Hole-spin qubits in germanium beyond the single-particle regime

Abstract

The intense simulation efforts on hole-spin qubits in germanium have so far focused primarily on singly occupied quantum dots. Here, we theoretically investigate three-hole qubits in germanium and demonstrate that their performance can rival that of single-hole qubits in both strained and unstrained systems. In particular, we find that -- in the widely used quasi-circular geometry -- a three-hole qubit encoding can yield enhancements of the Rabi frequencies of up to two orders of magnitude and a large advantage also in terms of quality factors.

Figures (5)

• Figure 1: (a) Pictorial representation of a single-hole state (left) and a three-hole state (right) in a QD elongated along the $x$ direction. (b) Schematic view of a MOS device, where the Ge channel has a width $L$ along the $z$ direction. The bias applied to the metal top gate determines the in-plane confining potential $V_\parallel$ and the linear (in $z$) contribution in the out-of-plane confining potential $V_\perp$.
• Figure 2: In-plane particle distribution $N_z(x,y)$ for the ground state $|\Phi_0\rangle$ of the three-hole system, obtained by integrating the particle density $n({\bf r})$ along the $z$ direction. The four panels display: (a) the function $N_z(x,y)$; (b,c) the heavy-hole contributions; (d) the sum of the two light-hole contributions. All the plotted functions are normalized to 1, the isolines correspond to values that are multiples of 0.1. The values of the parameters are: $\hbar\omega_x=5\,$meV, $\hbar\omega_y=6\,$meV, $B=0.05\,$T and $\theta=0$ (parallel field orientation).
• Figure 3: Rabi frequencies $f^{(n)}_{{\rm R}, \alpha}$ of the $n$-hole qubit as functions of $\hbar \omega_x$, with $n=1$ (red curves) and $n=3$ (blue symbols), when the oscillating electric field (of amplitude $\left| \delta \boldsymbol{E}_{\rm R} \right| = 1$ mV$/$nm) is oriented along the $x$ (a, c) or $y$ direction (b, d). The magnetic field amplitude is $| \boldsymbol{B} | = 0.05$ T; its direction is given by $\theta = 0 \degree$ (a, b) or $\theta = 90 \degree$ (c, d), with $\phi = 45 \degree$.
• Figure 4: Dephasing time scale $\tau$ as a function of $\hbar \omega_x$ (with $\hbar \omega_y = 6\,$meV), for a magnetic field of intensity $B = 0.05\,$T oriented (a) parallel ($\theta = 0^\circ$) or (b) perpendicular ($\theta = 90^\circ$) to the $z$ axis. The charge-noise induced electric field is oriented along the $z$ direction, with $\left| \boldsymbol{E}_{\rm cn} \right| = 10^{-2}$ mV/nm.
• Figure 5: Quality factor $Q_x$ for the THQ as a function of $\hbar\omega_x$ (with $\hbar\omega_y=6\,$meV), for a magnetic field intensity $B=0.05\,$T oriented out fo plane (parallel to the $z$ axis). Yellow and purple symbols correspond to the strained and to the unstrained dots, respectively. Inset: $Q_x$vs$\hbar\omega_x$ for the SHQ, with the same units and the same color code as in the main plot.