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arXiv:2605.07942

Carrier Revival in Long Trapped-Ion Chains

Florian Egli, Chris Shanks, James Bounds, Jorge Moreno, Muhammad Thariq, Erdem Yilmaz, Theodor W. Hänsch, Thomas Udem, Akira Ozawa

physics.atom-phquant-phSubmitted May 8, 2026• Announced May 11, 2026• 0 citations
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Abstract

For a single trapped ion, the excitation spectrum of a narrow optical transition consists of a Doppler- and recoil-free carrier accompanied by motional sidebands, which are equally spaced by the trap secular frequency and lie under a Doppler-broadened envelope that is shifted by the photon recoil. Outside the Lamb-Dicke regime, the large photon recoil distributes the line strength across many sidebands and suppresses excitation of the carrier. With multiple ions, the motional spectrum becomes dense, and the carrier is further weakened. Here, we predict a counterintuitive revival effect: increasing the number of ions in a linear chain can restore strong carrier excitation even under trapping conditions far from the single-ion Lamb-Dicke regime. Using a quantum-mechanical model of the excitation dynamics in linear ion chains, we find that sufficiently long chains concentrate the spectrum into the carrier. This effect enables efficient excitation of light ions at short wavelengths. It may also benefit multi-ion optical clocks and mixed-species quantum-logic spectroscopy.

Comments
11 pages, 3 figures

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For a single trapped ion, the excitation spectrum of a narrow optical transition consists of a Doppler- and recoil-free carrier accompanied by motional sidebands, which are equally spaced by the trap secular frequency and lie under a Doppler-broadened envelope that is shifted by the photon recoil. Outside the Lamb-Dicke regime, the large photon recoil distributes the line strength across many sidebands and suppresses excitation of the carrier. With multiple ions, the motional spectrum becomes dense, and the carrier is further weakened. Here, we predict a counterintuitive revival effect: increasing the number of ions in a linear chain can restore strong carrier excitation even under trapping conditions far from the single-ion Lamb-Dicke regime. Using a quantum-mechanical model of the excitation dynamics in linear ion chains, we find that sufficiently long chains concentrate the spectrum into the carrier. This effect enables efficient excitation of light ions at short wavelengths. It may also benefit multi-ion optical clocks and mixed-species quantum-logic spectroscopy.
First order Doppler effects are usually ignored in laser driven trapped ions when the recoil frequency is much smaller than the trapping frequency (Lamb-Dicke regime). This means that the central, carrier excitation band is supposed to be unaffected by vibronic transitions in which the vibrational number changes. While this is strictly true in the Lamb-Dicke limit (infinitely tight confinement), the vibronic transitions do play a role in the Lamb-Dicke regime. In this paper we quantify the asymptotic behavior of their effect with respect to the Lamb-Dicke parameter. In particular, we give analytical expressions for the frequency shift, ``pulling'' or ``pushing,'' produced in the carrier absorption band by the vibronic transitions both for Rabi and Ramsey schemes. This shift is shown to be independent of the initial vibrational state.
The spectrum of resonance fluorescence of a single trapped and laser-cooled ion is studied theoretically. The quantum motion of the trapped particle manifests itself in the form of narrow motional sidebands in the fluorescence spectrum. For our calculations it is assumed that the ion is confined to dimensions much smaller than the optical wavelength (Lamb-Dicke limit) and the approach is valid for multilevel systems, general trapping potentials, and for both traveling-wave and standing-wave configurations. The motional sidebands in the spectrum have asymmetric amplitudes and this asymmetry is shown to depend on the ion energy, the detector position, and the choice of standing- or traveling-wave laser excitation.
The spectrum of resonance fluorescence of a single trapped and laser-cooled ion is studied theoretically. The quantum motion of the trapped particle manifests itself in the form of narrow motional sidebands in the fluorescence spectrum. For our calculations it is assumed that the ion is confined to dimensions much smaller than the optical wavelength (Lamb-Dicke limit) and the approach is valid for multilevel systems, general trapping potentials, and for both traveling-wave and standing-wave configurations. The motional sidebands in the spectrum have asymmetric amplitudes and this asymmetry is shown to depend on the ion energy, the detector position, and the choice of standing- or traveling-wave laser excitation.
We reflect on the prospect of exploiting the recoil associated with absorption and emission of photons to perform spectroscopy of a single molecular ion. For this recoil to be detectable, the molecular ion is sympathetically cooled by a laser-cooled atomic ion to near their common quantum-mechanical ground state within a trapping potential. More specifically, we present a general framework for simulating the expected photon recoil spectra in regimes where either the natural transition linewidth ${\mathrm{\ensuremath{\Gamma}}}_{t}$ of the molecular ion or the spectral width ${\mathrm{\ensuremath{\Gamma}}}_{L}$ of the exciting light source exceeds the motional frequencies of the two-ion system. To exemplify the framework, we present two complementary cases: spectroscopy of the broad $3\mathrm{s} ^{2}\mathrm{S}_{1/2}\ensuremath{-}3\mathrm{p} ^{2}\mathrm{P}_{3/2}$ electronic transition (${\mathrm{\ensuremath{\Gamma}}}_{t}/2\ensuremath{\pi}=41.8\phantom{\rule{0.16em}{0ex}}\mathrm{M}\mathrm{Hz}$) of a single 24Mg+^{24}\mathrm{Mg}^{+}24Mg+ ion at $\ensuremath{\lambda}=279.6$ nm by a narrow laser source $({\mathrm{\ensuremath{\Gamma}}}_{L}/2\ensuremath{\pi}\ensuremath{\lesssim}1\phantom{\rule{0.16em}{0ex}}\mathrm{MHz})$ and mid-infrared vibrational spectroscopy of the very narrow $|v=0,J=1\ensuremath{\rangle}\ensuremath{-}|{v}^{\ensuremath{'}}=1,{J}^{\ensuremath{'}}=0\ensuremath{\rangle}$ transition $({\mathrm{\ensuremath{\Gamma}}}_{t}/2\ensuremath{\pi}=2.50\phantom{\rule{0.16em}{0ex}}\mathrm{Hz})$ at $\ensuremath{\lambda}=6.17\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{m}$ in the $^{1}\mathrm{\ensuremath{\Sigma}}^{+}$ electronic ground state of 24MgH+^{24}\mathrm{MgH}^{+}24MgH+ by a broadband laser source $({\mathrm{\ensuremath{\Gamma}}}_{L}/2\ensuremath{\pi}\ensuremath{\gtrsim}50\phantom{\rule{0.16em}{0ex}}\mathrm{MHz})$. The atomic ion 24Mg+^{24}\mathrm{Mg}^{+}24Mg+ has been picked to introduce a simple system to make comparisons with experimental results while still capturing most of the physics involved in electronic excitations of molecular ions.
We study laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasicontinuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of dark states which may slow down the cooling is considerably reduced. We show that cooling to the ground state is also possible outside the Lamb-Dicke regime. We develop the model and use quantum Monte Carlo calculations for specific examples. We show that a rate equation treatment is a good approximation in all cases.
An experimental study of sideband laser cooling of trapped ions in miniature radio-frequency traps is presented. In+{\mathrm{In}}^{+}In+ ions are laser cooled by exciting the narrow intercombination line ${5s}^{2}{}^{1}{S}_{0}\ensuremath{\rightarrow}5s5p{}^{3}{P}_{1}$, which shows resolved sidebands due to the vibrational motion of the ions in the trap. The influence of the micromotion at the frequency of the oscillating trapping field is investigated. Heating processes through coupling with the trapping field occur for certain regions of negative laser detuning. A method of bichromatic sideband cooling is introduced that combines efficient cooling from high temperatures with the possibility of reaching low mean vibrational quantum numbers 〈n〉<1〈n〉<1〈n〉<1 by using narrow optical transitions. This technique allows recording of high-resolution excitation spectra of the cooling transition in the Lamb-Dicke regime and thus determination of 〈n〉〈n〉〈n〉. Single ions as well as two-ion Coulomb crystals have been cooled to temperatures below $100 \ensuremath{\mu}\mathrm{K}$, and population of the ground state of motion to more than 50% over long time scales is demonstrated. These values are close to the lower limit for the detection of vibrational excitation with the present method.