the cyclotron energy is consistent with our
model for the scattering because the latter
involves the overlap between electronic wave
functions, whose area naturally scales with the
square of the magnetic lengthl^2 band therefore
as 1/B. As a result, the dependence on mag-
netic field is indeed expected to be the same for
the even and odd plateaus, although their over-
all strengths differ because the Fermi energy is
closer to the center of the broadened Landau
band for the odd plateaus ( 27 ). Indeed, the
Zeeman gap is ~20% of the cyclotron gap ( 19 ).
To further illustrate the consistency of the
results, we report in Fig. 2C another set of data
forrnxxtaken from a different resonator on
the same heterostructure material, having the
same shape but with a different geometry of
the voltage leads (blue dots). The results, al-
though not identical, still exhibit the same
behavior. Finally, on the same graph we also
report the measured data for an identical reso-
nator fabricated on a third high-mobility 2D
electron gas (EV2124) with the same electron
density but grown in a different molecular
beam epitaxy reactor and exhibiting a mobility
reduced by a factor of 8. Here again, compa-
rable values are obtained, hence the scattering
mechanism has a moderate dependence on
the disorder.
The very strong increase of the cavity-
mediated hopping as the Fermi energy is
brought closer to the center of the Landau
level, predicted theoretically ( 27 ), explains the
strong difference in measured scattering rate
between odd and even plateaus. Because our
measurements of cavity-mediated hopping are
performed with the chemical potential lying
in the middle between the spin-split Landau
bands, a modification of the relative magni-
tude of the Zeeman splitting with respect to
the cyclotron energy should lead to a strong
decrease of the cavity-mediated hopping as
the Fermi energy is displaced farther away
from the bulk Landau band. Such behavior
has been tested experimentally by measuring
the sample in a tilted field configuration ( 23 ).
As predicted by Eq. 1, cavity-mediated hop-
ping is expected to increase strongly with the
normalized light-matter coupling strength
W~R=wcav. We investigated this dependence
by designing a series of three cavities with
the fundamental mode having the same fre-
quency, but exhibiting different coupling
strengthsW~R=wcav= 0.17, 0.2, and 0.22 ( 23 ).
As expected, a strong increase of the resistance
is observed as a function of the light-matter
coupling strengthWR, further demonstrating
that the scattering originates from the coupling
to the vacuum fields.
Finally, another possibility to study the de-
pendence of the cavity-mediated hopping on
the coupling is to modify the field distribution
of the resonator modes inside the gap. Doing
this in situ by approaching a metallic“tip”has
the advantage that, unlike the previous experi-
ment, the experiment does not rely on com-
paring the transport data between different
physical samples. We implemented this idea
by approaching, in a controlled and repeatable
way, a plane metallic surface in close proxim-
ity to the sample during the measurement of
the magnetotransport (Fig. 3). Although the
clear advantage of this experiment is that no
reference sample is needed, it is difficult to
exactly quantify the effect of the metallic plate
on the light-matter coupling because it changes
the local field profile and gradients in a non-
trivial way. In Fig. 3 we plot the difference in
the longitudinal and transverse Hall resistances
when a piezoelectric positioner brings a gold
metallic plane in close proximity to the reso-
nator (200 nm) or farther away (3 μm) while
the sample is kept at the base temperature.
To achieve a better signal/noise ratio and to
ensure that the changes were reproducible,
we repeated the experiment 60 times. The
averaged traces are shown in Fig. 3. Overall,
the results show an increase of the cavity-
induced scattering as the metallic plate is
brought in proximity to the resonator. This
shows that the dominant mechanism is the
increase of the field gradients as the gap is
reduced, rather than a change of the coupling
strengthWR, as the latter would decrease with
the gap.
Our results show that the vacuum field in a
deeply subwavelength electromagnetic reso-
nator produces a breakdown of the topological
protection of the integer quantum Hall effect.
The strong effect reported here on the quan-
tum Hall plateaus and longitudinal resist-
ance is such that one can legitimately wonder
whether the much weaker vacuum fluctua-
tions in free space might be what ultimately
limits the extreme metrological precision of
quantum Hall resistance standards. We note
also that an additional vacuum field–induced
mechanism for the breakdown of the topo-
logical quantization of conductance was re-
cently proposed ( 31 ). The results of the present
work demonstrate that vacuum field fluctu-
ations can have a strong effect on transport
even on topologically protected states. This is
shown here in the case of the quantum Hall
effect, but these results might have implica-
tions for a larger class of topologically pro-
tected states ( 32 ). An interesting possibility is
the engineering of photon modes with larger
field gradients, which would allow electro-
magnetic vacuum fields to be coupled more
efficiently to fractional quantum Hall phases
and would enable the study of cavity-mediated
electron-electron interactions.REFERENCES AND NOTES- P. Milonni,The Quantum Vacuum: An Introduction to Quantum
Electrodynamics(Academic Press, 1994). - C. Rieket al.,Science 350 , 420–423 (2015).
3. I.-C. Benea-Chelmus, F. F. Settembrini, G. Scalari, J. Faist,
Nature 568 , 202–206 (2019).
4. F. J. Garcia-Vidal, C. Ciuti, T. W. Ebbesen,Science 373 ,
eabd0336 (2021).
5. H. Hübeneret al.,Nat. Mater. 20 , 438–442 (2021).
6. E. Orgiuet al.,Nat. Mater. 14 , 1123–1129 (2015).
7. P. Bhatt, K. Kaur, J. George,ACS Nano 15 , 13616– 13622
(2021).
8. S. Smolkaet al.,Science 346 , 332–335 (2014).
9. T. Smoleńskiet al.,Nature 595 , 53–57 (2021).
10. Y. Todorovet al.,Phys. Rev. Lett. 105 , 196402 (2010).
11. G. Scalariet al.,Science 335 , 1323–1326 (2012).
12. P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, E. Solano,Rev. Mod. Phys.
91 , 025005 (2019).
13. A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta, F. Nori,
Nat. Rev. Phys. 1 , 19–40 (2019).
14. C. Ciuti, G. Bastard, I. Carusotto,Phys. Rev. B 72 , 115303
(2005).
15. F. Schlawin, A. Cavalleri, D. Jaksch,Phys. Rev. Lett. 122 ,
133602 (2019).
16. A. Thomaset al., arXiv 1911.01459 (2019).
17. Y. Ashidaet al.,Phys. Rev. X 10 , 041027 (2020).
18. A. Thomaset al.,Nano Lett. 21 , 4365–4370 (2021).
19. S. M. Girvin, K. Yang,Modern Condensed Matter Physics
(Cambridge Univ. Press, 2019).
20. N. Bartolo, C. Ciuti,Phys. Rev. B 98 , 205301 (2018).
21. G. L. Paravicini-Baglianiet al.,Nat. Phys. 15 , 186– 190
(2018).
22. M. Büttiker,Phys. Rev. B 38 , 9375–9389 (1988).
23. See supplementary materials.
24. H.-T. Chenet al.,Opt. Express 15 , 1084–1095 (2007).
25. C. Maissenet al.,Phys. Rev. B 90 , 205309 (2014).
26. G. L. Paravicini-Baglianiet al.,Phys. Rev. B 95 , 205304
(2017).
27. C. Ciuti,Phys. Rev. B 104 , 155307 (2021).
28. H. L. Stormer, D. C. Tsui, A. C. Gossard,Rev. Mod. Phys. 71 ,
S298–S305 (1999).
29. W. Kohn,Phys. Rev. 123 , 1242–1244 (1961).
30.P.L.McEuenet al.,Phys. Rev. Lett. 64 , 2062– 2065
(1990).
31. V. Rokajet al., arXiv 2109.15075 (2021).
32. R. M. Lutchynet al.,Nat. Rev. Mater. 3 , 52–68 (2018).
33. F. Appugliese, ETH Research Collection entry http://www.
research-collection.ethz.ch/handle/20.500.11850/519837
(2021).
ACKNOWLEDGMENTS
We thank J. Andberger for help in the initial stage of the project,
P. Märki for technical support, and A. Imamoglu for discussions.
Funding:Supported by the ERC Advanced grant Quantum
Metamaterials in the Ultra Strong Coupling Regime (MUSiC)
(grant 340975); the Swiss National Science Foundation (SNF)
through the National Centre of Competence in Research Quantum
Science and Technology (NCCR QSIT) as well as from the
individual grant 200020_192330; and ANR project TRIANGLE
(ANR-20-CE47-0011) and FET FLAGSHIP Project PhoQuS (grant
agreement 820392) (C.C.).Author contributions:F.A. performed
measurements, fabricated samples, analyzed the data, and
wrote the paper; J.E. performed measurements, fabricated
samples, contributed to data analysis, and wrote the paper. G.L.P.-B.
performed early measurements and analyzed early results; C.R.
and W.W. grew the epitaxial 2DEGs D and F; M.B. grew the epitaxial
2DEG EV; G.S. supported and designed experiments, setups, and
samples and wrote the paper; C.C. developed the theory of cavity-
induced electron hopping and wrote the paper; J.F. designed the
experiments, analyzed the data, developed the model of the effect of
the vacuum field on the resistance quantization, supervised the
whole work, and wrote the paper.Competing interests:The authors
declare no competing interests.Data and materials availability:
Data that support the findings of this article and codes are available
in the ETH Research Collection ( 33 ).SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abl5818
Materials and Methods
Supplementary Text
Figs. S1 to S14
Tables S1 and S2
References ( 34 – 38 )23 July 2021; accepted 20 December 2021
10.1126/science.abl58181034 4 MARCH 2022•VOL 375 ISSUE 6584 science.orgSCIENCE
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