Experimental and theoretical investigation of a multimode cooling scheme using multiple electromagnetically-induced-transparency resonances

We introduce and demonstrate double-bright electromagnetically-induced-transparency (D-EIT) cooling as an extension to EIT cooling. By involving an additional ground state, two bright states can be shifted individually into resonance for cooling of motional modes of frequencies that may be separated by more than the width of a single EIT cooling resonance. This allows three-dimensional ground-state cooling of a ${}^{40}\mathrm{Ca}^{+}$ ion trapped in a linear Paul trap with a single cooling pulse. Measured cooling rates and steady-state mean motional quantum numbers for this D-EIT cooling are compared with those of standard EIT cooling as well as concatenated standard EIT cooling pulses for multimode cooling. Experimental results are compared to full-density matrix calculations. We observe a failure of the theoretical description within the Lamb-Dicke regime that can be overcome by a time-dependent rate theory. Limitations of the different cooling techniques and possible extensions to multi-ion crystals are discussed.

Figure 1

(a) Schematic of the beam setup with respect to the trap and magnetic field $B$. (b) Energy levels of ${}^{40}\mathrm{Ca}^{+}$. (c) Relevant levels for single EIT cooling. (d) Relevant levels for D-EIT. In panels (c) and (d), the pump beam(s) are drawn with thicker lines.

Figure 2

Evolution of $\overline{n}\left(t\right)$ in radial and axial direction, measured after a delay time $t$ after D-EIT cooling, during which all lasers have been switched off. The background heating rate is derived from a linear fit of $\overline{n}\left(t\right)$ as $(45.0\pm 2.8){\mathrm{s}}^{-1}$ in the axial direction and $(28.2\pm 3.4){\mathrm{s}}^{-1}$ in the radial direction.

Figure 3

Fluorescence plot of dark resonances between $866\text{−}\mathrm{nm}$ laser and $397\text{−}\mathrm{nm}\pi$ beam, where the latter is red detuned by $21.34\mathrm{MHz}$ from the unperturbed ${}^{2}P_{1/2}$ resonance. Dashed lines indicate theoretically expected frequencies of dark resonances for the given polarizations and magnetic field.

Figure 4

Typical dependency of $\overline{n}$ on cooling time, here for the axial direction after D-EIT cooling at $\mathrm{\Delta }=3\mathrm{\Gamma }$. The solid line shows the exponential fit. The horizontal dashed line indicates where the exponential fit approaches the fitted $\overline{n}$ to within $|n_{\infty }-\overline{n}\left(t\right)|\le 0.005$, and the dash-dotted horizontal line represents the resulting $n_{\mathrm{ss}}$, which is calculated by averaging over all $\overline{n}\left(t\right)$ after the fit crosses below the dashed line.

Figure 5

Cross section through the center of the ion trap, showing the orientation of radial modes with respect to rf electrodes, schematically represented as gray shaded areas.

Figure 6

Simulated scattering rate during D-EIT cooling versus detuning $\mathrm{\Delta }$ from the virtual level $P_{+}$. Vertical straight lines indicate motional frequencies. The inset shows the single EIT spectrum in the respective direction optimized for axial cooling. D-EIT has a bright state at both radial and axial frequency, while for single EIT there is only one bright state. The position of the radial (axial) bright state is dominated by the Rabi frequency of the $397\text{−}\mathrm{nm}\sigma$ ($866\text{−}\mathrm{nm}{\sigma }^{-}$) beam.

Figure 7

Time-dependent rate $R\left(t,{\overline{n}}_0\right)$ in units of ${10}^3$ phonon ${\mathrm{s}}^{-1}$ for axial single EIT at $\mathrm{\Delta }=3\mathrm{\Gamma }$ as a function of the initial occupation number ${\overline{n}}_0$. The Lamb-Dicke prediction for the rate corresponds to $52.2\times {10}^3$ phonon ${\mathrm{s}}^{-1}$, whereas the measured value is $38.2\times {10}^3$ phonon ${\mathrm{s}}^{-1}$.

Figure 8

False-color map of the residual axial RSB excitation, depending on the Rabi frequencies of the pump and probe cooling beams after axial single EIT cooling. Areas of low RSB excitation correspond to pump-probe Rabi frequency combinations leading to low ${\overline{n}}_{\mathrm{a}}$. The horizontal line indicates the pump Rabi frequency along which the measurements shown in Fig. 9 were taken. A constant combined Stark shift of the bright state corresponding to the axial trap frequency due to the pump and probe lasers is given the by red dashed line.

Figure 9

Dependency of $R_{\mathrm{a}}$ and $n_{\mathrm{ss},\mathrm{a}}$ on the probe Rabi frequency after axial single EIT cooling at $\mathrm{\Delta }=3.4\mathrm{\Gamma }$, while the pump Rabi frequency is kept fixed at $14.6\mathrm{MHz}$.

Figure 10

$\overline{n}\left(t\right)$ of the radial (black circles) and the axial (blue crosses) modes during single EIT cooling of the axial mode at $\mathrm{\Delta }=3.4\mathrm{\Gamma },{\mathrm{\Omega }}_{\pi }=3.8\mathrm{MHz}$, ${\mathrm{\Omega }}_{\sigma }=11.4\mathrm{MHz}$, of the radial mode (black circles) and axial mode (blue crosses). The dashed and dotted lines represent exponential decay functions using the simulation results. Both curves employ the same cooling rate of $439{\mathrm{s}}^{-1}$, illustrating that the time to reach $n_{\mathrm{ss},\mathrm{ax}}$ is limited by the radial cooling rate. Radial mode: $n_{\mathrm{ss}}=1.36$, initial temperature $\overline{n}=10$. Axial mode: $n_{\mathrm{ss}}=0.05$, initial temperature of $\overline{n}=0.15$.

Figure 11

Cooling rates and equilibrium $\overline{n}$ for EIT and D-EIT as a function of the blue detuning $\mathrm{\Delta }$. (a) Radial modes. (b) Axial mode. The top panels display the cooling rates and the bottom panels show the $n_{\mathrm{ss}}$. The solid and dashed lines are quartic polynomials fitted to the data points to guide the eye. The error bars incorporate systematic and statistical uncertainties.

Figure 12

Temperature of the precooled mode during the second cooling pulse of 3D single EIT cooling. For sufficiently long cooling times of the second cooling pulse, both precooled modes will approach $n_{\mathrm{ss}}\approx 1.3$.

Figure 13

Axial cooling rates for D-EIT as a function of the blue detuning $\mathrm{\Delta }$. Compared are the measured rates with the simulated values obtained by the full master equation approach with limited Fock states and the Lamb-Dicke approximation. The dash-dotted line indicates the limit for the validity of the Lamb-Dicke approximation, see Eq. (12), the background gradient represents the gradual failure of the Lamb-Dicke approximation.