The electron transport in the integer quan-
tum Hall effect proceeds via edge states that
are protected against backscattering by the
lifting of the time-reversal symmetry provided
by the magnetic field (Fig. 1A). As shown in a
recent study ( 27 ), in the presence of disorder
a cavity-mediated long-range hopping can be
achieved via the exchange of a virtual cavity
photon (Fig. 1C). For an electron occupying a
disordered eigenstatefðÞln
E
in thenth Landauband, the coupling to another disordered
eigenstatef
ðÞn
l′
E
in the same Landau band
occurs via light-matter coupling to an inter-
mediate state, consisting of an electron in
the statefðÞln′′þ^1
E
in the (n+1)thbandwiththesamespinandonecavityphoton.Atthe
lowest order in perturbation theory, the effec-
tive coupling between disordered statesland
l′in the samenth Landau band is given by
the expression
~GðÞn
l;l′¼Xl′′~gðÞln;l;n′′þ^1 ~gðÞ ln′;;ln′′þ^1
en;lenþ 1 ;l′′ħw~cavð 1 Þand can be construed as a cavity-mediated
hopping. Note that this intermediate virtual
process is due to the counterrotating terms of
the light-matter interaction and is quantified
by the single-electron vacuum Rabi frequency
~glðÞn;l;′nþ^1 , as detailed in ( 27 ). The number of in-
termediate states is equal to the Landau de-
generacyNdeg=Nel/n, whereNelis the number
of electrons andnis the filling factor. The de-
nominator of the expression represents the
energy penalty associated to the excitation of
the virtual intermediate state described above.
Such cavity-mediated hopping involves all the
disordered eigenstates, affecting both edge
and bulk states. For the quantitative impact of
the process, a crucial role is played by the col-
lective vacuum Rabi frequencyW~R¼~gffiffiffiffiffiffiffi
Nelp
,
where the tilde indicates the diamagnetic re-
normalization ( 27 ). Given the anti-resonant
nature of the process, additional electromag-
netic modes with higher frequencies, such as
those that can be observed in Fig. 1B, can
contribute in the same qualitative way and
reinforce quantitatively the effective cavity-
mediated hopping. In addition, the strong elec-tromagnetic field gradients present at the edge
of the metal resonator, as apparent in Fig. 1A,
play a role similar to that of impurity disorder
in enabling cavity-mediated hopping ( 27 ).
The transverse and longitudinal magneto-
resistances of a reference sample and of a Hall
bar coupled to a cavity are compared in Fig. 2.
The reference sample and the cavity-coupled
one are fabricated on the same chip and share
the same source and drain ( 23 ). The sample
exhibits a mobility of 16 × 10^6 cm^2 V–^1 s–^1 and a
density of 2 × 10^11 cm–^2 and is kept in the dark
in a dilution fridge at the nominal temper-
ature of 10 mK; the electron temperature is
estimated to be close to 50 mK. Both samples
are measured in the same cooling run and are
physically located on the same chip. Indeed,
identifying unambiguously the effects of vac-
uum fluctuations experimentally requires that
other possible effects be ruled out. [See ( 23 ) for
details of the precautions taken during sam-
ple fabrication and measurement to avoid un-
wanted artifacts.]
For the reference sample (no cavity), all
the integer quantum Hall plateaus are well1032 4 MARCH 2022•VOL 375 ISSUE 6584 science.orgSCIENCE
Fig. 2. Breakdown of the quantum Hall effect by vacuum fluctuations.
(A) Longitudinal and transverse resistances for a reference Hall bar (black lines)
and for a cavity-embedded Hall bar (blue lines). The calculated transverse resistance
(orange dots) computed from the measured longitudinal resistance is overlaid. The
prediction is in excellent agreement with the measured transverse resistance (blue
curves). For the reference sample with no cavity, the longitudinal resistance is
negligible. (B) Magnetic field region from 0 to 0.5 T, reproduced for better visibility,
showing that both the longitudinal and transverse resistances with and without cavity
converge to the same value in the limit of zero magnetic field. Inset: Schematic
drawing of the electrical configuration allowing the comparison of the resonator and
the reference Hall bars. (C) The extracted resistivityrnxxas a function of cyclotron
energy for two different samples having very similar densities.RESEARCH | REPORTS