Dynamical control of electron spin coherence in a quantum dot: A theoretical study

We investigate the performance of dynamical decoupling methods at suppressing electron spin decoherence from a low-temperature nuclear spin reservoir in a quantum dot. The controlled dynamics is studied through exact numerical simulation, with emphasis on realistic pulse delays and the long-time limit. Our results show that optimal performance for this system is attained by a periodic protocol exploiting concatenated design, with control rates substantially slower than expected from the upper spectral cutoff of the bath. For a known initial electron spin state, coherence can saturate at long times, signaling the creation of a stable “spin-locked” decoherence-free subspace. Analytical insight into saturation is obtained for a simple echo protocol, in good agreement with numerical results.

Figure 1

(Color online) Minimum fidelity vs time in the logical frame with $\tau =0.1$. Hamiltonian parameters are $H_0=0$, ${\Gamma }_0=0$, and $N=15$. For deterministic DD, data points are acquired at the completion of each cycle, while for NRD and FID this is done after every $\tau$ and for SRPD after every $8\tau$. Random protocols are averaged over ${10}^2$ control realizations.

Figure 2

(Color online) (a) ${\mathrm{PCDD}}_2$ (solid lines) and ${\mathrm{PCDD}}_4$ (crosses) for $\tau =0.3,0.4,0.5$, top to bottom. (b) and (c) $T_{90\%}$ vs $\tau$ for ${\mathrm{PCDD}}_2$; different bath sizes (b) and intrabath interactions (c). $H_0=0$ in all panels, $N=15$ in (a) and (c).

Figure 3

(Color online) Fidelity saturation for CPMG, PDD, SDD, and ${\mathrm{PCDD}}_2$ starting from an initial state along the half-cycle direction. Hamiltonian parameters as in Fig. 1. The inset shows how the asymptotic value $F_{\mathit{sat}}$ is reached. In the main panel, a number of pulses sufficient to reach saturation and close the cycle of each protocol is chosen, $n_p\sim 50$.

Figure 4

Fidelity saturation vs pulse delay for CPMG from a known initial state. $H$ as in Fig. 1. Circles, QSA results; solid line, classical random field model; dashed line with plus signs, exact numerical simulations.