Measurement of finite-frequency current statistics in a single-electron transistor

7.5 MB·English

by  Niels Ubbelohde

Measurement of finite-frequency current statistics in a single-electron transistor
Niels Ubbelohde,1 Christian Fricke,1 Christian Flindt,2,3 Frank Hohls,1,4 and Rolf J. Haug1
1Institut fu¨r Festk¨orperphysik, Leibniz Universita¨t Hannover, 30167 Hannover, Germany
2D´epartement de Physique Th´eorique, Universit´e de Gen`eve, 1211 Gen`eve, Switzerland
3Department of Physics, Harvard University, Cambridge, MA 02138, USA
4Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany
(Dated: January 11, 2012)
Electron transport in nano-scale structures is acterize the transport process. In a zero-frequency mea-
2 strongly influenced by the Coulomb interaction surement, correlation effects are integrated over a long
1
which gives rise to correlations in the stream period of time and important information about charac-
0
of charges and leaves clear fingerprints in the teristictimescalesarelost. Toobservethedynamical fea-
2
fluctuations of the electrical current. A complete tures of the correlations, finite-frequency measurements
n understanding of the underlying physical pro- arerequired.2,3,28–34 Ithasevenbeenshowntheoretically
a
cesses requires measurements of the electrical that the frequency-dependent third cumulant (the skew-
J
fluctuations on all time and frequency scales, ness) of the current may contain additional information
0
but experiments have so far been restricted to about correlations and internal time scales of a system
1
fixed frequency ranges as broadband detection comparedtothefinite-frequencynoisealone,forinstance

] of current fluctuations is an inherently difficult in chaotic cavities29 and diffusive conductors.30
l
l experimental procedure. Here we demonstrate Figure 1a-c shows a typical time trace of the currents
a
h that the electrical fluctuations in a single electron inournano-scaleSETconsistingofaquantumdot(QD)
- transistor (SET) can be accurately measured on coupled via tunneling barriers to source and drain elec-
s all relevant frequencies using a nearby quantum trodes. TheQDisoperatedclosetoachargedegeneracy
e
m point contact for on-chip real-time detection of point in the Coulomb blockade regime, where only a sin-
the current pulses in the SET. We have directly gleadditionalelectronatatimecanenterfromthesource
.
t measured the frequency-dependent current electrode and leave via the drain electrode. The applied
a
m statistics and hereby fully characterized the voltagebiaseV islargerthantheelectronictemperature
fundamental tunneling processes in the SET. k T which prevents electrons from tunneling in the op-
- B
d Our experiment paves the way for future inves- posite direction of the mean current. While a stream
n tigations of interaction and coherence induced of electrons is driven through the QD, a separate cur-
o correlation effects in quantum transport. rentispassedthroughthenearbyquantumpointcontact
c
(QPC)whoseconductanceishighlysensitivetothepres-
[
The electrical fluctuations in a nano-scale conduc- ence of additional electrons (∆q = 0,1) on the QD.35,36

1 tor reveal a wealth of information about the physical By monitoring the switches of the current through the
v processes inside the device compared to what is avail- QPCwecanthusdetectinreal-timesingleelectronstun-
3 able from a conductance measurement alone.1–3 Zero- neling through the left (L) and right (R) barriers and
6
frequency noise measurements are now routinely per- therebydeterminethecorrespondingpulsecurrentsI (t)
1 L
2 formedusingstandardtechniques, butmorerecentlythe and IR(t). The time trace of the currents illustrates the
. detection of higher-order current correlation functions, electrical fluctuations of interest here.
1
0 or cumulants, has attracted considerable attention in We now analyze the current fluctuations obtained
2 experimental4–19 and theoretical20–22 studies of charge from data measured over 24 hours during which 300
1 transport in man-made sub-micron structures. Measure- million electrons passed through the SET. Our high-
: ments have for example been encouraged by the intrigu- quality measurements allow us to develop a complete
v
i ing connections between current fluctuations and entan- picture of the noise properties of the SET, which does
X glement entropy in solid-state systems23,24 as well as by not only focus on the zero-frequency components of the
r the possibility to test fluctuation theorems25–27 at the fluctuations, but contains the full frequency-resolved
a nano-scale which are now a topic at the forefront of non- information about the noise and higher-order corre-
equilibrium statistical physics. lation functions. From the measured finite-frequency
Experiments on current fluctuations have mainly fo- noise spectrum we extract the correlation time of the
cused on zero-frequency current correlations4–16 while current fluctuations. The noise spectrum, however, is
measurements of finite-frequency cumulants of the cur- a one-frequency quantity only which does not reflect
rent have remained an outstanding experimental chal- correlations between different spectral components of
lenge. However, noise detection in a restricted frequency the current. To observe such correlations, we employ
band gives only partial information about correlations bispectral analysis and consider the finite-frequency
and measurements that cover the full frequency range skewness (or bispectrum) of the current. The skewness
are necessary to access all relevant time scales that char- shows that the current fluctuations are non-gaussian
2
a c d
∆t = 40 µs
QPC
∆q1 1.00
0
6 L R RR
∆t)4 0.75 1/ τ
I (e/2 QD F(2) C
0
0.50
b 6
5 1.0
 
4 0.8I (e 0.25 L R
∆t)3 0.6/2
e/2 0.40∆ LR
I (1 0.2t) 0.00
0 0.0
102 103 104
0.00 0.02 0.04 0.06 0.08 0.10
/2π (Hz)
t (s)
FIG.1: Single electron transistor (SET) and finite-frequency noise spectra. a,b,Fluctuationsofthecurrentthrough
the left (red curve) and right (blue curve) tunneling barriers of the SET. c, Atomic force microscope topography of the SET
consisting of a Coulomb blockade quantum dot (QD) coupled via tunneling barriers to left (L) and right (R) electrodes. The
scale bar corresponds to 250 nm. A bias difference between the electrodes drives a stream of electrons through the QD from
L to R. An electron entering the QD from L causes a suppression of the current through the nearby quantum point contact
(QPC)untiltheelectrontunnelsoutoftheQDtoR. TheoccupationoftheQD(∆q=0,1)isinferredfromthetime-dependent
current through the QPC (shown with black in a). The corresponding pulse currents through the left (red) and right (blue)
barriers are discretized in time steps of ∆t=40 µs and shown displaced for clarity, a. In b, the same currents are shown on a
much longer time scale and discretized in time steps of ∆t (grey spikes) and 20∆t (full lines), respectively. d, Noise spectrum
(bluecurve)ofthecurrentthroughtherightbarrier(RR)andcross-correlations(greencurve)betweenthetwocurrents(LR).
Thethicknessofthelinesindicateexperimentalerrorestimates. Modelcalculations(dashedlines)ofthespectraareinexcellent
agreement with the experiment using Γ = 13.23 kHz and Γ = 4.81 kHz for the tunneling rates across the barriers and a
L R
detector rate of Γ =0.3 MHz. The mean dwell time of electrons on the QD is τ =Γ−1 (cid:39)210 µs and the correlation time is
D d R
τ =(Γ +Γ )−1 (cid:39)55 µs. The inset shows a schematic model of the charge transport process.
c L R
on all relevant time and frequency scales due to the (cid:104)I (t)(cid:105) being constant in the stationary state. The noise
R
non-equilibrium conditions imposed by the applied is symmetric in frequency, S(2)(ω) = S(2)(−ω), and re-
RR RR
voltagebias. Ourmeasurementsaresupportedbymodel sults are shown for positive frequencies only. For uncor-
calculations which are in excellent agreement with the related transport the noise spectrum would be white, i.
experimental data. The results presented here provide a e., F(2)(ω) = 1 on all frequencies, corresponding to a
fundamental understanding of the electrical fluctuations Poisson process. Our measurements, in contrast, show a
in SETs which are expected to constitute the basic clear suppression of fluctuations below the Poisson value
building blocks of future nano-scale electronics. atlowfrequencies. ThisisduetothestrongCoulombin-
teractionsontheQDwhichintroducecorrelationsinthe
Results stream of electrons propagating through the SET: each
electron spends a finite time on the QD during which it
Noise spectrum. The measured noise spectrum for blocks the next electron entering the QD. The measured
the right tunneling barrier is presented in Fig. 1d. The noise spectrum has a Lorentzian shape whose width is
finite-frequency current-correlation function S(2)(ω) is determined by the inverse dynamical time scale of the
αβ
defined in terms of the noise power as1 correlations. Our measurements of the finite-frequency
noise thereby enable a direct observation of the correla-
(cid:104)(cid:104)Iˆα(ω)Iˆβ(ω(cid:48))(cid:105)(cid:105)=2πδ(ω+ω(cid:48))Sα(2β)(ω), tion time τc (cid:39)55 µs of the transport, Fig. 1d.
where Iˆ (ω), α = L,R, is the Fourier transformed cur-
α
rent and double brackets (cid:104)(cid:104)...(cid:105)(cid:105) denote cumulant aver- To corroborate our experimental results we calculate
aging over many experimental realizations. Figure 1d the noise spectrum of the schematic model in the inset
showstheFanofactorF(2)(ω)=S(2)(ω)/eI fortheright ofFig.1d. Singleelectronstunnelfromtheleftelectrode
RR
tunneling barrier with the mean current I = (cid:104)I (t)(cid:105) = ontotheQDatrateΓ andleaveitviatherightelectrode
L L
3
a b
1.0
experiment
10
0.8 11 12 LRR
10
5 1
) RRR
z
H 0.6 9 2
k
π ( 0 F(3)
/22 0.4 8 3
-5 7 4
5
0.2 6
-10
theory
0.0
-10 -5 0 5 10
 /2π (kHz)
1
c d e
1.0 1.0
0.8 0.8
F(2)(0)
F(3)0.6 0.6
0.4 0.4
0.2 0.2
-10 -5 0 5 10 -10 -5 0 5 10 -10 -5 0 5 10
 /2π (kHz)  /2π (kHz)  /2π (kHz)
1 1 1
FIG.2: Finite-frequency skewness. a,ExperimentalresultsandmodelcalculationsofthethirdFanofactor(thefrequency-
dependent skewness) F(3)(ω ,ω )=S(3)(ω ,ω )/e2I. Experimental results are shown above the diagonal ω =ω and model
1 2 1 2 1 2
calculationsbelow. ContourlinesindicateF(3) =0.3,0.4,0.5,0.6,0.7,and0.8. Theparametersforthemodelcalculationsare
given in the caption of Fig. 1. b, Symmetries of the bispectrum S(3)(ω ,ω ). Several important symmetry conditions follow
1 2
fromthedefinition,40 includingsymmetrywithrespecttointerchangeoffrequenciesS(3)(ω ,ω )=S(3)(ω ,ω ). Knowledgeof
1 2 2 1
the bispectrum in any of the regions (cid:13)1 −(cid:13)12 is sufficient for a complete description of the bispectrum. We have measured the
skewness in region (cid:13)2 and the cross-bispectrum (Fig. 3) in regions (cid:13)1 −(cid:13)3. c,d,e, Finite-frequency skewness along the three
blue lines in a. Full lines are experimental results, while dashed lines show model calculations. The thickness of the full lines
indicate the experimental error estimates. The third order Fano factor F(3)(ω ,ω ) approaches the zero-frequency shot noise
1 2
F(2)(0) for ω =0 and |ω | being much larger than the inverse correlation time τ−1 as seen in c.
2 1 c
at rate Γ . The Fano factor is then1 extractedfromthe distributionof waitingtimesbetween
R
detectedtunnelingevents.14,38,39 Themodelcalculations
F(2)(ω)=1− 2ΓLΓR (seeMethods)areinexcellentagreementwithmeasure-
(Γ +Γ )2+ω2 ments over the full range of frequencies, demonstrating
L R
the high quality of our experimental data.
where theoretically τ =(Γ +Γ )−1 is identified as the
c L R Cross-correlations. Cross-correlations between
correlation time. This expression qualitatively explains
the left I (t) and the right I (t) currents can also
the measured noise spectrum. Quantitative agreement L R
be measured, Fig. 1d. For the schematic model,
is obtained by also taking into account the finite detec-
tionrateΓD oftheQPCchargesensingscheme.13,37 The the (cross-correlation) Fano factor reads FL(2R)(ω) ≡
three parameters Γ , Γ , and Γ can be independently Re[S(2)(ω)]/eI = (Γ2 +Γ2)/([Γ +Γ ]2 +ω2), which
L R D LR L R L R
4
a b  /2π (kHz)
1
F(3) -10 -5 0 5 10
1.0
0.8
1.00
F(3)0.6
0.4
0.75
0.2
0.0
0.50
c
-12.5
-10.0 0.25
-7.5 1.0
-5.0
-2.5 0.8
0.00
 /2π 0 0.6
1 2.5 F(3)
(kHz) 0.4
5.0
7.5 7.5 10.0 12.5 0.2
10.0 0 2.5 5.0 0.0
-2.5
-5.0 -10 -5 0 5 10
12.5 -12.5 -10.0 -7.5 2/2π (kHz)  /2π (kHz)
2
FIG.3: Finite-frequency cross-bispectrum. a,Measurementsofthefrequency-dependentcross-bispectrumF(3) (ω ,ω ).
LRR 1 2
Thecross-bispectrumshowsareducedsymmetrycomparedtothebispectrumandmeasurementsintheregions(cid:13)1-(cid:13)3,Fig.2b,
arerequiredforacompletecharacterizationofthecross-bispectrum. Forcomparison,thebispectrumF(3)(ω ,ω )fromFig.2a
1 2
isshownasasemi-transparentsurfaceabovethecross-bispectrum. b,Bispectrumandcross-bispectrumalongthelineω =0,
2
where they coincide. c, Bispectrum (full blue line) and cross-bispectrum (dashed red line) along the line ω =0.
1
inthe zero-frequencylimitcoincideswiththe noisespec- tries following from the definition, Fig. 2b. We exploit
trum, F(2)(0)=F(2)(0), asaconsequenceofcharge con- the mirror symmetry with respect to interchange of fre-
LR
servation on the QD. The two currents are clearly cor- quencies, S(3)(ω1,ω2) = S(3)(ω2,ω1), to compare mea-
related at frequencies that are lower than the inverse surement and model calculations: experimental results
correlation time τc−1, but the cross-correlator eventu- are presented above the diagonal ω1 = ω2, while model
ally reaches zero at higher frequencies. Interestingly, calculations are shown below, Fig. 2a. The nonzero
the cross-correlations of the detected pulse currents are bispectrum indicates non-Gaussian statistics on all fre-
slightlynegativeathighfrequencies. Thisisduetothefi- quencies and shows strong correlations between different
niteresolutionoftheQPCchargesensingprotocolwhich spectral components of the current. Importantly, these
is not able to distinguish current pulses that are sepa- correlations are not a consequence of non-linearities in
rated in time by an interval which is shorter than the the detection scheme, but are solely due to the physical
inverse detector rate Γ−1. non-equilibrium conditions imposed by the applied volt-
D
Higher-order cumulants. Wenowturntomeasure- age bias. The pulse currents are directly derived from
ments of higher-order finite-frequency cumulants. The the tunneling events and the influence of external noise
m’thfinite-frequencycurrentcorrelatorcorrespondingto sources, including the amplification of the QPC current,
a time-dependent current I(t) is defined as is thereby explicitly avoided. Intuitively, one would ex-
pect the correlations to vanish, if the observation fre-
(cid:104)(cid:104)Iˆ(ω )···Iˆ(ω )(cid:105)(cid:105)=2πδ(ω +...+ω )S(m)(ω ,...,ω ) quency is larger than the average frequency of the trans-
1 m 1 m 1 m−1
port. Surprisingly, however, a certain degree of corre-
where translational invariance in time implies frequency
lation persists even if one frequency is large, while the
conservationasindicatedbytheDiracdeltafunctionδ(ω)
other is kept finite. This is in stark contrast to the sec-
and S(m)(ω ,...,ω ) is the polyspectrum.40,41 In the
1 m−1 ond Fano factor F(2)(ω) which approaches unity in the
case m = 2, the polyspectrum yields the noise power
high-frequency limit, Fig. 1d, corresponding to uncorre-
spectrum S(2)(ω), while the skewness (or bispectrum)
lated tunneling events.
is given by m = 3 with the corresponding Fano factor
FortheschematicmodelinFig.1d,thefinite-frequency
F(3)(ω ,ω ) = S(3)(ω ,ω )/e2I. We focus here on the
1 2 1 2 skewness reads32
frequency-dependentskewnessS(3)(ω ,ω ),althoughour
experimental data in principle allows1us2also to extract F(3)(ω ,ω )=1−2Γ Γ (cid:81)2j=1(γj2+ω32−ω1ω2),
cumulants of even higher orders. 1 2 L R(cid:81)3 [(Γ +Γ )2+ω2]
The measured skewness, Fig. 2a, shows a much richer i=1 L R i
structure and frequency dependence compared to the having defined γ2 = Γ2 +Γ2, γ2 = 3(Γ +Γ )2, and
1 L R 2 L R
noise spectrum. The skewness obeys several symme- ω = ω + ω . This expression qualitatively explains
3 1 2
5
the measured finite-frequency skewness and shows that Methods
the skewness has a complex structure which is not just
a simple Lorentzian-shaped function of frequencies. We Device fabrication. Thedevicewasfabricatedbylocal
note that the third Fano factor F(3)(ω ,ω ) reduces to anodicoxidationtechniquesusinganatomicforcemicro-
1 2
the zero-frequency limit of the noise F(2)(0) in the limit scope on the surface of a GaAs/AlGaAs heterostructure
ω = 0 and |ω | (cid:29) τ−1. This is immediately visible withelectrondensity4.6×1015 m−2 andamobilityof64
2 1 c
in Fig. 2c. Again, quantitative agreement is achieved m2/Vs. Thetwo-dimensionalelectrongasresiding34nm
by including the finite detection rate Γ in the model belowtheheterostructuresurfaceisdepletedunderneath
D
calculations as illustrated explicitly in Figs. 2c-e. The the oxidized lines, allowing us to define the quantum dot
analyticexpressionshowsthatthefrequencydependence (QD) and the quantum point contact (QPC).
of the skewness, unlike the auto- and cross-correlation Measurements. The experiment was carried out in a
noise spectra, is not only governed by the correlation 3He cryostat at 500 mK with an applied bias of 900 µV
time τ = (Γ +Γ )−1. The skewness has an involved acrosstheQDinordertoensureunidirectionaltransport
c L R
frequency dependence, given by several frequency scales, and to avoid the influence of thermal fluctuations. The
whichcannotbededucedfromthenoisespectrumalone. QPC detector was tuned to the edge of the first conduc-
Higher-order cross-correlations. In Fig. tion step. The current through the QPC was measured
3a, we finally consider the cross-bispectrum with a sampling frequency of 500 kHz. The tunneling
F(3) (ω ,ω ) ≡ Re[S(3) (ω ,ω )]/e2I, measuring eventswereextractedfromtheQPCcurrentusingastep
LRR 1 2 LRR 1 2 detection algorithm and converted into time-dependent
the third-order correlations between tunneling electrons
pulse currents: Time was discretized in steps of ∆t=40
entering and leaving the SET. The three non-redundant
µsandineachstepthenumberoftunnelingevents∆nin
permutations of the left and right pulse currents lead
(out) of the QD was recorded. The current into (out of)
to a reduced symmetry of the cross-bispectrum, as
the QD at a given time step is then I(t)=(−e)∆n/∆t.
visualized in comparison with the auto-correlation
Error estimates. We estimated the errors of the mea-
bispectrum in Fig. 3a. The cross-bispectrum coincides
sured spectra by dividing the experimental data into 30
with the auto-bispectrum on the line ω = 0, where
2
separate batches. The spectra were determined for each
they approach the zero-frequency correlation of the shot
noise for |ω | > τ−1, Fig. 3b. In contrast, for |ω | =(cid:54) 0, batchindividuallyandtheirstandarddeviationwasused
1 c 2
as a measure of the experimental accuracy.
the cross-bispectrum shows a frequency dependence
Finite-frequency cumulants. To define the
similar to the cross-correlation shot noise and the
finite-frequency cumulants of the current we con-
anti-correlations due to the detection process become
sider the m-time probability distribution32,33
visible, Fig. 3c.
P(m)(n ,t ;...;n ,t ) that n electrons have been
1 1 m m k
transferred during the time span [0,t ] for k =1,...,m.
Discussion k
The corresponding cumulant generating function is
We have measured the current statistics of charge (cid:40) (cid:41)
(cid:88)
transportinanSETanddirectlydeterminedthedynam- F(m)(χ,t)≡log P(m)(n;t)ein·χ
ical features and time scales of the current fluctuations.
n
From the measured frequency-dependent noise power we
extracted the correlation time of the fluctuations. The with χ = (χ1,...,χm), t = (t1,...,tm), and n =
noise power is a single-frequency quantity only and in (n1,...,nm). (Anequivalentdefinitionusesonlyasingle
order to investigate the correlations between different buttime-dependentcountingfield.45)Them-timecumu-
spectral components of the current using bispectral lants of P(m)(n;t) are then
analysis we measured the frequency-dependent third
(cid:104)(cid:104)n(t )···n(t )(cid:105)(cid:105)=∂ ···∂ F(m)(χ,t)| ,
order correlation function (the skewness). The skewness 1 m iχ1 iχm χ→0
shows that the current statistics are non-gaussian on all
with the corresponding m-time cumulants of the current
frequencies due to the applied voltage bias. Our experi-
mental results are supported by model calculations that
(cid:104)(cid:104)I(t )···I(t )(cid:105)(cid:105)=∂ ···∂ (cid:104)(cid:104)n(t )···n(t )(cid:105)(cid:105).
are in excellent agreement with measurements. The re- 1 m t1 tm 1 m
sultspresentedhereareimportantforfutureapplications
These equations define the cumulant averages denoted
of SETs in nano-scale electrical circuits operating with
by double brackets (cid:104)(cid:104)...(cid:105)(cid:105). In the Fourier domain, the
single electrons. Our accurate and stable experiment
current cumulants can be expressed as
also facilitates several promising directions for basic
research on nano-scale quantum devices. These include (cid:104)(cid:104)Iˆ(ω )···Iˆ(ω )(cid:105)(cid:105)=2πδ(ω +...+ω )S(m)(ω ,...,ω )
1 m 1 m 1 m−1
experimental investigations of fluctuation relations at
finite frequencies and higher-order noise detection of sincethesumoffrequenciesiszerointhestationarystate.
interaction42 and coherence43 induced correlation effects The Fourier transformed current is denoted as Iˆ(ω) and
in quantum transport under non-equilibrium conditions. S(m)(ω ,ω ,...,ω ) is the polyspectrum,40 which for
1 2 m−1
m = 2 and m = 3 yields the noise spectrum S(2)(ω)
6
(second cumulant) and the bispectrum (third cumulant) where M =M(0,0) and I is the super operator for
0 L(R)
S(3)(ω1,ω2) respectively. We note that the noise spec- the detected current through the left (right) barrier.46
trumS(2)(ω)inthestationarystateisasingle-frequency Additionally, we need the stationary state |0(cid:105)(cid:105), found by
quantity which can be related directly to the one- solving M |0(cid:105)(cid:105)=0 and normalized such that (cid:104)(cid:104)˜0|0(cid:105)(cid:105)=1,
0
time probability distribution P(1)(n,t) via MacDonald’s where (cid:104)(cid:104)˜0| = (1,1,1,1). We then define the projectors46
formula44 S(2)(ω)= ω(cid:82)∞dtsin(ωt)d(cid:104)(cid:104)n2(cid:105)(cid:105)(t). The bis- P = |0(cid:105)(cid:105)(cid:104)(cid:104)˜0| and Q = 1 − P and the frequency-
0 dt
pectrumS(3)(ω ,ω )incontrastisatwo-frequencyquan- dependent pseudoinverse47 R(ω) = Q[iω + M ]−1Q,
1 2 0
tity which reflects correlations of the current beyond which is well-defined even in the zero-frequency limit
what is captured by P(1)(n,t) alone. ω → 0, since the inversion is performed only in the
Theoreticalmodel. Weconsidertheprobabilityvector subspace spanned by Q, where M is regular. These
0
|p(t)(cid:105)=[p (t),p (t),p (t),p (t)]T, where the first in- objects constitute the essential building blocks for
00 10 01 11
dex denotes the number of (additional) electrons on the our calculations. The frequency-dependent second
QD, i=0,1, and the second index denotes the detected and third cumulants, S(2)(ω) and S(3)(ω ,ω ), can then
1 2
number of (additional) electrons on the QD as inferred beevaluatedfollowingRefs.[47,48]and[33],respectively.
from the current through the QPC, j =0,1. The proba-
bility vector evolves according to the rate equation
Acknowledgments
We thank M. Bu¨ttiker, C. Emary, and Yu. V. Nazarov
d
|p(χ ,χ ;t)(cid:105)=M(χ ,χ )|p(χ ,χ ;t)(cid:105), for instructive discussions. W. Wegscheider (Regens-
dt L R L R L R
burg, Germany) provided the wafer and B. Harke
having introduced separate counting fields χ and χ (Hannover, Germany) fabricated the device. The work
L R
thatcoupletothenumberofdetectedelectronsthathave was supported by BMBF via nanoQUIT (C. Fr., N.
passedtheleftandtherightbarriers,respectively.45 The U., F. H., and R. J. H.), DFG via QUEST (C. Fr., N.
matrix M(χ ,χ ) reads13,14,37 U., F. H., and R. J. H.), the Villum Kann Rasmussen
L R
Foundation (C. Fl.), and the Swiss NSF (C. Fl.).
−ΓL ΓR ΓDeiχR 0 
Γ −(Γ +Γ ) 0 0
 L R D , Author contributions
 0 0 −(Γ +Γ ) Γ 
L D R All authors conceived the research. N. U., C. Fr., and
0 ΓDeiχL ΓL −ΓR F. H. carried out the experiment and analyzed data. All
authors discussed the results. C. Fl. developed theory
where factors of eiχL(R) in the off-diagonal elements cor-
and performed calculations. N. U., C. Fr., and C. Fl.
respond to processes that increase by one the number
wrote the manuscript. F. H. and R. J. H. supervised the
of detected electrons that have tunneled across the left
research. All authors contributed to the editing of the
(right) barrier, see Fig. 1c,d. The detector rate Γ of
D manuscript.
the QPC detector scheme tends to infinite for an ideal
detector only, but is finite otherwise.
Calculations. For the calculations of the finite- Additional information
frequency cumulants it is useful to write the matrix as Corresponding author. Correspondence and requests
for materials should be addressed to R. J. H. (email:
M(χ ,χ )=M +(eiχL −1)I +(eiχR −1)I , [email protected]).
L R 0 L R
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