Rapid Single-Shot Measurement of a Singlet-Triplet Qubit

We report repeated single-shot measurements of the two-electron spin state in a GaAs double quantum dot. The readout allows measurement with a fidelity above 90% with a $\sim 7\mu \mathrm{s}$ cycle time. Hyperfine-induced precession between singlet and triplet states of the two-electron system are directly observed, as nuclear Overhauser fields are quasistatic on the time scale of the measurement cycle. Repeated measurements on millisecond to second time scales reveal the evolution of the nuclear environment.

Figure 1

(a) Charge occupancy (left, right) of the double dot, detected using rf-QPC reflectometer voltage $v_{\mathrm{rf}}$ in continuous-sensing mode [7] rather than single-shot readout. The triangle in (0,2) indicates where charge state (1,1) is metastable. Markers indicate gate voltages used in single-shot mode. Preparation of (0,2) singlet ($P$); separation for $S-T_0$ mixing ($S$) and $S-T_{+}$ mixing ($I$); measurement ($M$); operating point with 0 V pulse amplitude ($D$). (b) Two-electron energy levels as a function of detuning $ϵ$ from (0,2)–(1,1) degeneracy. (c) Micrograph of device identical to measured device, indicating Ohmic contacts (boxes), fast gate lines, reflectometry circuit, grounded contacts, and field direction. (d) Pulse sequence of $ϵ$, controlled by $V_R$ and $V_L$, cycling through the points $P$, $S$, $M$. Sensor signal $v_{\mathrm{rf}}$ indicates triplet (green, marked $T$) or singlet (blue, marked $S$) outcome for ${\tau }_S=100\mathrm{ns}$. Integration subinterval time ${\tau }_M$ chosen in postprocessing.

Figure 2

(a) 7000 consecutive single-shot measurements of $V_{\mathrm{rf}}$ using the pulse sequence in Fig. 1d with integration subinterval ${\tau }_M=7\mu \mathrm{s}$ and separation time, ${\tau }_S$, incremented every 200 cycles [26]. (b) Histogram of the outcomes in (a), along with model (solid curve) [18]. (c) Instantaneous rf-QPC output $v_{\mathrm{rf}}(\tau )$ at time $\tau$ following pulsing to $M$, averaged over all cycles, along with a fit to the model, giving $T_1=34\mu \mathrm{s}$ [18]. (d) Histograms $N(V_{\mathrm{rf}})$ (gray scale) for varying ${\tau }_M$. (e) Horizontal cuts through (d) along with model, with values of the parameters $V_{\mathrm{rf}}^S$, $V_{\mathrm{rf}}^T$ from a fit to the ${\tau }_M=15\mu \mathrm{s}$ data [18]. (f) Fidelity of singlet, $F_S$, and triplet, $F_T$, and visibility $V=F_S+F_T-1$ as a function of threshold, $V_{\mathrm{T}}$, for data in (b). (g) Maximum visibility, $V^{\max }$, and optimal threshold, $V_{\mathrm{T}}$, as a function of measurement time, ${\tau }_M$.

Figure 3

(a) 6000 consecutive single-shot $S-T_0$ measurements, $V_{\mathrm{rf}}$, with separation times ${\tau }_S$ stepped by $\sim 17\mathrm{ns}$ every 200 cycles, as a function of overall measurement time, $t$ (bottom axis). Threshold $V_{\mathrm{T}}$ separates outcomes identified as singlet ($V_{\mathrm{rf}}<V_{\mathrm{T}}$) or triplet ($V_{\mathrm{rf}}>V_{\mathrm{T}}$). Oscillations due to Overhauser fields are evident, with slightly evolving period. (b) Single-shot outcomes (gray markers) and triplet probabilities, $P_T$, (black circles) over ${\tau }_S$, for three nominally identical runs taken 10 min apart. (c) Rapid acquisition of 108 $P_T$ traces at times $t$. Probabilities $P_T$ are determined from 400 measurements per ${\tau }_S$.

Figure 4

(a) Triplet probability $P_T$ as a function of separation time ${\tau }_S$ based on 1600 single-shot measurements per separation time. Fit (blue or gray curve) gives $T_2^{*}=27\mathrm{ns}$, $P_T(0)=1-F_S=0.08$, $V=0.28$ (see text). (b) $P_T$ as a function of separation time ${\tau }_I$ to the $S-T_{+}$ anticrossing at point $I$. Probabilities (gray circles) based on 400 single-shot binary measurements. At each ${\tau }_I$, 50 values of $P_T$ are taken 15 s apart and averaged (open black squares). Theory curve, Eq. 6, yields $B^{\mathrm{nuc}}=0.7\mathrm{mT}$ ($T_2^{*}=37\mathrm{ns}$), with relatively poor agreement between experiment and theory (see text).