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Journal of Experimental and Theoretical Physics Letters

Determination of the Heating Rate and Temperature of an Ion Chain in a Linear Paul Trap by the Dephasing of Rabi Oscillations

Nikita V. Semenin, A. S. Borisenko, Ilia V. Zalivako, Ilya A. Semerikov, Mikhail A. Aksenov, Ksenia Yu. Khabarova, N. Kolachevsky

Published Jul 1, 2022• 13 citations
Publisher

Abstract

The optimization of the parameters of laser cooling and the investigation of the heating rate in ion traps require the measurement of the temperature of ion chains for which the Lamb–Dicke regime is satisfied. A novel method based on the investigation of the dynamics of Rabi oscillations at a narrow electron transition in an individual ion of the chain has been suggested for such measurement. An analytical expression for the population of the upper state as a function of the excitation time is derived taking into account the thermal distribution of phonons over the vibrational modes of a chain with an arbitrary number of ions. The method is tested experimentally for the chain of five $$^{{171}}{\text{Y}}{{{\text{b}}}^{ + }}$$ ions using the quadrupole transition at 435 nm, as well as for a single ion. The heating rate measured for the axial vibrational mode in the implemented trap is 8 × 10 3 phonons/s.

DOI
10.1134/s0021364022601099

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Optical Spectroscopy and Measurement of Ion Heating Rates in a Planar Trap

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The experiment including the trapping of a single $$^{{171}}{\text{Y}}{{{\text{b}}}^{ + }}$$ ion into a planar Paul trap, spectroscopy of the quadrupole $${{{\text{|}}}^{2}}{{S}_{{1/2}}},F = 0\rangle \leftrightarrow {{{\text{|}}}^{2}}{{D}_{{3/2}}},F = 2\rangle $$ transition, observation of Rabi oscillations at this transition, and measurement of secular frequencies has been reported. The ion heating rates have been determined to be $$\bar {n}$$ = (840 ± 100) s –1 . This result is an important step towards a scalable ion quantum computer.

Journal of Experimental and Theoretical Physics Letters•Jul 1, 2026•0 citations
The optimization of the parameters of laser cooling and the investigation of the heating rate in ion traps require the measurement of the temperature of ion chains for which the Lamb–Dicke regime is satisfied. A novel method based on the investigation of the dynamics of Rabi oscillations at a narrow electron transition in an individual ion of the chain has been suggested for such measurement. An analytical expression for the population of the upper state as a function of the excitation time is derived taking into account the thermal distribution of phonons over the vibrational modes of a chain with an arbitrary number of ions. The method is tested experimentally for the chain of five 171Yb+^{{171}}{\text{Y}}{{{\text{b}}}^{ + }}171Yb+ ions using the quadrupole transition at 435 nm, as well as for a single ion. The heating rate measured for the axial vibrational mode in the implemented trap is 8 × 10 3 phonons/s.
Doppler cooling with lasers is essential to ions' trapping and also a preliminary step towards achievement of ultracold ions. Due to lack of effective tools, experimentally monitoring the ions' temperature and the laser-ion coupling is difficult in Doppler cooling. Here we analytically explore the Doppler cooling process of trapped ions, exemplified by 40^{40}40Ca+^{+}+, by solving the friction coefficient in the Doppler cooling with respect to a thermal bath, particularly, to a bath with large heating rate. We show four regions for cooling and heating induced by the three-level electromagnetically induced transparency and propose a practical method for measuring the Rabi frequency by the Doppler cooling window. In addition, the final temperature of the laser-cooled ions can be obtained analytically in the case of a weak thermal bath, whereas for the strong thermal bath this requires numerical treatment due to involvement of the Doppler shift and large Lamb-Dicke parameter. Our analytic results would help for understanding many experimental observations, such as configuration phase transition, phonon laser and thermodynamics regarding hot trapped ions.
Doppler cooling with lasers is essential to ion trapping and is also a preliminary step toward achieving ultracold ions. It requires substantial technological effort to experimentally monitor the ion temperature in the cooling process. Based on the simplified three-level $\mathrm{\ensuremath{\Lambda}}$ configuration, we analytically explore an one-dimensional Doppler cooling process of trapped ions, exemplified by 40Ca+{}^{40}{\mathrm{Ca}}^{+}40Ca+, by solving the friction coefficient in Doppler cooling with respect to a thermal bath; in particular, for a bath with large heating rate. We propose a practical method for measuring the Rabi frequency by the Doppler cooling window, which helps for monitoring the final temperature of the Doppler-cooled ions. With our approach, the obtained results are analytical in the case of a weak thermal bath, whereas for the strong thermal bath numerical treatment is partially needed due to the involvement of the Doppler shift and a large Lamb-Dicke parameter. Our analytic treatments are applied to understand many experimental observations, such as configuration phase transition, phonon lasers, and thermodynamics regarding hot trapped ions, and could help for further our understanding three-dimensional Doppler cooling in realistic ion-trap experiments.
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.
A single 40Ca+{}^{40}{\mathrm{Ca}}^{+}40Ca+ ion is confined in a linear Paul trap and Doppler-cooled on the S1/2{S}_{1/2}S1/2​ to P1/2{P}_{1/2}P1/2​ dipole transition. Then the narrow quadrupole S1/2{S}_{1/2}S1/2​ to D5/2{D}_{5/2}D5/2​ transition at 729 nm is probed. The observed spectrum is interpreted in terms of the Zeeman substructure superimposed with oscillation sidebands due to the harmonic motion in the trap. The height of the motional sidebands provides a sensitive method to determine the ion's temperature and thus allows us to test sub-Doppler laser cooling schemes needed for quantum state preparation and quantum computation. We also observe the dynamics induced by Rabi oscillations on a carrier transition and interpret it in terms of the thermal state which is reached after Doppler cooling.
Abstract We study heating of motional modes of a single ion and of extended ion crystals trapped in a linear radio frequency (rf) Paul trap with a precision of Δ n ̄ ̇ ≈ 0.1 phonons s−1. Single-ion axial and radial heating rates are consistent and electric field noise has been stable over the course of four years. At a secular frequency of ω sec = 2π × 620 kHz, we measure n ̄ ̇ = 0.56 ( 6 ) phonons s−1 per ion for the center-of-mass (com) mode of linear chains of up to 11 ions and observe no significant heating of the out-of-phase (oop) modes. By displacing the ions away from the nodal line, inducing excess micromotion, rf noise heats the com mode quadratically as a function of radial displacement r by n ̄ ̇ ( r ) / r 2 = 0.89 ( 4 ) phonons s−1 μm−2 per ion, while the oop modes are protected from rf-noise induced heating in linear chains. By changing the quality factor of the resonant rf circuit from Q = 542 to Q = 204, we observe an increase of rf noise by a factor of up to 3. We show that the rf-noise induced heating of motional modes of extended crystals also depends on the symmetry of the crystal and of the mode itself. As an example, we consider several 2D and 3D crystal configurations. Heating rates of up to 500 ph s−1 are observed for individual modes, giving rise to a total kinetic energy increase and thus a fractional time dilation shift of up to −0.3 × 10−18 s−1 of the total system. In addition, we detail how the excitation probability of the individual ions is reduced and decoherence is increased due to the Debye–Waller effect.
The experiment including the trapping of a single 171Yb+^{{171}}{\text{Y}}{{{\text{b}}}^{ + }}171Yb+ ion into a planar Paul trap, spectroscopy of the quadrupole |2S1/2,F=0⟩↔|2D3/2,F=2⟩{{{\text{|}}}^{2}}{{S}_{{1/2}}},F = 0\rangle \leftrightarrow {{{\text{|}}}^{2}}{{D}_{{3/2}}},F = 2\rangle |2S1/2​,F=0⟩↔|2D3/2​,F=2⟩ transition, observation of Rabi oscillations at this transition, and measurement of secular frequencies has been reported. The ion heating rates have been determined to be nˉ\bar {n}nˉ = (840 ± 100) s –1 . This result is an important step towards a scalable ion quantum computer.