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arXiv:1401.0672 · Physical Review A

Spin dephasing as a probe of mode temperature, motional state distributions, and heating rates in a two-dimensional ion crystal

Brian C. Sawyer, J. Britton, J. J. Bollinger

Published Mar 10, 2014• 27 citations
arXivHTMLPDF

Abstract

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 ${}^{9}$Be${}^{+}$. 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 $\overline{n}$ is the mean occupation of quantized COM motion in the axial harmonic well.

DOI
10.1103/physreva.89.033408

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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 dn‾dt\ensuremath∼5\frac{d\overline{n}}{dt}\ensuremath{\sim}5dtdn​\ensuremath∼5 s\ensuremath−1{}^{\ensuremath{-}1}\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.
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We report phase-coherent Doppler detection of optical dipole forces using large ion crystals in a Penning trap. The technique is based on laser Doppler velocimetry using a cycling transition in 9Be+ near 313 nm and the center-of-mass (COM) ion motional mode. The optical dipole force is tuned to excite the COM mode, and measurements of photon arrival times synchronized with the excitation potential show oscillations with a period commensurate with the COM motional frequency. Experimental results compare well with a quantitative model for a driven harmonic oscillator. This technique permits characterization of motional modes in ion crystals; the measurement of both frequency and phase information relative to the driving force is a key enabling capability--comparable to lockin detection - providing access to a parameter that is typically not available in time-averaged measurements. This additional information facilitates discrimination of nearly degenerate motional modes.
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We demonstrate spectroscopy and thermometry of individual motional modes in a mesoscopic 2D ion array using entanglement-induced decoherence as a method of transduction. Our system is a $\ensuremath{\sim}400\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$-diameter planar crystal of several hundred 9Be+^{9}\mathrm{Be}^{+}9Be+ ions exhibiting complex drumhead modes in the confining potential of a Penning trap. Exploiting precise control over the 9Be+^{9}\mathrm{Be}^{+}9Be+ valence electron spins, we apply a homogeneous spin-dependent optical dipole force to excite arbitrary transverse modes with an effective wavelength approaching the interparticle spacing ($\ensuremath{\sim}20\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$). Center-of-mass displacements below 1 nm are detected via the entanglement of spin and motional degrees of freedom.
Improving coherence is a fundamental challenge in quantum simulation and sensing experiments with trapped ions. Here we discuss, experimentally demonstrate, and estimate the potential impacts of two different protocols that enhance, through motional parametric excitation, the coherent spin-motion coupling of ions obtained with a spin-dependent force. The experiments are performed on two-dimensional crystal arrays of approximately 100 9Be+{}^{9}{\mathrm{Be}}^{+}9Be+ ions confined in a Penning trap. By modulating the trapping potential at close to twice the center-of-mass mode frequency, we squeeze the motional mode and enhance the spin-motion coupling while maintaining spin coherence. With a stroboscopic protocol, we measure $5.4\ifmmode\pm\else\textpm\fi{}0.9\phantom{\rule{0.28em}{0ex}}\mathrm{dB}$ of motional squeezing below the ground-state motion, from which theory predicts a 10-dB enhancement in the sensitivity for measuring small displacements using a recently demonstrated protocol [K. A. Gilmore et al., Science 373, 673 (2021)]. With a continuous squeezing protocol, we measure and accurately calibrate the parametric coupling strength. Theory suggests this protocol can be used to improve quantum spin squeezing, limited in our system by off-resonant light scatter. We illustrate numerically the trade-offs between strong parametric amplification and motional dephasing in the form of center-of-mass frequency fluctuations for improving quantum spin squeezing in our setup.
We have studied the dynamics of one and two laser-cooled trapped 40Ca+ ions by applying electric fields of different nature along the axial direction of the trap, namely, driving the motion with a harmonic dipolar field, or with white noise. These two types of driving induce distinct motional states of the axial modes: a coherent oscillation with the dipolar field, or an enhanced Brownian motion due to an additional contribution to the heating rate from the electric noise. In both scenarios, the sensitivity of an isolated ion and a laser-cooled two-ion crystal has been evaluated and compared. The analysis and understanding of this dynamics is important towards the implementation of a novel Penning trap mass-spectroscopy technique based on optical detection, aiming at improving precision and sensitivity.