Spin qubit gates via phonon buses in electron nanowires

Abstract

Scalable architectures for quantum computing using semiconductor quantum dots require interactions between qubits beyond adjacent quantum dots. Here, we propose using nanowires of electrons to mediate the interaction between two quantum dots. Virtual phonons in the linear chain of electrons can mediate an interaction that gives rise to effective spin-spin coupling of the electrons in distant quantum dots. We find coupling strengths of more than 30 MHz for experimentally realisable parameters in GaAs quantum dots.

Figures (5)

• Figure 1: Illustrations of the electron bus. (a) Schematic of the electrons, labelling the end quantum dots as $1$ and $N$, corresponding to the number of electrons in the system. The electric field $E(t)$ in the $z$ direction is shown influencing quantum dots $1$ and $N$. The electric field is time-dependent in that it can be switched on, $E(t)=E_0$, and off, $E(t) = 0$, to switch on and off the Rashba effect. The axes are shown with $z$ as the out-of-plane axis. Coulomb repulsion between the electrons is depicted as springs and gives rise to the phonon modes. (b) Depicts the confining potential for the electrons and that they are in their ground state. The barrier height $V_B$ is determined by the confinement of the quantum dots and the nanowire. $d$ is the width of a quantum dot, which is typically around 100 nm in GaAs. (c) An example of a 2-dimensional planar architecture is given with electron nanowires (black rectangle boxes), and quantum dots, (blue and red). The red quantum dots can be used for measurement of the blue quantum dots.
• Figure 2: (a) Shows the positions of 8 electrons that minimise the confining potential of the quantum dots and nanowire (shown rescaled in red) for $\omega_{\textrm{w},y} = \omega_\textrm{GS} / 10$ with $\omega_\textrm{GS} = 0.6\times10^{11}~\textrm{rad/s}$, with nanowire trap at the origin and dot centres at $\bm{r}_\textrm{a} = (0, -1000,0)~\textrm{nm}$ and $\bm{r}_\textrm{b} = (0, 1000,0)~\textrm{nm}$. (b) Shows the $y$-$z$ projection of the electron positions and the rescaled confining potential. (c) Plots the resulting phonons, with the axial phonon modes (underlined in blue) and the transverse phonon modes (underlined in green). The red line is at the frequency of the Zeeman splitting, $\omega_0 = 1.765 \times10^{11}~\textrm{rad/s}$, induced by a magnetic field along $z$ with strength $5.151~\textrm{T}$, with $g=0.39$ and $m^*=0.067 m_\textrm{e}$. (d) Gives the corresponding phonon modes of the 8 electrons for the first 6 of the axial modes, mode 2 and 3 show the strongest and approximately equal contribution of the first and last electrons -- we chose mode 2 because it has a lower frequency.
• Figure 3: Ratio of maximum coupling to detuning by changing the axial trapping frequency $\omega_y$ and distance between the end quantum dots $d = |\bm{r}_b - \bm{r}_a|$ for $N=10$ electrons. The colour indicates the maximum coupling over detuning ratio value, which for phonon mode 2 is $g_{y,12}/\Delta_2$. Where the detuning $\Delta_2 = \omega_0 - \omega_{y,2}$, depends on the trapping frequency via the mode frequency $\omega_{y,2}$, and $\omega_0$ is kept constant. The green indicates a ratio less than 0.1, which defines the dispersive regime which gives our effective XY model coupling $J_{1N}$, see App. \ref{['sec:effective_hamiltonian']}. The grey region is where the dispersive regime does not apply. The blue circle marks the maximum coupling $J_{1N}$ within the dispersive regime, where $g_{y,12}/\Delta_2 < 0.1$. The blue circle is for trapping frequency $\omega_y = 54.68~\mathrm{GHz}$ and distance between quantum dots $d = 2.066~\mathrm{\mu m}$, giving a maximum coupling strength $J_{1N} = 50.80~\mathrm{MHz}$.
• Figure 4: Maximum coupling strength $J_{1N}$ is plotted against the number of electrons $N$. At each point the trapping frequency $\omega_y$ and distance between end quantum dots $d = |\bm{r}_b - \bm{r}_a|$ is given. The maximum coupling value for $N=10$ is found using the parameter search in Fig. \ref{['fig:parameter_sweep_n_10']}. In App. \ref{['sec:parameter_optimisation']}, we show the parameters searches for the other values of $N$.
• Figure 5: Ratio of maximum coupling to detuning by changing trapping frequency $\omega_y$ and distance between the end quantum dots $d = |\bm{r}_b - \bm{r}_a|$. The colour indicates the maximum coupling over detuning ratio value, which for phonon mode 2 is $g_{y,12}/\Delta_2$. Where the detuning $\Delta_2 = \omega_0 - \omega_{y,2}$, depends on the trapping frequency via the mode frequency $\omega_{y,2}$, and $\omega_0$ is kept constant. The green indicates a ratio less than 0.1, which defines the dispersive regime that gives the effective XY model coupling $J_{1N}$, see App. \ref{['sec:effective_hamiltonian']}. The grey region is where the dispersive regime does not apply. The blue circle marks the maximum coupling $J_{1N}$ within the dispersive regime, where $g_{y,12}/\Delta_2 < 0.1$. The result of the blue circle gives the maximum $J_{1N}$ used in Fig. \ref{['fig:max_J_vs_N_electrons']} in the main text.