Sympathetic Ground State Cooling and Time-Dilation Shifts in an Al27+ Optical Clock

Abstract

We report on Raman sideband cooling of 25 Mg + to sympathetically cool the secular modes of motion in a 25 Mg + -27 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 ± 0.1) × 10 −18 for an 27 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 Al + clocks. Trapped and laser-cooled ions are useful for many applications in quantum information processing and quantum metrology because of their isolation from the ambient environment and mutual Coulomb interaction [1][2][3][4][5][6]. The Coulomb interaction establishes normal modes of motion, called secular modes, which enable information exchange and entanglement between ions. For many operations in quantum information processing and metrology, these secular modes should ideally be prepared in their ground state. For example, in state-of-the-art trapped-ion optical clocks [7], uncertainty in the Doppler shift due to the residual excitation of these modes is a dominant contribution to the total clock uncertainty [8][9][10].All trapped-ion optical clocks to date have been operated with the ions' motion near the Doppler cooling limit [7][8][9][10]. For clocks based on quantum-logic spectroscopy of the 1 S 0 ↔ 3 P 0 transition in 27 Al + [11], the smallest uncertainty has been achieved by performing continuous sympathetic Doppler cooling on the logic ion ( 25 Mg + ) during the clock interrogation [8]. Due to the difficulty of performing accurate temperature measurements of trapped ions near the Doppler cooling limit, the uncertainty of the secular motion temperature was limited to 30 % [8].To reduce the secular motion time-dilation shift and its uncertainty, sub-Doppler cooling techniques can be employed [12][13][14][15][16][17]. These schemes have not previously been implemented in ion-based clock experiments due to high ambient motional heating rates, the need for extra cooling laser beams, and the difficulty of characterizing the resulting motional state distribution, which can be nonthermal [8,10,18]. For example, in sideband cooling, a non-thermal distribution can result from zero-crossings in the cooling transition Rabi rate as a function of the Fock state [19,20] and from cooling durations insufficient to reach equilibrium. Although sideband cooling can reach extremely low energies, a small non-thermal component of the distribution may contribute more than 90 % of the total energy [21], rendering the temperature measurement technique using motional sideband ratios unsuitable for determining the energy [12,13].Here we demonstrate sympathetic cooling of a 25 Mg + -27 ...

Trapped Ions Stopped Cold

“…This is accomplished with Raman sideband cooling, selected for its low cooling limit and experimental convenience. The ion trap used here has motional heating rates less than 12 quanta=s [21], 2 orders of magnitude lower than those achieved in previous 27 Al þ clocks [8]. The evolution of the Fock-state distribution during the cooling process is investigated by using rateequation simulations, which is verified by comparison with experimental measurements.…”

“…For example, in sideband cooling, a nonthermal distribution can result from zero crossings in the cooling transition Rabi rate as a function of the Fock state [19,20] and from cooling durations insufficient to reach equilibrium. Although sideband cooling can reach extremely low energies, a small nonthermal component of the distribution may contribute more than 90% of the total energy [21], rendering the temperature measurement technique using motional sideband ratios unsuitable for determining the energy [12,13].…”

“…Numerical simulations indicate that the second order RSB pulses are significantly more efficient at reducing populations in high Fock states, depending on the Lamb-Dicke parameter and the time for repumping during each cooling cycle. Multiple second order RSB cooling pulses are applied before the first order RSB pulses for the two axial modes in our experiment [21]. After sideband cooling, all RSB signals nearly vanish, indicating the two-ion pair is close to the 3D motional ground state [21].…”

“…Along each axis there are two motional modes, the "common" (COM) mode where mode vectors of the two ions are in the same direction and the "stretch" (STR) mode where they are opposed. Our trap is operated with axial mode frequencies of 2.7 MHz (COM) and 4.8 MHz (STR) and transverse mode frequencies in the range of 5 to 7.5 MHz [21]. We begin each experiment with 2 ms of Doppler cooling on the 25 …”

“…After a RSB cooling transition, 25 Mg þ is in j↑i, which is then optically pumped to j↓i by applying a pulse sequence with three pulses that drive the j↑i to a j 2 P 3=2 ; F ¼ 3; m F ¼ −3i transition, between which are inserted two Raman carrier pulses that drive the j 2 S 1=2 ; F ¼ 3; m F ¼ −2i to an j↑i transition by applying RR and BR-co simultaneously. The 25 Mg þ atomic state is determined at the end of each experiment by collecting photons scattered from the cycling transition and converting the photon count histogram to the transition probability [21].…”