Optimized Noise Filtration through Dynamical Decoupling

Recent studies have shown that applying a sequence of Hahn spin-echo pulses to a qubit system at judiciously chosen intervals can, in certain noise environments, greatly improve the suppression of phase errors compared to traditional dynamical decoupling approaches. By enforcing a simple analytical condition, we obtain sets of dynamical decoupling sequences that are designed for optimized noise filtration, but are independent of the noise spectrum up to a single scaling factor set by the coherence time of the system. These sequences are tested in a model qubit system, ions in a Penning trap. Our combined theoretical and experimental studies show that in high-frequency-dominated noise environments with sharp high-frequency cutoffs this approach may suppress phase errors orders of magnitude more efficiently than comparable techniques can.

Figure 1

OFDD sequences for (a) $n=6$ and (b) $n=10$ pulses. The horizontal axis indicates the pulse locations, ${\delta }_j$, relative to the total sequence duration ${\tau }^{\prime }$ (vertical axis) that minimize $A_F({\tau }^{\prime })$. The horizontal dashed line at (a) ${\tau }^{\prime }=15.8$ and (b) ${\tau }^{\prime }=24.2$ indicates when $F_n({\tau }^{\prime })=1$. For longer sequence durations the system is largely dephased.

Figure 2

Comparison of decoherence, $[1-W({\tau }^{\prime })]/2$, due to phase errors when using $n=6$ (left) and $n=10$ (right) pulse CPMG (dotted lines), UDD (dashed lines), OFDD (solid lines), and spectrum dependent LODD (dot-dashed lines) in an (a),(b) Ohmic and (c),(d) $1/f$ noise environment, respectively. We assumed ${\tau }_{\pi }=0$ and used $\alpha =1$ throughout.

Figure 3

Time evolution of the normalized fluorescence counts, equivalent to $[1-W({\tau }^{\prime })]/2$, measured using ${}^9\mathrm{Be}^{+}$ qubits in a Penning ion trap in (a) the ambient noise environment and a synthesized Ohmic noise environment with (b) a frequency cutoff near 500 Hz and (c) a frequency cutoff near 250 Hz. We used 6 $\pi$ pulses in each sequence and open circles represent CPMG, triangles UDD, and diamonds OFDD. The data are well described by the fits (solid lines) which use the analytical expression for the coherence, the measured noise spectra in both environments, and the noise strength as a free parameter. We are at this time unsure of the cause of the minor discrepancy between the fits and the UDD and OFDD data in (c). This may be the consequence of external noise influences not accounted for in the synthesized spectrum.