Free-induction decay and envelope modulations in a narrowed nuclear spin bath

We evaluate free-induction decay for the transverse components of a localized electron spin coupled to a bath of nuclear spins via the Fermi-contact hyperfine interaction. Our perturbative treatment is valid for special (narrowed) bath initial conditions and when the Zeeman energy of the electron $b$ exceeds the total hyperfine coupling constant $A$: $b>A$. Using one unified and systematic method, we recover previous results reported at short and long times using different techniques. We find an unexpected modulation of the free-induction-decay envelope, which is present even for a purely isotropic hyperfine interaction without spin echoes and for a single nuclear species. We give subleading corrections to the decoherence rate, and show that, in general, the decoherence rate has a nonmonotonic dependence on electron Zeeman splitting, leading to a pronounced maximum. These results illustrate the limitations of methods that make use of leading-order effective Hamiltonians and re-exponentiation of short-time expansions for a strongly interacting system with non-Markovian (history-dependent) dynamics.

Figure 1

Contour used to evaluate the Bromwich inversion integral [in the rotating frame defined by Eq. (13)]. The dynamics of the coherence factor $x_t$ are determined by a single pole at $s_0=-1/T_2$ and three branch cuts (see main text). The pole is offset from the branch cuts by the Lamb shift $\Delta \omega$ and the excitation bandwidth (separation between branch points) is given by the size of the hyperfine coupling to a single nucleus at the center of the quantum dot, $A/N$.

Figure 2

(Color online) Comparison of the contribution from the pole at $s_0$ in Fig. 1 with exponentially decaying residue (blue dashed line) with the full fourth-order result, obtained numerically (red solid line). For comparison, the re-exponentiated short-time quadratic decay is also shown (black dotted line) with a parabolic decay having the same time scale (orange dash-dotted line). We take the initial condition $x_0=1$, assume an unpolarized nuclear spin system [with ${\omega }_n=b$, $c_{\pm }=\frac{2}{3}I\left(I+1\right)$, which follows from $⟪m^2⟫=I\left(I+1\right)/3$ if all Zeeman levels have equal population], and have chosen $I=3/2$ and $A/b=1/3$.

Figure 3

(Color online) Long-time decay. At long times, the exponential decay envelope is modulated by branch-cut contributions at a frequency given by the Lamb shift $\Delta \omega$. Parameters are as in Fig. 2.

Figure 4

(Color online) Comparison of the full numerical branch-cut integral $-{\sum }_i\text{Re}\left[{\beta }_i\left(t\right)\right]$ with the long-time asymptotic expression given by Eq. (38). Parameters are as in Fig. 2.

Figure 5

(Color online) Decoherence rates ${\Gamma }_i$ from Eq. (50) and total decoherence rate $1/T_2={\sum }_i{\Gamma }_i$ for an electron spin in a GaAs quantum dot containing $N={10}^5$ nuclei with $g$ factor $\left|g\right|=0.4$. The decoherence rate shows a nonmonotonic behavior, reaching a pronounced maximum. This is in contrast to the leading-order result (Ref. 25) and in contrast to the results from other higher-order expansions (Ref. 26). We have used hyperfine coupling constants $A^{{}^{69}\text{G}\text{a}}=74\mu e\text{V}$, $A^{{}^{71}\text{G}\text{a}}=96\mu e\text{V}$, and $A^{{}^{75}\text{A}\text{s}}=86\mu e\text{V}$ and relative abundances ${\nu }_{{}^{69}\text{G}\text{a}}=0.3$, ${\nu }_{{}^{71}\text{G}\text{a}}=0.2$, and ${\nu }_{{}^{75}\text{A}\text{s}}=0.5$, which have been estimated in Ref. 27 (see Table I of Ref. 28).

Figure 6

(Color online) Total decoherence rate $1/T_2={\sum }_i{\Gamma }_i$ (solid line) with ${\Gamma }_i$ from Eq. (50) for an electron spin in a ${\text{In}}_x{\text{Ga}}_{1-x}\text{As}$ quantum dot containing $N={10}^5$ nuclei with $g$ factor $\left|g\right|=0.5$ and In doping $x=0.3$. We show individual contributions from ${\Gamma }_{{}^{75}\text{A}\text{s}}$ (dotted) and ${\Gamma }_{\text{In}}$ (dashed). Contributions from the gallium isotopes are not visible on this scale. For this plot, we have taken hyperfine coupling constants for the gallium and arsenic isotopes as in Fig. 5 and have used $A^{{}^{113}\text{I}\text{n}}=A^{{}^{115}\text{I}\text{n}}=A^{\text{In}}=110\mu e\text{V}$ from Ref. 32 (see Table I of Ref. 28).

Figure 7

(Color online) Decoherence rate $1/T_2$ for an electron spin in a ${\text{In}}_x{\text{Ga}}_{1-x}\text{As}$ quantum dot containing $N={10}^5$ nuclei with $g$ factor $\left|g\right|=0.5$ and In doping of $x=0$, $x=0.3$, and $x=0.5$. Hyperfine coupling constants are as given in the caption of Fig. 6. The exponential decay rate shown here will be an accurate description of the full decay in the Markovian regime $T_2\gtrsim {\tau }_c$, where ${\tau }_c$ is the bath correlation time. This regime is reached without nuclear polarization $\left(p=0\right)$ whenever the perturbative self-energy expansion is valid (Ref. 25): $A\lesssim b$ (${\tau }_c\simeq N/A$ for $A\lesssim b$). At finite polarization, $1/T_2$ will be further reduced (e.g., $1/T_2\propto 1-p^2$ for a system with nuclear spin $I=1/2$), resulting in a dominant exponential decay even at lower electron spin splitting. Thus, at least for large nuclear polarization $p\to 1$, the qualitative behavior shown here will accurately describe the magnetic-field dependence of coherence decay. The behavior at $b\lesssim A$ cannot be accurately determined for $p=0$ without including higher-order corrections.

Figure 8

(Color online) Decay envelopes calculated from numerical evaluation of branch-cut integrals and pole contributions for a two-dimensional quantum dot with Gaussian envelope function (solid lines). Dashed lines show the contributions from the dominant pole at $s=s_0$. When the electron Zeeman splitting $b$ is below some critical value $b<b_c\sim A$, a second exponentially decaying pole appears, leading to a biexponential decay with strong envelope modulations (top, $b=2A$). When $b$ is decreased further, the dominant pole moves to the real axis (middle, $b=A$). At still smaller values of $b$, the subdominant exponential pole has a vanishing decay rate (bottom, $b=A/2$), leading to sustained oscillations. All plots correspond to an unpolarized narrowed nuclear bath with $I=3/2$. Insets illustrate the approximate relative positions of poles (circles) and branch points (crosses) in each case.