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The connection between statistical mechanics and the dynamics of
isolated many-body quantum systems is typically formulated in terms
of the eigenvalue thermalization hypothesis1-5 (ETH), the idea that
complexity in interacting systems prompts ergodicity at the quantum
level. Disorder and quantum interference can stymie thermalization,
often leading to regimes of sub-diffusive dynamics or suppressed
transport, as already identified in a broad class of systems
ranging from acoustic and optical waves to cold atomic gases6-8.
The breakdown of ergodicity can extend even to strongly interacting
quantum systems, a regime known as many-body localization9-11
(MBL). While most MBL experiments thus far have focused on isolated
arrays of trapped atoms12,13, recent theoretical work argued that
the interplay between diffusion and localization also influences
the out-of-equilibrium dynamics of driven open systems14,15. In
particular, these studies built on the extensive body of work on
dynamic nuclear polarization16,17 (DNP) to show that the concept of
spin temperature is directly connected (namely, relies on) quantum
ergodicity and ETH.
Indeed, ensembles of electron and nuclear spins in solids
provide a practical experimental platform to investigate
thermalization because disorder and long-range interactions compete
in ways that can be exposed using alternative spin control
techniques. For example, recent experimental work studied spin
depolarization in ensembles of nitrogen-vacancy (NV) centers in
diamond and revealed sub-exponential, disorder-dependent relaxation
associated with critical
thermalization dynamics18,19. Along similar lines, dynamic
nuclear polarization of carbon spins in diamond was exploited to
expose electron-spin-mediated nuclear spin diffusion exceeding the
value expected for naturally abundant 13C spins by nearly two
orders of magnitude20.
Here, we resort to nuclear spins in diamond to demonstrate
control over the localization/delocalization dynamics of
hyperfine-coupled carbons upon variation of the applied magnetic
field. We formally capture our observations by considering a model
electron-nuclear spin chain featuring magnetic-field-dependent spin
transport. Further, we show the dynamics at play can be cast in
terms of well-defined dynamic phases that can be accessed by tuning
the magnetic field strength and paramagnetic content. The spin
state hybridization emerging from the intimate connection between
electron and nuclear spins gives rise to otherwise forbidden
low-frequency transitions, whose presence underlies the system’s
singular spectral response to RF excitation of variable
amplitude.
In our experiments, we dynamically polarize and probe 13C spins
in a [100] diamond crystal (3×3×0.3 mm3) grown in a
high-pressure/high-temperature chamber (HPHT). The system is
engineered to host a large (~10 ppm) concentration of NV centers,
spin-1 paramagnetic defects that polarize efficiently under green
illumination. Coexisting with the NVs is a more abundant group of
P1 centers (~50 ppm), spin-1/2 defects formed by substitutional
nitrogen atoms. We tune an externally applied magnetic field " in
and out of the ‘energy
Magnetic-field-induced delocalization in hybrid electron-nuclear
spin ensembles
Daniela Pagliero1,*, Pablo R. Zangara6,7,*, Jacob Henshaw1,
Ashok Ajoy3, Rodolfo H. Acosta6,7, Neil Manson8,
Jeffrey A. Reimer4,5, Alexander Pines3,5, Carlos A.
Meriles1,2,†
1Department. of Physics, CUNY-City College of New York, New
York, NY 10031, USA. 2CUNY-Graduate Center, New York, NY 10016,
USA.
3Department of Chemistry, University of California at Berkeley,
Berkeley, California 94720, USA. 4Department of Chemical and
Biomolecular Engineering, University of California at Berkeley,
Berkeley, California 94720,
USA. 5Materials Science Division Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA.
6Facultad de Matemática, Astronomía, Física y Computación,
Universidad Nacional de Córdoba, Ciudad Universitaria, CP:X5000HUA
Córdoba, Argentina.
7Instituto de Física Enrique Gaviola (IFEG), CONICET, Medina
Allende s/n, X5000HUA, Córdoba, Argentina. 8Laser Physics Centre,
Research School of Physics and Engineering, Australian National
University, Canberra, A.C.T. 2601,
Australia *Equally contributing authors. †Corresponding author.
E-mail: [email protected]
We use field-cycling-assisted dynamic nuclear polarization to
demonstrate magnetic-field-dependent activation of nuclear spin
transport from otherwise isolated strongly-hyperfine-coupled
sites. With the help of a toy model comprising electron and nuclear
spins, we recast our observations in terms of a dynamic phase
diagram featuring zones of active or forbidden nuclear spin
current separated by boundaries defined by the interplay between
spin Zeeman, dipolar, and hyperfine couplings. Analysis of the
polarization transport as a function of the driving field reveals
the presence of many-body excitations stemming from the hybrid
electron-nuclear nature of the system. These findings could
prove relevant in applications to spin-based quantum information
processing and nanoscale sensing.
Keywords: Nuclear spin diffusion | Dynamic nuclear polarization
| Nitrogen-vacancy centers | Thermalization | Many-body
localization
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matching’ condition at ~51 mT, where the Zeeman splitting of the
P1 spins coincides with the frequency gap between the 0 and −1
states of the NVs. Following electron and
nuclear spin manipulation, we monitor the bulk 13C polarization
via high-field nuclear magnetic resonance (NMR) upon shuttling the
sample into the bore of a 9 T magnet (additional details can be
found in Ref. (21)).
Fig. 1a shows a typical experimental protocol: We continuously
illuminate the sample with a green laser (1 W at 532 nm) during a
time interval &
'(= 5 s while
simultaneously applying radio-frequency (RF) excitation. Figs.
1b and 1c show the resulting spectrum obtained as we measure the
bulk 13C NMR signal for different RF frequencies +
,- within the range 0.5-160 MHz. Besides the
dip at 551 kHz — corresponding to the Larmor frequency of bulk
13C at " = 51.5 mT — we find several RF absorption bands,
indicative of polarization transport from electron spins to bulk
carbons via select groups of strongly hyperfine-coupled
nuclei20.
Rather than relying on NV–P1 cross-relaxation, one can
dynamically polarize nuclear spins via the use of chirped
micro-wave (MW) pulses, consecutively applied during &
'(
22 (Fig. 1d). Upon simultaneous RF excitation at variable
frequencies, the spectrum that emerges shows the polarization
transport process is now fundamentally distinct. This is shown in
Figs. 1e through 1g, where we set the magnetic field to 47.1 mT, a
shift of only ~4 mT from the energy matching condition. In
particular, we find that the RF impact is mostly limited to a ~1.3
MHz band adjacent to the
13C Larmor frequency (~0.5 MHz at 47 mT, insert in Fig. 1e). The
differences are most striking near 40 MHz and 97 MHz where the dips
observed at 51 mT (Figs. 1b and 1c) virtually vanish (Figs. 1e and
1g). Similarly, the small RF dip at ~11 MHz (Figs. 1e and 1f)
amounts to only a small fraction of the broad absorption band
centered at that frequency, previously apparent under field
matching (Fig. 1b).
To qualitatively understand these observations, we start with
the toy model in Fig. 2a, a chain comprising an interacting pair of
NV–P1 electron spins, each of them coupled to a neighboring carbon
via hyperfine tensors of magnitude 0
1 with 2 = 1, .2; for illustration purposes, we
focus on the ‘hyperfine-dominated’ regime where 04~ 0
6> ℐ
9> :
;, where ℐ
9 is the NV–P1 dipolar
coupling constant, and :; is the nuclear Larmor frequency.
In the absence of hyperfine couplings to the host nitrogen
nuclei (a condition assumed here for simplicity) and using <
4
(<6) to denote the vector spin operator of the nuclear
spin
coupled to the NV (P1), the 13C spins in the chain are governed
by the effective Hamiltonian23
>?@@= A
?@@B4
C− A
?@@B6
C+ E
?@@B4
FB6
G+ B
4
GB6
F,(1)
where A?@@= 2J
?" − "
K is the effective nuclear spin
frequency offset relative to the matching field "K
, and J? is
the electron spin gyromagnetic ratio. Further, the effective
coupling between nuclear spins is given by E
?@@=
−:;ℐLsin
P
6
QR
ST
QR
SS RF Q
R
ST R, where tan W ≈ 0
4
CY04
CC, and
01
CC (01
CY) denotes the secular (pseudo-secular) hyperfine
Figure 1 | The role of P1 centers. (a) Static matching field
(SMF) protocol. (b) 13C NMR signal amplitude as a function of +
,-; the
external magnetic field is "(F) = 51.5 mT. (c) Zoomed SMF
response around ~97 MHz. (d) Dynamic nuclear polarization via
micro-wave sweeps (MWS). (e) Same as in (b) but using the MWS
protocol to induce nuclear polarization; the external magnetic
field is " = 47.1 mT. (f–g) Zoomed 13C response using the MWS
protocol. Unlike (b), we see no high-frequency dips. In all
experiments, &'(= 5 s, the total number of repeats per point is
8, the driving field amplitude is Ω
,-= 4 kHz, and the laser power is 1 W; solid
traces are guides to the eye. In (d) through (g), the MW power
is 300 mW, the sweep range is 25.2 MHz, the sweep rate is 15 MHz
ms-1 corresponding to a total of 8333 sweeps during &
'(.
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coupling constant for nuclear spin 2 = 1,2. Note that, for a
given set of hyperfine couplings, there are two matching fields
yielding analogous dynamics in the four-spin chain23; additional
fields are likely in more complex systems.
Eq. (1) is a nuclear-spin-only Hamiltonian where paramagnetic
interactions manifest in the form of field-dependent shifts and
effective couplings largely exceeding the intrinsic 13C-13C dipolar
couplings (of order 100 Hz for naturally abundant carbon). For
example, for the present 50 ppm nitrogen concentration, we have
ℐ
9~3 MHz and thus
E?@@~30 kHz for 0
1
CC~0
1
CY~10 MHz, 2 = 1,2. A numerical
example demonstrating good agreement between the exact and
effective nuclear spin evolution is presented in Fig. 2b for three
different magnetic fields. It is worth highlighting the amplified
sensitivity to field detuning " − "
K, impacting
the offset terms in Eq. (1) via the electronic (not the nuclear)
spin gyromagnetic ratio. Naturally, comparable ideas apply to the
case of larger electron and nuclear spin groups, even if deriving
accurate effective Hamiltonians becomes increasingly
difficult23,24.
To more generally capture the nuclear spin dynamics prompted by
NV–P1 couplings, we resort to the nuclear spin current operator ^ =
1 2_ B
4GB6F− B
4FB6G
, here serving
as a measure of delocalization25. Fig. 2c shows an explicit
calculation for different combinations of hyperfine couplings as a
function of " and ℐ
9. We find non-zero transport within
a small region of the parameter space, hence revealing the
equivalent of a dynamic phase transition controlled via the applied
magnetic field. Two discrete regions emerge in the limit of strong
hyperfine couplings (upper plot in the stack), corresponding to
conditions where carbons polarize positively or negatively20.
Similar to the case of carriers subjected to a Hubbard
Hamiltonian9, the dynamics in the present spin system can be cast
in terms of an interplay between ‘disorder’ — here expressed as
site-selective nuclear Zeeman frequencies — and the amplitude of
13C–13C ‘flip-flop’ couplings E
?@@ — also
referred to as the ‘hopping’ term in charge transport studies.
This is summarized in Fig. 2d where we compute a weighted average
that takes into account the known set of carbon hyperfine couplings
with the NV and P1 centers23,26-29, and find non-zero current in
the region where E
?@@≳ A
?@@. We warn
this latter condition must be understood in a distributional
sense, i.e., for a given concentration of paramagnetic centers
represented by ℐ
9, there is a magnetic field range where spin
diffusion channels become available to the most likely spin
arrays in the crystal.
Interestingly, field-insensitive spin transport between carbons
can be mediated via electron spins of the same type through the
effective coupling20,23 E
?@@
a≈
bcd
RQe
STQR
STℐf
g
∆e
R∆R
R, where
ℐ9
a denotes the homo-electron-spin coupling constant and ∆1
6= 0
1
CC6
+ 01
CY6
for 2 = 1,2. Since the number of NVs is typically tied to the P1
concentration, nuclear spin transport away from "
K can take place in samples with sufficient
paramagnetic content via NV–NV coupling. For example, assuming
an NV concentration of 10 ppm, we have ℐ
9
a~0.6
MHz thus yielding E?@@
a~6 KHz for the case 0
1
CC~0
1
CY~10
MHz, 2 = 1,2. Building on the above ideas, we can now interpret
the markedly different frequency response in Figs. 1b and 1e as the
manifestation of two complementary spin transport channels, one
relying on (field-dependent) matching between NV and P1 resonances,
the other emerging from field insensitive interactions between NVs.
In particular, we hypothesize this latter mechanism underlies the
appearance of the (weak) dip at ~11 MHz in the RF absorption
spectrum at 47.1 mT (Figs. 1e and 1f). Similar considerations apply
to the regime of moderate and weak hyperfine couplings ℐ
9> 0
1, :
; (corresponding to RF
frequencies ≲1–2 MHz), required to describe the flow of
polarization from and to more weakly-coupled carbons20.
Additional information on the dynamics at play can be obtained
through the experiments in Fig. 3, where we measure the DNP
response under the protocols of Figs. 1a and 1d using RF excitation
of variable power. Besides the anticipated gradual growth of the
absorption dips, we observe an overall spectral broadening, greatly
exceeding that expected from increased RF power alone. This
behavior is clearest in the range 5–15 MHz and near 40 MHz (Fig.
3a),
Figure 2 | Magnetic-field-dependent spin transport. (a) Model
spin chain (top) and schematic NV–P1 energy diagram when the
magnetic field and NV symmetry axis are aligned, i.e., k=0. (b)
Inter-carbon polarization transfer for the chain in (a). The solid
(faint) traces in each plot show the calculated evolution under the
effective (exact) Hamiltonian. (c) Nuclear spin current ^ for the
chain in (a) as a function of " and ℐ
9 for
different hyperfine couplings. (d) Same as in (c) but after a
weighted average over various configurations of hyperfine couplings
(see Ref. 23). In (b) we assume ℐ
9= 0.7 MHz,
04
YC= 13 MHz, 0
6
YC= 4 MHz, and 0
1
CC
= 01
CY for 2 = 1,2.
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4
where all absorption dips grow to encompass several MHz even
when the RF Rabi field Ω
,- never exceeds 10 kHz.
To interpret these observations, we resort one more time to the
electron–nuclear spin chain of Fig. 2a and model the system
dynamics in the presence of a driving RF field under the condition
" = "
K. Figures 3b and 3c show a schematic
of the energy diagram along with the calculated spectral overlap
l between carbons in the chain as a function of +
,-
for variable Rabi amplitude Ω,-
. Using m1: , 2 = 1,2 to
denote the Fourier transform of the polarization in each carbon,
the spectral overlap l ∝ o:m
4: m
6
∗: serves
as a convenient measure of synchronic nuclear spin dynamics, and
hence quantifies the disruption caused by the application of RF
through the appearance of ‘dips’ at select frequencies (Fig. 3c). A
detailed inspection shows that some of these resonances can be
associated to ‘zero-quantum’ (i.e., intra-band) transition
frequencies in the electron bath. Normally forbidden, these
transitions are activated here due to the hybrid, nuclear–electron
spin nature of the chain (e.g., transitions (3) and (4) in Fig. 3b,
see also Ref. 23). The separation between consecutive dips is
determined by the inter-electron and hyperfine couplings, thus
leading to complex spectral responses spanning several MHz.
Fig. 3d shows the calculated spectral overlap change Al ≡ l
Ω
,-, ν,-
− 1 as a function of Ω,-
at select excitation frequencies ν
,-: Interestingly, we find that all dips
— both nuclear and hybrid — grow at comparable rates, a
counter-intuitive response given the presumably hindered nature of
the zero-quantum transitions23. On the other hand, the transport of
nuclear spin polarization — faster for chains featuring greater
ℐ
9 — is more difficult to disrupt if Ω
,-≲
:?@@
, thus leading to slower growth rates for more strongly coupled
chains (Fig. 3e). Correspondingly, the response expected for spins
in a crystal (vastly more complex than our toy model) is one where
RF excitation of increasing amplitude gradually induces new dips
through the perturbation of faster polarization transport channels.
The result is an increasingly broader-looking absorption spectrum,
in qualitative agreement with our observations.
Despite the present limitations, our model suggests we should
view these hybrid spin systems as a single whole, where nominally
forbidden ‘hybrid’ excitations applied locally propagate spectrally
to impact groups of spins not directly addressed. Therefore,
besides the fundamental aspects, an intriguing practical question
is whether, even in the absence of optical pumping, Overhauser-like
DNP at higher fields could be simply attained via low-frequency
(i.e., RF) manipulation of the electron spins. More generally,
these results could prove useful in quantum applications relying on
spin platforms, for example, to transport information between
remote nuclear qubits or to develop enhanced nanoscale sensing
protocols.
Figure 3 | Dependence with RF power. (a) 13C NMR signal
amplitude as a function of the excitation frequency +,-
using the DNP protocols in Figs. 1a and 1d (respectively, left
and right panels) for various RF amplitudes (bottom right in each
panel). Horizontal dashed lines indicate the 13C NMR amplitude in
the absence of RF excitation and solid traces are guides to the
eye. (b) Schematic energy diagram for the electron-nuclear spin
chain in the cartoon. States are denoted using projection numbers
for the electronic spin and up/down arrows for nuclear spins with
primes indicating a dominating hyperfine field. Numbers illustrate
some nuclear and electron-nuclear spin transitions; energy
separations are not to scale. (c) Spectral overlap l as a function
of +
,- for different Rabi amplitudes
Ω,-
in the case of a spin chain with couplings ℐ9= 30 kHz, 0
4
YC= 9 MHz, 0
6
YC= 2.5 MHz, and 0
1
CC
= 01
CY for 2 = 1,2. (d) Spectral overlap change Al at select
frequencies (bottom right) as a function of Ω
,- for the spin chain in (c). (e) Same as in (d) but for a
spin
chain with couplings ℐ9= 800 kHz, 0
4
YC= 9 MHz, 0
6
YC= 2.5 MHz, and 0
1
CC
= 01
CY for 2 = 1,2.
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5
D.P., J.H., and C.A.M. acknowledge support from the National
Science Foundation through grant NSF-1903839, and from Research
Corporation for Science Advancement through a FRED Award; they also
acknowledge access to the
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1
I. The four-spin model
A simple spin system to study the main features of the
magnetic-field controlled spin transport is presented in Fig. S1.
Two 13C nuclear spins are hyperfine-coupled to two paramagnetic
impurities, one of them a NV center and the other a P1 center.
These two electronic spins, in turn, interact by means of a dipolar
coupling. Given the typically large mismatch between the resonance
frequencies of hyperfine-coupled and bulk carbons — and thus the
corresponding quenching of inter-nuclear flip-flop processes — the
4-spin model above provides a rationale for understanding the
dynamics of nuclear polarization in a real crystal. Following the
arguments discussed in Ref. [1], these mechanisms lead to an
effective, purely nuclear, description of spin diffusion among a
large number of 13Cs nuclear spins.
The Hamiltonian for the four-spin system is given by
!" = −%&'() − %&'*) + %,-) + %,-.) + / -) * + 0())-)'()
+ 0()1-)'(1 + 0*))-.)'*) + 0*)1-.)'*1
+ℐ42 -
6-.7 + -7-.6 . A. 1
Here, ;< = = 1,2 stands for the nuclear 13C spin operator, ?
is the NV electronic spin operator (- =1), ?. is the P1 electronic
spin operator (-. = 1/2), %& = A&B, %, = A, B, and /
corresponds to the NV zero-field splitting. Coefficients 0((*)))
and 0((*))1 respectively denote the hyperfine tensor components
coupling the left (right) 13C and the NV (P1) and ℐ4 stands for the
dipolar coupling strength between the NV and the P1 centers. In Eq.
(A.1) both hyperfine interactions have been already
secularized.
We restrict our analysis to the vicinity of B = 51mT, where the
spin states 0 ↑ for the NV-P1 pair is almost degenerate with −1 ↓ .
We focus the analysis of the nuclear spin dynamics
in the subspaces spanned by these two electronic states, since
all other electronic configuration remain energetically
inaccessible. Table S1 shows the matrix representation of the
Hamiltonian in Eq. (A.1) in such a subspace.
With the purpose of developing a model of effective 13C-13C
interactions, we perform a partial diagonalization of each 13C spin
in a local basis given by the hyperfine coupling. More precisely,
the quantization axis of each nuclear spin is now determined by the
vectors
Supporting Information
Magnetic-field-induced delocalization in hybrid electron-nuclear
spin ensembles
Daniela Pagliero1,*, Pablo R. Zangara6,7,*, Jacob Henshaw1,
Ashok Ajoy3, Rodolfo H. Acosta6,7, Neil
Manson8, Jeffrey A. Reimer4,5, Alexander Pines3,5, Carlos A.
Meriles1,2
1 Department. of Physics, CUNY-City College of New York, New
York, NY 10031, USA. 2 CUNY-Graduate Center, New York, NY 10016,
USA. 3 Department of Chemistry, University of California at
Berkeley, Berkeley, California 94720, USA. 4
Department of Chemical and Biomolecular Engineering, University
of California at Berkeley, Berkeley, California 94720, USA. 5
Materials Science Division Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA. 6 Facultad de Matemática,
Astronomía, Física y Computación, Universidad Nacional de Córdoba,
Ciudad Universitaria, CP:X5000HUA Córdoba, Argentina. 7 Instituto
de Física Enrique Gaviola (IFEG), CONICET, Medina Allende s/n,
X5000HUA, Córdoba,
Argentina. 8 Laser Physics Centre, Research School of Physics
and Engineering, Australian National University, Canberra, A.C.T.
2601, Australia.
*Equally contributing authors
-
2
K( = 0()1L + 0()) + %& M A. 2
K*± = 0*)1L + 0*)) ± 2%& M(A. 3)
for 13Cs 1 and 2, respectively. Notice that K( defines the
quantization axis only in the subspace corresponding to the
electronic states −1 ↓ , since in the subspace with electronic
configuration 0 ↑ the quantization axis remains defined by the
external magnetic field (no hyperfine interaction). Additionally,
the two possible axes K*± for the second 13C are defined for the 0
↑ subspace (Eq. (A.3), negative sign) and the −1 ↓ subspace
(Eq. (A.3), positive sign). The corresponding rotation angles
required to transform into such a representation are respectively
defined by
tan S( = 0(TU 0()) + %& (A. 4)
tan S*± = 0*TU (0*)) ± 2%&).(A. 5)
The rotations of each 13C quantization axis transform the
original Hamiltonian !" into a new one !", whose matrix
representation is shown in Table S2. In this new basis, primed
labels for the nuclear states indicate the change in the
quantization axis when required.
II. The effective 13C-13C Hamiltonian
It is clear from the matrix representation of Hamiltonian !" in
Table S2 that nuclear spin flip-flops are possible provided that
the appropriate energy-matching condition is achieved. For
instance, states ↑ 0 ↑↓. and ↓ −1 ↓↑. are degenerate if
−%&2 +
%,2 −
∆*7
4 = / −3%,2 −
∆*6
4 +∆(2 ,(A. 6)
which defines an equation for the ‘matching’ magnetic field B =
BY((). When this condition is met, the
dynamics of the pair of nuclear spins is essentially a flip-flop
process ↑↓. ↔ ↓↑. . The time-scale for such a process is given by
the matrix element
↓ −1 ↓↑. !" ↑ 0 ↑↓. = −ℐ42 [*[((A. 7)
An equivalent situation can be found when the condition
%&2 +
%,2 +
∆*7
4 = / −3%,2 +
∆*6
4 −∆(2 (A. 8)
is satisfied. In this case, states ↓ 0 ↑↑. and ↑ −1 ↓↓. are
degenerate, and the coupling matrix element is the same as in Eq.
(A.7). This energy-matching condition corresponds to a different
magnetic field B = BY
(*) as Eq. (A.6) is not equivalent to (A.8). At any of these
‘matching’ fields, an effective description can be proposed for the
nuclear spins,
Figure S1. The four-spin system. Two 13Cs interact with two
dipolarly coupled electrons, one NV and one P1 center.
-
3
| ↑0↑↑⟩
| ↑0↑↓⟩
| ↓0↑↑⟩
| ↓0↑↓⟩
| ↑−1↓↑⟩
| ↑−1↓↓⟩
| ↓−1↓↑⟩
| ↓−1↓↓⟩
⟨ ↑0↑↑|
−)*+) , 2
+. /0
0 4
. /02 4
0 0
ℐ 4/2
0
0 0
⟨ ↑0↑↓|
. /02 4
) , 2−. /0
0 4
0 0
0 ℐ 4/2
0
0
⟨ ↓0↑↑|
0 0
) , 2+. /0
0 4
. /02 4
0 0
ℐ 4/2
0
⟨ ↓0↑↓|
0 0
. /02 4
)*+) , 2
−. /0
0 4
0 0
0 ℐ 4/2
⟨ ↑−1↓↑|
ℐ 4/2
0
0 0
−)*+6−3)
, 2−. /0
0 4−. 80
0 2
−. /0
2 4
−. 80
2 2
0
⟨ ↑−1↓↓|
0 ℐ 4/2
0
0 −. /0
2 4
6−3)
, 2+. /0
0 4−. 80
0 2
0 −. 80
2 2
⟨ ↓−1↓↑|
0 0
ℐ 4/2
0
−. 80
2 2
0 6−3)
, 2−. /0
0 4+. 80
0 2
−. /0
2 4
⟨ ↓−1↓↓|
0 0
0 ℐ 4/2
0
−. 80
2 2
−. /0
2 4
)*+6−3)
, 2+. /0
0 4+. 80
0 2
Tabl
e S1
. Mat
rix
repr
esen
tatio
n of
Ham
ilton
ian 9:
. The
two
subs
pace
s co
rres
pond
ing
to e
ach
NV
-P1
elec
troni
c co
nfig
urat
ion
have
bee
n sh
adow
ed
with
gre
en (|0↑⟩
) and
gre
y (| −
1↓⟩
).
-
4
| ↑0↑↑; ⟩
| ↑0↑↓; ⟩
| ↓0↑↑; ⟩
| ↓0↑↓; ⟩
| ↑−1↓↑; ⟩
| ↑−1↓↓; ⟩
| ↓−1↓↑; ⟩
| ↓−1↓↓; ⟩
⟨ ↑0↑↑; |
−)* 2+) , 2
+∆ /= 4
0
0 0
ℐ 4 2> /> 8
−ℐ 4 2? /> 8
−ℐ 4 2> /? 8
ℐ 4 2? /? 8
⟨ ↑0↑↓; |
0 −)* 2+) , 2
−∆ /= 4
0
0 ℐ 4 2? /> 8
ℐ 4 2> /> 8
−ℐ 4 2? /? 8
−ℐ 4 2> /? 8
⟨ ↓0↑↑; |
0 0
)* 2+) , 2
+∆ /= 4
0
ℐ 4 2> /? 8
−ℐ 4 2? /? 8
ℐ 4 2> /> 8
−ℐ 4 2? /> 8
⟨ ↓0↑↓; |
0 0
0 )* 2+) , 2
−∆ /= 4
ℐ 4 2? /? 8
ℐ 4 2> /? 8
ℐ 4 2? /> 8
ℐ 4 2> /> 8
⟨ ↑−1↓↑; |
ℐ 4 2> /> 8
ℐ 4 2? /> 8
ℐ 4 2> /? 8
ℐ 4 2? /? 8
6−3)
, 2−∆ /@ 4
−∆ 8 2
0
0 0
⟨ ↑−1↓↓; |
−ℐ 4 2? /> 8
ℐ 4 2> /> 8
−ℐ 4 2? /? 8
ℐ 4 2> /? 8
0
6−3)
, 2+∆ /@ 4
−∆ 8 2
0
0
⟨ ↓−1↓↑; |
−ℐ 4 2> /? 8
−ℐ 4 2? /? 8
ℐ 4 2> /> 8
ℐ 4 2? /> 8
0
0 6−3)
, 2−∆ /@ 4
+∆ 8 2
0
⟨ ↓−1↓↓; |
ℐ 4 2? /? 8
−ℐ 4 2> /? 8
−ℐ 4 2? /> 8
ℐ 4 2> /> 8
0
0 0
6−3)
, 2+∆ /@ 4
+∆ 8 2
Tabl
e S2
. Mat
rix
repr
esen
tatio
n fo
r th
e H
amilt
onia
n 9A:
. Ela
bora
ting
on E
qns.
(A2-
A5)
, the
not
atio
n is
giv
en b
y B C⃗8B=∆ 8
, BC⃗ /±B =
∆ /±, >8=cos(K 8/2),
? 8=sin(K 8/2), > /=cos[(K/@−K /=)/2]
, and
?/=sin[(K/@−K /=)/2]
. Off
-dia
gona
l mat
rix e
lem
ents
that
indu
ce n
ucle
ar fl
ip-f
lops
are
hig
hlig
hted
in y
ello
w.
-
5
!"## = %"##&'( − %"##&*
( + ,"## &'-&*
. + &'.&*
- , A. 9
where
%"## = 2 5 − 56(',*)
9:(A. 10)
,"## =ℐ>2sin(B'/2) sin[(B*
- − B*.)/2].(A. 11)
Figure S2(a-b) exemplifies the comparison of the 13C
polarization dynamics for both the complete !F and the effective
!"## Hamiltonians.
From Eq. (A.7) we conclude that the effective 13C-13C coupling
,"## embodies a four-body transition matrix element. This is
equivalent to the “hyperfine-dominated” effective coupling
described in Ref. [1], where two P1 centers were used to mediate
the 13C spins. As a matter of fact, we can use here the same
assumption of a ‘hyperfine dominated’ regime to derive a more
explicit form for ,"##. Thus, we consider cases where the hyperfine
energy is much larger than the Zeeman energy at each 13C, so we can
analyze the limit B*- − B*. →0,
sinB*- − B*
.
2≈1
2tan B*
- − B*. =
1
2
tan B*- − tan B*
.
1 + tan B*- tan B*
. =1
2
K*(L
(K*(( + 2MN)
−K*(L
(K*(( − 2MN)
1 +K*(L *
K*(( * − 2MN *
.(A. 12)
Then,
sinB*- − B*
.
2≈
−2MNK*(L
K*(( * − 2MN * + K*
(L *.(A. 13)
If we again use the fact that the hyperfine energy dominates
over the Zeeman energy, we can approximate ∆*-~∆*.~ K*(( * + K*(L *
≡ ∆*. Then, the effective flip-flop matrix element can be written
as
,"## ≈ −ℐ> sinB'2
MNK*(L
K*(( * + K*
(L *,(A. 14)
which is analogous to the matrix element for the case of a pair
of P1 centers in the ‘mediating’ role,
,"## ≈ −ℐ>MNK'
(L
K'(( * + K'
(L *
MNK*(L
K*(( * + K*
(L *,(A. 15)
as derived in Ref. [1]. We warn, however, that the P1-P1
mechanism of electron-mediated nuclear interaction is not dependent
on the magnetic field (more precisely, it does not require the
field-matching condition), so it is insufficient to provide for a
rationale for our experimental observations.
III. The spin current and the delocalization diagram
A complementary analysis of the polarization dynamics can be
done in terms of the polarization current [2]
U = 1 2V &'.&*
- − &'-&*
. ,(A. 16)
which provides an observable to quantify the flow of
polarization between the two 13C spins. For example, we show in
Fig. S2 the time-dependence of the polarization at each 13C along
with U for a given set of couplings and satisfying the matching
condition 5 = 56
(').
-
6
Being our model strictly finite, both the spin polarization and
the current keep oscillating as the two nuclear spins undergo the
flip-flop dynamics. A more realistic description would encompass a
large number of nuclear spins effectively interacting by means of
an appropriate extension of Hamiltonian !"## in Eq. (A.9). In such
a case, the polarization would jump from the second 13C to a nearby
13C (also coupled by mediating NV-P1 or P1-P1 pairs), ultimately
diffusing away. This physical picture of spin-diffusion from
strongly hyperfine-shifted 13Cs to gradually more weakly
hyperfine-shifted 13Cs (and then finally to bulk 13Cs) has been
extensively discussed Ref. [1]. In Fig. S3(a) we evaluate U as a
function of time and the magnetic field 5 for a given choice of
hyperfine couplings and ℐ>. In order to capture the dynamical
trends at both ‘matching’ fields 56
(') and 56(*),
we consider the initial state
XY =1
2↑ 0 ↑↓\ ↑ 0 ↑↓\ +
1
2↓ 0 ↑↑\ ↓ 0 ↑↑\ .(A. 17)
Figure S3(b) shows a cross-section of the amplitude of the
oscillations in U as a function of 5. Notice the two dominant peaks
corresponding to 56
(') and 56(*), whose widths are associated to higher order
processes.
Figure S3(c) shows the same quantity, the amplitude of U as a
function of 5, but here calculated for different ℐ> (Figure 2(c)
in the main text shows the same simulation for different sets of
hyperfine couplings).
Figure S2. Polarization and spin current dynamics. (a) Time
dependence of polarization '̂ and *̂ at 13Cs C1
and C2 respectively, using the complete four-spin model evolved
with Hamiltonian !_F and initial state |↑ 0 ↑↓\⟩. (b) Same as in
(a), but using the effective nuclear interaction given by
Hamiltonian !"##, and initial state |↑↓⟩. (c) Time dependence of
the spin current for the dynamics induced by !_F as in case (a).
(d) Time dependence of the spin current for the effective
interaction !"##. In all the cases, we consider K'(( = K'(L = 9MHz,
K*(( = K*(L =2.5MHz, ℐ> = 800kHz, 5 = 56
(')= 51.3085mT.
-
7
The amplitude of U is used here as a quantitative indicator for
localization/delocalization of nuclear spin polarization. As the
amplitude of U increases, the polarization is more efficiently
transferred and it can therefore diffuse. This leads us to build a
dynamical phase-diagram as shown in Fig. 2(d) of the main text,
where we average over many different possible choices of hyperfine
couplings (configurations in our 4-spin system). Since the possible
NV-13C and P1-13C hyperfine couplings are well known [3–5], we
performed a sampling of the most relevant combinations of couplings
from 1 to ~14 MHz. In particular, Fig. S4(a) shows the actual
bivariate distribution of possible hyperfine configurations and
Fig. S4(b) shows the cases we evaluated and accounted for to mimic
such a distribution.
IV. The 4-spin system in the presence of RF driving
In order to analyze the spectral broadening observed in the DNP
signal under RF excitation, we study the dynamics of polarization
transfer in our 4-spin model adding the time-dependent
perturbation
i j = Ωlm cos νlmj &'L + &*
L ,(A. 18)
Figure S4. Distributions of hyperfine couplings. (a) Bivariate
distribution of NV-13C and P1-13C hyperfine couplings as reported
in Refs. [3-4] and [5], respectively. (b) Bivariate ‘toy’
distribution employed to mimic (a) and compute the averaged case
shown in Fig. 2(d) of the main text.
Figure S3. Dynamics of spin current q. (a) Time evolution of U
as a function of the magnetic field. (b) Amplitude of the
oscillations in U (i.e., maximum along the time-axis in (a)) as a
function of the magnetic field. (c) Same as in (b), but for a
variable dipolar coupling strength ℐ>. In all the cases, we
consider K'(( = K'(L =2.9MHz, K*(( = K*(L = 3.2MHz; the initial
state is given in Eq. (A.17). In (a-b), ℐ> = 800kHz.
-
8
where Ωlm stands for RF amplitude and νlm denotes its frequency.
The system’s dynamical response is computed without any explicit
approximation using the QuTiP [6] time-dependent solver. Fig.
S5(a-b) shows the time dependence of the polarization of each
carbon r̂ j (s = 1,2), as a function of the irradiation frequency
νlm. Here, we assume resonant polarization transfer (i.e., 5 =
56
(')) and a weak effective interaction ,"##~1kHz (K'(( = K'(L =
9MHz, K*(( = K*(L = 2.5MHz, ℐ> = 30kHz). The amplitude Ωtu is
kept constant at 25kHz.
Figure S5. Polarization dynamics in the presence of RF
irradiation. (a-b) Time response r̂(j) as a function of the
irradiation frequency νlm for C1 (a) and C2 (b). (c-d) Fourier
spectra '̂
(Y)(M) and *̂(Y)(M) for C1 and
C2 respectively (no RF irradiation). (e-f) Fourier spectra '̂(M;
νlm) and *̂(M; νlm) for C1 and C2 respectively, in the presence of
RF irradiation with νtu ≈ 2.16MHz and Ωtu = 25kHz. In all cases,
K'(( = K'(L = 9MHz, K*(( = K*
(L = 2.5MHz, and ℐ> = 30kHz.
-
9
Four possible transitions can be observed as distortions in the
synchronized flip-flop dynamics shown in Fig. S5(a-b), each of them
distinguished with a label (1,2,3,4). Using the Hamiltonian
representation in Table S2, these transitions can be identified
according to Table S3. In order to assess the effect of RF
excitation, we compute the Fourier transform of r̂ j for a given
(Ωlm, νlm). Each normalized Fourier spectrum r̂ M can be then
compared to quantify how the two
13C spins decouple in the presence of a time-dependent
perturbation. Figs. S5(c-d) show the unperturbed spectra ̂ '
(Y)M
and *̂(Y)
M for C1 and C2, respectively, obtained in the absence of RF
irradiation. Figs. S5(e-f) show the perturbed spectra '̂ M; νlm and
*̂ M; νlm for C1 and C2 respectively, obtained in the presence of
RF excitation at a frequency νlm corresponding to transition #3 and
a given (fixed) RF amplitude Ωtu. This approach provides a
systematic way to evaluate the decoupling of the two 13C spins, by
computing the overlap between the Fourier spectra r̂ M . Indeed, in
cases (c-d) the spectral overlap is ~1 (as in the case of
non-resonant RF excitation). Quite on the contrary, the overlap is
less than 1 in cases (e-f), which means that the two 13C nuclei
are, to some extent, decoupled. This magnitude is ultimately
associated to the ‘dip’ in the obtained signal, as explicitly shown
in Fig. 3 of the main text. References [1] D. Pagliero, P. R.
Zangara, J. Henshaw, A. Ajoy, R. H. Acosta, J. A. Reimer, A. Pines,
and C. A.
Meriles, Sci. Adv. 6, eaaz6986 (2020). [2] A. De Luca, M.
Collura, and J. De Nardis, Phys. Rev. B 96, 1 (2017). [3] A. Dréau,
J. R. Maze, M. Lesik, J. F. Roch, and V. Jacques, Phys. Rev. B -
Condens. Matter Mater.
Phys. 85, 1 (2012). [4] B. Smeltzer, L. Childress, and A. Gali,
New J. Phys. 13, (2011). [5] C. V. Peaker, M. K. Atumi, J. P. Goss,
P. R. Briddon, A. B. Horsfall, M. J. Rayson, and R. Jones,
Diam. Relat. Mater. 70, 118 (2016). [6] J. R. Johansson, P. D.
Nation, and F. Nori, Comput. Phys. Commun. 184, 1234 (2013).
Label States involved Frequency
1 |↑ 0 ↑↓\⟩ ↔ |↓ 0 ↑↓\⟩ νlm~Mx + y(ℐ>*)
2 |↑ 0 ↑↓\⟩ ↔ |↑ 0 ↑↑\⟩ νlm~∆*./2 + y(ℐ>*)
3 |↑ 0 ↑↓\⟩ ↔ |↓ −1 ↓↓\⟩ νlm~∆*-/2 + y(ℐ>*)
4 |↑ 0 ↑↓\⟩ ↔ |↑ −1 ↓↑\⟩ νlm~∆' + y(ℐ>*)
Table S3. Transition frequencies excited by the RF irradiation
(see Fig. S5).
