Fast transport of two ions in an anharmonic trap

We design fast trajectories of a trap to transport two ions using a shortcut-to-adiabaticity technique based on invariants. The effects of anharmonicity are analyzed first perturbatively, with an approximate, single relative-motion mode, description. Then, we use classical calculations and full quantum calculations. This allows us to identify discrete transport times that minimize excitation in the presence of anharmonicity. An even better strategy to suppress the effects of anharmonicity in a continuous range of transport times is to modify the trajectory using an effective trap frequency shifted with respect to the actual frequency by the coupling between relative and center-of-mass motions.

Figure 1

Fidelity of the anharmonic system vs final time $t_f$ following the inverse engineering trajectory using second-order perturbation theory (blue thick line), 1D dynamics for the initial ground state of the harmonic oscillator (red dotted line), 1D dynamics for the initial ground state of the perturbed 1D Hamiltonian (green dashed line), 2D dynamics for the initial ground state of the 2D Hamiltonian (filled triangles). $M=2m=29.93\times {10}^{-27}$ kg corresponding to ${}^9$Be${}^{+}$ ions, $\omega /\left(2\pi \right)=20$ kHz, $d=370$ $\mu$m, $r_e=62$ $\mu$m, and $\beta ={10}^6$ m${}^{-2}$.

Figure 2

Fidelity vs final time ($t_f$) for the second-order perturbation theory, indistinguishable from an exact 1D quantum dynamical calculation (the initial state is the ground state of the perturbed harmonic oscillator) for the quadratic perturbation (blue thick line) in Eq. (14), and the quartic perturbation (black dashed line) in Eq. (14). Same parameters as in Fig. 1.

Figure 3

We plot $E_0/(E_0+E_{ex})$, where $E_0$ is the ground-state energy for the 1D Hamiltonian and $E_{ex}$ is the excitation energy after the transport, for the 1D quantum evolution (blue solid line), 2D quantum evolution (black triangles), and a single classical trajectory (green dashed line). Same parameters as in Fig. 1.

Figure 4

Fidelity vs final time $t_f$ for adjusted trap trajectories $Q_0(t;\tilde{\omega })$. The initial condition is the ground state. 1D: blue solid line; 2D: filled triangles. Same parameters as in Fig. 1.

Figure 5

Motional excitation vs final time. $M=2m=29.93\times {10}^{-27}$ kg, $\omega /\left(2\pi \right)=2$ MHz, $d=370$ $\mu$m for the three cases and $\beta =6.4\times {10}^9$ m${}^{-2}$, $r_e=2.807\mu$m (solid blue line), $\beta ={10}^9$ m${}^{-2}$, $r_e=2.883$ $\mu$m (red dashed line), and $\beta ={10}^{10}$ m${}^{-2}$, $r_e=2.764$ $\mu$m (green dotted line). The middle value of $\beta$ ($6.4\times {10}^9$) is chosen so that at $t_f=8$ $\mu$s the excitation is similar to the one seen experimentally in [12]. The trap trajectory is given by Eq. (4). If instead the adjusted trajectory $Q_0(t;\tilde{\omega })$ is used, then $E_{ex}\left(t_f\right)=0.$