Sympathetic Ground State Cooling and Time-Dilation Shifts in an ${{}^{27}\mathrm{Al}}^{+}$ Optical Clock

We report on Raman sideband cooling of ${{}^{25}\mathrm{Mg}}^{+}$ to sympathetically cool the secular modes of motion in a ${{}^{25}\mathrm{Mg}}^{+}\text{−}{{}^{27}\mathrm{Al}}^{+}$ two-ion pair to near the three-dimensional (3D) ground state. The evolution of the Fock-state distribution during the cooling process is studied using a rate-equation simulation, and various heating sources that limit the efficiency of 3D sideband cooling in our system are discussed. We characterize the residual energy and heating rates of all of the secular modes of motion and estimate a secular motion time-dilation shift of $-(1.9\pm 0.1)\times {10}^{-18}$ for an ${{}^{27}\mathrm{Al}}^{+}$ clock at a typical clock probe duration of 150 ms. This is a 50-fold reduction in the secular motion time-dilation shift uncertainty in comparison with previous ${{}^{27}\mathrm{Al}}^{+}$ clocks.

Figure 1

(a) Geometry of laser beams. (b) ${{}^{25}\mathrm{Mg}}^{+}$ energy level diagram (not to scale). Quantum states $|F,m_F⟩$ are labeled by the total angular momentum $F$ and the projection along the quantization direction $m_F$. The Raman laser detuning from the transition $|\downarrow (\uparrow )⟩\leftrightarrow |{{}^{2}P}_{3/2},F=3,m_F=-3⟩$ is approximately 50 GHz, and the magnetic field is about 1 G. $D$, Doppler cooling; $R$, repumping; RR, red Raman; BR-$a$, axial blue Raman; BR-$t$, transverse blue Raman; BR-co, copropagating blue Raman. (Clock) ${{}^{27}\mathrm{Al}}^{+}$ laser beam driving the ${{}^{1}S}_0\leftrightarrow {{}^{3}P}_0$ clock transition.

Figure 2

Rabi oscillations of the first RSB (red, circle), the first BSB (blue, down triangle), and carrier (green, up triangle). Raman transitions of the axial mode for a single ${{}^{25}\mathrm{Mg}}^{+}$ after different numbers of pulses in the sideband cooling sequence. Carrier Rabi-oscillation curves are shifted by $-0.5$ for visibility. The cooling sequence consists of 17 second order RSB pulses followed by eight first order RSB pulses to prepare the ion near the motional ground state. Solid lines are given by the rate-equation simulation without any free parameters [21]. The simulated population evolution of the first ten Fock states during the cooling process are shown in the insets. We include 150 Fock states in the simulation.

Figure 3

Red-sideband Rabi oscillations on the $|\downarrow ⟩\to |\uparrow ⟩$ transition of ${{}^{25}\mathrm{Mg}}^{+}$ for the six ${{}^{25}\mathrm{Mg}}^{+}\text{−}{{}^{27}\mathrm{Al}}^{+}$ motional modes after Raman sideband cooling. The blue data points are the average of about 50 000 experiments, and the error bars are the standard deviation of the mean. Solid line, double-thermal distribution fit; dotted line, single-thermal distribution fit; dashed line, rate-equation simulation. The red shaded regions represent the range of the off-resonant carrier transitions of the double-thermal distribution fit, which is significant for the $z$-COM mode, but not the other motional modes [21]. The insets represent the average occupation numbers from fits and the simulation. Diamond, rate-equation simulation; circle, single-thermal distribution fit; square, double-thermal distribution fit. Circles overlay diamonds to within less than the size of the symbol. The green error bars represent the experimental uncertainties of average occupation numbers after sideband cooling. The upper bound of energy is given by the 95% confidence interval of the double-thermal distribution fit.

Figure 4

(a) Rabi oscillation of the ${{}^{27}\mathrm{Al}}^{+}{{}^{1}S}_0\leftrightarrow {{}^{3\hspace{0.17em}\hspace{0.17em}}P}_0$ clock transition. Red circles, continuous sympathetic Doppler cooling, ${\mathrm{\Omega }}_{\text{Doppler}}=2\pi \times 1.126(3)\mathrm{kHz}$; blue squares, 3D sideband cooling, ${\mathrm{\Omega }}_{\mathrm{sc}}=2\pi \times 1.258(6)\mathrm{kHz}$. Lines are from fitting to an exponential decaying sinusoidal function. (b) Comparison of the secular motion time-dilation shifts in two different clock operating conditions. The solid line and the dash-dotted line represent the fractional frequency shifts, while the shaded regions are uncertainties. For a typical clock interrogation time of 150 ms, the secular motion time-dilation shift of a sideband cooled optical clock is $-(1.9\pm 0.1)\times {10}^{-18}$. The uncertainty of a Doppler cooled optical clock is the sum of the secular motion time-dilation shift, $-(16.3\pm 5.0)\times 10^{-18}$, and the ac Stark shift from the Doppler cooling light, $-(3.6\pm 1.5)\times 10^{-18}$ [8].