Dephasing due to Nuclear Spins in Large-Amplitude Electric Dipole Spin Resonance

We analyze effects of the hyperfine interaction on electric dipole spin resonance when the amplitude of the quantum-dot motion becomes comparable or larger than the quantum dot’s size. Away from the well-known small-drive regime, the important role played by transverse nuclear fluctuations leads to a Gaussian decay with characteristic dependence on drive strength and detuning. A characterization of spin-flip gate fidelity, in the presence of such additional drive-dependent dephasing, shows that vanishingly small errors can still be achieved at sufficiently large amplitudes. Based on our theory, we analyze recent electric dipole spin resonance experiments relying on spin-orbit interactions or the slanting field of a micromagnet. We find that such experiments are already in a regime with significant effects of transverse nuclear fluctuations and the form of decay of the Rabi oscillations can be reproduced well by our theory.

Figure 1

(a) Plot of Eqs. (5) and (6), which characterize the longitudinal and transverse nuclear fluctuations (upper and lower curves, respectively). (b) Numerical result for the Rabi oscillations $\overline{P_{\downarrow }(t)}$, averaged over nuclear fluctuations (the thick blue curves). We used $\eta =0.05$, $\mathrm{\Delta }=0$, and $\delta R/\delta x=0.1,0.2,\dots ,0.6$ (from bottom to top). For clarity, the curves are shifted vertically. The red dashed curves are the asymptotic power-law decay of Ref. [11], valid when $\delta R/\delta x\lesssim \sqrt{2\eta }=0.3$. The thin gray curves are from Eq. (9). (c) Plot of Eq. (12). Dots are decay times from Ref. [10], rescaled using $T_2^{*}\simeq 9\mathrm{ns}$ and $\eta \simeq 0.09$. (d) Error in realizing a $\pi$ rotation, plotted as a function of $\delta R/\delta x$ at $\eta =0.01,0.03,0.05$ (from bottom to top). Numerical results (the thick solid lines) are compared to the asymptotic result (the thick dashed lines) and the upper bound (the thin red lines) given in Eq. (13) [19]. The thin red lines also practically coincide with the ESR result, as seen by taking $\delta h_z=1$ and $\delta h_{xy}=0$ in Eq. (13).

Figure 2

Rabi oscillations at finite detuning. (a) and (b) represent $\delta R/\delta x=0.8$ and 0.1, respectively. In both panels, $\eta =0.05$. (c) shows the line cuts of (a) at $\mathrm{\Delta }/\sigma =0,2.5,5,\dots ,12.5$ (from bottom to top). For clarity, the curves are shifted vertically. The numerical results are virtually indistinguishable from Eq. (9), plotted as gray dashed curves. In (d) we plot Eq. (10) as a function of detuning for several values of $\delta R/\delta x=0.4,0.8,1.2,1.6$ (from bottom to top).

Figure 3

(a) Fit of our theory to Rabi oscillations of Ref. [9]. For each curve, the fit parameters are $\delta R/\delta x$, a conversion factor to current, and $b=4\pi \hslash f_R$, while $\sigma =0.22\pm 0.03\mu \mathrm{eV}$ is from the experiment [9]. (b) $\delta R/\delta x$ and $f_R$ obtained from the fits in (a). The error bars are from the uncertainty on $\sigma$. The dashed line is a fit to $\delta R/\delta x=C{f}_R$.