Scaling of Dynamical Decoupling for Spin Qubits
We investigate the scaling of coherence time ${T}_{2}$ with the number of $\ensuremath{\pi}$ pulses ${n}_{\ensuremath{\pi}}$ 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 ${T}_{2}\ensuremath{\propto}({n}_{\ensuremath{\pi}}{)}^{{\ensuremath{\gamma}}_{e}}$, with ${\ensuremath{\gamma}}_{e}=0.72\ifmmode\pm\else\textpm\fi{}0.01$, 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, $S(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}\ensuremath{\beta}}$, yields $\ensuremath{\beta}={\ensuremath{\gamma}}_{e}/(1\ensuremath{-}{\ensuremath{\gamma}}_{e})=2.6\ifmmode\pm\else\textpm\fi{}0.1$. Model values for ${T}_{2}\left({n}_{\ensuremath{\pi}}\right)$ using $\ensuremath{\beta}=2.6$ for CPMG with both even and odd ${n}_{\ensuremath{\pi}}$ up to 32 and CDD orders 3 through 6 compare very well with the experiment.