Abstract

In stochastic resonance, the combination of a weak signal with noise leads to its amplification and optimization1. This phenomenon has been observed in several systems in contexts ranging from palaeoclimatology, biology, medicine, sociology and economics to physics1,2,3,4,5,6,7,8,9. In all these cases, the systems were either operating in the presence of thermal noise or were exposed to external classical noise sources. For quantum-mechanical systems, it has been theoretically predicted that intrinsic fluctuations lead to stochastic resonance as well, a phenomenon referred to as quantum stochastic resonance1,10,11, but this has not been reported experimentally so far. Here we demonstrate tunnelling-controlled quantum stochastic resonance in the a.c.-driven charging and discharging of single electrons on a quantum dot. By analysing the counting statistics12,13,14,15,16, we demonstrate that synchronization between the sequential tunnelling processes and a periodic driving signal passes through an optimum, irrespective of whether the external frequency or the internal tunnel coupling is tuned.

Main

Stochastic resonance is known to arise in noisy bistable systems with proper periodic modulation of the switching rates. It manifests itself as an optimal synchronization of the driving signal with the switching process, as a function of the driving frequency f as well as the noise level. As a rule of thumb, this optimum occurs at1,13

$$2f \approx \Gamma_{\mathrm{S}}$$
(1)

when the doubled driving frequency 2f approximately corresponds to the unperturbed switching rate ΓS. The switching rate ΓS thereby characterizes the noise spectrum of the system. Because stochastic resonance does not depend on the concrete physical realization of the system, this timescale matching condition should also be applicable to the tunnelling process between two quantum states. The noise source in that case is the intrinsic shot noise, which stems directly from the random quantum mechanical tunnelling dynamics. To observe stochastic resonance in an a.c.-driven quantum dot, one needs to precisely control the tunnelling process between two quantum states of the quantum dot and measure the temporal fluctuations.

Figure 1a presents a scanning electron microscopy (SEM) image of our device structure as well as a schematic of the experimental set-up. The quantum dot (green e island) is formed electrostatically by nanosized gate electrodes and behaves physically like an artificial atom, which can be charged or discharged via two tunnel-coupled electron reservoirs17. This single-electron charging process is sensed directly with the capacitively coupled quantum point contact (QPC), operated as a time-resolved charge detector15,16. The number Ne of bound electrons on the quantum dot is set in a controlled manner via the three gate voltages \({\mathbf{V}}_{\mathrm{G}} = \left( {V_{{\mathrm{d}}1},V_{{\mathrm{d}}2},V_{{\mathrm{d}}3}} \right)\). Figure 1b presents a gate-dependent charge stability diagram of the quantum dot in the few-electron regime. Along the visible charging lines, a quantum dot charge state μN is energetically close to the Fermi level μF of the tunnel-coupled reservoirs, allowing electrons to tunnel back and forth between the quantum dot and the reservoirs. The thermal energy kBT = 130 μeV at T = 1.5 K is much smaller than the charging energy Ec ≈ 2.5 meV. Ec is the difference between the electrochemical potential of the Nth and (N − 1)st electron. Therefore, only one additional electron can occupy the quantum dot at any given time (Coulomb blockade), causing sequential ‘in’ and ‘out’ tunnelling.

Fig. 1: Experimental set-up, device operation and statistical analysis.
Fig. 1

a, SEM image of the device structure with a schematic of the experimental set-up. The quantum dot (green e island) is defined electrostatically by the red gates Vd1,d2,d3 and yellow gates Vd4,d5. The QPC charge detector is formed by the blue gate Vqpc. The quantum dot and QPC current paths are galvanically isolated from each other; the visible gap between the yellow middle gates is closed electrostatically and has the purpose of enhancing detector sensitivity. The quantum dot can be charged and discharged by electrons tunnelling in from (red arrow) and tunnelling out to (blue arrow) the coupled electron reservoirs. The tunnelling process is controlled and driven periodically by the three gate voltages \({\mathbf{V}}_{\mathrm{G}} = \left( {V_{{\mathrm{d}}1},V_{{\mathrm{d}}2},V_{{\mathrm{d}}3}} \right)\). Crossed squares indicate the ohmic contacts of the sample. b, Charge stability diagram of the quantum dot in the few-electron regime. The operation point \({\mathbf{V}}_{{\mathrm{G}},{\mathrm{OP}}}\) is indicated by a circle with a dot at the charging line Ne = 1. The gate voltages \({\mathbf{V}}_{\mathrm{G}}\) are periodically modulated along the highlighted direction \({\mathbf{e}}_{\mathrm{d}}\). c, A typical time-resolved current trace Iqpc(t) of the QPC charge detector, revealing the sequential charging and discharging of the quantum dot. The shown trace was recorded with a drive of f = 800 Hz and an amplitude of A = 10 mV. d, Extracted ‘in’ (red) and ‘out’ (blue) tunnelling events relative to the external a.c. drive. The times tin,tout and the residence times τ are extracted from the detected events. e, Counting statistics P(n) for the number of tunnelling events within one driving period Tf. The Fano factor F = σ2/〈n〉, equation (3), of the distribution is calculated from the mean 〈n〉 and variance σ2 = 〈n2〉 − 〈n2 of the distribution.

We set our experimental operation point \({\mathbf{V}}_{\mathrm{OP}}\) at the first charging line, indicated as Ne = 1 in Fig. 1b, where the electron number on the quantum dot fluctuates between zero and one. For the driving we periodically modulated the three gate voltages

$${\mathbf{V}}_{\mathrm{G}}(t) = {\mathbf{V}}_{\mathrm{OP}} + {\mathbf{e}}_{\mathrm{d}}{\kern 1pt} A\,{\mathrm{sin}}(2\pi ft)$$
(2)

with frequency f, amplitude A and in the direction of the unit vector \({\mathbf{e}}_{\mathrm{d}}\).

To study the synchronization between the deterministic external a.c. drive and internal stochastic tunnelling process, the detector current Iqpc(t) was monitored. A short snapshot of a typical detector trace is displayed in Fig. 1c, which directly reveals the sequential charging and discharging of the quantum dot. Whenever an electron tunnels into the quantum dot the current Iqpc jumps down, and it jumps up again when the electron tunnels out. From the detector traces we extract the times tin,out of all ‘in’ and ‘out’ tunnelling events (Fig. 1d).

The fluctuations in the occupation are characterized by the single-electron counting statistics14. We thus counted the total number n(Tf) of ‘in’ and ‘out’ tunnelling events within one driving period Tf. From all possible periods and phases of the a.c. drive we finally obtained a counting probability P(n) (Fig. 1e). A quantitative measure for the fluctuations is given by the Fano factor

$$F = \frac{{\sigma ^2}}{{\left\langle n \right\rangle }} = \frac{{\left\langle {n^2} \right\rangle - \left\langle n \right\rangle ^2}}{{\left\langle n \right\rangle }}$$
(3)

calculated from the variance σ2 = 〈n2〉 − 〈n2 and the mean 〈n〉 of the counting probability P(n). For a strictly Poissonian random process the Fano factor equals F = 1. Without external a.c. drive the Fano factor always exceeds or equals this Poissonian limit (that is, F ≥ 1) due to the bunching of tunnelling-in and tunnelling-out events (Supplementary Fig. 1). The driving-dependent antibunching of tunnelling events, signalled by a suppression of the Fano factor below the Poissonian limit (that is, F < 1) thus provides an unambiguous measure of the synchronization strength14.

The red circles in Fig. 2 depict the experimentally determined Fano factor F as a function of external driving frequency f. Additionally, the inset shows the corresponding mean counts 〈n〉 per period. The amplitude A = 10 mV was kept constant for all frequencies. The experimental Fano factor F has a minimum at f = 800 Hz, providing evidence for stochastic resonance. At the minimum, essentially two tunnelling events (one ‘in’ and one ‘out’) occur on average per driving period (〈n〉 ≈ 2), corroborating the optimal synchronization between external a.c. drive and internal tunnelling. If the a.c. drive is too slow \(\left( {f \ll 800\,{\mathrm{Hz}}} \right)\), more than two tunnelling events occur within one period \(\left( {\left\langle n \right\rangle \gg 2} \right)\) and the Fano factor F increases above the Poissonian limit. If the drive is instead too fast \(\left( {f \gg 800\,{\mathrm{Hz}}} \right)\), most electrons need multiple periods to tunnel in and out of the quantum dot \(\left( {\left\langle n \right\rangle \ll 2} \right)\) and the Fano factor F approaches the Poissonian limit from below.

Fig. 2: Frequency-dependent stochastic resonance.
Fig. 2

Experimental (red circles) and theoretical (black line) Fano factor F as a function of driving frequency f. The experimental data have a minimum at f = 800 Hz, in good agreement with the theoretical resonance frequency. Inset, Corresponding average number of tunnelling events per period 〈n〉. At the resonance frequency (vertical red line) nearly two tunnelling events (〈n〉 ≈ 2) occur on average per period.

Because the ‘in’ and the ‘out’ tunnelling events are alternating, and hence their occurrences are strictly dependent, stochastic resonance can be equally characterized on the basis of the occurrences of the specific tunnel events (Supplementary Fig. 2) or on the basis of the set of all transitions independent of their direction, as we do here. To demonstrate, in the presently investigated system, that the general mechanisms of stochastic resonance are at play, we extracted the tunnelling rates Γin(t) and Γout(t) with which an electron enters and leaves the quantum dot at time t, respectively, from the experimental data (see Methods).

Figure 3a displays the tunnelling-in and tunnelling-out rates for a quantum dot with driving frequency f = 800 Hz and amplitude A = 10 mV in the direction \({\mathbf{e}}_{\mathrm{d}}\), as indicated in Fig. 1b. The in rate (red) instantly follows the external a.c. driving voltage, as given by equation (2), without any visible delay, corroborated by the fact that the rate at half of the period agrees with its initial value, Γin(0) = Γin(Tf/2). The out rate (blue) is shifted by a half period Tf/2 relative to the in rate and also agrees at half of the period with its initial value, Γout(0) = Γout(Tf/2). This asymmetric modulation of the rates is caused by the periodic shift of the charging state μ0(t) around the symmetry level μS, as illustrated in Fig. 3b. At the symmetry level μS = μ0 \(\left( {t = 0,\frac{1}{2}T_f,T_f, \ldots } \right)\) the in and out rates are equal. In the first half of a period (0 < t < Tf/2) the charging state is pushed below the symmetry level, that is μ0 < μS, causing an enhancement of the in rate and a suppression of the out rate. An electron tunnels from the reservoir most probably into the quantum dot when the charge state is at its energetically lowest position \(\left( {t = \frac{1}{4}T_f,\frac{5}{4}T_f, \ldots } \right)\). In the second half of a period (Tf/2 < t < Tf) the charging state is instead pushed above the symmetry level, that is μ0 > μS. There, the in rate is suppressed and the out rate is enhanced. The best opportunity to tunnel out occurs at the energetically highest position \(\left( {t = \frac{3}{4}T_f,\frac{7}{4}T_f, \ldots } \right)\). The transition rates Γin(t) and Γout(t) specify a two-state Markovian process for which the counting probability P(n) can be determined13,14 (see Methods). The theoretical results for the first moment of the number of transitions and the Fano factor are in good agreement with the respective direct experimental outcomes, as can be seen from Figs. 2 and 4.

Fig. 3: Temporal modulation of the tunnelling process.
Fig. 3

a, Experimentally time-dependent tunnelling-in rate Γin(t) (red dots) and tunnelling-out rate Γout(t) (blue dots). The rates were extracted for a drive with f = 800 Hz and A = 10 mV. The fits (solid lines) are based on a Fourier expansion of the logarithm of the rates. The horizontal black line marks the tunnel coupling ΓS = 1,675 Hz at symmetry level μS. b, Asymmetric modulation of the rates is caused by the periodic shift of the quantum dot charging state around the symmetry level μS. At the symmetry level μS (t = 0, Tf/2), the two tunnelling rates are identical. An electron tunnels from reservoirs most probably into the quantum dot when the charge state is at its lowest position (t = Tf/4). The best opportunity to tunnel out of the quantum dot occurs when the charging state reaches its energetically highest position (t = 3Tf/4). c, Periodic modulation of the tunnelling process is also observable in the residence time probability density ρ1(τ). For a fast driving frequency f = 10 kHz, the residence time probability density function (p.d.f.) ρ1t) displays a train of maxima at odd integer multiples of the half period τk = (2k + 1)Tf/2, k = 0,1,2,… At the lower driving frequency f = 800 Hz, the maximum at Tf/2 is accompanied by considerably suppressed satellites. In both cases the agreement between experiment (dots) and theory (solid lines) is striking.

Fig. 4: Tunnel coupling-dependent stochastic resonance.
Fig. 4

a, The tunnel coupling ΓS of the quantum dot depends on the size of the tunnel barriers, which can be altered by choosing a different operation point \({\mathbf{V}}_{{\mathrm{G}},{\mathrm{OP}}}\) on the first charging line. This way, the tunnel coupling can be tuned from values smaller than to values much larger than the double driving frequency 2f. b, Experimental (green diamonds) and theoretical (black line) Fano factor F as a function of tunnel coupling ΓS. The experimental data have a minimum at ΓS = 1.75 kHz, in good agreement with the minimum of the theoretical model. The driving parameters were f = 1 kHz and A = 10 mV. Inset, Corresponding average number of tunnelling events per period 〈n〉 as a function of tunnel coupling. At the resonance frequency (vertical green line) nearly two tunnelling events (for example, 〈n〉 ≈ 2; one ’in’ and one ’out’) occur on average per period.

The resonance frequency f = 800 Hz at which the Fano factor attains its minimum also conforms well with the stochastic rule of thumb. The double resonance frequency 2f ≈ 1,600 Hz approximately matches the tunnel coupling Γs ≈ 1,675 Hz at the symmetry level μS. When the driving frequency is much faster than the tunnel coupling (that is, \(\Gamma_{\mathrm{s}} \ll 2f\), an electron easily ‘misses’ the first good opportunity to tunnel and will wait until another good opportunity occurs, one or several periods later. These findings show that the variation of the frequency f conforms well with the idea of stochastic resonance as a resonance phenomenon.

The so-called ‘residence time’ is given by the time span during which a quantum dot is occupied without interruption1. As a random variable it is characterized by a probability density function (p.d.f.), denoted as ρ1(τ). The blue dots in Fig. 3c display the experimentally determined residence time p.d.f. The residence time p.d.f. for fast driving with f = 10 kHz consists of a train of maxima at odd integer multiples of the half period1, that is τk = (2k + 1)Tf/2, k = 0,1,2,…, demonstrating the waiting of the electron for a good tunnelling opportunity. At the resonance frequency f = 800 Hz instead, obeying 2f ≈ ΓS, ρ1(τ) quickly decreases with increasing residence times with a strong shoulder at Tf/2 and a weak shoulder at 3Tf/2, indicating optimal synchronization between internal tunnelling and the external a.c. drive.

Under the assumption that the time-periodic rates Γin/out(t) govern a Markovian process of alternate visits of the occupied (1) and empty (0) quantum dot states, the residence time p.d.f., ρ1(τ), can be calculated14 (see Methods). Figure 3c demonstrates that the residence time p.d.f.s from the Markovian model (solid lines) agree nicely with experimental statistics.

So far, we have succeeded in showing that the synchronization can be optimized by tuning the driving frequency f. Alternatively, one may aim at synchronizing a signal at fixed frequency f. Thereby, the tunnel coupling ΓS needs to be adapted to the double driving frequency 2f. The coupling ΓS depends on the size of the tunnel barriers, as illustrated in Fig. 4a. We tuned the coupling by shifting the operation point \({\mathbf{V}}_{{\mathrm{G}},{\mathrm{OP}}}\) along the charging line (Fig. 1b). The driving frequency f = 1 kHz and the amplitude A = 10 mV were kept constant. The experimentally determined Fano factor F is plotted as a function of the tunnel coupling ΓS in Fig. 4b and the inset shows the corresponding mean 〈n〉 of the counting statistics. The experimental Fano factor F has a minimum at 1.75 kHz, which is close to the double driving frequency 2f = 2 kHz. At the minimum we also find nearly two tunnelling events per period (〈n〉 ≈ 2), corroborating the optimal synchronization of the Rice frequency defined as the rate of transition events into the occupied state12,13. Furthermore, we compared the experimental results with the outcome of the Markovian two-state model. According to our empirical finding the rates at different operation points are proportional to the tunnel coupling ΓS. Therefore, the rates are given by \(\Gamma _{{\mathrm{in/out}}}\left( {t,\Gamma _{\mathrm{S}}} \right) = \left( {\Gamma _{\mathrm{S}}/\Gamma _{\mathrm{S}}^ \ast } \right)\Gamma_{{\mathrm{in/out}}} \left( {t,\Gamma _{\mathrm{S}}^ \ast } \right)\). As reference we chose the extracted tunnelling rates (see Methods) with \(\Gamma _{\mathrm{S}}^ \ast\) = 1,675 Hz. Figure 4 displays good agreement between the resulting theoretical Fano-factors and the average numbers of transitions with the according experimental findings.

We emphasize that there is a crucial difference between the variation of the external driving frequency f and the internal tunnel coupling ΓS. With a change of frequency f, the time averages of the transition rates as well as the values of their minima and maxima remain the same, while both rate characteristics alter upon a variation of the tunnel coupling ΓS. Hence the variation of ΓS corresponds more closely to the original characterization of the stochastic resonance phenomena, indicating that an increase of the noise up to a certain level can lead, counter-intuitively, to an improved signal-to-noise ratio.

Such a.c.-driven single-electron tunnelling has also been studied intensively in the form of turnstiles18, ratchets19 and pumps20, the latter being promising candidates for the redefinition of the ampere21. The functionality of these devices is based on the locking of the mean counts 〈n〉 = const., meaning that the number of transferred electrons per period is frequency independent and robust against noise. For the stochastic resonance discussed here with a relatively weak driving amplitude A = 10 mV, locking is not present, as is evident from the inset of Fig. 2. The observed synchronization is a consequence of the timescale matching of the external a.c. drive and the internal tunnelling process. At a larger amplitude, A = 30 mV (Supplementary Fig. 3), however, the average number of transitions starts to develop plateaux around the respective resonance values of frequency and tunnel coefficient, in accordance with theoretical12,13,14 and experimental22 observations for strongly a.c.-driven stochastic resonance systems. This confirms that the locking in a.c.-driven single-electron devices is subject to the same statistical physics and can be seen as a special case of stochastic resonance in the limit of strong a.c. driving.

Stochastic resonance in a.c.-driven single-electron tunnelling should also occur in direct shot-noise measurements23 or corresponding heat and work distributions24.The optimal working point for on-demand single-electron sources25,26, which provide an important toolbox for quantum electronics, is defined by tunnelling-driven stochastic resonance. Quantum dots have been utilized successfully as displacement sensors for nanomechanical oscillators27,28. Their resolution is thought to be limited by the standard quantum limit, due to the stochastic backaction of the single-electron tunnelling process. The phenomenon of tunnelling-driven stochastic resonance provides a way to overcome this limit.

Methods

Experimental set-up

Our quantum dot device is based on a GaAs/AlGaAs heterostructure, which forms a two-dimensional electron gas (2DEG) 100 nm below the surface. The 2DEG charge carrier density is ne = 2.4 × 10−11 cm, and the mobility is μe = 5.1 × 105 cm2 V−1 s−1. On the surface we patterned nanosized metallic top gates (7 nm Cr, 30 nm Au) using electron-beam and optical lithography. The quantum dot and QPC are formed electrostatically. By applying negative voltages to the gates, we deplete the 2DEG below.

All measurements were carried out on a low-noise d.c. transport set-up in a 4He cryostat at 1.5 K. Filtering and signal amplification were fully carried out outside the cryostat at room temperature. All gates were filtered by a 1 MHz low-pass filter. The QPC source was filtered by a 10 Hz low-pass filter and a 1:1,000 voltage divider was used to increase the resolution. The QPC charge detector current was amplified with a low-noise FEMTO transimpedance amplifier (100 MV A−1 gain, 100 kHz bandwidth), connected to the QPC drain by a 25 pF low-capacity coaxial line. An Adwin Pro2 real-time system was used (1 GHz ADSP T12) to supply the voltage (16-bit digital-to-analog convertor card) and to record the QPC detector signal (18-bit analog-to-digital convertor card) for the statistical analysis. The a.c. signals were also generated by the ADwin system. The input and output sampling rates were ΓS = 400 kHz.

To minimize the crosstalk with the drive, the detector was kept at a constant working point by periodically adjusting the QPC gate voltage Vqpc(t).

The detector current Iqpc(t) was monitored with a temporal resolution of Δts = 2.5 μs. For significant statistics we always recorded traces for a duration of 10 min, typically extended over 105–107 driving periods and containing approximately 106 tunnelling events.

Extraction of tunnelling rates

Starting from a long detector trace of alternating in and out states the numbers Nin(te) (Nout(te)) with which an electron has entered (left) the quantum dot within a bin of width Δts around te = t mod Tf and the numbers N0(te) (N1(te)) representing how often the quantum dot is empty (occupied) within the same bin are determined. On the basis of these numbers one can estimate the conditional probabilities of a transition to the occupied state within the interval Δts as p(1, t + Δts|0, t) = Nin(te)/N0(te) and, similarly, of a transition to an empty dot as p(0, t + Δts|1,t) = Nout(te)/N1(te). For a sufficiently small bin width Δts, the conditional probabilities can be expanded as p(1, t + Δts|0) ≈ Γin(tts and (0, t + Δts|1) ≈ Γout(tts, yielding for the time-periodically varying rates

$$\begin{array}{l}\Gamma _{{\mathrm{in}}}(t) = \frac{1}{{{\mathrm{\Delta }}t_{\mathrm{s}}}}\frac{{N_{{\mathrm{in}}}\left( {t_e} \right)}}{{N_0\left( {t_e} \right)}}\\ \Gamma _{{\mathrm{out}}}(t) = \frac{1}{{{\mathrm{\Delta }}t_{\mathrm{s}}}}\frac{{N_{{\mathrm{out}}}\left( {t_e} \right)}}{{N_1\left( {t_e} \right)}}\end{array}$$
(4)

The dots in Fig. 3a represent the results of equation (4) for bin width Δts = Tf/500. Solid lines are fits based on a Fourier expansion of the logarithm of the rate with at most three higher harmonics. The experimental in and out rates do not cross exactly at t = 0, Tf/2, …, because the charging state was not perfectly adjusted to the symmetry level μS. Also, the experimental rates differ slightly in their form. This has two main causes. First, the two tunnelling processes are differently affected due the twofold spin degeneracy of the first charging state29,30, which slightly shifts the symmetry level \(\mu _{\mathrm{S}} = \mu _{\mathrm{F}} + k_{\mathrm{B}}T{\mathrm{/}}\sqrt 2\) above the Fermi level μF. Second, the a.c. drive also modulates the size of the tunnelling barriers.

Generally, the in and out rates exhibit a non-trivial dependence on the driving frequency f. Because, in the present case, the driving frequency f is much smaller than any electronic or phononic timescale relevant for the tunnelling process of an electron, the rates depend adiabatically on the frequency f, that is Γin/out(t, f) = Γin/out(tf/f*, f*), where f* is a reference frequency, which we chose as f* = 800 Hz.

Counting statistics

For the two-state Markov process with periodically time-dependent rates Γin(t) and Γout(t), the probabilities pα(n;t, s) to find the quantum dot at time s in the empty or the occupied state α = 0, 1, respectively, and to observe n transitions until the time t > s can be calculated as the solution of a hierarchy of first-order differential equations with respect to time t (refs. 13,14). This hierarchy can be solved successively starting at n = 0. Choosing in pα(n;t, s) the later time as t = Tf + s with 0 ≤ s < Tf one obtains the counting statistics P(n;s) = p0(n, Tf + s, s) + p1(n, Tf + s, s) for the number of transitions within a period, which still depend on the initial phase 2πs/Tf at which the counting window begins. For a comparison with the experimental, phase-averaged results the mean over this phase is performed, yielding the averaged counting statistics \(P(n) = {\int}_0^{T_f} {\kern 1pt} {\mathrm{d}}sP(n;s){\mathrm{/}}T_f\), from which the moments of n and the Fano factor can be determined. The theoretical results for the first moment of the number of transitions and the Fano factor are in good agreement with the respective direct experimental outcomes, as can be seen from Figs. 2 and 4. The average number of transitions per period at the minimum is only slightly larger than two, in accordance with the matching condition of the driving frequency with the average Rice frequency defined as the rate of transition events into the occupied state12,13.

Residence time p.d.f

The residence time p.d.f. ρ1(τ) can be obtained in terms of the conditional p.d.f. ρ(τ|s) = Γout(τ + s)P1(τ + s|s) to find the quantum dot occupied without interruption during the time span (s, s + τ), where \(P_1(\tau + s{\mathrm{|}}s) = {\mathrm{exp}}\left\{ { - {\int}_s^{\tau + s} {\kern 1pt} {\mathrm{d}}t\Gamma _{{\mathrm{out}}}(t)} \right\}\) denotes the probability for a quantum dot being permanently occupied from s to τ + s. To obtain the residence time p.d.f. the conditional p.d.f. ρ1(τ|s) must be averaged with respect to the time s at which an electron occupies the quantum dot. These events occur according to the p.d.f. \(\rho _{{\mathrm{in}}}(s) = \Gamma _{{\mathrm{in}}}(s)p_0(s){\mathrm{/}}{\int}_0^{T_f} {\kern 1pt} {\mathrm{\Gamma }}_{{\mathrm{in}}}(s)p_0(s)\). Because we disregard initial transients, the probability p0(s) to find the quantum dot unoccupied at time s is determined as the asymptotic, and hence periodic, solution of the master equation \(\dot p_0(t) = - \left[ {\Gamma _{{\mathrm{in}}}(t) + \Gamma _{{\mathrm{out}}}(t)} \right]p_0(t) + \Gamma _{{\mathrm{out}}}(t)\). This yields, for the residence time, the result p.d.f.14

$$\rho _1(\tau ) = \frac{{{\int}_0^{T_f} {\kern 1pt} {\mathrm{d}}s\Gamma _{{\mathrm{out}}}(\tau + s{\mathrm{|}}s)P_1(\tau + s{\mathrm{|}}s)\Gamma _{{\mathrm{in}}}(s)p_0(s)}}{{{\int}_0^{T_f} {\kern 1pt} {\mathrm{d}}s\Gamma _{{\mathrm{in}}}(s)p_0(s)}}$$
(5)

Because the two tunnelling rates Γin/out(t) differ in phase but only little in form, the residence time probability density ρ0(τ) for the unoccupied (intervals between ‘out’ and ‘in’ events) quantum dot is almost identical to that of the occupied quantum dot.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.

Additional information

Journal peer review information: Nature Physics thanks Christian Flindt and the other anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Acknowledgements

This work was supported financially by the Research Training Group 1991 (DFG), the School for Contacts in Nanosystems (NTH), the Center for Quantum Engineering and Space-Time Research (QUEST), the Laboratory for Nano and Quantum Engineering (LNQE) and the ‘Fundamentals of Physics and Metrology’ initiative (T.W, J.C.B., E.R. and R.J.H.).

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Affiliations

  1. Institut für Festkörperphysik, Leibniz Universität Hannover, Hanover, Germany

    • Timo Wagner
    • , Johannes C. Bayer
    • , Eddy P. Rugeramigabo
    •  & Rolf J. Haug
  2. Institut für Physik, Universität Augsburg, Augsburg, Germany

    • Peter Talkner
    •  & Peter Hänggi
  3. Nanosystems Initiative Munich, Munich, Germany

    • Peter Hänggi

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Contributions

T.W. carried out the experiments, analysed the data and wrote the manuscript. J.C.B. and T.W. fabricated the device. E.P.R. grew the wafer material. P.T. and P.H. provided theory support. T.W., P.T., P.H. and R.J.H discussed the results. R.J.H. supervised the research. All authors contributed to editing the manuscript.

Competing interests

The authors declare no competing interests.

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Correspondence to Timo Wagner.

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https://doi.org/10.1038/s41567-018-0412-5