Measurement of the Instantaneous Velocity of a Brownian Particle

Transcription

1 Measurement of the Instantaneous Velocity of a Brownian Particle Tongcang Li, Simon Kheifets, David Medellin, Mark G. Raizen* Center for Nonlinear Dynamics and Deartment of Physics, University of Texas at Austin, Austin, TX 787, USA. *To whom corresondence should be addressed. raizen@hysics.utexas.edu Brownian motion of articles imacts many branches of science. We reort on the Brownian motion of micronsized beads of glass held in air in an otical tweezer, over a wide range of ressures, and measure the instantaneous velocity of a Brownian article. Our results rovide direct verification of the energy equiartition theorem for a Brownian article. For short times, the ballistic regime of Brownian motion is observed, in contrast to the usual diffusive regime. We discuss the alications of these methods towards cooling the center of mass motion of a bead in vacuum to the quantum ground motional state. In 907, Albert Einstein ublished a aer in which he considered the instantaneous velocity of a Brownian article (,). By measuring this quantity, one could rove that the kinetic energy of the motion of the centre of gravity of a article is indeendent of the size and nature of the article and indeendent of the nature of its environment. This is one of the basic tenets of statistical mechanics, known as the equiartition theorem. However, due to the very raid randomization of the motion, Einstein concluded that the instantaneous velocity of a Brownian article would be imossible to measure in ractice. We reort here on the measurement of the instantaneous velocity of a Brownian article in a system consisting of a single, micrometer-sized SiO bead held in a dual-beam otical tweezer in air, over a wide range of ressures. The velocity data was used to verify the Maxwell-Boltzmann velocity distribution, and the equiartition theorem for a Brownian article. The ability to measure instantaneous velocity enables new fundamental tests of statistical mechanics of Brownian articles, and is also a necessary ste towards cooling of a article to the quantum ground motional state in vacuum. The earliest quantitative studies of Brownian motion were focused on measuring velocities, and generated enormous controversy (3,4). The measured velocities of Brownian articles (3) were almost 000-fold smaller than redicted by the energy equiartition theorem. Recent exeriments on Brownian motion in liquid (5 7) and gaseous environments (8 0) with fast detectors have observed nondiffusive motion of a Brownian article. Einstein's theory redicts that [ Δ x( = Dt, where [ Δx( is the mean-square dislacement (MSD) in one dimension of a free Brownian article during time t, and D is the diffusion constant (). The diffusion constant can be calculated by D = kbt / γ, where k B is Boltzmann's constant, T is the temerature, and γ is the Stokes friction coefficient. The mean velocity measured over an interval of [ ] t time t is v Δx( t) / t = D /. This diverges as t aroaches 0, and therefore does not reresent the real velocity of the article (,). The equation [ Δ x( = Dt, however, is only valid when t >> τ, i.e., in the diffusive regime. Here τ = m /γ is the momentum relaxation time of a article with mass m. At very short time scales (t << τ ), the dynamics of a article is dominated by its inertia, and the motion is ballistic. The dynamics of a Brownian article over all time scales can be described by a Langevin equation (). The MSD of a Brownian article at very short time scales is redicted to be [ Δ x( = ( kbt / m) t, and its instantaneous velocity can be measured as v = Δx( t) / t, when t << τ (3). For a μm diameter silica (SiO ) shere in water, τ is about 0. μs and the root mean square (rms) velocity is about mm/s in one dimension. To measure the instantaneous velocity with 0% uncertainty, one would require m satial resolution in 0 ns, far beyond what is exerimentally achievable today (7). Due to the lower viscosity of gas, comared to liquid, the τ of a article in air is much larger. This lowers the technical demand for both temoral and satial resolution. The main difficulty of erforming high recision measurements of a Brownian article in air, however, is that the article will fall under the influence of gravity. We overcome this roblem by using otical tweezers to simultaneously tra and monitor a silica bead in air and vacuum, allowing long duration ultra-high-resolution measurements of its motion. For small dislacements, the effect of otical tweezers on the bead s motion can be aroximated by a harmonic otential. The MSD of a Brownian article in an underdamed harmonic tra in air can be obtained by solving the Langevin equation (4): / / 0 May 00 / Page / 0.6/science.89403

2 kbt t / τ sinω t [ Δx( = e cosω t + () mω0 ωτ where ω 0 is the resonant frequency of the tra, and ω ω0 /(τ ). The normalized velocity autocorrelation function (VACF) of the article is (4) = t / τ sinω t ψ ( t) e cosω t () ωτ In the simlified scheme of our otical tra and vacuum chamber (Fig. ), the tra is formed inside a vacuum chamber by two counter-roagating laser beams focused to the same oint by two identical asheric lenses with focal length of 3. mm and numerical aerture of 0.68 (5). The two 064 nm laser beams are orthogonally olarized, and their frequencies differ by 60 MHz to avoid interference. The scattering forces exerted on the bead by the two beams cancel, while the gradient forces near the center of the focus create a three dimensional harmonic otential for the bead. When the bead deviates from the center of the tra, it deflects both traing beams. The osition of the bead is monitored by measuring the deflection of one of the beams, which is slit by a mirror with a shar edge. The difference between the two halves is measured by a fast balanced detector (7,6). The lifetime of a bead in our tra in air is much longer than our measurement times over a wide range of ressures and tra strengths. We have tested it by traing a 4.7 μm bead in air continuously for 46 hours, during which the ower of both laser beams was reeatedly changed from 5 mw to.0 W. The tra becomes less stable in vacuum. The lowest ressure at which we have traed a bead without extra stabilization is about 0. Pa. For studying the Brownian motion of a traed bead, unless otherwise stated, the owers of the two laser beams are 0.7 mw and 4. mw (5), the diameter of the bead is 3 μm, the temerature of the system is 97 K, and the air ressure is 99.8 kpa or.75 kpa. The traing is stable, and the heating due to laser absortion is negligible under these conditions. In tyical samles of osition and velocity traces of a traed bead (Fig. ), the osition traces of the bead at these two ressures aear very similar. On the other hand, the velocity traces are clearly different. The instantaneous velocity of the bead at 99.8 kpa changes more frequently than that at.75 kpa, because the momentum relaxation time is shorter at higher ressure. Figure 3 shows the mean square dislacements of a 3 μm silica bead as a function of time. The measured MSD's fit with Eq. over three decades of time for both ressures. The calibration factor α osition/ voltage of the detection system is the only fitting arameter of Eq. for each ressure. τ and ω 0 are obtained from the measured normalized VACF. The two α's obtained for these two ressures differ by 0.8%. This is because the vacuum chamber is distorted slightly when the ressure is decreased from 99.8 kpa to.75 kpa. The measured MSD's are comletely different from those redicted by Einstein s theory of Brownian motion in a diffusive regime. The sloes of measured MSD curves at short time scales are double those of the MSD curves of diffusive Brownian motion in the loglog lot (Fig. 3A). This is because the MSD is roortional to t for ballistic Brownian motion, while it is roortional to t for diffusive Brownian motion. Another imortant feature is that the MSD curves are indeendent of air ressure at short time scales, as is redicted by [ Δ x( = ( kbt / m) t for ballistic Brownian motion, whereas the MSD in the diffusive regime does deend on the air ressure. At long time scales, the MSD saturates at a constant value because of the otical tra. Fig. 3B dislays more detail of the Brownian motion at short time scales. It clearly demonstrates that we have observed ballistic Brownian motion. The distributions of the measured instantaneous velocities (Fig. 4A) agree very well with the Maxwell-Boltzmann distribution. The measured rms velocities are = 0.4 mm/s at 99.8 kpa and = 0.45 mm/s at.75 kpa. These are very close to the rediction of the energy equiartition theorem, v = rms kbt / m, which is 0.49 mm/s. As exected, the velocity distribution is indeendent of ressure. The rms value of the noise signal is 0.0 mm/s, which means we have.0 satial resolution in 5 μs. This measurement noise is about 4.8% of the rms velocity. Fig. 4A reresents direct verification of the Maxwell-Boltzmann distribution of velocities and the equiartition theorem of energy for Brownian motion. For a Brownian article in liquid, the inertial effects of the liquid become imortant. The measured rms velocity of the article will be * = k B T / m in the ballistic regime, where the effective mass m * is the sum of the mass of the article and half the mass of the dislaced fluid (7). This is different from the equiartition theorem. In order to measure the true instantaneous velocity in liquid as redicted by the equiartition theorem, the temoral resolution must be much shorter than the time scale of acoustic daming, which is about ns for a μm article in liquid (7). Figure 4B shows the normalized VACF of the bead at two different ressures. At.75 kpa, one can see the oscillations due to the otical tra. Eq. is indeendent of the calibration factor α of the detection system. The only indeendent variable is time t, which we can measure with high recision. Thus the normalized VACF rovides an accurate method to measure τ and ω 0. Fitting the normalized VACF with Eq., we obtained τ = 48.5 ± 0. μs, ω 0 = π (3064 ± 4) Hz at 99.8 kpa, and τ = 47.3 ± 0. μs, ω 0 = π (368 ± 0.5) Hz at.75 kpa. The traing frequency changed by 3% due to the distortion of the vacuum chamber at different ressures. For a article at a certain ressure and temerature, τ should be indeendent of the traing frequency. We verified this by changing the total ower of the two laser beams from 5 mw / / 0 May 00 / Page / 0.6/science.89403

3 to 0 mw. The measured τ changed less than.3% for both ressures, thus roving that the fitting method is accurate, and the heating due to the laser beams (which would change the viscosity and affect τ ) is negligible. We can also calculate the diameter of the silica bead from the τ value at 99.8 kpa (8). The obtained diameter is.79 μm. This is within the uncertainty range given by the sulier of 3.0 μm silica beads. We use this value in the calculation of MSD and normalized VACF. The ability to measure the instantaneous velocity of a Brownian article will be invaluable in studying nonequilibrium statistical mechanics (9,0) and can be used to cool Brownian motion by alying a feedback force with a direction oosite to the velocity (,). In vacuum, our otically traed article romises to be an ideal system for investigating quantum effects in a mechanical system (6,3 5), due to its near-erfect isolation from the thermal environment. Combining feedback cooling and cavity cooling, we exect to cool the Brownian motion of a bead starting from room temerature to the quantum regime, as redicted by recent theoretical calculations (4,5). We have directly verified the energy equiartition theorem of Brownian motion. However, we also exect to observe deviation from this theorem when the bead is cooled to the quantum regime. The kinetic energy of the bead will not aroach zero even at 0 K because of its zero-oint energy. The rotational energy of the bead should also become quantized. References and Notes. A. Einstein, Theoretische Bemerkungen über die Brownsche Bewegung. Zeit. f. Elektrochemie 3, 4 (907).. A. Einstein, Investigations on the Theory of the Brownian Movement, R. Fürth, Ed., A. D. Cower, Transl. (Methuen, London, 96), F. M. Exner, Ann. d. Phys., 843 (900). 4. M. Kerker, Brownian movement and molecular reality rior to 900. J. Chem. Educ. 5, 764 (974). 5. B. Lukić et al., Direct observation of nondiffusive motion of a Brownian article. Phys. Rev. Lett. 95, 6060 (005). 6. Y. Han et al., Brownian motion of an ellisoid. Science 34, 66 (006). 7. I. Chavez, R. Huang, K. Henderson, E.-L. Florin, M. G. Raizen, Develoment of a fast osition-sensitive laser beam detector. Rev. Sci. Instrum. 79, 0504 (008). 8. P. D. Fedele, Y. W. Kim, Direct measurement of the velocity autocorrelation function for a Brownian test article. Phys. Rev. Lett. 44, 69 (980). 9. J. Blum et al., Measurement of the translational and rotational Brownian motion of individual articles in a rarefied gas. Phys. Rev. Lett. 97, 3060 (006). 0. D. R. Burnham, P. J. Reece, D. McGloin, Brownian dynamics of otically traed liquid aerosols. htt://arxiv.org/abs/ (009).. A. Einstein, Ann. d. Phys. 7, 549 (905).. P. Langevin, C. R. Acad. Sci. (Paris), 46, 530 (908). 3. G. E. Uhlenbeck, L. S. Ornstein, On the theory of the Brownian motion. Phys. Rev. 36, 83 (930). 4. M. C. Wang, G. E. Uhlenbeck, On the theory of the Brownian motion II. Rev. Mod. Phys. 7, 33 (945). 5. Materials and methods are available as suorting material on Science online. 6. K. G. Libbrecht, E. D. Black, Toward quantum-limited osition measurements using otically levitated microsheres. Phys. Lett. A 3, 99 (004). 7. R. Zwanzig, M. Bixon, Comressibility effects in the hydrodynamic theory of Brownian motion. J. Fluid Mech. 69, (975). 8. A. Moshfegh, M. Shams, G. Ahmadi, R. Ebrahimi, A novel surface-sli correction for microarticles motion. Colloids and Surfaces A: Physicochem. and Eng. Asects 345, (009). 9. R. Kubo, Brownian motion and nonequilibrium statistical mechanics. Science 33, 330 (986). 0. G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, D. J. Evans, Exerimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys. Rev. Lett. 89, (00).. A. Hokins, K. Jacobs, S. Habib, K. Schwab, Feedback cooling of a nanomechanical resonator. Phys. Rev. B 68, 3538 (003).. D. Kleckner, D. Bouwmeester, Sub-kelvin otical cooling of a micromechanical resonator. Nature 444, 75 (006). 3. A. Ashkin, J. M. Dziedzic, Otical levitation in high vacuum. Al. Phys. Lett. 8, 333 (976). 4. D. E. Chang et al., Cavity oto-mechanics using an otically levitated nanoshere. Proc. Natl. Acad. Sci. USA 07, 005 (00). 5. O. Romero-Isart, M. L. Juan, R. Quidant, J. Ignacio Cirac, Towards quantum suerosition of living organisms. New J. Phys., (00). 6. MGR acknowledges suort from the Sid W. Richardson Foundation and the R. A. Welch Foundation grant no. F- 58. DM acknowledges suort from CONACYT for his graduate fellowshi (0649). The authors would also like to thank E.-L. Florin and Zhangqi Yin for helful discussions, and I. Poov for his hel in the exeriment. Suorting Online Material Materials and Methods 0 March 00; acceted 0 May 00 Published online 0 May 00; 0.6/science Include this information when citing this aer Fig. Simlified schematic showing the counter-roagating dual-beam otical tweezers, and a novel detection system that has 75 MHz bandwidth and ultra-low noise. The s-olarized beam is reflected by a olarizing beam slitter cube after it asses through a traed bead inside a vacuum chamber. For detection, it is slit by a mirror with a shar edge. The - olarized beam asses through the cube. Fig. One-dimensional trajectories of a 3 μm diameter silica bead traed in air at 99.8 kpa (A) and.75 kpa (B). The / / 0 May 00 / Page 3 / 0.6/science.89403

4 instantaneous velocities of the bead corresonding to these trajectories are shown in (C) and (D). Fig 3. (A) The mean square dislacements of a 3 μm silica bead traed in air at 99.8 kpa (red square) and.75 kpa (black circle). They are calculated from 40 million osition measurements for each ressure. The noise signal (blue triangle) is recorded when there is no article in the otical tra. The solid lines are theoretical redications of Eq.. The rediction of Einstein's theory of free Brownian motion in the diffusive regime is shown in dashed lines for comarison. (B) MSD's at short time scales are shown in detail. The dash-dot line indicates ballistic Brownian motion of a free article. Fig 4. (A) The distribution of the measured instantaneous velocities of a 3 μm silica bead. The statistics at each ressure is calculated from 4 million instantaneous velocities. The solid lines are Maxwell-Boltzmann distributions. We obtained = 0.4 mm/s at 99.8 kpa (red square) and = 0.45 mm/s at.75 kpa (black circle) from the measurements. The rms value of the noise (blue triangle) is 0.0 mm/s. (B) The normalized velocity autocorrelation functions of the 3 μm bead at two different ressures. The solid lines are fittings with Eq.. / / 0 May 00 / Page 4 / 0.6/science.89403

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