Fast and optimal transport of atoms with nonharmonic traps

We investigate the fast transport of an atom or a packet of atoms by different kinds of nonharmonic traps including power-law traps. The study is based on the reverse-engineering method. Exact results are obtained and applied to design robust transport protocols. The optimization of the transport trajectory is performed with classical trajectories and remains valid for the transport of a wave packet.

Figure 1

Robustness of the transport in quartic potential $\left(n=2\right)$ against fluctuations of the final time. The relative error $|x\left(t\right)/x\left(t_f\right)-1|$ is plotted as a function of $t/t_f$ for a polynomial interpolation of order 5 (solid line), 7 (dashed line), and 9 (dotted line).

Figure 2

Residual energy, $\mathrm{\Delta }E$, after transport normalized to $\hslash {\omega }_0$ as a function of the anharmonicity parameter $\xi /d$ and for different transport time $t_f$: (a) cubic anharmonicity and (b) quartic anharmonicity. The calculations are performed in dimensionless units and are therefore generically valid. The prefactor for the quantitative estimate of the excess of energy has been estimated with the following parameters: ${\omega }_0=2\pi \times 1.41\times {10}^5$ Hz, $m=40\times 1.667\times {10}^{-27}$ kg $\left({}^{40}{\mathrm{Ca}}^{+}\right)$, $a_0={[\hslash /\left(m{\omega }_0\right)]}^{1/2}$, and $d=20.2a_0$.

Figure 3

Relative excess of final energy after the transport as a function of the transport duration $t_f$ and the strength of the anharmonicity parameter $\xi$: (a) cubic anharmonicity and (b) quartic anharmonicity. Dark vertical lines correspond to “magic times” for which the transport of the packet is optimal.

Figure 4

Transport of a quantum wave packet. Time evolution of the total energy (kinetic + potential), $E\left(t\right)$, with respect to the initial energy, $E_i$, in the course of the transport for three different time transports $t_f=0.75t_f^{*}$ (dotted line), $t_f=t_f^{*}=13.335/{\omega }_0$ (solid line), and $t=1.25t_f^{*}$ (dashed line), where $t_f^{*}$ has been obtained from the results of Sec. 2c based on classical mechanics.