Trapping and counting ballistic non-equilibrium electrons

We demonstrate the trapping of electrons propagating ballistically at far-above-equilibrium energies in GaAs/AlGaAs heterostructures in high magnetic field. We find low-loss transport along a gate-modified mesa edge in contrast to an effective decay of excess energy for the loop around a neighboring, mesa-confined node, enabling high-fidelity trapping. Measuring the full counting statistics via single-charge detection yields the trapping (and escape) probabilities of electrons scattered (and excited) within the node. Energetic and arrival-time distributions of captured electron wave packets are characterized by modulating tunnel barrier transmission.

The ability to prepare and subsequently detect discrete particles constitutes a crucial component for applications ranging from metrology over sensing to quantum information technologies, for example in utilizing single ions [1] or electrons [2] for quantum computation, or photons for quantum cryptography [3]. In electron quantum optics (EQO), the solid-state analogue to quantum optics, the recent introduction of on-demand single-electron sources is advancing experiments that have previously been realized with continuous electron sources [4][5][6][7][8] by offering an inherent time-control, for instance in mesoscopic capacitors [9][10][11], leviton injections [12], singleelectron pumps (SEPs) [13][14][15], or the application of surface acoustic waves (SAWs) [16]. However, implementing a single-electron detector for an EQO experiment with ballistic electrons poses the challenge to detect ballistic electrons either on-the-fly or to trap them prior to detection. Combining single-particle detection with an ondemand single-particle source provides direct access to the full counting statistics, independent of the bosonic, fermionic, or atomic nature of the observed particles, and ensures high signal fidelities even in circuits with simultaneous multi-particle injection. Additionally, multiple detectors can easily be combined and inherently record coincidence correlations.
Here, we present a single-electron circuit, operated in single-shot mode, demonstrating the trapping of ballistic electrons emitted at non-equilibrium energies from an SEP-source originating from high-precision metrological applications, combining it with ballistic wave guides and a quantum-dot charge detector. This technique resembles "sample and hold"-circuits in analog electronics, where a voltage is sampled by charging a capacitor that retains the voltage for readout. The single-electron counterpart consists of four functional elements: A ballistic electron is emitted on-demand (Source) and traverses a low-loss wave guide (WG) which directs it through the circuit. At the circuit's output, the ballistic electron is then trapped inside a node by energy relaxation (Capture) and read out (Detect). Following single-photon experiments, the fidelity, in quantum optics denoted as detection efficiency η to register a photon at the detector, is one of the key parameters for applicability of any circuit [17]. Similarly, we seek to determine the fidelity η Circuit of our single-electron circuit as a basis for its future application in EQO. Following the functional segmentation, η Circuit splits up into the product of η Source , η WG , η Capture , and η Detect .
The two circuit geometries investigated (samples A and B, Fig. 1) have been realized [18] in the same GaAs/AlGaAs heterostructure with 97 nm nominal 2DEG depth, charge carrier density 1.9 × 10 11 cm −2 , mobility 1.15 × 10 6 cm 2 V −1 s −1 , and quantum life time 2.9 ps. Components are formed by Cr/Au gates upon a shallow etched channel. Measurements are carried out in a dry dilution refrigerator with base temperature below 50 mK and at 10 T perpendicular magnetic field (ν ≈ 1).
A non-adiabatic single-parameter single-electron pump [13,19,20] is used as an on-demand source of nonequilibrium electrons with typical excess energies of several ten meV above the Fermi level [14,21], separating transport from the Fermi sea. It is confined by three gates (pump entrance and exit barriers G P1 and G P2 , side gate G P3 ) and excited by a f = 300 MHz sinusoidal pulse on G P1 to produce a single electron. In continuous operation, this would correspond to a generated current I = ef ≈ 48 pA with elementary charge e. SEPs have been reported with sub-ppm accuracy [22] and hence do not currently limit the circuit fidelity. Here, η Source ≈ 0.999 is estimated from the difference of measured current to ef .
The electron is emitted into a magnetic-field-induced chiral edge state [23] that serves as a low-scattering, ballistic wave guide transporting the non-equilibrium electron through the circuit. An edge depletion gate G D is placed on top, which was introduced for suppression of electron-electron interactions [24] and modulates and suppresses the emission of LO phonons [24,25]. Here, V GD = −350 mV, set for strongest suppression of relaxation, generates a gate-modified mesa edge for the wave guide. (Stated voltages are referring to the signal's DC parts.) Sample A, shown in Fig. 1a, is specifically designed to determine the trapping-capability (i.e. the achievable η Capture ), with a short wave guide length of l ≈ 1.25 µm to minimize losses due to scattering within the wave guide. The probability of an electron being emitted and reaching the detector node is obtained from transmission-dependent current measurements as (η Source η WG ) ≈ 0.997, resulting in η WG ≈ 0.998. A characteristic scattering length λ can be estimated in a simple, exponential model [25,26] from the transfer coefficient α = exp (−l/λ) for length l. With α WG = η WG , this yields λ WG ≈ 624 µm and exceeds l by far.
Sample B (Fig. 1b) integrates more components and an increased wave guide length of l ≈ 3 µm to demonstrate the functionality in more complex circuitry (data taken from sample B is marked explicitly).
Electrons that scattered while propagating through the wave guide are reflected on the node entrance barrier G N1 and sunk into an ohmic side contact, i.e., only unscattered electrons enter the node. In contrast to transport through the wave guide, this capture-node relies on the controlled relaxation of the non-equilibrium electrons. Relaxation and, thus, trapping of the electrons is ensured by defining a sufficiently large node size where the electrons are, in the absence of a depletion gate, subjected to material intrinsic scattering processes. Therefore, the node exit barrier G N2 is placed 4.2 µm apart from G N1 , forming a large, elongated quantum dot with a lithographic circumference of 10.4 µm and a typical charging energy of few ten µeV. Assuming drift velocities of magnitude 5 × 10 4 m s −1 [24,25], the node transit time is supposed to exceed the typical temporal wave packet extent [27][28][29].
The detector dot is formed by split-gates against the edge of a separate mesa structure, with an additional floating gate to increase capacitive coupling to the node. Separation of the two mesas avoids the strong electrostatic screening and the resulting reduction in detector resolution that would accompany a complete split-gate realization [30]. The detector dot is operated in the Coulomb blockade regime to record charge changes on the node, with two readouts per repetition of the counting cycle, following the sequence shown in Fig. 2a: after completing the first readout [of the initial detector state, (i)], emission of an electron is triggered (ii), while after the second readout [final detector state, (iii)], the node state is reset (iv). The typical cycle repetition rate is 24 Hz.
To estimate the circuit's capture and readout fidelities, many cycles are recorded and the data binned into a 2D histogram, depicting final over initial detector state. Exemplary data can be seen in Fig. 2b for 10 5 cycles, where counts on the diagonal represent a constant detector signal (i.e. node's charge state remains unchanged), while an off-diagonal signature reflects a change in the deposited charge. Multiple clusters emerge, clearly separated in x-direction (different initial node charge states after reset) and in y-direction (change in the deposited charge within a cycle). The ≥ 5σ peak separation proves the excellent single-electron resolution of the charge detector, despite the large node size of > 4 µm 2 , and therefore the approximately-unity detection fidelity η Detect ≈ 1 is not limiting η Circuit . All counts with sufficient detector sensitivity are summed together (cyan lines in Fig. 2b) and lead to the full counting statistics P n of a change in the node's charge by n electrons. P T denotes the transmission probability over the barrier (here, over G N1 , P T = P 1 ), which is P T ≈ 0.996 (P 0 + P 1 = 1). The high probability P T proves the successful trapping of electrons in this circuit and is equivalent to the achieved maximal circuit fidelity, η Circuit ≈ 0.996, revealing the almost-unity trapping fidelity η Capture ≈ 0.999.
Sample B reveals a slightly reduced circuit fidelity, η Circuit,B ≈ 0.96 (mainly limited by η WG,B ), that agrees well with (η Source,B η WG,B ) as obtained from transmission-dependent current characterization measurements, indicating that the trapping-and detectionfidelities are comparably high as in sample A, The achievable fidelity of these circuits, which employ and capture ballistic electrons, is therefore largely limited by transport properties of the wave guide. Corrected for errors of the electron source, this yields η Circuit,A /η Source,A ≈ 0.997 and η Circuit,B /η Source,B ≈ 0.96 which compares favorably to the fidelity 0.919 of (non-ballistic) electron transport inside SAW-induced moving quantum dots (N 1001 /N 100x ) [16] across a distance matching that of SEP sample B. We note that initialization of the detector node in the SEP-driven circuit achieves unity fidelity, i.e. does not contribute to η Detect,A/B .
For application of this circuit in an EQO experiment, for example a collision of two electron wave packets from independent sources, the wave packets need to overlap in energy, time, and space to enable interference effects, demanding the ability to observe the emitted wave packet's characteristics. Therefore, by utilizing the energy-dependent transmission of the node-defining barriers, energy-and time-resolved measurements can be performed to gain insights on the wave packet's energetic and temporal distributions [14,15,24,27,29,31]. Figure 3a shows an energy-dependent probability distribution in a case where the barrier height of G N1 is changed from low (right) to high (left), while the node exit barrier is high. The node is formed below V GN1 ≈ −255 mV, in the first segment (marked in orange) no clear charge state is observable, hinting at nonlinear transmission properties of G N1 . Detector evaluation becomes possible below V GN1 ≈ −315 mV. The energy-resolved detection is demonstrated by the transition P 1 ≈ 1 to P 0 = 1 for increasing barrier heights.
A relative energy-scale can be derived from current characterization measurements, where barrier height is related to electron emission energy, and is shown on the top axis of Fig. 3a (see Supplemental Material for more details [32]; zero set for maximum electron energy arriving at G N1 ). The energetic width of the distribution is evaluated as the transition's full width half maximum (FWHM) to δE ≈ 4 meV and agrees well with other experiments [14,27,29], providing an upper bound to the energetic width of the wave packet [27].
The energy loss due to relaxation while crossing of the node can be scanned by fixing G N1 at a low barrier level and sweeping G N2 (Fig. 3b). In the occurring transition, G N2 has a non-linear energy-dependence due to the nearby barrier pinch-off, thus energy-scaling and δE cannot be derived. Contrary to the scenario discussed before, transmission over the barrier results in n = 0, thus P T = P 0 . The maximal transmission probability is P T,max ≈ 0.52, which corresponds to a scattering length λ GN2 ≈ 6.5 µm for transport between G N1 and G N2 , already two orders of magnitude smaller than λ WG . The very small fraction of electrons traversing the full circumference of the node and exiting over  GP1 and GP2). An additional region of missing electrons (compared to Fig. 3a) is marked by blue stripes. Top axis energy scale is extended from Fig. 3a. (b) Time-resolved capturing at E0 by raising GN1 for 5 ns at a time delay τ relative to the pump excitation. Measured data on left axis (markers), exemplary AC excitations for t exmpl = 3 ns on right axis (lines). G N1 , 1 − η Capture ≈ 0.001, implies an even further reduction in scattering length. The strong discrepancy of more than two orders of magnitude when compared to the wave guide relates to strongly differing relaxation in directly neighboring segments of this circuit, which we attribute to the different edge-potential profiles, where relaxation is significantly suppressed in the gate-modified edge state in the depleted wave guide [24,25], as opposed to the node's unmodified mesa-edge. Additional manipulations of transport across the node, aimed at a supplementary manipulation of the node's edge-potential profile by in-plane gating, or of the whole node potential, did not further enhance the capture-probability.
The scattering mechanism with the most distinct signature, considered as the main mechanism of relaxation in this type of devices [25,[33][34][35], is the emission of LO phonons with a 36 meV signature. Additionally, the presence of electron-electron scattering is anticipated, but is not as clearly identifiable as the LO phonon signature. More detailed discussions on relaxation processes are given in Refs. [34][35][36], in part including further scattering processes.
We estimate the SEP's emission energy E 0 of the measurements shown in Fig. 3 to be of comparable magnitude to the LO phonon energy [32]. In general, the emission energies of SEPs can span a large range, exceeding E > 100 meV [14,21]. The energy-resolved data at an increased emission energy ∆E 1 ≈ 20 meV above E 0 [32], shown in Fig. 4a (compare the transition's position to Fig. 3a), demonstrates that a high trapping fidelity η ∆E1 Capture ≈ η E0 Capture is maintained despite the increased emission energy, as apparent from η ∆E1 Circuit ≈ 0.992. The energetic width δE ∆E1 ≈ 5 meV is also close to the lowerenergy value. A further increased energy ∆E 2 ≈ 50 meV above E 0 (data not shown) yields δE ∆E2 ≈ 9.5 meV and the reduced circuit fidelity η ∆E2 Circuit ≈ 0.90, which however is mainly due to deteriorating fidelity of the source, η ∆E2 Source . There is no discernible indication for a decrease in η Capture over the inspected range of 50 meV in emission energy.
In the energy-resolved measurements at increased emission energies, additional signatures appear in the limit of small node entrance barrier heights (blue stripes in Fig. 4a, [37]), where P 1 drops for decreasing barrier heights, while P 0 increases and in some cycles the charge on the node is unexpectedly reduced, P −1 > P −2 > 0. The appearance of P −1 , P −2 > 0 cannot be plausibly explained as a modulation of source-, wave guide-or detector-functionality and is directly related to electrons emitted from the SEP [38]. We interpret P −1 , P −2 > 0 as a process of energy-redistribution from relaxation of the hot electron, similar to the observations in Ref. [39], enabling thereby excited electrons to escape the node. Including P −2 > 0, four non-zero probabilities exist, highlighting the benefit of continued access to the full counting statistics provided by the single-electron resolution.
The wave packet's arrival-time distribution is accessible by time-resolved modulation of the barrier transmission [14,24,27], by applying in interval (ii) of the cycle (Fig. 2a) an additional, rectangular pulse to G N1 , whose time delay τ is tuned relative to the pump excitation (cf. lines in Fig. 4b). For τ → 0 ns, G N1 is raised at the beginning of the cycle (presumably entirely blocking electrons from entering the node), while for large τ , when wave packet transport is finished prior to barrier elevation, the system should behave the same as without the pulse. Figure 4b demonstrates this time-resolved capturing of electron wave packets, representing their arrivaltime distribution at G N1 . The FWHM of the distribution, δt ≈ 250 ps, is an upper bound to the temporal width of the electron wave packet [14] due to an unknown barrier slew rate. Arrival-time distributions with δt an order of magnitude smaller have been reported [27][28][29], however the temporal width of the emitted wave packet strongly depends on the SEP's operating point [31,40], which has not been investigated further in this case. This time-resolved technique constitutes the minimal module for single-shot voltage sampling [29].
In conclusion, we have successfully realized a complete single-electron circuit with a combined fidelity up to 0.996, where a single ballistic electron was emitted on-demand and with tunable excess energy, traveled ballistically along a wave guide and was ultimately captured. The full counting statistics was collected, including additional excitation products. This functionality was enabled by the controlled manipulation of energy relaxation by more than two orders of magnitude in directly neighboring parts of the same device. Energy-and timeresolved trapping of electrons allowed first insights on the characteristics of single-electron wave packets in singleshot measurements. Additionally, we demonstrated the detection's robustness against variations of the electron's emission energy. This consistently high capture fidelity is especially important for future collision and interferometry experiments that require an energetic overlap between electron wave packets emitted from independent non-equilibrium sources. The sample's transport properties are characterized by performing energy-resolved transmission measurements, similar to [1], by measuring transmitted (I T ) and reflected (I R ) currents in dependence of current emission energy, separately versus both node barrier heights (with only one node barrier formed at a time). For generation of the fixed current, two different methods are applied, i) the SEP as described in the main article, with a limited range of emission energies (here, the quantized 1efplateau is typically extended over ≤ 30 meV and η Source is not constant throughout) and ii) a stochastic constant current hot-electron source, driving 200 pA by applying a bias over a fixed barrier (G P2 ) across an extended range of emission energies. In both methods, the emission energy scales with G P2 , data for the latter case is shown in Fig. S1.
In method ii), monitoring the bias applied to drive the current provides an energy-scaling for G P2 that is confirmed by the 36 meV repetitive pattern of LO phonon replica [1,2] (red arrows in Fig. S1). This energy-scaling can be mapped onto a node barrier via the relation between the node barrier's current pinch-off and the emission energy, yielding a relative energy scale for G N1 , as demonstrated in the top axis in Fig. S1 (though not applicable in this article, this in principle works for G N2 as well).
Since the node defining barriers are largely independent of the method used for generation of the current, the energy-scaling for method i) can be obtained by conversely matching the node barrier-dependent pinch-off to that of method ii) (vertical lines in Fig. S1, taken from single-shot measurements) and, consequently, to the initial energy-scaling (horizontal lines). This calibration can be performed for arbitrary SEP configurations/working points and allows a reliable comparison of their relative emission energies (∆E 1 and ∆E 2 , Fig. S1), but does not simultaneously provide an absolute energy-scale (i.e. specify the exact value of E 0 as well) due to the fact that the barrier pinch-off is non-linear and not well enough defined.
FIG. S1. ∂IT /∂VG N 1 , taken with method ii), in dependence of applied bias voltage and node entrance barrier height VG N 1 (grayscale, logarithmic scaling). Red arrows denote the 36 meV LO phonon signature. Vertical lines indicate wave packet pinch-off voltage by GN1 as apparent from single-shot measurements at three SEP working points (compare transitions in Figs. 3a and 4a in the main text), horizontal lines highlight the difference in emission energies between the working points. Top axis energy scale matches Fig. 4a in main text.