Optimized dynamical decoupling for power-law noise spectra

We analyze the suppression of decoherence by means of dynamical decoupling in the pure-dephasing spin-boson model for baths with power law spectra. The sequence of ideal $\pi$ pulses is optimized according to the power of the bath. We expand the decoherence function and separate the canceling divergences from the relevant terms. The proposed sequence is chosen to be the one minimizing the decoherence function. By construction, it provides the best performance. We analytically derive the conditions that must be satisfied. The resulting equations are solved numerically. The solutions are very close to the Carr-Purcell-Meiboom-Gill sequence for a soft cutoff of the bath while they approach the Uhrig dynamical-decoupling sequence as the cutoff becomes harder.

Figure 1

Schematic visualization of the filter function (black dashed line) and of the spectral function $S(\omega )$ for a soft and for a hard cutoff. ${\omega }_D$ stands for the ultraviolet cutoff. The plot illustrates that the overlap between the filter function and the spectral density remains significant for a soft cutoff, even if $|y_n(\omega t)|{}^2$ shifts to the right. This is in contrast to the situation for a hard cutoff where the overlap becomes extremely small for $n\to \infty$ or $t\to 0$.

Figure 2

Comparison of the optimized sequences PLODD and UDD and the sequence CPMG for $\alpha =4$ and for various numbers of pulses $n$. For clarity only the first half of the sequences is plotted but the symmetry about $\delta =0.5$ (${\delta }_{n+1-j}=1-{\delta }_j$) allows for the straightforward reconstruction of the second half. The concatenated sequence CDD is shown only for $n=10$ because it does not exist for $n=20$ or $n=30$. Note that we refer here to the CDD for pure dephasing [14].

Figure 3

Log-log plot of the integral $I_n$ as a function of the total number of pulses $n$ for $\alpha =2$ and $\alpha =4$. Different sequences are compared, including the CDD sequence for pure dephasing [14]. The latter takes only the values $n=\{0,1,2,5,10,21,42,\dots \}$.

Figure 4

Evaluation of PLODD sequences for various exponents for $n=10$. For clarity only the ${\delta }_j$ from $j=1$ to 5 are plotted. The second half of the sequence is symmetric to the first one (${\delta }_{n+1-j}=1-{\delta }_j$). The PLODD approaches UDD for increasing $\alpha$.

Figure 5

The maximum difference between the $\{{\delta }_j\}$ for PLODD and UDD is shown for two sequences with $n=10$ and $n=12$ pulses. The PLODD sequences tend to approach the UDD sequences for harder cutoffs. In the inset the dependence of $I_n$ on $\alpha$ is depicted. Note that the $y$ axis of the inset is in logarithmic scale.