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jump after the push pulse exhibited oscilla- Fig. 4. Measured and simulated spin quantum beats. (A) Experimental pump-push TA signals taken at
tory behavior—i.e., spin quantum beats. The 19,600 cm−1 for the formation of 3PDI* product at 0 mT. (B) Experimental postpush signal profiles at
observed pump-push signals of the triplet magnetic fields of 0, 1, 7, and 500 mT. (C) Theoretical pump-push signals at zero field, calculated
product and their magnetic field dependence with J = 0.35 mT. (D) Theoretical results simulating the curves in (B); for details, see the text. The vertical
were simulated quantum theoretically as men- line in (A) marks the position of the maximum of the pump pulse. The black dots in (A) and (C) represent the
tioned above (Fig. 4C). To obtain a realistic positions of the maxima of the push pulses for the pertinent signals, and the red points represent the
representation of the observed signals, the positions of the ends of the push pulses, taken 17 ns later. The end of that push pulse with the same
finite width [full width at half maximum maximum as the pump pulse is taken as time zero in (B) and (D).
(FWHM); 9.5 ns total extension with fringes up
to 40 ns; fig. S11] of the laser pulses has been CIDNP experiments (42). The pump-push with strong external magnetic fields. These
considered. The width of the pump pulse was spectroscopic method, which has also been rules apply for any method of observation.
considered as well. In the region of overlap of exploited for the fluorescence signal yielding However, Molin’s method requires nonpolar
pump and push pulse, the initial bleaching by complementary results (figs. S13 to S16), of- solvents, high energy excitation, and efficient
the pump pulse and the rise partially com- fers a new approach with a much better time fluorophores with short fluorescence lifetimes.
pensated each other, making it difficult to resolution. The pump-push method with absorption detec-
separate the contribution of the positive jump tion introduced here requires neither of these
because of the push pulse within the first Discussion and conclusions conditions and therefore is more generally ap-
~20 ns. However, this situation can be treated plicable. However, our method is best suited
with a theoretical simulation of the pump-push In this paper, we have established pump-push for RPs with fixed distance. In flexibly linked
signals, such as shown in Fig. 4C (for more spectroscopy as an optical technique for real- or freely diffusing RPs, the fluctuating dis-
examples at higher fields, see fig. S17). time observation of electron spin motion in tance modulates the recombination efficiency
CSSs. Although the spin motion in such states after the push pulse to a large extent, which
Electron spin motion is determined by in- is usually hidden for optical probing absorp- will probably attenuate the push pulse effect
ternal and external magnetic fields. To extract tion techniques, the proposed method allows substantially.
the essential information about the spin quan- us to look behind the scenes. This realization
tum beats, we plotted the signal intensities was achieved through push pulses triggering Compared with the detection of quantum
at the end of each push pulse as a function of quasi-instantaneous spin measurements by beats by transient EPR, our method provides
the delay time in Fig. 4B. In addition to zero two spin-selective CR channels exhibiting the advantage that it can be applied at any
field, the behavior at fields of 1, 7, and 500 mT— drastically enhanced rates in the electronically magnetic field, including the notable J-resonant
representing the cases of level crossing, middle excited CSS. This method has the potential fields. The EPR technique requires microwaves
field, and high field—has also been determined to fully map the coherent and incoherent spin and a magnetic field resonant with the micro-
(see fig. S17 for the signal representations). The dynamics in a CSS, including eventual quan- waves, so only a discrete set of fields is acces-
pump-push time profiles obtained, although tum beats, as in the present case. sible for which microwave frequency bands
somewhat noisy, directly revealed the oscilla- are available.
tory structure of spin motion in the first ~50 ns. As has been shown in the work of Molin and
In Fig. 4D, the theoretically simulated pump- co-workers (16), the best conditions for quan- As far as the time resolution of the pump-
push time profiles are shown. The positions of tum beats prevail if radicals have either (i) push method is concerned, the laser pulse
the oscillation maxima and minima as well simple hyperfine structure (i.e., small numbers length and the recombination time in the
as the general absolute scaling of the effects of preferably equivalent nuclear spins, as is the CSS* state could be limiting. In the present
were well reproduced, although for 1 mT, the case for the TAA-An-PDI dyad studied here) or case, the recombination time is much shorter
amplitude of the first maximum is obtained (ii) small hyperfine coupling (e.g., because of than the laser pulse because the time deriva-
higher than in experiments. In fitting the perdeuteration) and large Dg in combination tive of the signal jumps induced by the push
theoretical results, it was found that the sign
of the exchange interaction J caused a sig-
nificant difference in the field-dependent
pump-push profiles at low fields. Without
carrying out the fitting of all parameters in the
quantum calculation to a final optimum—
because in this work we wanted to focus on a
proof of principle of the pump-push spectros-
copy—it can be concluded that only a positive
exchange interaction (i.e., 3CSS below 1CSS)
led to satisfactory results (figs. S18 to S20). For
small absolute values of the exchange energy
parameter J on the order of the intrinsic
hyperfine couplings of the system, the usual
rule of a magnetic field effect resonance at
a field of Bmax = |2J| is no longer valid. A
diagram of the resonance field Bres versus the
value of J for positive and negative values is
not symmetric around J = 0, where the so-
called low-field effect is dominated by the
hyperfine coupling constants. So far, the sign of J
could only be determined for spin-correlated
RPs by EPR (38–41) or by field-dependent

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pulses closely matches the laser profile. Here, 37. K. Maeda et al., Chem. Commun. 47, 6563–6565 (2011). (N.N.L.). Author contributions: Conceptualization: C.L., U.E.S.,
38. C. D. Buckley, C. N. Hunter, P. J. Hore, K. A. McLauchlan, and D.M. Formal analysis: U.E.S. and N.N.L. Funding acquisition:
the time resolution was limited by the ~10-ns C.L. and N.N.L. Investigation: D.M., J.H., and U.E.S. Methodology:
Chem. Phys. Lett. 135, 307–312 (1987). D.M. and U.E.S. Project administration: C.L. Software: N.N.L.
width of the laser pulses, which already led to 39. G. L. Closs, M. D. E. Forbes, J. R. Norris, J. Phys. Chem. 91, and U.E.S. Supervision: C.L. and U.E.S. Writing – original draft:
D.M. and U.E.S. Writing – review and editing: C.L., D.M., U.E.S., and
considerable damping of the quantum beats. 3592–3599 (1987). N.N.L. Competing interests: The authors declare no competing
40. P. J. Hore, D. A. Hunter, C. D. McKie, A. J. Hoff, Chem. Phys. interests. Data and materials availability: All experimental data
Shorter pulse widths of 0.1 to 1 ns would allow shown in the main text or the supplementary materials are
Lett. 137, 495–500 (1987). accessible at the Dryad repository (43).
better resolution and would be desirable to 41. M. D. E. Forbes, S. R. Ruberu, J. Phys. Chem. 97, 13223–13233
SUPPLEMENTARY MATERIALS
follow faster oscillations occurring, for exam- (1993). science.org/doi/10.1126/science.abl4254
ple, because of higher values of J or because 42. I. Zhukov et al., J. Chem. Phys. 152, 014203 (2020). Materials and Methods
of fast Zeeman mixing at large Dg values and 43. C. Lambert, D. Mims, J. Herpich, N. Lukzen, U. Steiner, Data Supplementary Text
higher fields. Figs. S1 to S31
for spin quantum beats in pump push spectroscopy, Table S1
The pump-push technique reported here Dryad (2021); https://doi.org/10.5061/dryad.4mw6m90b1. Synthesis Protocols
References (44–61)
represents an extension of the arsenal of spin ACKNOWLEDGMENTS
14 July 2021; accepted 19 October 2021
chemical techniques. It can be also expected This work is dedicated to the memory of our late colleague 10.1126/science.abl4254
Konstantin Ivanov, who died 5 March 2021 of COVID-19 at the age
to open applications for molecular electronics of 44. C.L. and D.M. thank A. Schmiedel for assisting with the
pump-push setup. Funding: This work was supported by Deutsche
and quantum information science. Forschungsgemeinschaft grant LA991/20-1 (C.L. and D.M.); the
SolTech Initiative of the Bavarian State Ministry of Education,
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Fig. 1. Programmable spin-based quantum simulator. (A) We program an sequentially read out through the NV electronic spin using electron-nuclear and
effective 1D chain of nine spins in an interacting cluster of 27 13C nuclear spins
nuclear-nuclear two-qubit gates (see text). Colored boxes with “I” denote re-
( Jojrka n>ge1):c5loHsze, to a single NV center. Connections indicate nuclear-nuclear couplings initialization into the given state. (C) Coupling matrix for the nine-spin chain.
and blue (red) lines represent negative (positive) nearest-neighbor
(D) Average coupling magnitude as a function of site distance across the chain. Orange
couplings within the chain (29). Magnetic field: Bz $ 403 G. (B) Experimental line: least-squares fit to a power-law function J0=jj À kja, giving J0 ¼ 6:7ð1Þ Hz
and a ¼ 2:5ð1Þ, confirming that the chain maps to an effective 1D system.
rsoetqauteionncse:oTf htheespfoinrsmaRreðϑin;itϕiaÞliz¼edebxypÂaÀppi lϑ2yiðnsgintðhϕe ÞPsuxlsþePcool ssðeϕqÞuseny ÞcÃe. (34), followed by
After evolution (E) Measured expectation values hsjzi after initializing the state j↑↑↑↑↑↑↑↑↑i. The
data are corrected for measurement errors (27).
under N cycles of the Floquet unitary UF ¼ UintðtÞ Á UxðqÞ Á UintðtÞ, the spins are

many-body localization (MBL). Specifically ability to prepare arbitrary initial states with electron-nuclear hyperfine interaction induces
in this context, DTC order is present across
the full eigenspectrum of the Floquet system, site-resolved measurements, we observe the a frequency shift hj for each nuclear spin,
translating into time-crystalline dynamics for DTC response for a variety of initial states up which—combined with an applied magnetic
generic initial states (4, 5, 9–11, 24, 25). The to N ¼ 800 Floquet cycles. This robustness for field Bz in the z direction—reduces the dipolar
demonstration of such robust DTC order has generic initial states provides a key signature interactions to Ising form (27). We addition-
remained an outstanding challenge (4, 26). to distinguish the many-body–localized DTC ally apply a radio-frequency (rf) driving field to
phase from prethermal responses, which only
Here, we present an observation of the hall- implement nuclear-spin rotations. The nuclear-
mark signatures of the many-body–localized show a long-lived response for particular states
DTC phase. We develop a quantum simulator (13, 21, 22, 26). spin Hamiltonian is then given by H ¼ Hint þ
based on individually controllable and detect- Hrf, where Hint and Hrf describe the interaction
able 13C nuclear spins in diamond, which can Our experiments are performed on a system and rf driving terms respectively:
be used to realize a range of many-body Ham- of 13C nuclear spins in diamond close to a
iltonians with tunable parameters and dimen- XX
sionalities. We implement a Floquet sequence nitrogen-vacancy (NV) center at 4 K (Fig. 1A). Hint ¼ ðB þ hj Þsjz þ Jjksjz skz
in a one-dimensional (1D) chain of L ¼ 9 spins The nuclear spins are well-isolated qubits with
and observe the characteristic period doubling coherence times up to tens of seconds (27, 28). Xj j<k ð1Þ
associated with the DTC. By combining the They are coupled via dipole-dipole interac- Hrf ¼ WðtÞsxj

tions and are accessed through the optically j
addressable NV electronic spin (28, 29). With
the electronic spin in the ms ¼ À1 state, the Here, sjb (b ¼ x; y; z) are the Pauli matrices
for spin j , B ¼ gcBz=2 is the magnetic field

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splitting, gc is the 13C gyromagnetic ratio, Jjk terized in detail (27, 29); for 27 13C spins, the UspinitnðtrÞo¼taetixopnðÀs UiHxiðnqttÞÞ,¼inetexrple aÀveiqdXwitjLhsgxjl=o2b al.
is the zz component of the dipole-dipole inter- hyperfine shifts hj, the spatial coordinates, and
action between spins j and k, and WðtÞ is the the 351 interaction terms Jjk are known. To realize the global rotations, we develop

applied time-dependent rf field, and we set To investigate the DTC phase, we apply a multifrequency rf pulses that simultaneously
rotate a chosen subset of spins (Hrf in Eq. 1)
ℏ ¼ 1. The system has previously been charac- Floquet unitary consisting of free evolution (27). We symmetrize the Floquet unitary such
that UF ¼ UintðtÞ Á UxðqÞ Á UintðtÞ, and apply
Fig. 2. Isolating spin N cycles of this basic sequence (Fig. 1B). For
chains. (A) We test the q ∼ p, this decouples the targeted spins from
programming of interacting their environment while preserving the inter-
spin chains for the first nal interactions (27).
four spins of the nine-spin
chain (Fig. 1, A, C, and D). To stabilize MBL, the Floquet sequence½UFŠN
For q $ p, the Floquet should satisfy two requirements. First, the sys-
sequence ½UFŠN decouples
the spin chain from its tem should be low-dimensional and short-range
environment but preserves interacting (26, 30–33). This requirement is
the internal interactions. not naturally met in a coupled 3D spin system
(B) Measured expectation
values hsxj i after initializing (Fig. 1A). To resolve this discrepancy, we program
the state jþþþþi and
applying ½UFŠN with q ¼ p. an effective 1D spin chain using a subset of nine
Here t ¼ 2tN is varied by spins [Fig. 1, A, C, and D (27)]. Second, because
fixing t ¼ 3:5 ms and vary- the periodic rotations approximately cancel
ing N. The blue (orange) the on-site disorder terms hj, the system must
points show the evolution exhibit Ising-even disorder to stabilize MBL in
with (without) spin-spin interactions (27). Blue lines: numerical simulations of only the four-spin system (27). the Floquet setting (4, 11, 26). This is natural-
Measurements in this figure and hereafter are corrected for state preparation and measurement errors. ly realized in our system because the Ising
couplings, Jjk, inherit the positional disorder of
the nuclear spins. The disorder in the magni-

tude of the nearest-neighbor couplings is dis-
tributed over a range W ∼ 10 Hz. The ratio of
disorder to average nearest-neighbor coupling

Fig. 3. Discrete time two-point correlation c (F) and coherence C (G) after preparing the superposition
crystal in the nine-spin state ½cosðp=8Þj↑i þ sinðp=8Þj↓iŠ 9 and applying ½UFŠN with t ¼ 5 ms. The
chain. (A) Sketch of the subharmonic response in c is preserved, whereas C quickly decays because of
phase diagram as a function
of t and q when applying interaction-induced local dephasing. The dashed line in (G) indicates a reference
the Floquet sequence ½UFŠN
(Fig. 1B) (4). The yellow value for C measured after preparing the state j↑i 9 (27).
region indicates the many-
bodyÐlocalized DTC phase.
The colored points mark
three combinations of fq; tg
that illustrate the DTC
phase transition. Additional
data for other values are
given in the supplementary
materials (27). (B) Aver-
aged two-point correlation c
as a function of the number
of Floquet cycles N for
q ¼ 0:95p and initial state
j↑↑↑↑↑↑↑↑↑i. Without
interactions [purple (27)],
c decays quickly. With small
interactions (t ¼ 1:55 ms,
green), the system is on the
edge of the transition to
the DTC phase. With strong interactions (t ¼ 5 ms, blue), the subharmonic
response is stable and persists over all 100 Floquet cycles. (C) The
corresponding Fourier transforms show a sharp peak at f ¼ 0:5 emerging as the
system enters the DTC phase. (D and E) Individual spin expectation values hszj i
for interaction times t ¼ 1:55 ms (D) and t ¼ 5 ms (E). (F and G) Averaged

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