## Abstract

In stochastic resonance, the combination of a weak signal with noise leads to its amplification and optimization^{1}. This phenomenon has been observed in several systems in contexts ranging from palaeoclimatology, biology, medicine, sociology and economics to physics^{1,2,3,4,5,6,7,8,9}. 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 resonance^{1,10,11}, 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 statistics^{12,13,14,15,16}, 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 at^{1,13}

when the doubled driving frequency 2*f* approximately corresponds to the unperturbed switching rate *Γ*_{S}. The switching rate *Γ*_{S} 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 reservoirs^{17}. This single-electron charging process is sensed directly with the capacitively coupled quantum point contact (QPC), operated as a time-resolved charge detector^{15,16}. The number *N*_{e} 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 *μ*_{N} is energetically close to the Fermi level *μ*_{F} of the tunnel-coupled reservoirs, allowing electrons to tunnel back and forth between the quantum dot and the reservoirs. The thermal energy *k*_{B}*T* = 130 μeV at *T* = 1.5 *K* is much smaller than the charging energy *E*_{c} ≈ 2.5 meV. *E*_{c} is the difference between the electrochemical potential of the *N*th 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.

We set our experimental operation point \({\mathbf{V}}_{\mathrm{OP}}\) at the first charging line, indicated as *N*_{e} = 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

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 *I*_{qpc}(*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 *I*_{qpc} jumps down, and it jumps up again when the electron tunnels out. From the detector traces we extract the times *t*_{in,out} of all ‘in’ and ‘out’ tunnelling events (Fig. 1d).

The fluctuations in the occupation are characterized by the single-electron counting statistics^{14}. We thus counted the total number *n*(*T*_{f}) of ‘in’ and ‘out’ tunnelling events within one driving period *T*_{f}. 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

calculated from the variance *σ*^{2} = 〈*n*^{2}〉 − 〈*n*〉^{2} 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 strength^{14}.

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.

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 *Γ*_{in}(*t*) and *Γ*_{out}(*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, *Γ*_{in}(0) = *Γ*_{in}(*T*_{f}/2). The out rate (blue) is shifted by a half period *T*_{f}/2 relative to the in rate and also agrees at half of the period with its initial value, *Γ*_{out}(0) = *Γ*_{out}(*T*_{f}/2). This asymmetric modulation of the rates is caused by the periodic shift of the charging state *μ*_{0}(*t*) around the symmetry level *μ*_{S}, as illustrated in Fig. 3b. At the symmetry level *μ*_{S} = *μ*_{0} \(\left( {t = 0,\frac{1}{2}T_f,T_f, \ldots } \right)\) the in and out rates are equal. In the first half of a period (0 < *t* < *T*_{f}/2) the charging state is pushed below the symmetry level, that is *μ*_{0} < *μ*_{S}, 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 (*T*_{f}/2 < *t* < *T*_{f}) the charging state is instead pushed above the symmetry level, that is *μ*_{0} > *μ*_{S}. 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 *Γ*_{in}(*t*) and *Γ*_{out}(*t*) specify a two-state Markovian process for which the counting probability *P*(*n*) can be determined^{13,14} (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.

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 2*f* ≈ 1,600 Hz approximately matches the tunnel coupling *Γ*_{s} ≈ 1,675 Hz at the symmetry level *μ*_{S}. 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 interruption^{1}. As a random variable it is characterized by a probability density function (p.d.f.), denoted as *ρ*_{1}(*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 period^{1}, that is *τ*_{k} = (2*k* + 1)*T*_{f}/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 2*f* ≈ *Γ*_{S}, *ρ*_{1}(*τ*) quickly decreases with increasing residence times with a strong shoulder at *T*_{f}/2 and a weak shoulder at 3*T*_{f}/2, indicating optimal synchronization between internal tunnelling and the external a.c. drive.

Under the assumption that the time-periodic rates *Γ*_{in/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., *ρ*_{1}(*τ*), can be calculated^{14} (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 *Γ*_{S} needs to be adapted to the double driving frequency 2*f*. The coupling *Γ*_{S} 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 *Γ*_{S} 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 2*f* = 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 state^{12,13}. 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 *Γ*_{S}. 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 *Γ*_{S}. 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 *Γ*_{S}. Hence the variation of *Γ*_{S} 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 turnstiles^{18}, ratchets^{19} and pumps^{20}, the latter being promising candidates for the redefinition of the ampere^{21}. 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 theoretical^{12,13,14} and experimental^{22} 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 measurements^{23} or corresponding heat and work distributions^{24}.The optimal working point for on-demand single-electron sources^{25,26}, 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 oscillators^{27,28}. 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 *n*_{e} = 2.4 × 10^{−11} cm, and the mobility is *μ*_{e} = 5.1 × 10^{5} cm^{2} 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 ^{4}He 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 *Γ*_{S} = 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 *V*_{qpc}(*t*).

The detector current *I*_{qpc}(*t*) was monitored with a temporal resolution of Δ*t*_{s} = 2.5 μs. For significant statistics we always recorded traces for a duration of 10 min, typically extended over 10^{5}–10^{7} driving periods and containing approximately 10^{6} tunnelling events.

### Extraction of tunnelling rates

Starting from a long detector trace of alternating in and out states the numbers *N*_{in}(*t*_{e}) (*N*_{out}(*t*_{e})) with which an electron has entered (left) the quantum dot within a bin of width Δ*t*_{s} around *t*_{e} = *t* mod *T*_{f} and the numbers *N*_{0}(*t*_{e}) (*N*_{1}(*t*_{e})) 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 Δ*t*_{s} as *p*(1, *t* + Δ*t*_{s}|0, *t*) = *N*_{in}(*t*_{e})/*N*_{0}(*t*_{e}) and, similarly, of a transition to an empty dot as *p*(0, *t* + Δ*t*_{s}|1,*t*) = *N*_{out}(*t*_{e})/*N*_{1}(*t*_{e}). For a sufficiently small bin width Δ*t*_{s}, the conditional probabilities can be expanded as *p*(1, *t* + Δ*t*_{s}|0) ≈ *Γ*_{in}(*t*)Δ*t*_{s} and (0, *t* + Δ*t*_{s}|1) ≈ *Γ*_{out}(*t*)Δ*t*_{s}, yielding for the time-periodically varying rates

The dots in Fig. 3a represent the results of equation (4) for bin width Δ*t*_{s} = *T*_{f}/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, *T*_{f}/2, …, because the charging state was not perfectly adjusted to the symmetry level *μ*_{S}. 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 state^{29,30}, which slightly shifts the symmetry level \(\mu _{\mathrm{S}} = \mu _{\mathrm{F}} + k_{\mathrm{B}}T{\mathrm{/}}\sqrt 2\) above the Fermi level *μ*_{F}. 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 *Γ*_{in/out}(*t*, *f*) = *Γ*_{in/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 *Γ*_{in}(*t*) and *Γ*_{out}(*t*), the probabilities *p*_{α}(*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 *p*_{α}(*n*;*t*, *s*) the later time as *t* = *T*_{f} + *s* with 0 ≤ *s* < *T*_{f} one obtains the counting statistics *P*(*n*;*s*) = *p*_{0}(*n*, *T*_{f} + *s*, *s*) + *p*_{1}(*n*, *T*_{f} + *s*, *s*) for the number of transitions within a period, which still depend on the initial phase 2π*s*/*T*_{f} 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 state^{12,13}.

### Residence time p.d.f

The residence time p.d.f. *ρ*_{1}(*τ*) can be obtained in terms of the conditional p.d.f. *ρ*(*τ*|*s*) = *Γ*_{out}(*τ* + *s*)*P*_{1}(*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. *ρ*_{1}(*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 *p*_{0}(*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}

Because the two tunnelling rates *Γ*_{in/out}(*t*) differ in phase but only little in form, the residence time probability density *ρ*_{0}(*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.).

## Author information

### Affiliations

#### Institut für Festkörperphysik, Leibniz Universität Hannover, Hanover, Germany

- Timo Wagner
- , Johannes C. Bayer
- , Eddy P. Rugeramigabo
- & Rolf J. Haug

#### Institut für Physik, Universität Augsburg, Augsburg, Germany

- Peter Talkner
- & Peter Hänggi

#### 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.

### Corresponding author

Correspondence to Timo Wagner.

## Supplementary information

### Supplementary Information

Supplementary Figures and Methods

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