Numerical analysis of optimized coherent control pulses

Numerically we simulate the effect of optimized coherent control pulses with a finite duration on a qubit in a bath of spins. The pulses of finite duration are compared with ideal instantaneous pulses. In particular, we show that properly designed short pulses can approximate ideal instantaneous pulses up to a certain order in the shortness of the pulse. We provide examples of such pulses, quantify the discrepancy from the ideal case, and compare their effect for various ranges of the coupling constants.

Figure 1

(Color online) Examples of $\pi$ pulses implemented in the simulations. The ideal pulse is given by a $\delta$ peak operating at the instant ${\tau }_s$. SGLPi is the standard pulse of constant amplitude without optimization of the pulse shape. For details see Table .

Figure 2

(Color online) Examples of $\pi ∕2$ pulses implemented in the simulations. The ideal pulse is given by a $\delta$ peak operating at the instant ${\tau }_s$. SGLPi2 is the standard pulse of constant amplitude without optimization of the pulse shape. For details see Table .

Figure 3

(Color online) Examples of $\pi ∕2$ pulses for which also some of the second order corrections vanish, namely, ${\eta }_{21}=0$ and ${\eta }_{22}=0$ (see the main text). For details see Table .

Figure 4

(Color online) Sketch of the spin chain representing the qubit and the spin bath. The qubit 1 is coupled to the spin 2 of the chain. The coupling between the qubit and the spin bath is given by $J$ while the internal exchange coupling within the chain is $\alpha J$.

Figure 5

(Color online) Deviation $d$ as a function of $J∕B_m$ ($B_m$ is the maximum amplitude of the pulse) for various lengths $N$ of the spin chain at $\alpha =5.0$. The data refers to the ASY1 pulse.

Figure 6

(Color online) Case of $\pi$ pulses. The deviation $d$ is plotted as a function of $J∕B_m$ for $N=10$ and various values of $\alpha$. For an unbiased comparison of the pulses, $J$ is normalized to the maximum amplitude $B_m$ of the pulses. The notation for the pulses refers to Table . The dashed lines ease the comparison with pure power laws.

Figure 7

(Color online) Case of $\pi$ pulses. Plot of the prefactors $a_{\alpha }$ in $d=a_{\alpha }{J}^2+O\left(J^3\right)$ for $N=7$. The solid lines represent the analytical prediction in Eq. (18).

Figure 8

(Color online) Case of $\pi ∕2$ pulses. The deviation $d$ is plotted as a function of $J∕B_m$ for $N=10$ and various values of $\alpha$. For an unbiased comparison of the pulses, $J$ is normalized to the maximum amplitude $B_m$ of the pulses. The notation for the pulses refers to Table . The dashed lines ease the comparison with pure power laws.

Figure 9

(Color online) Case of $\pi ∕2$ pulses. Plot of the prefactors $a_{\alpha }$ in $d=a_{\alpha }{J}^2+O\left(J^3\right)$ for $N=7$. The solid lines represent the analytical prediction in Eq. (18).

Figure 10

(Color online) Case of $\pi ∕2$ pulses with partly vanishing quadratic order. The deviation $d$ is plotted as a function of $J∕B_m$ for $N=10$ and various values of $\alpha$. For an unbiased comparison of the pulses, $J$ is normalized to the maximum amplitude $B_m$ of the pulses. The notation for the pulses refers to Table . The dashed lines ease the comparison with pure power laws.