Science - USA (2021-12-17)

pulses closely matches the laser profile. Here,
the time resolution was limited by the ~10-ns
width of the laser pulses, which already led to
considerable damping of the quantum beats.
Shorter pulse widths of 0.1 to 1 ns would allow
better resolution and would be desirable to
follow faster oscillations occurring, for exam-
ple, because of higher values ofJor because
of fast Zeeman mixing at largeDgvalues and
higher fields.
The pump-push technique reported here
represents an extension of the arsenal of spin
chemical techniques. It can be also expected
to open applications for molecular electronics
and quantum information science.

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ACKNOWLEDGMENTS

This work is dedicated to the memory of our late colleague

Konstantin Ivanov, who died 5 March 2021 of COVID-19 at the age

of 44. C.L. and D.M. thank A. Schmiedel for assisting with the

pump-push setup.Funding:This work was supported by Deutsche

Forschungsgemeinschaft grant LA991/20-1 (C.L. and D.M.); the

SolTech Initiative of the Bavarian State Ministry of Education,

Culture, Science, and the Arts; and Ministry of Science and Higher

Education of the Russian Federation grant 075-15-2020-779

(N.N.L.).Author contributions:Conceptualization: C.L., U.E.S.,

and D.M. Formal analysis: U.E.S. and N.N.L. Funding acquisition:

C.L. and N.N.L. Investigation: D.M., J.H., and U.E.S. Methodology:

D.M. and U.E.S. Project administration: C.L. Software: N.N.L.

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

N.N.L.Competing interests:The authors declare no competing

interests.Data and materials availability:All experimental data

shown in the main text or the supplementary materials are

accessible at the Dryad repository ( 43 ).

SUPPLEMENTARY MATERIALS

science.org/doi/10.1126/science.abl4254

Materials and Methods

Supplementary Text

Figs. S1 to S31

Table S1

Synthesis Protocols

References ( 44 – 61 )

14 July 2021; accepted 19 October 2021

10.1126/science.abl4254

REPORTS

◥

QUANTUM SIMULATION

Many-bodyÐlocalized discrete time crystal with a

programmable spin-based quantum simulator

J. Randall1,2†, C. E. Bradley1,2†, F. V. van der Gronden1,2, A. Galicia1,2, M. H. Abobeih1,2, M. Markham^3 ,

D. J. Twitchen^3 , F. Machado4,5, N. Y. Yao4,5, T. H. Taminiau1,2*

The discrete time crystal (DTC) is a nonequilibrium phase of matter that spontaneously breaks time-

translation symmetry. Disorder-induced many-body localization can stabilize the DTC phase by breaking

ergodicity and preventing thermalization. Here, we observe the hallmark signatures of the many-

bodyÐlocalized DTC using a quantum simulation platform based on individually controllable carbon-13

nuclear spins in diamond. We demonstrate long-lived period-doubled oscillations and confirm that they

are robust for generic initial states, thus showing the characteristic time-crystalline order across

the many-body spectrum. Our results are consistent with the realization of an out-of-equilibrium

Floquet phase of matter and introduce a programmable quantum simulator based on solid-state spins

for exploring many-body physics.

A

time crystal spontaneously breaks time-

translation symmetry ( 1 – 3 ). Understand-

ing the consequences and limitations of

this idea has motivated intense explo-

ration across a multitude of physical set-

tings ( 4 – 8 ). A particularly fruitful case has been

that of isolated periodically driven (Floquet)

quantum systems, in which the stability of time

crystals is connected to fundamental ques-

tions in nonequilibrium statistical mechanics

( 4 , 5 , 9 – 11 ).

In a Floquet system, the spontaneous break-

ing of the time-translation symmetry asso-

ciated with the periodic drive results in a

discrete time crystal (DTC) ( 9 – 11 ). The DTC is a

new phase of matter that locks onto a period

that is a multiple of that of the drive and is

stable against perturbations. Experiments have

revealed signatures of discrete time-crystalline

behavior in a range of systems including trap-

ped ions ( 12 , 13 ), spin ensembles ( 14 – 17 ), ultra-

cold atoms ( 18 , 19 ), and superconducting

qubits ( 20 ).

A key challenge is that the stability of the

DTC, as well as any type of order in a Floquet

many-body system, requires a mechanism to

avoid thermalization and heating from the

periodic drive. Although there exist a num-

ber of strategies for exponentially delaying

thermalization ( 7 , 13 , 21 – 23 ), the only known

mechanism for breaking ergodicity in a ge-

neric interacting system is disorder-induced

1474 17 DECEMBER 2021¥VOL 374 ISSUE 6574 science.orgSCIENCE

(^1) QuTech, Delft University of Technology, PO Box 5046, 2600
GA Delft, Netherlands.^2 Kavli Institute of Nanoscience Delft,
Delft University of Technology, P.O. Box 5046, 2600 GA
Delft, Netherlands.^3 Element Six Innovation, Fermi Avenue,
Harwell Oxford, Didcot, Oxfordshire OX11 0QR, UK.
(^4) Department of Physics, University of California, Berkeley,
CA 94720, USA.^5 Materials Sciences Division, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA.
*Corresponding author. Email: [email protected]
†These authors contributed equally to this work.
RESEARCH