Memory effects in a dynamical decoupling process

TL;DR

This work addresses how environmental memory (non-Markovianity) influences dynamical decoupling performance. By tailoring non-Markovianity measures to DD and deriving a commutation-based bound, the authors show that the DD-induced change in the final map or state is bounded by the sum or difference of memory strengths with and without control, provided kicks commute with the uncontrolled map. They demonstrate the theory with a dissipative quantum Rabi system under parity kicks, showing the bounds hold across one and many cycles and that, in several regimes, the control gain tracks memory strength. The results offer a practical, calculable link between memory effects and control effectiveness, with implications for optimizing DD in non-Markovian environments and potential experimental validation in trapped-ion or related platforms.

Abstract

We establish a simple quantitative relationship between the environmental memory effects and the characteristics in a dynamical decoupling process. In contrast to previous works, our measures of non-Markovianity are tailored and extended to evaluate the strength of memory effects in dynamical decoupling. We find that if each kick commutes with the dynamical map of the uncontrolled system, then the change of the final dynamical map or the final state brought by the control (called the "effect of control") is upper (lower) bounded by the summation (difference) of the strengths of memory effects with and without control. We propose sufficient conditions for the commutation relation for parity kicks and illustrate our finding with a dissipative quantum Rabi model by numerical simulations where one or many cycles of parity kicks are implemented on the qubit. Besides, the results show that under certain conditions, the effect of control or the increase of performance by the control may be simply proportional to the strength of memory effects with or without control.

Figures (7)

• Figure 1: Performances (with and without control), effects of control and their bounds as a function of the final time $t_f$ for 1 cycle of parity kicks ($t_f=t_c=2\tau$). The parameters are set as $\omega_E=\omega_S$, $g/\omega_S=0.01$, $\gamma/g=0.1$ and $\bar{n}=0.1$. (a) and (c) represent those for the final dynamical maps, while (b) and (d) for the final states. The system initial state in (b) and (c) is $1/\sqrt{2}(|g\rangle+|e\rangle)$. The difference of the two lines in (a) is plotted in (c) by the dotted line for comparisons with $E$, similarly for (b) and (d).
• Figure 2: Performances (with and without control), effects of control and their bounds as a function of the final time $t_f$ for 1 cycle of parity kicks. The parameters and settings are the same as those in Fig. \ref{['FIG:PicA']} except that $\gamma/g=100$.
• Figure 3: Performances (with and without control), effects of control and their bounds as a function of the final time $t_f$ for 1 cycle of parity kicks. The parameters and settings are the same as those in Fig. \ref{['FIG:PicA']} except that $g/\omega_S=1$ and $\gamma/g=0.01$.
• Figure 4: Performances (with and without control), effects of control and their bounds as a function of the final time $t_f$ for 1 cycle of parity kicks. The parameters and settings are the same as those in Fig. \ref{['FIG:PicC']} except that $\omega_E=10\omega_S$.
• Figure 5: Performances (with and without control), effects of control and their bounds as a function of the final time $t_f$ for 1 cycle of parity kicks. The parameters and settings are the same as those in Fig. \ref{['FIG:PicA']} except that $\bar{n}=5$.