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arXiv:1201.4415 · Physical Review Letters

Spectroscopy and Thermometry of Drumhead Modes in a Mesoscopic Trapped-Ion Crystal Using Entanglement

Brian C. Sawyer, J. Britton, Adam C. Keith, C.‐C. Joseph Wang, James K. Freericks, Hermann Uys, Michael J. Biercuk, J. J. Bollinger

Published May 24, 2012• 60 citations
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Abstract

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 $^{9}\mathrm{Be}^{+}$ ions exhibiting complex drumhead modes in the confining potential of a Penning trap. Exploiting precise control over the $^{9}\mathrm{Be}^{+}$ 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.

DOI
10.1103/physrevlett.108.213003

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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∼400  \ensuremathμm\ensuremath{\sim}400\text{ }\text{ }\ensuremath{\mu}\mathrm{m}\ensuremath∼400  \ensuremathμm-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∼20  \ensuremathμm\ensuremath{\sim}20\text{ }\text{ }\ensuremath{\mu}\mathrm{m}\ensuremath∼20  \ensuremathμm). Center-of-mass displacements below 1 nm are detected via the entanglement of spin and motional degrees of freedom.
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.
We experimentally study electromagnetically induced transparency cooling of the drumhead modes of planar two-dimensional arrays with up to N≈190 Be^{+} ions stored in a Penning trap. Substantial sub-Doppler cooling is observed for all N drumhead modes. Quantitative measurements for the center-of-mass mode show near ground-state cooling with motional quantum numbers of n[over ¯]=0.3±0.2 obtained within 200 μs. The measured cooling rate is faster than that predicted by single particle theory, consistent with a quantum many-body calculation. For the lower frequency drumhead modes, quantitative temperature measurements are limited by frequency instabilities, but near ground-state cooling of the full bandwidth is strongly suggested. This advance will greatly improve the performance of large trapped ion crystals in quantum information and metrology applications.
Two-dimensional crystals of ions stored in Penning traps are a leading platform for quantum simulation and sensing experiments. For small amplitudes, the out-of-plane motion of such crystals can be described by a discrete set of normal modes called the drumhead modes, which can be used to implement a range of quantum information protocols. However, experimental observations of crystals with Doppler-cooled and even near-ground-state-cooled drumhead modes reveal an unresolved drumhead-mode spectrum. In this work, we establish in-plane thermal fluctuations in ion positions as a major contributor to the broadening of the drumhead-mode spectrum. In the process, we demonstrate how the confining magnetic field leads to unconventional in-plane normal modes, whose average potential and kinetic energies are not equal. This property, in turn, has implications for the sampling procedure required to choose the in-plane initial conditions for molecular-dynamics simulations. For current operating conditions of the NIST Penning trap, our study suggests that the two-dimensional crystals produced in this trap undergo in-plane potential-energy fluctuations of the order of 10mK. Our study therefore motivates the need for designing improved techniques to cool the in-plane degrees of freedom.
Penning traps, with their ability to control planar crystals of tens to hundreds of ions, are versatile quantum simulators. Thermal occupations of the motional drumhead modes, transverse to the plane of the ion crystal, degrade the quality of quantum simulations. Laser cooling using electromagnetically induced transparency (EIT cooling) is attractive as an efficient way to quickly initialize the drumhead modes to near ground-state occupations. We numerically investigate the efficiency of EIT cooling of planar ion crystals in a Penning trap, accounting for complications arising from the nature of the trap and from the simultaneous cooling of multiple ions. We show that, in spite of challenges, the large bandwidth of drumhead modes (hundreds of kilohertz) can be rapidly cooled to near ground-state occupations within a few hundred microseconds. Our predictions for the center-of-mass mode include a cooling time constant of tens of microseconds and an enhancement of the cooling rate with increasing number of ions. Successful experimental demonstrations of EIT cooling in the NIST Penning trap [Jordan et al., [Phys. Rev. Lett. 122, 053603 (2019)] validate our predictions.
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.