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arXiv:2012.10336 · Quantum Science and Technology

Motional heating of spatially extended ion crystals

D Kalincev, L S Dreissen, A P Kulosa, C-H Yeh, Henning A. Fürst, T E Mehlstäubler

Published Mar 15, 2021• 16 citations
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Abstract

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.

DOI
10.1088/2058-9565/abee99

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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.
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 ṅ ≈ 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 ṅ = 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 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.
Ions in radio-frequency traps are widely used in various fields of applied and fundamental physics, such as metrology and quantum computing. One of the important tasks required for modern experiments is deep cooling of ion crystals. The results of simulations of an increase in the efficiency of deep cooling of linear ion crystals by the method of electromagnetically induced transparency (EIT cooling) by imposing an optical lattice on the radio-frequency trap have been reported. It has been shown that this method makes it possible to narrow the frequency range occupied by various vibrational modes of ions and to increase their axial frequencies of motion without violating the linear configuration of the crystal. Thus, for a crystal of eight ions in a Paul trap with secular frequencies ω z = 2 π × 100 kHz and ω r = 2 π × 650 kHz, the application of an optical lattice allows the reduction of the frequency range occupied by vibrational modes by a factor of 2. The dependence of the optimal power of the optical lattice for narrowing the vibrational spectrum on the number of particles in the trap and its parameters has been investigated.
We present a novel ultrastable superconducting radio-frequency (RF) ion trap realized as a combination of an RF cavity and a linear Paul trap. Its RF quadrupole mode at 34.52 MHz reaches a quality factor of Q ≈ 2.3 × 105 at a temperature of 4.1 K and is used to radially confine ions in an ultralow-noise pseudopotential. This concept is expected to strongly suppress motional heating rates and related frequency shifts that limit the ultimate accuracy achieved in advanced ion traps for frequency metrology. Running with its low-vibration cryogenic cooling system, electron-beam ion trap, and deceleration beamline supplying highly charged ions (HCIs), the superconducting trap offers ideal conditions for optical frequency metrology with ionic species. We report its proof-of-principle operation as a quadrupole-mass filter with HCIs and trapping of Doppler-cooled 9Be+ Coulomb crystals.
We investigate the energy dynamics of noncrystallized (melted) ions, confined in a Paul trap. The nonperiodic Coulomb interaction experienced by melted ions forms a medium for nonconservative energy transfer from the radio-frequency (rf) field to the ions, a process known as rf heating. We study rf heating by analyzing numerical simulations of noncrystallized ion motion in Paul trap potentials, in which the energy of the ions' secular motion changes at discrete intervals, corresponding to ion-ion collisions. The analysis of these collisions is used as a basis to derive a simplified model of rf heating energy dynamics, from which we conclude that the rf heating rate is predominantly dependent on the rf field strength. We confirm the predictability of the model experimentally: Two trapped 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ ions are deterministically driven to melt, and their fluorescence rate is used to infer the ions' energy. Under these experimental conditions, we investigate which Doppler cooling parameters are required for efficient recrystallization, and compare our results to simulations.
Trapped ions are sensitive detectors of weak forces and electric fields that excite ion motion. Here measurements of the center-of-mass motion of a trapped-ion crystal that are phase coherent with an applied weak external force are reported. These experiments are conducted far from the trap motional frequency on a two-dimensional trapped-ion crystal of approximately 100 ions, and determine the fundamental measurement imprecision of our protocol free from noise associated with the center-of-mass mode. The driven sinusoidal displacement of the crystal is detected by coupling the ion crystal motion to the internal spin degree of freedom of the ions using an oscillating spin-dependent optical dipole force. The resulting induced spin precession is proportional to the displacement amplitude of the crystal, and is measured with near-projection-noise-limited resolution. A 49pm49\phantom{\rule{4pt}{0ex}}\mathrm{pm}49pm displacement is detected with a signal-to-noise ratio of 1 in a single experimental determination, which is an order-of-magnitude improvement over prior phase-incoherent experiments. This displacement amplitude is 40 times smaller than the zero-point fluctuations. With our repetition rate, an 8.4pm/Hz8.4\phantom{\rule{4pt}{0ex}}\mathrm{pm}/\sqrt{\mathrm{Hz}}8.4pm/Hz​ displacement sensitivity is achieved, which implies 12(yN/ion)/Hz12\phantom{\rule{0.16em}{0ex}}(\mathrm{yN}/\mathrm{ion})/\sqrt{\mathrm{Hz}}12(yN/ion)/Hz​ and $77\phantom{\rule{0.16em}{0ex}}(\ensuremath{\mu}\mathrm{V}/\mathrm{m})/\sqrt{\mathrm{Hz}}$ sensitivities to forces and electric fields, respectively. This displacement sensitivity, when applied on-resonance with the center-of-mass mode, indicates the possibility of weak force and electric field detection below ${10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}\mathrm{yN}$/ion and 1nV1\phantom{\rule{4pt}{0ex}}\mathrm{nV}1nV/m, respectively.
We employ spin-dependent optical dipole forces to characterize the transverse center-of-mass (COM) motional mode of a two-dimensional Wigner crystal of hundreds of 9{}^{9}9Be+{}^{+}+. By comparing the measured spin dephasing produced by the spin-dependent force with the predictions of a semiclassical dephasing model, we obtain absolute mode temperatures in excellent agreement with both the Doppler laser cooling limit and measurements obtained from a previously published technique [B. C. Sawyer et al., Phys. Rev. Lett. 108, 213003 (2012)]. Furthermore, the structure of the dephasing histograms allows for discrimination between initial thermal and coherent states of motion. We also apply the techniques discussed here to measure the ambient-heating rate of the COM mode of a 2D Coulomb crystal in a Penning trap. This measurement places an upper limit on the anomalous single-ion heating rate due to electric field noise from the trap electrode surfaces of $\frac{d\overline{n}}{dt}\ensuremath{\sim}5$ s${}^{\ensuremath{-}1}$ for our trap at a frequency of 795 kHz, where n‾\overline{n}n is the mean occupation of quantized COM motion in the axial harmonic well.