Measurement Efficiency and $n$-Shot Readout of Spin Qubits

We consider electron spin qubits in quantum dots and define a measurement efficiency $e$ to characterize reliable measurements via $n$-shot readouts. We propose various implementations based on a double dot and a quantum point contact (QPC) and show that the associated efficiencies $e$ vary between 50% and 100%, allowing single-shot readout in the latter case. We model the readout microscopically and derive its time dynamics in terms of a generalized master equation, calculate the QPC current, and show that it allows spin readout under realistic conditions.

Figure 1

Electron spin readout setup consisting of a double dot. The right “reference” dot is coupled capacitively to a QPC shown on the right. (a) Readout using different Zeeman splittings. For $\uparrow$, the electron tunnels between the two dots. For $\downarrow$, tunneling is suppressed by the detuning and the stationary state has a large contribution of the left dot since it has lower energy. This allows single-shot readout, i.e., $e=100\%$. (b) Spin-dependent tunneling amplitudes, $t_d^{\downarrow }<t_d^{\uparrow }$, also enable efficient readout. (c) Readout with the singlet state. Tunneling of spin $\uparrow$ to the reference dot is blocked due to the Pauli principle. (d) Schematic current vs time during a single measurement. Here, ${\tau }_{dd}$ is the time scale for tunneling and we assume ${\Gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}}>t_d$, i.e., that the tunneling events can be resolved in the current.