Main

Our single-electron source is a single-electron pump with a tunable-barrier quantum dot (QD)18, which consists of the entrance (Fig. 1a, left) and exit (right) potential barriers, formed by applying an entrance gate voltage Vent + Va.c.(t) and an exit gate voltage Vexit, respectively. The a.c. voltage Va.c.(t) = Vampcos(2πfint) with frequency fin and amplitude Vamp added to dynamically tune the entrance barrier. The QD energy level is also tuned owing to the cross coupling. When the energy level \(E_n^{{\mathrm{QD}}}\) (n = 1, 2, …) of the QD with n electrons is lower than the Fermi level EF, electrons can be loaded from the left lead (loading stage). After that, when \(E_n^{{\mathrm{QD}}}\) is lifted and becomes higher than EF, the loaded electrons can escape to the left lead. However, when the escape rate is much slower than the barrier-rise rate, the electrons can be dynamically captured by the QD (capture stage)19. Finally, the captured electrons can be ejected to the right lead (ejection stage). This gives the pumping current as IP = nefin, where e is the elementary charge. Detailed models are given in Supplementary Section I.

Fig. 1: Internal coherent dynamics in the single-electron source.
figure 1

a, Schematic potential diagrams of a QD electrically formed by applying d.c. voltages Vent and Vexit. The left and right potential barriers are referred to as the entrance and exit barriers, respectively. \(E_n^{{\mathrm{QD}}}\) (n = 1, 2, ...) is the energy level of the QD with n electrons. The entrance barrier and QD energy level are dynamically tuned by a high-frequency signal Va.c.(t), which leads to the three stages. EF, Eadd and ΔE are the Fermi level, charge addition energy and energy gap between the ground and first orbital excited states, respectively. A single electron is transferred from the left to right leads. b, Schematic diagram of the rise of the QD potential (black solid curves). In the QD, the electron forms a wave packet that coherently moves between the left and right sides (\(\left| {\psi _{\mathrm{L}}} \right\rangle = \sqrt {1 - p} \left| {\psi _{\mathrm{G}}} \right\rangle - \sqrt p \left| {\psi _{\mathrm{E}}} \right\rangle\) and \(\left| {\psi _{\mathrm{R}}} \right\rangle = \sqrt {1 - p} \left| {\psi _{\mathrm{G}}} \right\rangle + \sqrt p \left| {\psi _{\mathrm{E}}} \right\rangle\)). The spatial distribution |ψL(R)|2 of the wave packet is drawn in red (blue) when it is located at the left (right) side. \(\dot V\) is the rising speed of the QD bottom. The red line in the exit barrier depicts the resonant level Eres with broadening Δres. ΓL(R) is the coupling energy between the QD energy level (the right lead) and the resonant level. The wave packet is eventually ejected to the right lead via the resonant level with probability PT. c, Calculated probability density |ψs|2 of the electron wave packet as a function of time and position at fin = 4 GHz. The red line is the trajectory of the QD bottom. Inset: acceleration of the horizontal movement of the QD as a function of time. The purple line is a critical acceleration for non-adiabatic excitation. The dashed line determines t0 for the simulation in Fig. 5. Note that p ≈ 0.1 here.

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When fin is high (typically in the gigahertz regime), non-adiabatic excitation can occur in the QD15. Then, electrons can be in a superposition of the ground state and orbital excited states. In the case that only one electron is captured in the QD, the electron forms a wave packet14,16 that moves coherently back and forth between the entrance and exit barriers in the QD, as shown in Fig. 1b. The coherent spatial oscillations can be approximately described by a time-dependent superposition |ψS(t)〉 between the instantaneous ground state |ψG(t)〉 and the first orbital excited state |ψE(t)〉 of the QD:

$$\left| {\psi _{\mathrm{S}}} \right\rangle \simeq \sqrt {1 - p} \left| {\psi _{\mathrm{G}}} \right\rangle + e^{ - i\left( {2\uppi \frac{t}{{\tau _{{\mathrm{coh}}}}} - \theta } \right)}\sqrt p \left| {\psi _{\mathrm{E}}} \right\rangle$$
(1)

where p is the probability that the orbital excited state is occupied and θ is the initial phase. The oscillation period τcoh is determined by the energy gap ΔE between the ground and first orbital excited states, and it is written as τcoh = h/ΔE when ΔE is approximately time independent. Here, h is the Planck constant. Note that the valley degree of freedom is not the origin of the spatial oscillations because the valley excitation20 only affects the wave-function distribution perpendicular to the direction of the spatial oscillations.

To check the feasibility of the non-adiabatic excitation and coherent dynamics described by equation (1), we numerically solved the time-dependent Schrödinger equation with a realistic potential profile (Supplementary Sections II and III). Figure 1c shows a calculated wave-packet distribution as a function of time and position at fin = 4 GHz. The entrance barrier starts to push the QD away from it at around 40 ps, at which the acceleration aQD of the horizontal movement of the QD bottom rapidly increases (inset of Fig. 1c). Around this time, the acceleration is higher than the critical value \(l_{{\mathrm{QD}}}/\tau _{{\mathrm{coh}}}^2\) above which the non-adiabatic excitation occurs, where \(l_{{\mathrm{QD}}} = h/\left(2\uppi \sqrt {m\Delta E}\right)\) is the confinement length of the QD and m is the electron effective mass. After that there occurs, following equation (1), the spatial oscillations at the picosecond scale, which is beyond the currently available bandwidth for measurements of coherent charge oscillations6,12,13.

We propose that such fast coherent oscillations can be detected using a resonant level Eres formed in the exit barrier (Fig. 1b). Although the electron moves back and forth, the potential energy of the QD increases from the value \(\overline {E_{{\mathrm{ini}}}^{{\mathrm{QD}}}}\) at initial time t0 (onset of the non-adiabatic excitation). The energy becomes aligned with Eres at the detection time t1, at which the electron can be ejected through the exit barrier via the resonant level21, to generate current. We formulate the ejection probability PT based on scattering theory (Supplementary Section IV) as:

$$P_{\mathrm{T}} \simeq \overline {P_{\mathrm{T}}} \left[ {1 + 2\sqrt {p(1 - p)} {\mathrm{cos}}\left( {2\uppi \frac{{t_1 - t_0}}{{\tau _{{\mathrm{coh}}}}} - \theta } \right)} \right]$$
(2)

PT depends on the time difference \(t_1 - t_0 = (E_{{\mathrm{res}}} - \overline {E_{{\mathrm{ini}}}^{{\mathrm{QD}}}} )/\dot V\), which is tuned by changing the gate voltages or the rising speed \(\dot V\) of the QD bottom. The probability becomes maximal (minimal) when the tuning makes the electron wave packet be located near the exit (entrance) barrier at t1, which results in gate-dependent current oscillations. The mean probability \(\overline {P_{\mathrm{T}}} = 2\uppi T_{{\mathrm{max}}}\Delta _{{\mathrm{res}}}/(\tau _{{\mathrm{coh}}}\dot V)\) depends on the transmission probability Tmax = 4ΓLΓR/(ΓL + ΓR)2 through the resonant level and on the ratio of the energy broadening Δres = ΓL + ΓR of the resonant level to the energy rise \(\tau _{{\mathrm{coh}}}\dot V\) in one period of the spatial oscillations, where ΓL(R) is the coupling energy between the resonant level and QD (right lead).

The conditions for observing the gate-dependent current oscillations are:

$$\Delta E \,\lesssim\, \Delta _{{\mathrm{res}}} \,\lesssim\, \tau _{{\mathrm{coh}}}\dot V$$
(3)

Under the left inequality, the energy uncertainty ΔE of the electron wave packet is smaller than the resonance energy that broadens Δres so that the electron can fully pass the resonant level16. The right inequality is also required; in the opposite limit, \(\Delta _{{\mathrm{res}}}/\dot V \gg \tau _{{\mathrm{coh}}}\), the current becomes gate independent because the wave packet reaches the exit barrier many times within the time window \(\Delta _{{\mathrm{res}}}/\dot V\) and the electron is allowed to pass the resonant level (the limit \(\Delta _{{\mathrm{res}}}/\dot V \ll \tau _{{\mathrm{coh}}}\) is also not acceptable because the oscillation amplitude becomes too small, as expected from the expression of \(\overline {P_{\mathrm{T}}}\)).

To observe the coherent dynamics, we measured a device with a double-layer gate structure on a non-doped silicon wire22,23 at 4.2 K (Fig. 2). The fabrication process and measurement detail are described in Methods. This kind of device often has a resonant level in the exit barrier, which most probably originates from an interface trap level24,25,26. Such a resonant level can be identified by investigating a map of IP as a function of Vent and Vexit (Fig. 3a). In the map, there are several threshold voltages indicated by dashed lines. Along the red solid line, we observe an efin current plateau (Fig. 3b). The signature of the resonant level appears in a wide region with a current less than efin, indicated by the green parallelogram, in which the direct tunnelling through the exit barrier is suppressed and the resonant and inelastic tunnelling27 via the resonant level can be resolved.

Fig. 2: Schematic three-dimensional (left) and top (right) images of the device structure.
figure 2

The high-frequency signal Va.c.(t) with frequency fin is attenuated by 3 dB and is combined with gate voltage Vent in the bias tee. The combined signal is applied to the left lower gate (entrance gate). Vexit is applied to the central lower gate (exit gate). To turn on the channel under the gate, 1.5 V is applied to the right lower gate in all the experiments in this article. Vupper is applied to the upper gate. The current through the silicon channel is measured using an ammeter. The red oval indicates the position of the QD.

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Fig. 3: Experimental observation of current oscillations related to the coherent dynamics.
figure 3

a, The pumping current IP normalized by efin as a function of Vent and Vexit at fin = 1 GHz and T = 4.2 K, where Vupper = 2.5 V and the power P of the high-frequency signal Va.c.(t) is 10 dBm. The red and black lines and the blue dashed lines are threshold voltages determined by the loading, capture and ejection stages, respectively, described in Fig. 1a. When the highest QD energy level is lower than Eres, the ejection through the resonant level is suppressed to give another threshold-voltage line (trap-ejection line), indicated by the yellow dashed line. The green parallelogram indicates the region in which the current oscillations appear. b, IP (left axis) and IP/efin (right axis) as a function of Vexit at fin = 1 GHz and T = 4.2 K. The red and purple curves are cuts along the red and purple lines indicated by the red (Vent = −1.3 V) and purple (Vent = −0.915 V), respectively, triangles in a.

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We observed current oscillations in the region with the resonant-level signature (line cut in Fig. 3b). In this region, the current (normalized by efin) through the resonant level increases with increasing fin, which indicates a non-adiabatic excitation (Supplementary Section V). dIP/dVexit oscillates as a function of Vent and Vexit (Fig. 4a–d). The oscillation period increases with increasing fin. This is expected from equation (2) because \(\dot V\) increases with increasing fin. In addition, the period becomes shorter when the trap-ejection line (the yellow dashed line in Fig. 4a–d) approaches and the oscillation lines slightly bend towards the bottom right.

Fig. 4: Frequency dependence of the current oscillations.
figure 4

a–d, First derivative of IP with respect to Vexit as a function of Vent and Vexit at T = 4.2 K and fin = 1 GHz (a), 2 GHz (b), 3 GHz (c) and 4 GHz (d), where Vupper = 2.5 V and P = 10 dBm. The current oscillations are mainly observed in the region highlighted by red lines. The tilt of the capture line at 3 and 4 GHz indicated by the black dashed lines results from the cross talk of the high-frequency signal. The trap-ejection lines are indicated by the yellow dashed lines. e–h, IP as a function of t1 − t0 at (fin, Vexit) = (1 GHz, −0.96 V) (e), (2 GHz, −1.03 V) (f), (3 GHz, −1.09 V) (g) and (4 GHz, −1.13 V) (h). These are obtained from the two-dimensional maps in a–d, respectively. The reference voltages (\(V_{{\mathrm{ent}}}^ \ast\)) used in equation (5) are −0.58 V (e), −0.82 V (f), −0.73 V (g) and −0.76 V (h). The length of the black arrows equals 4 ps. Inset: t1 − t0 as a function of Vent obtained using equations (4) and (5).

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Our measurement is effectively a time-resolved detection of the wave-packet spatial oscillations. The current IP is larger when the wave packet is located closer to the exit barrier (namely, to the spatial location of the resonant level) at the detection time t1, which depends on Vent and Vexit. We converted the dependency between IP and Vent at a value of Vexit in Fig. 4a–d to a dependency between IP and t1 (measured from the initial time t0) using the following relations between the times and the gate voltages. t0 is related to Vent by:

$$t_0 = \frac{1}{{2\uppi f_{{\mathrm{in}}}}}{\mathrm{cos}}^{ - 1}\left( { - \frac{{V_{{\mathrm{ent}}}}}{{V_{{\mathrm{amp}}}}}} \right)$$
(4)

At t0, the entrance barrier starts to push the QD away from the barrier, hence Vent + Va.c.(t0) = 0 (as in the inset of Fig. 1c). The detection time t1 is given by (Supplementary Section VI):

$$t_1 = \frac{1}{{2\uppi f_{{\mathrm{in}}}}}{\mathrm{cos}}^{ - 1}\left( { - 1 - \frac{{V_{{\mathrm{ent}}} - V_{{\mathrm{ent}}}^ \ast }}{{V_{{\mathrm{amp}}}}}} \right)$$
(5)

where \(V_{{\mathrm{ent}}}^ \ast\) is the value of Vent on the trap-ejection line (yellow dashed line in Fig. 3a) at the selected value of Vexit; Vamp, \(V_{{\mathrm{ent}}}^ \ast\), Vent and fin all are directly accessible in the experiment. The result of the conversion in Fig. 4e–h shows the oscillation of the current IP with t1 − t0, as expected.

From the oscillation period of IP (black arrows in Fig. 4e–h), we found that the period of the coherent wave-packet oscillations is τcoh ≈ 4 ps. This value of τcoh corresponds to a ΔE of about 1 meV or QD size (~2lQD) of about 40 nm. The estimated QD size is about half of the lower-gate spacing, which is consistent with the previous estimation in the silicon pump28. Moreover, τcoh is independent of fin (Fig. 4e–h), as it should be. We highlight that τcoh ≈ 4 ps (1/τcoh ≈ 250 GHz) is far in excess of currently achievable bandwidths1,6,12,13. The resolution of our effective time-resolved detection, which is of the order of picoseconds, is determined not by the bandwidth (or a fine tuning of the timing of the a.c. gate voltage), but by the energy change and broadening of the resonant level Δres.

We then simulated the position of the current oscillations in the active parameter space of Vent and Vexit using a simplified version of equation (2) (with p ≃ 0.5, \(\overline {P_{\mathrm{T}}}\) ≃ 0.5 and θ ≃ π) as:

$$P_{\mathrm{T}} = \frac{1}{2}\left[ {1 - {\mathrm{cos}}\left( {2\uppi \frac{{t_1 - t_0}}{{\tau _{{\mathrm{coh}}}}}} \right)} \right]$$
(6)

By estimating the factors that convert the gate voltages to the energy of the electron (Supplementary Section VII) and by using ΔE = 1 meV (τcoh ≈ 4 ps) obtained above, we simulated the current-oscillation maps for fin = 1–4 GHz (Fig. 5). The peak positions with respect to the trap-ejection line are well reproduced, including the shorter period at the trap-ejection line and the curvature of the oscillation lines; the measured oscillation amplitude is also in agreement with estimation based on equation (2) (Supplementary Sections VIII and IX). We emphasize that the current oscillations are reproduced based on reasonably estimated parameters (not on any fitting parameters). These simulations corroborate our interpretation that the coherent wave-packet dynamics of ~250 GHz is detected in our experiment.

Fig. 5: Calculated frequency dependence of the current oscillations.
figure 5

a–d, The calculated first derivative of IP with respect to Vexit as a function of Vent and Vexit at fin = 1 GHz (a), 2 GHz (b), 3 GHz (c) and 4 GHz (d). The regions highlighted by red lines are the same as those shown in Fig. 4 at the same fin. ΔE = 1 meV in this calculation. The other parameters are estimated from the experimental results (Supplementary Section II).

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Next, we roughly evaluated equation (3) using the above results to investigate the validity of the experimental observation (Supplementary Section X), and found that 1 meV \(\lesssim \Delta _{{\mathrm{res}}} \lesssim 6.4\) meV for ΔE = 1 meV and fin = 1 GHz. The range of Δres is acceptable, in comparison with the energy difference (>10 meV, estimated from the result in Fig. 3a) between the top of the exit barrier and the resonant level; the energy difference should be larger than Δres to observe the current oscillations.

We can exclude other possible mechanisms for the observed current oscillations, such as the phonon density of states29 or the Fabry–Perot interference through an unintentionally-formed QD30, because they should be independent of fin. Given the position of the oscillations in the current-oscillation maps, we also rule out Landau–Zener–Stückelberg interference31 and resonant tunnelling through a trap level located in the entrance barrier (Supplementary Sections XI and XII).

Our results imply a protocol for measuring fast electron dynamics in an effective time-resolved fashion of picosecond resolution (Supplementary Section XIII). We suggest that when any kind of dynamic control of a particle in a cavity, which includes its initialization, excitation and coherent oscillations, is repeated by frequency f, the coherent dynamics can be detected as oscillations of the current of the particle through the resonant level by coupling the cavity to any kind of a resonant level driven by an a.c. signal with f. This detection protocol could be useful, for example, to realize an ultra-high-speed charge qubit. Note that such a time-resolved detection of a picosecond dynamics in a cavity is difficult to achieve using a time-dependent tunnel barrier8 or a surface-acoustic-wave QD14. Although there are some similarities between our proposal and the method that uses a surface-acoustic-wave QD14, this point is a marked advantage of our device (Supplementary Section XIV). In addition, although the current approach relies on an uncontrolled trap level, it could be improved using a precisely positioned single atom32. Finally, we stress that the understanding of the picosecond internal dynamics may lead to innovation in quantum technologies based on submicrometre electron devices under a non-equilibrium drive. For example, the spatial electron motion is useful to engineer a wave packet emitted from a single-electron source, for example, to be of a Gaussian form16 (Supplementary Section XV). A Gaussian wave packet has a narrow width in terms of energy or time (achieving the Heisenberg uncertainty limit), and such a narrow wave packet could lead to ultimately high-speed and high-resolution quantum sensing8 and enhancement of the visibility of electron quantum optics experiments2. Furthermore, as such a narrow emitted wave packet (as well as timing control of the emission) is important for the operation of flying qubits1,5, this understanding could contribute to the improvement of the fidelity. In addition, our results indicate that control of the horizontal movement of the QD, for example, by using an additional gate pulse, could suppress non-adiabatic excitation, which would lead to an improvement of the accuracy of single-electron pumping towards the realization of a high-accuracy current source.

Methods

Device fabrication

A silicon wire was patterned using electron-beam lithography and dry etching on a non-doped silicon-on-insulator wafer with a buried-oxide thickness of 400 nm. After formation of thermally grown silicon dioxide with a thickness of 30 nm, three lower gates made of heavily doped polycrystalline silicon were formed using chemical vapour deposition, electron-beam lithography and dry etching. The gate length and spacing of the adjacent lower gates were 40 and 100 nm, respectively. Then, interlayer silicon dioxide was deposited using chemical vapour deposition. Next, an upper gate made of heavily doped polycrystalline silicon was formed using chemical vapour deposition, optical lithography and dry etching. Next, the left and right leads were heavily doped using ion implantation, during which the upper gate was used as an implantation mask. Finally, aluminium pads were formed to obtain an Ohmic contact. The width and thickness of the silicon wire were 10 and 15 nm, respectively.

Measurement detail

Measurements were performed in liquid He at 4.2 K. A Keithley 213 voltage source and a Keysight 83623B signal generator were used to apply the d.c. and a.c. voltages, respectively. The pumping current was measured using the Keithley 6514 electrometer. The d.c. voltage Vupper was applied to the upper gate to induce electrons in the silicon wire. Vent + Va.c.(t) and Vexit were applied to the entrance and exit gates, respectively.