# Hyperpolarisation of nuclear spins: polarisation blockade

O.T. Whaites  Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom  C.I. Ioannou  QuTech, Delft
University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands  Kavli
Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046,
2600 GA Delft, The Netherlands  B.J.Pingault  QuTech, Delft University of
Technology, PO Box 5046, 2600 GA Delft, The Netherlands  Kavli Institute of
Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600 GA Delft,
The Netherlands  G.L. van de Stolpe  QuTech, Delft University of Technology,
PO Box 5046, 2600 GA Delft, The Netherlands  Kavli Institute of Nanoscience
Delft, Delft University of Technology, PO Box 5046, 2600 GA Delft, The
Netherlands  T.H. Taminiau  QuTech, Delft University of Technology, PO Box
5046, 2600 GA Delft, The Netherlands  Kavli Institute of Nanoscience Delft,
Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands
T.S. Monteiro  Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom

###### Abstract

Efficient hyperpolarisation of nuclear spins via optically active defect
centers, such as the nitrogen vacancy (NV) center in diamond, has great
potential for enhancing NMR based quantum information processing and nanoscale
magnetic resonance imaging. Recently, pulse-based protocols have been shown to
efficiently transfer optically induced polarisation of the electron defect
spin to surrounding nuclear spins- at particular resonant pulse intervals. In
this work, we investigate the performance of these protocols, both
analytically and experimentally, with the electronic spin of a single NV
defect. We find that whenever polarisation resonances of nuclear spins are
near-degenerate with a ‘blocking’ spin, which is single spin with stronger
off-diagonal coupling to the electronic central spin, they are displaced out
of the central resonant region- without, in general, significant weakening of
the resonance. We analyse the underlying physical mechanism and obtain a
closed form expression for the displacement. We propose that spin blocking
represents a common but overlooked effect in hyperpolarisation of nuclear
spins and suggest solutions for improved protocol performance in the presence
of (naturally occurring) blocking nuclear spins.

##  I Introduction

There is significant current interest in techniques for the control of nuclear
spins using solid-state defects like nitrogen vacancy (NV) centers in diamond
Degen2014 ; Renbao2014 . Many of these techniques rely on protocols of
periodically applied microwave pulses. Although they were originally developed
to dynamically decouple the electron spin from the environment BarGill ;
Zhao2011 ; Zhao2012a , it was subsequently found that when pulses are applied
at intervals resonant with surrounding 13C precession frequencies, the
resulting entanglement between and individual nuclear spin and the electronic
spin of the defect offers a very effective technique for sensing and
controlling nuclear spin states Degen2014 ; NV3 ; Neumann2010 .

Such pulse-based control has been exploited for nuclear polarisation and state
initialisation with applications ranging from quantum error correction and
quantum information NV3 ; Neumann2010 ; Bernien2013 ; Taminiau2014 ; Retzker ;
Registers , to nanoscale NMR and other sensing applications Wrachtrup2017 ;
Ajoy2022 . Dynamical nuclear polarisation (DNP) Abragam1978 , the transfer of
polarisation from electrons to nuclear spins, originally developed for NMR is
also being developed in this context. Recently proposed pulse-based DNP
protocols explicitly aimed at nuclear polarisation with NVs, PulsePol PulsePol
; PulsePol1  and PolCPMG PolCPMG  have been demonstrated to polarise 13C
nuclei in diamond. Polarisation of spins external to the diamond sample using
PulsePol has been achieved using an ensemble of NV centers Healey2021 .

Figure 1: Illustration of nuclear spin polarisation in the presence of a
blocking spin: (a) shows an NV center in diamond. A pulse-based polarisation
protocol, characterised by a pulse interval $\tau$, is applied to polarise a
distant cluster of nuclear spins (in light green) (b) Efficient polarisation
of the weakly-coupled spins (green dots) is expected near the resonant pulse
period $T=T_{r}$. However, in the presence of a spin that is near-degenerate
but interacts more strongly with the NV (blocking spin B), the cluster’s
resonances are displaced, to $T\simeq T^{B}_{r}$ (solid green line). The
resonance of spin B (red line) is unperturbed. We term this effect
‘polarisation blockade,’ in analogy to blockade effects encountered in other
fields of physics blockade2009 .

However, while these protocols were designed in the setting of polarisation
transfer to a single spin, in a realistic setting, the central spin couples to
multiple spins. In this study, we investigate polarization transfer from a
single spin simultaneously coupled to several environmental spins. In
particular, we find that polarisation transfer, at the expected polarisation
resonance $T=T_{r}$, can be suppressed by a blocking spin, which is a
simultaneously coupled environmental spin, with similar precession frequencies
but with stronger coupling. This effect is illustrated in Fig.1: the presence
of the blocking spin expels the polarisation resonances of the weaker-coupled
spins from the central $T\approx T_{r}$ region. We analyse the underlying
physical mechanism and clarify its relation to dark states, which are known to
suppress polarisation of nuclear spins TimeCrystal2021 ; DarkState . Spin
blocking is quite distinct and, to our knowledge, not previously investigated:
while dark-states suppress polarisation by decoupling a subspace of states
from the dynamics, spin-blocking acts by shifting a subset of spins off-
resonance. We experimentally verify this analysis using a single NV whose
microscopic spin environment has been precisely characterised Abobeih2019 ;
vandS2023 .

In section II we review pulse-based control of nuclear spins and introduce the
theoretical Floquet-based models used to analyse the joint dynamics of the NV
and nuclear spins and simulate the experiments. In section III we present our
results, including theory-experimental comparisons as well as an expression
for the resonance displacement, equation 6, a key new result of the work. In
Sec.IV we discuss the implications for nuclear polarisation and more efficient
control of nuclear registers.

##  II Methods: Pulse-based control

An NV electron spin system surrounded by $N_{\mathrm{nuc}}$ nuclear spins may
be described by the Hamiltonian ${\hat{H}}(t)={\hat{H}}_{p}(t)+{\hat{H}}_{0}$
where :

${\hat{H}}_{0}=\omega_{\text{L}}\sum^{N_{\mathrm{nuc}}}_{n=1}{\hat{I}}^{(n)}_{z}+{\hat{S}}_{z}\sum^{N_{\mathrm{nuc}}}_{n=1}\textbf{A}^{(n)}\cdot{\bf
I}^{(n)}.$ (1)

The operators for the electronic spin in subspace $\\{|0\rangle,|-1\rangle\\}$
and the nuclear spins are labelled $\hat{S}$, $\hat{I}$ respectively.
$\omega_{\text{L}}$ is the nuclear Larmor frequency; the hyperfine field
$\textbf{A}^{(n)}$ acting on the nuclear spin has components
$A^{(n)}_{\perp},A^{(n)}_{z}$ relative to the $z$-axis; we take
${A^{(n)}_{\perp}}\equiv A^{(n)}_{x}$, without loss of generality.
${\hat{H}}_{p}(t)=\Omega(t){\hat{S}}_{k}$ is the pulse control Hamiltonian.
$\Omega(t)$ is set by the microwave control field, while $k\equiv x,y$ for
common protocols.

For pulse-based control, there is a resonant pulse spacing $\tau=\tau_{r}$ for
which the electron and nuclear spin states selectively interact, allowing
efficient control of the nuclear states. For example, in the well-known CPMG
sequence Schweiger , microwave pulses are applied along the $x$-axis at
regular intervals, $\tau$; its resonant pulse spacing is
$\tau_{r}=j\pi/\omega_{I}$, where $j$ is an odd integer and the resonant
frequency of a given nucleus is $\omega_{I}\simeq\omega_{L}-A_{z}/2$. For
common protocols, the full protocol period $T$ is an integer multiple of
$\tau$. For CPMG specifically, it is $T_{r}=2\tau_{r}$. Note that we omit the
$n$ superscript for single nuclear spin calculations.

Figure 2:  Comparisons with experiment and spectral analysis (a) Polarisation
of weakly coupled nuclear spin C21 with $(A_{x},A_{z})\equiv(\approx
5.0,-9.7)$ kHz $\times 2\pi$ employing the PulsePol protocol, with
$N_{p}=4,R=100$. The NV environment contains a blocking spin C3 with
$(A^{B}_{x},A^{B}_{z})\equiv(59.,-11.3)$ kHz $\times 2\pi$. Nuclear spin C21
shows the asymmetric displaced resonance expected for $N_{p}=4$. Lower panels
show experiment (blue), single-spin simulation of C21 (black dash line),
simulation of C21 and C3 (orange line). The upper panel shows the
corresponding Floquet spectra, and offers an intuitive spectroscopic
understanding of spin blocking. The wide, broad avoided crossing is associated
primarily with C3. It overlaps with the much narrower avoided crossing
corresponding to the polarisation resonance of C21. This means the narrow C21
crossing is pushed away from $T_{r}$ to lower $T^{\prime}_{r}<T_{r}$. (b) For
C16, with $(A_{x},A_{z})\equiv(5.3,-19.8)$ kHz $\times 2\pi$, the overlap with
the strong C3 avoided crossing results in the narrow C16 crossing being pushed
towards larger $T^{\prime}_{r}>T_{r}$. Eq.6 quantifies the magnitude and
clarifies that the sign of the displacement depends on $A^{B}_{z}-A_{z}$.

Polarisation protocols: recently, new pulse-based protocols were identified
PulsePol ; PolCPMG  that split the resonance, such that each component
selectively addresses one nuclear spin state, allowing polarisation. In the
present work we focus on the DNP protocol PulsePol, commonly used due to its
robustness to detuning PulsePol . PulsePol combines a series of $x$\- and
$y$-directional MW pulses to map the NV state onto a nuclear spin. It has
pulse period $T=4\tau$ where $\tau$ is the pulse interval. Its resonant pulse
interval is $\tau_{r}=j\pi/(2\omega_{L})$, for $j=1,2...$ and the 3rd harmonic
($j=3$) is often selected for its effectiveness PulsePol . By averaging over
the period, an effective, time-independent single nuclear-spin Hamiltonian
${\hat{H}}^{(n)}\equiv
g^{(n)}({\hat{S}}_{+}{\hat{I}}^{(n)}_{-}+{\hat{S}}_{-}{\hat{I}}^{(n)}_{+})$
can be obtained, corresponding to a flip-flop type interaction between the NV
spin ($S$) and the $n$-th nuclear spin ($I$). The corresponding flip-flop rate
at this resonance was found to be $g^{(n)}=A_{x}^{(n)}(\sqrt{2}+2)/(6\pi)$,
PulsePol .

Repetitions: in general, in order to achieve high levels of polarisation,
repetitions of the polarisation protocol are required. For each repetition,
$N_{p}$ cycles are applied at each value of $\tau$. This is the periodic
component, where the NV-nuclear evolution is largely coherent. However, the NV
electronic spin is reinitialised optically after $N_{p}$ cycles to its $m=0$
state. The reduced nuclear bath evolution is (ideally) uninterrupted. The
$N_{p}$ sequence is then repeated.

A series of NV reinitialisations, interspersed with $N_{p}$ protocol cycles,
is repeated $R$ times. Typically, protocols employ a short run $N_{p}=2-8$
that yields appreciable polarisation for only the most strongly coupled spins;
but if this is followed by many $R\gg 1$ repetitions, polarisation of even
nuclear spins with weak coupling is gradually achieved. Corresponding
theoretical simulation involves $R$ sets of coherent Hamiltonian evolution for
$t=N_{p}T$ interspersed with calculation steps where the NV states are traced
out in order to simulate re-initialisation in $m_{s}=0$.

###  II.1 Experimental set-up

We study nuclear polarisation dynamics surrounding a single NV center at
cryogenic temperatures ($4\,\mathrm{K}$). The NV electron spin is initialised
and read out via resonant optical excitation. The NV sample employed here was
previously characterised in detail, allowing for accurate modelling of the
microscopic nuclear environment Abobeih2019 ; vandS2023 . The individual
nuclear spins are labelled as C1, C2…Cn and their hyperfine coupling strengths
$A^{(n)}_{x},A^{(n)}_{z}$, taken from Abobeih2019 , are tabulated in the
appendix. The nuclear spin expectation values are read out by applying a
combination of nuclear-nuclear and electron-nuclear gates, and subsequently
reading out the electron spin state as detailed in TimeCrystal2021 . As in
previous work, TimeCrystal2021 , we systematically correct for pulse errors
and amplitude damping during the readout pulse sequences, in order to get a
best estimate for the spin expectation values.

###  II.2 Theoretical methods: Floquet methods

We analyse the coherent dynamics with Floquet theory, a general framework for
periodically driven physical systems that has found wide applicability,
ranging from NMR continuous driving FloquetNMR  to also pulse-based control of
NV centers FloSpec  . However, Floquet theory encompasses several different
analytical tools. Floquet engineering (FE), Shirley1965  where a system driven
by a typically strong or high frequency (non-resonant) field can be shown to
correspond to an effective, static Hamiltonian with renormalised parameters,
by averaging over the period of the driving. Varying the amplitude of the non-
resonant drive, one may tune over the effective Hamiltonian to polarise the
bath PolChicago . A common and widely used approach is the Fourier series
decomposition of the one-period Hamiltonian, in a suitable rotating frame,
which has also been employed for NV pulse-based control of a nuclear-spin bath
FloApp2017 . Floquet spectroscopy FloSpec  has been introduced in this
context: resonances for pulse-based protocols were shown to correspond to
avoided crossings of the underlying Floquet quasi-energies of the pulse-
protocol unitary. Thus the morphology of these single or multiple avoided
crossings has proved insightful for analysis of NV-nuclear entanglement and
polarisation in terms of Landau-Zener dynamics FloAdiab2022 .

Here we employed both Fourier analysis and Floquet spectroscopy to analyse our
results.
Floquet spectroscopy: for a system with a temporally periodic Hamiltonian,
${\hat{H}}(t+T)={\hat{H}}(t)$, Floquet’s theorem allows one to write solutions
of the Schrödinger equation in terms of quasi-energy states
$|\psi_{l}(t)\rangle=\exp{(-i{\epsilon}_{l}t)}|\Phi_{l}\rangle$ where
${\epsilon}_{l}$ is the quasi energy.
$|\Phi_{l}(t)\rangle=|\Phi_{l}(t+T)\rangle$, $T$ is the period while
$l=1,..,D$ ($D$ is the dimension of the state space).

One may also obtain eigenstates of the one-period unitary evolution operator
${\hat{U}}(T)\equiv{\hat{U}}(T,0)$. The Floquet states $|\Phi_{l}\rangle$,
obey the eigenvalue equation:

${\hat{U}}(T)|\Phi_{l}\rangle=\lambda_{l}|\Phi_{l}\rangle\equiv\exp{(-i{\mathcal{E}}_{l})}|\Phi_{l}\rangle$
(2)

where
${\mathcal{E}}_{l}(T)\equiv\tan^{-1}{\operatorname{Im}{\lambda_{l}}/\operatorname{Re}{\lambda_{l}}}$
is the eigenphase (the Floquet phase). For Floquet spectroscopy numerics, we
diagonalise the full state space of the NV plus a cluster of $N_{nuc}\sim 1-7$
nuclear spins. Thus we can readily calculate and plot ${\mathcal{E}}_{l}(T)$
as a function of period $T$, to investigate resonances and gain insight on the
role played here by overlapping avoided crossings.

For Fourier series analysis, a transformation to the toggling frame (the frame
of the pulses, see Appendix for details) is widely used, including for
analysis of polarisation protocols PulsePol ; PolCPMG  and their resonances; a
key step is to average over a single period. However, in order to understand
experimental traces as a function of $\tau$, we must in addition consider also
off-resonant behavior (away from $\tau=\tau_{r}$), as shown below.

##  III Results

###  III.1 Single spin polarisation: off-resonant behavior

Details of our analysis are given in the Appendix and here we summarise the
key steps. Away from $\tau=\tau_{r}$, we introduce a small nuclear detuning,
slightly altering the PulsePol Hamiltonian to:

${\hat{H}}\equiv\sum^{N_{nuc}}_{n=1}g^{(n)}({\hat{S}}_{+}{\hat{I}}^{(n)}_{-}+{\hat{S}}_{-}{\hat{I}}^{(n)}_{+})+(\omega_{I}^{(n)}-\omega){\hat{I}}^{(n)}_{z}$
(3)

where the detuning of each nucleus corresponds to
$\delta_{n}(\tau)=\omega_{I}^{(n)}-\omega\ll\omega_{L}$ and the protocol
frequency $\omega=6\pi/T=3\pi/(2\tau)$. The resonant nuclear precession
frequency is
$\omega_{I}^{(n)}=\sqrt{(\omega_{L}-A^{(n)}_{z}/2)^{2}+(A_{x}^{(n)}/2)^{2}}$.
For coherent evolution over $N_{p}$ pulses, we can readily show that the
polarisation $2\langle{\hat{I}}^{(n)}_{z}\rangle$ of a single nuclear spin,
for moderate detuning, takes the simple form:

$\mathcal{P}(N_{p}T)=\left(\frac{2g}{\Omega_{r}}\right)^{2}\sin^{2}\left(\frac{\Omega_{r}N_{p}T}{2}\right)$
(4)

where the generalised Rabi frequency $\Omega_{r}=\sqrt{\delta^{2}+(2g)^{2}}$.
Hence, the maximum population transfer into this state is
$\mathcal{P}_{max}=1/(1+(\delta/2g)^{2})$ at the integer closest to the pulse
number $N_{p}=2\pi/(\Omega_{r}T)$. At resonance, $\delta=0$ and the maximum
saturation $\mathcal{P}_{max}=1$ and the Rabi frequency is $\Omega_{r}=2g$.
For $N_{p}$ greater than this maximal value, the polarisation oscillates
cyclically. Here, by convention $\mathcal{P}\in[-1,1]$. We adopt
$\mathcal{P}\in[-1/2,1/2]$, where results can be recovered with the
appropriate rescaling.

Asymptotic behavior: we note the above result is for a single repetition,
$R=1$, and experimental results range from $R\sim 10^{2}-10^{4}$. One may show
that single spin polarisation behaviour tends to an asymptotic envelope in
$R\to\infty$ limit. This is illustrated in Fig.1 (right panel, red solid
line). Stronger-coupled spins attain the asymptotic form after a few
repetitions. For some weaker coupled blocked spins, simulations indicate that
even $R=10000$ may be insufficient to reach the asymptotic limit. A notable
feature of the polarisation traces is that they exhibit sharp ‘dips’ at period
$T=T_{dip}$, seen in Fig.1 (red solid line) and also seen in the experiments.
Here we show that these dips (see Appendix for further details) occur for:

$T_{\mathrm{dip}}\simeq\frac{T_{r}}{(1+\mu^{2})}\left[1\pm\frac{n}{3N_{p}}\sqrt{1+\mu^{2}\left(\frac{9N_{p}^{2}}{n^{2}}-1\right)}\right]$
(5)

where $n\in\mathbb{Z}^{+}$, $n>0$, $T_{r}=6\pi/\omega_{I}$ and
$\mu=2g/\omega_{I}\ll 1$ thus $T_{\mathrm{dip}}\simeq
T_{r}\left[1\pm\frac{n}{3N_{p}}\right]$. The experimental traces also contain
additional fine-structure due to dephasing arising from the instrumental
waiting time $\sim 10\,\mu\mathrm{s}$ in between repetitions.

Figure 3:  Upper panel tests the analytical expression for the position of the
resonance Eq.6 against numerical simulations. Simulations for this colour map
use C3 as a blockade spin and spin with $A_{x}/(2\pi)=5$ kHz and a variable
$A_{z}$. Although agreement with the displaced peak is not exact, the
expression tracks the displacement quite well. The lower panel compares
experimental polarisation of C16 (blue) in the presence of blockade spin C3,
demonstrating good agreement with simulation (orange) of C3 and C16. For
comparison, the undisplaced single C16 simulation is shown (black dash line).
Eq.6 is shown to give reasonable agreement with the displaced peak position.
All simulations and experiments in this figure use parameters $N_{p}=8$,
$R=100$ which results in a displaced resonance rather than the ‘wedge’ profile
obtained for $N_{p}=4$.

###  III.2 Blockade spins: theory and experiment

If an NV center has a proximate 13C nuclear spin, at relative orientation such
that the spin will have $A^{B}_{z}\sim 0$ but reasonably strong off-diagonal
coupling $A^{B}_{x}$, we label this spin (see Fig.1(a))with superscript $B$ as
this ‘blockade’ spin in effect expels the resonances of weakly-coupled spins
(with $A^{s}_{z}\sim 0$ with $s=1,2...$) from the region around
$T_{r}=4\tau_{r}^{(B)}$, the expected resonance. The resonance of the more
strongly coupled blocking spin is unperturbed and remains at $T=T_{r}$. The
effect was illustrated in Fig.1.

Although full numerical simulation of clusters of 5-8 nuclei is feasible, for
insight, our analysis of spin blockade requires consideration of the NV
electron spin as well as the nuclear spin pair comprising B and one more
nuclear spin. Pair dynamics involves consideration of an 8-state space.
However, two states are largely decoupled and analysis reduces to two triplets
of coupled states (numerics involve full diagonalisation but, for insight, a
simpler model is analysed).

From the Floquet spectroscopy, this scenario corresponds to two sets of
avoided crossings in the Floquet eigenphases. This is illustrated in the upper
panels of Fig.2. It shows the pair of avoided crossings: a very broad crossing
of width $\sim A^{B}_{x}$ for the case of a strong-coupled spin and, within
it, a very narrow crossing due to the weaker coupled spin since $A^{B}_{x}\gg
A^{s}_{x}$. In other words the strong avoided crossing involves a pair of
states which mostly overlap with states of the single-spin avoided crossing of
spin $B$; while the overlapping narrow crossing involves states that mostly
overlap with the weaker spin states.

The lower panels show the corresponding experimental profiles for the weakly
coupled nuclear spins C21 with $(A_{x},A_{z})\equiv(\approx 5.0,-9.7)$
kHz$\times 2\pi$ and C16, with $(A_{x},A_{z})\equiv(5.3,-19.8)$ kHz$\times
2\pi$ respectively. Their resonances are displaced by the stronger blockade
spin spin C3 which has experimentally measured couplings
$(A^{B}_{z},A^{B}_{x})=(-11,-59)$ kHz$\times 2\pi$. There is excellent
agreement with simulations. As $N_{p}=4$ the resonance takes a characteristic
wedge shape whereas for $N_{p}=8$ it is predicted to be fully displaced.

A striking result is that while the C21 resonance is displaced to lower $T$,
for the C16 resonance, the converse is true. Analysis of the 3 state matrix
and its eigenvalues gives the position of the weak spin resonance and
magnitude of the displacement (see Appendix for derivation):

$\Delta
T_{r}/T_{r}\simeq-\frac{{(A_{x}^{B}})^{2}}{\omega_{I}^{(s)}(\omega_{I}^{(B)}-\omega_{I}^{(s)})}$
(6)

Eq.6 is a key result of this work. Fig.3 tests this expression against
numerics and experimental data. In the upper panel, the numerical colour map
shows polarisation as a function of $T$ and detuning
$\omega_{I}^{(B)}-\omega_{I}^{(s)}$. The overlaid white dots are from Eq.6 and
demonstrate that it provides a robust estimate of the magnitude of the
displacement of the polarisation resonance peak. The lower panel illustrates
an example of the displacement for $N_{p}=8$ and spin C16.

In contrast to the observed resonance displacement $\Delta T_{r}$, the Rabi
frequency, or width of the avoided Floquet crossing, is not strongly affected
by the blockade spin provided that
$|\omega_{I}^{(B)}-\omega_{I}^{(s)}|/A_{x}^{B}\ll 1$ (see appendix for
details). In general the measured polarisation is not very significantly
reduced, but rather is shifted to a different period $T_{r}+\Delta T_{r}$.
There are particular exceptions, such as the case of experimental data for a
spin simultaneously perturbed by two blockade spins (discussed in the
appendix).

Figure 4: Compares the conventional polarisation method of applying PulsePol
at a constant $T$ (or $\tau$) to our proposed adaptation of applying two
different regimes of $T$ in the presence of blocking spin C3. (a) Simulated
polarisation against periodicity of PulsePol, $T$, with parameters $Np=8$,
$R=100$ of both C3 in blue and C16 with in red. The two regimes are
highlighted as $T_{1}\simeq T_{r}+\Delta T_{r}\simeq 1.85$ $\mu$s which is a
periodicity at the displaced resonance and $T_{2}\simeq T_{r}\simeq 1.74$
$\mu$s near the original resonance. (b) Shows simulations polarisation of both
spins with total time. The solid line is the application of 200 repetitions
(grey region) at $T_{1}$ followed by 200 repetitions at $T_{2}$, the dashed
line is the application of 400 repetitions at $T_{2}$. A higher level of
polarisation in less time is achieved for C16. For the polarisation of C3,
although polarisation rises rapidly for both values of $T$, driving close to
its resonant value at $T=T_{2}$, rather than far off resonance at $T=T_{1}$ is
important for ensuring robust polarisation, to the 0.5 limit.

##  IV Discussion

As natural diamond contains of order 1.1% of 13C, we estimate that of order
20% of NV defects will have a nuclear spin with reasonably strong $A_{x}$
coupling but with $A_{z}\sim 0$, thus is able to produce a blocking effect on
distant, weakly coupled nuclear spins.

However, Eq.6 makes clear that the blocking effect is more generic and will
occur wherever a weaker coupled spin is near-degenerate with a stronger
coupled spin, thus can occur for arbitrary $A_{x},A_{z}$, provided
$A^{(B)}_{x}\gg A_{x}$ and $A^{(B)}_{z}\sim A_{z}$. Thus it should be a
relatively common feature in such studies, and spin-blocking is identifiable
via its distinctive spectral profiles, such as the ‘wedge’ shape for
$N_{p}=4$. The scenario of two blocking spins acting simultaneously on a
weaker spin is less common; but in the present data, we observed the case
where two blocking spins act to provide displacements of opposite signs
(presented in the Appendix). The result is a sort of destructive cancellation
that fully suppresses the resonance peak of the weaker spin. While the present
study considered PulsePol, our simulations show that similar behavior occurs
also for PolCPMG.

Understanding of the blocking spin mechanism allows one to propose approaches
to improve polarisation of weak spins. Figure 4 demonstrates a method for
drastically improving the polarisation efficiency by employing two different
$T$. The upper panel highlights the shifted resonance of spin C16 in the
presence of blockade spin C3. First, PulsePol is applied with $T\simeq
T_{r}+\Delta T_{r}$ in the region with the shifted resonance to polarise weak
spins. Following this, PulsePol with $T\simeq T_{r}$ is applied to maximise
polarisation: the asymptotic, saturated polarisation is maximal for $T\simeq
T_{r}$. The lower panel compares the effectiveness of the two $T$ method
compared to the standard technique of applying the protocol at $T=T_{r}$ only.
The two spin system of C16 (weak spin) and C3 (blocking spin) was used. The
initial stage (grey region) with $T\simeq T_{r}+\Delta T_{r}$, shows rapid
polarisation of both spins (solid lines); the second stage with $T\simeq
T_{r}$ then yields an improvement on overall polarisation relative to the
single $T$, on-resonance polarisation (dashed line).

Spin blocking effects are relevant to DNP of 13C in the diamond crystal and,
potentially, external nuclear spins as well. Even for different spin species
with different Larmor frequencies accidental resonances Loretz2015  such as
those that occur between harmonics for 1H and 13C might come into play, but
this has not been investigated here.

#### Spin blocking versus dark modes.

Dark-bright modes are a ubiquitous effect occurring in the physics of 3-level
systems DB1 ; DB2 : if we consider two degenerate modes, with eigenvalues
$\epsilon_{1,2}\approx\epsilon$, independently interacting with a third mode
$\epsilon_{3}\sim\epsilon$ with finite coupling strengths $g_{1}=g_{2}\equiv
g$, but not with each other, then modes $1,2$ hybridize such that one of the
hybrid modes fully decouples from mode 3 (dark mode, effective $g=0$) while
the others acquire enhanced coupling $\sqrt{2}g$ (bright modes). The spectral
signature is generic DB1 : instead of two independent avoided crossings of
width $g_{1}$ and $g_{2}$, the mixing/hybridisation produces a a single wider
avoided crossing of width $\sqrt{2}g$ and a completely decoupled state. The
role of dark states in impeding polarisation has been noted TimeCrystal2021
and investigated in a many body context PolChicago .

Spin blocking is a distinct polarisation suppression mechanism. It occurs for
similar regimes, but for the case where the coupling is highly anisotropic
$g_{1}\gg g_{2}$. The spectral signature is also generic. There are once again
two separate crossings, of width not far from the unperturbed widths $g_{1}$
and $\sim g_{2}$, like the unhybridised case. However only mode 1 remains at
$\epsilon\simeq\epsilon_{1}$. The weaker coupled mode has its crossing pushed
out of the $\epsilon\approx\epsilon_{1,2}$ region. For the polarisation
protocols, the weaker spin in effect is pushed off-resonance, which is
different from having its effective coupling suppressed, as is the case for a
dark state.

Off-resonant driving in general does not fully suppress polarisation, and in
principle, all spins should eventually tend to the $R\to\infty$ limit, which
is only maximal on-resonance. However it may make it extremely inefficient and
slow, potentially allowing imprecisions and decoherent processes to perturb
the protocol in a real experiment. However, unlike dark-mode suppression, the
effect may be mitigated by adjusting the protocol period to account for the
shifted resonance.

##  V Conclusions

In conclusion: in the present work we introduce, and theoretically and
experimentally investigate, the spin-blockade effect, thus named in analogy to
blockade effects blockade2009  encountered in other fields of physics. We show
that nuclear spins with strong interactions to the central spin can block
polarisation transfer by detuning weaker-coupled spins away from resonance.
This many-body effect, detrimental for polarisation efficiency, can be
mitigated by pulse sequences that are tailored to the microscopic
configuration of the spin system. Polarisation transfer to a complex spin
system can be highly dependent on the microscopic configuration of the spins
and our results thus provide an opportunity for optimization of dynamical
nuclear polarisation in various settings.

Acknowledgements Oliver Whaites acknowledges support from an EPSRC DTP grant.
This work was supported by the Netherlands Organisation for Scientific
Research (NWO/OCW) through a Vidi grant. This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement No. 852410). This work
is part of the research programme NWA-ORC (NWA.1160.18.208 and
NWA.1292.19.194), (partly) financed by the Dutch Research Council (NWO). This
work was supported by the Dutch National Growth Fund (NGF), as part of the
Quantum Delta NL programme. We acknowledge funding from the Dutch Research
Council (NWO) through the project “QuTech Phase II funding: Quantum Technology
for Computing and Communication” (Project No. 601.QT.001). B.P. acknowledges
financial support through a Horizon 2020 Marie Sklodowska-Curie Actions global
fellowship (COHESiV, Project Number: 840968) from the European Commission. We
thank M.Markham and D.J.Twitchen from Element Six for providing the diamond.

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##  VI Appendix

###  VI.1 Nuclear spin parameters

Table 1: List of parallel and perpendicular couplings between the ${}^{13}\mathrm{C}$ spins and the NV. In Abobeih2019  the same spins are labelled under a different numbering; the numbering system is included in the table for consistency. Label |  $A_{z}$/$2\pi$ (kHz) |  $A_{x}$/$2\pi$ (kHz) |  $\omega_{I}^{(i)}$ (rad/$\mu$s) | M Label
---|---|---|---|---
C0 | 213.153 | 3 | 2.04 | C9
C1 | -36.308 | 26.62 | 2.83 | C18
C2 | 20.569 | 41.51 | 2.65 | C12
C3 | -11.346 | 59.21 | 2.75 | C5
C4 | 8.029 | 21.0 | 2.69 | C13
C5 | 24.399 | 24.81 | 2.64 | C19
C6 | -48.58 | 9.0 | 2.86 | C6
C7 | 14.58 | 10 | 2.67 | C11
C8 | 7.683 | 4 | 2.69 | C22
C9 | -20.72 | 12 | 2.78 | C1
C10 | -23.22 | 13 | 2.78 | C2
C11 | -13.961 | 9 | 2.75 | C15
C12 | -31.25 | 8 | 2.81 | C3
C13 | -14.07 | 13 | 2.76 | C4
C15 | -5.62 | 5 | 2.73 | C17
C16 | -19.815 | 5.3 | 2.77 | C14
C17 | -4.66 | 7 | 2.73 | C16
C18 | 17.643 | 8.6 | 2.66 | C10
C20 | -8.32 | 3 | 2.74 | C7
C21 | -9.79 | 5.0 | 2.74 | C8
C22 | 1.212 | 13 | 2.71 | C21
C23 | 2.69 | 11 | 2.70 | C20
C24 | -3.177 | 2 | 2.72 | C23
C25 | -4.039 | 0.5 | 2.72 | C27
C26 | -4.225 | 0.771 | 2.72 | C24
C27 | -3.873 | 1.247 | 2.72 | C25
C28 | -3.618 | 9.472 | 2.72 | C26

###  VI.2 PulsePol polarisation protocol

In this section we briefly review the well-studied PulsePol protocol. Consider
a Nitrogen Vacancy (NV) defect in diamond coupled to $N_{nuc}$
${}^{13}\mathrm{C}$ spins in a global magnetic field, $B_{0}$. Under a
microwave control field, the pure dephasing Hamiltonian in the rotating frame
of the NV is:

$\hat{H}(t)=\left[\omega_{L}\hat{I}_{z}^{(n)}+\hat{S}_{z}(A^{(n)}_{z}\hat{I}_{z}^{(n)}+A_{x}^{(n)}\hat{I}_{x}^{(n)})\right]+\Omega(t)\hat{S}_{\varphi(t)}$
(7)

for a single ($n$-th) nuclear spin and where $\omega_{L}=-\gamma_{C13}B_{0}$
with a gyromagnetic ratio for ${}^{13}\mathrm{C}$ $\gamma_{C13}$,
$A^{(n)}_{i}$ are $i$th components of the hyperfine coupling strength relative
to the $z$-axis between the $n$th nuclei to the NV, $\Omega(t)$ is the
waveform of the microwave control field, $\hat{S}_{i}$ are in the qubit basis
$m_{s}=\\{0,-1\\}$, $\hat{I}^{(n)}_{i}$ are the $n$th nuclear spin-1/2
operators. and
$\hat{S}_{\varphi(t)}=\cos(\varphi(t))\hat{S}_{x}+\sin(\varphi(t))\hat{S}_{y}$.
The experimental external magnetic field strength is aligned with the NV
$z$-axis and measured to be $B_{0}=403\,\mathrm{G}$. The derivation of this
Hamiltonian has assumed the rotating wave approximation. For brevity, we
assume $\hbar=1$ in this appendix, unless considering experimental results in
SI units.

The waveform of the microwave control field is dependent on the dynamical
decoupling (DD) protocol that is applied. PulsePol is a DD protocol used for
dynamic nuclear polarisation (DNP), as it is state selective. The form of this
pulse protocol is
$\left[\left(\frac{\pi}{2}\right)_{Y}\left({\frac{\tau}{2}}\right)\left(\pi\right)_{-X}\left({\frac{\tau}{2}}\right)\left(\frac{\pi}{2}\right)_{Y}\left(\frac{\pi}{2}\right)_{X}\left({\frac{\tau}{2}}\right)\left(\pi\right)_{Y}\left({\frac{\tau}{2}}\right)\left(\frac{\pi}{2}\right)_{X}\right]^{2N_{p}}$
where $(\theta)_{\varphi}$ is a pulse with duration $T_{p}=\theta/\Omega$ and
phase $\varphi$ and $\rightarrow_{t}$ is free evolution for duration $t$. The
Rabi frequency due to the microwave drive is denoted $\Omega$. This protocol
is periodic with period $T=4\tau$ and it is applied for a total time of
$T_{tot}=4N_{p}\tau$. Using the waveform of the DD control field, the system
can be transformed into the frame of the microwave control field, also known
as the toggling frame, such that the Hamiltonian in this frame is

$\hat{H}(t)=\left[\omega^{(n)}_{I}\hat{I}_{z}^{(n)}+(f_{1}(t)\hat{S}_{x}+f_{2}(t)\hat{S}_{y})(A^{(n)}_{z}\hat{I}_{z}^{(n)}+A_{x}^{(n)}\hat{I}_{x}^{(n)})\right]$
(8)

where $f_{i}(t)$ are known as the modulation functions with period
$f_{i}(t+T)=f_{i}(t)$ and are step functions of the form

$f_{1}(t)=\begin{cases}1\text{ for }0<t<\tau/2\text{ and }5\tau/2<t<3\tau\\\
-1\text{ for }\tau/2<t<\tau\text{ and }2\tau<t<5\tau/2\\\ 0\text{ otherwise
}\end{cases}$ (9)

and

$f_{2}(t)=\begin{cases}1\text{ for }\tau/2<t<3\tau/2\text{ and
}7\tau/2<t<4\tau\\\ -1\text{ for }3\tau/2<t<2\tau\text{ and
}3\tau<t<7\tau/2\\\ 0\text{ otherwise }\end{cases}$ (10)

These functions are periodic and so a Fourier decomposition can be used, where
for $f_{1}(t)$

$f_{1}(t)=\sum^{\infty}_{k=0}\left[a_{k}^{(1)}\cos\left(\frac{k\pi
t}{2\tau}\right)+b_{k}^{(1)}\sin\left(\frac{k\pi t}{2\tau}\right)\right]$ (11)

where

$\displaystyle a_{k}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{k\pi}\frac{1-(-1)^{k}}{2}\left[4\sin\left(\frac{k\pi}{4}\right)-2\sin\left(\frac{k\pi}{2}\right)\right]$
$\displaystyle b_{k}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{k\pi}\frac{1-(-1)^{k}}{2}\left[-4\cos\left(\frac{k\pi}{4}\right)+2\right]$
(12)

and the remaining modulation function $f_{2}(t)$ can be found in a similar
fashion, only with coefficients $a_{k}^{(2)}=-(-1)^{k}b_{k}^{(1)}$ and
$b_{k}^{(2)}=(-1)^{k}a_{k}^{(1)}$ due to the shift of $t-\tau/2$ in their time
dependence. If it is assumed that $A_{x}<<\pi/\tau$, then the time dependence
of the Hamiltonian can be resolved in a perturbative fashion using the Magnus
expansion. To first order, the evolution operator is
$\hat{U}(T)=\exp[-i\hat{H}_{avg}T]$, where $\hat{H}_{avg}$ is the time period
averaged Hamiltonian constructed as

$\hat{H}_{avg}=\frac{1}{T}\int_{0}^{T}\hat{H}(t)\,dt$ (13)

hence removing the time dependence in place of time averaged field strengths.
This assumption of small times scales is fulfilled by transforming into the
interaction frame of the nuclear spin of interest and assuming that $A_{x}\ll
1/T$. Choosing the resonance $\tau_{r}=3\pi/(2\omega_{L})$ or $k=3$, and
taking the time average for the first order perturbation, the first order
Hamiltonian can be found to be:

$\hat{H}_{avg}=\sum_{n=0}^{N_{nuc}}\left[g^{(n)}(\hat{S}_{+}\hat{I}_{-}^{(n)}+\hat{S}_{-}\hat{I}_{+}^{(n)})+\delta^{(n)}\hat{I}_{z}^{(n)}\right]$
(14)

where we assume the Hamiltonian is an independent sum of $N_{nuc}$ spins, with
all interactions mediated by the central electronic spin. We define
$g^{(n)}=A_{x}^{(n)}\alpha/4$, $\alpha=2(\sqrt{2}+2)/3\pi$ and
$\delta^{(n)}=\omega^{(n)}_{I}-\omega_{p}$ ($\simeq A_{z}^{(n)}/2$ for
$\omega_{p}=6\pi/T\simeq\omega_{L}$) and resonance harmonic $k$. For
simplicity, as stated previously $k=3$ is chosen and will be absorbed into the
relationship between protocol frequency and period, such that
$\omega_{p}=2k\pi/T=6\pi/T$. The nuclear precession frequency is
$\omega^{(n)}_{I}=\sqrt{(\omega_{L}-A_{z}^{(n)}/2)^{2}+(A_{x}^{(n)}/2)^{2}}$.
This Hamiltonian is appropriate for small detuning from the nuclear precession
frequency, or that $\delta\ll\omega_{L}$. If this is not satisfied, higher
order terms may be needed.

##  VII Single Spin Polarisation

We now review the simplest NV plus single nuclear spin case. While this is
well-known, it is important for our study to keep track of the detuning
$\delta$ as we are interested in the behaviour away from the resonant pulse
period. Equation 14, for the case of a single spin, takes the matrix form:

$\hat{H}_{avg}=\begin{matrix}|\uparrow\uparrow\rangle\\\
|\downarrow\uparrow\rangle\\\ |\uparrow\downarrow\rangle\\\
|\downarrow\downarrow\rangle\end{matrix}\begin{pmatrix}\delta/2&0&0&0\\\
0&\delta/2&g&0\\\ 0&g&-\delta/2&0\\\ 0&0&0&-\delta/2\end{pmatrix}$ (15)

where the superscript in the previous section is removed in the single spin
case for simplicity. For ease of notation, the NV states are re-labelled as
$|0/-1\rangle=|\uparrow/\downarrow\rangle$.

From whence we see that the stretched states are decoupled thus:

$\displaystyle|\chi_{1}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\uparrow\rangle,\quad\epsilon_{1}=\delta/2$ (16)
$\displaystyle|\chi_{2}\rangle$ $\displaystyle=$
$\displaystyle|\downarrow\downarrow\rangle,\quad\epsilon_{2}=-\delta/2$ (17)

while the anti-aligned states form a 2x2 subspace of
$\\{|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle\\}$, or a pseudo-
spin 1/2 Hamiltonian of the form

$\hat{H}=\delta\hat{I}_{z}+2g\hat{I}_{x}=\mathbf{h}\cdot\hat{\mathbf{I}}$ (18)

where $\mathbf{h}=\omega(\sin\theta_{p},0,\cos\theta_{p})$ with eigenstates
and eigenvalues:

$\displaystyle|\chi_{3}\rangle$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\theta_{p}}{2}\right)|\downarrow\uparrow\rangle+\sin\left(\frac{\theta_{p}}{2}\right)|\uparrow\downarrow\rangle,\quad\epsilon_{3}=\frac{\omega}{2}$
$\displaystyle|\chi_{4}\rangle$ $\displaystyle=$
$\displaystyle\sin\left(\frac{\theta_{p}}{2}\right)|\downarrow\uparrow\rangle-\cos\left(\frac{\theta_{p}}{2}\right)|\uparrow\downarrow\rangle,\quad\epsilon_{4}=-\frac{\omega}{2}$

defining $\omega=\sqrt{\delta^{2}+4g^{2}}$ and $\tan\theta_{p}=2g/\delta$. The
detuning from the nuclear resonance creates an effective magnetic field in
this subspace which is not aligned with the $x$-axis. This misalignment, means
that maximum transfer between these two states is not possible. To see this,
consider an initial state of $|\psi(0)\rangle=|\uparrow\downarrow\rangle$.
With the unitary $\hat{U}(t)=\exp(-i\hat{H}_{av}t)$ and
$|\psi(t)\rangle=\hat{U}(t)|\psi(0)\rangle$ we readily obtain:

$\displaystyle|\psi(T_{tot}=N_{p}T)\rangle$ $\displaystyle=$
$\displaystyle\left[\cos(\varphi(N_{p}T))+i\cos(\theta_{p})\sin(\varphi(N_{p}T))\right]$
$\displaystyle|\uparrow\downarrow\rangle$ $\displaystyle-$ $\displaystyle
i\sin(\theta_{p})\sin(\varphi(N_{p}T))|\downarrow\uparrow\rangle$ (20)

where $\varphi(N_{p}T)=\omega N_{p}T/2$. The population from the
$|\uparrow\downarrow\rangle$ state will transfer into the corresponding flip
flop state $|\downarrow\uparrow\rangle$. The population in this state is then

$P_{\downarrow\uparrow}(N_{p}T)=|\langle\downarrow\uparrow|\psi(N_{p}T)\rangle|^{2}=\sin^{2}(\theta_{p})\sin^{2}\left(\frac{\omega
N_{p}T}{2}\right)$ (21)

and hence, the maximum population transfer into this state is
$P_{\downarrow\uparrow}^{max}=\sin^{2}\theta_{p}=1/(1+(\delta/2g)^{2})$ at
$N_{p}=2\pi/(\omega T)$. On resonance, or $\delta=0$, the maximum saturation
$P_{\downarrow\uparrow}^{max}=1$ as expected. The system undergoes
oscillations between these two states characterised by the frequency
$\Omega_{r}=\omega=\sqrt{\delta^{2}+4g^{2}}$, the generalised Rabi-frequency.

####  VII.0.1 Mixed states and repetitions

In general, nuclear spins such as ${}^{13}\mathrm{C}$ are not naturally in a
pure state but instead are in thermal mixture of equal parts
$|\uparrow\rangle$ and $|\downarrow\rangle$. This is often represented as a
density matrix, where the thermal mixture is initially

$\rho_{I}(0)=\frac{1}{2}\begin{matrix}|\uparrow\rangle\\\
|\downarrow\rangle\end{matrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}=\frac{\rho_{\uparrow}}{2}+\frac{\rho_{\downarrow}}{2}.$ (22)

However, the NV is initialised into one of the basis states, where here it is
taken that the NV is initialised into the $|\uparrow\rangle$ state such that
$\rho_{NV}(0)=\rho_{\uparrow}$. As with the state representation, the density
matrix of the product space of NV-C13 is constructed through the tensor
product $\rho(0)=\rho_{NV}(0)\otimes\rho_{I}(0)$. The initial density matrix
then evolves in time according to the transformation

$\rho(N_{p}T)=\hat{U}(N_{p}T)\,\rho(0)\,\hat{U}^{\dagger}(N_{p}T).$ (23)

For this system, the evolved density matrix is found to be

$\displaystyle\rho(N_{p}T)=\frac{1}{2}$
$\displaystyle\begin{matrix}|\uparrow\uparrow\rangle\\\
|\downarrow\uparrow\rangle\\\ |\uparrow\downarrow\rangle\\\
|\downarrow\downarrow\rangle\end{matrix}\begin{pmatrix}1&0&0&0\\\
0&|\beta(N_{p}T)|^{2}&\alpha^{*}(N_{p}T)\beta(N_{p}T)&0\\\
0&\alpha(N_{p}T)\beta^{*}(N_{p}T)&|\alpha(N_{p}T)|^{2}&0\\\
0&0&0&0\end{pmatrix}$ (24)

where
$\alpha(N_{p}T)=\cos(\varphi(N_{p}T))+i\cos(\theta_{p})\sin(\varphi(N_{p}T))$
and $\beta(N_{p}T)=-i\sin(\theta_{p})\sin(\varphi(N_{p}T))$. In order to
simplify this and explicitly show the evolution of the ${}^{13}\mathrm{C}$
spin, a partial trace can be performed on this density matrix in order to
reduce the 4x4 matrix to a 2x2 matrix and trace out the NV states. A partial
trace over one subspace, $\mathcal{H}_{B}$, with basis states
$\\{|b_{l}\rangle\\}$ in a dual system between this subspace and
$\mathcal{H}_{A}$ with basis states $\\{|a_{i}\rangle\\}$ is defined as

$\rho_{A}=\text{tr}_{B}(\rho_{AB})=\sum_{ijlk}c_{ijlk}|a_{i}\rangle\langle
a_{j}|\langle b_{l}|b_{k}\rangle$ (25)

where the dual space density matrix is decomposed as
$\rho_{AB}=\sum_{ijlk}c_{ijlk}|a_{i}\rangle\langle
a_{j}|\otimes|b_{l}\rangle\langle b_{k}|$. By performing this operation over
the NV basis states on the full density matrix in Eq.(24), the dynamics of the
nuclear spin can be reduced to

$\rho_{I}(N_{p}T)=\frac{1}{2}\begin{matrix}|\uparrow\rangle\\\
|\downarrow\rangle\end{matrix}\begin{pmatrix}1+\mathcal{P}(N_{p}T)&0\\\
0&1-\mathcal{P}(N_{p}T)\end{pmatrix}$ (26)

where
$\mathcal{P}(N_{p}T)=|\beta(N_{p}T)|^{2}=\sin^{2}(\theta_{p})\sin^{2}\left(\Omega_{r}N_{p}T/2\right)$.
This density matrix can be represented in the Pauli basis as
$\rho(N_{p}T)=(\mathbb{I}+\mathcal{P}(N_{p}T)\hat{\sigma}_{z})/2$. The
polarisation, defined as
$\langle\hat{\sigma}_{z}(t)\rangle/2=\text{tr}(\rho(t)\hat{\sigma}_{z})/2$ is
found to be

Figure 5: (a) Simulation showing that with increasing repetitions $R$, of the
polarisation sequence, in principle all spins tend to an asymptotic
polarisation envelope. Isolated strong spins attain the $R\to\infty$ limit for
$R\sim 10$, while isolated very weak coupled spins may require $R\gg 1000$.
Comparisons of single spin simulations of polarisation in (red) to
experimental data (blue) for three weak coupled spins in a cluster C6 (b), C8
(c) and C21 (d). The parameters used here are $R=1000$ and $N_{p}=4$. The
modest-coupling strength C6 is close to asymptotic. The weak coupled spins C8
and C21 are far from the limit, not only because of the weak coupling, but
also because of the many-body spin blocking investigated here.

$\langle\hat{I}_{z}(N_{p}T)\rangle=\frac{\mathcal{P}(N_{p}T)}{2}=\frac{1}{2}\sin^{2}(\theta_{p})\sin^{2}\left(\frac{\Omega_{r}N_{p}T}{2}\right)$
(27)

and completely equivalent to the result of the previous section.

However, in the experiment the NV is repeatedly optically re-initialised into
the $m=0$ state, with a repetition number $R$. All single-spins simulations
saturate to an asymptotic envelope as $R\to\infty$ although the rate at which
the limit is approached differs significantly, as illustrated in Fig.5. Weaker
spins evidently polarise more slowly, even in the isolated spin case. In
addition there are the many-body effects, such as dark state formation and the
new ‘spin-blocking’ that we investigated here that can significantly reduce
the polarisation rate.

Even in the single-spin case, both the simulated and experimental polarisation
traces show a detailed structure of ‘polarisation dips’. In the following
sections we analyse a key mechanism which causes sharp polarisation dips,
prior to discussing many-spin effects.

####  VII.0.2 Polarisation envelope side-dips

In Fig.5 and other figures one sees extremely large ‘side-dips’, symmetrically
distributed about the resonant $T_{r}$ where the average Hamiltonian model
predicts zero polarisation, but full numerics allow even negative
polarisation.

Within the average Hamiltonian model, the polarisation in Eq.(27) is zero when
the condition

$\sin\left(\frac{\Omega_{r}N_{p}T}{2}\right)=0$ (28)

is met. Solving this equation for $T$ yields the position for a series of such
side ‘dips’, $T_{\mathrm{dip}}$, to be

$\displaystyle T_{\mathrm{dip}}$ $\displaystyle=$
$\displaystyle\frac{T_{r}}{(1+\mu^{2})}\left[1\pm\frac{n}{kN_{p}}\sqrt{1+\mu^{2}\left(\frac{k^{2}N_{p}^{2}}{n^{2}}-1\right)}\right]$
(29) $\displaystyle\simeq$ $\displaystyle
T_{r}\left[1\pm\frac{n}{kN_{p}}\right]$

where $n\in\mathbb{Z}^{+}$, $n>0$ is the $n$-th side dip.

$k$ is the resonance harmonic and , $T_{r}=2k\pi/\omega_{I}$ while
$\mu=2g/\omega_{I}$. For harmonic $k=3$, the side dips are approximately

$T_{\mathrm{dip}}\simeq T_{r}\left(1\pm\frac{n}{3N_{p}}\right)$ (30)

assuming $\mu\ll 1$ since the nuclear spins and magnetic field used here
$g\ll\omega_{I}$.

##  VIII Two nuclear spins

The average Hamiltonian for the two-spin system is a straightforward extension
of the single-spin case:

$\displaystyle\hat{H}_{avg}$ $\displaystyle=$ $\displaystyle
g_{1}(\hat{S}_{+}\hat{I}^{(1)}_{-}+\hat{S}_{-}\hat{I}^{(1)}_{+})$ (31)
$\displaystyle+$
$\displaystyle\delta_{1}\hat{I}^{(1)}_{z}+g_{2}(\hat{S}_{+}\hat{I}^{(2)}_{-}+\hat{S}_{-}\hat{I}^{(2)}_{+})+\delta_{2}\hat{I}^{(2)}_{z}$

and corresponds to a $8\times 8$ matrix that may be decomposed into 4
independent subspaces. As with the single spin model, the stretched states
$|\uparrow\uparrow\uparrow\rangle$ and
$|\downarrow\downarrow\downarrow\rangle$ are eigenstates of $\hat{H}_{avg}$
with

$\displaystyle|\chi_{1}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\uparrow\uparrow\rangle,\quad\epsilon_{1}=\frac{\delta_{1}+\delta_{2}}{2}=\frac{\delta_{+}}{2}$
(32) $\displaystyle|\chi_{2}\rangle$ $\displaystyle=$
$\displaystyle|\downarrow\downarrow\downarrow\rangle,\quad\epsilon_{2}=-\frac{\delta_{1}+\delta_{2}}{2}=-\frac{\delta_{+}}{2}$
(33)

The remaining states form two decoupled subspaces of
$\\{|\uparrow\downarrow\uparrow\rangle,|\downarrow\uparrow\uparrow\rangle,|\uparrow\uparrow\downarrow\rangle\\}$
with overall magnetisation $M_{j}=1/2$ and
$\\{|\downarrow\downarrow\uparrow\rangle,|\uparrow\downarrow\downarrow\rangle,|\downarrow\uparrow\downarrow\rangle\\}$
with overall magnetisation $M_{j}=-1/2$. For the case of the $M_{j}=1/2$
subspace, the Hamiltonian matrix takes the form:

$\hat{H}_{avg}=\begin{matrix}|\uparrow\downarrow\uparrow\rangle\\\
|\downarrow\uparrow\uparrow\rangle\\\
|\uparrow\uparrow\downarrow\rangle\end{matrix}\begin{pmatrix}-\delta_{-}/2&g_{1}&0\\\
g_{1}&\delta_{+}/2&g_{2}\\\ 0&g_{2}&\delta_{-}/2\end{pmatrix}$ (34)

where $\delta_{-}=\delta_{1}-\delta_{2}$.

###  VIII.1 Dark and bright states

For the completely degenerate case

$\hat{H}_{avg}\equiv\delta\hat{I}+\begin{matrix}|\uparrow\downarrow\uparrow\rangle\\\
|\downarrow\uparrow\uparrow\rangle\\\
|\uparrow\uparrow\downarrow\rangle\end{matrix}\begin{pmatrix}0&g_{1}&0\\\
g_{1}&0&g_{2}\\\ 0&g_{2}&0\end{pmatrix}$ (35)

where $\delta$ is arbitrary and $\hat{I}$ is the $3X3$ identity matrix, there
are well-studied eigenstates:

$\displaystyle|\psi_{B_{-}}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\rangle_{NV}[\cos\varphi|\downarrow\uparrow\rangle+\sin\varphi|\uparrow\downarrow\rangle-|\downarrow\uparrow\uparrow\rangle]$
$\displaystyle|\psi_{B_{+}}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\rangle_{NV}[\cos\varphi|\downarrow\uparrow\rangle+\sin\varphi|\uparrow\downarrow\rangle+|\downarrow\uparrow\uparrow\rangle]$
$\displaystyle|\psi_{D}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\rangle_{NV}[\cos\varphi|\uparrow\downarrow\rangle-\sin\varphi|\downarrow\uparrow\rangle]$
(36)

where $|\psi_{D}\rangle$ is termed a ‘dark state’, with eigenvalue $\epsilon$,
and $|\psi_{B_{\pm}}\rangle$ so-called ‘bright’ states.
$g_{rms}=\sqrt{g_{1}^{2}+g_{2}^{2}}$, with $\cos\varphi=g_{1}/g_{rms}$,
$\sin\varphi=g_{2}/g_{rms}$

The dark state $|\psi_{D}\rangle$ has no overlap with the fully polarised
state $|\downarrow\uparrow\uparrow\rangle$, thus any state that acquires a
significant dark state component may not polarise. For the bright states, the
converse is true. For the case $g_{1}=g_{2}=g$ the problem is even simpler as
the matrix reduces to $g\hat{S}_{x}$, the spin 1 angular momentum $x$ matrix.

However, we need to consider the behaviour in the presence of detuning, hence
away from resonance, as well as spins that are not perfectly degenerate.
Neither the uncoupled Zeeman states nor the dark-bright states are eigenstates
of the general $\hat{H}_{avg}$ case. For insight, we represent $\hat{H}_{avg}$
in a slightly modified alternative ‘dark-bright’ basis
$\\{|\psi_{pol}\rangle,|\psi_{B}^{(1)}\rangle,|\psi_{D}^{(1)}\rangle\\}$ where

$\displaystyle|\psi_{pol}\rangle$ $\displaystyle=$
$\displaystyle|\downarrow\uparrow\uparrow\rangle$ (37)
$\displaystyle|\psi_{B}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\rangle_{NV}[\cos\varphi|\downarrow\uparrow\rangle+\sin\varphi|\uparrow\downarrow\rangle]$
(38) $\displaystyle|\psi_{D}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\rangle_{NV}[\cos\varphi|\uparrow\downarrow\rangle-\sin\varphi|\downarrow\uparrow\rangle]$
(39)

for $\tan\varphi=g_{2}/g_{1}$. In this basis,

$\hat{H}_{av}^{\prime}=\begin{matrix}|\psi_{pol}\rangle\\\
|\psi_{B}^{(1)}\rangle\\\
|\psi_{D}^{(1)}\rangle\end{matrix}\begin{pmatrix}\delta_{+}/2&g_{rms}&0\\\
g_{rms}&-\delta_{-}\cos(2\varphi)/2&\delta_{-}\sin(2\varphi)/2\\\
0&\delta_{-}\sin(2\varphi)/2&\delta_{-}\cos(2\varphi)/2\end{pmatrix}$ (40)

that we use to analyse two important cases.

###  VIII.2 Dark-bright states in the presence of detuning

First we consider the case where the two nuclear spins are degenerate
($A^{(1)}_{z}=A^{(2)}_{z})$ so $\delta_{-}=0$ and
$\delta_{+}=2\delta_{1}=2\delta$, a detuning which varies as the experiment
sweeps $T$ through the resonance $T=T_{r}$. In practice, we require
$\delta_{-}/2\ll g_{rms}$. The dark state $|\psi_{D}^{(1)}\rangle$ decouples
from the other bright states. The other two states form a separate subspace
$\\{|\psi_{pol}\rangle|\psi_{B}^{(1)}\rangle\\}$, such that the Hamiltonian is
$\hat{H}=\delta/2\mathbb{I}+\delta\hat{I}_{z}+2g_{rms}\hat{I}_{x}$. This
Hamiltonian is similar to the single spin model with $g\to g_{rms}$ and an
extra global energy shift. Hence:

$\displaystyle|\chi_{+}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\theta}{2}\right)|\psi_{pol}\rangle+\sin\left(\frac{\theta}{2}\right)|\psi_{B}^{(1)}\rangle,\quad\epsilon^{(1)}_{+}=\frac{\delta+\omega}{2}$
$\displaystyle|\chi_{-}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle\sin\left(\frac{\theta}{2}\right)|\psi_{pol}\rangle-\cos\left(\frac{\theta}{2}\right)|\psi_{B}^{(1)}\rangle,\quad\epsilon^{(1)}_{-}=\frac{\delta-\omega}{2}$
$\displaystyle|\chi_{D}^{(1)}\rangle$ $\displaystyle=$
$\displaystyle|\psi_{D}^{(1)}\rangle,\quad\epsilon_{0}=0$ (41)

where $\tan\theta=2g_{rms}/\delta$ and $\omega=\sqrt{\delta^{2}+4g_{rms}^{2}}$
are dependent on the detuning. The dark state is $\delta-$ independent and is
fully decoupled (effective zero coupling to the polarisation state) even for
$g_{1}=g_{2}$ . As $\delta$ is varied, $|\chi_{\pm}^{(1)}\rangle$ sweep
through a single avoided crossing with enhanced coupling $g_{rms}$.

If we take the initial state of the system to be $|\uparrow\rangle$ for the NV
and the nuclear spins in a thermal mixture, thus
$|\psi(0)\rangle=|\uparrow\downarrow\uparrow\rangle$, temporal evolution will
yield

$\displaystyle|\psi(N_{p}T)\rangle$ $\displaystyle=$ $\displaystyle
e^{-i\delta
N_{p}T/2}\cos\varphi\left[\alpha(N_{p}T)|\psi_{B}^{(1)}\rangle+\beta(N_{p}T)|\psi_{pol}\rangle\right]$
(42) $\displaystyle-$ $\displaystyle\sin\varphi|\psi_{D}^{(1)}\rangle$

where $\alpha(t)=\cos(\omega t/2)+i\cos(\theta)\sin(\omega t/2)$,
$\beta(t)=-i\sin(\theta)\sin(\omega t/2)$ and
$\cos\varphi=\frac{g_{1}}{g_{rms}}$. This is similar to the previous single
spin only with an extra, time independent, term involving the dark state. The
population of the state in the polarised state is then

$P_{pol}(N_{p}T)=\cos^{2}\varphi\sin^{2}\theta\sin^{2}\left(\frac{\omega
N_{p}T}{2}\right).$ (43)

Figure 6: Illustration of polarisation saturation for spin C8 due to spin-pair
dark states formed with a nearly degenerate spin C4. Simulations of single
spin C8 (dashed line) is shown to polarise fully after $R=1000$ of PulsePol
with $N_{p}=4$. This is not seen in experiment (blue) which only reaches a
polarisation of 0.2. This experimental saturation is reproduced to good effect
by simulation of the spin-pair C8 and C4 (orange).

This population has the same off-resonance envelope with $\sin\theta$ blocking
population transfer if $\delta\neq 0$, but also has an extra term involving
$\varphi$, which quantifies overlap with the dark state and suppresses
population transfer to the polarised state, as $\cos^{2}\varphi\leq 1$. This
is seen in Eq.(43) at arbitrary $\delta(\tau)$. An equivalent approach can be
used to analyse the $M_{j}=-1/2$ subspace
$\\{|\downarrow\downarrow\uparrow\rangle,|\uparrow\downarrow\downarrow\rangle,|\downarrow\uparrow\downarrow\rangle\\}$.

In summary, for degenerate spins, dark states cause a proportion of the spin’s
population to remain unpolarised , even for a large number of repetitions $R$.
Perfect degeneracy is not typical in a realistic setting, and in most cases
slow mixing between dark and bright states allows slow polarisation. An
example of dark state polarisation suppression is shown in Fig.6 between C8
and C4, which are almost degenerate, as in table.1. There is a significant
reduction in polarisation between the single spin simulation of C8 and a two
spin simulation with only C8 and C4, which better captures the experimental
behavior.

###  VIII.3 Spin Blockade

In this section we present the details of the spin blocking effect which is
introduced in this work. We consider the case of anisotropic coupling
strengths thus $g_{1}\gg g_{2}$. We relabel $g_{1}\equiv G$ and $g_{2}\equiv
g$, hence $G\gg g$, to clearly distinguish the strong from the weak coupling
spin. However, the spins are not fully degenerate, or $\delta_{-}\neq 0$, but
$\delta_{-}\ll G$. Such a scenario may be common when using NV centres to
polarise a cluster of distant nuclear spins. There are typically a small
number of proximate or near-proximate spins such as spin C3 where this
condition is satisfied by a larger number of weaker coupled spins such as
C16,C21,C25 etc. The anisotropy requirement is not too stringent: blockade
spins can displace the resonance of ‘local’ spins with moderate perpendicular
coupling, an effect which may need to be considered when using local clusters
as memory registers. Fig.7 demonstrates an example: spin C6 has reasonable
coupling $A_{x}/(2\pi)=9$ kHz, but still experiences a small resonance shift
from blockade spin C3, suggesting that the blocking effect is quite common.

For this case, since we write the effective Hamiltonian in Eq.(34)
$\hat{H}_{avg}=\hat{H}_{0}+\hat{V}$ as a zeroth order Hamiltonian plus
perturbation matrix where:

$\hat{H}_{0}+\hat{V}=\begin{matrix}|\uparrow\downarrow\uparrow\rangle\\\
|\downarrow\uparrow\uparrow\rangle\\\
|\uparrow\uparrow\downarrow\rangle\end{matrix}\begin{pmatrix}-\delta_{-}/2&G&0\\\
G&\delta_{+}/2&0\\\ 0&0&\delta_{-}/2\end{pmatrix}+\begin{pmatrix}0&0&0\\\
0&0&g\\\ 0&g&0\end{pmatrix}$ (44)

The detunings are relabelled to $\delta_{1}\to\delta_{B}$ and
$\delta_{2}\to\delta_{s}$ for a blockade spin and ‘small’ spin respectively.
The unperturbed Hamiltonian $\hat{H}_{0}$ can be reduced into two subspaces.
One subspace only contains the state
$\\{|\uparrow\uparrow\downarrow\rangle\\}$ which decouples from the others.
Therefore the eigenstate and eigenvalue can be read from the matrix to be
$|\chi^{(0)}_{3}\rangle=|\uparrow\uparrow\downarrow\rangle,\quad\epsilon_{3}^{(0)}=\delta_{-}/2$.
The remaining states form their own subspace
$\\{|\uparrow\downarrow\uparrow\rangle,|\downarrow\uparrow\uparrow\rangle\\}$
with Hamiltonian

$\hat{\tilde{H}}_{0}=\begin{matrix}|\downarrow\uparrow\uparrow\rangle\\\
|\uparrow\downarrow\uparrow\rangle\end{matrix}\begin{pmatrix}\delta_{+}/2&G\\\
G&-\delta_{-}/2\end{pmatrix}=\frac{\delta_{s}}{2}\mathbb{I}+\delta_{B}\hat{I}_{z}+2G\hat{I}_{x}\\\
$ (45)

Figure 7: Polarisation of C6 displaced by blocking spin C3. Simulations of C6
polarisation with C3 (orange line) after $R=100$ of PulsePol with $N_{p}=4$
has a slightly displaced polarisation peak from the single C6 spin (dashed
black-line) despite having a stronger perpendicular coupling of
$A_{x}=7.68\,\mathrm{kHz}/2\pi$. This displacement is seen in experimental
data for C6 polarisation (blue).

As expected, this takes the same form as the single spin case with an extra
global energy term from the second spin. The corresponding eigenstates and
eigenvalues are:

$\displaystyle|\chi^{(0)}_{1}\rangle$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\theta_{p}}{2}\right)|\downarrow\uparrow\uparrow\rangle+\sin\left(\frac{\theta_{p}}{2}\right)|\uparrow\downarrow\uparrow\rangle,\quad\epsilon^{(0)}_{1}=\frac{\delta_{s}+\omega}{2}$
$\displaystyle|\chi^{(0)}_{2}\rangle$ $\displaystyle=$
$\displaystyle\sin\left(\frac{\theta_{p}}{2}\right)|\downarrow\uparrow\uparrow\rangle-\cos\left(\frac{\theta_{p}}{2}\right)|\uparrow\downarrow\uparrow\rangle,\quad\epsilon^{(0)}_{2}=\frac{\delta_{s}-\omega}{2}$
$\displaystyle|\chi^{(0)}_{3}\rangle$ $\displaystyle=$
$\displaystyle|\uparrow\uparrow\downarrow\rangle,\quad\epsilon^{(0)}_{3}=\frac{\delta_{-}}{2}$
(46)

where $\omega=\sqrt{\delta_{B}^{2}+4G^{2}}$ and
$\tan\theta_{p}=2G/\delta_{B}$. The superscript $(0)$ is used to denote that
these are the unperturbed eigenstates and eigenvalues of $\hat{H}_{0}$. In the
basis of $\hat{H_{0}}$ eigenstates, the full Hamiltonian is
$\hat{H}^{\prime}_{avg}=\hat{H}^{\prime}_{0}+\hat{V}^{\prime}$ with

$\hat{H}^{\prime}_{0}=\begin{matrix}|\chi^{(0)}_{1}\rangle\\\
|\chi^{(0)}_{2}\rangle\\\
|\chi^{(0)}_{3}\rangle\end{matrix}\begin{pmatrix}(\delta_{s}+\omega)/2&0&0\\\
0&(\delta_{s}-\omega)/2&0\\\ 0&0&\delta_{-}/2\end{pmatrix}$ (47)

and the perturbation

$\hat{V}^{\prime}=\begin{pmatrix}0&0&gc_{p}\\\ 0&0&gs_{p}\\\
gc_{p}&gs_{p}&0\end{pmatrix}$ (48)

where $s_{p}=\sin(\theta_{p}/2)$ and $c_{p}=\cos(\theta_{p}/2)$. For a
$2\times 2$ single spin system, PulsePol is resonant with the spin when the
eigenvalues of $\hat{H}_{0}$ are degenerate, allowing $g$ to lift the
degeneracy and couple the two eigenstates. Ignoring the blockade spin ($G=0$),
the resonance of the second spin occurs when $\delta_{+}=\delta_{-}$, or
$\delta_{s}=0$. This is satisfied when the frequency of the PulsePol protocol
$\omega_{p}$ is resonant with the spin’s precession frequency, or

$\omega_{p}=\omega_{I}^{(s)}$ (49)

Using the definition of the protocol frequency $\omega_{p}=6\pi/T$, the period
of PulsePol, which is resonant with the nuclear spin, is
$T_{r}=6\pi/\omega_{I}^{(s)}$.

The same principle can be applied to the blockade spin system, where the
degeneracy is lifted by the perturbation, $\hat{V}^{\prime}$, when two
eigenvalues of the $3\times 3$ Hamiltonian, $\hat{H}^{\prime}_{0}$, are
degenerate. Explicitly, $\hat{V}^{\prime}$ weakly couples the state
$|\chi_{3}^{(0)}\rangle$ to either $|\chi_{1}^{(0)}\rangle$ or
$|\chi_{2}^{(0)}\rangle$. The eigenvalues of $\hat{H}^{\prime}_{0}$ are
different depending on the sign of $\delta_{-}$. For example, if
$\delta_{-}<0$, then the eigenvalues $\epsilon_{1}^{(0)}$ and
$\epsilon_{3}^{(0)}$ are never be degenerate, and $\hat{V}^{\prime}$ is not
strong enough to couple the corresponding states. However, there is a value of
detuning in which $\epsilon_{2}^{(0)}=\epsilon_{3}^{(0)}$, giving an updated
condition for the weaker spin’s resonance. For, $\delta_{-}>0$ this is
reversed and instead $\epsilon_{1}^{(0)}=\epsilon_{3}^{(0)}$. Initially, we
will only consider systems where $\delta_{-}<0$ and therefore the resonance
satisfies $\delta_{s}-\sqrt{\delta_{B}^{2}+4G^{2}}=\delta_{B}-\delta_{s}$.
Using the definition that $\delta_{i}=\omega_{I}^{(i)}-\omega_{p}$, then

$\omega_{p}=\tilde{\omega}_{I}^{(s)}=\omega_{I}^{(s)}+\frac{G^{2}}{\delta_{-}}$
(50)

which is shifted by $G^{2}/\delta_{-}$ from the original single spin resonant
frequency $\omega_{I}^{(s)}$. Assuming that the coupling $g\ll
G\ll\omega_{L}$, then the relative displacement of the resonant period is

$\frac{\Delta
T_{r}}{T_{r}}\simeq-\frac{G^{2}}{\omega_{I}^{(s)}(\omega_{I}^{(B)}-\omega_{I}^{(s)})}$
(51)

where the new resonance is at $T_{r}^{\prime}\simeq T_{r}+\Delta T_{r}$. The
robustness of Eq.(51) is tested against both two spin simulations and
experimental polarisation of C21 in Fig.3 of the main paper.

Figure 8:  Effect of two competing blockade spins: shows the polarisation of
C8 at a higher harmonic ($k=11$). (a) shows the Floquet phases and
polarisation for a two spin system of C4 and C8. A displaced avoided crossing
for C8 is present in the Floquet phases and a corresponding displaced
polarisation peak from the single spin (black) to the two spin (orange).
However, experimental data (blue) instead simply has significantly lower
polarisation (b) shows Floquet phases and polarisation for a three spin system
of C3, C4 and C8. The original displaced avoided crossing of C8 from (b) is
now nearly degenerate with the avoided crossing of C3, which now acts as a
blockade spin, pushing the crossing back towards its original position. This
three spin simulation (green) demonstrates the flattening of the two spin peak
and fits the experimental data better, showing the effect of higher order
multi-spin effects. Simulations and data here are for $R=1000$ repetitions of
$N_{p}=8$ PulsePol cycles.

Furthermore, degenerate perturbation theory can be used to find the Rabi-
frequency of this resonance by solving the $2\times 2$ matrix in the subspace
$\\{|\chi_{2}^{(0)}\rangle,|\chi_{3}^{(0)}\rangle\\}$, which is of the form
$\hat{\tilde{H}}^{\prime}=\epsilon\mathbb{I}+2gs_{p}\hat{I}_{x}$ where
$\epsilon=\epsilon_{2}^{(0)}=\epsilon_{3}^{(0)}$. The corresponding Rabi
frequency is found to be:

$\Omega_{r}\simeq 2g\sin\left(\frac{\theta_{p}}{2}\right)$ (52)

This Rabi frequency is attenuated by a factor $\sin\theta/2$ relative to the
one spin case. In fact, if $|\Delta T_{r}/T_{r}|\gg 1$ then the coupling
between the two states will be completely suppressed and $\Omega_{r}=0$.
However, for the typical scenarios studied here, the displacement is less
drastic and $\Omega_{r}\sim 2g$. We note that in the $\delta_{-}\to 0$ limit,
the shift tends to infinity. In this scenario, the second small crossing
vanishes and the dark mode behaviour is regained.

###  VIII.4 Competing Blockade Spins

While the focus here was primarily on the case of a single spin (eg C3),
simultaneously and independently blocking multiple weaker-coupled spins, we
can consider the case where two blockade spins affect the same weaker coupled
spin. This results in a competing or combined blockade effect. Such a scenario
is identifiable in the experimental cluster here, when looking at higher
harmonic resonances of PulsePol at larger $T$.

In previous sections, a PulsePol resonance of harmonic $k=3$ was studied.
Consider now a higher harmonic, $k=11$, at a much greater periodicity $T$. At
higher harmonics, single spin resonances have a greater spacing, meaning that
C8 and C4 are relatively close but no longer degenerate. Hence, the
polarisation of C8 no longer saturates due to dark states, and is displaced by
$\Delta T_{r}\simeq-0.32\,\mu\mathrm{s}$. A simulation of this higher harmonic
displacement of C8 due to C4 is shown in the first panel of Fig.8. However,
this is not seen in the experimental data, where there is significantly less
polarisation than expected in this region of $T$. In fact, by considering the
initial blockade spin C3 in simulations, as is done in the second panel of
Fig.8, polarisation levels closer to the experimental data are obtained.

As in the Floquet states in Fig.8, the expected displacement of the avoided
crossing for C8 due to blockade spin C4 is nearly degenerate with the avoided
crossing of C3, which was previously shown to act as a blockade spin for
others with small $A_{x}$. Therefore, this shift closer to C3 causes a
secondary blockade effect on C8, displacing the resonance back towards its
original position and ‘smearing’ the polarisation across $T$. This double-
blockade effect on C8 is a three-spin effect, further highlighting the
importance of considering many-body effects when performing polarisation of
clusters.


Choreographies, Concurrency, $\lambda$-calculus, Type Systems, Polymorphism

Cambridge University Press

Alice or Bob? Process Polymorphism in Choreographies

Eva Graversen
Department of Mathematics and Computer Science, University of Southern Denmark

Andrew K. Hirsch
Department of Computer Science and Engineering, University at Buffalo, SUNY

Fabrizio Montesi
Department of Mathematics and Computer Science, University of Southern Denmark

We present , a language for higher-order functional choreographic programming—an emerging paradigm by which programmers write the desired cooperative behaviour of a system of communicating processes and then compile it into distributed implementations for each process, a translation called endpoint projection.there was some repetition here (programming... programming), does it read ok now?
Unlike its predecessor, Chor$\lambda$, has both type and process polymorphism inspired by System F$_\omega$.
That is, is the first (higher-order) functional choreographic language which gives programmers the ability to write generic choreographies and determine the participants at runtime.
This novel combination of features also allows processes to communicate distributed values, leading to a new and intuitive way to write delegation.
While some of the functional features of give it a weaker correspondence between the semantics of choreographies and their endpoint-projected concurrent systems than some other choreographic languages, we still get the hallmark end result of choreographic programming: projected programs are deadlock-free by design.

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§ FULL TYPING RULES

Θ;Γ⊢N M:[2]

Θ;Γ'⊢::  for all  ∈

Θ∩(∪([1]) ∪([2])∪([1])∪([2]));Γ,x:[1]⊢M:[2]

Θ;Γ⊢[1] ::

Θ;Γ⊢[2] ::

Θ;Γ⊢M :
Θ;Γ⊢[1][2]ℓM: *[left=Tcom]

Θ;Γ⊢[1]:: (()∪())

Θ;Γ⊢[1][2]:( [1] [2])

Θ;Γ⊢M[2] : []X[2]
Θ';Γ', X :: ⊢M :

if ∃, . =
thenΓ' = (Γ+ X)X
elseΓ' = Γ+ X

if = or ∃.  =
then Θ' = Θ, X
else Θ' = Θ
Θ, X;Γ+ X & ∖{X}, X::  ⊢M :

Θ, X; Γ+ X, X :: ⊢M :

Θ; Γ+ X & ∖{X}, X ::  ⊢M :

Θ, X; Γ+ X, X :: ⊢M :

∄, .  =

[1] ≡[2]

∀f∈𝖽𝗈𝗆𝖺𝗂𝗇(D).  f : ∈Γ∅;Γ⊢D(f) :
Θ;Σ;Γ⊢x: *[left=Tcase]

Θ;Γ⊢NxM'x'M”: *[left=Tfun]
Θ;Γ⊢f: *[left=Tpair]

Θ;Γ⊢MN: [1][2] *[left=Tproj1]
Θ;Γ⊢M:[2] *[left=Tinl]
Θ;Σ;Γ⊢M[2]:[1][2] *[left=Tinr]

§ FULL OPERATIONAL SEMANTICS

(x[]M)V →_D MxV

M_1 →_D M_2

M_1N →_D M_2N
N_1 →_D N_2

VN_1 →_D VN_2

(X[]M) →_D MX
M_1 →_D M_2

M_1 →_D M_2

[1] →_D [2]

V[1] →_D V[2]

M_1 →_D M_2

M_1 →_D M_2
M_1 →_D M_2

M_1 →_D M_2
N_1 →_D N_2

N_1xM_1yM_2 →_D N_2xM_1yM_2

VxM_1yM_2 →_D M_1xV

VxM_1yM_2 →_D M_1xV

M_1 →_D M_2

M_1N →_D M_2N
N_1 →_D N_2

VN_1 →_D VN_2
M_1 →_D M_2

M_1 →_D M_2
M_1 →_D M_2

M_1 →_D M_2

V_1V_2 →_D V_1
V_1V_2 →_D V_2

*[left=Def] f →_D D(f)

*[left=Sel] ℓM →_D M *[left=Com]
V →_D V

§ FULL NETWORK AND LOCAL PROGRAM OPERATIONAL SEMANTICS

V_ V'
V_ V'
L_1_ M'_1

L_2_ M'_2

[] [M_1] |[] [M_2]_ [] [M'_1] |[] [M'_2]
ℓM_ M
{ℓ_1: M_1,…,ℓ_n: M_n}_ M_i
L_1_ M'_1

L_2_ M'_2

[M_1] | [M_2]_  [M'_1] | [M'_2]
MM' _ M
MM' _ M'
(x[]M)V _MxV
V _V
M _M'

[M]_  [M']
(XM) _ MX

M_1 _M_2

M_1M' _M_2M'

M _M'

M _M
M _M'

M _M'
M _M'

M _M'
M _M'

M _M'
M _M'

VxM_1yM_2 _M_1xV
VxM_1yM_2 _M_2xV
M_1 _ M_1'

M_1M_2 _M_1'M_2
M_2 _ M_2'

M_1M_2 _M_1M_2'
M_1 →_D M_2

M_1 →_D M_2
M_1 →_D M_2

M_1 →_D M_2
V_1V_2 _V_1
V_1V_2 _V_2
f _(f)
𝒩_1 _ 𝒩_2

𝒩_1 |𝒩' _ 𝒩_2 |𝒩'