Competing interactions in semiconductor quantum dots

We introduce an integrability-based method enabling the study of semiconductor quantum dot models incorporating both the full hyperfine interaction as well as a mean-field treatment of dipole-dipole interactions in the nuclear spin bath. By performing free-induction decay and spin-echo simulations we characterize the combined effect of both types of interactions on the decoherence of the electron spin, for external fields ranging from low to high values. We show that for spin-echo simulations the hyperfine interaction is the dominant source of decoherence at short times for low fields, and competes with the dipole-dipole interactions at longer times. On the contrary, at high fields the main source of decay is due to the dipole-dipole interactions. In the latter regime an asymmetry in the echo is observed. Furthermore, the nondecaying fraction previously observed for zero-field free-induction decay simulations in quantum dots with only hyperfine interactions, is destroyed for longer times by the mean-field treatment of the dipolar interactions.

Figure 1

A schematic picture of a semiconductor quantum dot with an electron central spin confined in a harmonic trap. The interactions between the closest nuclear spins and the central spin are described by the full hyperfine interaction including flip-flop terms. For the nuclear spins with hyperfine couplings smaller than or equal to $1/e$, the dipolar interactions among themselves is the dominant contribution to the dynamics. The dynamics of these spins is treated with a mean-field approximation with an effective time-dependent magnetic field in the $z$ direction.

Figure 2

The expectation value of $S_0^x$ is plotted for different pulse times $\tau$ for $N=14$ nuclear spins with couplings between 1 and $1/e$. In (a)-(c) the closest nuclear spins with hyperfine couplings in the range $\left[1,1/e\right)$ are initially in a Néel-like pure state. The magnetic fields are given by $B_z=0.5B_{\mathrm{fluct}}$ in (a), $B_z=1.5B_{\mathrm{fluct}}$ in (b), and $B_z=2B_{\mathrm{fluct}}$ in (c). On the contrary, in (d)-(f) the closest nuclear spins are initially in a random initial state. The same magnetic fields as in (a)-(c) are considered. All curves are offset from one another by $1/2$ for clarity and are averaged over 100 OU processes to mimic the dynamics of the mean-field bath. Furthermore, in each run in (d)-(f) a different random initial state was generated. The time step used for the Suzuki-Trotter decomposition is $dt=0.005$, the decay rate of the fluctuating field is fixed to $R=0.01$ and the dispersion $b$ is given by Eq. (5) and has an approximate value of $b\approx 1.008$. The arrows indicate the expected arrival time of the echo at $t=2\tau$. For magnetic fields $B_z=1.5B_{\mathrm{fluct}}$ and $B_z=2B_{\mathrm{fluct}}$ the echo signal is pushed down in an asymmetric fashion. This leads to a shift in the echo for (b) and (c), but no visible shift for (e) and (f). For $B_z=0.5B_{\mathrm{fluct}}$ there is no clear distinction between the plots for the two initial states and no shift present.

Figure 3

For $N=14$ and $B_z=2B_{\mathrm{fluct}}$, the expectation value $\langle S_0^x\left(2\tau \right)\rangle$ is plotted for the Néel pure initial state and an average over initial random states. The red and green curves show the results for the integrable model with only hyperfine interactions, corresponding to a static mean-field bath with zero average polarization. The purple and blue curves show the echo envelope for a slow mean-field bath with $R=0.01$, averaged over 40 OU processes for each point (every point thus requires 40 time evolution computations of which only the value $\langle S_0^x\left(2\tau \right)\rangle$ is saved).

Figure 4

Echo envelopes for $N=14$ nuclear spins treated with the full hyperfine coupling, an external field of $B_z=2B_{\mathrm{fluct}}$ and varying mean-field bath correlation decay rates $R$. All curves corresponding to a non static mean-field bath have been computed by averaging over 50 OU processes. The mean-field bath fluctuations are most destructive for $R=b$, and show a slower decay for $R=0.1b$ and $R=10b$.

Figure 5

The free induction decay of $\langle S_0^x\rangle$ for different mean-field bath speeds $R$. The results have been averaged over 100 OU processes, with a newly generated initial random state for each run. For comparison the integrable case is shown as well. Similar to the case of spin echo simulations, the mean-field bath with $R=b$ shows the fastest decay of the correlation function.