Interplay of Zeeman Splitting and Tunnel Coupling in Coherent Spin Qubit Shuttling

Abstract

Spin shuttling offers a promising approach for developing scalable silicon-based quantum processors by addressing the connectivity limitations of quantum dots. In this work, we demonstrate high-fidelity bucket-brigade spin shuttling in a silicon MOS device, utilizing Pauli-spin-blockade readout. We achieve an average shuttling fidelity of \SI{99.8}{\percent}. The residual shuttling error is highly sensitive to the ratio between interdot tunnel coupling and Zeeman splitting, with tuning of these parameters enabling up to a 20-fold variation in error rate. An appropriate four-level Hamiltonian model supports our findings. These results provide valuable insights for optimizing high-performance spin-shuttling systems in future quantum architectures.

Figures (11)

• Figure 1: The device and polarization shuttling.a, A false-color scanning-electron-micrograph (SEM) image of a device nominally identical to the one used in this work. The three plunger gates, $\mathrm{P_1}$, $\mathrm{P_2}$ and $\mathrm{P_3}$ are colored in magenta, and the QDs under them are indicated in red, blue, and cyan. The barrier gate $\mathrm{J_1}$ between $\mathrm{Q_1}$ and $\mathrm{Q_2}$ is colored in sienna. The direction of the dc external magnetic field (ac control microwave field) is indicated by white (yellow) arrows. The single-electron transistor (SET) [electron spin resonance (ESR) line] is marked in light pink (green). b, A pulse schematic for the pulsed electron spin-orbital spectroscopy (PESOS) map. A constant ESR control pulse is applied after ramping. c, The PESOS map near the yellow star in (d) under a 0.89-T external field. The inset is an enlarged image near 24.735 GHz, where $f_{\mathrm{Q_1}}$ and $f_{\mathrm{Q_3}}$ are located. d, The charge-stability diagram (CSD) as a function of the voltage detuning $\varepsilon_{\mathrm{P_2 \text{-} P_1}}$, which is used in the $\mathrm{Q_1 \text{-} Q_2}$ transition, and the gate voltage $V_\mathrm{P_3}$, which is used in the $\mathrm{Q_2 \text{-} Q_3}$ transition. The current difference at transitions of the $\mathrm{Q_1 \text{-} Q_2}$ is clear in red, while those of the $\mathrm{Q_2 \text{-} Q_3}$ transition are unclear after removing the background and are indicated by the dotted blue lines (see Fig. \ref{['fig:CSD']}a in the Supplemental Material). The control and readout points are labeled as C and RO, respectively. The cyan path indicates our shuttling protocol from the (110) to the (011) charge state. e, The pulse schematic for the polarization-shuttling experiment. Depending on the states prepared or the measurement projections, $\pi$ rotations are applied before or after the consecutive shuttling. f, The probabilities of finding spin-up or spin-down postshuttling if spin-up or spin-down are prepared under a 0.17-T external field. The solid curves are fits to the data (crosses); the shuttling depolarizing rates $r^{\downarrow(\uparrow)}$ and their errors are calculated from the exponential fits in Fig. \ref{['fig:eigen']} in the Supplemental Material.
• Figure 2: Phase-coherent shuttling spectroscopy under an 0.17-T external field.a, The pulse schematic for the shuttling spectroscopy. The spin in $\mathrm{Q_1}$ is first rotated to the equatorial state at the control point. After ramping the voltage, the spin accumulates a phase during the wait time $t_\mathrm{wait}$ before ramping back to the (110) state. A second $X(\pi/2)$ gate is applied to project the phase onto the polarization in the measurement basis. b, Shuttling spectroscopy near the $\mathrm{Q_1 \text{-} Q_2}$ change transition. The continuous fringe evolution demonstrates the phase coherence when shuttling. c, A line-cut of (b) at $\varepsilon_\mathrm{P_2 \text{-} P_1} = \pm 40mV$. d, The pulse schematic for the consecutive shuttling spectroscopy. Between two single-qubit gates at the control point, the voltage is ramped back and forth between $\mathrm{Q_1}$ and $\mathrm{Q_2}$ repeatedly in a total evolution time $T_\mathrm{evol}$. e, The consecutive-shuttling spectroscopy near the $\mathrm{Q_1 \text{-} Q_2}$ charge transition. f, The shuttle characterization at $\varepsilon_{\mathrm{P_2 \text{-} P_1}} = 40mV$. The stable oscillation period demonstrates the consistency of each shuttling operation.
• Figure 3: The tunnel-coupling dependence.a, A schematic of the energy diagram when $\mathop{\mathrm{\overline{\textit{E}_\mathrm{Z}}}}\nolimits \gg \mathop{\mathrm{2\textit{t}_\mathrm{c}}}\nolimits$; e.g., at the point indicated by the purple up arrow in (d). The eigenenergies of these four states are calculated from diagonalizing the Hamiltonian in Appendix C. The dotted magenta lines indicate the transitions between two charge states for spin-up (cyan) and spin-down (blue) states. Two degenerate points of the $\ket{\mathrm{g},\uparrow}$ (cyan) and $\ket{\mathrm{e}, \downarrow}$ (gray) states are indicated by red circles. b, A schematic of the energy diagram when $\mathop{\mathrm{\overline{\textit{E}_\mathrm{Z}}}}\nolimits \ll \mathop{\mathrm{2\textit{t}_\mathrm{c}}}\nolimits$, e.g., the point indicated by the green-yellow down arrow in (d). In this case, the energy difference between spin-up and spin-down changes slowly compared to the case in whch $\mathop{\mathrm{\overline{\textit{E}_\mathrm{Z}}}}\nolimits \gg \mathop{\mathrm{2\textit{t}_\mathrm{c}}}\nolimits$ in (a). Furthermore, the chemical potential of the orbital excited states is always higher than that of the ground states. c, The tunnel couplings as a function of the gate voltage $\mathop{\mathrm{\textit{V}_\mathrm{J_1}}}\nolimits$ are determined from both charge-transition broadening and spin exchange-coupling (see Appendix D)(see also Fig. S5 in the Supplemental Meterial). From the fits, we can calculate the tunnel couplings $\mathop{\mathrm{\textit{t}_\mathrm{c}}}\nolimits = (0.32 \pm 0.01)\exp[(23.56 \pm 0.3)(\mathop{\mathrm{\textit{V}_\mathrm{J_1}}}\nolimits -1.072)]$ as a function of $\mathop{\mathrm{\textit{V}_\mathrm{J_1}}}\nolimits$. d, The dephasing rate $p$ of the shuttling process as a function of the barrier gate voltage $\mathop{\mathrm{\textit{V}_\mathrm{J_1}}}\nolimits$. The corresponding tunnel coupling ($\mathop{\mathrm{\textit{t}_\mathrm{c}}}\nolimits$) is calculated from fits of the experiment results (c). The tunnel couplings $\mathop{\mathrm{\textit{t}_\mathrm{c}}}\nolimits$ that equal half of the Zeeman splitting $\mathop{\mathrm{\overline{\textit{E}_\mathrm{Z}}}}\nolimits$ under 0.17- and 0.89- T magnetic fields are indicated by dotted sienna lines. We characterize these dephasing rates by varying the number of shuttles (see Fig. \ref{['fig:LF']} and Fig. \ref{['fig:HF']} in the Supplemental Material).
• Figure 4: Spin-state assessment. The postshuttling states are measured on various projections and the results are fitted by sinusoidal functions to determine the amplitude and the phase. The different-colored curves correspond to the projections measured after 20 (blue), 52 (cyan), and 84 (green) shuttling events, respectively.
• Figure S1: The charge-stability diagram (CSD) and pulsed electron spin-orbital spectroscopy (PESOS) Map.a, The original CSD with background noise. The dark red vertical lines correspond to the $\mathrm{Q_1 \text{-} Q_2}$ charge transitions. The tilted blue lines correspond to the $\mathrm{Q_2 \text{-} Q_3}$ charge transitions. The cotunnling event between (110) and (011) states are also shown in the plot. The two tilted blurred red transition in the (110) and (020) states correspond to the inter-dot orbital transition in $\mathrm{Q_2}$ because of the weak voltage confinement. By carefully tuning the voltage, we did not cross these transitions in our experiments. b, A false-color CSD. The regime of each charge state is marked by one color and the transitions between them are indicated by the red lines. The transitions of $\mathrm{Q_1 \text{-} Q_3}$ cotunneling are not seen in our experiments (a) and the lines are only guess. c, The PESOS map near the (101)-(011) transition under an 0.17-T external field. The Larmor frequencies difference between the spins in $\mathrm{Q_1}$ and $\mathrm{Q_3}$ is too small under this external magnetic field strength, and hence their resonance frequencies almost overlap with each other at 4.72G in the plot.