Abstract
There is an intimate connection between the acquisition of information and how this information changes the remaining uncertainty in the system. This trade-off between information and uncertainty plays a central role in the context of detection. Recent advances in the ability to make accurate, on-chip measurements of individual-electron current through a quantum dot1,2,3,4,5,6,7,8 (QD) have been enabled by exploiting the sensitivity of a second current, passing through a nearby quantum point contact (QPC), to the fluctuating charge on the QD4,5,6,7,8. An important characteristic of QPC detectors is their minimal influence on the systems they probe. Here we show that even the operation of an effectively non-invasive QPC detector can statistically alter the system’s behaviour. By observing a particular QPC current, the statistical distribution of the QD conditional current undergoes a substantial change in comparison to that expected for unconditional shot noise9. These results are in almost perfect agreement with a theoretical model we develop to predict the joint current probability distribution and conditional transport statistics of interacting nanoscale systems.
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Main
Noise is generally due to randomness, which can be classical or quantum in nature. Telegraph noise, where there is random switching between two stable states10, originates from such diverse phenomena as thermal activation of an unstable impurity11,12,13, non-equilibrium activation of a bistable system14,15,16,17, switching of magnetic domain orientation18,19,20, or a reversible chemical reaction in a biological ion channel21.
In nanoscale conductors, where charge motion is quantum coherent over distances comparable to the system size, shot noise and telegraph noise have recently been shown to be two sides of the same coin6,7,22,23. A quantum dot (QD) is sufficiently small that it is effectively zero dimensional, and behaves as an artificial atom, holding a small number of electrons. Figure 1a shows the sample used in the experiment reported here. The QD is marked by the dotted circle24. An extra electron can tunnel into the QD from the source lead (S), stay in the QD for a random amount of time and then tunnel out into the drain lead (D) if the applied voltage bias exceeds the temperature. This single-electron transport produces a fluctuating electrical current. In order to detect the statistical properties of this current, a sensitive electrometer with a bandwidth much higher than the tunnelling rates is required. The electrometer is a nearby quantum point contact (QPC) that is capacitively coupled to the QD via the Coulomb interaction. The voltage-biased QPC detector transports many electrons through a narrow constriction in the surrounding two-dimensional electron gas (represented with an arrow). The resistance of the QPC is susceptible to changes in the surrounding electrostatic environment, and can therefore be used to sense the presence (or absence) of an extra electron on the QD4. When the extra electron tunnels into or out of the QD, the current I flowing through the QPC switches between two different values (see Fig. 1c). Therefore, the shot noise of the QD current J (randomness in the number of switches in a given time interval) is intimately linked with telegraph noise in the QPC current I (randomness of duration time in each current value).
In the following, we develop a theoretical model for this experiment. Ideal switching of the measured detector signal between two noiseless values I1,2 is identified with the two current levels experienced by the QPC. When an electron enters the QD the current switches from I1 to I2, and when the electron leaves the QD the reverse switch happens. The number of ‘down’ or ‘up’ switches M in a given time trace of duration t is identified with the number of different transport electrons that occupy the QD in that time interval, naturally defining a QD current variable J=M/t (we set e=1 to count in single-electron charge units). The analogous number of electrons N passed by the QPC in this same time interval defines the QPC current variable I=N/t. The assumption of noiseless current levels implies that I1≤I(t)≤I2, and the unidirectional nature of the QD transport implies that (see the Supplementary Information for justification). Stochastic, statistically independent quantum tunnelling into and out of the QD is described with rates Γ1,2 (ref. 22). For later convenience, we define the average and difference variables I0=(I1+I2)/2,Γ0=(Γ1+Γ2)/2 and ΔI=(I2−I1)/2,ΔΓ=(Γ2−Γ1)/2. This model is capable of describing a host of phenomena in many fields of science.
Taken alone, each side of the random process may be characterized mathematically with the probability distributions P(I,t) and P(J,t) of finding a given number of electrons transmitted in a given time, or equivalently all current cumulants 〈〈In〉〉 and 〈〈Jm〉〉 (see the Supplementary Information for a discussion of these statistical quantities). This catalogue of cumulants gives a unique signature of the particular electronic conductor, and is also known as full counting statistics25. For example, the first two cumulants are the average current 〈I〉 and the shot noise power , where δI=I(t)−〈I〉. For the QPC and QD, the average currents 〈I〉 and 〈J〉 and shot noise powers 〈〈I2〉〉 and 〈〈J2〉〉 are given respectively by
In the limit of small switching rates Γ1,2→0 the current and noise of the QD vanish, whereas the noise of the QPC actually diverges.
However, a simple specification of the counting statistics of each conductor individually misses the important fact that the two conductors are strongly correlated by the Coulomb interaction between them. The simplest measure of the correlation between the systems is the cross-correlation , given by
This correlator is approximately constant under scaling of the switching rates, compromising between the behaviour of the two noises individually (1). Result (2) implies that the two currents may be either positively or negatively correlated, depending on the sign of ΔΓΔI. This effect has a simple physical interpretation: taking ΔI>0, if Γ2>Γ1, then the system typically spends more time in state 1 than in state 2. The current J is increased by adding another ‘up’ and ‘down’ switch to the current trace. This new event typically divides a long time interval spent in I1 into two segments, subtracting a short interval from I1 and adding it to I2, thereby increasing the current I. The same argument applied with Γ2<Γ1 leads to a decrease in I given an increase in J, explaining the sign of the cross-correlation function.
In order to specify the statistical correlation between the conductors, we introduce the joint counting statistics of both conductors. More specifically, the correlations may be quantified by the joint probability distribution P(I,J,t) of finding current I and current J in a time t (equivalently, all cross-cumulants 〈〈InJm〉〉), or may also be specified by the conditional distribution functions P(I|J) or P(J|I), the probability of observing one current given an observation of the other. These distributions are all related to one another by P(I,J)=P(I|J)P(J)=P(J|I)P(I), where the last equality is an expression of Bayes’s theorem. The conditional distributions (and their associated conditional cumulants) play a key role in the informational approach to detection26.
From the model described above, the complete characterization of the system/detector fluctuation statistics may be obtained from conditional master equation formalism. The statistical cumulants of the current fluctuations are given with the help of a generating function H(λ,χ), such that the cross-cumulants are given simply by taking derivatives, 〈〈InJm〉〉=∂λn∂χmH(0,0). From the mathematical derivation given in the Supplementary Information, we find that the joint generating function is given by
generalizing previous results27,28. Results (1) and (2) follow from (3). The function H(0,χ) generates the current cumulants of the QD27 (the first few of which have been experimentally verified22,29), and H(λ,0) generates the current cumulants of the QPC28. The full QPC current distribution was predicted to have an elliptical shape28: logP(I)/t=−(G1−G2)2/(2ΔI), where . This prediction is experimentally confirmed in Fig. 1b for data sets A (blue asterisks) and B (red squares) with no fitting (see the Methods section). The joint generating function (3) is directly related to the joint probability distribution of measuring current I and current J (see Supplementary Information, equation (6)). The logarithm of this distribution has been measured and given in Fig. 2a,c for configurations A and B respectively. The theoretical prediction for this quantity is given in Fig. 2b,d with striking agreement.
Having described the joint statistical properties of both currents, we now return to the detection question. It is important to distinguish between physical back-action and statistical–informational back-action. The non-invasive QPC detector changes its physical current state depending on whether or not the QD is occupied by an extra electron, whereas the physical dynamics of the QD is unaffected by the state of the QPC. However, by observing a particular outcome of the detector variable (I), this leads to conditional (Bayesian) back-action of the detector on the system variable (J). This informational back-action, or constrained randomness, manifests itself in a variety of novel effects. We introduce the concept of conditional counting statistics: the statistical current fluctuations of one system, given the observation of a given current in the other. These statistics may be calculated from the joint generating function (3) (see the Supplementary Information), yielding a mixed generator H1(I,χ) of the (normalized) conditional statistics of the QD, given the observation of a current I,
Taking derivatives, all conditional cumulants are given by 〈〈Jn〉〉c=Ω/2n, yielding a set of universal semicircles as a function of the current I. At the endpoints of the interval, the QPC current is observed to remain in I1 or I2, and therefore there cannot be any QD current, or any associated noise. The conditional QD current cumulants all have a maximum at I0. The conditional current maximum is always larger than the unconditional current 〈J〉. Figure 3a,b compares the experimental values of the first two conditional cumulants to the theoretical semicircles, with excellent agreement. The distribution described by the generating function (4) is a poissonian distribution with a generalized rate Ω, and effective charge e*=e/2, showing a radical change when compared with the unconditional distribution27.
Turning our perspective around, we can pose similar questions about the conditional detector statistics, given an observation of the system current J. Although the generating function H2(λ,J) for these statistics may be found analytically (see the Supplementary Information), we focus on the conditional current 〈I〉c, and the conditional noise 〈〈I2〉〉c, given by
where . These conditional cumulants are experimentally calculated and compared with equation (5) in Fig. 3c,d. As a function of the QD current J, the conditional current tends to either I1 or I2 as J→0, depending on the sign of ΔΓ. This corresponds to the most likely detector current configuration in the event of no switches being observed: the QPC current stays at one value, also implying that the system becomes noiseless in this limit. This is easily seen in (5) because 〈〈I2〉〉c is proportional to J. The exception to this rule is the perfectly symmetric situation Γ1=Γ2, where the QPC conditional average current is I0. This situation corresponds to rare symmetric switching between the states, whose effective rate is the conditional QD current J. The corresponding QPC conditional noise actually diverges in this limit, because the effective switching rate is vanishing. This effect, where the noise in one system (monitored by another) can be dramatically larger than the unmonitored noise, we refer to as conditional noise enhancement. The same effect persists in the asymmetric situation, and the maximum of the conditional noise occurs at . For the conditional noise peak to exceed the unconditional noise, the ratio R=|ΔΓ|Γ1Γ2/Γ03 must be less than . For data sets A and B, RA≈0.38 and RB≈0.19, so only data set B shows conditional noise enhancement. In the opposite limit, , the conditional current tends to I0, and the noise tends to zero. This situation corresponds to rapid symmetric switching between the current states, whose effective rate is again controlled by J. In both limits, the typical dynamics of the telegraph process is completely taken over by the transport condition.
Methods
We now discuss the experimental procedure. From a measurement such as shown in Fig. 1c, we can directly determine the rates Γ1,2 for electrons tunnelling into and out of the QD6,30. The tunnelling rates are controlled by tuning the voltages applied to gates G1 and G2. The data presented here were taken at two different gate voltage configurations. Configuration A is characterized by Γ1=160 Hz, Γ2=586 Hz, and configuration B is characterized by Γ1=512 Hz, Γ2=345 Hz. For each configuration, we collected traces of length T=700 s, containing around 105 tunnelling events.
Sample fabrication
Figure 1a shows the sample used in the experiments reported here. The structure was fabricated using scanning probe lithography on a GaAs/Al0.3Ga0.7As heterostructure containing a two-dimensional electron gas 34 nm below the surface.
Sample characteristics
The measurements were carried out in a 3He/4He dilution refrigerator with an electron temperature of about 190 mK, as determined from the width of thermally broadened Coulomb blockade resonances. The charging energy of the QD is 2.1 meV and the mean level spacing is 200–300 μeV.
Experimental method
The conductance of the QPC, GQPC, was tuned close to 0.5×2e2/h. We apply a d.c. bias voltage between the source and the drain of the QPC, VQPC=250 μV, and measure the current through the QPC, IQPC, which depends on the number of electrons N in the QD. The current signal was digitized with a sampling frequency of 100 kHz, thereafter software filtered at 4 kHz using an eighth-order Butterworth filter. To avoid tunnelling due to thermal fluctuations, the QD was voltage biased with a bias much larger than the temperature.
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Acknowledgements
We acknowledge the support of the Swiss National Science Foundation. We thank M. Reinwald and W. Wegscheider at the Institut für experimentelle und angewandte Physik Universität Regensburg, Germany, for fabricating the wafers used in this experiment.
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Sukhorukov, E., Jordan, A., Gustavsson, S. et al. Conditional statistics of electron transport in interacting nanoscale conductors. Nature Phys 3, 243–247 (2007). https://doi.org/10.1038/nphys564
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DOI: https://doi.org/10.1038/nphys564
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