Experimental Test of the High-Frequency Quantum Shot Noise Theory in a
  Quantum Point Contact
Experimental test of the high frequency quantum shot noise theory in a Quantum
Point Contact
E. Zakka-Bajjani, J. Ségala, F. Portier,∗ P. Roche, and D. C. Glattli†
Nanoelectronic group, Service de Physique de l’Etat Condensé,
CEA Saclay, F-91191 Gif-Sur-Yvette, France
A. Cavanna and Y. Jin
CNRS, Laboratoire de Photonique et Nanostructures ,
Route de Nozay, F-91460 Marcoussis, France
(Dated: November 9, 2018)
We report on direct measurements of the electronic shot noise of a Quantum Point Contact
(QPC) at frequencies ν in the range 4-8 GHz. The very small energy scale used ensures energy
independent transmissions of the few transmitted electronic modes and their accurate knowledge.
Both the thermal energy and the QPC drain-source voltage Vds are comparable to the photon
energy hν leading to observation of the shot noise suppression when Vds < hν/e. Our measurements
provide the first complete test of the finite frequency shot noise scattering theory without adjustable
parameters.
PACS numbers: 73.23.-b,73.50.Td,42.50.-p,42.50.Ar
Pauli’s exclusion principle has striking consequences on
the properties of quantum electrical conductors. In an
ideal quantum wire, it is responsible for the quantization
of the conductance by requiring that at most one electron
(or two for spin degeneracy) occupies the regularly time-
spaced wave-packets emitted by the contacts and propa-
gating in the wire [1]. Concurrently, at zero temperature,
the electron flow is noiseless [2, 3] as can be observed in
ballistic conductors [4, 5, 6]. In more general quantum
conductors, static impurities diffract the noiseless elec-
trons emitted by the contacts. This results in a partition
of the electrons between transmitted or reflected states,
generating quantum shot noise [1, 2, 3, 7, 8]. However,
Pauli’s principle possesses more twists to silence elec-
trons. At finite frequency ν, detection of current fluctu-
ations in an external circuit at zero temperature requires
emission of photons corresponding to a finite energy cost
hν [9]. For drain-source contacts biased at voltage Vds,
a sharp suppression is expected to occur when the pho-
ton energy hν is larger than eVds as an electron emit-
ted by the source can not find an empty state in the
drain to emit such a photon [9, 10, 11]. Another striking
consequence of Pauli’s principle is the prediction of non-
classical photon emission for a conductor transmitting
only one or few electronic modes. It has been shown that
in the frequency range eVds/2h < ν < eVds/h, the popu-
lation of a photon mode obeys a sub-Poissonian statistics
inherited from the electrons [12]. Investigating quantum
shot noise in this high frequency regime using a Quan-
tum Point Contact (QPC) to transmit few modes is thus
highly desirable.
The first step is to check the validity of the above pre-
diction based on a non-interacting picture of electrons.
For 3D or 2D wide conductors with many quantum chan-
nels which are good Fermi liquids, one expects this non-
interacting picture to work well. Indeed, the eVds/h sin-
gularity has been observed in a 3D diffusive wire in the
shot noise derivative with respect to bias voltage [13].
However, for low dimensional systems like 1D wires or
conductors transmitting one or few channels, electron in-
teractions give non-trivial effects. Long 1D wires defined
in 2D electron gas or Single Wall Carbon Nanotubes be-
come Luttinger liquids. Long QPCs exhibit a 0.7 con-
ductance anomaly [14], and a low frequency shot noise
[15] compatible with Kondo physics [16]. Consequently,
new characteristic frequencies may appear in shot noise
reflecting electron correlations. Another possible failure
of the non-interacting finite frequency shot noise model
could be the back-action of the external circuit. For high
impedance circuits, current fluctuations implies potential
fluctuations at the contacts [17]. Also, the finite time re-
quired to eliminate the sudden drain-source charge build-
up after an electron have passed through the conductor
leads to a dynamical Coulomb blockade for the next elec-
tron to tunnel. A peak in the shot noise spectrum at the
electron correlation frequency I/e is predicted for a tun-
nel junction connected to a capacitive circuit [18]. Other
timescales may also be expected which affect both con-
ductance [19] and noise [20] due to long range Coulomb
interaction or electron transit time. This effects have
been recently observed for the conductance [21] .
The present work aims at giving a clear-cut test of
the non-interacting scattering theory of finite frequency
shot noise using a Quantum Point Contact transmitting
only one or two modes in a weak interaction regime. It
provides the missing reference mark to which further ex-
periments in strong interaction regime can be compared
in the future. We find the expected shot noise suppres-
sion for voltages ≤ hν/e in the whole 4-8 GHz frequency
range. The data taken for various transmissions perfectly
http://arxiv.org/abs/0704.0907v4
agree with the finite temperature, non-interacting model
with no adjustable parameter. In addition to provide
a stringent test of the theory, the technique developed
is the first step toward the generation of non-classical
photons with QPCs in the microwave range [12]. The
detection technique uses cryogenic linear amplification
followed by room temperature detection. The electron
temperature much lower than hν/kB, the small energy
scale used (eVds ≪ 0.02EF ) ensuring energy independent
transmissions, the high detection sensitivity, and the ab-
solute calibration allow for direct comparison with the-
ory without adjustable parameters. Our technique differs
from the recent QPC high frequency shot noise measure-
ments using on-chip Quantum Dot detection in the 10-
150 GHz frequency range [22]. Although most QPC shot
noise features were qualitatively observed validating this
promising method, the lack of independent determina-
tion of the QPC-Quantum Dot coupling, and the large
voltage used from 0.05 to 0.5EF making QPC transmis-
sions energy dependent, prevent quantitative comparison
with shot noise predictions. However, Quantum Dot de-
tectors can probe the vacuum fluctuations via the stim-
ulated noise while the excess noise detected here only
probes the emission noise [9, 10].
The experimental set-up is represented in fig. 1. A
two-terminal conductor made of a QPC realized in a
2DEG in GaAs/GaAlAs heterojunction is cooled at 65
mK by a dilution refrigerator and inserted between two
transmission lines. The sample characteristics are a 35
nm deep 2DEG with 36.7 m2V−1s−1 mobility and 4.4
1015 m−2 electron density. Interaction effects have been
minimized by using a very short QPC showing no sign
of 0.7 conductance anomaly. In order to increase the
sensitivity, we use the microwave analog of an optical re-
flective coating. The contacts are separately connected
to 50 Ω coaxial transmission lines via two quarter wave
length impedance adapters, raising the effective input
impedance of the detection lines to 200 Ω over a one
octave bandwidth centered on 6 GHz. The 200 Ω elec-
tromagnetic impedance is low enough to prevent dynam-
ical Coulomb blockade but large enough for good cur-
rent noise sensitivity. The transmitted signals are then
amplified by two cryogenic Low Noise Amplifiers (LNA)
with Tnoise ≃ 5K. Two rf-circulators, thermalized at
mixing chamber temperature protect the sample from
the current noise of the LNA and ensure a circuit en-
vironment at base temperature. After further amplifi-
cation and eventually narrow bandpass filtering at room
temperature, current fluctuations are detected using two
calibrated quadratic detectors whose output voltage is
proportional to noise power. Up to a calculable gain
factor, the detected noise power contains the weak sam-
ple noise on top of a large additional noise generated by
the cryogenic amplifiers. In order to remove this back-
ground, we measure the excess noise ∆SI(ν, T, Vds) =
SI(ν, T, Vds) − SI(ν, T, 0). Practically, this is done by
4.2 K 
adjustable 
filters
VDrain-Source
50/200 Ω
λ/4 adapter
65 mK
circulator
FIG. 1: Schematic diagram of the measurement set-up. See
text for details.
applying a 93 Hz 0-Vds square-wave bias voltage on the
sample through the DC input of a bias-T, and detecting
the first harmonic of the square-wave noise response of
the detectors using lock-in techniques. In terms of noise
temperature referred to the 50 Ω input impedance, an
excess noise ∆SI(ν, T, Vds) gives rise to an excess noise
temperature
∆T 50Ωn (ν, T, Vds) =
ZeffZ
sample∆SI(ν, T, Vds)
(2Zeff + Zsample)2
. (1)
Eq. 1 demonstrates the advantage of impedance match-
ing : in the high source impedance limit Zsample ≫ Zeff ,
the increase in noise temperature due to shot noise is
proportional to Zeff . Our set up (Zeff = 200Ω) is thus
four times more efficient than a direct connection of the
sample to standard 50 Ω transmission lines. Finally, the
QPC differential conductance G is simultaneously mea-
sured through the DC input of the bias-Tee using low
frequency lock-in technique.
The very first step in the experiment is to characterize
the QPC. The inset of fig. 4 shows the differential con-
ductance versus gate voltage when the first two modes
are transmitted. As the experiment is performed at zero
magnetic field, the conductance exhibits plateaus quan-
tized in units of G0 = 2e
2/h. The short QPC length
(80 nm) leads to a conductance very linear with the
low bias voltage used (δG/G ≤ 6% for Vds ≤ 80µV for
G ≃ 0.5G0). It is also responsible for a slight smooth-
ing of the plateaus. Each mode transmission is extracted
from the measured conductance (open circles) by fitting
with the saddle point model (solid line) [23].
We then set the gate voltage to obtain a single mode
at half transmission corresponding to maximum electron
partition (G ≃ 0.5G0). Fig. 2 shows typical excess noise
measured at frequencies 4.22 GHz and 7.63 GHz and
bandwidth 90 MHz and 180 MHz. We note a striking
suppression of shot noise variation at low bias voltage,
and that the onset of noise increases with the measure-
ment frequency. This is in agreement with the photon
suppression of shot noise in a non-interacting system.
-80 -60 -40 -20 0 20 40 60 80
2 X V
(4.22 GHz)
2 X V
(7.63 GHz)
Drain Source Voltage V
 (µV)
 4.22 GHz
 7.63 GHz
FIG. 2: Color Online. Excess noise temperature as a func-
tion of bias voltage, measured at 4.22 GHz (open circles) and
7.63 GHz (open triangles). The dashed lines represent the
linear fits to the data, from which the threshold V0 is de-
duced. The solid lines represent the expected excess noise
SI(ν, Te(Vds), Vds) − SI(ν, Te(0), 0), using Te(Vds) obtained
from eq. 5. The frequency dependent coupling is the only
fitting parameter.
The expected excess noise reads
∆SI(ν, T, Vds) = 2G0
Di(1−Di)
hν − eVds
e(hν−eVds)/kBT − 1
hν + eVds
e(hν+eV )/kBT − 1
ehν/kBT − 1
. (2)
It shows a zero temperature singularity at eVds = hν :
∆SI(ν, T, Vds) = 2G0
i Di(1−Di)(eVds−hν) if eVds >
hν and 0 otherwise. At finite temperature, the singular-
ity is thermally rounded. At high bias (eVds ≫ hν, kBT ),
equation 2 gives an excess noise
∆SI(ν, T, Vds) = 2G0
iDi(1−Di) (eVds − eV0) (3)
with eV0 = hν coth (hν/2kBT ) . (4)
In the low frequency limit, the threshold V0 charac-
terizes the transition between thermal noise and shot
noise (eV0 = 2kBT ), whereas in the low temperature
limit, it marks the onset of photon suppressed shot noise
(eV0 = hν). As shown on fig. 2, V0 is determined by the
intersection of the high bias linear regression of the mea-
sured excess noise and the zero excess noise axis. Fig. 3
shows V0 for eight frequencies spanning in the 4-8 GHz
range for G ≃ 0.5G0 . Eq. 4 gives a very good fit to
the experimental data. The only fitting parameter is the
electronic temperature Te = 72 mK, very close to the
fridge temperature Tfridge = 65 mK. We will show that
electron heating can account for this small discrepancy.
To get a full comparison with theory, we now inves-
tigate the influence of the transmissions of the first two
electronic modes of the QPC. To do so, we repeat the
same experiment at fixed frequency (here we used a 5.4-
5.9 GHz filter) for different sample conductances. The
0 5 10 15 20 25 30 35
: Experiment
 Fit to theory
          yields T
 = 72mK
          (fridge temp = 65 mK)
hν/e (µV)
0 2 4 6 8
 Observation Frequency ν (GHz)
2 /k T e
asymptote
V h eν=
FIG. 3: Onset V0 as a function of the observation frequency.
The experimental uncertainty corresponds to the size of the
symbols. The dashed lines correspond to the low (eV0 =
2kBT ) and high (eV0 = hν) frequency limits, and the solid
line is a fit to theory, with the electronic temperature as only
fitting parameter.
0,0 0,5 1,0 1,5 2,0
-0,5 -0,4 -0,3
Gate Voltage(V)
FIG. 4: Open circles: d∆SI/d(eVds) deduced from ∆T
Full line : theoretical prediction. The only fitting parameter
is the microwave attenuation. The experimental uncertainty
corresponds to the size of the symbols. Inset : Open circles :
conductance of the QPC as a function of gate voltage. Solid
Line : fit with the saddle point model [23].
noise suppression at Vds ≤ hν/e is the only singularity
we observe, independently of the QPC conductance G.
Fig. 4 shows the derivative with respect to eVds of the
excess noise d∆SI/d(eVds) deduced from the excess noise
temperature measured between 50 µV and 80 µV. This
energy range is chosen so that eVds is greater than hν
by at least 5kBTfridge over the entire frequency range.
The data agree qualitatively with the expected D(1−D)
dependence of pure shot noise, showing maxima at con-
ductances G = 0.5G0, and G = 1.5G0, and minima at
conductances G = G0 and G = 2G0. The short QPC
is responsible for the non zero minima as, when the sec-
ond mode starts to transmit electrons, the first one has
not reached unit transmission (inset of fig. 4). How-
ever, eq. 2 is not compatible with a second maximum
higher than the first one, which is due to electron heating.
The dimensions of the 2-DEG being much larger than
the electron-electron energy relaxation length, but much
smaller than electron-phonon energy relaxation length,
there is a gradient of electronic temperature from the
QPC to the ohmic metallic contacts assumed at the fridge
temperature. Combining the dissipated power IVds with
the Wiedemann-Franz law, one gets [5, 24]
T 2e = T
fridge +
where Gm stands for the total conductance of the 2D
leads, estimated from measurements to be 12 mS ±20%.
The increased noise temperature is then due to both
shot noise and to the increased thermal noise. For a
fridge temperature of 65 mK and G = G0/2, the elec-
tronic temperature will increase from 69 mK to 77 mK
as Vds increases from 50 µV to 80 µV. This accounts
for the small discrepancy between the fridge temper-
ature and the electron temperature deduced from the
variation of V0 with frequency. As G increases, the ef-
fect is more important, as can be seen both in fig. 4
and eq. 5. The solid line in figure 4 gives the av-
erage derivative with respect to eVds of the total ex-
pected excess noise SI(ν, Te(Vds), Vds) − SI(ν, Te(0), 0),
using the attenuation of the signal as a free parame-
ter. The agreement is quite satisfactory, given the ac-
curacy of the saddle point model description of the QPC
transmission. We find a 4.7 dB attenuation, which is
in good agreement with the expected 4 ±1 dB deduced
from calibration of the various elements of the detection
chain. Moreover, the voltage dependent electron temper-
ature obtained from eq. 5 can also be used to evaluate
SI(ν, Te(Vds), Vds) − SI(ν, Te(0), 0) as a function of Vds
at fixed sample conductance G = 0.5G0. The result, as
shown by the solid lines of fig. 2, is in excellent agreement
with experimental observations.
In conclusion, we performed the first direct measure-
ment of the finite frequency shot noise of the simplest
mesoscopic system, a QPC. Accurate comparison of the
data with non-interacting shot noise predictions have
been done showing perfect quantitative agreement. Even
when a single mode is transmitted, no sign of devia-
tion related to interaction was found, as expected for
the experimental parameters chosen for this work. We
have also shown that accurate and reliable high frequency
shot noise measurements are now possible for conductors
with impedance comparable to the conductance quan-
tum. This opens the way to high frequency shot noise
characterization of Carbon Nanotubes, Quantum Dots
or Quantum Hall samples in a regime where microscopic
frequencies are important and will encourage further the-
oretical work in this direction. Our set-up will also allow
to probe the statistics of photons emitted by a phase co-
herent single mode conductor.
It is a pleasure to thank D. Darson, C. Ulysse, P.
Jacques and C. Chaleil for valuable help in the construc-
tion of the experiments, P. Roulleau for technical help,
and X. Waintal for useful discussions.
∗ Electronic address: fabien.portier@cea.fr
† Also at LPA, Ecole Normale Supérieure, Paris.
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mailto:fabien.portier@cea.fr
http://arxiv.org/abs/cond-mat/0703123
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http://arxiv.org/abs/0705.3166
