Double-Electromagnetically-Induced-Transparency Ground-State Cooling of Stationary Two-Dimensional Ion Crystals

We theoretically and experimentally investigate double-electromagnetically-induced transparency (double-EIT) cooling of two-dimensional ion crystals confined in a Paul trap. The double-EIT ground-state cooling is observed for ${{}^{171}\mathrm{Yb}}^{+}$ ions with a clock state, for which EIT cooling has not been realized like many other ions with a simple $\mathrm{\Lambda }$ scheme. A cooling rate of $\dot{\overline{n}}=34(\pm 1.8){\mathrm{ms}}^{-1}$ and a cooling limit of $\overline{n}=0.06(\pm 0.059)$ are observed for a single ion. The measured cooling rate and limit are consistent with theoretical predictions. We apply double-EIT cooling to the transverse modes of two-dimensional (2D) crystals with up to 12 ions. In our 2D crystals, the micromotion and the transverse mode directions are perpendicular, which makes them decoupled. Therefore, the cooling on transverse modes is not disturbed by micromotion, which is confirmed in our experiment. For the center of mass mode of a 12-ion crystal, we observe a cooling rate and a cooling limit that are consistent with those of a single ion, including heating rates proportional to the number of ions. This method can be extended to other hyperfine qubits, and near ground-state cooling of stationary 2D crystals with large numbers of ions may advance the field of quantum information sciences.

Figure 1

(a) Relevant energy levels of ${{}^{171}\mathrm{Yb}}^{+}$ for EIT cooling. (b) Fano-like profile of double EIT. The spectrum is calculated by steady-state solution of the master equation or by scattering amplitude [43]. In the simulation, we set ${\mathrm{\Delta }}_d/(2\pi )=55.6\mathrm{MHz}$, ${\delta }_B/(2\pi )=4.6\mathrm{MHz}$, ${\mathrm{\Omega }}_{\sigma ,\pm }/(2\pi )=17\mathrm{MHz}$, ${\mathrm{\Omega }}_{\pi }/(2\pi )=4\mathrm{MHz}$, and $\nu /(2\pi )=1.5\mathrm{MHz}$, where $\nu$ is the frequency of the mode we intend to cool down. The bottom one shows the spectrum in a large range, while the red lines represent theoretical predictions of the positions of dressed states. The top one shows the spectrum around the peak we use for cooling, and the blue (red) line represents the position of the motional sideband, while the dark line represents carrier transition. The red sideband has lower energy, which corresponds to a higher detuning. (c) Optical configuration. The EIT beam is first separated from the Doppler cooling beam with a 14 GHz sideband and then is split into the driving beam and the probe beam by a polarizing beam splitter (PBS). The relative detuning ${\mathrm{\Delta }}_d-{\mathrm{\Delta }}_p$ is controlled by two AOMs acted on the driving beam. The first-order diffraction of the 270 MHz AOM and the negative first order of the variable AOM are used. The net propagating vectors $\mathrm{\Delta }k$ of both the EIT and the Raman beams are along the direction of the transverse mode. A quarter-wave plate (QWP) is used to adjust the polarization of the driving beam.

Figure 2

(a) Experimental sequence for exploring EIT cooling of a single trapped ion. (b) Cooling dynamics for the transverse mode along the $y$ axis. Red points are experimental data obtained by fitting blue-sideband transitions shown in (c) and (d). Error bars denote fitting errors. The black line is an exponential fit. The horizontal dashed line indicates $1/e$ of initial phonon number. (c),(d) The blue-sideband transition after (c) Doppler cooling and (d) EIT cooling of $200\mu \mathrm{s}$. (e) Average phonon number $\overline{n}$ at the end of double-EIT cooling versus the relative detuning between the probe beam and the driving beam. The black line is the numerical simulation result obtained by solving the master equation [43].

Figure 3

The final mean phonon numbers (circular points) and the cooling rates (square points) are plotted versus the power of (a) the driving beam and (b) the probe beam. Error bars denote the fitting uncertainties of blue-sideband evolutions, similar to Figs. 2 and 2. Solid lines are numerical simulation results obtained by solving the master equation [43].

Figure 4

(a),(b) Blue-sideband (blue curve) and red-sideband (red curve) spectrum after (a) Doppler cooling and (b) EIT cooling. The vertical axis represents the count globally collected by the photomultiplier tube (PMT). The horizontal axis ${\mu }_R={\omega }_R-{\omega }_0$ is the detuning of the Raman transition from the qubit transition. Vertical lines indicate the locations of 12 motional modes perpendicular to the 2D-crystal plane. (c) Pulse sequence for the ODF thermometry. (d) ODF spectrum with different average phonon numbers. The dashed black line indicates the position we choose for the cooling rate measurement. (e) Cooling dynamics for a single ion (green) and a 2D crystal with 12 ions (red). The dots are experimental data. The error bars represent the standard deviation induced by the quantum projection noise. Solid lines are fitting curves by exponential decay functions.