Sideband excitation of trapped ions by rapid adiabatic passage for manipulation of motional states

We describe an analysis and experimental results of the manipulation of motional states of a single trapped ${}^{40}$Ca${}^{+}$ ion based on sideband excitation by rapid adiabatic passage. When the sideband transition is excited by rapid adiabatic passage, adiabaticity may be affected by ac Stark shifts. We investigate the influence of ac Stark shifts and compensate for these shifts with an additional laser field. This makes the population transfer by rapid adiabatic passage more robust with respect to experimental parameters. Finally, we manipulate the motional states and generate motional Fock states of a single ${}^{40}$Ca${}^{+}$ ion by rapid adiabatic passage with ac Stark compensation.

Figure 1

Calculated energies of the dressed states and $|{\alpha }_{ji}\left(t\right)/{\omega }_{ji}\left(t\right){|}^2$ for the case in which the blue sideband is excited with RAP. (a) Time dependence of $\Omega \left(t\right)/2\pi$ (solid red curve) and $\Delta \left(t\right)/2\pi$ (dotted blue line). (b) Variation of the energies of the dressed states against time. The bold red curve (thin blue curve) approximately corresponds to $|\downarrow ,0\rangle$ ($|\uparrow ,1\rangle$) at $t$ = 0 ms. The dotted curves represent the cases in which Hamiltonian matrix elements for carrier transitions are set to be zero. (c) Variation of $|{\alpha }_{ji}\left(t\right)/{\omega }_{ji}\left(t\right){|}^2$ against time.

Figure 2

Population transfer by blue-sideband RAP. The population of $|\uparrow \rangle$ is plotted for different Ω${}_0$, as described in Eq. (1). The experimental parameters are σ = $\frac{0.3}{2\sqrt{2}}T,$ as obtained by Eq. (1), and $D$ = 400 kHz, as obtained by Eq. (2). Data are presented for four pulse times $T$, namely, 768, 384, 192, and 144 μs.

Figure 3

Rabi oscillations on the blue sideband (points) for the case of preparation in (a) $|\downarrow ,0\rangle$ and (b) $|\downarrow ,1\rangle$. The solid curves are exponentially decaying sinusoids that are fitted to the points. (c) Rabi frequency |Ω${}_{n,n+1}|/$2π vs motional quantum number $n$. The experimental parameters are $T$ = 512 μs, $D$/2π = 200 kHz, σ = 54 μs, and ${\Omega }_0/2\pi =8.9$ kHz. The curve shows the calculated values for the Lamb-Dicke parameter of the trap η = 0.082 with a common proportionality factor that gives the least-χ${}^2$ fit to the measured values.

Figure 4

Blue-sideband RAP excitation. The population of $|\uparrow \rangle$ is plotted for different RAP pulse times. The solid red, dashed blue, and dotted green curves are for Γ/2π = 0, 100, and 300 Hz, respectively. Here the maximum Rabi frequency of the RAP pulse is 200 kHz and the chirp width is 100 kHz.