Finite bias visibility of the electronic Mach-Zehnder interferometer
Preden Roulleau, F. Portier, D. C. Glattli,∗ and P. Roche†
Nanoelectronic group, Service de Physique de l’Etat Condensé,
CEA Saclay, F-91191 Gif-Sur-Yvette, France
A. Cavanna, G. Faini, U. Gennser, and D. Mailly
CNRS, Phynano team,
Laboratoire de Photonique et Nanostructures,
Route de Nozay, F-91460 Marcoussis, France
(Dated: November 28, 2018)
We present an original statistical method to measure the visibility of interferences in an electronic
Mach-Zehnder interferometer in the presence of low frequency fluctuations. The visibility presents
a single side lobe structure shown to result from a gaussian phase averaging whose variance is
quadratic with the bias. To reinforce our approach and validate our statistical method, the same
experiment is also realized with a stable sample. It exhibits the same visibility behavior as the
fluctuating one, indicating the intrinsic character of finite bias phase averaging. In both samples,
the dilution of the impinging current reduces the variance of the gaussian distribution.
PACS numbers: 85.35.Ds, 73.43.Fj
Nowadays quantum conductors can be used to per-
form experiments usually done in optics, where electron
beams replace photon beams. A beamlike electron mo-
tion can be obtained in the Integer Quantum Hall Effect
(IQHE) regime using a high mobility two dimensional
electron gas in a high magnetic field at low temperature.
In the IQHE regime, one-dimensional gapless excitation
modes form, which correspond to electrons drifting along
the edge of the sample. The number of these so-called
edge channels corresponds to the number of filled Lan-
dau levels in the bulk. The chirality of the excitations
yields long collision times between quasi-particles, mak-
ing edge states very suitable for quantum interferences
experiments like the electronic Mach-Zehnder interfer-
ometer (MZI) [1, 2, 3]. Surprisingly, despite some ex-
periments which show that equilibrium length in chiral
wires is rather long [4], very little is known about the
coherence length or the phase averaging in these ”per-
fect” chiral uni-dimensional wires. In particular, while in
the very first interference MZI experiment the interfer-
ence visibility showed a monotonic decrease with voltage
bias, which was attributed to phase noise [1], in a more
recent paper, a surprising non-monotonic decrease with
a lobe structure was observed [5]. A satisfactory expla-
nation has not yet been found, and the experiment has
so far not been reported by other groups to confirm these
results.
We report here on an original method to measure the
visibility of interferences in a MZI, when low frequency
phase fluctuations prevent direct observation of the peri-
odic interference pattern obtained by changing the mag-
netic flux through the MZI. We studied the visibility at
finite energy and observed a single side lobe structure,
which can be explained by a gaussian phase averaging
whose variance is proportional to V 2, where V is the
FIG. 1: SEM view of the electronic Mach-Zehnder with a
schematic representation of the edge state. G0, G1, G2 are
quantum point contacts which mimic beam splitters. The
pairs of split gates defining a QPC are electrically connected
via a Au metallic bridge deposited on an insolator (SU8). G0
allows a dilution of the impinging current, G1 and G2 are the
two beam splitters of the Mach-Zehnder interferometer. SG
is a side gate which allows a variation of the length of the
lower path (b).
bias voltage. To reinforce our result and check if low
frequency fluctuation may be responsible for that behav-
ior, we realized the same experiment on a stable sample
: we also observed a single side lobe structure which can
be fitted with our approach of gaussian phase averaging.
This proves the validity of the results, which cannot be
an artefact due to the low frequency phase fluctuations
in the first sample. In both samples, the dilution of the
impinging current has an unexpected effect : it decreases
the variance of the gaussian distribution.
The MZI geometry is patterned using e-beam lithogra-
phy on a high mobility two dimensional electron gas in a
GaAs/Ga1−xAlxAs heterojunction with a sheet density
nS = 2.0×1011 cm−2 and a mobility of 2.5×106 cm2/Vs.
http://arxiv.org/abs/0704.0746v3
The experiment was performed in the IQHE regime at
filling factor ν = nSh/eB = 2 (magnetic field B =5.2
Tesla). Transport occurs through two edge states with an
extremely large energy redistribution length [4]. Quan-
tum point contacts (QPC) controlled by gates G0, G1
and G2 define electronic beam splitters with transmis-
sions T0, T1 and T2 respectively. In all the results pre-
sented here, the interferences were studied on the outer
edge state schematically drawn as black lines in Fig.(1),
the inner edge state being fully reflected by all the QPCs.
The interferometer consists of G1, G2 and the small cen-
tral ohmic contact in between the two arms. G1 splits
the incident beam into two trajectories (a) and (b), which
are recombined with G2 leading to interferences. The
two arms defined by the mesa are 8 µm long and en-
close a 14 µm2 area. The current which is not transmit-
ted through the MZI, IB = ID − IT , is collected to the
ground with the small ohmic contact. An additional gate
SG allows a change of the length of the trajectory (b).
The impinging current I0 can be diluted thanks to the
beam splitter G0 whose transmission T0 determines the
diluted current dID = T0 × dI0. We measure the differ-
ential transmission through the MZI by standard lock-in
techniques using a 619 Hz frequency 5 µVrms AC bias
VAC superimposed to the DC voltage V . This AC bias
modulates the incoming current dID = T0 × h/e2 ×VAC ,
and thus the transmitted current in an energy range close
to eV , giving the transmission T (eV ) = dIT /dI0.
Using the single particle approach of the Landauer-
Büttiker formalism, the transmission amplitude t
through the MZI is the sum of the two complex
transmission amplitudes corresponding to paths (a)
and (b) of the interferometer; t = t0{t1 exp(iφa)t2 −
r1 exp(iφb)r2}. This leads to a transmission probabil-
ity T (ǫ) = T0{T1T2 + R1R2 +
T1R2R1T2 sin[ϕ(ǫ)]},
where ϕ(ǫ) = φa − φb and Ti = |ti|2 = 1 −Ri. ϕ(ǫ) cor-
responds to the total Aharonov-Bohm (AB) flux across
the surface S(ǫ) defined by the arms of the MZI, ϕ(ǫ) =
2πS(ǫ) × eB/h. The surface S depends on the energy ǫ
when there is a finite length difference ∆L = La−Lb be-
tween the two arms. This leads to a variation of the phase
with the energy, ϕ(ǫ+EF ) = ϕ(EF )+ǫ∆L/(~vD), where
vD is the drift velocity. When varying the AB flux, the
interferences manifest themselves as oscillations of the
transmission; in practice this is done either by varying
the magnetic field or by varying the surface of the MZI
with a side gate [1, 5, 6]. The visibility of the interfer-
ences defined as V = (TMAX−TMIN )/(TMAX+TMIN ), is
maximum when both beam splitter transmission are set
to 1/2. In the present experiment the MZI is designed
with equal arm lengths (∆L = 0) and the visibility is not
expected to be sensitive to the coherence length of the
source ~vD/max(kBT, eVAC). Thus the visibility pro-
vides a direct measurement of the decoherence and/or
phase averaging in this quantum circuit.
In Ref.[1], 60% visibility was observed at low tem-
-0.76 -0.72 -0.68 -0.64
-0.4 -0.2 0.0 0.2 0.4
2 x Visibility
δT / T
0.0 0.2 0.4 0.6 0.8 1.0
Transmission T
gate voltage (V
 or V
) (Volt)
FIG. 2: Sample #1 a)Transmission T = dIT /dI0 as a func-
tion of the gate voltages V 1 and V 2 applied on G1 and G2.
(◦) T = T1 versus V 1. (•) T = T2 versus V 2. The solid
line is the transmission T obtained with T1 fixed to 1/2 while
sweeping V 2 : transmission fluctuations due to interferences
with low frequency phase noise appears. b) Stack histogram
on 6000 successive transmission measurements as a function
of the normalized deviation from the mean value. The solid
line is the distribution of transmission expected for a uniform
distribution of phases. c)Visibility of interferences as a func-
tion of the transmission T2 when T1 = 1/2. The solid line is
T2(1− T2) dependence predicted by the theory.
perature, showing that the quantum coherence length
can be at least as large as several micrometer at 20 mK
(and probably larger if phase averaging is the limiting
factor). At finite energy (compared to the Fermi en-
ergy), the visibility was also found decreasing with the
bias voltage[1, 5, 6]. This effect is not due to an increase
of the coherence length of the electron source which re-
mains determined by eVAC or kBT [7]. In a first exper-
iment, a monotonic visibility decrease was found, which
was attributed to phase averaging, as confirmed by shot
noise measurements [1]. Nevertheless, it remains unclear
why and how the phase averaging increases with the bias.
In a recent paper, instead of a monotonic decrease of the
visibility, a lobe structure was observed for filling factor
less than 1 in the QPCs [5]. No non-interacting electron
model was found to be able to explain this observation,
and although interaction effects have been proposed [8],
a satisfactory explanation has not yet been found to ac-
count for all the experimental observations. So far, two
experiments have shown to two different behaviors, rais-
ing questions about the universality of these observations.
Here, we report experiments where different samples give
consistent results, with a fit to the data clearly demon-
strating that our MZI suffers from a gaussian phase av-
eraging whose variance is proportional to V 2, leading to
the single side lobe structure of the visibility.
We have used the following procedure to tune the MZI.
We first measure independently the two beam splitters’
transparencies versus their respective gate voltages, the
inner edge state being fully reflected. This is shown in
figure (2a) where the transmission (T1 or T2) through
one QPC is varied while keeping unit transparency for
the other QPC. This provides the characterization of the
transparency of each beam splitter as a function of its
gate voltage. The fact that the transmission vanishes
for large negative voltages means that the small ohmic
contact in between the two arms can absorb all incom-
ing electrons, otherwise the transmission would tend to
a finite value. This is very important in order to avoid
any spurious effect in the interference pattern. In a sec-
ond step we fix the transmission T1 to 1/2 while sweep-
ing the gate voltage of G2 (solid line of figure (2a)).
Whereas for a fully incoherent system the T should be
1/2× (R2 + T2) = 1/2, we observe large temporal trans-
mission fluctuations around 1/2. We show in the fol-
lowing that they result from the interferences, expected
in the coherent regime, but in presence of large low fre-
quency phase noise. This is revealed by the probabil-
ity distribution of the transmissions obtained when mak-
ing a large number of transmission measurements for the
same gate voltage. Figure (2b) shows a histogram of T
when making 6000 measurements (each measurement be-
ing separated from the next by 10 ms). The histogram of
the transmission fluctuations δT = T − Tmean displays
two maxima very well fitted using a probability distribu-
tion p(δT /Tmean) = 1/(2π
1− (δT /Tmean)2/V2) (the
solid line of figure (2b)). This distribution is obtained
assuming interferences δT = Tmean×Vsin(ϕ) and a uni-
form probability distribution of ϕ over [−π,+π]. Note
that the peaks around |δT /Tmean| = V have a finite
width. They correspond to the gaussian distribution as-
sociated with the detection noise which has to be convo-
luted with the previous distribution.
Although no regular oscillations of transmission can
be observed due to phase noise, we can directly extract
the visibility of the interferences by calculating the vari-
ance of the fluctuations (the approach is similar to mea-
surements of Universal Conductance Fluctuations via the
amplitude of 1/f noise in diffusive metallic wires) [10].
As expected when T1 = 1/2, the visibility extracted by
our method is proportional to
T2(1− T2), definitively
showing that fluctuations results from interference: we
are able to measure the visibility of fluctuating interfer-
ences (see figure (2c)).
The visibility depends on the bias voltage with a lobe
structure shown in figure (3), confirming the pioneer-
ing observation [5]. Nevertheless, there are marked dif-
ferences. The visibility shape is not the same as that
in ref.[5]. We have always seen only one side lobe, al-
though the sensitivity of our measurements would be high
enough to observe a second one if it existed. Moreover,
-100 -50 0 50 100
 = 0.02
 = 0.14
Drain-Source Voltage (µV)
FIG. 3: (Color online) Sample #1 : Visibility of the inter-
ferences as a function of the drain-source voltage I0h/e
2 for
three different values of T0. The curves are shifted for clar-
ity. The energy width of the lobe structure is modified by
the dilution whereas the maximum visibility at zero bias is
not modified. Solid lines are fits using equation (1). From
top to bottom, T0 = 0.02 and V0 = 31 µV, T0 = 0.14 and
V0 = 22 µV, T0 = 1 and V0 = 11.4 µV.
the lobe width (see figure (3)) can be increased by di-
luting the impinging current with G0, whereas no such
effect is seen for G1 and G2. This apparent increase of
the energy scale cannot be attributed to the addition of
a resistance in series with the MZI because G0 is close to
the MZI, at a distance shorter than the coherence length.
An almost perfect fit for the whole range of T0 (dilu-
tion), is
V = V0e−V
2/2V 2
0 |1− V ID
dID/dV
|, (1)
where V0 is a fitting parameter. Equation (1) is obtained
when assuming a gaussian phase averaging with a vari-
ance < δϕ2 > proportional to V 2 and a length difference
∆L small enough to neglect the energy dependence of
the phase in the observed energy range eV ≪ ~vD/∆L.
In such a case, the interfering part of the current I∼ is
thus proportional to ID sin(ϕ). The gaussian distribu-
tion of the phase leads to I∼ ∝ ID sin(< ϕ >)e−<δϕ
2>/2,
where < ϕ > is the mean value of the phase distribu-
tion. The measured interfering part of the transmission,
T∼ = h/e2 dI∼/dV gives a visibility corresponding to for-
mula (1) when < δϕ2 >= V 2/V 2
. Such behavior gives
a nul visibility accompanied with a π shift of the phase
when V ID/(V
dID/dV ) = 1. When T0 ∼ 1, ID is pro-
portional to V and the width of the central lobe is simply
equal to 2V0. However in the most general case, dID/dV
varies with V . One can see in figure (3) that the fit with
Equation (1) is very good, definitively showing that the
-20 0 20
0.4 0.6 0.8
 / dI
0.5 0.6 0.7
 / dI
 V  (µV)
-40 -20 0 20 40
d)  T
 = 0.06
Drain Source Voltage (µV)
FIG. 4: (Color online) Sample #2 : a) Gray plot of the
transmission T as a function of the bias voltage V and the
side gate voltage VSG. Note the π shift of the phase when
the visibility reaches 0. b) & c) T as a function of the side
gate voltage for two different values of the drain source volt-
age corresponding to the dashed line of a) (0 and 16 µV
respectively). d) Lobe structure of the visibility fitted using
equation (1) for a diluted and an undiluted impinging current.
existence of one side lobe, as observed in the experiment
of ref.[5] at ν = 2 (for the highest fields) and at ν = 1,
can be explained within our simple approach. Concern-
ing multiple side lobes, we cannot yet conclude if they do
arise from long range interaction as recently proposed by
ref.[8]. Our geometry is different from the one used in the
earlier experiment [5] and the coupling between counter
propagating edge states, thought to be responsible for
multiple side lobe [8], should be less efficient here.
To check if low frequency fluctuations have an impact
on the finite bias phase averaging, we have studied an-
other sample, with the same geometry and fabricated
simultaneously (sample #2), which exhibits clear inter-
ference pattern (see Figure (4a,b,c)). As one can remark
on figure (4d), the lobe structure is well fitted with our
theory, definitively showing that the gaussian phase av-
eraging is not associated with low frequency phase fluc-
tuations.
It is noteworthy that V0 increases (see figure (5)) with
the dilution, namely when the transmission T0 at zero
bias decreases. An impact of the dilution was already
observed as it suppressed multiple side lobes [9] (arXiv
version of Ref.[5]), but the conclusion was that the width
of the central lobe was barely affected. Here, dilution
plays a clear role whose T0 dependence is the same for
the two studied samples, once normalized to the not di-
luted case. This dilution effect is nevertheless not easy
to explain. For example, mechanisms like screening, in-
tra edge scattering and fluctuations mediated by shot
noise should have maximum effect at half transmission,
in contradiction with figure (5). More generally, it is
difficult to determine if the process responsible for the
phase averaging introduced in our model is located at
the beam splitters, or is uniformly distributed along the
interfering channels. However, setting T1 = 0.02 or 0.05,
keeping T2 = 0.5, leaves the lobe width unaffected. This
shows that, if located at the Quantum Point Contacts,
the phase averaging process is independent of transmis-
sion.
0.0 0.2 0.4 0.6 0.8 1.0
 Sample #1 : V
(1) = 13.7 µV
 Sample #2 : V
(1) = 10.6 µV
FIG. 5: (Color online) V0 obtained by fitting the visibility
with equation (1), normalized to V0 at T0 = 1, as a function
of T0 at zero bias.
To summarize, we propose a statistical method to mea-
sure the visibility of ”invisible” interferences. We observe
a single side lobe structure of the visibility on stable and
unstable samples which is shown to result from a gaus-
sian phase averaging whose variance is proportional to
V 2. Moreover, this variance is shown to be reduced by
diluting the impinging current. However, the mechanism
responsible for such type of phase averaging remains yet
unexplained.
The authors would like to thank M. Büttiker for fruit-
ful discussions. This work was supported by the French
National Research Agency (grant n◦ 2A4002).
∗ Also at LPA, Ecole Normale Supérieure, Paris.
† Electronic address: patrice.roche@cea.fr
[1] Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu,
and H. Shtrikman, Nature 422, 415 (2003).
[2] P. Samuelsson, E. V. Sukhorukov, and M. Büttiker, Phys.
Rev. Lett. 92, 026805 (2004).
[3] I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu,
and V. Umansky, arXiv:0705.0173 (2007).
[4] T. Machida, H. Hirai, S. Komiyama, T. Osada, and Y.
Shiraki, Solid State Commun. 103, 441 (1997).
mailto:patrice.roche@cea.fr
http://arxiv.org/abs/0705.0173
[5] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V.
Umansky, Phys. Rev. Lett. 96, 016804 (2006).
[6] L. V. Litvin, H.-P. Tranitz, W. Wegscheider, and C.
Strunk, Phys. Rev. B 75, 033315 (2007).
[7] V. S.-W. Chung, P. Samuelsson, and M. Büttiker, Phys.
Rev. B 72, 125320 (2005).
[8] E. V. Sukhorukov and V. V. Cheianov,
cond-mat/0609288 .
[9] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V.
Umansky, cond-mat/0508024 (2005).
[10] All the results on the visibility reported here on sam-
ple #1 have been obtained using the following proce-
dure : we measured N = 2000 times the transmission
and calculated the mean value Tmean and the variance
< δT 2 >. It is straightforward to show that the visi-
bility is V =
< δT 2 > − < δT 2 >0/Tmean, where
< δT 2 >0 is the measurement noise which depends on
the AC bias amplitude, the noise of the amplifiers and
the time constant of the lock-in amplifiers (fixed to 10
ms), measured in absence of the quantum interferences.
http://arxiv.org/abs/cond-mat/0609288
http://arxiv.org/abs/cond-mat/0508024
0.2 0.4 0.6
0.6 d)
Magnetic Field - 4.6 T (mT)
0.2 0.4 0.6
0.6 c)
0.2 0.4 0.6

  We present an original statistical method to measure the visibility of
interferences in an electronic Mach-Zehnder interferometer in the presence of
low frequency fluctuations. The visibility presents a single side lobe
structure shown to result from a gaussian phase averaging whose variance is
quadratic with the bias. To reinforce our approach and validate our statistical
method, the same experiment is also realized with a stable sample. It exhibits
the same visibility behavior as the fluctuating one, indicating the intrinsic
character of finite bias phase averaging. In both samples, the dilution of the
impinging current reduces the variance of the gaussian distribution.

Finite bias visibility of the electronic Mach-Zehnder interferometer
Preden Roulleau, F. Portier, D. C. Glattli,∗ and P. Roche†
Nanoelectronic group, Service de Physique de l’Etat Condensé,
CEA Saclay, F-91191 Gif-Sur-Yvette, France
A. Cavanna, G. Faini, U. Gennser, and D. Mailly
CNRS, Phynano team,
Laboratoire de Photonique et Nanostructures,
Route de Nozay, F-91460 Marcoussis, France
(Dated: November 28, 2018)
We present an original statistical method to measure the visibility of interferences in an electronic
Mach-Zehnder interferometer in the presence of low frequency fluctuations. The visibility presents
a single side lobe structure shown to result from a gaussian phase averaging whose variance is
quadratic with the bias. To reinforce our approach and validate our statistical method, the same
experiment is also realized with a stable sample. It exhibits the same visibility behavior as the
fluctuating one, indicating the intrinsic character of finite bias phase averaging. In both samples,
the dilution of the impinging current reduces the variance of the gaussian distribution.
PACS numbers: 85.35.Ds, 73.43.Fj
Nowadays quantum conductors can be used to per-
form experiments usually done in optics, where electron
beams replace photon beams. A beamlike electron mo-
tion can be obtained in the Integer Quantum Hall Effect
(IQHE) regime using a high mobility two dimensional
electron gas in a high magnetic field at low temperature.
In the IQHE regime, one-dimensional gapless excitation
modes form, which correspond to electrons drifting along
the edge of the sample. The number of these so-called
edge channels corresponds to the number of filled Lan-
dau levels in the bulk. The chirality of the excitations
yields long collision times between quasi-particles, mak-
ing edge states very suitable for quantum interferences
experiments like the electronic Mach-Zehnder interfer-
ometer (MZI) [1, 2, 3]. Surprisingly, despite some ex-
periments which show that equilibrium length in chiral
wires is rather long [4], very little is known about the
coherence length or the phase averaging in these ”per-
fect” chiral uni-dimensional wires. In particular, while in
the very first interference MZI experiment the interfer-
ence visibility showed a monotonic decrease with voltage
bias, which was attributed to phase noise [1], in a more
recent paper, a surprising non-monotonic decrease with
a lobe structure was observed [5]. A satisfactory expla-
nation has not yet been found, and the experiment has
so far not been reported by other groups to confirm these
results.
We report here on an original method to measure the
visibility of interferences in a MZI, when low frequency
phase fluctuations prevent direct observation of the peri-
odic interference pattern obtained by changing the mag-
netic flux through the MZI. We studied the visibility at
finite energy and observed a single side lobe structure,
which can be explained by a gaussian phase averaging
whose variance is proportional to V 2, where V is the
FIG. 1: SEM view of the electronic Mach-Zehnder with a
schematic representation of the edge state. G0, G1, G2 are
quantum point contacts which mimic beam splitters. The
pairs of split gates defining a QPC are electrically connected
via a Au metallic bridge deposited on an insolator (SU8). G0
allows a dilution of the impinging current, G1 and G2 are the
two beam splitters of the Mach-Zehnder interferometer. SG
is a side gate which allows a variation of the length of the
lower path (b).
bias voltage. To reinforce our result and check if low
frequency fluctuation may be responsible for that behav-
ior, we realized the same experiment on a stable sample
: we also observed a single side lobe structure which can
be fitted with our approach of gaussian phase averaging.
This proves the validity of the results, which cannot be
an artefact due to the low frequency phase fluctuations
in the first sample. In both samples, the dilution of the
impinging current has an unexpected effect : it decreases
the variance of the gaussian distribution.
The MZI geometry is patterned using e-beam lithogra-
phy on a high mobility two dimensional electron gas in a
GaAs/Ga1−xAlxAs heterojunction with a sheet density
nS = 2.0×1011 cm−2 and a mobility of 2.5×106 cm2/Vs.
http://arxiv.org/abs/0704.0746v3
The experiment was performed in the IQHE regime at
filling factor ν = nSh/eB = 2 (magnetic field B =5.2
Tesla). Transport occurs through two edge states with an
extremely large energy redistribution length [4]. Quan-
tum point contacts (QPC) controlled by gates G0, G1
and G2 define electronic beam splitters with transmis-
sions T0, T1 and T2 respectively. In all the results pre-
sented here, the interferences were studied on the outer
edge state schematically drawn as black lines in Fig.(1),
the inner edge state being fully reflected by all the QPCs.
The interferometer consists of G1, G2 and the small cen-
tral ohmic contact in between the two arms. G1 splits
the incident beam into two trajectories (a) and (b), which
are recombined with G2 leading to interferences. The
two arms defined by the mesa are 8 µm long and en-
close a 14 µm2 area. The current which is not transmit-
ted through the MZI, IB = ID − IT , is collected to the
ground with the small ohmic contact. An additional gate
SG allows a change of the length of the trajectory (b).
The impinging current I0 can be diluted thanks to the
beam splitter G0 whose transmission T0 determines the
diluted current dID = T0 × dI0. We measure the differ-
ential transmission through the MZI by standard lock-in
techniques using a 619 Hz frequency 5 µVrms AC bias
VAC superimposed to the DC voltage V . This AC bias
modulates the incoming current dID = T0 × h/e2 ×VAC ,
and thus the transmitted current in an energy range close
to eV , giving the transmission T (eV ) = dIT /dI0.
Using the single particle approach of the Landauer-
Büttiker formalism, the transmission amplitude t
through the MZI is the sum of the two complex
transmission amplitudes corresponding to paths (a)
and (b) of the interferometer; t = t0{t1 exp(iφa)t2 −
r1 exp(iφb)r2}. This leads to a transmission probabil-
ity T (ǫ) = T0{T1T2 + R1R2 +
T1R2R1T2 sin[ϕ(ǫ)]},
where ϕ(ǫ) = φa − φb and Ti = |ti|2 = 1 −Ri. ϕ(ǫ) cor-
responds to the total Aharonov-Bohm (AB) flux across
the surface S(ǫ) defined by the arms of the MZI, ϕ(ǫ) =
2πS(ǫ) × eB/h. The surface S depends on the energy ǫ
when there is a finite length difference ∆L = La−Lb be-
tween the two arms. This leads to a variation of the phase
with the energy, ϕ(ǫ+EF ) = ϕ(EF )+ǫ∆L/(~vD), where
vD is the drift velocity. When varying the AB flux, the
interferences manifest themselves as oscillations of the
transmission; in practice this is done either by varying
the magnetic field or by varying the surface of the MZI
with a side gate [1, 5, 6]. The visibility of the interfer-
ences defined as V = (TMAX−TMIN )/(TMAX+TMIN ), is
maximum when both beam splitter transmission are set
to 1/2. In the present experiment the MZI is designed
with equal arm lengths (∆L = 0) and the visibility is not
expected to be sensitive to the coherence length of the
source ~vD/max(kBT, eVAC). Thus the visibility pro-
vides a direct measurement of the decoherence and/or
phase averaging in this quantum circuit.
In Ref.[1], 60% visibility was observed at low tem-
-0.76 -0.72 -0.68 -0.64
-0.4 -0.2 0.0 0.2 0.4
2 x Visibility
δT / T
0.0 0.2 0.4 0.6 0.8 1.0
Transmission T
gate voltage (V
 or V
) (Volt)
FIG. 2: Sample #1 a)Transmission T = dIT /dI0 as a func-
tion of the gate voltages V 1 and V 2 applied on G1 and G2.
(◦) T = T1 versus V 1. (•) T = T2 versus V 2. The solid
line is the transmission T obtained with T1 fixed to 1/2 while
sweeping V 2 : transmission fluctuations due to interferences
with low frequency phase noise appears. b) Stack histogram
on 6000 successive transmission measurements as a function
of the normalized deviation from the mean value. The solid
line is the distribution of transmission expected for a uniform
distribution of phases. c)Visibility of interferences as a func-
tion of the transmission T2 when T1 = 1/2. The solid line is
T2(1− T2) dependence predicted by the theory.
perature, showing that the quantum coherence length
can be at least as large as several micrometer at 20 mK
(and probably larger if phase averaging is the limiting
factor). At finite energy (compared to the Fermi en-
ergy), the visibility was also found decreasing with the
bias voltage[1, 5, 6]. This effect is not due to an increase
of the coherence length of the electron source which re-
mains determined by eVAC or kBT [7]. In a first exper-
iment, a monotonic visibility decrease was found, which
was attributed to phase averaging, as confirmed by shot
noise measurements [1]. Nevertheless, it remains unclear
why and how the phase averaging increases with the bias.
In a recent paper, instead of a monotonic decrease of the
visibility, a lobe structure was observed for filling factor
less than 1 in the QPCs [5]. No non-interacting electron
model was found to be able to explain this observation,
and although interaction effects have been proposed [8],
a satisfactory explanation has not yet been found to ac-
count for all the experimental observations. So far, two
experiments have shown to two different behaviors, rais-
ing questions about the universality of these observations.
Here, we report experiments where different samples give
consistent results, with a fit to the data clearly demon-
strating that our MZI suffers from a gaussian phase av-
eraging whose variance is proportional to V 2, leading to
the single side lobe structure of the visibility.
We have used the following procedure to tune the MZI.
We first measure independently the two beam splitters’
transparencies versus their respective gate voltages, the
inner edge state being fully reflected. This is shown in
figure (2a) where the transmission (T1 or T2) through
one QPC is varied while keeping unit transparency for
the other QPC. This provides the characterization of the
transparency of each beam splitter as a function of its
gate voltage. The fact that the transmission vanishes
for large negative voltages means that the small ohmic
contact in between the two arms can absorb all incom-
ing electrons, otherwise the transmission would tend to
a finite value. This is very important in order to avoid
any spurious effect in the interference pattern. In a sec-
ond step we fix the transmission T1 to 1/2 while sweep-
ing the gate voltage of G2 (solid line of figure (2a)).
Whereas for a fully incoherent system the T should be
1/2× (R2 + T2) = 1/2, we observe large temporal trans-
mission fluctuations around 1/2. We show in the fol-
lowing that they result from the interferences, expected
in the coherent regime, but in presence of large low fre-
quency phase noise. This is revealed by the probabil-
ity distribution of the transmissions obtained when mak-
ing a large number of transmission measurements for the
same gate voltage. Figure (2b) shows a histogram of T
when making 6000 measurements (each measurement be-
ing separated from the next by 10 ms). The histogram of
the transmission fluctuations δT = T − Tmean displays
two maxima very well fitted using a probability distribu-
tion p(δT /Tmean) = 1/(2π
1− (δT /Tmean)2/V2) (the
solid line of figure (2b)). This distribution is obtained
assuming interferences δT = Tmean×Vsin(ϕ) and a uni-
form probability distribution of ϕ over [−π,+π]. Note
that the peaks around |δT /Tmean| = V have a finite
width. They correspond to the gaussian distribution as-
sociated with the detection noise which has to be convo-
luted with the previous distribution.
Although no regular oscillations of transmission can
be observed due to phase noise, we can directly extract
the visibility of the interferences by calculating the vari-
ance of the fluctuations (the approach is similar to mea-
surements of Universal Conductance Fluctuations via the
amplitude of 1/f noise in diffusive metallic wires) [10].
As expected when T1 = 1/2, the visibility extracted by
our method is proportional to
T2(1− T2), definitively
showing that fluctuations results from interference: we
are able to measure the visibility of fluctuating interfer-
ences (see figure (2c)).
The visibility depends on the bias voltage with a lobe
structure shown in figure (3), confirming the pioneer-
ing observation [5]. Nevertheless, there are marked dif-
ferences. The visibility shape is not the same as that
in ref.[5]. We have always seen only one side lobe, al-
though the sensitivity of our measurements would be high
enough to observe a second one if it existed. Moreover,
-100 -50 0 50 100
 = 0.02
 = 0.14
Drain-Source Voltage (µV)
FIG. 3: (Color online) Sample #1 : Visibility of the inter-
ferences as a function of the drain-source voltage I0h/e
2 for
three different values of T0. The curves are shifted for clar-
ity. The energy width of the lobe structure is modified by
the dilution whereas the maximum visibility at zero bias is
not modified. Solid lines are fits using equation (1). From
top to bottom, T0 = 0.02 and V0 = 31 µV, T0 = 0.14 and
V0 = 22 µV, T0 = 1 and V0 = 11.4 µV.
the lobe width (see figure (3)) can be increased by di-
luting the impinging current with G0, whereas no such
effect is seen for G1 and G2. This apparent increase of
the energy scale cannot be attributed to the addition of
a resistance in series with the MZI because G0 is close to
the MZI, at a distance shorter than the coherence length.
An almost perfect fit for the whole range of T0 (dilu-
tion), is
V = V0e−V
2/2V 2
0 |1− V ID
dID/dV
|, (1)
where V0 is a fitting parameter. Equation (1) is obtained
when assuming a gaussian phase averaging with a vari-
ance < δϕ2 > proportional to V 2 and a length difference
∆L small enough to neglect the energy dependence of
the phase in the observed energy range eV ≪ ~vD/∆L.
In such a case, the interfering part of the current I∼ is
thus proportional to ID sin(ϕ). The gaussian distribu-
tion of the phase leads to I∼ ∝ ID sin(< ϕ >)e−<δϕ
2>/2,
where < ϕ > is the mean value of the phase distribu-
tion. The measured interfering part of the transmission,
T∼ = h/e2 dI∼/dV gives a visibility corresponding to for-
mula (1) when < δϕ2 >= V 2/V 2
. Such behavior gives
a nul visibility accompanied with a π shift of the phase
when V ID/(V
dID/dV ) = 1. When T0 ∼ 1, ID is pro-
portional to V and the width of the central lobe is simply
equal to 2V0. However in the most general case, dID/dV
varies with V . One can see in figure (3) that the fit with
Equation (1) is very good, definitively showing that the
-20 0 20
0.4 0.6 0.8
 / dI
0.5 0.6 0.7
 / dI
 V  (µV)
-40 -20 0 20 40
d)  T
 = 0.06
Drain Source Voltage (µV)
FIG. 4: (Color online) Sample #2 : a) Gray plot of the
transmission T as a function of the bias voltage V and the
side gate voltage VSG. Note the π shift of the phase when
the visibility reaches 0. b) & c) T as a function of the side
gate voltage for two different values of the drain source volt-
age corresponding to the dashed line of a) (0 and 16 µV
respectively). d) Lobe structure of the visibility fitted using
equation (1) for a diluted and an undiluted impinging current.
existence of one side lobe, as observed in the experiment
of ref.[5] at ν = 2 (for the highest fields) and at ν = 1,
can be explained within our simple approach. Concern-
ing multiple side lobes, we cannot yet conclude if they do
arise from long range interaction as recently proposed by
ref.[8]. Our geometry is different from the one used in the
earlier experiment [5] and the coupling between counter
propagating edge states, thought to be responsible for
multiple side lobe [8], should be less efficient here.
To check if low frequency fluctuations have an impact
on the finite bias phase averaging, we have studied an-
other sample, with the same geometry and fabricated
simultaneously (sample #2), which exhibits clear inter-
ference pattern (see Figure (4a,b,c)). As one can remark
on figure (4d), the lobe structure is well fitted with our
theory, definitively showing that the gaussian phase av-
eraging is not associated with low frequency phase fluc-
tuations.
It is noteworthy that V0 increases (see figure (5)) with
the dilution, namely when the transmission T0 at zero
bias decreases. An impact of the dilution was already
observed as it suppressed multiple side lobes [9] (arXiv
version of Ref.[5]), but the conclusion was that the width
of the central lobe was barely affected. Here, dilution
plays a clear role whose T0 dependence is the same for
the two studied samples, once normalized to the not di-
luted case. This dilution effect is nevertheless not easy
to explain. For example, mechanisms like screening, in-
tra edge scattering and fluctuations mediated by shot
noise should have maximum effect at half transmission,
in contradiction with figure (5). More generally, it is
difficult to determine if the process responsible for the
phase averaging introduced in our model is located at
the beam splitters, or is uniformly distributed along the
interfering channels. However, setting T1 = 0.02 or 0.05,
keeping T2 = 0.5, leaves the lobe width unaffected. This
shows that, if located at the Quantum Point Contacts,
the phase averaging process is independent of transmis-
sion.
0.0 0.2 0.4 0.6 0.8 1.0
 Sample #1 : V
(1) = 13.7 µV
 Sample #2 : V
(1) = 10.6 µV
FIG. 5: (Color online) V0 obtained by fitting the visibility
with equation (1), normalized to V0 at T0 = 1, as a function
of T0 at zero bias.
To summarize, we propose a statistical method to mea-
sure the visibility of ”invisible” interferences. We observe
a single side lobe structure of the visibility on stable and
unstable samples which is shown to result from a gaus-
sian phase averaging whose variance is proportional to
V 2. Moreover, this variance is shown to be reduced by
diluting the impinging current. However, the mechanism
responsible for such type of phase averaging remains yet
unexplained.
The authors would like to thank M. Büttiker for fruit-
ful discussions. This work was supported by the French
National Research Agency (grant n◦ 2A4002).
∗ Also at LPA, Ecole Normale Supérieure, Paris.
† Electronic address: patrice.roche@cea.fr
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mailto:patrice.roche@cea.fr
http://arxiv.org/abs/0705.0173
[5] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V.
Umansky, Phys. Rev. Lett. 96, 016804 (2006).
[6] L. V. Litvin, H.-P. Tranitz, W. Wegscheider, and C.
Strunk, Phys. Rev. B 75, 033315 (2007).
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Rev. B 72, 125320 (2005).
[8] E. V. Sukhorukov and V. V. Cheianov,
cond-mat/0609288 .
[9] I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V.
Umansky, cond-mat/0508024 (2005).
[10] All the results on the visibility reported here on sam-
ple #1 have been obtained using the following proce-
dure : we measured N = 2000 times the transmission
and calculated the mean value Tmean and the variance
< δT 2 >. It is straightforward to show that the visi-
bility is V =
< δT 2 > − < δT 2 >0/Tmean, where
< δT 2 >0 is the measurement noise which depends on
the AC bias amplitude, the noise of the amplifiers and
the time constant of the lock-in amplifiers (fixed to 10
ms), measured in absence of the quantum interferences.
http://arxiv.org/abs/cond-mat/0609288
http://arxiv.org/abs/cond-mat/0508024
0.2 0.4 0.6
0.6 d)
Magnetic Field - 4.6 T (mT)
0.2 0.4 0.6
0.6 c)
0.2 0.4 0.6
