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arXiv:1212.4685 · Physical Review A

Dark-state suppression and optimization of laser cooling and fluorescence in a trapped alkaline-earth-metal single ion

Thomas Lindvall, Mikko Merimaa, Ilkka Juhani Tittonen, Alan A. Madej

Published Sep 4, 2012• 16 citations
arXivHTMLPDF

Abstract

We study the formation and destabilization of dark states in a single trapped ${}^{88}{\mathrm{Sr}}^{+}$ ion caused by the cooling and repumping laser fields required for Doppler cooling and fluorescence detection of the ion. By numerically solving the time-dependent density matrix equations for the eight-level system consisting of the sublevels of the $5s{\phantom{\rule{0.16em}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{S}_{1/2}$, $5p{\phantom{\rule{0.16em}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2}$, and $4d{\phantom{\rule{0.16em}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{D}_{3/2}$ states, we analyze the different types of dark states and how to prevent them in order to maximize the scattering rate, which is crucial for both the cooling and the detection of the ion. The influence of the laser linewidths and ion motion on the scattering rate and the dark resonances is studied. The calculations are then compared with experimental results obtained with an endcap ion trap system located at the National Research Council of Canada and found to be in good agreement. The results are applicable also to other alkaline-earth-metal ions and isotopes without hyperfine structure.

DOI
10.1103/physreva.86.033403

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We study the formation and destabilization of dark states in a single trapped 88Sr+{}^{88}{\mathrm{Sr}}^{+}88Sr+ ion caused by the cooling and repumping laser fields required for Doppler cooling and fluorescence detection of the ion. By numerically solving the time-dependent density matrix equations for the eight-level system consisting of the sublevels of the 5s2S1/25s{\phantom{\rule{0.16em}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{S}_{1/2}5s2S1/2​, 5p2P1/25p{\phantom{\rule{0.16em}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{P}_{1/2}5p2P1/2​, and 4d2D3/24d{\phantom{\rule{0.16em}{0ex}}}^{2}\phantom{\rule{-0.16em}{0ex}}{D}_{3/2}4d2D3/2​ states, we analyze the different types of dark states and how to prevent them in order to maximize the scattering rate, which is crucial for both the cooling and the detection of the ion. The influence of the laser linewidths and ion motion on the scattering rate and the dark resonances is studied. The calculations are then compared with experimental results obtained with an endcap ion trap system located at the National Research Council of Canada and found to be in good agreement. The results are applicable also to other alkaline-earth-metal ions and isotopes without hyperfine structure.
Many ion species commonly used for laser-cooled ion-trapping studies have a low-lying metastable 2D3/2{}^{2}{D}_{3/2}2D3/2​ state that can become populated due to spontaneous emission from the 2P1/2{}^{2}{P}_{1/2}2P1/2​ excited state. This requires a repumper laser to maintain the ion in the Doppler cooling cycle. Typically, the 2D3/2{}^{2}{D}_{3/2}2D3/2​ state, or some of its hyperfine components if the ion has nuclear spin, has a higher multiplicity than the upper state of the repumping transition. This can lead to dark states, which have to be destabilized by an external magnetic field or by modulating the polarization of the repumper laser. We propose using unpolarized, incoherent amplified spontaneous emission (ASE) to drive the repumping transition. An ASE source offers several advantages compared to a laser. It prevents the buildup of dark states without external polarization modulation even in zero magnetic field, it can drive multiple hyperfine transitions simultaneously, and it requires no frequency stabilization. These features make it very compact and robust, which is essential for the development of practical, transportable optical ion clocks. We construct a theoretical model for the ASE radiation, including the possibility of the source being partially polarized. Using 88Sr+{}^{88}{\mathrm{Sr}}^{+}88Sr+ as an example, the performance of the ASE source compared to a single-mode laser is analyzed by numerically solving the eight-level density-matrix equations for the involved energy levels. Finally, a reduced three-level system is derived, yielding a simple formula for the excited-state population and scattering rate, which can be used to optimize the experimental parameters. The required ASE power spectral density can be obtained with current technology.
Incoherent repumping scheme in the 88^{88}88Sr+^{+}+ five-level manifold
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^{88}88Sr+^{+}+ ion involves several electronic levels and repumping channels that strongly influence fluorescence. In this work, we study a repumping scheme for the 88^{88}88Sr+^{+}+ 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.
We demonstrate simple and robust methods for Doppler cooling and obtaining high fluorescence from trapped 43 Ca + ions at a magnetic field of 146 Gauss. This field gives access to a magnetic-field-independent 'atomic clock' qubit transition within the ground level hyperfine structure of the ion, but also causes the complex internal structure of the 64 states relevant to Doppler cooling to be spread over many times the atomic transition line-width. Using a time-dependent optical Bloch equation simulation of the system we develop a simple scheme to Doppler-cool the ion on a two-photon dark resonance, which is robust to typical experimental variations in laser intensities, detunings and polarizations. We experimentally demonstrate cooling to a temperature of 0.3 mK, slightly below the Doppler limit for the corresponding two-level system, and then use Raman sideband laser cooling to cool further to the ground states of the ion's radial motional modes. These methods will enable two-qubit entangling gates with this ion, which is one of the most promising qubits so far developed.
Single, rf-trapped ions find various applications ranging from metrology to quantum computation. High-resolution interrogation of an extremely weak transition under best observation conditions requires an ion almost at rest. To avoid line-broadening effects such as the second order Doppler effect or rf heating in the absence of laser cooling, excess micromotion has to be eliminated as far as possible. We have tested a novel method to probe the motional state of a confined three-level ion. Actually, dark resonances observed with two counterpropagating laser beams are very sensitive to the trapped ion's velocity. Their contrast increases with reduced micromotion and allows in this way to control the oscillation amplitude of the stored ion. The influence of different parameters such as laser linewidth and power has been investigated experimentally and numerically.
We develop a unified many-body theory of two-photon dark-state laser cooling, the workhorse for preparing trapped ions close to their motional quantum ground state. For ions with a ΛΛΛ level structure, driven by Raman lasers, we identify an ion-number-dependent crossover between weak and strong coupling where both the cooling rate and final temperature are simultaneously optimized. We obtain simple analytic results in both extremes: In the weak coupling limit, we show a Lorentzian spin-absorption spectrum determines the cooling rate and final occupation of the motional state, which are both independent of the number of ions. We also highlight the benefit of including an additional spin dependent force in this case. In the strong coupling regime, our theory reveals the role of collective dynamics arising from phonon exchange between dark and bright states, allowing us to explain the enhancement of the cooling rate with increasing ion number. Our analytic results agree closely with exact numerical simulations and provide experimentally accessible guidelines for optimizing cooling in large ion crystals, a key step toward scalable, high-fidelity trapped-ion quantum technologies.