Fast shuttling of a trapped ion in the presence of noise

We theoretically investigate the motional excitation of a single ion caused by spring-constant and position fluctuations of a harmonic trap during trap shuttling processes. A detailed study of the sensitivity on noise for several transport protocols and noise spectra is provided. The effect of slow spring-constant drifts is also analyzed. Trap trajectories that minimize the excitation are designed combining invariant-based inverse engineering, perturbation theory, and optimal control.

Figure 1

Comparison of trap trajectories $x$ between the initial and final trap positions (yellow solid segments). Polynominal protocol (red dotted line); bounded optimal [black solid line, only in (a)]; unbounded optimal [blue dashed line, the jump at the boundary times is $6d/\left({\omega }^2{T}^2\right)$]; bang-bang (dot-dashed purple line). In (a) $T=T_{\omega }/2$ ($T_{\omega }$ is the oscillation period); in (b) $T=9T_{\omega }/2$. $\delta =0.5$ $d$, mass of ${}^{40}$Ca${}^{+}$, initial state in $n=0$, $\omega =2\pi \times 1.4$ MHz, $d=280$ $\mu$m.

Figure 2

$G(T;0)$ versus final time. Polynomial ansatz (red dotted line), unbounded optimal (blue dashed line), bounded optimal [black solid line; the vertical dashed lines delimitate the time window in Eq. (34)], and bang-bang (purple dot, $T=T_{\omega }/2$; see the magnification in the inset). (a) $\delta =0.5d$; (b) $\delta =0.02d$; other parameters are the same as in Fig. 1.

Figure 3

$G(T;0)$ for Ornstein-Uhlenbeck noise versus correlation time. Polynomial ansatz (red dotted line), unbounded optimal (blue dashed line), and bounded optimal (black solid line). $\delta =0.005d$, $T=5T_{\omega }$, and other parameters are the same as in Fig. 1.

Figure 4

$G(T;0)$ for flicker noise versus upper limit of the interval of correlation times ${\tau }_2$. Polynomial ansatz (red dotted line), unbounded optimal (blue dashed line), and bounded optimal (black solid line). $\delta =0.5d$, ${\tau }_1=1\times {10}^{-10}$ $s$, $T=5T_{\omega }$, and other parameters are the same as in Fig. 1.

Figure 5

(a) $x_c\left(t\right)$ versus $t$; (b) excitation energy versus $\Lambda$. Dashed red line, quintic polynomial (29); solid blue line, seventh order polynomial in Eq. (62). $T=6.5T_{\omega }$ and other parameters are the same as in Fig. 1.