A sound-driven single-electron circuit

The sample is realised via surface electrodes forming a depleted potential landscape in the two-dimensional electron gas (2DEG) of a GaAs/AlGaAs heterostructure. An interdigital transducer (IDT) is used to send a finite SAW train towards our single-electron circuit as shown schematically in Fig. 1a. A scanning-electron-microscopy (SEM) image of the investigated single-electron circuit is shown in Fig. 1b. The device consists of two 22-µm-long quantum rails that are coupled along a region of 2 µm by a tunnel-barrier, which is defined by a 20 nm -wide surface gate. The SAW train allows the transport of a single electron from one gate-defined QD (source) to another stationary QD (receiver) through the circuit of coupled quantum rails (QR). Figure 1c shows a zoom on the lower receiver QD with indications of the electrical connections. To detect the presence of an electron, a quantum point contact (QPC) is placed next to each QD. By biasing this QPC at a sensitive working point, an electron leaving or entering the QD can be detected by a jump in the current \({I}_{{\rm{QPC}}}\)41.

Fig. 1 Sound-driven circuit of coupled quantum rails. a Schematic of the experimental set-up. An interdigital transducer (IDT) launches a SAW train towards the single-electron circuit, which is realised via metallic surface gates in a GaAs/AlGaAs heterostructure. b SEM image of the quantum rails (QR) with indications of the transport paths, U and L, and the voltages to control the coupling region. c SEM image of the lower receiver quantum dot (QD) with indication of the coupled quantum rail (QR) and the close-by quantum point contact (QPC). d Jumps in QPC current, \({I}_{{\rm{QPC}}}\), at the upper receiver QD from thousand SAW-driven single-shot transfers with (red) and without (grey) initial loading of a solitary electron at the source QD Full size image

Transfer efficiency

Let us first quantify the efficiency of SAW-driven single-electron transfer along a single quantum rail. For this purpose, we decouple the two transport channels by setting a high tunnel-barrier potential using a gate voltage of \({V}_{{\rm{T}}}=-1.2\ {\rm{V}}\). To quantify the errors of loading, sending and catching, we repeat each SAW-driven transfer sequence with a reference experiment where we initially do not load an electron at the source QD. Figure 1d shows the jump in QPC current, \(\Delta {I}_{{\rm{QPC}}}\), after SAW transmission at the upper receiver QD for an exemplary set of thousand single-electron transfer experiments in an optimised configuration. The grey data points stem from the reference experiments without initial loading at the source QD. The distinct peaks in the histograms of the events with (red) and without (grey) initial loading show that the presence of an electron in the QD is clearly distinguishable. Analysing 70,000 successive experiments of this kind in a single optimised configuration of the quantum rail, we quantify the efficiency of SAW-driven single-electron transport. Thanks to the low error rates of loading (0.07%) and catching (0.18%), we deduce a transfer efficiency along our 20 µm-long quantum rail of 99.75%. A similar single-shot transfer efficiency has recently been obtained with single-electron pumps emitting high-energy ballistic electrons42.

Partitioning an electron in flight

Having established highly efficient single-electron transport, we now couple the two channels to partition an electron in flight between the two quantum rails. The aim of this directional coupling is to prepare a superposition state of a flying electron qubit. We find that we can finely control the partitioning of the electron by detuning the double-well potential as indicated in Fig. 2a, b. To achieve this effect, we sweep the voltages applied to the side electrodes of the coupling region, \({V}_{{\rm{U}}}\) and \({V}_{{\rm{L}}}\), in opposite directions while keeping \({V}_{{\rm{T}}}\) constant. With a potential detuning, \(\Delta ={V}_{{\rm{U}}}-{V}_{{\rm{L}}}=0\) V, the quantum rails are aligned in electric potential. Setting a voltage configuration where \(\Delta \ < \ 0\), the potential of the lower quantum rail (L) is decreased with respect to the upper path (U). For \(\Delta \,> \,0\), the situation is reversed. Deducing the transfer probabilities to the receiver QDs from a thousand single-shot experiments per data point, we measure the partitioning of the electrons for different values of \(\Delta\) as shown in Fig. 2c. Here, we sweep \({V}_{{\rm{U}}}\) and \({V}_{{\rm{L}}}\) in opposite directions from \(-1.26\ {\rm{V}}\) to \(-0.96\ {\rm{V}}\) while keeping \({V}_{{\rm{T}}}=-0.75\ {\rm{V}}\). The data shows a gradual transition of the electron transfer probability from the upper (U) to the lower (L) detector QD while the total transfer efficiency stays at \(99.5\pm 0.3 \%\).

Fig. 2 Directional-coupler operation. a Schematic slices along the double-well potential, \(U\). The horizontal lines represent the eigenstates in the moving QD, whereas the grey shading of the energy levels indicates the exponentially decreasing occupation. b Schematic showing the QDs that are formed by the SAW in the coupling region with additional indications of the surface gates and the transport paths. The black vertical bar indicates the positions of the aforementioned potential slices. c Probability, \(P\), to end up in the upper (U) or lower (L) quantum rail for different values of potential detuning, \(\Delta\). The lines show a fit by a Fermi function providing the scale parameter, \(\sigma\). d Transition widths, \(\sigma\), for different values of the tunnel-barrier voltage, \({V}_{{\rm{T}}}\). The line shows the course of a stationary, one-dimensional model of the partitioning process Full size image

An interesting feature of the observed probability transition is that it follows the course of a Fermi–Dirac distribution:

$${P}_{{\rm{U}}}(\Delta )\approx \frac{1}{\exp (-\Delta /\sigma )+1}$$ (1)

Fitting the experimental data with such a function (see lines in Fig. 2c), we can quantify the width of the probability transition via the scale parameter, \(\sigma\). To test the dependencies of the directional-coupler transition on the different properties of the device, we experimentally investigated if the width of the probability transition changes as we sweep the gate voltage configurations on different surface electrodes of the nanostructure. We find a significant narrowing of the probability transition (see Fig. 2d) as we increase the tunnel-barrier potential.

The role of excitation

To obtain a better understanding of our experimental observations, we first investigate the partitioning process by means of a stationary model. We consider a one-dimensional cut of the double-well potential in the tunnel-coupling region. In this region, we have a sufficiently flat potential landscape, \(U({\boldsymbol{r}},t)\approx U(y)+{U}_{{\rm{SAW}}}(x,t)\), such that the eigenstate problem becomes separable in the \(x\) and \(y\) coordinates. The electronic wave function \({\phi }_{i}(y)\) along the transverse \(y\) direction satisfies the one-dimensional Schrödinger equation:

$$\frac{{\hslash }^{2}}{2{m}^{* }}\frac{{\partial }^{2}{\phi }_{i}(y)}{\partial {y}^{2}}+U(y)\cdot {\phi }_{i}(y)={E}_{i}{\phi }_{i}(y)$$ (2)

where \(U(y)\) is a the electrostatic double-well potential for a given set of surface-gate voltages \({V}_{{\rm{U}}}\), \({V}_{{\rm{L}}}\) and \({V}_{{\rm{T}}}\). \({m}^{* }\) indicates the effective electron mass in a GaAs crystal. Here, we obtain \(U(y)\) for the specific geometry of the presently investigated device by solving the corresponding Poisson problem43,44.

To obtain the probability of finding the electron in the upper or lower potential well, we can now simply sum up the contributions of the wave function in the eigenstates for the respective region of interest. For the upper quantum rail, we integrate the modulus squared of the wave function over the spatial region of the upper quantum rail:

$${P}_{{\rm{U}}}={\sum }_{i}{p}_{i}\mathop{\int }\limits_{y> 0\ {\rm{nm}}}\left|\right.{\phi }_{i}\left(\right.y,U(y)\left)\right.{\left|\right.}^{2}\ {\rm{d}}y$$ (3)

where \({p}_{i}\) is the occupation of the eigenstate \({\phi }_{i}\). For a fixed tunnel-barrier height, we can detune the double-well potential by varying \(\Delta\), as in experiments. It is now straightforward to calculate the directional-coupler transition for the experimental setting with any imaginable occupation of the eigenstates.

Let us first consider the hypothetical situation where only the ground state is occupied. We evaluate Eq. (3) with mere ground state occupation (\({p}_{0}=1\)) and fixed barrier potential (\({V}_{{\rm{T}}}=-0.7\ {\rm{V}}\)) for different values of potential detuning, \(\Delta\), that are changed as in experiment. Doing so, we obtain a course of the probability transition having the shape of the aforementioned Fermi–Dirac distribution. Assuming ground state occupation in the double-well potential, we obtain however an extremely abrupt transition in transfer probability with a width, \(\sigma\), that is in the order of several microvolts what is much smaller than in our experiment.

Let us now investigate how the situation changes as we populate successively excited eigenstates of the double-well potential. For this purpose we define the occupation of the eigenstates, \({\phi }_{i}\), with eigenenergies, \({E}_{i}\), via an exponential distribution:

$${p}_{i}\propto \exp \left(\!\!-\frac{{E}_{i}-{E}_{0}}{\varepsilon }\right)$$ (4)

where \(\varepsilon\) is a parameter determining the occupation of higher energy eigenstates. This approach allows us to maintain the course of a Fermi distribution as we successively occupy excited states. Increasing the occupation parameter \(\varepsilon\), we find a broadening of the probability transition. For \(\varepsilon =3.5\ {\rm{meV}}\) we obtain simulation results showing very good agreement with the experimental data. Keeping \(\varepsilon\) constant, the one-dimensional model follows the experimentally observed transition width, \(\sigma\), over a wide range of \({V}_{{\rm{T}}}\) as shown by the line in Fig. 2d. Note, however, that \(\varepsilon\) only provides a rough estimate for the excitation energy that is present in our experiment due to the uncertainties that enter the model via the potential calculation. The model shows that the width of the directional-coupler transition, \(\sigma\), reflects the occupation of excited states and thus indirectly the confinement in the moving QDs that are formed by the SAW along the tunnel-coupled quantum rails.

Our analysis of the experimental data shows that the flying electron is significantly excited as it propagates through the coupling region of the present circuit. To find possible sources of charge excitation, we employed a more elaborate model to simulate the time-dependent SAW-driven propagation of the electron along different sections of our beam-splitter device45. For this purpose, we superimpose the static, two-dimensional potential landscape, \(U({\boldsymbol{r}})\), with the dynamic modulation of a SAW train, \({U}_{{\rm{SAW}}}(x,t)\), that we estimate from Coulomb-blockade measurements. Simulating the entrance of a flying electron from the injection channel into the tunnel-coupled region, we find significant excitation of the flying electron into higher energy states.

To quantify adiabatic transport of the flying charge qubit, we define the qubit fidelity, \(F\), as projection of the electron wave function on the two lowest eigenstates of the moving QD potential that is formed by the SAW along the coupled quantum rails. Figure 3a shows courses of the qubit fidelity, \(F\), of a flying electron state that propagates along the tunnel-coupled region for different values of peak-to-peak SAW amplitude, \(A\). For the present experiment, we estimate \(A\) as 17 meV. For this value (red solid line), the simulation data shows an abrupt reduction of the qubit fidelity, \(F\), due to the aforementioned excitation of the SAW-transported electron at injection from a single-quantum rail into the tunnel-coupled region. In congruence with the stationary, one-dimensional model that we applied before, the coupling into higher energy states leads to a spreading over both sides of the double-well potential as shown in Fig. 3b and Supplementary Movie 1. The simulation thus shows up a major source of excitation. When the electron passes from the strongly confined injection channel into the wide double-well potential it experiences an abrupt reconfiguration of the eigenstates in the moving QD what causes Landau–Zener transitions in higher energy states.

Fig. 3 Time-dependent simulation of electron propagation. a Course of the qubit fidelity, \(F\), for SAW-driven single-electron transport along the coupling region for different values of SAW amplitude, \(A\). b Trace of the electron wave function, \(\Psi\), along the coupled quantum rails for \(A=17\ {\rm{meV}}\) at selected times, \(t\), indicated via the vertical dashed lines. The grey regions indicate the surface gates. c Trace of \(\Psi\) for \(A=45\ {\rm{meV}}\) Full size image

Towards adiabatic transport

Let us now investigate if we can reduce the probability for charge excitation by increasing the longitudinal confinement via the SAW amplitude. For \(A=30\ {\rm{meV}}\)—see red dashed line in Fig. 3a—charge excitation is already strongly mitigated. The qubit fidelity vanishes however also in this case, since the electron still occupies low-energy states above the two-level system we are striving for. Despite non-adiabatic transport, we can already recognise coherent tunnel oscillations when looking at the trace of the wave function as shown in Supplementary Movie 2. This shows that also excited electron states can undergo coherent tunnelling processes as previously expected in magnetic-field-assisted experiments on continuous SAW-driven single-electron transport through a quantum rail that is tunnel-coupled to an electron reservoir46. Increasing the SAW amplitude further to \(A=45\ {\rm{meV}}\) (blue solid line), the transport of the electron gets nearly adiabatic and clear coherent tunnel oscillations occur as shown in Fig. 3c and Supplementary Movie 3. The simulations show that stronger SAW confinement can indeed prohibit charge excitation and maintain adiabatic transport. In experiment, one can increase the SAW confinement via many ways such as reduced attenuation of the IDT signal, longer IDT geometries, impedance matching or the implementation of more advanced SAW generation approaches47,48,49. We anticipate therefore the experimental observation of coherent tunnel oscillations in follow-up investigations.

Triggering single-electron transfer

Achieving adiabatic single-electron transport, a SAW train could also be employed to couple a pair of flying electrons in a beam-splitter set-up. In the long run, this coupling could enable entanglement of single-flying electron qubits through their Coulomb interaction14 or spin15. For this purpose, electrons must be sent simultaneously from different sources in a specific position of the SAW train. Let us now investigate if we can achieve such synchronisation by using a fast voltage pulse as trigger for the sending process with the SAW9. After loading an electron from the reservoir, we bring the particle into a protected configuration where it cannot be picked up by the SAW. To load the electron into a specific minimum of the SAW train, we then apply a voltage pulse at the right moment to the plunger gate of the QD as schematically indicated in Fig. 4a, b. This pulse allows the electron to escape the stationary source QD into a specific moving QD formed by the SAW along the quantum rail.

Fig. 4 Pulse-triggered single-electron transfer. a SEM image of the source quantum dot (QD) showing the pulsing gate highlighted in yellow. A fast voltage pulse on this gate allows one to trigger SAW-driven single-electron transport along the quantum rail (QR) as schematically indicated. b Measurement scheme showing the modulation, \(\delta U\), of the electric potential at the stationary source QD: the delay of a fast voltage pulse, \(\tau\), is swept along the arrival window of the SAW. c Measurement of the probability, \(P\), to transfer a single electron with the SAW from the source to the receiver QD for different values of \(\tau\). d Zoom in on a time frame of four SAW periods, \({T}_{{\rm{SAW}}}\) Full size image

To demonstrate the functioning of this trigger, we use a very short voltage pulse of 90 ps corresponding to a quarter SAW period50. Sweeping the delay of this pulse, \(\tau\), over the arrival window of the SAW at the source QD, we observe distinct fringes of transfer probability as shown in Fig. 4c and more detailed in Fig. 4d. The data shows that the fringes are exactly spaced by the SAW period. The periodicity of the transmission peaks indicates that there is a particular phase along the SAW train where a picosecond pulse can efficiently transfer an electron from the stationary source QD into a specific SAW minimum. As the voltage pulse overlaps in time with this phase, the sending process is activated and the transfer probability rapidly goes up from 2.7 ± 0.5% to 99.0 ± 0.4%. The finite background transfer probability is due to limited pulse amplitude in the present set-up. The envelope of the transfer fringes is consistent with the expected SAW profile. Comparing the directional-coupler measurement with and without triggering of the sending process, we find no change in the transition width what indicates that excitation at the source QD is comparably small or not present. By reduction of pulse attenuation along the transmission lines and optimisation of the QD structure, we anticipate further enhancements in the efficiency of the voltage-pulse trigger. The present pulsing approach allows us to synchronise the SAW-driven sending process along parallel quantum rails and represents thus an important milestone towards the coupling of single-flying electrons.