Lens-Free Optical Detection of Thermal Motion of a Submillimeter Sphere Diamagnetically Levitated in High Vacuum

Letter

Lens-Free Optical Detection of Thermal Motion of a Submillimeter Sphere Diamagnetically Levitated in High Vacuum

Fang Xiong, Peiran Yin, Tong Wu, Han Xie, Rui Li, Yingchun Leng, Yanan Li, Changkui Duan, Xi Kong, Pu Huang, and Jiangfeng Du

Phys. Rev. Applied 16, L011003 – Published 13 July 2021

Abstract

Levitated oscillators of millimeter or submillimeter size are particularly attractive due to their potential role in studying various fundamental problems and practical applications. One of the crucial issues towards these goals is to achieve efficient measurements of oscillator motion, although this remains a challenge. Here we theoretically propose a lens-free optical detection scheme, which can be used to detect the motion of a millimeter or submillimeter levitated oscillator with a measurement efficiency close to the standard quantum limit with a modest optical power. We demonstrate experimentally this scheme on a 0.5-mm-diameter microsphere that is diamagnetically levitated under high vacuum and room temperature, and the thermal motion is detected with high precision. Based on this system, an estimated acceleration sensitivity of $9.7 \times 10^{- 10} g / \sqrt{Hz}$ is achieved, which is an improvement of more than 1 order of magnitude over the best value reported for a levitated mechanical system. Due to the stability of the system, the minimum resolved acceleration of $3.5 \times 10^{- 12} g$ is reached with measurement times of $10^{5}$ s. This result is expected to have potential applications in the study of exotic interactions in the millimeter or submillimeter range and the realization of compact gravimeters and accelerometers.

Received 9 February 2021
Revised 7 April 2021
Accepted 27 May 2021

DOI:https://doi.org/10.1103/PhysRevApplied.16.L011003

Physics Subject Headings (PhySH)

Force sensing Quantum metrology Quantum sensing

Micromechanical & nanomechanical oscillators

Cooling & trapping Magnetic force microscopy Precision measurements

Atomic, Molecular & OpticalAccelerators & BeamsCondensed Matter, Materials & Applied PhysicsGravitation, Cosmology & AstrophysicsNonlinear DynamicsPhysics Education ResearchQuantum Information, Science & TechnologyFluid DynamicsInterdisciplinary PhysicsNuclear PhysicsPlasma PhysicsStatistical Physics & ThermodynamicsPhysics of Living SystemsGeneral PhysicsNetworksParticles & FieldsPolymers & Soft Matter

Authors & Affiliations

Fang Xiong ^1,§, Peiran Yin^1,§, Tong Wu ¹, Han Xie¹, Rui Li ^2,3,4, Yingchun Leng ¹, Yanan Li ¹, Changkui Duan^2,3,4, Xi Kong^1,*, Pu Huang ^1,†, and Jiangfeng Du^2,3,4,‡

¹National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
²CAS Key Laboratory of Microscale Magnetic Resonance and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
³Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
⁴Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China

^*kongxi@nju.edu.cn
^†hp@nju.edu.cn
^‡djf@ustc.edu.cn
^§These authors contributed equally to this paper.

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Vol. 16, Iss. 1 — July 2021

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Images

Figure 1
Schematic of our method. (a) Lens-free optical measurement system. A 1550-nm laser with a power of $P_{in}$ illuminates the levitated microsphere via the left incident fiber. The microsphere works as a spherical lens to refocus the light into the detection fiber on the right. The optical fiber is positioned at $x_{fib}$ relative to the center in the $x$ -axis direction, and the power detected by the detection optical fiber is $P_{out}$ . (b) The calculated transmission coefficient $T = P_{out} / P_{in}$ as a function of the detecting fiber position $x_{fib}$ . The red curve represents where the microsphere is in the central equilibrium position, and the light red curve corresponds to the response curve a small displacement $δ x$ along the $x$ axis. The black line represents the response of signal $δ T$ of the measured signal to the displacement $δ x$ .
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Figure 2
Estimated measurement noise of acceleration. (a) The red curve (left axis) is the minimum measurement noise of acceleration $\sqrt{S_{a a}^{mea}}$ versus the shifted detection fiber position $x_{fib}$ . We assume that the refractive index of the microsphere $n = 1.4$ , the diameter $2 R = 0.5$ mm, the acceleration noise is estimated using the mode that oscillates along the $x$ axis, the resonant frequency is 10 Hz, and the mechanical dissipation is $10^{- 6}$ Hz. The red dashed lines are the thermal noise of acceleration $\sqrt{S_{a a}^{mea}}$ and SQL as labeled. The blue curve (right axis) is the corresponding optimized laser power $P_{in}^{opt}$ . (b) Same as (a) but versus the size of the microsphere (radius $R$ in unit of wavelength) with a fixed $x_{fib} = 5.8 μ m$ . (c) Same as (a) but versus the incident laser power, and with a fixed $x_{fib} = 5.8 μ m$ .
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Figure 3
Experimental measurement of thermal movement of a submillimeter microsphere. (a) An optical image of a microsphere of $0.5$ mm in diameter levitated in a magnetic gravitational trap. A 1550-nm laser transmits via the left fiber to the right one passing the submillimeter microsphere. The fibers are fixed on the magnetic trap by UV glues. Vibration mode 1 and mode 2 on the horizontal plane are indicated by arrows in the inset, and mode 1 is used in the current experiment. The angle between mode 1 and the optical axis is about $85^{\circ} \pm 5^{\circ}$ . (b) The measured power spectral density of motion $S_{x x}^{tot} (ω)$ of oscillation mode 1 under a pressure of $4 \times 10^{- 3}$ mbar (red circles). The data are fitted to the Lorentz curve (red line). The blue line shows the fitted imprecision noise power spectral density $S_{x x}^{imp}$ . The light-red area represents the contribution of effective thermal Brownian motion at an ambient temperature of 298 K.
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Figure 4
The minimum resolvable acceleration. The blue circles are the measured acceleration noise as a function of measurement time at pressure $1.2 \times 10^{- 5}$ mbar. To build up the statistics, a total experiment time of about $7 \times 10^{5}$ s (about 190 h) is used. The red curve is the theoretical curve assuming the temperature fluctuations are eliminated.
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