Universal Phase Shift and Nonexponential Decay of Driven Single-Spin Oscillations

We study, both theoretically and experimentally, driven Rabi oscillations of a single electron spin coupled to a nuclear-spin bath. Because of the long correlation time of the bath, two unusual features are observed in the oscillations. The decay follows a power law, and the oscillations are shifted in phase by a universal value of $\sim \pi /4$. These properties are well understood from a theoretical expression that we derive here in the static limit for the nuclear bath. This improved understanding of the coupled electron-nuclear system is important for future experiments using the electron spin as a qubit.

Figure 1

Rabi oscillations for four different driving fields $B_{\mathrm{ac}}$ [$B_z=55\mathrm{mT}$, $g=0.355$ and $\sigma =g{\mu }_B(1.4\mathrm{mT})$]. The gray circles represent the experimentally measured dot current (averaged over 15 s for each value of $t$), which reflects the probability to find an odd spin-parity state after the rf burst that generates $B_{\mathrm{ac}}$. The dotted, solid, and dashed lines represent the best fit to the data of an exponentially decaying cosine function and the derived analytical expressions for $P_{\mathrm{odd}}(t)$ and $P_{\mathrm{odd}}^{(1)}(t)$ [Eqs. (4, 7)], respectively. For clarity, the dashed line is shown only for the top two panels. The fit was carried out for the range 60 to 900 ns and the displayed values for $B_{\mathrm{ac}}$ were obtained from the fit with $P_{\mathrm{odd}}(t)$ [Eq. (4)]. We fit the data with an exponentially decaying cosine with a tunable phase shift that is zero at $t=0$: $a_1{e}^{-t/a_2}[\cos (\varphi )-\cos (2\pi t/a_3+\varphi )]+a_4(1-e^{-t/a_2})$. The last term was added such that the saturation value is a fit parameter as well. We note that the fit is best for $\varphi =\pi /4$, as discussed in the text.

Figure 2

(a) Dot current after an rf burst of 950 ns as a function of $B_{\mathrm{ac}}$, approximately representing the steady-state value. The solid curve is the best fit with $a_1(\frac{1}{2}-2C^2)$: the steady-state expression of Eq. (4) with $a_1$ and $\sigma$ as fit parameters. We find, for the 95% confidence interval, $\sigma =g{\mu }_B(1.0–1.7\mathrm{mT})$. (b) Decay power obtained from the best fit of the data (partially shown in Fig. 1) with the expression $a_1+a_2\cos (2\pi t/a_3+\pi /4)/t^d$, where $a_{1,2,3}$ and $d$ are fit parameters.

Figure 3

(a) The dot current (represented in color scale) is displayed over a wide range of $B_{\mathrm{ac}}$ (the sweep axis) and burst durations. The green and blue lines correspond, respectively, to the maxima of a cosine with and without a phase shift of $\pi /4$. The current-to-field conversion factor $K$ is fitted for both cases separately ($K=0.568\mathrm{mT}/\mathrm{mA}$ and $K=0.60\mathrm{mT}/\mathrm{mA}$ for, respectively, with and without phase shift; the fit range is $t=60–500\mathrm{ns}$ and $I_s=3.6–6.3\mathrm{mA}$). (b) Phase shift for a wide range of $B_{\mathrm{ac}}$, displayed as a function of stripline current $I_s$. Values obtained from a fit of each trace of the data in (a) (varying burst time, constant $B_{\mathrm{ac}}$) to a damped cosine $a_1-a_2\cos (\frac{1}{2}K{I}_{s}g{\mu }_{B}t+a_3\pi )/\sqrt{t}$, where $a_{1,2,3}$ are fit parameters and $K=0.568\mathrm{mT}/\mathrm{mA}$. $I_s$ is a known value in the experiment, extracted from the applied rf power. The gray dashed lines represent the 95% confidence interval.