developed, showing zero longitudinal resist-
ance down ton= 11. By contrast, the cavity-
coupled sample displays deviations from the
integer quantization, which are especially
strong for all the odd plateaus. As expected
from the anti-resonant origin of cavity-mediated
electron hopping, the modifications of the
quantized transport occur on a very wide mag-
netic field range starting fromB≈0.25 T. At
the same time (Fig. 2B, inset), it is important
to note that the resistance in the limit of van-
ishing magnetic field is essentially unchanged
by the cavity and that the transport features
associated with the fractional quantum Hall
regime are only weakly affected. This is an
important sanity check because it shows that
the mobility and overall quality of the sample
are unchanged by the complementary resona-
tor (indeed, the metallic gold layer defining
the resonator is deposited around the Hall bar
and spatially separated from it) and are also
remarkable because the fractional quantum
Hall features are usually the most fragile with
respect to perturbations ( 28 ). This is consistent
with the fact that fractional Hall excitations
couple extremely weakly to electromagnetic
fields. Indeed, a consequence of Kohn’s theorem
( 29 ) is that for a spatially homogeneous electric
field and no disorder, light couples only to the
center-of-mass motion of the electrons, which is
not affected by the electron-electron interac-
tions responsible for the fractional quantum Hall
states. Note also that for moderate magnetic
fields (0.1 <B<0.25T),thecavitysampledis-
plays a reduction of the amplitude of Shubnikov–
de Haas oscillations with respect to the reference
sample, as already evidenced and discussed in
our previous study ( 21 ) (Fig. 2B, inset).
The experiments shown in Fig. 2 were re-
peated for a number of temperatures up to 1 K,
which allowed us to extract the activation
energies of the main transport features. The
results ( 23 ) confirm the relative immunity
toward vacuum fluctuations of the fractional
quantumHallfeaturesascomparedtothe
integer ones.
Our experiments revealed that vacuum fluc-
tuations, having been strongly enhanced in
the gap of a metallic resonator, generate a
long-range hopping that breaks the quan-
tization of the resistance. The quantitative
characterization of this phenomenon can be
conveniently performed by adopting the for-
malism based on the Landauer-Büttiker edge
state picture at integer filling factors ( 30 ). As
shown schematically in Fig. 1A, in this theoret-
ical framework, the deviation from quantiza-
tion is interpreted by a finite transmissionti
of the highest populated edge state of section
iof the conductor due to scattering of a frac-
tion (1–ti) to the other edge. The strength of
the scattering for the edge statenis related
to the lengthLiand the widthwiof the con-
ductor, and to resistivityrnxx, by the relation
ti¼ 1 =½ 1 þrnxxðÞLi=wi. This model was suc-
cessfully used to explain how the longitudi-
nal resistance in the region between quantized
plateaus depends on the geometry of the volt-
age probes ( 30 ).
In our case, we interpretrnxxas the“resis-
tivity”originating from the vacuum fluctua-
tions, which concerns only the fraction of
the conductor exposed to this field. Such an
interpretation, however, requires that the rest
of the conductor is in the quantum Hall re-
gime,andasaresultthisanalysisisonlyperformed in the middle of a quantum Hall
plateau.
We implemented and solved the model in
the case of a standard Hall bar with six con-
tacts ( 23 ). The experimental inputs are the
three length-to-width ratios describing the
interaction of the vacuum field with the cur-
rent, voltage probes as well as the main part
of the Hall bar (designatedG 1 ,G 2 , andG 3 ,
respectively, in Fig. 1A). Using these parame-
ters, the measured values of the longitudinal
resistancesforthecavitysampleareusedtofit
the relevant value ofrnxx, whereas the predicted
values of the correspondingRxy, calculated
with our model, are displayed along with the
experimental data in Fig. 2A.
The magnetic field values are chosen in the
middle of the plateau measured on the ref-
erence sample. The impressive quantitative
agreement between theory (orange points)
and experiment (blue lines) for the Hall re-
sistance is an indication that this model cap-
tures well the physics of our experiments and
also has predictive power. The value ofrnxxis
displayed as a function of the cyclotron energy
of the corresponding plateau in Fig. 2C for
values of the magnetic field corresponding
to the center of the plateaus of the reference
sample. The data align according to two groups
of points. As is already obvious from the trans-
verse and longitudinal Hall resistance data
showninFig.2,thevaluesofrnxxfor odd
plateaus are larger (by about two orders of
magnitude) than the ones for even plateaus.
However, both groups of points show a striking
exponential dependence on the cyclotron
energy, with a similar slope of ~0.4 meV. This
observed overall exponential dependence onSCIENCEscience.org 4 MARCH 2022•VOL 375 ISSUE 6584 1033
Fig. 3. Modifying the cavity vacuum fields in situ by a metallic tip.Differential longitudinal and transverse resistances are shown for the resonator at 140 GHz
with the tip 200 nm and 3 μm away from the surface. Inset: A pictorial representation of the tip brought into the near field of the complementary split ring resonator.
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