins. We
derive a basis transformation such that in the new basis the PHIP system can
be written as a set of pseudo-spins whose dynamics is governed by a
Hamiltonian that takes the form of eq. 2. A summary of the resulting
correspondences between PHIP and DNP is given in Tab. 1.

| PHIP | DNP
---|---|---
Definitions of | $2\hat{I}_{z}=\ket{T_{0}}\bra{T_{0}}-\ket{S_{0}}\bra{S_{0}}$ | $2\hat{I}_{z}=\ket{\uparrow}\bra{\uparrow}-\ket{\downarrow}\bra{\downarrow}$
Spin-I operators | $2\hat{I}_{x}=\ket{S_{0}}\bra{T_{0}}+\ket{T_{0}}\bra{S_{0}}$ | $2\hat{I}_{x}=\ket{\downarrow}\bra{\uparrow}+\ket{\uparrow}\bra{\downarrow}$
Spin $I$ |  $\\{\ket{T_{0}},\ket{S_{0}}\\}$ effective spin |  $\\{\ket{\downarrow},\ket{\uparrow}\\}$ nuclear spin
| $J\ \hat{I}_{z}$ | $\omega_{I}\ \hat{I}_{z}$
Spin $S$ | heteronuclear spin | electron spin
| $\omega_{S}\ \hat{S}_{z}$ | $\omega_{S}\hat{S}_{z}$
$I$-$S$ coupling | $(J^{(1)}-J^{(2)})\hat{I}_{x}\hat{S}_{z}$ | $A_{\perp}\hat{I}_{x}\hat{S}_{z}$
Initial state |  $I$ polarized |  $S$ polarized
Scheme: | SLIC [17] | NOVEL [18] (III.2.1)
| S2hM [12, 19] | NV nuc. spin init. [20]
| PulsePol (this work, III.2.2) | PulsePol [13]
| ADAPT [21] (III.2.2) | TOP-DNP [22]
| adiab. $B_{1}$ sweeps [14] (III.2.2) | RA-NOVEL [23] (III.2.1)
Table 1: A tabular correspondence for polarization schemes in PHIP and DNP.
The full names of the polarization schemes are Spin-Lock Induced Crossing for
SLIC, Nuclear Spin Orientation via Electron Spin Locking for NOVEL, Singlet to
Heteronuclear Magnetization Transfer for S2hM, NV nuclear spin initialization,
Alternating Delays Achieve Polarization Transfer for ADAPT, Time-Optimized
Pulsed DNP for TOP-DNP, adiabatic $B_{1}$ sweeps and Ramped-amplitude NOVEL,
for RA-NOVEL.

To this end, we consider a system consisting of two hydrogen nuclear spins
$I^{(1)}$ and $I^{(2)}$ and a carbon nuclear spin $S$, all spin-$1/2$, which
are all coupled via J-coupling (cf. Fig. 1a). Following [12], the
corresponding Hamiltonian is given by

$\displaystyle H$
$\displaystyle=\omega^{0}_{I}\hat{I}^{(1)}_{z}+\omega^{0}_{I}\hat{I}^{(2)}_{z}+\omega_{S}\hat{S}_{z}$
(4)
$\displaystyle\quad+J\,\hat{\vec{I}}^{(1)}\cdot\hat{\vec{I}}^{(2)}+J^{(1)}\hat{S}_{z}\hat{I}^{(1)}_{z}+J^{(2)}\hat{S}_{z}\hat{I}^{(2)}_{z},$

where $\omega^{0}_{I}$ ($\omega_{S}$) is the hydrogen (carbon) Larmor
frequency, $J$ is the inter-hydrogen $J$-coupling strength, and $J^{(1)}$,
$J^{(2)}$ are the heteronuclear coupling strengths between the respective
hydrogen spin and the carbon spin. Here, all J-couplings are defined as
angular frequencies. The singlet-triplet basis

$\displaystyle\ket{S_{0}}$ $\displaystyle:=$
$\displaystyle(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2}$
$\displaystyle\ket{T_{-}}$ $\displaystyle:=$
$\displaystyle\ket{\downarrow\downarrow}$ $\displaystyle\ket{T_{0}}$
$\displaystyle:=$
$\displaystyle(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})/\sqrt{2}$
$\displaystyle\ket{T_{+}}$ $\displaystyle:=$
$\displaystyle\ket{\uparrow\uparrow}$

in the Hilbert space of hydrogen nuclei, that satisfies

$\displaystyle\vec{I}^{(1)}\cdot\vec{I}^{(2)}\ket{S_{0}}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\ket{S_{0}},$
$\displaystyle\vec{I}^{(1)}\cdot\vec{I}^{(2)}\ket{T_{\alpha}}$
$\displaystyle=$ $\displaystyle+\frac{1}{4}\ket{T_{\alpha}}$

is the eigenbasis of the inter-hydrogen $J$-coupling term. This basis allows
us to re-partition the hydrogen manifold into the direct sum of two pseudo-
spins: We define the first as the singlet-triplet pseudo-spin $I$ with
$2\hat{I}_{z}:=\ket{T_{0}}\bra{T_{0}}-\ket{S_{0}}\bra{S_{0}}$ and the
corresponding $2\hat{I}_{x}:=\ket{T_{0}}\bra{S_{0}}+\ket{S_{0}}\bra{T_{0}}$.
The second is the complementary pseudo-spin $\hat{\tilde{I}}$ with
$2\hat{\tilde{I}}_{z}:=\ket{T_{+}}\bra{T_{+}}-\ket{T_{-}}\bra{T_{-}}$. If we
use the spin labels to denote the respective Hilbert spaces, this basis change
can be regarded as switching from the the true-spin basis $I^{(1)}\otimes
I^{(2)}\otimes S$ to the pseudo-spin basis $(I\oplus\tilde{I})\otimes S$ which
both span the same combined Hilbert space. With this new basis, we can use the
identities

$\displaystyle\hat{I}^{(1)}_{z}$
$\displaystyle=\hat{\tilde{I}}_{z}+\hat{I}_{x}$ (5)
$\displaystyle\hat{I}^{(2)}_{z}$
$\displaystyle=\hat{\tilde{I}}_{z}-\hat{I}_{x}$ (6)
$\displaystyle\hat{\vec{I}}^{(1)}\cdot\hat{\vec{I}}^{(2)}$
$\displaystyle=\hat{\tilde{I}}_{z}^{2}-\hat{I}_{z}^{2}+\hat{I}_{z}$ (7)
$\displaystyle\hat{\tilde{I}}^{2}_{z}$
$\displaystyle=\frac{1}{4}\hat{P}_{\tilde{I}}$ (8)

to transform the Hamiltonian. Here, operators such as $\hat{I}_{z}$ need to be
regarded as acting on the combined 4-level Hydrogen manifold, and
$\hat{P}_{I}$ and $\hat{P}_{\tilde{I}}$ are the corresponding projectors onto
the basis states spanned by $I$ and $\tilde{I}$. Using these definitions, we
arrive at

$\displaystyle H$
$\displaystyle=J\hat{I}_{z}+\omega_{S}\hat{S}_{z}+(J^{(1)}-J^{(2)})\hat{I}_{x}\hat{S}_{z}$
$\displaystyle\quad+2\omega^{0}_{I}\hat{\tilde{I}}_{z}+\frac{J}{4}\hat{P}_{\tilde{I}}-\frac{J}{4}\hat{P}_{I}+\dfrac{J^{(1)}+J^{(2)}}{2}\hat{P}_{\tilde{I}}\hat{S}_{z}$
(9)

The first line corresponds exactly to the terms of the DNP Hamiltonian eq. 2
albeit without the driving term. The latter can be added to both (4) and
(II.2) without modifications.

The second line describes the level-splitting of the third pseudospin
$\tilde{I}$ and energy shifts of the $I$ and $\tilde{I}$ sub-spaces, all of
which commute with all the terms in the first line. Thus, the PHIP Hamiltonian
is indeed equivalent to the DNP setting and thus allows for the mapping of
polarization sequences when making the correspondence

$\displaystyle J$ $\displaystyle\leftrightarrow$ $\displaystyle\omega_{I}\,,$
(10) $\displaystyle J^{(1)}-J^{(2)}$ $\displaystyle\leftrightarrow$
$\displaystyle A_{\perp}\,,$ (11) $\displaystyle\omega_{B1}=\omega_{S}$
$\displaystyle\leftrightarrow$ $\displaystyle\omega_{B1}=\omega_{S}.$ (12)

Here, the suitable resonant angular frequency for $B_{1}(t)$ is defined by
$\omega_{B1}$. The energy hierarchy condition eq. 1 implies that we need to be
in the near-equivalence regime [12] where $J\gg J^{(1)}-J^{(2)}$. The full
translation table is given in Table 1.

##  III Results

###  III.1 Comparing polarization schemes

A wide variety of polarization schemes with different advantages and drawbacks
have been proposed in the literature, both for PHIP and for DNP. Here, we aim
to give an overview of the underlying mechanism and main properties of a
number of commonly used polarization schemes in each setting. For each scheme
we emphasize the similarity and differences between the counterparts in DNP or
PHIP. In practice, robustness properties and transfer schemes are relevant
also in situations where the Hamiltonian of eq. 2 does not apply or additional
unwanted contributions are present. The reader is referred to [16, 24, 15] for
treatments that expand on the scope of the sequence properties discussed in
this work.

The largest difference between the PHIP and the DNP settings is the initial
state of the two (pseudo-)spins: Whereas in PHIP it is the pseudo-spin $I$,
not subject to external control, which is initially polarized (given by the
singlet-state), in DNP it is the electronic spin $S$, subject to external
control, which starts out in a polarized state. For polarization schemes, this
is of importance for the specific choice of the initial or final pulses in the
polarization sequences that are driving the spin $S$: In principle, most
schemes need both initial and final pulses to ensure a polarization exchange
along $\hat{S}_{z}$ and $\hat{I}_{z}$ in the lab frame. This is due to the
central part of all the schemes discussed here which creates a slightly
different effective interaction of the form
$(A_{\ast}/2)(\hat{I}_{x}\hat{S}_{z}+\hat{I}_{y}\hat{S}_{y})$ in the frame
rotating with both the constant field $B_{0}$ and the drive $B_{1}$. This
interaction transfers polarization along $\hat{S}_{x}$ and $\hat{I}_{z}$
instead. If there is no initial polarization on spin $S$ (PHIP), the initial
pulse has no effect and becomes optional. Correspondingly, for an initially
polarized spin $S$ (DNP), the state of $S$ after polarization transfer is
unpolarized, such that any final pulse becomes optional. In the figures
representing the polarization schemes, we always include both the initial and
the final pulse, which makes the schemes applicable for both the PHIP and DNP
setting. In practice, of course, it will often be expedient to remove the
respective optional pulse. For pulsed schemes, any stated values for the
resonance condition for $\tau$ refers to the limit
$A_{\ast}/\omega_{I}\rightarrow 0$. This condition will experience corrections
that scale in the ratio of $A_{\ast}/\omega_{I}$[25].

####  III.1.1 SLIC and NOVEL

The conceptually simplest polarization scheme drives the spin $S$ with a
constant Rabi frequency that matches the level-splitting of $I$, i.e.
$\Omega(t)=\omega_{I}$ in $\gamma_{s}B_{1}(t)=\Omega(t)\,\cos(\omega_{S}t)$,
which results in a polarization transfer at the maximally possible rate
$A_{\ast}=A_{\perp}$ (cf. eq. 3). In the field of PHIP, this scheme is known
as Spin-Locking Induced Crossing (SLIC) [17], in the field of DNP as Nuclear
spin Orientation via Electron spin Locking (NOVEL) [18] (see [9] for an
implementation and [26] for a numerical analysis based on NV centers) and is
known as Hartmann-Hahn resonance in nuclear spin double resonance experiments
[27]. If the constant intensity drive is of phase $X$, by itself it would
transfer polarization along the $\hat{S}_{x}$ and $\hat{I}_{z}$ axes. Thus,
suitable $\pi/2$ pulses are needed to rotate any initial (final) polarization
along $\hat{S}_{x}$ from (to) $\hat{S}_{z}$ as indicated in Fig. 2a.

Discussion of Robustness Properties – Limitations of this method are that the
acceptable error in the Rabi frequency $\Omega_{error}$ is proportional to
$A_{\ast}$, while the acceptable resonance offset $\Delta$ is proportional to
$\omega_{I}$. The continuous wave drive is easy to implement in experiment,
however accurately matching the Rabi frequency $\Omega(t)$ to $\omega_{I}$ can
be difficult due to magnetic field fluctuations and power fluctuations of the
source. The latter may be addressed to some extent by concatenation schemes
which add suitably chosen sidebands to the driving field that protect against
power fluctuations [28].

####  III.1.2 S2hM and NV nuclear spin initialization

Trains of $n$ regularly spaced $\pi$ pulses are commonly used to dynamically
decouple the driven spin from its environment or to make it susceptible to a
specific resonant frequency.

In our case, an interpulse temporal separation of $\tau=\pi/\omega_{I}$
creates the effective interaction Hamiltonian

$\displaystyle H^{\text{eff}}_{1}=A_{\ast}\hat{I}_{x}\hat{S}_{z},$ (13)

with $A_{\ast}=(2/\pi)\ A_{\perp}\approx 0.64\ A_{\perp}$ in the frame co-
rotating with both $B_{0}$ and $B_{1}$.

A second train of $\pi$ pulses can induce

$\displaystyle H^{\text{eff}}_{2}=A_{\ast}\hat{I}_{y}\hat{S}_{y}$ (14)

with a suitable transition: Shifting $\hat{S}_{z}\rightarrow\hat{S}_{y}$ can
be achieved with a $\pi/2$ pulse of phase $X$, and a waiting time of
$(\pi/2)/\omega_{I}=\tau/2$ induces $\hat{I}_{x}\rightarrow\hat{I}_{y}$ in the
co-rotating frame.

If the total duration of the two pulse trains is shorter than the polarization
transfer ($2n\tau\ll(2\pi)/A_{\ast}$), the effective Hamiltonians
$H^{\text{eff}}_{1}$ and $H^{\text{eff}}_{2}$ averages to
$(A_{\ast}/2)(\hat{I}_{x}\hat{S}_{z}+\hat{I}_{y}\hat{S}_{y})$. As in the case
of SLIC and NOVEL, suitable initial or final pulses can now be added in order
to reach $\hat{S}_{z}$-$\hat{I}_{z}$-transfer (cf. Fig. 2c,h).

In Singlet to heteronuclear Magnetization transfer (S2hM) [12, 19], $n$ is
chosen such that the polarization is transferred already after the second
pulse train. As in this regime Hamiltonian averaging cannot be used to reach
the transfer Hamiltonian eq. 3 exactly, the reachable polarization is slightly
below 100 % (cf. Table 1 in [19]). For DNP, an almost identical scheme has
been used for NV nuclear spin initialization [20].

The robustness properties of S2hM can be significantly improved with the use
of phase cycling [29, 30] and well-chosen pulse timing to refocus all waiting
times using the $\pi$ pulses. Due to the variable number of $\pi$ pulses per
train, this creates the need to use different phase cycles for different
system parameters $A_{\perp}$ and $\omega_{I}$. This can be avoided by using a
relatively small and fixed $n$ for each $\pi$-train and appending the $\pi/2$
pulse and the $\pi/(2\omega_{I})$ waiting time after each of a variable number
of $\pi$ trains. The pulse trains are now repeated until polarization is fully
transferred. Fig. 2c,h) shows one such choice and a detailed description as
well as the robustness properties of S2hM without phase cycling can be found
in the S.I. . This approach has been used in [13] as motivation for developing
the even more robust PulsePol sequence, which will be described in the next
section.

Discussion of Robustness Properties – For sequences based on pulse trains, the
acceptable error in the resonance offset $\Delta$ is proportional to the Rabi
frequency $\Omega$. The acceptable errors on the Rabi frequency typically
remain tied to $A_{\ast}$, but the use of error-correcting phase-cycles or
composite pulses [19] can improve this to a scaling with the Rabi frequency
$\Omega$ itself. Compared to SLIC and NOVEL, the improved robustness
properties come at the cost of moderately reduced speed.

####  III.1.3 PulsePol

PulsePol is a polarization sequence developed for DNP in [13]. It consists of
a phase-shifted pair of three pulses each, with a central $\pi$ pulse
surrounded by symmetric waiting times, and $\pi/2$ pulses at both ends (cf.
Fig 2d,i). With regards to how the effective Hamiltonian arises, PulsePol can
be considered as a minimally short variation of a S2hM-type sequence as
described in the previous section: In PulsePol, each pulse train consists of
only a single pulse. A pair of $\pi/2$ pulses between each pulse train
reorients the current $S$ polarization along $\pm\hat{S}_{z}$ at their
intersection, and a shift $\hat{I}_{x}\rightarrow-\hat{I}_{y}$ between pulse
trains is reached by uniformly reducing all inter-pulse delays $\tau$ instead
of inserting an additional waiting time of $\pi/(2\omega_{I})$. The uniform
pulse delays strengthen the refocusing properties and the changed waiting time
improves the effective coupling by roughly $14\,\%$ compared to S2hM. The
robustness properties of PulsePol become clear if we consider that the
composition of a $\pi$ pulse surrounded by the two 90 degree phase-shifted
$\pi/2$ pulses creates the effect of a composite $\pi$ pulse, while the
intended effective Hamiltonian is created during the waiting times as
described above. Additionally, every set of two repetitions corresponds to the
$XY4$-sequence based on this composite pulse which provides a second layer of
error correction.

With a total duration $\tau$ for one repetition, the optimal resonance is
given by $\tau=3\pi/\omega_{I}$, which results in
$A_{\ast}=[2\,(2+\sqrt{2})/(3\pi)]A_{\perp}\approx 0.72\,A_{\perp}$.

Discussion of Robustness Properties – In PulsePol, both the acceptable Rabi
amplitude error and resonance offset scale with the Rabi frequency $\Omega$.
Together with its high degree of robustness, the slightly increased speed of
transfer compared to S2hM makes PulsePol a very attractive sequence. However,
in setups with a limited bandwidth it can be challenging to implement the
consecutive $\pi/2$ pulses. This problem can be alleviated by PulsePol
variations which include a waiting time in between the $\pi/2$ pulses [15].

Figure 2: Drawings of the different polarization schemes introduced in the
results section (a,b,c,d,e) and respective heatmaps showing the numerical
results for the transferred polarization for varying B0- and B1-errors at time
$t_{fin}$ (f,g,h,i,j). The parameters used in the simulation are
$\Omega=100\,A_{\perp}$ and $\omega_{I}=24\,A_{\perp}$. In the sequence plots,
$\pi/2$ pulses and continuous-wave pulses are filled, whereas $\pi$ pulses are
not. a,f) show the SLIC sequence (NOVEL in DNP), b,g) show the S2hM sequence
in the case of PHIP and the NV-nuclear spin initialization in DNP with the
slight adjustment of using a fixed n=8 XY8 phase cycle and the waiting time
distributed equally before the first and fourth $\pi$ pulse after the $\pi/2$
pulse. The corresponding results for S2hM using fixed phases are given in the
S.I. c,h) show PulsePol, d,i) show the ADAPT sequence in the case of PHIP and
TOP-DNP in the case of DNP. The chosen pulse duration corresponds to a $\pi/2$
pulse and the resonance condition equivalent to SLIC/NOVEL was used. e,j) show
a linear $B_{1}$ sweep with $\Omega\in[0.6\omega_{I},\ 1.4\omega_{I}]$. The
duration of the swept pulse in (e) is not to scale.

####  III.1.4 ADAPT and TOP-DNP

Both Time-optimized DNP (TOP-DNP) [22] and Alternating Delays Achieve
Polarization Transfer (ADAPT) [21] are families of sequences that can be
described as pulsed versions of SLIC or NOVEL: By exchanging the CW pulse with
$\Omega=\omega_{I}$ with an alternating sequence of short pulses and waiting
times, the same effective interaction can be approximated while improved
robustness properties and additional freedom in the choice of the waiting time
are gained. In [22], the latter is used to achieve polarization transfer also
in the regime $\Omega<\omega_{I}$, which however comes at the cost of lowering
the speed of polarization transfer ($A_{\ast}$). A similar approach is also
used in [31] to enable transfer in the $\Omega<\omega_{I}$ regime which can
lead to an overall reduction in average pulse energy.

Using the ’natural’ resonance where $\langle\Omega(t)\rangle_{t}=\omega_{I}$,
the coupling strength can approach that of SLIC/NOVEL $A_{\ast}\approx
A_{\perp}$ when using small rotation angles in each pulse. Fig. 2g,h use this
condition together with $\pi/2$ pulses as a representative example of the
properties of this scheme in the $\Omega>\omega_{I}$ regime.

Discussion of Robustness Properties – As visible in the numerical results
shown in Fig. 2h, the robustness to resonance-offset $\Delta$ gains side-bands
with successful polarization transfer in a region scaling with $\Omega$ where
each band has robustness properties very similar to SLIC or NOVEL. Thus, in
each band, acceptable Rabi errors scale with $A_{\ast}$ and the acceptable
resonance offset scales with $\omega_{I}$. As $\Omega$ defines the achievable
control strength, it provides an upper bound for the scaling behavior of
robustness in any sequence. Thus, the limitation of the robustness to
resonance offset to values $\Delta<\omega$ stops being of relevance in the
regime of $\Omega<\omega$.

####  III.1.5 adiabatic $B_{1}$ sweeps

Swept polarization methods [23, 32, 33, 14, 34], can only partially be
described with the effective Hamiltonian picture used in this work: They can
usually be regarded as a variation of SLIC or NOVEL in which one of the
parameters such as the drive amplitude $\Omega$ relevant for the resonance is
swept at a well-chosen speed. In words, the condition is that for every value
of the swept parameter, resonance to spins with this value is approximately
fulfilled for an amount of time that allows polarization to be transferred.
However, the effect on the driven spin itself is now more complicated. As
there are no repetitions in the sequence, not only the induced effective
interaction Hamiltonian, but also the state evolution of the driven spin can
be non-trivial. Often, slow adiabatic sweeps are used to achieve a well-
defined evolution. Schemes which sweep the resonance offset $\Delta$ include
the integrated solid effect [34] for DNP.

Fig. 2e,j show the sequence and robustness for an adiabatic $B_{1}$ sweep
similar to [14] (PHIP) and [23] (DNP). For simplicity, the sweep rate is
chosen as $\dot{\Omega}=A_{\perp}^{2}/(2\pi)$. Compared to SLIC and NOVEL
(Fig. 2a,f), the sweep significantly enhances the robustness to amplitude
errors. However this increased robustness comes at the cost of correspondingly
increasing the duration of the sequence. In a similar fashion, frequency-swept
pulses can reach very high bandwidths with successful polarization transfer.

Discussion of Robustness Properties – If the total duration of the scheme is
not a limiting factor, swept polarization methods can provide very high
robustness to errors in the swept parameter, while often being simpler to
implement than pulsed schemes.

###  III.2 Experimental translation between DNP and PHIP

We can now utilize the correspondence between PHIP and DNP illustrated in
Table 1 to apply the most fitting polarization transfer sequences to our
experimental conditions. We consider two experimental systems - polarization
of ${}^{1}H$ spins in naphthalene using triplet-DNP and achieving high
polarization in the ${}^{13}C$ spin of dimethyl maleate using PHIP. In the
former, using an amplitude sweep allows for a significant increase in the
transferred polarization without significantly more complicated $B_{1}$
drives. The latter demonstrates the applicability of the highly robust
PulsePol to PHIP.

Figure 3: Comparison of the achieved polarization rates in a pentacene doped
naphthalene crystal when applying the NOVEL and the adiabatic amplitude sweep.
The spectra were taken after polarizing for 20 s. We see a significant
increase in polarization rates for the amplitude sweep, which will allow
higher final polarization values.

####  III.2.1 adiabatic $B_{1}$ sweeps with PETS

Photo excited triplet states (PETS) can be used as a tool for DNP, where the
optically induced polarization of the PETS is transferred to close by nuclear
spins. Here, a pentacene doped naphthalene crystal is used, where the
pentacene acts as PETS and the protons in naphthalene are the nuclear spins
for which macroscopic polarization is desired. Various sequences for this
polarization transfer are known and have been shown to work, e.g. NOVEL and
the integrated solid effect (ISE) [35, 36, 34].

The experimental setup is described in [37]. We performed experiments at a B0
field of 175 mT, corresponding to a hydrogen Larmor frequency of around 7.5
MHz. Using NOVEL, successful polarization transfer was achieved (cf. Fig.
3(a)), although the presence of amplitude errors proved to be a limiting
factor: Using a linear adiabatic $B_{1}$-sweep/RA-NOVEL, the transfer rate
over a duration of 20 s could be improved by up to 60 % when using an
optimized sweep rate (cf. Fig. 3(b)).

Figure 4: Hyperpolarized ${}^{13}\mathrm{C}$ spectra of 15 mM
(1-${}^{13}\mathrm{C}$,$\mathrm{d}_{6}$)-dimethyl maleate acquired at 1.8 T,
obtained by using different polarization transfer methods operating at 100
$\mathrm{\mu}$T bias field. (a) Transfer induced by ${}^{13}\mathrm{C}$-ADAPT
consisting of 5 [90∘-$\mathrm{\tau}$] loops with free evolution time of 21.4
ms. (b) adiabatic $B_{1}$ sweep with transverse field amplitude swept from 0
to 25 Hz in 2 seconds. (c) Application of a single PulsePol echo block with
free evolution time set to 20.5 ms. (d) Conventional ${}^{13}\mathrm{C}$ NMR
spectrum obtained on a 13C-methanol at thermal equilibrium sample (averaged
over 4 transients). Estimated ${}^{13}\mathrm{C}$ polarization levels are
indicated.

####  III.2.2 Pulsepol for PHIP

(1-${}^{13}\mathrm{C}$,$\mathrm{d}_{6}$)-dimethyl maleate provides a PHIP
system with strong coupling $A_{\perp}$ which strains the condition of near-
equivalence (1). Nonetheless, polarization sequences can still be applied
successfully while the strong coupling allows for a correspondingly increased
robustness to detuning or amplitude errors (cf. Fig. 5) which in our case is
sufficient to reach high absolute polarization for all of the regarded
sequences ADAPT, adiabatic $B_{1}$ sweeps and PulsePol. We used solutions of
20 mM (1-${}^{13}\mathrm{C}$,$\mathrm{d}_{6}$)-dimethyl acetylenedicarboxylate
and 5 mM [Rh(dppb)(COD)]BF4 dissolved in acetone-d6. For each experiment,
hydrogenation of the precursor solutions at a pressure of 10 bar for 2 s was
applied with a subsequent purging with nitrogen gas for 18 s at room
temperature. The 92%-para-enriched hydrogen was produced at 21 K by an
Advanced Research Systems generator. To sustain singlet-order during this
process, continuous-wave decoupling of 1H was applied.

The polarization schemes were applied at a bias field of $B_{0}=100\,\mu$T
(Fig.4)and a drive amplitude of $\Omega=(2\pi)$ 50 Hz for pulses. The
adiabatic $B_{1}$ sweep in Fig.4(b) involved a transverse field amplitude
sweep from 0 Hz to 25 Hz performed in 2 seconds followed by a 90 degrees
pulse.

After the polarization transfer, the level of ${}^{13}\mathrm{C}$ polarization
was assessed by rapidly inserting the sample into a Bruker benchtop 1.8 T NMR
magnet and acquiring the ${}^{13}\mathrm{C}$ NMR spectrum after the excitation
by a single pulse. The polarization was estimated by comparing spectra against
a reference spectrum of thermally polarized neat ${}^{13}\mathrm{C}$-methanol.
For the absolute polarization levels, the concentration of was product was
calculated by recording 1H spectra of both the product solutions and an
aqueous solution of 492 mM DMSO. The error margins in Fig. 4 were established
by repeating each experimental procedure three times. The highly similar
polarization levels reached indicate that all three schemes provide sufficient
robustness to cover the bulk of the distribution of detuning and amplitude
errors in the system, such that the strong robustness of PulsePol provides
only a small advantage.

##  IV Discussion

Many insights in physics can be obtained by the translation of tools and
techniques from one field to another. In the same sense, the direct
translation between the near-equivalence regime in PHIP and DNP enables a
cross-pollination between the fields. First, this straightforwardly provides
new insights for existing control protocols. For example, the case of long MW
pulses studied thoroughly in TOP-DNP can shed light on the comparable case of
long RF pulses in the ADAPT protocol, as will be encountered for example in
implementations of ADAPT at very low magnetic fields. Second, this translation
allowed us to identify sequences developed for one physical system and apply
them in the other system, where they exceed the performance of the current
state of the art. This includes the use of PulsePol for PHIP polarization
transfer, and the use of adiabatic $B_{1}$ sweeps for the transfer of
polarization from optically polarised electron spins to nuclear spins in
pentacene:naphthalene crystals.

Another interesting area of application would be in the refinement of
polarization transfer in SABRE, where polarization in the near-equivalence
regime has been recently shown to lead to superior enhancements of ${}^{13}C$
nuclear spins in partially deuterated molecules [38]

We expect that additional insights can be gained from the above equivalence,
not only in the field of polarization transfer but other instances where a
similar pseudospin formalism is used. For example, methods and ideas from
singlet-NMR could be applied to quantum information processing and nanoscale
sensing using NV centers in diamond and vice-versa, as already brought to
light in the recent work by Sabba et al. [39].

## Acknowledgements

The authors acknowledge financial support by the German Federal Ministry of
Education and Research (BMBF) under the funding program quantum technologies -
from basic research to market via the project QuE-MRT (FKZ: 13N16447). MBP and
MK additionally acknowledge financial support by the ERC Synergy grant HyperQ
(grant no. 856432), and the Center for Integrated Quantum Science and
Technology (IQST).

## Declaration of competing interests

The authors declare the following competing financial interest(s): Laurynas
Dagys, Tim R. Eichhorn, Stefan Knecht, Christoph Müller, Alon Salhov, and Ilai
Schwartz are employees of NVision Imaging Technologies GmbH (NVision). Martin
B. Plenio and Ilai Schwartz are co-founders of NVision. NVision is
commercializing products in the field of magnetic resonance hyperpolarization.

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### Details of the Hamiltonian equivalence

In Section II, we introduce the pseudo-spin basis $(I\oplus\tilde{I})\otimes
S$ to arrive at the Hamiltonian given in Section II.2 which, in turn, is
equivalent to the Hamiltonian in the DNP setting if restricted to the
$I\otimes S$ subspace. Here, we provide the intermediate steps of the required
calculations.

We define the pseudo-spin operators
$I_{z}=(\ket{T_{0}}\bra{T_{0}}-\ket{S_{0}}\bra{S_{0}})/2$,
$I_{x}=(\ket{T_{0}}\bra{S_{0}}+\ket{S_{0}}\bra{T_{0}})/2$ and
$\tilde{I}_{z}=(\ket{T_{+1}}\bra{T_{+1}}-\ket{T_{-1}}\bra{T_{-1}})/2$. For
their squares we find

$\displaystyle
I_{z}^{2}=\dfrac{1}{4}(\ket{T_{0}}\bra{T_{0}}+\ket{S_{0}}\bra{S_{0}})=:\frac{1}{4}\hat{P}_{\tilde{I}}$
(15)

and

$\displaystyle\tilde{I}_{z}^{2}=\dfrac{1}{4}(\ket{T_{+1}}\bra{T_{+1}}+\ket{T_{-1}}\bra{T_{-1}})=:\frac{1}{4}\hat{P}_{I}$
(16)

with the respective projectors $\hat{P}_{I}$ and $\hat{P}_{\tilde{I}}$ onto
subsystems $I$ and $\tilde{I}$ in the Hydrogen-manifold. Note that in this
pseudo-spin basis, all operators in $I\oplus\tilde{I}$ need to be considered
as acting on the full 4-level Hydrogen-manifold. In contrast, the original
basis $I^{(1)}\otimes I^{(2)}$ allows the corresponding operators to be
considered as true single-spin operators due to the tensor-product structure.

The $J$-coupling contribution can now be expressed in the new basis, resulting
in

$\displaystyle\hat{\vec{I}}^{(1)}\cdot\hat{\vec{I}}^{(2)}$
$\displaystyle=\dfrac{1}{4}(\ket{T_{+1}}\bra{T_{+1}}+\ket{T_{0}}\bra{T_{0}}+\ket{T_{-1}}\bra{T_{-1}})$
$\displaystyle\quad-\dfrac{3}{4}\ket{S_{0}}\bra{S_{0}}$
$\displaystyle=\dfrac{1}{4}(\ket{T_{+1}}\bra{T_{+1}}+\ket{T_{-1}}\bra{T_{-1}})$
$\displaystyle\quad-\dfrac{1}{4}(\ket{T_{0}}\bra{T_{0}}+\ket{S_{0}}\bra{S_{0}})$
$\displaystyle\quad+\dfrac{1}{2}(\ket{T_{0}}\bra{T_{0}}-\ket{S_{0}}\bra{S_{0}})$
$\displaystyle=\frac{1}{4}\hat{P}_{\tilde{I}}-\frac{1}{4}\hat{P}_{I}+\hat{I}_{z}.$
(17)

Finally the $\hat{I}^{(i)}_{z}$ operators become

$\displaystyle\hat{I}^{(1)}_{z}\otimes\hat{\mathbb{1}}^{(2)}$
$\displaystyle=\dfrac{1}{2}(\ket{S_{0}}\bra{T_{0}}+\ket{T_{0}}\bra{S_{0}}+\ket{T_{+1}}\bra{T_{+1}}-\ket{T_{-1}}\bra{T_{-1}})$
$\displaystyle=\hat{I}_{x}+\hat{\tilde{I}}_{z},$ (18)
$\displaystyle\hat{\mathbb{1}}^{(1)}\otimes\hat{I}^{(2)}_{z}$
$\displaystyle=\dfrac{1}{2}(-\ket{S_{0}}\bra{T_{0}}-\ket{T_{0}}\bra{S_{0}}+\ket{T_{+1}}\bra{T_{+1}}-\ket{T_{-1}}\bra{T_{-1}})$
$\displaystyle=-\hat{I}_{x}+\hat{\tilde{I}}_{z}.$ (19)

With these results, we can directly replace all contributions to the
Hamiltonian from Eq. 4 to reach

$\displaystyle H$
$\displaystyle=\omega^{0}_{I}\hat{I}^{(1)}_{z}+\omega^{0}_{I}\hat{I}^{(2)}_{z}+\omega_{S}\hat{S}_{z}+J\,\hat{\vec{I}}^{(1)}\cdot\hat{\vec{I}}^{(2)}$
$\displaystyle\quad+J^{(1)}\hat{S}_{z}\hat{I}^{(1)}_{z}+J^{(2)}\hat{S}_{z}\hat{I}^{(2)}_{z}$
(20)
$\displaystyle=2\omega^{0}_{I}\hat{\tilde{I}}_{z}+\omega_{S}\hat{S}_{z}+\frac{J}{4}\,(\hat{P}_{\tilde{I}}-\hat{P}_{I}+4\hat{I}_{z})$
$\displaystyle\quad+\frac{1}{2}(J^{(1)}+J^{(2)})S_{z}\hat{P}_{\tilde{I}}+(J^{(1)}-J^{(2)})S_{z}I_{x}$
(21)
$\displaystyle=J\hat{I}_{z}+\omega_{S}\hat{S}_{z}+(J^{(1)}-J^{(2)})\hat{S}_{z}\hat{I}_{x}$
$\displaystyle\quad+2\omega^{0}_{I}\hat{\tilde{I}}_{z}+\frac{J}{4}\hat{P}_{\tilde{I}}-\frac{J}{4}\hat{P}_{I}+(\dfrac{J^{(1)}+J^{(2)}}{2})\hat{P}_{\tilde{I}}\hat{S}_{z},$
(22)

where all terms in the second line commute with all terms in the first line
and the former becomes equivalent to Eq. 2 if restricted to the $I\otimes S$
subspace.

### Details of the numerical simulation

The numerical simulations in this work use the pseudo-spin basis $I\otimes S$
for the Hamiltonian. The drive is parametrized via a time-dependent amplitude
$\Omega(t)$ and phase $\varphi(t)$:

$\displaystyle H(t)$
$\displaystyle=\omega_{I}\hat{I}_{z}+A_{\perp}\hat{S}_{z}\hat{I}_{x}$
$\displaystyle+(1+\dfrac{\Omega_{error}}{\Omega})\cdot\Omega(t)\left(\cos(\varphi(t))\hat{S}_{x}+\sin(\varphi(t))\hat{S}_{y}\right)+\Delta\hat{S}_{z}.$

Here, $\Omega_{error}/\Omega$ is the relative error in the Rabi frequency
created by the drive on spin $S$, and $\Delta$ parametrizes resonance-offset
errors of the drive. The system state starts in the state

$\rho=|+\frac{1}{2}\rangle\langle+\frac{1}{2}|\otimes\frac{1}{2}\hat{\mathbb{1}}$

where $\hat{I}_{z}|+\frac{1}{2}\rangle=\frac{1}{2}|+\frac{1}{2}\rangle$.

The integration in time of the dynamics is governed by the piece-wise constant
Hamiltonian $H(t)$. It proceeds by direct exponentiation of the Hamiltonian in
each interval where it is constant and chooses, where applicable, the number
of sequence repetitions $N$ such that the first maximum of the
$S$-magnetization is reached for the error-free case of
$\Omega_{error}=\Delta=0$. For schemes which do not rely on sequence
repetition such as amplitude sweeps and the non-repeating variant of S2hM,
$N=1$ is chosen whereas the duration of the sequence, i.e. the sweep duration
or the lengths of the pulse trains, is given by the theoretically optimal
values. For the robustness-plots, this value of $N$ is used for calculating
the polarization reached for a variety of errors in $\Omega_{error}/\Omega$
and $\Delta/\Omega$ and the final polarization is calculated via the
expectation value $\langle S_{z}\rangle$.

The timing of pulses is calculated in two steps: First, the total duration of
a sequence iteration is calculated as a sequence-dependent multiple of the
Larmor period $\tau$. Secondly, the individual pulses of rotation angle
$\alpha_{i}$ and durations $T_{i}=\alpha_{i}/\Omega_{i}$ are distributed in
the time interval such that pulses begin and end together with their
corresponding section defined by $\tau$. Unless specified differently by the
sequence, the pulses have equal interpulse waiting times to optimize
refocussing. Unless specified otherwise by the polarization scheme, the
maximum Rabi frequency $\Omega(t)=\Omega_{i}=\Omega$ is assumed to be used for
the duration of each pulse $i$.

### Description of robustness properties

The intended effective Hamiltonian for polarization transfer between the spins
$S$ and $I$ is given by Eq. 3 and contains only the Larmor precession of the
two spins together with a flip-flop contribution of strength $A_{\ast}\leq
A_{\perp}$.

For a full polarization transfer, it is necessary that any accumulating errors
which alter the spin state remain small over the whole duration
$T=2\pi/A_{\ast}$. For non-error-correcting sequences, unwanted Hamiltonian
contributions such as a resonance offset $\Delta\hat{S}_{z}$ will accumulate
at an error rate $\alpha=\Delta$, whereas first order error-correction due to
a drive of Rabi frequency $\Omega$ can suppress this to
$\alpha\propto(\Delta/\Omega)^{2}\cdot\omega_{I}$ where $(\Delta/\Omega)^{2}$
estimates the scaling behavior of the error accumulated during a single
repetition of the error-correcting part of the sequence and $\omega_{I}\propto
1/\tau$ estimates the duration of that sequence part.

Thus, for an error of strength $\beta$ and a non-correcting sequence,
successful polarization transfer is assured for

$\displaystyle\beta=\alpha\ll A_{\ast},$

and in the case of a (first-order) robust sequence for

$\displaystyle c\,(\beta/\Omega)^{2}\cdot\omega_{I}=\alpha\ll
A_{\ast}\Leftrightarrow\beta\ll c\,\Omega\sqrt{\dfrac{A_{\ast}}{\omega_{I}}},$

where $c$ is a proportionality constant depending on the sequence and the
details of the type of the error. In the main text, we refer to the former as
"maximum acceptable error scaling with $A_{\perp}$" and to the latter as
"maximum acceptable error scaling with $\Omega$". Note that the latter case
still includes an additional square root scaling with $A_{\perp}$.

The errors we regard in this work are detuning errors
$H_{err}=\Delta\hat{S}_{z}$ of the driving field with $\beta=\Delta$, and Rabi
errors

$H_{err}(t)=\dfrac{\Omega_{error}}{\Omega}\cdot\Omega(t)\left(\cos(\varphi(t))\hat{S}_{x}+\sin(\varphi(t))\hat{S}_{y}\right)$

with $\beta=\Omega_{error}.$

### Details of the regarded polarization schemes

Here, we provide further details about the sequences and robustness properties
of the schemes under consideration in their respective parameter regimes.

The more general robustness plots in Fig. 5 show the robustness properties of
the sequences for different regimes of the scale hierarchy
$\Omega,\omega_{I}\gg A_{\perp}$. From left to right, the coupling between the
spins $A$ decreases; from top to bottom $\omega_{I}$ decreases. All results
shown remain in the regime $\Omega>\omega_{I}$. Still, one observes that
different sequences display comparative advantages in different parameter
regimes. All of the robustness plots show the duration $t_{fin}$ of the
sequence compared to a reference given by the ideal application of SLIC
$t_{fin}^{SLIC}=2\pi/A_{\perp}$. Note that these times are not necessarily the
optimal choices as the algorithm for choosing $N$ is quite simple, and
additionally for SLIC/NOVEL and ADAPT/TOP-DNP the duration for a single
repetition was chosen to correspond to a total rotation angle of $2\pi$
although the sequences allow for a finer decomposition. Due to this, the
resulting values for $t_{fin}$ as well as the calculated robustness properties
are less representative of the corresponding sequences in the regime
$A_{\perp}\approx\omega_{I}$, i.e. the lower-left corner of Fig. 5b,d,f,h,j
and m.

##### SLIC and NOVEL.

This scheme (cf. Fig. 5 a,b) consists of initial and final $\pi/2$ pulses to
switch from the $+z$-orientation to the $+x$-orientation (and back) and an
intermediate spin-locking pulse with amplitude $\Omega=\omega_{I}$ and phase
$X$. In order to treat this scheme as a pulsed scheme, we restricted the spin-
locking pulse to $N$ repetitions of a pulse of rotation angle $2\pi$ (Fig.
5a). In Fig. 5b, one can clearly see that close to the error-free point of
$\Delta=0=\Omega_{error}$, acceptable amplitude errors $\Omega_{error}$ are
limited by $A_{\perp}$ , whereas detuning errors $\Delta$ may reach larger
values proportional to the amplitude of the spin-locking pulse $\omega_{I}$.
As the amplitude of this pulse is fixed by the spin system, it is not possible
to take advantage of larger possible values for $\Omega$.

##### ADAPT and TOP-DNP.

For simplicity and comparability, we only consider the versions with $\pi/2$
pulses and stay in the regime of $\Omega>\omega_{I}\gg A_{\perp}$. For the
regime of $\Omega<\omega_{I}$ or $\omega_{I}\sim A_{\perp}$ we refer to the
original works [21, 22]. Choosing $\tau=\pi/(2\omega_{I})$, each repetition of
the sequence consists of four $\pi/2$ pulses of phase $X$ with pulse-to-pulse
delay $\tau$. Again, initial and final pulses are $\pi/2$ pulses with phases
$Y$ and $-Y$ and suitable waiting times such that any two subsequent pulses
have equal waiting times (cf. Fig.5c). The robustness properties of this
sequence are shown in Fig.5d and show a central robust region very similar to
that of SLIC/NOVEL, however with additional sidebands in $\Delta$ repeating
over a region proportional to the Rabi amplitude $\Omega$.

##### S2hM and NV nuclear initialization

In the main text, we presented the properties of an S2hM variant with a fixed
pulse train length of $n=8$ $\pi$ pulses and adjusted waiting times in order
to enable phase cycling and full rephasing without adjustments to the specific
values of $\omega_{I}$ and $A$. Comparable robustness properties are still
achievable with a repetition-free version of S2hM (i.e. $N=1$) as long as
phase cycling is used and additionally all waiting times are refocussed by
$\pi$ pulses. S2hM uses the same initial and final $\pi/2$ pulses of phases
$Y$ and $-Y$ as SLIC/NOVEL, and the main part of the sequence consists of two
pulse trains of $n$ $\pi$-pulses, each surrounded by symmetric waiting times
of length $\tau/2$ (which together include the pulse-duration) and a
transitory $\pi/2$ pulse of phase $X$ together with an extra waiting time of
$\tau/2$. The time scale of $\tau$ is given by $\tau=\pi/\omega_{I}$. In the
robust repeating version of S2hM used in the main text, the $\pi$-pulse trains
each consist of four pulses, the first with phases $X,Y,X,Y$ and the second
with phases $Y,X,Y,X$. To achieve refocusing of all waiting times for any
number of repetitions $N$, the extra waiting time of $\tau/2-\pi/(2\Omega)$
after the transitory $\pi/2$ pulse is separated into two equal parts with the
second part being added to the waiting time between the third and fourth pulse
of the second pi train (cf. Fig. 5c). Note that for a single repetition $N=1$,
this sequence is fully equivalent to a usual S2hM sequence with $n=4$ $\pi$
pulses per train and slightly unusual waiting times. For more repetitions
($N>1$), the additional pulse trains per repetition each correspond to a
single longer pulse train of $n=8$ $\pi$ pulses with $XY8$ phases. As can be
seen in Fig. 5f, this choice allows for $\Omega$-scaling of acceptable errors
in both $\Omega_{error}$ and $\Delta$.

As an example for the robustness properties of S2hM without phase cycling, we
regard a simpler sequence of equal pulse trains which each consist of
$n=\lfloor\dfrac{\pi\omega_{I}}{2A}\rceil$ pulses of equal phase $X$, where
$\lfloor\cdot\rceil$ rounds to the closest integer. The waiting time after the
transitory $\pi/2$ pulse of phase $X$ is chosen such that the delay between
the two pulse trains is $\tau/2$. Finally, an equal waiting time is added
before the final $\pi/2$ pulse in order to refocus the intermediary waiting
times at least in the cases where $N$ is an odd integer (cf. Fig. 5k). As can
be seen in Fig. 5l, this version of S2hM is susceptible to Rabi errors
$\Omega_{error}$, and while the robustness to detuning errors $\Delta$ does
scale with $\Omega$, it is still significantly weaker than that of the phase
cycled version or that of PulsePol.

##### PulsePol

In PulsePol, the initial and final pulses are already included in the basic
blocks that are repeated in the sequence. This part consists of two equal
blocks which only differ by a phase shift of all pulses by $\varphi=\pi/2$.
The first block consists of a central $\pi$ pulse of phase $X$, surrounded by
two equal waiting times and begins and ends with a $\pi/2$ pulse of phase $Y$.
The total duration of a block is $\tau/2$ (cf. Fig. 5g), while the optimal
resonance condition is $\tau=3\pi/\omega_{I}$. As can be seen in Fig. 5h,
PulsePol retains successful polarization transfer in a large uniform region
scaling with $\Omega$ both for Rabi ($\Omega_{error}$) and detuning ($\Delta$)
errors. At a slight cost to the speed of the sequence, it is possible to
increase the robust region further by using $\varphi=\pi/4$ and
$\tau=3.5\pi/\omega_{I}$ instead [15].

##### Adiabatic $B_{1}$ sweeps

Similar to SLIC/NOVEL, the amplitude sweep consists of initial and final
$\pi/2$ pulses of phases $Y$ and $-Y$. The main part consists of a continuous
pulse with linearly increasing amplitude, here $\Omega\in[0.6\,\omega_{I},\
1.4\omega_{I}]$. In our case, this pulse is approximated by 150 pulses of
constant amplitude each (cf. Fig. 5i). The total duration of the swept pulse
is given by
$(1.4\omega_{I}-0.6\omega_{I})/\dot{\Omega}=\frac{(2\pi)}{A_{\perp}^{2}}\cdot
0.8\omega_{I}$. The robustness properties are very similar to the ones of
SLIC/NOVEL except for the significantly improved robustness to amplitude
errors (cf. Fig. 5j).

Figure 5: Detailed robustness plots for all sequences. The sequences are shown
in (a,c,e,g,i,k) and the corresponding robustness plots in different regimes
of $A_{\perp},\omega_{I},\Omega$ are shown in (b,d,f,h,j,l). In the robustness
plots, from left to right $A_{\perp}$ and from top to bottom $\omega_{I}$ is
decreased by a factor of 2 in each step and each heatmap shows the success of
polarization transfer over varying detuning $\Delta$ and amplitude errors
$\Omega_{error}$. a,b) show the SLIC sequence (NOVEL in DNP), c,d) show the
ADAPT sequence in the case of PHIP and TOP-DNP in the case of DNP. The chosen
pulse duration corresponds to a $\pi/2$ pulse and the resonance condition
equivalent to SLIC/NOVEL was used. e,f) show the S2hM sequence with a fixed
n=8 XY8 phase cycle and the waiting time distributed equally before the first
and fourth $\pi$ pulse after the $\pi/2$ pulse. The corresponding results for
S2hM with fixed phases are given in the k,l). g,h) show PulsePol, i,j) show a
linear $B_{1}$ sweep with $\Omega\in[0.6\omega_{I},\ 1.4\omega_{I}]$. The
duration of the swept pulse in (e) is not to scale. e,f) show the S2hM
sequence with without phase cycling. The corresponding results for S2hM with
XY8 phases are given in the c,f).