Raman velocity filter as a tool for collinear laser spectroscopy

The velocity distribution of a hot ionic beam can be filtered with a narrow stimulated Raman process to prepare a colder subensemble, as substantiated in this theoretical analysis. Using two counterpropagating far-detuned lasers, we can define a $\pi$ pulse for the resonant velocity to transfer atoms within the linewidth of the Raman resonance between the ground states of a $\mathrm{\Lambda }$ system. Spontaneous emission from the two single-photon resonances, as well as the ground-state decoherence induced by laser noise, diminishes the efficiency of the filter. From a comprehensive master equation, we obtain conditions for the optimal frequency pair of lasers and evaluate the filter performance numerically as well as analytically. If we apply this analysis to current ${}^{40}{\text{Ca}}^{+}$ ion experiments, we obtain a sensitivity for measuring high ion acceleration voltages on the ppm level or below.

Figure 1

Three-level energy diagram for ${}^{40}{\text{Ca}}^{+}$. Laser 1 induces $eg$ transitions with ${\omega }_{eg}={\omega }_e-{\omega }_g$ and laser 2 couples the $em$ transition with ${\omega }_{em}={\omega }_e-{\omega }_m$. In the inertial rest frame of the ions, we define a one-photon detuning $\mathrm{\Delta }={\omega }_2^{\prime }-{\omega }_{em}$ and a two-photon detuning $\delta ={\omega }_1^{\prime }-{\omega }_2^{\prime }-{\omega }_{mg}$ with respect to the Doppler-shifted frequencies ${\omega }_1^{\prime }$ and ${\omega }_2^{\prime }$ given by Eq. (2). The spontaneous decay rates ${\mathrm{\Gamma }}_{eg}$ and ${\mathrm{\Gamma }}_{em}$ couple the excited state $|e\rangle$ to the ground state $|g\rangle$ and metastable state $|m\rangle$. Further, laser noise induces ground-state decoherence with rates ${\mathrm{\Gamma }}_{gg}$ and ${\mathrm{\Gamma }}_{mm}$.

Figure 2

Two counterpropagating lasers with wave number vectors ${\mathit{k}}_1=-k_1{\mathit{e}}_z$ and ${\mathit{k}}_2=k_2{\mathit{e}}_z$ interact with the ions, which move with velocity $\mathit{v}=v{\mathit{e}}_z$ parallel to laser 2. Wave numbers and scalar velocities are positive quantities $k_i,v>0$.

Figure 3

Ionic velocity distribution $f\left(v\right)$ vs velocity $v$ with mean velocity $\overline{v}$ and width $\mathrm{\Delta }{v}_i$ (dash-dotted line). Superimposed are the initial growth rate $r_m$ (16) (dotted line) of the metastable-state population from perturbation theory and the exact stationary solution ${\rho }_{mm}^{\infty }$ (B4) and (B5) (solid line), which exhibit resonances at $v_1$, $v_2$, and $v_{\text{R}}$. Note that the velocity distribution emerging from the accelerator is rather flat topped.

Figure 4

Linkage pattern for two-photon transitions connecting the ground state $|g\rangle$ with the metastable state $|m\rangle$.

Figure 5

(a) Velocity distances ${\beta }_1$ (solid line) and ${\beta }_2$ (dashed line) between the Raman resonance and the rogue resonances versus detuning ${\mathrm{\Delta }}_{1,0}$. The inadmissible range ${\omega }_2<0$ is shaded in gray. (b) Real velocities $c{\beta }_i$ and the Doppler-shifted one-photon detuning $\mathrm{\Delta }$ (dotted line) on a small scale. The detunings for parameter sets $\text{A}$ and $\text{B}$ are marked with vertical lines (Table 2).

Figure 6

Rabi oscillations of the metastable-state population $\mathfrak{m}(t,v_{\text{R}})$ (red solid line) for the resonant velocity, together with the velocity averaged population ${\langle \mathfrak{m}(t,v)\rangle }_v$ for $\mathrm{\Delta }{v}_i=50\text{m}{\text{s}}^{-1}$ (blue dashed line). Three parameter sets (Table 2) are compared: $\text{A}$ ($\times$), ${\text{B}}_1$ ($\circ$), and ${\text{B}}_2(▵$).

Figure 7

The velocity-dependent metastable-state population $\mathfrak{m}\left(t_{\pi }\right)$ after a $\pi$ pulse ($t_{\pi }^{\text{A}}=t_{\pi }^{{\text{B}}_1}=4.62\mu \mathrm{s}$ and $t_{\pi }^{{\text{B}}_2}=0.68\mu \mathrm{s}$) is indiscernible for parameter sets $\text{A}$ and ${\text{B}}_1$. For maximal laser power ${\text{B}}_2$ the resonance is broadened as well as shifted. The approximation ${\mathfrak{m}}_0$ (33) matches the full solution.

Figure 8

Velocity-dependent population of the metastable state after applying a $\pi$ pulse $\mathfrak{m}\left(t_{\pi }\right)$, conforming with the analytical approximation (47). The narrow Raman resonance at ${\nu }_{\text{R}}\approx 0\text{m}{\text{s}}^{-1}$, shown in detail in Fig. 7, and the broad resonance of laser 1 at ${\nu }_1^{\text{A}}=400\text{m}{\text{s}}^{-1}$ and ${\nu }_1^{\text{B}}=1200\text{m}{\text{s}}^{-1}$ are apparent.

Figure 9

Velocity-dependent difference between the metastable-state populations, considering ($\mathfrak{m}$) and neglecting (${\mathfrak{m}}_0$) spontaneous emission effects, after $t_{\pi }$.

Figure 10

Time evolution of the velocity averaged metastable-state population ${\langle \mathfrak{m}(t,v)\rangle }_v$ for two chracteristic ion velocity widths (a) $\mathrm{\Delta }{v}_i=10\text{m}{\text{s}}^{-1}$ and (b) $\mathrm{\Delta }{v}_i=50\text{m}{\text{s}}^{-1}$ and the parameters $\text{A}$ ($\times$), ${\text{B}}_1$ ($\circ$), and ${\text{B}}_2$ ($▵$). The numerical solutions considering incoherent effects (solid line), well predicted by the analytical approximation (47) (black dashed line), are compared to the analytical ones, indicating the transferred population purely due to the Raman transition (30) (dotted line).

Figure 11

Velocity width $\mathrm{\Delta }v$ (FWHM) of the metastable-state population depending on the laser powers $P_1$ and $P_2$ for different times: (a) and (b) after applying a $\pi$ pulse and (c) and (d) $\overline{\tau }=4.62\mu \mathrm{s}$. The FWHM of ${\mathfrak{m}}_{\text{ana}}$ (47) (a) $\mathrm{\Delta }{v}_{\text{ana}}\left(t_{\pi }\right)$ and (c) $\mathrm{\Delta }{v}_{\text{ana}}\left(\overline{\tau }\right)$ is compared to the approximations (b) $\mathrm{\Delta }{v}_{\text{R}}\left(t_{\pi }\right)$ (39) and (d) $\mathrm{\Delta }{v}_{\text{RE}}\left(\overline{\tau }\right)$ (43). The pluses highlight the laser powers used with parameter set ${\text{B}}_1$ and the border between under- and overdamping is highlighted (dashed white line).

Figure 12

Velocity-dependent population of the metastable state after $t_{\pi ,0}$ for different spatial distributions: (a) no intensity variations according to $\sigma \ll w_0$ and (b) averaged over the spatial intensity variations for an ion beam width $\sigma =w_0$. Considering finite laser linewidths, $\gamma =300\mathrm{kHz}$ (dashed line) is compared to $\gamma =0$ (solid line). The analytic approximation (47) (dotted line) matches the simulation results.

Figure 13

(a)–(d) Time evolution of the metastable-state population summed over a certain velocity width. (e)–(h) Time-dependent ratio $\mu$ (48) of the population transferred into the metastable state via the Raman transition to the whole transferred population, including spontaneous population transfer. Initial ion velocity widths (a), (b), (e), and (f) $\mathrm{\Delta }{v}_i=10\text{m}{\text{s}}^{-1}$ and (c), (d), (g), and (h) $\mathrm{\Delta }{v}_i=50\text{m}{\text{s}}^{-1}$ are considered. Two spatial intensity distributions are analyzed: (a), (c), (e), and (g) neglecting spatial variations according to $\sigma \ll w_0$ and (b), (d), (f), and (h) averaging over the spatial distributions for an ion beam width $\sigma =w_0$ for the parameter sets $\text{A}$ ($\times$), ${\text{B}}_1$ ($\circ$), and ${\text{B}}_2$ ($▵$), the latter scaled with a factor of 0.5 in (a)–(d). (a)–(d) The numerical results for $\mathrm{\Gamma }\ne 0$ and $\gamma =300\mathrm{kHz}$ (solid line) are well predicted by the analytic approximation for the full solution (47) (black dashed line). The analytic approximation (36) (dash-dotted line) gives the populations purely transferred via the Raman transition. The numerical solutions for $\gamma =0$ (dotted line) are given for the sake of completeness. (e)–(h) Considering a finite laser linewidth $\gamma =300\mathrm{kHz}$ (solid line) is compared to neglecting it $\gamma =0$ (dotted line).