Soft-pulse dynamical decoupling with Markovian decoherence

We consider the effect of broadband decoherence on the performance of refocusing sequences, having in mind applications of dynamical decoupling in concatenation with quantum error correcting codes as the first stage of coherence protection. Specifically, we construct cumulant expansions of effective decoherence operators for a qubit driven by a pulse of a generic symmetric shape and for several sequences of $\pi$ and $\pi /2$ pulses. While, in general, the performance of soft pulses in decoupling sequences in the presence of Markovian decoherence is worse than that of the ideal $\delta$ pulses, it can be substantially improved by shaping.

Figure 1

Assumed spectral function of a generic oscillator bath (schematic). Dynamical decoupling can be very effective in preserving coherence in the presence of (a) featureless low-frequency bath [21] and also (b) sharp resonances [33]. In this work we consider the effect of Markovian noise (c) on DD of a single qubit. We assume $F\left(\omega \right)=\text{const}$ up to a large cut-off frequency ${\omega }_{\text{max}}\equiv \Lambda$ so that the qubit dynamics associated with the modes (c) can be modeled by the Linblad Eq. (11). When the high-frequency spectrum is not flat but is, e.g., a power law $\propto 1/{\omega }^{\alpha }$ as in Refs. [30, 32], the decomposition into (a) low-frequency and (c) Markovian parts becomes approximate, with the Markovian decoherence rates $\tilde{\gamma }$ determined by $F\left(\omega \right)$ near $\omega =\Omega \equiv 2\pi /\tau$ determined by the sequence duration $\tau$.

Figure 2

(Color online) Pulse shapes used in the simulations. The Gaussian shape $G_x\left(y\right)={\left[{\left(2\pi \right)}^{1/2}x{\tau }_p\right]}^{-1}\text{exp}\left(-0.5y^2/x^2\right)$, with $x=0.10$ and $y=t/{\tau }_p-0.5$ is $\sqrt{2}$ wider than that used in Ref. [33]. The shapes $S_1$, $Q_1$ were originally constructed in Ref. [19]. The shapes $F_1$, $W_{11}$, and $W_{21}$ are constructed in this work, see Table for the coefficients.

Figure 3

(Color online) Single-qubit average fidelity for a particular realization of time-dependent fields and Markovian dephasing with the rate ${\gamma }_{\varphi }=2\pi \times {10}^{-3}/{\tau }_p$. Open boxes connected with the dashed line marked “no control” correspond to the absence of decoupling. Other symbols are numerical data for the indicated decoupling sequences of Gaussian inversion pulses. The black solid line marked “ideal” corresponds to maximum achievable average fidelity. Dashed line corresponds to Eq. (63) with ${\upsilon }_2=-0.7086$ as appropriate for the pulse $G_{0.10}$ (see Table ).

Figure 4

(Color online) Single-qubit average fidelity loss due to redistribution of the decoherence rates over directions in the absence of low-frequency fields. Symbols are numerical data for pulses and sequences as indicated, and lines are computed analytically using Eq. (63) with corresponding $v_2$ from Table . This contribution to infidelity would be zero for hard ($\delta$ function) pulses with ${\upsilon }_2=-1$.

Figure 5

(Color online) Single-qubit decoupling error for the eight-pulse sequence $\mathbf{8}\mathbf{s}$ with several pulse shapes, as indicated. The decoupling error is small for the second-order pulse $W_{21}\left(\pi \right)$ and, somewhat surprisingly, for the symmetrizing shape $F_1$. With ${\gamma }_{\varphi }=0$, the infidelity for this pulse shape with sequence $\mathbf{8}\mathbf{s}$ grows similarly to that of first-order pulses $S_1$ and $W_{11}$ (Fig. 7).

Figure 6

(Color online) As in Fig. 5 but for 16-pulse sequence $\mathbf{1}\mathbf{6}\mathbf{a}$. For this sequence and with time-independent fields $B_{\mu }$, there would be no contribution from the first two orders of the effective decoherence operator [Eqs. (56, 57)]; the decoupling error is determined by the nonadiabaticity and by the terms of higher order. Generic pulse shapes $G_{0.10}$, $S_1$, and $Q_1$ show the largest decoupling error, while it is the smallest for the symmetrizing shape $F_1$ (technically, zeroth order, $\upsilon \ne 0$; ${\upsilon }_2\approx 1/3$). For shapes $F_1$, $W_{11}$, and $W_{21}$, the decoupling error here is actually smaller than the infidelity in Fig. 8. We believe this is an artifact of the definition of the decoupling error related to the full average fidelity approaching saturation at 1/2 (Fig. 3).

Figure 7

(Color online) As in Fig. 5 but in the absence of Markovian dephasing, ${\gamma }_{\varphi }=0$. Infidelity curves with second-order pulses $Q_1$ and $W_{21}$ have a finite slope, unlike the plots in Ref. [21] where a smaller rms value $B_0$ of the slow fields was used. We checked that the slopes for the second-order pulse shapes are greately reduced if $B_0$ is reduced by a factor of two (not shown).

Figure 8

(Color online) As in Fig. 7 but for the sequence $\mathbf{1}\mathbf{6}\mathbf{a}$. The slopes for all the pulse shapes excluding $G_{0.10}$ are greately reduced if $B_0$ is reduced by a factor of two (not shown). For the Gaussian pulses $G_{0.10}$, the second-order cancellation is achieved only after the complete sequence with the duration $\tau =16{\tau }_p$; the slope reduction happens for slower fields $B_{\mu }\left(t\right)$ with ${\tau }_c=16{\tau }_p$ (not shown).