Shuttling an Electron Spin through a Silicon Quantum Dot Array

Coherent links between qubits separated by tens of micrometers are expected to facilitate scalable quantum computing architectures for spin qubits in electrically defined quantum dots. These links create space for classical on-chip control electronics between qubit arrays, which can help to alleviate the so-called wiring bottleneck. A promising method of achieving coherent links between distant spin qubits consists of shuttling the spin through an array of quantum dots. Here, we use a linear array of four tunnel-coupled quantum dots in a ${}^{28}$$\mathrm{Si}/\mathrm{Si}\mathrm{Ge}$ heterostructure to create a short quantum link. We move an electron spin through the quantum dot array by adjusting the electrochemical potential for each quantum dot sequentially. By pulsing the gates repeatedly, we shuttle an electron forward and backward through the array up to $250$ times, which corresponds to a total distance of approximately $80\text{μ}\mathrm{m}$. We make an estimate of the spin-flip probability per hop in these experiments and conclude that this is well below $0.01\mathrm{\%}$ per hop.

Popular Summary

Individual electron spins confined in semiconductor quantum dots have gained traction as a viable route to large-scale quantum computation. A large-scale device will likely consist of a network of spin qubit registers on a single chip, interconnected via quantum coherent links. A powerful and versatile realization of a quantum link is a spin shuttler, whereby electrons are physically moved around the chip in such a way that their spin state is preserved. Shuttling implementations in silicon, the most widely studied host for quantum-dot qubits today, have been limited to double dots. Here we shuttle an electron spin through up to four dots and probe the impact of shuttling on the spin polarization.

Electrons are passed on from one quantum dot to the next, controlled by gate-voltage pulses, as in a bucket brigade. In this way, we shuttle an electron spin back and forth across up to four quantum dots over a cumulative distance of up to 80 micrometers. We test to what extent the spin polarization is affected as we increase the number of shuttle rounds through the array and see no sign of a loss of spin polarization from shuttling.

Follow-up studies will be aimed at quantifying how well spin coherence is preserved while shuttling through extended arrays, as not only the polarization but also the phase must be preserved in future quantum links.

Figure 1

Shuttling through a $\mathrm{Si}/\mathrm{Si}\mathrm{Ge}$ device. (a) A scanning-electron microscope image of a device nominally identical to the device used for this work. The plunger and barrier gates are interleaved and have been fabricated in separate steps. The quantum dots used in this work are indicated by the small circles. The sensing dots on either side of the array are indicated by the large circles. The left-hand sensor is connected to an rf readout circuit via one of the ohmic reservoirs, indicated by squares. This sample does not contain a micromagnet. (b)–(d) Charge-stability diagrams for (b) quantum dots 1–2 while quantum dots 3 and 4 are empty, (c) quantum dots 2–3 while quantum dots 1 and 4 are empty, and (d) quantum dots 4–3 while quantum dots 1 and 2 are empty (right-hand sensing-dot response). The charge-stability diagrams are measured by means of virtual gates. For reference, 25 mV in the virtual gate voltage $\mathrm{\Delta }{V}_{\mathrm{VP}1}$ corresponds to 3.75 meV in the electrochemical potential of dot 1. The electron occupation per charge region and the direction of the shuttling events are indicated in the figures. (e),(f) The shuttling procedure through three quantum dots. An electron with random spin is injected during the loading stage: (e) and (f) show the case of a spin up and spin down electron, respectively. At the end of the shuttling sequence, the spin state is read out using energy-selective tunneling, where a spin-up electron will tunnel out of the quantum dot (e), whereas for a spin-down electron this is energetically forbidden (f). The sensing-dot signal reveals whether or not the electron tunneled to the reservoir, from which the spin state is inferred.

Figure 2

Charge shuttling over many rounds. The single-shot sensor response for $100$ shuttling traces. The spin is loaded in dot $1$ and then shuttled forth and back to dot $3$ for $78$ rounds. Each single-shot trace has an offset with respect to the previous trace of $0.04$ mV in $\mathrm{\Delta }{V}_{\mathrm{VP}1}$. The inset shows an enlargement of part of the plot to better visualize the sensor response. We see that for each trace, the electron shuttles back and forth repeatedly as expected.

Figure 3

Shuttling through (a) three and (b) four quantum dots. The colored circles represent the spin-up probability for different times $t_{\mathrm{array}}$ after shuttling the spin back and forth through the array for $n$ rounds (bottom axis). Each data point is averaged $500$ times. The dashed colored lines represent the average of the data points for each $t_{\mathrm{array}}$ and are a guide to the eye. This average is also plotted as a function of total time spent in the array (top axis, open circles) with an error bar of one standard deviation (the error bars sometimes fall behind the circles). The black circles show the spin-up probability after shuttling back and forth through the quantum dot array once as a function of the total time spent in the array (top axis). Each data point is averaged (a) $750$ and (b) $500$ times. Fitting an exponential to the data yields a weighted $T_1$ of (a) $170\pm 18$ ms and (b) $156\pm 33$ ms.

Figure 4

Shuttling for $500$ rounds through an array of three quantum dots ($2000$ hops). Each data point is the average of $1000$ single-shot traces, taken for three different total times in the quantum dot array: $25$, $50$, and 100 ms. As there is a shift in the readout position during the data collection for the 25-ms case, the first three data points have a calibration error. These outliers are plotted as open circles, yet are not included in further data analysis (see also Fig. 8).

Figure 5

Charge-shuttling data showing the approximately $11$ first shuttling rounds of a shuttling sequence of 78 rounds, showing the sensor response for $100$ repetitions (each trace has an offset with respect to the previous trace of 0.04 mV in $\mathrm{\Delta }{V}_{\mathrm{VP}1}$). The time between hops is about $100\text{μ}\mathrm{s}$ for dots 1 and 3 ($50\text{μ}\mathrm{s}$ for dot 2), roughly 4 times shorter compared to the data in Fig. 2. By examining an enlargement of the first $11$ shuttling rounds, latching effects during loading from the reservoir can be seen in a subset of the traces. Even during the shuttling rounds, it can be seen that in some cases the electron passes from one dot to the next with a small delay. We observe that this jitter only occurs while shuttling from dot 1 to dot 2 and is not symmetric. Moreover, the jitter in this figure is on average $7\text{μ}\mathrm{s}$ long, while this jitter does not appear in other transitions that we monitor, where the electron spends only $10\text{μ}\mathrm{s}$ per quantum dot. While we do not understand the origin of the small delay in electron transfer for the transition from dot 1 to dot 2, such small delays are not a generic limitation of this type of device. Earlier work on devices with similar geometries has also demonstrated fast and controlled transfer of electrons through quantum dot arrays [40], as will be necessary for phase coherent spin shuttling in future experiments.

Figure 6

Relaxation times for the separate quantum dots. The spin-up probability of a randomly loaded spin as a function of the waiting time in (a) dot 1, (b) dot 2, (c) dot 3, and (d) dot 4 at a magnetic field of $1.3$ T. Each data point is averaged $300$, $500$, $400$, and $250$ times for dots $1$, $2$, $3$, and $4$, respectively. Fitting the data to an exponential decay yields relaxation times of $T_{1,1}=129\pm 33$ ms, $T_{1,2}=257\pm 79$ ms, $T_{1,3}=152\pm 48$ ms, and $T_{1,4}=145\pm 71$ ms. The relatively large error bars are a result of the sparsity of data points for longer waiting times.

Figure 7

Tunnel coupling. Excess charge (in units of e) as a function of detuning at the interdot transitions of (a) dots 1 and 2 (1,0,0,0)-(0,1,0,0), (b) dots 2 and 3 (0,1,0,0)-(0,0,1,0) and (c) dots 3 and 4 (0,0,1,0)-(0,0,0,1), measured with the right-hand sensing dot. The extracted tunnel couplings are presented in microelectronvolt.

Figure 8

Single-shot traces for two different data points in Fig. 4. Single-shot traces showing the sensing-dot response of the readout segment after shuttling the electron (a) $500$ and (b) $250$ rounds through the array, respectively, for $t_{\mathrm{array}}=50$ ms. The pink lines indicate a blip, meaning that a tunneling event has happened (spin up). A purple line indicates the absence of a blip and thus represents a spin-down electron. (a) The digitizer timing is slightly offset with respect to the readout pulse; the last two shuttling steps are still visible on the left-hand side of each trace. (b) The digitizer timing is offset with respect to the readout pulse; the start of the compensation pulse is seen on the right-hand side of each trace.

Figure 9

The simulation and fitting of possible spin-flip models. (a),(d) Simulation of the spin-up probability at the end of a shuttling sequence through three dots for (a) symmetric spin flips and (d) asymmetric spin flips, considering in both cases a $0.01\mathrm{\%}$ probability of a spin flip per hop and $T_1$ decay during the wait times in the dots, for different $t_{\mathrm{array}}$. (b),(c) The fit of the numerical spin-flip model considering a symmetric spin flip to the shuttle data of shuttling (b) $250$ rounds and (c) $500$ rounds through three quantum dots. (e),(f) The fit of the numerical spin-flip model considering an asymmetric spin flip to the shuttle data of shuttling (e) $250$ rounds and (f) $500$ rounds through three quantum dots. For all four panels, the fit parameters are the spin-flip probability and the initial spin-up probability. These two parameters are fitted simultaneously for all five (three) values of $t_{\mathrm{array}}$ in a panel. The fit parameters per panel are as follows: (b) initial spin-up probability $53.1\pm 0.8\mathrm{\%}$, symmetric spin-flip probability per hop $0.00\pm 0.005\mathrm{\%}$; (c) initial spin-up probability $56.9\pm 0.6\mathrm{\%}$, symmetric spin-flip probability per hop $0.00\pm 0.008\mathrm{\%}$; (e) initial spin-up probability $53.2\pm 1\mathrm{\%}$, asymmetric spin-flip probability per hop $0.00\pm 0.004\mathrm{\%}$, (f) initial spin-up probability $56.9\pm 0.8\mathrm{\%}$, asymmetric spin-flip probability per hop $0.00\pm 0.002\mathrm{\%}$. In all cases, the error bars are one standard deviation around the mean. Given that both the symmetric and asymmetric model fit the data very well, the fit quality in itself does not allow us to conclude that one model captures the physics better than the other.