to be larger than in the standard MZI configu-
ration of Fig. 2A (see black symbols in Fig. 2E).
This increase is opposite to the reduction that
wouldbeexpectedfromtheAharonov-Bohm
period with the larger surface enclosed by the
outer channel path and the inner boundary
of the floating metallic island [see ( 25 ) for a
graphical representation;S≃ 18 : 4 mm^2 would
correspond to an Aharonov-Bohm period of
225 mT≃h=eS]. Such opposite evolution and
relatively important discrepancy (36%) es-
tablish that the MZI phase does not reduce
to the usual Aharonov-Bohm phase acquired
by a single electron propagating along two
different edge paths. Even a naïve applica-
tion of the surface modulation that cancels
the magnetic field dependence in Fabry-Pérot
interferometers dominated by Coulomb in-
teraction ( 29 , 30 ) would only compensate
for the period reduction by the added area
enclosed between the metallic island and
the barring gate. Instead, the larger period
corroborates the transfer of the electrons’
state across the island. Indeed, in the pres-
ence of an electron path amputated from a
section (the 2DEG/metal interface), the closed
surface involved in the Aharonov-Bohm phase
is no longer well-defined. Whether one can
still speak of an Aharonov-Bohm phase with a
smaller effective surface, or whether another
period reduction mechanism is at play when
going through the floating island, is not estab-
lished. This question calls for further investi-
gations, both theoretical and experimental,
with devices implementing different injection-
emission distances.
The blue continuous line in Fig. 2D was mea-
suredwithoneMZIarmgoingthroughthe
floating island and in the presence of a second
electronic channel connected to it (configura-
tion schematically displayed in Fig. 2C). We
find strongly suppressedconductance oscilla-
tions corresponding to a full decoherence of
the electrons going through the island. The
residual visibilityV≲ 0 :2 is consistent with the
separately obtained proportion 1tisland≲3%
of reflected electrons, not penetrating into the
island ( 25 ). Indeed, the MZI contribution of
the reflected electrons readsVð 1 tisland≪ 1 Þ≃
4 V 0
3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 tislandp
≲ 0 :21, withV 0 ≈90% the MZI
visibility in the standard configuration ( 25 , 27 ).
The magnetic field period of 246T 4 mT for
these smaller oscillations [see blue symbols
in Fig. 2E; a standard fast Fourier transform
analysis of these residual oscillations yields
the less resolved value 237T 16 mT( 25 )] is 9%
longer than the 225mT nominally expected,
and is close to the period observed in the
standard MZI configuration shown in Fig.
2A. It is plausible that the reflected elec-
trons are those propagating the furthest away
from the edge and from the semiconductor-metal interface, effectively corresponding to
a smaller surface compatible with the ob-
served longer period. Other possibilities in-
clude that the residual electrons’reflection
takes place at the level of the barring gate
(colored blue, left of island in Fig. 1) or, even-
tually, a Coulomb-induced compensation in
the spirit of Coulomb dominated Fabry-Pérot
interferometers ( 29 , 30 ). Also, the average
htMZIi≃ 0 :39 is shifted below 0.5. This is
simply because part of the injected current is
evacuated toward a remote electrical ground
through the second (right) channel connected
to the floating island (the valuehtMZIi¼ 0 : 375
is expected from current conservation for a
floating island and a central ohmic contact
both perfectly connected).
We now investigate the relation between
the island’s charge and the electron phase shift
associated with the quantum state transfer.
For this purpose, Fig. 3 focuses on the in-
fluence ontMZIof the voltageVplapplied to a
plunger gate (colored purple in Fig. 1), which
is relatively far from the MZI outer quantum
Hall channel but close to the island. The equiv-
alent role on the MZI phase ofVplandBis first
directly established, in Fig. 3A, with the device
set in the floating island MZI configuration
[schematic in Fig. 2B, see also, e.g., ( 31 , 32 )for
the influence of an electrostatic field on quan-
tum interferences through different mecha-
nisms]. Figure 3B displays Coulomb diamond
measurements of the conductance across the
island as a function of the same plunger gate
voltageVpl. For this purpose, the island is
here weakly connected through tunnel barriers,
thereby implementing asingle-electron tran-
sistor of quantized chargeQin units ofe(Qis
not quantized for the strongly coupled floating
island in the MZI configurations). The left
MZI arm was disconnected during this mea-
surement, as schematically illustrated in fig.
S2 ( 25 ). Notably, the MZI gate voltage period
in Fig. 3A precisely matches the Coulomb
diamonds’period in Fig. 3B, as can be seen
by directly comparing the two panels plotted
using the sameVplscale. In the floating MZI
limit of strongly connected channels,Q¼
eVpl=D,withD≃ 1 :7 mV being the Coulomb
diamond period ( 11 – 13 ). A quantum phase
shift of 2pQ=etherefore applies to the trans-
ferred electrons, as specifically predicted the-
oretically ( 4 , 5 ) and in agreement with Friedel’s
sum rule. Figure 3C shows a comparison of
tMZIoscillations in the standard MZI config-
uration (red line) and with one arm going
through the metallic island (black line); both
have maximum visibilityV≃90% as also seen
versus magnetic field in Fig. 2D. However,
theVplperiod is increased by a large factor of
160, from 1:7 to 270 mV, when the island is
disconnected; this reflects the weak cou-
pling of the plunger gate voltage to the MZI
outer edge channel [see fig. S3 ( 25 )foranDuprezet al.,Science 366 , 1243–1247 (2019) 6 December 2019 3of4
Fig. 3. Quantum phase
versus island charge.
(A) Color plot of
tMZIðB;VplÞin the floating
island MZI configuration
(schematic in Fig. 2B),
with the larger values
shown brighter, which
establishes the
equivalent role ofB
andVpl.(B) Coulomb
diamond characterization
of the floating island
(larger differential con-
ductancetSETshown
brighter; in this con-
figuration, the island is
weakly coupled on both
sides andVdcis the
applied dc bias voltage).
A comparison with (A),
plotted using the same
Vplscale, reveals that
the addition of a charge
ofeon the island
precisely corresponds, in the floating island MZI configuration, to an electron quantum phase of 2p
(one quantum oscillation period). (C) The top and bottom panels display measurements oftMZIðVplÞ
with the device set in the floating island MZI configuration (black line) and in the standard MZI
configuration (red line, schematic in Fig. 2A). The MZI oscillations’period inVplis shorter by a factor
of 1=160 when the island is connected. An additional modulation of fixed period (≈15 mV) appears
in both configurations.
RESEARCH | REPORT
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