Zeeman Degenerate Sideband Cooling in $^{176}$Lu$^+$

TL;DR

The paper introduces degenerate Raman sideband cooling (DRSC) as a fast, robust method to prepare trapped ions in the motional ground state by coupling neighboring Zeeman states within a fixed hyperfine level via a two-photon Raman transition. DRSC enables removal of multiple vibrational quanta in a single pulse, demonstrated in $^{176}$Lu$^+$ where an initial mean occupancy $ar{n}$ of $6.9$ is reduced to $0.118$ with 10 pulses and further to $0.0129$ with Raman Dark Preparation. A theoretical framework based on a population-transfer matrix $W(t)$ and analysis of $F=7$ and $F=8$ manifolds is supported by ab initio simulations that agree with experimental results. The authors also propose a heuristic protocol that achieves near-optimal cooling with reduced design complexity and quantify heating contributions from Raman and optical pumping. Overall, DRSC provides a scalable pathway to rapid, high-fidelity ground-state preparation for multi-level ions with broad relevance to optical clocks and quantum information processing.

Abstract

We explore degenerate Raman sideband cooling in which neighboring Zeeman states of a fixed hyperfine level are coupled via a two-photon Raman transition. The degenerate coupling between $|F,m_F\rangle\rightarrow |F,m_F-1\rangle$ facilitates the removal of multiple motional quanta in a single cycle. This method greatly reduces the number of cooling cycles required to reach the ground state compared to traditional sideband cooling. We show that near ground state cooling can be achieved with a pulse number as low as $\bar{n}$ where $\bar{n}$ is the average phonon number in the initial thermal state. We demonstrate proof-of-concept in $^{176}\mathrm{Lu}^+$ by coupling neighboring Zeeman levels on the motional sideband for the $F=7$ hyperfine level in $^3D_1$. Starting from a thermal distribution with an average phonon number of 6, we demonstrate near ground-state cooling with $\sim10$ pulses. A theoretical description is given that applies to any $F$ level and demonstrates how effective this approach can be.

Figures (9)

• Figure 1: Schematic of Raman transitions for $F=8$ (a) and $F=7$ (b). $\pi$ (red) and either $\sigma^+$ or $\sigma^-$ (orange) polarized fields couple adjacent $m_F$ states while lowering $n$.
• Figure 2: Graphical representation of the first $50\times50$ elements of the population transfer matrix. (a) through (d) depict the population transfer matrices for pulse durations $t = [0.2,\, 0.4, \,0.6 ,\,0.8] \times T_f$.
• Figure 3: Evolution of the motional state distribution under a fixed DRSC pulse sequence from an initial thermal distribution with $\bar{n}=15$. (a) DRSC with F=7; colored lines show the distribution after every 5 pulses (15 pulses in total). (b) DRSC with F=8; colored lines show the distribution after every 2 pulses (7 pulses in total). (c) Two level SC; colored lines show the distribution after every 10 pulses (72 pulses in total). The dashed lines represent a suppressed thermal distribution fit to the asymptote of the final distributions.
• Figure 4: (a) Relative Raman coupling between adjacent $m_F$ states in $F=8$ and $F=7$ manifold. (b) Left (right) panels show the evolution of the pure state $\ket{-8,8,n=20}$ ($\ket{7,0,n=20}$) for a single cooling pulse in the $F=8$ ($F=7$) scheme. Pulse times $t$ are scaled by the $\pi$ time of the transition $\ket{8,-8,1}$ to $\ket{8,-7,0}$ ($\ket{7,0,1}$ to $\ket{7,-1,0}$).
• Figure 5: (a) atomic-level structure of $^{176}$Lu$^+$ showing the wavelengths of repump (blue), cooling/detection (red), and clock (orange) transitions used. (b) and (c) ion trap geometry with laser orientations and polarizations, where relevant.