Waiting time distributions in a two-level fluctuator coupled to a superconducting charge detector

We analyze charge ﬂuctuations in a parasitic state strongly coupled to a superconducting Josephson-junction-based charge detector. The charge dynamics of the state resembles that of electron transport in a quantum dot with two charge states, and hence we refer to it as a two-level ﬂuctuator. By constructing the distribution of waiting times from the measured detector signal and comparing it with a waiting time theory, we extract the electron in-and out-tunneling rates for the two-level ﬂuctuator, which are severely asymmetric. DOI: 10.1103/PhysRevResearch.1.033163

In this work, we investigate electronic waiting times between charge transitions in and out of a parasitic state to 2643-1564/2019/1(3)/033163 (6) 033163-1 Published by the American Physical Society directly extract the time scales of such TLF.We employ a superconducting single-electron transistor (SSET) to monitor the switching events on the TLF and apply a continuous electrostatic feedback on the detector to maintain a constant charge sensitivity, which tends to fasten an asymmetry in the detected in-and out-tunneling rates.This paper is organized as follows: In Sec.II, we provide a short overview of the waiting time theory [20] and illustrate these concepts by evaluating the distribution of electron waiting times for a TLF in Secs.II A and II B. We treat the two-level fluctuator as a potential well that is in tunnel contact with a charge reservoir as illustrated in Fig. 1(a).In Sec.III, we discuss our experimental setup.In Sec.IV, we present our measurement results and compare them with the waiting time distribution theory.Since our model assumes sequential in-and out-tunneling, we calculate the FCS of the switching events and compare the results with the waiting times.Finally, in Sec.V we conclude our work.

II. ELECTRON WAITING TIMES
The time that passes between two subsequent singleelectron tunneling events of the same type is usually referred to as the electron waiting time τ [20,37,50].The single-electron tunneling process has a stochastic nature and therefore is described by a waiting time distribution (WTD) function W (τ ).For stationary transport problems, the WTD relates to the idle-time probability (τ ) as [37,38] where (τ ) is the probability of having no tunneling events during a time span τ .The mean waiting time τ can be expressed in terms of the idle-time probability as [37,38] ).The statistics of single-electron tunneling events is captured by the probability P(n, t ) of having n tunneling events of the chosen type during the time span [t 0 , t 0 + t] [55][56][57].However, we only need to know the idle-time probability (τ ) = P(n = 0, τ ) to obtain the WTD.In FCS, the moment generating function provides us with all the moments of n as (2) we observe that M(i∞, t ) = P(n = 0, t ) is exactly the idle-time probability.Next, we utilize these concepts by evaluating the WTDs for a two-level fluctuator.

A. Waiting times in a two-level fluctuator
We describe the parasitic state as a single-electron box consisting of a nanoscale island, the charge dynamics of which is governed by the master equation where the vector p(t ) = [p 0 (t ), p 1 (t )] T contains the probabilities p 0 (t ) and p 1 (t ) for the island to be empty or occupied by 1 electron, respectively, and the rate matrix L describes the transitions between 0 and 1 charge states of the island.We partition the rate matrix as L = L 0 + J + + J − with jump operators J ± describing charge transfers to and from the island, respectively [58].We resolve the probability vector p(n, t ) such that it accounts for the number of tunneling events n.The n-resolved equations of motion, d dt p(n, t ) = L 0 p(n, t ) + J + p(n − 1, t ) + J − p(n + 1, t ), are decoupled by introducing the counting field χ via the definition p(χ, t ) ≡ n p(n, t )e inχ .We then arrive at a modified master equation for p(χ, t ) For χ = 0 in Eq. ( 4), we recover the original master equation (3).Further on, we focus on the waiting times between the into-the-island tunneling events, and hence set J + = 0.In this case, the modified rate matrix L(χ ) assumes the form We have included the counting factor e iχ in the lower offdiagonal element together with + , corresponding to counting the number of tunneling events into the parasitic state [55][56][57].
The solution of the modified master equation formally reads The idle-time probability follows as (τ ) = j p j (i∞, t ), where p j (i∞, t ) is an jth component of the vector p(χ, t )| iχ→i∞ .For a given rate matrix (5) the solution is analytic and we are able to evaluate the WTD for a two-level fluctuator using relation (1) as where τ is the waiting time between single-electron subsequent tunneling events from the charge reservoir to the parasitic state.Counting the out-of-the-island tunneling events yields an identical result for the distribution of waiting times.We continue with another example of WTD, where we take into account the effect of finite detection time.

B. Residence times in a two-level fluctuator coupled to a detector
Let us focus on the time that the electron spends in the parasitic state known as the residence time τ * [59].We use the term residence time to avoid confusion with the waiting time discussed in the previous section.When the residence time becomes comparable to the inverse of the detector bandwidth, 2π/ D , we have to take it into account when evaluating the distribution of residence times.
Due to the finite bandwidth of the detector, we may fail to detect some of the tunneling events.Therefore, we add two more possible states when formulating the rate equation [60].The revised probability vector reads p * (t ) = [p 00 (t ), p 10 (t ), p 01 (t ), p 11 (t )] T .Probabilities p 01 (t ) and p 10 (t ) correspond to the situation when the parasitic state is empty or occupied, respectively, but it has not been detected.The detected empty and occupied states are denoted by p 00 (t ) and p 11 (t ), respectively.

033163-2
While actual charge transitions between the occupied state p 11 (t ) to the undetected empty state p 01 (t ) may occur several times before the empty state p 00 (t ) gets detected, we obtain the experimentally observed residence time distribution by counting the events p 01 (t ) → p 00 (t ).After including the effect of the finite detector bandwidth D /(2π ) in the rate equation, we arrive at the following rate matrix [61,62]: In analogy to the previous section, we solve the modified rate equation given the rate matrix Eq. ( 8) in the limit χ → i∞ with an initial condition p * (i∞, 0) = [0, 0, 0, 1] T and obtain the idle-time probability (τ * ).We evaluate the detected residence time distribution as a conditioned distribution where = ( α α ) 2 − 4 D − and α = +, −, D. If we assume the detector to be perfect D → ∞, we recover the exponential decay of the occupied parasitic state W ∞ (τ * ) = − e −τ * − .

III. EXPERIMENTAL SETUP
We employ a superconducting charge detector which consists of two Al/Al 2 O 3 /Al Josephson junctions in series with resistances and capacitances R i , C i , i = 1, 2 and a superconducting gap , which form a charge island with a charging energy of E ch e 2 /C , where C = C 1 + C 2 + C c + C g , e is the elementary charge, C g is the capacitive coupling between the gate and the island, and C c is the mutual capacitance between the TLF and the detector island.The device is fabricated on a 500-μm-thick high-resistivity silicon wafer with 5-nm-thick high-purity field oxide.The measurements are carried out in a cryostat that has a base temperature of T = 100 mK. Figure 1(b) shows the top view of a detector that is nominally identical to the device that is used to carry out the presented experiments.The Al leads provide galvanic contacts between the tunnel junctions, and the gate voltage V g is used to tune the charge sensitivity of the SSET.The bias voltage V dc controls the Fermi level of the lead, whereas the other lead is connected to a room temperature transimpedance amplifier.At the output of the amplifier, we measure the voltage and transmit the signal to a proportional-integral-derivative (PID) feedback circuit that maintains the gate charge point, and hence the charge sensitivity of the detector.
The charge stability of the detector, shown in Fig. 1(c), reveals how to associate the measured dc current with the gate-tunable charge point of the superconducting island.To minimize the detector backaction on the two-level fluctuator, we choose the bias voltage that corresponds to the double Josephson-quasiparticle process [63,64].

IV. RESULTS
The signal produced by the TLF is read out using the detector such that the PID controller is utilized and tuned to a set point indicated in Fig. 1(d).In addition to the white noise, we observe a systematic signal, as exemplified in Fig. 2(a).The jumps are distinguished from the white noise when the amplitude of the detector signal exceeds the 5σ white-noise level.Furthermore, the detector signal is filtered with a twopoint moving averaging window, which results in a detector bandwidth of D /(2π ) = 11.25 Hz.
First, a jump with a negative sign appears and the detector current offset takes a negative value.Then, the PID steers back to the set point using the detector gate.After τ * residence time, another jump appears, but always with the opposite, positive sign compared to the first jump.The consecutive jump occurs at τ counted from the previous negative jump.The probability distribution of the jump amplitudes, shown in Fig. 2 7).(b) Measured (circles) and fitted residence time distribution of the charge jumps in the case of finite detector bandwidth (line) and ideal detector (dashed line) according to Eq. ( 9).The data have been recorded as illustrated in Fig. 2(a).
Since the switching between the two states is associated with single-electron transitions, τ and τ * can be extracted from traces such as that shown in Fig. 2(a).The total duration of the measured time trace is 16 h, which approximately 1000 back-and-forth tunneling events, giving rise to the WTD shown in Fig. 3(a).
We obtain the parameters + and − by fitting Eq. ( 7) to the data in Fig. 3(a).The fitted waiting time distribution, where the tunneling rates are + = 15.8 × 10 −3 s −1 and − = 473.2× 10 −3 s −1 , agrees well with the measured distribution.The significant difference between + and − originates from the energy splitting of the two charge states and from the feedback; for every switching event the detector gate induces a compensating electrostatic field that not only affects the operation of the SSET, but also tends to polarize the TLF towards the opposite state.Since the average residence time τ * is significantly shorter than τ and comparable to 1/ D , we fit Eq. ( 9) to the residence time distribution, shown in Fig. 3(b), by first fixing + and − according to the WTD.Thus, the only fitting parameter in the model is D , with the estimated D /(2π ) = 10.4Hz based on the fit.The fitted D deviates roughly 7.6% from the nominal detector bandwidth.
From the measured state transitions we can also construct the counting statistics P(n, t ) used in Eq. ( 2), where n denotes the number of switching events regardless of the direction.The limited number of total jumps requires a moving time window to achieve the FCS with different time window lengths.Therefore, consecutive time windows are shifted by 1 s and used in the average over the 16-h time trace.The Markovian single-electron tunneling in a dot coupled to a charge reservoir follows the Poisson distribution [65].If we assume an effective tunneling rate eff , we can fit a Poisson distribution to the data shown in Fig. 4. The extracted effective tunneling rate is eff = 31.5 × 10 −3 s −1 , which can be explained by the branching of jumps observed in Fig. 2(a).More precisely, we effectively observe two events with a rate of + , since after each in-tunneling event there is an out-tunneling event with a brief delay.

V. CONCLUSIONS
We have demonstrated that WTD is useful in extracting the time scales of a TLF.Including the finite bandwidth of the detector in the waiting time theory provides us with an accurate distribution of the waiting times in the charge trap.In contrast to WTD, full counting statistics covers long time scales and delivers mean values, noise, and high-order moments.Both approaches are powerful tools and they are connected [20].From the observed probability distribution P(n, t ), we conclude that charge transitions in the TLF constitute a Poisson process.The distribution of waiting times, Eq. ( 7), indicates that there are two Poisson processes with different rates + and − in the two-level fluctuator.It requires less steps to evaluate the WTD than to evaluate the full counting statistics P(n, t ).Moreover, the waiting time formalism allows us to evaluate the WTD analytically where as an analytic expression for P(n, t ) is not currently available.Waiting times belong to the short-time-scale statistics and are sensitive to the finite detector bandwidth.Here, we have shown how the distribution of waiting times has obvious advantages in extracting the short time scales of the system.

FIG. 1 .
FIG. 1.(a) Schematic diagram of the investigated system which consists of a Josephson-junction-based detector (top) and a two-level system (bottom) that is in tunnel contact with a charge reservoir.All symbols are described in the main text.(b) False-color scanning electron micrograph of the charge detector similar to that in the measurements together with a circuit diagram of the experimental setup.(c) Detector current as a function of the normalized bias voltage eV dc /(2 ) and the gate charge n g = V g C g /e, where is the superconducting gap.The dashed line indicates the bias point for the measurement, which (d) is plotted along, where a Lorentzian function (blue line) is fitted to the data (blue circle) that is used to estimate the sensitivity (red) in arbitrary units.The black arrow indicates the PID set point.The parameters of the measured device are = 195 μeV, R T = R 1 + R 2 = 180 k , and E ch = 160 μeV.

FIG. 2 .FIG. 3 .
FIG. 2. (a)Electric current offset of the detector as a function of time (blue line).The two characteristic parameters, the waiting time (τ ) and the residence time (τ * ) are indicated.The red dashed lines correspond to ±5σ white-noise levels and the red solid line depicts the PID set current, which is at the positive slope on the dashed line in Fig.1(c).The shaded areas indicate the signal corresponding to the jump events.(b) Probability density of the charge signal, ρ, obtained by monitoring the jump amplitudes.The colors match with those of (a).

FIG. 4 .
FIG.4.Measured (circles) and fitted (lines) FCS of sequential single-electron tunneling events for three different time windows.