Ground-State Cooling of a Trapped Ion Using Long-Wavelength Radiation

We demonstrate ground-state cooling of a trapped ion using radio-frequency (rf) radiation. This is a powerful tool for the implementation of quantum operations, where rf or microwave radiation instead of lasers is used for motional quantum state engineering. We measure a mean phonon number of $\overline{n}=0.13(4)$ after sideband cooling, corresponding to a ground-state occupation probability of 88(7)%. After preparing in the vibrational ground state, we demonstrate motional state engineering by driving Rabi oscillations between the $|n=0⟩$ and $|n=1⟩$ Fock states. We also use the ability to ground-state cool to accurately measure the motional heating rate and report a reduction by almost 2 orders of magnitude compared with our previously measured result, which we attribute to carefully eliminating sources of electrical noise in the system.

Figure 1

Population in $F=1$ after (a) a frequency scan over the red sideband and (b) a frequency scan over the blue sideband, both after sideband cooling. The red and blue lines are the result of a numerical simulation of the system where we have set ${\nu }_z/2\pi =426.7\mathrm{kHz}$, ${\mathrm{\Omega }}_{\mathrm{dr}}/2\pi =32\mathrm{kHz}$, and $\mathrm{\Omega }/2\pi =61.2\mathrm{kHz}$, and used a probe pulse time of $1270\mu \mathrm{s}$ with $\overline{n}$ as the only free parameter. A least-squares fit gives $\overline{n}=0.13(4)$, corresponding to a ground-state occupation probability of $p_0=0.88(7)$.

Figure 2

To measure the heating rate $\dot{\overline{n}}$ a variable delay is introduced after sideband cooling. For each of the three delay times a scan over the red and blue sidebands is taken, as in Fig. 1, allowing $\overline{n}$ to be determined. The black line is a fit to the data, which gives a heating rate of $\dot{\overline{n}}=41(7){\mathrm{s}}^{-1}$, or one phonon in 24 ms, for the axial secular frequency of ${\nu }_z/2\pi =426.7(1)\mathrm{kHz}$.

Figure 3

Population in $F=1$ after ground-state cooling and applying an rf pulse for a variable time, resonant with the blue (circles) and red (triangles) motional sidebands of the $|0^{\prime }⟩\leftrightarrow |D⟩$ transition. Coherent Rabi oscillations between $|0^{\prime },n=0⟩$ and $|D,n=1⟩$ can be observed lasting for over 10 ms, which is an order of magnitude longer than gate times achievable in this setup. The blue and red solid lines are the results of integrating a master equation describing the dynamics of a two-level atom coupled by either the red or blue sideband to a harmonic oscillator mode undergoing heating. The heating rate and initial thermal state are set to the measured values of $\dot{\overline{n}}=41{\mathrm{s}}^{-1}$ and $\overline{n}=0.13$, respectively, and the ground-state sideband Rabi frequency is set to ${\eta }_{\text{eff}}\mathrm{\Omega }/2\pi =0.35\mathrm{kHz}$.