Coherent long-distance displacement of individual electron spins

Controlling nanocircuits at the single electron spin level is a possible route for large-scale quantum information processing. In this context, individual electron spins have been identified as versatile quantum information carriers to interconnect different nodes of a spin-based semiconductor quantum circuit. Despite extensive experimental efforts to control the electron displacement over long distances, maintaining electron spin coherence after transfer remained elusive up to now. Here we demonstrate that individual electron spins can be displaced coherently over a distance of 5 µm. This displacement is realized on a closed path made of three tunnel-coupled lateral quantum dots at a speed approaching 100 ms−1. We find that the spin coherence length is eight times longer than expected from the electron spin coherence without displacement, pointing at a process similar to motional narrowing observed in nuclear magnetic resonance experiments. The demonstrated coherent displacement will open the route towards long-range interaction between distant spin qubits.

W hile it is clear by now that the spin degree of freedom of an electron is an interesting building block for processing and storing quantum information [1][2][3][4][5][6] , important questions concerning the system scalability remain to be addressed before building a large-scale spin-based quantum processor. Ultimately, the problem reduces to the ability to transfer quantum information on a chip. Following the work on superconducting qubits, significant experimental efforts are currently focusing on the possibility to couple distant electron spins via a quantum mediator [7][8][9] . An alternative way consists in displacing the electron spin itself 2 . One possibility is to convey the electron in moving quantum dots defined by surface acoustic waves, where it is trapped and propagates isolated from the surrounding electrons at the speed of sound [10][11][12][13] . Even though electron and spin transfer have been demonstrated, the technology of moving quantum dots at the single electron level is not yet controlled well enough to investigate coherence properties 12 . A more conventional strategy consists in displacing the electron in an array of tunnel-coupled quantum dots [14][15][16][17][18][19][20][21][22] . So far, only a classical spin shuttle over linear arrays of three and four dots has been demonstrated, whereas slow electron displacement on a closed loop has been demonstrated in a fourquantum-dot system 17 .
In this article, we demonstrate the coherent spin displacement of individual electrons in investigating the spin dynamics of two electrons initially prepared in a singlet spin state and displaced in an array of three lateral quantum dots defined in a circular geometry (a similar demonstration was recently realized in linear dot arrays 18 ). When the electrons are displaced and explore a larger surface area than without motion, the main decoherence mechanism for static electron spins is averaged away and the coherence time of the singlet state is observed to be enhanced. Moreover, displacement-induced spin-flip processes are revealed with the dependence of the coherence time with the externally applied magnetic field and limit the distance over which electron spin coherence can be preserved.

Results
Electron displacement and spin measurement. The triple-dot system is defined by Schottky gates with standard split-gate techniques in a two-dimensional electron gas (2DEG) residing at the interface of AlGaAs-GaAs heterostructure (see Fig. 1a and "Methods" section). The two electrons are loaded into the system via the bottom dot. With large gate µs-movement, they are brought above the Fermi energy in the isolated configuration where the coupling to the electron reservoir can be ignored (see Supplementary Note 1) 23 . The resulting charge response of the electrometer when changing the chemical potentials of the three dots in the isolated configuration is presented in Fig. 1b. As expected, only six possible charge configurations are observed (see Supplementary Note 1 and Supplementary Fig. 1), rendering electron displacement in the circular triple dot system straightforward to implement. The tunnelling rates between the three dots are tunable up to the gigahertz regime. Nanosecond control of the gate voltages permits therefore adiabatic electron transfer between the dots faster than the spin coherence time 23 .
The two-electron spin state after displacement can be inferred by bringing the electrons in the bottom dot, where exchange of electrons with the reservoir is possible 23 . In the two-electron case, the ground singlet (S) and the three excited triplet (T + , T 0 , T − ) states are distinguished using the tunnel-rate spin read-out method with a single-shot fidelity of 80% (see Supplementary Note 2 and Supplementary Fig. 2) 24,25 . Unless specified, the experiments are repeated a thousand times to obtain the singlet probability.
Electron spin coherence without displacement. First, we specifically focus on the spin dynamics when the two electrons are static in two different dots. Initialization in the singlet ground state is performed by relaxation in the bottom dot. By rapidly pulsing the gate voltages, the electrons are separated into two different dots for a controlled duration τ s . In this way, we probe how long the phase coherence initially present in the singlet state can be preserved when the electrons are separated 16 . If the phase coherence is maintained, the system will remain in the singlet state. Otherwise, the final spin state will be a mixture of singlet and triplet states. Figure 2a presents where singlet-triplet spin mixing is occurring in the charge stability diagram. Each point is obtained by initialization to the singlet state at point R of Fig. 1b. Pulses of 50 ns duration and amplitude V 1 and V 2 are then simultaneously applied on the V M,1 and V M,2 gates, respectively. The resulting two-electron spin state population for each (V 1 , V 2 ) is then averaged over 150 single shot measurements. Three distinct regions where the spin mixing is efficient are observed. They correspond to the three charge configurations where the  electrons are separated in two dots. In these regions, the exchange interaction between the two electrons can be neglected and the system is dominated by the coupling to the nuclear spins of the heterostructure via hyperfine interaction 16,23,26 . At a magnetic field of 150 mT, the spin mixing occurs only between S and T 0 (see Fig. 2b). By varying τ s , we observe a Gaussian decay e Àτ s 2 =T Ã2 2 of the singlet probability characterized by a timescale Τ 2 * close to 10 ns, very similar in each mixing region 16,23 . In addition to the S-T 0 mixing regions, we notice four additional mixing lines, clearly separated from the other mixing regions. We attribute them to the mixing of S and T + states and their observation is a signature of large and coherent tunnel-coupling between the dots (see Supplementary Note 3) 23 .
Electron spin coherence with displacement. We proceed to the investigation of the two-electron phase coherence while the two electrons are individually displaced on the closed loop formed by the three quantum dots. More specifically, the electrons are initially prepared in the singlet state of the (2, 0, 0) charge configuration. The system is then pulsed fast to the region (1, 1, 0), where the electrons are separated in two dots and rotated repeatedly between the spin mixing regions of the (1, 1, 0), (0, 1, 1) and (1, 0, 1) charge configurations with voltage pulse sequence on V 1 and V 2 (see Figs. 2a and 3a). It leads to a series of quantum dot displacements and single electron tunnelling events schematically shown in Fig. 3b. Arbitrary long displacements can be implemented by repeating the loop. We control the number of loops N t performed by the electrons and the duration τ r spent in each charge configuration. With displacement, the time τ s where the electrons are separated and experience singlet-triplet mixing is equal to 3 N t τ r . Finally, the system is tuned back from the (1, 1, 0) to the (2, 0, 0) charge configuration, where spin read-out is performed. With τ r equal to 1.7 ns, the resulting singlet probability is decaying as a function of τ s . These results demonstrate coherent electron spin transfer in an array of quantum dots in a circular geometry. It is worth noting that the decoherence law follows an exponential decay e Àτs=τDecay with displacement (see Supplementary Note 4 and Supplementary Fig. 4). We fit a spin coherence time of 80 ns, almost eight times longer than for the static case at 200 mT (see Fig. 3c), only possible with a significant reduction of the influence of the hyperfine interaction during the electron displacement (for the  Fig. 1b, 50-ns pulses of amplitude V 1 and V 2 are simultaneously applied on V M,1 and V M,2 , respectively, before performing single-shot read-out of the two-electron spin states to extract the singlet probability. In the separated configurations, S mixes with T 0 . In between two of these regions, one electron is exchanged between two dots and the spin mixing is less effective due to the increase of the exchange interaction (see Supplementary Note 3 and Supplementary Fig. 3). The arrows represent the (V 1 , V 2 ) voltage pulses used to implement the separation and the displacement of the electrons and presented in Fig. 3a.  For τ r set to 2.5 ns, the time dependence of the spin mixing no longer exhibits a single exponential behaviour but is characterized by two timescales (see inset in Fig. 3c). First, the system decays fast on a timescale similar to the coherence time in static dots, and then it evolves on a longer timescale towards a mixed singlet-triplet state. Figure 4a shows the spin singlet probability as a function of N t for different values of τ r . The long decay is only dependent on N t , and the underlying decay mechanism must therefore depend dominantly on the number of tunnelling events or on the travelled distance of the electrons, rather than the separation time (the influence of the displacement geometry is discussed in Supplementary Note 7 and Supplementary Fig. 7). We interpret these observations as the consequence of two different phases during the displacement procedure: the static phase where the electrons are static in two different dots and the transfer phase where they are moving between two dots separated by~110 nm on a fixed timescale corresponding to the rise time of the pulse generator (0.9 ns).
When the electrons realize only one rotation (N t = 1) with increasing τ r (τ s = 3τ r ), the influence of the transfer phase is minimized. In this situation, the observed spin mixing time is only 1.74 ± 0.17 times longer than in the static configuration (see Fig. 4b). Additionally, the singlet probability decay is Gaussian hinting at a similar decoherence mechanism as in the case with no displacement.
For τ r set to 1.7 ns, the time spent in the static phase is minimized and the spin decoherence process is mainly occurring during the transfer phase (see "Methods" section). In this case, the spin dynamics is characterized by a single exponential decay of the singlet probability (see Fig. 5). As the magnetic field is increased from 0 to 200 mT, we observe a progressive reduction of the singlet mixing with T + and T − after a 250 ns evolution and a linear increase of the spin coherence time, from 12 ns to almost 80 ns. At 200 mT, only mixing between S and T 0 is to a good approximation observed. Considering the estimated distance of 110 nm between the dots (see Supplementary Note 1), we measure a maximal spin coherence length of 5 µm.

Discussion
The extremal cases of minimized static phase and minimized transfer phase allow us to address the important issue of the coherence limiting processes during the displacement. During the static phase, the electrons are experiencing a fast spin mixing similar to the case with no displacement. Moreover, the observed increase factor of the spin mixing time equal to 1.74 is very close to the ffiffi ffi 3 p -factor expected for an electron spin coupled to a three times larger nuclear spin bath 26,27 . Therefore, these observations support a scenario where the main source of decoherence in the static phase is the longitudinal hyperfine interaction.
During the transfer phase, the electrons are displaced in moving quantum dots induced by the time-dependent potentials applied on the gates, before and after the tunnelling processes. As a consequence, the number of nuclei coupled to the electrons is drastically increased and a process similar to motional narrowing observed in liquid nuclear magnetic resonance experiments 27, 28 is expected to result in an increase of the electron spin coherence. This is in agreement with the observed decoherence law in Figs. 2b and 3c, which changes from Gaussian to exponential decays without and with displacement. Moreover, individual spin-flip processes, stimulated by the electron motion and resulting from The data are fitted with an exponential decay with a characteristic time τ Decay . Inset: Extracted τ Decay as a function of the magnetic field B. The solid line is a linear fit of the data either spin-orbit or transverse hyperfine couplings, are then a possible decoherence channel for the electron spins 28,29 . Due to the change of the dot position during the displacement procedure, both mechanisms are producing an effective time-dependent magnetic field ΔB on the displaced electrons leading eventually to a spin-flip. As a result, the two-electron system is able to leave the S-T 0 subspace (see Supplementary Note 5 and Supplementary  Fig. 5). The smaller the change of the dot position, the smaller is ΔB. The displacement procedure in moving quantum dots therefore implies that the amplitude of ΔB decreases with frequency. As only the ΔB component at the Larmor frequency leads to spin-flips, the spin-flip processes are expected to become less and less efficient as the magnetic field is increased from 0 to 200 mT. These considerations are in agreement with the experimental findings of an enhancement of the spin mixing time and a gradual increase of the singlet population after 250 ns evolution with magnetic fields presented in Fig. 5. At 200 mT, the system remains, to a good approximation, in the S-T 0 subspace and the spin coherence time is expected to be limited by the residual longitudinal hyperfine interaction after motional narrowing 28 .
In conclusion, we have demonstrated that the coherence of a two-electron singlet state is preserved when the electrons are separated and displaced over 5 µm around a closed loop in a three-dot system. Compared to the situation without displacement, the spin coherence time is increased by a factor of eight via a motional narrowing process to 80 ns. Furthermore, spin-flip processes stimulated by the electron motion are found to limit the spin coherence time. The demonstrated coherent spin displacement could be a viable route to interconnect quantum nodes in spin-based quantum processors. On the more fundamental side, increasing the speed of the closed-loop transfer with larger tunnel-couplings should allow the exploration of non-abelian and holonomic spin manipulation [30][31][32][33] in future experiments.

Methods
Multi-dot system and experimental set-up. The device is defined by Schottky gates in an n-Al 0.3 Ga 0.7 As/GaAs 2DEG-based heterostructure (the properties of the non-illuminated 2DEG are as follows: mobility μ ≈ 10 6 cm 2 V −1 s −1 , density n s ≈ 2.7 × 10 11 cm −2 , depth 100 nm) with standard split-gate techniques. It is anchored to a cold finger mechanically screwed to the mixing chamber of a dilution fridge with a base temperature of 70 mK. It is placed at the centre of the magnetic field produced by a solenoid. The coil allows to produce magnetic fields perpendicular to the 2DEG. The charge configuration of the triple-dot system is determined by measuring the conductance of the sensing dots biased with 300 μV; the current is measured using a current-to-voltage converter with a bandwidth of 10 kHz. The voltage on each gate can be varied on µs-timescales to allow exploration of the isolated configuration. Each green gate (V M,1 , V M,2 , V M,3 ) in Fig. 1a is connected through a low temperature home-made bias-T to both DC and high bandwidth coaxial lines allowing gigahertz manipulations.
Electron displacement procedure. The voltage pulses to induce electron displacement are generated by an arbitrary waveform generator Tektronix 5014 C with a typical rise time (20-80%) approaching 0.9 ns. For τ r equal to 1.7 ns, the pulse sequence presented in Fig. 3a is just reaching the programmed voltage amplitude. We can therefore assume that the electrons are only in the transfer phase during the displacement for τ r = 1.7 ns.
Data availability. All relevant data are available from the authors.