A schematic illustration of our sample structure and experimental setup is shown in Fig. 1a. A self-assembled InAs/GaAs dot is embedded in a p-i-n diode structure for electrical control (details in methods section and in supplementary note 1) and cooled in a bath cryostat to 4.2 K. The dot is charged and discharged by tunneling of electrons from or to an electron reservoir, i.e., a highly doped GaAs layer below the dot. A gate voltage V allows us to tune the occupation probability of the QD from zero to one, as the lowest electron state is energetically shifted through the Fermi edge of the electron reservoir (see upper insets in Fig. 1b). The charge state is detected in real-time by means of resonance fluorescence from the neutral exciton transition.
We use the dark-field technique (crossed excitation-collection polarization) to separate the reflected laser light from the QD signal. The light emitted from the QD is detected by an avalanche photo diode (APD) that serves as a single-photon detector (see methods section). The detected photons are accumulated over a binning time t to obtain a measure of the fluorescence brightness.
Figure 1b shows the resonance fluorescence (RF) of the exciton transition (X) with its fine-structure splitting as a function of gate voltage V and laser frequency. Due to the quantum confined-Stark effect, the applied gate voltage shifts the frequency of the exciton resonance towards higher energies as the gate voltage is increased. In addition, the energy of the quantum dot states will decrease with respect to the chemical potential μ of the electron reservoir until, around V = 0.5 V, an electron can tunnel into the s-state of the dot, marked by the dashed line in Fig. 1b and schematically illustrated by the inset in the upper right corner. This shifts the optical resonance, and the neutral exciton transition vanishes. Close to this gate voltage, the QD charge fluctuates in time and the exciton transition can be used as a sensitive optical detector to observe all tunneling events in real-time.
The gate voltage range between the QD being uncharged and charged is given by the reservoir temperature of 4.2 K and a gate-voltage-to-energy conversion factor ('lever arm') of 15.2 mVmeV. This is demonstrated in Fig. 2a, which displays the tunneling rates γ and γ for tunneling into and out of the QD, that were determined by means of the pulsed measurement scheme presented in refs. . For V < 0.47 V, γ vanishes and the QD is mostly empty, while for V > 0.52 V, γ vanishes, and the QD becomes occupied with a single electron. In between, the average QD occupation, given by γ/(γ + γ), changes continuously from 0 to 1 with increasing gate voltage. The energetic position of the QD state with respect to the chemical potential μ of the electron reservoir is shown in three schematic insets. At V = 494 mV, the tunneling rates γ and γ are approximately equal, with electrons randomly tunneling back and forth between the QD and reservoir. Note that here no Auger process is possible because only the exciton (and not the trion) transition is driven. A charged QD yields a low ("off") fluorescence signal while the uncharged dot leads to a bright ("on") fluorescence. The result is a random telegraph signal in the optical response of the QD.
To drive the system, we modulate the gate voltage with a square function, as shown at the bottom of Fig. 2b and also indicated by the modulation scale bars in Fig. 2a. Stochastic resonance can be qualitatively characterized by the switching behavior of the charge state: the switching becomes more regular at resonance while it is dominated by fluctuations away from it.
In Fig. 2b, a photon stream without modulation is shown, where the electron tunnels in and out randomly. With a modulation amplitude of 16 mV, though, even the unaided eye can discern that the low/high value of the gate voltage (white/grey region in Fig. 2b) is predominantly accompanied with a large/small RF signal in the telegraph stream. Hence, the charge switching is synchronized with the driving voltage, indicating stochastic resonance.
In Fig. 2b, center, the synchronization between voltage drive and charge-state switching is evident. A systematic analysis of quantum stochastic resonance, however, requires a quantitative tool not only to characterize how regular the switching occurs but also to identify the resonance condition, i.e. to find (for given gate voltage values) the modulation frequency f for which the switching is most regular. For this, a statistical analysis of the switching behavior needs to be performed.
To address the switching dynamics, we make use of the framework of full counting statistics, applied to the number of switching events within a given time span. The central quantity is the probability distribution P(Δt) that N switching events have taken place within an interval of length Δt. To be specific, we decide to count only tunneling-out events. The alternative choices of counting only tunneling-in events or counting both types of tunneling events (as it was done in ref. ) would render our conclusions about the stochastic resonance unchanged. An example for the distribution P(5 ms) for a modulation amplitude of 16 mV and frequency of f = 796 Hz is shown in Fig. 3. The distribution P(5 ms) (red dots) is broadened as compared to a completely regular behavior of one tunneling-out event per modulation cycle (gray). This reflects the stochastic nature of tunneling. The measured distribution is, however, much sharper than a Poisson distribution with the same mean value (blue line). The latter would be expected if all tunneling-out events were independent of each other. The reduced width of the measured distribution, thus, indicates that the switching behavior has become more regular. We remark that this is not yet a unique indicator of stochastic resonance, since even in the undriven system correlations (due to Pauli exclusion and electron-electron interactions) lead to sub-Poissonian statistics.
A more quantitative comparison of a given distribution, within the completely random and completely regular limits, also used by previous experimental studies, is provided by the signal-to-noise ratio or its inverse, the Fano factor
i.e. the ratio of the variance σ = 〈N〉 - 〈N〉 and the mean value λ = 〈N〉 for the number N of measured switching events. If the tunneling-out events occurred completely independent of each other, the full counting statistics would be Poissonian, which implies a Fano factor of 1. However, correlations are always present since after a tunneling-out event an electron has to tunnel in before the next tuneling-out can occur. For the undriven case, the Fano factor is known to be , which ranges between values of F = 1 for γ ≪ γ or γ ≪ γ and F = 0.5 for γ = γ. The measured value of F = 0.35 in Fig. 4, therefore, indicates a more regular switching behavior in the presence of driving. In the limit of completely regular switching, the Fano factor would be F = 0. With this tool at hand, stochastic resonance can be defined by the minimum of F as a function of the driving frequency for fixed switching rates or as a function of switching rates for fixed driving frequency.
The characterization of such a complex process as quantum stochastic resonance, using only two numbers λ and σ, however, seems to be limited. Deeper insight is expected from a more complete theoretical evaluation. As one possibility, the spectral properties of the noise, expressed in terms of a frequency-dependent Fano factor, has been employed for the analysis of the quantum stochastic resonance. Here, we suggest an alternative route that is based on the idea that a distribution function can be completely characterized in terms of cumulants. The mean value λ and the variance σ are nothing but the first- and second-order ordinary cumulants. They are sufficient to fully describe a Gauss distribution. Deviations from a Gauss distribution are expressed in terms of third- and higher-order ordinary cumulants. They have been measured in quantum-dot systems up to 20-th order. Our aim, however, is to measure deviations from a Poisson rather than a Gauss distribution. Therefore, we make use of factorial cumulants, which are better suited for discrete rather than continuous stochastic variables. They prevent unwanted universal oscillations and are resilient against detector imperfections. They have proven successful in determining the rates for spin relaxation as well as Auger recombination and spin-flip Raman scattering from random telegraph signals such as the ones investigated here.
The factorial cumulants C of order m can be calculated using the generating function
i.e., the logarithm of the z-transform of the probability distribution P, and taking their derivatives with respect to the counting variable z,
Explicit expressions for the first four factorial cumulants in terms of moments 〈N〉 are given in Eqs. (S2a-d), see supplementary note 5.
Using the factorial cumulants, the Fano factor can be alternatively expressed as
where the first-order factorial cumulant C = 〈N〉 = λ is the mean value of counts. Higher-order factorial cumulants vanish for a Poisson distribution of completely uncorrelated discrete events, for m≥2. In the opposite limit of fully regular switching behavior, the distribution is described by a Kronecker delta, which yields .
For a quantitative measure of the regularity of the switching behavior, it is useful to normalize the measured factorial cumulants C with . We, thus, define for m≥2 the normalized factorial cumulants
that are all 0 for a Poisson and 1 for a Kronecker distribution.
To infer the factorial cumulants (and Fano factor) from the measured random telegraph stream of 15 min for a given time interval length Δt, we first determine the probability distribution P(Δt) for the number N of tunneling events. Then, we calculate the moments 〈N〉 up to m = 4 and insert them into Eq. (1) for the Fano factor and Eqs. (S2a-d) shown in supplementary note 5 for the factorial cumulants, respectively.
Having established that normalized factorial cumulants x provide a quantitative measure how regular or random a process is, we define the quantum stochastic resonance frequency as the frequency that maximizes x. We will show below that the frequency that maximises x depends on the order m of the chosen factorial cumulant, raising the question of how rigorous a resonance frequency can be defined in the context of stochastic resonance.
We begin with a discussion of x. It is related to the Fano factor by F = 1 - x, i.e. a maximum in x is equivalent to a minimum in F. In the field of quantum optics, - x = C/C = : Q is also known as the Mandel Q-parameter to characterize deviations from Poissonian photon statistics.
The distribution P(Δt) depends on the length Δt of the time interval in which the tunneling-out events are counted. Furthermore, the accuracy with which the random telegraph signal reflects the actual charge switching of the system depends on the binning time t, which is the time interval used to group the photon arrival events. The binning time t is chosen large enough to distinguish the charged from the uncharged state by a clear separation of the two peaks in the histogram of the RF counts per bin, see Fig. S2 in supplementary note 2. Both times, Δt and t, can be chosen a posteriori from the measured stream of individual photon counts. They introduce two new time scales that are unrelated to stochastic resonance. We must therefore optimize them in our quest for stochastic resonance to avoid artifacts.
In Fig. 4a, we show the Fano factor as a function of Δt for different modulation frequencies ranging from 52 Hz up to 5 kHz. For all data sets that each comprise a time span of 15 min, we used a modulation amplitude of 16 mV and a binning time of t = 100 μs. Each data set contains 600k events on average. For very short intervals Δt, most intervals accommodate no or, with a small probability, only one tunneling-out event. This can be described by a Bernoulli distribution with 0 ≤ q ≪ 1, for which the Fano factor F = 1 - q approaches the Poissonian value F → 1 for Δt → 0 (which implies q → 0). As a result, operating in the short-time limit is inappropriate for unraveling quantum stochastic resonance. Instead, it is advantageous to go to the long-time limit, in which each interval contains many counts of tunneling-out events such that the Fano factor becomes independent of Δt. In the following, we choose Δt = 20 ms (indicated by the vertical dashed line in Fig. 4a) to ensure that, on the one hand, the long-time limit is already reached and that, on the other hand, slow, long-term fluctuation in the experimental setup will not become relevant yet.
Since the period T = 1/f of the gate modulation already introduces a time scale, one may be tempted to choose this time scale for Δt as well. This, however, may introduce artifacts as shown in Fig. 4a, where the choice Δt = T/2 is indicated by black dots. We find that, at least for low frequencies, the Fano factor is enhanced around this choice of Δt. To identify quantum stochastic resonance, we vary the modulation frequency f. Fig. 4a shows that for all values of Δt the Fano factor first decreases with increasing modulation frequency (from f = 52 Hz to f = 796 Hz), and subsequently increases when the modulation frequency is set to 5 kHz (or larger). This is shown more clearly in Fig. 4b, which depicts the Fano factor as a function of the modulation frequency f for different values of the binning time t. The interval length is fixed at the long-time limit of Δt = 20 ms. For a binning time of t = 100 μs, as chosen in Fig. 4a, the Fano factor has a minimum at f = 796 Hz.
The binning time t affects the accuracy of the charge-state switching detection. A short binning time provides high time resolution, however, increases noise, leading to false event detection. This explains why at t = 25 μs, the Fano factor shows no clear minimum. Increasing t reduces false detections. However, this comes at the expense of time resolution, which results in fast events being missed. At t = 200 μs, the Fano factor shows a clear minimum, however, slightly shifted. As a compromise, we choose t = 100 μs, the smallest value where the resonance remains well pronounced.
In Fig. 5a and b, we show the Fano factor as a function of the modulation frequency for five different modulation amplitudes. In all cases, the average gate voltage is chosen close to the value where the in- and out-tunneling rates are equal to each other (the exact values of the upper, lower and average value of the gate voltage is given in Table S1 in supplementary note 3). A modulation amplitude of 1 mV is not sufficient to identify stochastic resonance. With increasing amplitude, however, a pronounced minimum develops around f ≈ 800 Hz. For a modulation amplitude of 16 mV, the Fano factor drops below 0.35, see Fig. 5a.
In Fig. 5b, we compare the Fano-factor for a moderate modulation amplitude (16 mV) to that for a high amplitude (87 mV). For the high amplitude, we not only observe a deep minimum, down to F = 0.15. We also find that the position of the stochastic resonance is strongly shifted towards a lower frequency of about 500 Hz, which is discussed in the next section. We note an experimental detail here: Due to the larger modulation amplitude, the laser is no longer in resonance with the exciton X transition (see Fig. 2a) during the half cycle with the larger gate voltage. As a result, the electron dynamics by photon counting is only detected during the lower gate voltage phase.
For all voltage amplitudes, we compare the measured data (dots) with a theoretical model (lines) that takes the finite time resolution of the detector into account, see Methods. We find that both the position and the depth of the minimum are well reproduced, see Fig. 5. Missed events due to the finite time resolution of the detector reduce the measured C while C remains almost unaffected due to the resilience of the higher-order factorial cumulants against detector imperfections. This decreases the Fano factor but hardly moves the position of the minimum.
The value of the resonance frequency f = 1/T depends in a complicated way on all four rates γ, γ, γ and γ, where γ is the rate for charging the quantum dot during the low/high value of the gate voltage and, similarly γ, the rate for discharging. There are, however, two limits in which simple analytical expressions can be derived (shown in Fig. 5 as vertical lines). To achieve this, we compare the modulation period T = 1/f with the waiting times for the quantum dot being in a given state before it switches, i.e., the inverse of the corresponding rate.
For small modulation amplitudes, i.e. in the linear-response regime, we find the condition for stochastic resonance by requiring that there is exactly one tunneling-out event (and, therefore, also one tunneling-in event) per modulation cycle. For this, the modulation period T has to be equal to the sum of the waiting times of the quantum dot to be charged and discharged, respectively, where we can use the tunneling rates γ and γ of the undriven system and find
since is the waiting time for the QD being uncharged and for being charged. For the present experiment with γ = γ = : γ, this reduces to the condition 2f = γ. The value of this estimate for a modulation amplitude of 1 mV, where stochastic resonance is not yet visible, is shown as a dashed line in Fig. 5a. Already for 6 mV modulation amplitude, the linear-response estimate can no longer be used since the low- and high-voltage tunneling rates strongly differ from each other (and therefore also from the undriven case).
A simple analytic expression can also be found in the regime γ ≪ γ and γ ≪ γ, for which a large modulation amplitude is necessary (but not sufficient). In the extreme limit of γ = 0 and γ = 0, a perfectly regular switching behavior with exactly one discharging event per cycle is achieved in the limit T → ∞. This implies (for Δt = T) a vanishing Fano factor in the limit f → 0 and, thus, qualitatively explains why the minimum of the Fano factor is shifted towards lower frequencies for large modulation amplitudes. For small but finite γ and γ, the regular switching behavior is destroyed once the quantum dot can be charged during the low-voltage () or discharged during the high-voltage phase (). Taking the smaller of the two modulation periods and reducing the modulation period by a factor 1/2 to ensure that the destruction of the regular switching behavior has not yet set in, we arrive at
i.e. the larger of the two small rates determines the resonance condition. The vertical lines in Fig. 5b demonstrate a good agreement of the estimate with the measured data.
Going beyond the second-order normalized factorial cumulant x (or equivalently the Fano factor or the Mandel Q-parameter), we now also consider the higher-order terms x and x as a function of the modulation frequency f. This is shown in Fig. 6a for 16 mV modulation amplitude and a time interval length of Δt = 10 ms. We find good agreement between experimental data (dots) and theoretical model (lines) without any fitting parameter. In each order m, we find a clear resonance. This nicely shows that the influence of stochastic resonance affects the statistical distribution far beyond the second moment or the Fano factor.
We find that the resonance frequency decreases with increasing order m. While experimentally we can only access the first few orders (Fig. 6a), we can calculate them within our theoretical model up to very high order, see Fig. 6b, and find a pronounced shift with no obvious lower bound. A physical understanding of this intriguing behavior remains an open question. It demonstrates, however, that the intuitive assessment that stochastic resonance occurs at the frequency given by the switching rate is incomplete.
Finally, we remark that higher-order ordinary cumulants are not useful for identifying stochastic resonance. They vanish (C = 0 for m ≥ 2) for a Kronecker distribution (), but these zeroes of C as a function of the frequency are masked by trivial zeroes that arise due to universal oscillations, see supplementary note 9.