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Universal oscillations in counting statistics

Edited by Peter Zoller, University of Innsbruck, Innsbruck, Austria, and approved April 22, 2009 (received for review January 29, 2009)
Abstract
Noise is a result of stochastic processes that originate from quantum or classical sources. Higherorder cumulants of the probability distribution underlying the stochastic events are believed to contain details that characterize the correlations within a given noise source and its interaction with the environment, but they are often difficult to measure. Here we report measurements of the transient cumulants 〈〈n^{m}〉〉 of the number n of passed charges to very high orders (up to m = 15) for electron transport through a quantum dot. For large m, the cumulants display striking oscillations as functions of measurement time with magnitudes that grow factorially with m. Using mathematical properties of highorder derivatives in the complex plane we show that the oscillations of the cumulants in fact constitute a universal phenomenon, appearing as functions of almost any parameter, including time in the transient regime. These ubiquitous oscillations and the factorial growth are systemindependent and our theory provides a unified interpretation of previous theoretical studies of highorder cumulants as well as our new experimental data.
Counting statistics concerns the probability distribution P_{n} of the number n of random events that occur during a certain time span t. One example is the number of electrons that tunnel through a nanoscopic system (1–4). The first cumulant of the distribution is the mean of n, 〈〈n〉〉 = 〈n〉, the second is the variance, 〈〈n^{2}〉〉 = 〈n^{2}〉 − 〈n〉^{2}, the third is the skewness, 〈〈n^{3}〉〉 = 〈(n − 〈n〉)^{3}〉. With increasing order the cumulants are expected to contain more and more detailed information on the microscopic correlations that determine the stochastic process. In general, the cumulants 〈〈n^{m}〉〉 = S^{(m)}(z = 0) are defined as the mth derivative with respect to the counting field z of the cumulant generating function (CGF) S(z) = lnΣ_{n}P_{n}e^{nz}. Recently, theoretical studies of a number of different systems have found that the highorder cumulants oscillate as functions of certain parameters (5–9), however, no systematic explanation of this phenomenon has so far been given. Examples include oscillations of the highorder cumulants of transport through a MachZender interferometer as functions of the AharonovBohm flux (6), and in transport through a double quantum dot as functions of the energy dealignment between the two quantum dots (8). As we shall demonstrate, oscillations of the highorder cumulants in fact constitute a universal phenomenon which is to be expected in a large class of stochastic processes, independently of the microscopic details. Inspired by recent ideas of M. V. Berry for the behavior of highorder derivates of complex functions (10), we show that the highorder cumulants for a large variety of stochastic processes become oscillatory functions of basically any parameter, including time in the transient regime. We develop the theory underlying this surprising phenomenon and present the first experimental evidence of universal oscillations in the counting statistics of transport through a quantum dot.
We first present our experimental data. In our setup (Fig. 1A), single electrons are driven through a quantum dot and counted (11–16), using a quantum point contact. The quantum dot is operated in the Coulomb blockade regime, where only a single additional electron at a time is allowed to enter and leave. A large biasvoltage across the quantum dot ensures that the electron transport is unidirectional. Electrons enter the quantum dot from the source electrode at rate Γ_{S} = Γ and leave via the drain electrode with rate Γ_{D} = Γ(1 − a)/(1 + a), where −1 ≤ a ≤ 1 is the asymmetry parameter (14, 15). A nearby quantum point contact (QPC) is capacitively coupled to the quantum dot and used as a detector for realtime counting of the number of transferred electrons during transport: When operated at a conductance step edge, the QPC current is highly sensitive to the presence of localized electrons on the dot. By monitoring switches of the current through the QPC (see Fig. 1B) it is thus possible to detect single electrons as they tunnel through the quantum dot and thereby obtain the distribution of the number of transferred electrons P_{n}. In the experiment we fix the asymmetry parameter a and monitor the time evolution of the number of passed charges. This requires a single long timetrace of duration T during which a large number of tunneling events are counted. The timetrace is divided into a large number N of time segments of length t. From the number of electrons counted in each time segment we find the probability distribution P_{n} from which the cumulants as functions of measurement time are obtained. In this approach t can be varied continuously as it was recently shown experimentally up to the fifth cumulant (15). For the experiment considered here ≈670,000 electrons were counted during a time span of T = 770 s. This allowed us to estimate P_{n} for t in the range from 0 to ≈2 ms. In Fig. 2, we show the corresponding experimental results for the time evolution of the highorder cumulants up to order 15. Most remarkably, the cumulants show transient oscillations as functions of time that get faster and stronger in magnitude with increasing order of the cumulants.
Together with the experimental results, we show theoretical fits of the cumulants as functions of measurement time. The model is based upon a master equation (17) and takes into account the finite bandwidth of the detector (15, 18), which is not capable of resolving very short time intervals between electron tunneling events (see SI Appendix). The detector bandwidth Γ_{Q} and the rates Γ_{D} and Γ_{S} are the only input parameters of the model and they can be extracted from the distribution of switching times, shown in Fig. 1C, between a high and a low current through the quantum point contact. The calculated cumulants show excellent agreement with the experiment for the first 15 cumulants as seen in Fig. 2.
The master equation calculation per se does not provide any hints concerning the origin and character of the oscillations seen in the experiment. Similarly, in the theoretical studies of highorder cumulants mentioned previously (5–9), no systematic explanation of the origin of these oscillations has been proposed. Here, we point out the universal character of the oscillations of highorder cumulants and develop the underlying theory. The following analysis applies to a large class of stochastic processes, including the aforementioned theoretical studies and the experiment described above. We thus consider a general stochastic process with a CGF denoted by S(z,λ). Here, all relevant quantities related to the system—collectively denoted as λ—enter as real parameters. As it follows from complex analysis, the asymptotic behavior of the highorder cumulants is determined by the analytic properties of the CGF in the complex z plane. We consider the generic situation, where the CGF S(z,λ) has a number of singularities z_{j},j = 1, 2, 3, … , in the complex plane that can be either poles or branchpoints. Exceptions to this scenario do exist, e.g., the simple Poisson process, whose CGF is an entire function, i.e., it has no singularities. Such exceptions, however, are nongeneric and are excluded in the following, although they can be addressed by analogous methods (10). Close to a singularity z ≈ z_{j} the CGF takes the form S(z, λ) ≈ A_{j}/(z − z_{j})^{μj} for some A_{j} and μ_{j}. The corresponding derivatives for z ≈ z_{j} are ∂_{z}^{m}S(z, λ) ≈ (−1)^{m}A_{j}B_{m,μj}/(z − z_{j})^{m + μj} with B_{m,μj} = (μ_{j} + m − 1)(μ_{j} + m − 2) … μ_{j} for m ≥ 1. The approximation of the derivatives becomes increasingly better away from z ≈ z_{j} as the order m is increased. This is also known as Darboux's theorem (10, 19). For large m, the cumulants are thus wellapproximated as a sum of contributions from all singularities, This simple result determines the largeorder asymptotics of the cumulants. Generally, the singularities come in complexconjugate pairs, ensuring that the expression above is real. Although actual calculations of the highorder cumulants using this expression may be cumbersome, the general result displays a number of ubiquitous features. In particular, we notice that the magnitude of the cumulants grows factorially with the order m due to the factors B_{m,μj}. Furthermore, writing z_{j} = z_{j}e^{iArg(zj)} with z_{j} being the absolute value of z_{j} and Arg(z_{j}) the corresponding complex argument, we also see that the most significant contributions to the sum come from the singularities closest to z = 0. The relative contributions from other singularities are suppressed with the relative distance to z = 0 and the order m, such that they can be neglected for large m. Most importantly, we recognize that the highorder cumulants become oscillatory functions of any parameter among λ that changes Arg(z_{j}) as well as of the cumulant order m. This important observation shows that the highorder cumulants for a large class of CGFs will oscillate as functions of almost any parameter.
For a simple illustration of these concepts, we consider a charge transfer process described by a CGF reading
with w_{z,a} ≡
Finally, we now return to our experimental data presented in Fig. 2. In the transient regime we see that the highorder cumulants oscillate as functions of measurement time. This is due to the time dependence of the dominating singularities of the CGF. In the case of an ideal detector (Γ_{Q} → ∞) (see SI Appendix) the CGF at finite times in Eq. 2 has timedependent singularities when the argument of the logarithm is zero. These singularities have the form z_{k,j} = x_{k} + (2j + 1)iπ, k = 1,2, … , j = … −1,0,1, … , where x_{k} ≡ ln⌊(a^{2} + u_{k}^{2})/(1 − a^{2})⌋ and u_{k} solves the transcendental equation 2u_{k}Γt/(1 + a) − 4arctan(1/u_{k}) = 2π(k − 1) (see Fig. 4A). The derivatives of the logarithmic singularity encountered here can be treated with our theory by formally setting μ_{k,j} = 0 and μ_{k,j}A_{k,j} = 1. In Fig. 4B we see that the approximation for the 15th cumulant as function of time, using the two timedependent singularities z_{1,−1} and z_{1,0} closest to z = 0 for the given time interval, agrees well with exact calculations in the limit of an ideal detector (Γ_{Q} → ∞), taking a = −0.34 as in the experiment. The curves are also in good agreement with the experimental results in Fig. 2, showing that the oscillations cannot be dismissed as an experimental artifact due to, e.g., the finite bandwidth of the detector. Of course, in the longtime limit the cumulants relax to their linearintime asymptotics given by the first term of the CGF. The loworder cumulants (m = 4 − 7) seen in Fig. 2, normalized with respect to the first cumulant, clearly reach their longtime limits for t ≥ 1.5 ms. This does not contradict the fact that the cumulants oscillate as functions of time in a given finite time interval for high enough order.
The experimental and theoretical results presented in this work clearly demonstrate the universal character of the oscillations of highorder cumulants. In our experiment the highorder cumulants oscillate as functions of time in the transient regime. As our theory shows, such oscillations are however predicted to occur as functions of almost any parameter in a wide range of stochastic processes, regardless of the involved microscopic mechanisms. The universality of the oscillations stems from general mathematical properties of cumulant generating functions: as some parameter is varied, dominating singularities move in the complex plane, causing the oscillations. Oscillations of highorder cumulants have been seen also in other branches of physics, including quantum optics (20) and elementary particle physics (21), further demonstrating the universality of the phenomenon.
Methods
Device.
The quantum dot and the quantum point contact were fabricated using local anodic oxidation techniques with an atomic force microscope on the surface of a GaAs/AlGaAs heterostructure with electron density n = 4.6×10^{15}m^{−2} and mobility μ = 64m^{2}/Vs. With this technique the 2dimensional electron gas residing 34 nm below the heterostructure surface is depleted underneath the oxidized lines on the surface. A number of inplane gates were also defined, allowing for electrostatic tuning of the quantum point contact and electrostatic control of the tunneling barriers between the quantum dot and the source and drain electrodes.
Measurement.
The experiment was performed at an electron temperature of about 380 mK, as determined from the width of thermally broadened Coulomb blockade resonances. To avoid tunneling from the drain to the source contact of the quantum dot due to thermal fluctuations, we applied a bias of 330 μV across the quantum dot. The QPC detector was tuned to the edge of the first conduction step. The current through the QPC was measured with a sampling frequency of 100 kHz. This sufficiently exceeded the bandwidth of our experimental setup of about 40 kHz. The tunneling events were extracted from the QPC signal using a step detection algorithm.
Error Estimates.
To estimate the error of the experimentally determined cumulants we created an ensemble of simulated data using the same rates as observed in the experiment. We then extracted the cumulants for each simulated data set in the ensemble and determined the ensemble variance of the cumulants for each order m as function of time t. The error bars in Fig. 2 show the squareroot of the variance.
Acknowledgments
We thank N. Ubbelohde (Hannover, Germany) for his support in the development of the data analysis algorithms, W. Wegscheider (Regensburg, Germany) for providing the wafer, and B. Harke (Hannover, Germany) for fabricating the device. The work was supported by the Villum Kann Rasmussen Foundation (C. Flindt), the Czech Science Foundation Grant 202/07/J051 (to C. Flindt, T.N., and K.N.), Federal Ministry of Education and Research of Germany via nanoQuit (C. Fricke, F.H., and R.J.H.), German Excellence Initiative via the “Centre for Quantum Engineering and SpaceTime Research” (C. Fricke, F.H., and R.J.H.), research plan MSN0021620834 financed by the Ministry of Education of the Czech Republic (to T.N.), project AV0Z10100520 in the Academy of Sciences of the Czech Republic (to K.N.), and Deutsche Forschungsgemeinschaft Project BR 1528/5–1 (to T.B.).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: flindt{at}physics.harvard.edu

Author contributions: C. Flindt, C. Fricke, F.H., T.N., K.N., T.B., and R.J.H. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0901002106/DCSupplemental.
References
 ↵
 ↵
 Nazarov YV
 ↵
 Levitov LS,
 Lesovik GB
 ↵
 ↵
 ↵
 Förster H,
 Pilgram S,
 Büttiker M
 ↵
 Förster H,
 Samuelsson P,
 Pilgram S,
 Büttiker M
 ↵
 ↵
 Novaes M
 ↵
 ↵
 ↵
 ↵
 Fujisawa T,
 Hayashi T,
 Tomita R,
 Hirayama Y
 ↵
 Gustavsson S,
 et al.
 ↵
 Gustavsson S,
 et al.
 ↵
 ↵
 Andrews GE,
 Askey R,
 Roy R
 ↵
 Bagrets DA,
 Nazarov,
 Yu V
 ↵
 ↵
 ↵
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