Incoherent repumping scheme in the $^{88}$Sr$^{+}$ five-level manifold

TL;DR

The study addresses Doppler cooling of $^{88}$Sr$^{+}$ ions where metastable states prevent a closed two-level cycle. It combines precision spectroscopy with an $18$-level optical Bloch equation model driven by three lasers at $422$ nm, $1003$ nm, and $1033$ nm under a magnetic field to capture incoherent repumping dynamics. The results show fluorescence spectra with Lorentzian profiles but widths and amplitudes that cannot be captured by a two-level model, and identify an optimal repumping configuration that maximizes the scattering rate at resonance, approximately $R_{sc}\approx 6.3\times 10^{6}$ photons/s, with a width near $30~\mathrm{MHz}$. The findings provide quantitative guidance for multi-level Doppler cooling of Sr$^{+}$ and related ions, with broad implications for precision spectroscopy and quantum information experiments.

Abstract

Laser-cooled trapped ions are at the heart of modern quantum technologies and their cooling dynamics often deviate from the simplified two-level atom model. Doppler cooling of the $^{88}$Sr$^{+}$ ion involves several electronic levels and repumping channels that strongly influence fluorescence. In this work, we study a repumping scheme for the $^{88}$Sr$^{+}$ ion by combining precision single-ion spectroscopy with comprehensive numerical modeling based on optical Bloch equations including 18 Zeeman sublevels. We show that, although the observed fluorescence spectra retain a Lorentzian lineshape, their width and amplitude cannot be explained by a two-level atom description. Moreover, we find the optimal repumping conditions for maximizing the photon scattering rate.

Figures (11)

• Figure 1: The $^{88}\text{Sr}^{+}$ three-level manifold sketch with cooling and repumping lasers in the coherent repumping scheme. A $1092$ nm beam couples the $D_{3/2}$ and $P_{1/2}$ states with a Rabi frequency noted $\Omega_{1092}/2\pi$. The $422$ nm laser coupling the S and P states is referred as the probe laser of Rabi frequency $\Omega_{422}/2\pi$. Wavy arrows indicate spontaneous emission channels.
• Figure 2: The $^{88}\text{Sr}^{+}$ five-level manifold sketch with cooling and repumping lasers in the incoherent repumping scheme. Two infrared lasers at $1003$ nm and $1033$ nm couple respectively the $D_{3/2}$ and $D_{5/2}$ to the $P_{3/2}$ state with Rabi frequencies noted $\Omega_{1003}/2\pi$ and $\Omega_{1033}/2\pi$ respectively. Wavy arrows indicate spontaneous emission channels
• Figure 3: Example of a fluorescence spectrum obtained by varying the 422 nm laser frequency. At resonance, the detected scattering rate is 12000 photons per second. The lorentzian full width at half-maximum is 30 MHz. The collection efficiency of our setup is $2\times10^{-3}$. The total scattering rate at resonance for this single ion is then $6.5\times10^{6}$ photons/s
• Figure 4: Electronic states populations calculated with the following parameters : $\Omega_{422}/2\pi = 11\,\text{MHz}$, $\Omega_{1003}/2\pi = 150\,\text{MHz}$, $\Omega_{1033}/2\pi = 250\,\text{MHz}$. The 1003 nm laser is at resonance, while the 1033 nm laser is +400 MHz detuned from resonance. The magnetic field amplitude is $4 \times 10^{-4}$ T.
• Figure 5: Fluorescence spectrum calculated with the following parameters : $\Omega_{422}/2\pi = 11\,\text{MHz}$, $\Omega_{1003}/2\pi = 150\,\text{MHz}$, $\Omega_{1033}/2\pi = 250\,\text{MHz}$. The 1003 nm laser is at resonance, while the 1033 nm laser is +400 MHz detuned from resonance. The magnetic field amplitude is $4 \times 10^{-4}$ T.