Scaling of Dynamical Decoupling for Spin Qubits

We investigate the scaling of coherence time $T_2$ with the number of $\pi$ pulses $n_{\pi }$ in a singlet-triplet spin qubit using Carr-Purcell-Meiboom-Gill (CPMG) and concatenated dynamical decoupling (CDD) pulse sequences. For an even numbers of CPMG pulses, we find a power law $T_2\propto (n_{\pi }{)}^{{\gamma }_e}$, with ${\gamma }_e=0.72\pm 0.01$, essentially independent of the envelope function used to extract $T_2$. From this surprisingly robust value, a power-law model of the noise spectrum of the environment, $S(\omega )\sim {\omega }^{-\beta }$, yields $\beta ={\gamma }_e/(1-{\gamma }_e)=2.6\pm 0.1$. Model values for $T_2\left(n_{\pi }\right)$ using $\beta =2.6$ for CPMG with both even and odd $n_{\pi }$ up to 32 and CDD orders 3 through 6 compare very well with the experiment.

Figure 1

(a) Micrograph of a lithographically identical device with dot locations depicted. The gate voltages $V_{R(L)}$ set the charge occupancy of the right (left) dot as well as the detuning of the qubit. An rf-sensor quantum dot is indicated on the right. (b) Double-dot charge state mapped onto dc conductance change, $\Delta g$, with lettered pulse sequence gate voltages. The detuning axis is orthogonal to the (0,2)–(1,1) charge degeneracy through points $E$, $S$, and $M$.

Figure 2

(a) Energy level diagram along the detuning axis $ϵ$. The (0,2) singlet was prepared at ${ϵ}_P$, followed by separation to ${ϵ}_S$. $\pi$ pulses are performed at ${ϵ}_E$, allowing for subsequent rephasing at ${ϵ}_S$. There is a single-shot readout at ${ϵ}_M$ using a proximal sensor dot. The exchange energy $J_E$ that drives the $\pi$ pulses at ${ϵ}_E$ is indicated with a dashed line. (b)–(d) Schematics of detunings during CPMG and CDD pulse sequences with detuning points on the vertical axis.

Figure 3

(a) Experimental singlet return probabilities as a function of time for CPMG with $n_{\pi }=2,4,8$. Fits to $P_S({\tau }_{\mathrm{D}})=0.5+V/2\exp [-({\tau }_{\mathrm{D}}/T_2{)}^{\alpha }]$, with $\alpha$ constrained to 2 (solid curves), 3 (dotted curves), and 4 (dashed curves) [34]. It is difficult to determine $\alpha$ from these fits. (b) Extracted $T_2$ for even-$n_{\pi }$ CPMG sequences for $\alpha$ constrained to 1, 2, 3, 4, and 5. The circle size is proportional to the ${\chi }^2$ goodness of fit of $P_S({\tau }_{\mathrm{D}})$ in (a). A power-law fit to the form $\ln [T_2^{\mathrm{even}}(\alpha )]=\ln (T_2^0)+{\gamma }_e\ln (n_{\pi })$, shown for $\alpha =3$ (dashed line), gives ${\gamma }_e=0.72$. The fit value ${\gamma }_e$ depends only weakly on $\alpha$ in the range 2–5 (inset). The weighted average over $\alpha =2–5$ yields ${\gamma }_e=0.72\pm 0.01$.

Figure 4

$T_2$ for all measured $n_{\pi }$ for CPMG, extracted using $\alpha =2$, 3, and 4 (circles). The circle size for each $\alpha$ is proportional to the ${\chi }^2$ goodness of fit. Theory (black solid curve) for the integration of Eq. (1) with the CPMG filter functions and $\beta ={\gamma }_e/(1-{\gamma }_e)=2.6$. Note that Eq. (2) captures the even/odd effect quantitatively for small $n_{\pi }$. The black dashed line is the power-law fit to the even $n_{\pi }$ points.

Figure 5

(a) Experimental singlet return probabilities for orders 1 through 6 CDD (symbols), along with fits to $P_S({\tau }_{\mathrm{D}})=0.5+V/2\exp [-({\tau }_{\mathrm{D}}/T_2{)}^{\alpha }]$ using $\beta ={\gamma }_e/(1-{\gamma }_e)=2.6$ (solid curves). (b) Extracted $T_2$ for using different values of $\alpha$ (the circle size is proportional to the ${\chi }^2$ measure of goodness of fit). The theory (solid curve) is based on $\beta ={\gamma }_e/(1-{\gamma }_e)=2.6$ and the integration of Eq. (1) with the appropriate CDD filter function [12].