Five-dimensional cooling and nonlinear dynamics of an optically levitated nanodumbbell

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Five-dimensional cooling and nonlinear dynamics of an optically levitated nanodumbbell

Jaehoon Bang, T. Seberson, Peng Ju, Jonghoon Ahn, Zhujing Xu, Xingyu Gao, F. Robicheaux, and Tongcang Li

Phys. Rev. Research 2, 043054 – Published 9 October 2020

Abstract

Optically levitated nonspherical particles in vacuum are excellent candidates for torque sensing, rotational quantum mechanics, high-frequency gravitational wave detection, and multiple other applications. Many potential applications, such as detecting the Casimir torque near a birefringent surface, require simultaneous cooling of both the center-of-mass motion and the torsional vibration (or rotation) of a nonspherical nanoparticle. Here we report five-dimensional cooling of a levitated nanoparticle. We cool the three center-of-mass motion modes and two torsional vibration modes of a levitated nanodumbbell in a linearly polarized laser simultaneously. The only uncooled rigid-body degree of freedom is the rotation of the nanodumbbell around its long axis. This free rotation mode does not couple to the optical tweezers directly. Surprisingly, we observe that it strongly affects the torsional vibrations of the nanodumbbell. This work deepens our understanding of the nonlinear dynamics and rotation coupling of a levitated nanoparticle and paves the way towards full quantum control of its motion.

Received 17 April 2020
Accepted 16 September 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043054

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Classical mechanics Cooling & trapping Mechanical effects of light on material media Optomechanics

Atomic, Molecular & OpticalNonlinear Dynamics

Authors & Affiliations

Jaehoon Bang ¹, T. Seberson², Peng Ju², Jonghoon Ahn¹, Zhujing Xu², Xingyu Gao ², F. Robicheaux ^2,3, and Tongcang Li ^1,2,3,4,*

¹School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
²Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
³Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
⁴Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA

^*tcli@purdue.edu

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Vol. 2, Iss. 4 — October - December 2020

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Figure 1
(a) Schematic of 5D cooling of a levitated nanodumbbell. A 1064-nm laser is tightly focused with an NA = 0.85 objective lens to levitate a nanodumbbell in vacuum. The 1064-nm trapping laser is polarized in the $x$ direction and propagates along the $z$ axis as indicated by a red arrow. Cooling lasers and their directions of propagation are illustrated in green and orange. The polarization directions of the $y$ and $z$ cooling lasers are tilted to form an angle of about 10 degrees with respect to the polarization direction of the trapping laser. The cooling laser propagating in the $x$ direction is polarized along the $y$ axis. (b) Trapping potential as a function of the c.m. motion along the $y$ axis and the torsional vibration along $α$ direction. For direct comparison, the radius $r = 85$ nm is multiplied to the angle $α$ in the torsional potential case. The inset is a scanning electron microscope (SEM) image of a nanodumbbell. (c) Definition of the Euler angles $α$ , $β$ , $γ$ used in this paper for $z - y^{'} - z^{''}$ convention. For small libration amplitudes, $α$ represents the motion near the $x$ axis in the $x - y$ plane, and $β$ represents the motion in the $x - z$ plane. $γ$ stands for the free rotation around the $z^{''} = z^{'''}$ axis. $x^{''}$ , $y^{''}$ , $x^{'''}$ , $y^{'''}$ are not shown to simplify the figure. (d) Power spectral density (PSD) measured for a nanodumbbell, which consists of two 170-nm-diameter silica spheres. PSDs obtained from four different detectors for $x$ , $y$ , $z$ , and $α$ are shown. Note that the PSD for the $β$ motion could be obtained from the high frequency part of the signal obtained by the $x$ detector. These PSDs are taken at a pressure of $3 \times 10^{- 3}$ Torr with $x$ , $y$ and $z$ motion cooling. The data acquisition time is 1 s.
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Figure 2
(a) Relative frequency fluctuations for translational motions ( $ω_{x}$ , $ω_{y}$ ) and torsional motions at $3.33 \times 10^{- 3}$ Torr. (b) Relative frequency fluctuations of $ω_{+}$ (red), $ω_{-}$ (blue), and $ω_{y}$ (grey) at $3.33 \times 10^{- 3}$ Torr. The $ω_{+}$ and $ω_{-}$ frequencies are normalized by the average $〈 (ω_{+} + ω_{-}) / 2 〉$ . $ω_{y}$ is normalized by its average $〈 ω_{y} 〉$ . (c) Relative frequency fluctuations of $ω_{+}$ (red), $ω_{-}$ (blue), $ω_{y}$ (grey) at 5 Torr. In (a), (b), (c), each frequency is determined from a PSD corresponding to 2 ms of data in time. (d) Frequency correlations between different degrees of freedom. The correlations are plotted as a function of pressure. Each data point corresponds to an average of five sets of measurements. For each measurement, the correlations are calculated from 500 PSD data. The error bar shows the standard deviation.
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Figure 3
(a), (b) Measured PSDs of motions along $α$ (grey) and $β$ (red) directions when $ω_{c}$ is small (a) or large (b). The PSDs are taken at $3 \times 10^{- 3}$ Torr and the measurement time is 2 ms. (c), (d) Corresponding simulation results for $α$ (purple) and $β$ (yellow) motions when $ω_{c} / 2 π$ is 17 kHz (c) or 115 kHz (d). The simulated motion time is 1 ms. Random background noise is added to mimic the experiment.
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Figure 4
5D cooling of a levitated nanodumbbell. (a) Simplified schematic of the setup showing the sequence of cooling for a single direction. In the real experiment, signals from the particle are collected with four balanced detectors (D) and processed with five home-built circuits to generate the corresponding feedback signals. The signal obtained from the balanced detector for both $x$ and $β$ motions is split to two and fed into two different derivative circuits in order to generate the cooling signals for the motion in $x$ and $β$ directions. After processing, signals corresponding to $α$ and $β$ motions are added to the signals for $z$ and $y$ motions, respectively [Fig. 1]. These signals are then used to modulate the cooling lasers. The results of 5D feedback cooling are shown in (b) x, (c) y, (d) z, (e) $α$ , and (f) $β$ . The blue curves are PSDs before cooling, and the red curves are PSDs with cooling. The green spectra show the noise levels when there is no particle. The pressure is 1 Torr for blue curves (no cooling), and is $1.8 \times 10^{- 3}$ Torr for red curves (with cooling). The effective temperatures for $x$ , $y$ , and $z$ motions are calculated based on Lorentzian fittings (shown in grey dashed lines). The effective temperatures for $α$ and $β$ motions are calculated by comparing the areas below the PSDs.
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