Long-time electron spin storage via dynamical suppression of hyperfine-induced decoherence in a quantum dot

The coherence time of an electron spin decohered by the nuclear spin environment in a quantum dot can be substantially increased by subjecting the electron to suitable dynamical decoupling sequences. We analyze the performance of high-level decoupling protocols by using a combination of analytical and exact numerical methods, and by paying special attention to the regimes of large interpulse delays and long-time dynamics, which are outside the reach of standard average Hamiltonian theory descriptions. We demonstrate that dynamical decoupling can remain efficient far beyond its formal domain of applicability, and find that a protocol exploiting concatenated design provides best performance for this system in the relevant parameter range. In situations where the initial electron state is known, protocols able to completely freeze decoherence at long times are constructed and characterized. The impact of system and control nonidealities is also assessed, including the effect of intrabath dipolar interaction, magnetic field bias and bath polarization, as well as systematic pulse imperfections. While small bias field and small bath polarization degrade the decoupling fidelity, enhanced performance and temporal modulation result from strong applied fields and high polarizations. Overall, we find that if the relative errors of the control pulse flip angles do not exceed 3%, decoupling protocols can still prolong the coherence time by up to 2 orders of magnitude.

Figure 1

(Color online) Best-case performance for single-cycle DD under the CPMG (plus signs), PDD (circles), SDD (upward pointing triangles), and ${\mathrm{PCDD}}_2$ (downward pointing triangles) protocol in the small $\tau$ region. Bath size $N=5$. Solid lines are linear fits, with fitting parameters given in Table .

Figure 2

Best-case performance of (a) CPMG, (b) PDD, (c) SDD, and (d) ${\mathrm{PCDD}}_2$ starting from an initial state along the half-cycle direction [$z$ for (a) and (c); $y$ for (b) and (d), respectively]. Bath size: $N=15$. $\tau =0.1,0.2,0.3,0.4,0.5$, from top to bottom in all panels [in (d), the curves for $\tau =0.1$ and 0.2 are indistinguishable]. Decoherence freezes at sufficiently long time.

Figure 3

(Color online) Dependence of the coherence saturation value on pulse delay for CPMG (plus signs), PDD (circles), SDD (upward pointing triangles), and ${\mathrm{PCDD}}_2$ (downward pointing triangles). Dashed lines are linear fittings at small $\tau$.

Figure 4

(Color online) Worst-case single-cycle performance for PDD (circles), SDD (upward pointing triangles), and ${\mathrm{PCDD}}_2$ (downward pointing triangles) in the small $\tau$ region. Bath size: $N=5$. Solid lines are linear fittings, with fitting parameters given in Table .

Figure 5

(Color online) Worst-case DD performance in the logical frame (Refs. 48, 53) with $\tau =0.1$. Hamiltonian parameters are ${\omega }_0=0$, ${\Gamma }_0=0$, 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 $\tau$, for RPD after every $4\tau$, and for SRPD after every $8\tau$. Random protocols are averaged over 100 control realizations.

Figure 6

(Color online) Bath size effect. $T_{0.9}$ vs $1∕\left(2\sigma \tau \right)$ for ${\mathrm{PCDD}}_2$ with different bath sizes $N=15$ (black squares), 17 (blue circles), 20 (green upward pointing triangles), and 25 (red downward pointing triangles).

Figure 7

(Color online) Bath dynamics effect for ${\mathrm{PCDD}}_2$ with ${\Gamma }_0=0$ (black dashed lines), 0.01 (red solid lines), and 0.1 (blue crosses). The parameters are $N=20$ and $\tau =0.2$, 0.25, 0.3, and 0.4 from top to bottom.

Figure 8

(Color online) Effect of bias magnetic field for different evolution times (top) and effect of initial bath polarization for (bottom) on the worst-case performance of ${\mathrm{PCDD}}_2$. Parameters are $\tau =0.3$ and $N=15$. All energies are measured in units of $A$.

Figure 9

Effect of systematic rotation-angle errors on the worst-case ${\mathrm{PCDD}}_2$ performance, with $\tau =0.3$ and $N=15$. The relative flip-angle errors are $\epsilon =0$, 0.001 (indistinguishable from $\epsilon =0$), 0.003, 0.01, 0.03, from top to bottom.

Figure 10

Effect of finite pulse width on the worst-case ${\mathrm{PCDD}}_2$ performance, with $T_c=4.8=16\tau +20w$ and $N=15$. The pulse widths are $w=0$, 0.0024, 0.008, 0.024, 0.08, from top to bottom.