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Optically Levitated Nanodumbbell Torsion Balance and GHz Nanomechanical Rotor PRL

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Optically Levitated Nanodumbbell Torsion Balance and GHz Nanomechanical Rotor PRL
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Optically Levitated Nanodumbbell Torsion Balance and GHz Nanomechanical Rotor Jonghoon Ahn,1Zhujing Xu,2Jaehoon Bang,1Yu-Hao Deng,3Thai M. Hoang,2,*Qinkai Han,4, Ren-Min Ma,3,5,and Tongcang Li1,2,6,7, 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 3State Key Lab for Mesoscopic Physics and School of Physics, Peking University, Beijing 100871, China 4The State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China 5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 6Purdue Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA 7Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA (Received 18 April 2018; published 20 July 2018) Levitated optomechanics has great potential in precision measurements, thermodynamics, macroscopic quantum mechanics, and quantum sensing. Here we synthesize and optically levitate silica nanodumbbells in high vacuum. With a linearly polarized laser, we observe the torsional vibration of an optically levitated nanodumbbell. This levitated nanodumbbell torsion balance is a novel analog of the Cavendish torsion balance, and provides rare opportunities to observe the Casimir torque and probe the quantum nature of gravity as proposed recently. With a circularly polarized laser, we drive a 170-nm-diameter nanodumbbell to rotate beyond 1 GHz, which is the fastest nanomechanical rotor realized to date. Smaller silica nanodumbbells can sustain higher rotation frequencies. Such ultrafast rotation may be used to study material properties and probe vacuum friction. DOI: 10.1103/PhysRevLett.121.033603 Levitated optomechanical systems provide a powerful platform for precision measurements with great isolation from the thermal environment 16. Optically levitated nano- and microspheres have been used to demonstrate force sensing at the level of 1021N 7 and to search for interactions associated with dark energy 8. Optically trapped nanoparticles can also be driven to rotate at high speed. Previously, a rotation frequency of about 10 MHz had been achieved 911. It is desirable to increase the rotation frequency further for studying material properties under extreme conditions 12,13 and probing vacuum friction 1416. Recently, a novel ultrasensitive torsion balance with an optically levitated nonspherical nanoparticle was proposed 17, utilizing the coupling between the spin angular momentum of photons and the mechanical motion of the nanoparticle 4,1720. Torsion balances have enabled many breakthroughs in the history of modern physics 2124. For example, the Cavendish torsion balance Fig. 1(a) determined the gravitational constant and the density of the Earth 21. An optically levitated nanoscale torsion balance can provide a rare opportunity to detect the Casimir torque 2528, and test the quantum nature of gravity 2931. An essential step towards these goals is to optically trap a well-defined nonspherical nanoparticle in high vacuum. However, optically trapped nonspherical nanoparticles such as nanodiamonds and silicon nanorods in former experiments were lost at about 1 torr due to laser heating 10,17,3234. Additionally, several years ago, levitated nanodumbbells were theoretically proposed to study many-body phase transitions 19. To the best of our knowledge, however, there has been no report on optical levitation of a nanodumbbell in vacuum prior to this work. In this Letter, we synthesize silica nanodumbbells with two different methods and optically trap them in high vacuum. With a linearly polarized laser Fig. 1(b), we observe the torsional vibration of a levitated nanodumbbell in high vacuum, which is an important step towards probing the Casimir torque 2528 and the quantum nature of gravity 2931. With a circularly polarized laser, we drive the nanodumbbell to rotate beyond 1 GHz, which is the highest mechanical rotation frequency reported to date. In a linearly polarized optical tweezer, the long axis of a nanodumbbell will tend to align with the polarization direction of the trapping laser Fig. 1(b). This is because the polarizability of the nanodumbbell along its long axis is larger than the polarizability perpendicular to its long axis. If the nanodumbbell is not aligned with the polarization direction of the optical tweezer, it will twist the polarization of the optical tweezer Fig. 1(b), as an analog of twisting the torsion wire by the lead spheres in the original Cavendish torsion balance Fig. 1(a). If the optical tweezer is circularly polarized, the nanodumbbell will be driven to rotate at high speed. The torsional vibration or rotation of the nanodumbbell can be detected by monitoring the change of the polarization of the trapping laser Fig. 1(c). PHYSICAL REVIEW LETTERS 121, 033603 (2018) Featured in Physics 0031-9007=18=121(3)=033603(5)033603-1 2018 American Physical Society We have synthesized pure silica nanodumbbells using chemical and physical methods. The interpenetration of the two particles in a nanodumbbell is tunable. To synthesize nanodumbbells with diameter D and length L inset of Fig. 1(c), we first synthesize silica cores with a diameter of d L D by adding tetraethyl orthosilicate (TEOS) to a mixture of ammonia, pure water, and ethanol under stirring 35,36. Then acetone is added into the solution to induce aggregation. Next, a small amount of TEOS is added under stirring to grow the silica shells 35. This chemical method can synthesize a large quantity of silica nanodumbbells with a tunable aspect ratio 35, but is demanding. So we also develop a physical method to assemble nanodumb- bells. In this method,we first prepare a colloidal suspension of silica nanospheres in water. We then generate water microdroplets in air with an ultrasonic nebulizer 33. By controlling the concentration of silica nanospheres, a fraction of water microdroplets (5 m in diameter) con- tain 2 silica nanospheres in them. Two nanospheres in the same microdroplet will form a nanodumbbell as the water evaporates. Figures 2(a) and 2(b) show SEM images of our nanodumbbells in two different sizes. Their aspect ratio is between 1.9 and 2. To optically levitate a silica nanodumbbell in vacuum, a 500 mW, 1550 nm laser is tightly focused with an NA 0.85 objective lens in a vacuum chamber. The laser is initially linearly polarized, and its polarization can be controlled with a quarter wave plate Fig. 1(c). Silica nanodumbbells are delivered into the optical trap at atmospheric pressure with an ultrasonic nebulizer 33. Once a nanoparticle is trapped, we evacuate the vacuum chamber to below 0.01 torr, and then increase the pressure back to desired levels for measurements. This procedure removes extra nanoparticles in the chamber. To monitor the trapping process, a 532 nm laser is applied on the nano- particle and the scattered light is viewed using a camera. We verify the trapped nanoparticle is a nanodumbbell by checking the ratios of damping coefficients for translational motions along different directions. The motion of the levitated nanoparticle changes the direction and polariza- tion of the laser beam, which allows us to monitor the motion of a nanodumbbell with the same 1550 nm trapping laser Fig. 1(c) 17. Figure 2(c) shows the power spectrum density(PSD) of both torsional (TOR) vibrationand center- of-mass (c.m.) vibration of a levitated 170-nm-diameter nanodumbbell in vacuum at 5 104torr. This is an important step towards quantum ground state cooling of the torsional vibration and testing recent theoretical proposals 25,2931. FIG. 2.(a),(b): SEM images of silica nanodumbbells with two different sizes. The scale bar is 200 nm in (a) and 100 nm in (b). (c) Measured power spectrum densities (PSD) of the torsional vibration (labeled “Tor”) and the translational vibrations (labeled “X,” “Y,” “Z”) of a 170-nm-diameter nanodumbbell optically levitated at 5 104torr. (d) Measured PSD of the translational vibrations of the nanodumbbell levitated at 10 torr. The black curves are Lorentz fits. (c) (a)(b) FIG. 1.(a) A simplified diagram of the original Cavendish torsion balance that has two lead spheres suspended by a copper silvered torsion wire. (b) A nanodumbbell levitated by a linearly polarized optical tweezer in vacuum. The linearly polarized optical tweezer provides the restoring torque that confines the orientation of the nanodumbbell. xT, yT, zTare Cartesian coordinates of the trapping laser. xTis parallel to the electric field E of the incoming linearly polarized laser, and zTis parallel to the wave vector k of the laser. xNis parallel to the long axis of the nanodumbbell. The angle between xTand xNis . (c) A simplified diagram for detecting the center-of-mass (c.m.) mo- tion, the torsional (TOR) vibration, and the rotation (ROT) of a levitated nanodumbbell. The nanodumbbell is trapped at the focus of the lenses. The laser beam is initially linearly polarized. A quarter-wave (=4) plate is used to control its polarization. BS: non-polarizing beam splitter; PBS: polarizing beam splitter; =2: half-wave plate. Inset: A nanodumbbell with diameter D and length L created by attaching two identical nanospheres. PHYSICAL REVIEW LETTERS 121, 033603 (2018) 033603-2 A levitated nanodumbbell will have anisotropic damping rates for translational motions in air if its orientation is fixed. We use the direct simulation Monte Carlo (DSMC) method to obtain the damping force and damping torque of a nanodumbbell in the free molecular flow regime 3639. In the simulation, molecules with uniform spatial distribu- tion are launched from a spherical surface enclosing the nanodumbbell 36. The speeds of these molecules satisfy a shifted Maxwell distribution to include the effect of the motion of the nanodumbbell 39. Figure 3(a) shows the calculated ratio of damping rates of a nanodumbbell moving along (x) and perpendicular (y) to its axial direction. The calculated ratio is y=x z=x 1.276 when L=D 1.9, and y=x z=x 1.258 when L=D 2. The measured ratios for the data of a 170 nm nanodumbbell shown in Fig. 2(d) are y=x 1.25 ? 0.01 and z=x 1.27 ? 0.02, which agree excellently with our theoretical predictions. The DSMC method is also utilized to obtain the drag torque Tzon a nanodumbbell rotating at speed . We then calculate the ratio Tz=Tsphere, where Tsphere D4=11.976M is the drag torque on a single sphere with diameter D rotating at the same speed 39. Here is the viscosity of air, and Mis the mean free path of air molecules which is inversely proportional to air pressure pair. The calculated results are shown in Fig. 3(b). The damping rate of the rotation or torsional vibration around the z axis is Tz=Iz, where Izis its moment of inertia. When the size of a silica nanodumbbell is much smaller than the wavelength of the trapping laser, the dipole approximation can be applied. The complex amplitude oftheinduceddipoleofthenanodumbbellis p xEx xN yEy yN zEz zN,wherethecomplex amplitude of the electric field of the laser beam E is decomposed into components along the principle axes of the nanodumbbell. xNis in the direction along the long axis of the nanodumbbell. The components of the optical force Fjand the optical torque Mjacting on the nano- dumbbell can be expressed as 40 Fj 1 2Refp ? jEg and Mj 1 2Refp ? Egj. The quasistatic polarizability 0 j (j x, y, z) of a nanodumbbell can be calculated assuming the electric field is static 41. Figure 3(c) shows the effective susceptibilities x 0 x=0V, y 0y=0V of the nanodumbbell as a function of the aspect ratio. Here0isthe permittivityof vacuumandV is thevolumeof the nanodumbbell. x y=y 0.14 when L=D 1.9. The dipole approximation can be improved by including the effects of radiation reaction due to the oscillation of the electric field in a laser beam. Then the polarizability is 40,42,43 j 0 j=1 ik300j=60?, where k0 is the wave number. The real part of the polarizability Rej? 0 j is responsible for optical confinement and alignment, while the imaginary part Imj? is important for optically rotating a nanodumbbell. With calculated damping rates and polarizabilities, we can calculate the torque detection sensitivity of a levitated nanodumbbell. The minimum torque that it can measure limited by thermal noise is 44 Mth ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 4kBTenvIz=t p , where Tenvis the environmental temperature. In ultrahigh vacuum, the thermal noise from the residual air molecules becomes negligible and the minimum torque it can detect will be limited by the shot noise of the laser beam25:Mrad x yk2 0V ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Jp=3t p .Here, Jp Ilaser=L is the photon flux. Ilaseris the laser intensity, and Lis the angular frequency of the laser. As shown in Fig. 3(d), at pressures below 108torr, a nano- dumbbell with D 170 nm and D 50 nm trapped in a 500 mW laser will have a torque detection sensitivity of about 1027Nm= ffi ffi ffi ffi ffiffi Hz p and 1029Nm= ffi ffi ffi ffi ffiffi Hz p , respectively. Remarkably, a levitated nanodumbbell at 104torr is already much more sensitive than the current state-of- the-art nanofabricated torsion balance, which has achieved a torque sensitivity of 1022Nm= ffi ffi ffi ffi ffiffi Hz p at room temper- ature, and 1024Nm= ffi ffi ffi ffi ffiffi Hz p at 25 mK in a dilution refrigerator 45. The original Cavendish experiment mea- sured a torque of about 107Nm caused by gravitational attraction 21. FIG. 3.(a) The ratio of air damping coefficientsfor translational motions perpendicular or parallel to its long axis (xNaxis) as a function of the aspect ratio (L=D) of a nanodumbbell. (b) Calcu- lated normalized drag torque of the rotation of a levitated nanodumbbell around zNaxis as a function of the aspect ratio. (c) Effective susceptibilities of a silica nanodumbbell parallel (x) or perpendicular (y) to its long axis. (d) Calculated torque detection sensitivity of a levitated nanodumbbell with D 170 nm or D 50 nm as a function of air pressure. We assume L=D 1.9 in the calculations. The optical tweezer is assumed to be a focused 500 mW, 1550 nm laser with a waist of 820 nm. PHYSICAL REVIEW LETTERS 121, 033603 (2018) 033603-3 While a nanodumbbell levitated by a linearly polarized optical tweezer can be an ultrasensitive nanoscale torsion balance, it will become an ultrafast nanomechanical rotor in a circularly polarized optical tweezer Fig. 4(a). The frequency of the detected signal will be twice the rotation frequency of the nanodumbbell due to the symmetry of its shape. Figure 4(b) shows a PSD of the rotation of a 170-nm-diameter nanodumbbell at 7.9 105torr. The detected signal has a sharp peak near 2.2 GHz. This shows the nanodumbbell rotates at 1.1 GHz, which is much faster
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