Interlaced Dynamical Decoupling and Coherent Operation of a Singlet-Triplet Qubit

We experimentally demonstrate coherence recovery of singlet-triplet superpositions by interlacing qubit rotations between Carr-Purcell (CP) echo sequences. We then compare the performance of Hahn, CP, concatenated dynamical decoupling (CDD), and Uhrig dynamical decoupling for singlet recovery. In the present case, where gate noise and drift combined with spatially varying hyperfine coupling contribute significantly to dephasing, and pulses have limited bandwidth, CP and CDD yield comparable results, with $T_2\sim 80\mu \mathrm{s}$.

Figure 1

(a) Micrograph of lithographically identical device with contacts, dot locations, and GaAs crystal axes indicated. Gate voltages, $V_L$ and $V_R$, set the energy of left and right dot. An rf-coupled sensor dot measures the double dot charge state. (b) dc conductance change, $\Delta g$, with double dot charge state. Markers indicate gate voltages during experimental pulse sequences. Detuning $ϵ$ is controlled by $V_L$ and $V_R$ along the diagonal through points $S$ and $M$. (c) Energy level diagram as function of detuning $ϵ$, indicating locations of ${ϵ}_P$ where the (0,2) singlet is prepared, ${ϵ}_S$ of separated electrons for dephasing and $x$ rotations, ${ϵ}_E$ of exchange pulse and $\pi$ pulses, ${ϵ}_R$ of mapping ramps, and ${ϵ}_M$ of the measurement point. Exchange energy, $J_E$ used for $\pi$ pulses is indicated. (d) Bloch sphere of the singlet-triplet qubit, with mechanisms of rotation indicated, exchange energy $J$, and Zeeman field difference along the applied field direction $\Delta B_z$, drive rotations around the qubit $z$ axis and $x$ axis, respectively.

Figure 2

(a) Pulse sequence for $x$ rotations, cycling through preparation ($P$), separation ($S$), exchange-echo ($E$), and measurement ($M$) points in gate-voltage space. Evolution at $S$ for time ${\tau }_S$ produces $x$ rotations due to Overhauser gradient. (b) Singlet return probability $P_S$ as function of ${\tau }_S$ for $N_{\pi }=4$ (green diamonds) and $N_{\pi }=0$ (black circles), with ${\tau }_D=4\mu \mathrm{s}$, along with cosine fit (curve) to data, postselected to have a specific period (see text). Without echo pulses ($N_{\pi }=0$), no oscillations are seen. Error bar reflects statistics of 100 single-shot measurements per point. (c) Identical to (b) with ${\tau }_D=16\mu \mathrm{s}$, showing reduced visibility. (d) Pulse sequence for $z$ rotations: after an initial CP sequence, singlet is mapped to $\mid \uparrow \downarrow ⟩$ by pulsing to $R$ and ramping to $S$. Pulsing to $E$ for ${\tau }_E$ induces rotation around $z$. Pulsing to $S$ then ramping to $R$ then pulsing back to $S$ maps $\mid \uparrow \downarrow ⟩$ back into the singlet. After a second CP series, pulsing to $M$ yields single-shot measurement. (e) Singlet probability $P_S$ as function of ${\tau }_S$ for $N_{\pi }=8$ (blue diamonds) and $N_{\pi }=0$ (black circles), with ${\tau }_D=16\mu \mathrm{s}$, based on an average of $600\times 100$ single-shot measurements, along with cosine fit with Gaussian envelope, see text. Error bars from statistics are smaller than markers. (f) Identical to (e) with ${\tau }_D=64\mu \mathrm{s}$.

Figure 3

(a) Singlet recovery amplitude $P_S$ as a function of total dephasing time ${\tau }_D$ for Hahn, CP, CDD, and UDD sequences (see text), with fits of Eq. (1) with $\alpha =2$, yielding $T_2^{\mathrm{HE}}\sim 6\mu \mathrm{s}$, $T_2^{\mathrm{CDD}}\sim 80\mu \mathrm{s}$, $T_2^{\mathrm{CP}}\sim 80\mu \mathrm{s}$, $T_2^{\mathrm{UDD}}\sim 30\mu \mathrm{s}$ [27]. (b) Replotting data in (a) as $-\ln [2(P_S-1/2)/V]$ allows a fit giving the exponent $\alpha$ in Eq. (1). On a log-log plot, Eq. (1) and data appears linear with slope $\alpha$ [27]. Pulse sequences for (c) 5th order CDD and (d) 22nd order UDD.