Quantum kinetics of an open system in the presence of periodic refocusing fields

We consider quantum kinetics of an open quantum system in the presence of periodic fields designed to suppress the internal evolution and shield the system from a generic low-frequency environment (refocusing or dynamical decoupling in an application to multiqubit systems). Assuming that the refocusing has order $K$, that is, for a frozen environment the cumulant expansion of the evolution operator over the period $\tau$ begins with the term $\sim {\tau }^{K+1}$, we trace the associated cancellations in the kernel of the quantum kinetic equation in the Floquet formalism and characterize the remaining decoherence processes.

Figure 1

“Refocusing error” (specifically, three times the deviation of the average fidelity for the spin to remain in the initial state from one), in the absence of refocusing. The Gaussian random field is applied along the $x$ axis only. For different curves, it has the same rms amplitude $b_0$ but different values of the correlation time ${\tau }_0$ [see Eq. (54)]. Symbols show the results of simulation averaged over 900 samples of the random field; lines show the corresponding exact results (55, 56) which for $t⪢{\tau }_0$ are also very close to the QKE result (not shown). See text for other notations.

Figure 2

The Frobenius norm of the matrices ${\widehat{A}}_m$ [Fourier transform of the evolution matrices $\widehat{Q}\left(t\right)$, see Eq. (15)] with frequencies ${\Omega }_m=2\pi m∕{\tau }_p$ for sequences $n\mathrm{c}$, $n=4,8,\dots ,64$ with Gaussian $\pi$-pulses. The width of a pulse is chosen to be $0.05{\tau }_p$ so that the discontinuity at the ends of the interval is numerically negligible, which results in a steep cutoff at high frequencies. Vertical lines mark the spectral features of the parent sequence $4\mathrm{c}$, which dominate the spectrum of all higher-order sequences. Thin dotted lines guide the eye with the slope corresponding to power laws as indicated.

Figure 3

As in Fig. 2 for first-order self-refocusing pulses $S_1$.

Figure 4

As in Figs. 2, 3 with second-order pulses $Q_1$. Note a suppression of the low-frequency part of the spectrum compared with lower-order pulses.

Figure 5

As in Figs. 2, 3, 4 for second-order pulses $Q_2$. These shapes vanish at the ends of the interval along with the first three derivatives, which suppresses the high-frequency part of the spectrum.

Figure 6

Refocusing error with the fluctuating magnetic field along the $x$ axis as in Fig. 1, but now in the presence of refocusing sequence $8\mathrm{c}$ with pulses $Q_1$ (order 2, see Table ). Symbols represent the data averaged over 900 realizations of noise and dashes are the linear fits [Eq. (60)] for data further than $\Delta t=3{\tau }_0$ from the ends of the interval. The inset shows the fit coefficients for sequences $4\mathrm{c}$, $8\mathrm{c}$, and $16\mathrm{c}$ as a function of the correlation time ${\tau }_0$. Dotted lines on the inset indicate the slope corresponding to the power laws as indicated. The decoherence rate (proportional to the slopes $B$) is reduced dramatically for the correlation time ${\tau }_0$ exceeding the duration of the sequence, $\tau =n{\tau }_p$ for sequence $n\mathrm{c}$. Yet the refocusing error does not disappear altogether because of the initial decoherence (proportional to the intercept $A$) which does not vanish with increased noise correlation time ${\tau }_0$ for these nonsymmetric sequences [see Eq. (52)]. The data on the inset also show that the refocusing accuracy is not improved with the longer sequences of order above second, but it also does not worsen even for small ${\tau }_0∕\tau$ (the low-frequency harmonics ${\widehat{A}}_m$ are suppressed, see Fig. 4). The refocusing errors are strongly suppressed near the end of the interval because the fluctuating field used in the calculation is periodic with the period $T=256{\tau }_p$.

Figure 7

As in Fig. 6 but for the sequence $8\mathrm{p}$ (pulses $Q_1$) and for the magnetic field fluctuating in all three directions, Eq. (22). Amplitude $b_0$ along each direction and other simulation parameters are as in Fig. 6. Inset shows the linear fit coefficients for this sequenc with the magnetic field fluctuating in one $(8\mathrm{p}:1)$ and all three $(8\mathrm{p}:3)$ directions. The effective decoherence rate for the fastest fluctuations, ${\tau }_0=4{\tau }_p$, is bigger then those in the simulations with $n\mathrm{c}$ sequences. However, with ${\tau }_0>\tau =8{\tau }_p$, the decoherence rate (slope $B$) again goes down dramatically, while the initial decoherence also scales down quadratically with increasing ${\tau }_0$, resulting in a superior refocusing accuracy.