action in the Hamiltonian configuration is a
crucial feature of our coupling scheme. Inter-
ference of the two spin-light interactions re-
duces the spin’s quantum back-action rate to
gs,ba= (1–h^4 )Gs, whereas it isgm,ba=h^2 Gmfor
the membrane. Assuming thermal noise to be
negligible, the quantum cooperativityC=2g/
(gs,ba+gm,ba) can be optimized for a given one-
way transmissionh^2. We find an upper bound
C≤hð 1 þh^2 Þ=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 h^4p
,reaching2.7forour
current setup. The bound is achieved for an
optimal choice of measurement ratesGs/Gm=
h^2 /(1–h^4 ), balancing the back-action on both
systems. Further improvement is possible with
a double-loop coupling scheme that also sup-
presses quantum back-action on the membrane
( 28 ). In this case,C=h/(1–h^2 ) atGs=h^2 Gmis
inversely proportional to optical loss, scaling
more favorably at high transmission so that
C≈10 can be reached forh^2 = 0.9.
Our results demonstrate a comprehensive
and versatile toolbox for generating coherent
long-distance interactions with light and open
up a range of exciting opportunities for quan-
tum information processing, simulation, and
metrology. The coupling scheme constitutes
a coherent feedback network ( 33 ) that allows
quantum systems to directly exchange, pro-
cess, and feed back information without the
use of classical channels. The ability to create
coherent Hamiltonian links between sepa-
rate and physically distinct systems in a re-
configurable way substantially extends the
available toolbox not only for hybrid spin-
mechanical interfaces ( 9 , 31 ) but for quantum
networks ( 4 ) in general. It facilitates the faith-
ful processing of quantum information and
the generation of entanglement between spa-
tially separated quantum processors across a
room-temperature environment.
REFERENCES AND NOTES
- L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, P. Treutlein,
Rev. Mod. Phys. 90 , 035005 (2018). - C. Gross, I. Bloch,Science 357 , 995–1001 (2017).
- T. D. Laddet al.,Nature 464 , 45–53 (2010).
- H. J. Kimble,Nature 453 , 1023–1030 (2008).
- R. Blatt, D. Wineland,Nature 453 , 1008–1015 (2008).
- R.Hanson,D.D.Awschalom,Nature 453 , 1043– 1049
(2008). - L. F. Buchmann, D. M. Stamper-Kurn,Ann. Phys. 527 , 156– 161
(2015). - P. Treutlein, C. Genes, K. Hammerer, M. Poggio, P. Rabl, in
Cavity Optomechanics: Nano- and Micromechanical Resonators
Interacting with Light, M. Aspelmeyer, T. J. Kippenberg,
F. Marquardt, Eds. (Springer, 2014), pp. 327–351. - G. Kurizkiet al.,Proc. Natl. Acad. Sci. U.S.A. 112 , 3866– 3873
(2015). - C. Gardiner, P. Zoller,Quantum Noise: A Handbook of Markovian
and Non-Markovian Quantum Stochastic Methods with
Applications to Quantum Optics(Springer, 2004). - P. Lodahlet al.,Nature 541 , 473–480 (2017).
- D. E. Chang, J. S. Douglas, A. González-Tudela, C.-L. Hung,
H. J. Kimble,Rev. Mod. Phys. 90 , 031002 (2018). - K. Lalumièreet al.,Phys. Rev. A 88 , 043806 (2013).
- S. Ritteret al.,Nature 484 , 195–200 (2012).
- P. Campagne-Ibarcqet al.,Phys. Rev. Lett. 120 , 200501
(2018). - P. Kurpierset al.,Nature 558 , 264–267 (2018).
- B. Julsgaard, A. Kozhekin, E. S. Polzik,Nature 413 , 400– 403
(2001).
18. J. Hofmannet al.,Science 337 , 72–75 (2012).
19. R. Riedingeret al.,Nature 556 , 473–477 (2018).
20. H. Krauteret al.,Phys. Rev. Lett. 107 , 080503 (2011).
21. J. Majeret al.,Nature 449 , 443–447 (2007).
22. M. Mirhosseiniet al.,Nature 569 , 692–697 (2019).
23. K. Baumann, C. Guerlin, F. Brennecke, T. Esslinger,Nature 464 ,
1301 – 1306 (2010).
24. N. Spethmann, J. Kohler, S. Schreppler, L. Buchmann,
D. M. Stamper-Kurn,Nat. Phys. 12 , 27–31 (2015).
25. S. Katoet al.,Nat. Commun. 10 , 1160 (2019).
26. N. Leunget al.,Npj Quantum Inf. 5 , 18 (2019).
27. A. F. Kockum, G. Johansson, F. Nori,Phys. Rev. Lett. 120 ,
140404 (2018).
28. T. M. Karg, B. Gouraud, P. Treutlein, K. Hammerer,Phys. Rev. A
99 , 063829 (2019).
29. A. Jöckelet al.,Nat. Nanotechnol. 10 , 55–59 (2015).
30. P. Christophet al.,New J. Phys. 20 , 093020 (2018).
31. C. B. Mølleret al.,Nature 547 , 191–195 (2017).
32. R. A. Thomaset al., arXiv 2003.11310 [quant-ph] (25 March 2020).
33. J. Zhang, Y. Liu, R.-B. Wu, K. Jacobs, F. Nori,Phys. Rep. 679 ,
1 – 60 (2017).
34. See supplementary materials.
35. K. Hammerer, A. S. Sørensen, E. S. Polzik,Rev. Mod. Phys. 82 ,
1041 – 1093 (2010).
36. E. S. Polzik, K. Hammerer,Ann. Phys. 527 , A15–A20 (2015).
37. J. D. Thompsonet al.,Nature 452 , 72–75 (2008).
38. M. Aspelmeyer, T. J. Kippenberg, F. Marquardt,Rev. Mod. Phys.
86 , 1391–1452 (2014).
39. S. Gröblacher, K. Hammerer, M. R. Vanner, M. Aspelmeyer,
Nature 460 , 724–727 (2009).
40. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser,
T. J. Kippenberg,Nature 482 , 63–67 (2012).
41. H. Xu, D. Mason, L. Jiang, J. G. E. Harris,Nature 537 , 80– 83
(2016).
42. N. R. Bernier, L. D. Tóth, A. K. Feofanov, T. J. Kippenberg,Phys.
Rev. A 98 , 023841 (2018).
43. Y. Tsaturyan, A. Barg, E. S. Polzik, A. Schliesser,
Nat. Nanotechnol. 12 , 776–783 (2017).
44. T. M. Karget al., Zenodo doi:10.5281/zenodo.3779295 (2020).
ACKNOWLEDGMENTS
We thank G. Buser for setting up the dipole trap and M. Ernzer
for discussions.Funding:Supported by the project“Modular
mechanical-atomic quantum systems”(MODULAR) of the
European Research Council (ERC) and by the Swiss Nanoscience
Institute (SNI). K.H. acknowledges support through the cluster
of excellence“Quantum Frontiers”and from DFG through CRC
1227 DQ-mat, projects A06.Author contributions:T.M.K., B.G.,
K.H., and P.T. conceived the experiment; T.M.K. and B.G. developed
the theory, with input from K.H. and P.T.; T.M.K., B.G., C.T.N.,
and G.-L.S. built the experimental setup; T.M.K. took and analyzed
the data, discussing with P.T.; T.M.K., P.T., and K.H. wrote the
manuscript with input from other authors; and K.H. and P.T.
supervised the project.Competing interests:The authors declare
no competing interests.Data and materials availability:All
data needed to evaluate the conclusions in the paper are present in
the figures of the paper. The data can be accessed at ( 44 ).SUPPLEMENTARY MATERIALS
science.sciencemag.org/content/369/6500/174/suppl/DC1
Supplementary Text
Figs. S1 to S4
Table S1
References ( 45 – 56 )
27 January 2020; accepted 28 April 2020
Published online 7 May 2020
10.1126/science.abb0328REPORTS
◥TOPOLOGICAL MATTERObservation and control of maximal Chern numbers
ina chiral topological semimetal
Niels B. M. Schröter^1 *, Samuel Stolz2,3, Kaustuv Manna^4 , Fernando de Juan5,6, Maia G. Vergniory5,6,
Jonas A. Krieger1,7,8, Ding Pei^9 , Thorsten Schmitt^1 , Pavel Dudin^10 †, Timur K. Kim^10 , Cephise Cacho^10 ,
Barry Bradlyn^11 , Horst Borrmann^4 , Marcus Schmidt^4 , Roland Widmer^2 ,
Vladimir N. Strocov^1 , Claudia Felser^4 *Topological semimetals feature protected nodal band degeneracies characterized by a topological
invariant known as the Chern number (C). Nodal band crossings with linear dispersion are expected to
have at mostjCj¼4, which sets an upper limit to the magnitude of many topological phenomena in
these materials. Here, we show that the chiral crystal palladium gallium (PdGa) displays multifold band
crossings, which are connected by exactly four surface Fermi arcs, thus proving that they carry the
maximal Chern number magnitude of 4. By comparing two enantiomers, we observe a reversal of
their Fermi-arc velocities, which demonstrates that the handedness of chiral crystals can be used to
control the sign of their Chern numbers.T
opological invariants are mathematical
objects that can be used to classify Ham-
iltonians and have found widespread
applications in physics, chemistry, and
materials science. One of the best known
topological invariants in condensed matter
physics is the Chern number, which can be
defined as the flux of Berry curvature through
a closed two-dimensional surface. If this sur-
face is taken to be the whole Brillouin zone,
the Chern number classifies insulators in twodimensions, as first used in the context of the
quantum Hall effect by Thouless and co-workers
( 1 , 2 ) in the 1980s. More recently, Chern num-
bers have also been used to classify topo-
logical nodal semimetals ( 3 ), where point-like
energy degeneracies in their bulk electronic
structure act as sources and sinks of quan-
tized Berry flux through any local isoenergy
surface enclosing the node. For the simplest
case of a linear touching of two bands, which
can occur in any noncentrosymmetric orSCIENCEsciencemag.org 10 JULY 2020•VOL 369 ISSUE 6500 179
RESEARCH