Fast atomic transport without vibrational heating

We use the dynamical invariants associated with the Hamiltonian of an atom in a one dimensional moving trap to inverse engineer the trap motion and perform fast atomic transport without final vibrational heating. The atom is driven nonadiabatically through a shortcut to the result of adiabatic, slow trap motion. For harmonic potentials this only requires designing appropriate trap trajectories, whereas perfect transport in anharmonic traps may be achieved by applying an extra field to compensate the forces in the rest frame of the trap. The results can be extended to atom stopping or launching. The limitations due to geometrical constraints, energies, and accelerations involved are analyzed along with the relation to previous approaches based on classical trajectories or “fast-forward” and “bang-bang” methods, which can be integrated in the invariant-based framework.

Figure 1

Trap trajectories $q_0/d$ versus $s=t/t_f$ for $t_f=12.57/{\omega }_0=2T_0$ (blue line with dots), $t_f=2.505/{\omega }_0$ (the critical value; red line with triangles), and $t_f=2/{\omega }_0$ (brown line with squares). For all three cases, $q_c/d$ is the dashed line, which is hardly distinguishable from $q_0/d$ for the slowest case, i.e., for $t_f=12.57/{\omega }_0$.

Figure 2

Three stopping trajectories: $a=0.8$, $b={\omega }_0{t}_f=1.6$, and $t_f/T_0=0.25$ (brown line with squares); $b={\omega }_0{t}_f=1.9$ and $t_f/T_0=0.30$ (red line with triangles); and $b={\omega }_0{t}_f=3.0$ and $t_f/T_0=0.477$ (blue line with circles). They all share the same $q_c(s)/d$ (black dashed line) given by Eq. (26).

Figure 3

Stopping atoms. The shaded areas represent the values of $a=v_0{t}_f/d$ and $b/2\pi =t_f/T_0$ for which there is a $t$ in $(0,t_f)$ where (left) $q_0(t)<0$ or (right) $q_0(t)>d$. We have used the polynomial ansatz of Eqs. (26) and (27).

Figure 4

Dependences of time-averaged energies on $t_f$. The $\delta$ is the asymptotic exponent of $t_f$. Parameters are ${\omega }_0=2\pi \times 8$ Hz, $E_0=\hslash {\omega }_0/2$, $d=2.25$ mm, and rubidium-87 atoms that begin and end in the ground vibrational state. (a) Bound (35) (solid red line) and time average of $E_P$ for a polynomial trajectory (dotted blue line). (b) AA bound (dot-dashed magenta line) and time average of $\Delta H$ for a polynomial trajectory (dotted blue line).

Figure 5

Graphical representation of the lower bound of Eq. (37) for (top) the maximum velocity ${\mathrm{ẋ}}_{\text{max}}$ and (bottom) the maximum acceleration $\left|\mathrm{ẍ}\right|{}_{\text{max}}$ of Eqs. (38) and (39).