Fast expansions and compressions of trapped-ion chains

We investigate the dynamics under diabatic expansions or compressions of linear ion chains. Combining a dynamical normal-mode harmonic approximation with the invariant-based inverse-engineering technique, we design protocols that minimize the final motional excitation of the ions. This can substantially reduce the transition time between high and low trap-frequency operations, potentially contributing to the development of scalable quantum information processing.

Figure 1

Final excitation energy at $t_f$ for the expansion of two ${}^{40}{\mathrm{Ca}}^{+}$ ions, with respect to the final ground state (quantum) or the final equilibrium energy (classical). The initial state is the ground state (quantum) or the equilibrium state (classical) for the initial trap. The dashed red line is the excitation in the harmonic approximation, using Eq. (14) for the NM energies, with the protocol obtained by the shooting method; the solid blue line (classical) and black triangles (quantum) are for the same protocol but with the dynamics driven by the full Hamiltonian (1). The dotted green line is for the protocol (16) with the full Hamiltonian. The parameters used are ${\omega }_0/\left(2\pi \right)=1.2$ MHz and ${\gamma }^2=3$.

Figure 2

Values of the optimizing free parameters in the two-ion expansion $a_{10}$ (solid blue line) and $a_{11}$ (dashed red line) in the expansion of two ${}^{40}{\mathrm{Ca}}^{+}$ ions starting in the ground state. ${\omega }_0/\left(2\pi \right)=1.2$ MHz; ${\gamma }^2=3$.

Figure 3

Final excitation energy for the expansion of a ${}^9{\mathrm{Be}}^{+}{-}^{40}{\mathrm{Ca}}^{+}$ ion chain (solid blue line) and a ${}^9{\mathrm{Be}}^{+}{-}^{40}{\mathrm{Ca}}^{+}-{}^9{\mathrm{Be}}^{+}$ chain (dashed red line) starting in the equilibrium configuration. The protocols are optimized with four (for ${}^9{\mathrm{Be}}^{+}-{}^{40}{\mathrm{Ca}}^{+}$) and two (for ${}^9{\mathrm{Be}}^{+}-{}^{40}{\mathrm{Ca}}^{+}-{}^9{\mathrm{Be}}^{+}$) free parameters; see the main text. ${\omega }_0/\left(2\pi \right)=1.2$ MHz; ${\gamma }^2=3$.

Figure 4

Final excitation vs final times for expansions of two equal ions (solid blue line), four equal ions (red dots), and eight equal ions (dash-dotted black line). The simulations are performed according to the approximate protocol in Eq. (16) and by solving the classical equations of motion for ${}^{40}{\mathrm{Ca}}^{+}$ ions. The initial ion chain is at equilibrium. ${\omega }_0/\left(2\pi \right)=1.2$ MHz; ${\gamma }^2=3$.

Figure 5

Classical trajectories of eight expanding ${}^{40}{\mathrm{Ca}}^{+}$ ions. The evolution is performed according to Eq. (16). ${\omega }_0/\left(2\pi \right)=1.2$ MHz, ${\gamma }^2=3$, and $t_f=4.4\mu \mathrm{s}$.

Figure 6

Comparison of final excitation quanta (classical simulation as in Fig. 1) vs final time in the expansion of two ${}^{40}{\mathrm{Ca}}^{+}$ ions following the shooting protocol (solid blue), linear protocol in Eq. (24) (dotted green), and cosine protocol in Eq. (25) (dashed red). In (a) we plot in logarithmic scale up to times where only the shooting protocol reaches the level of $0.1$ excitation quanta. In (b) we extend the analysis up to longer final times, so that the best of the cosine protocol reaches also 0.1 final excitation quanta. ${\omega }_0/\left(2\pi \right)=1.2$ MHz; ${\gamma }^2=3$.