For small displacements, the effect of optical tweezers on the bead���s motion can be approxi- mated by a harmonic potential. The MSD of a Brownian particle in an underdamped harmon- ic trap in air can be obtained by solving the Langevin equation (14) ���[Dx(t)]2��� �� 2kBT mw0 2 1 ��� e���t=2tp cos w1t �� sin w1t 2w1tp ��1�� where w0 is the resonant frequency of the trap and w1��� w0 2 ��� 1/(2tp)2. q������������������������������������������������������������������������������ The normalized veloc- ity autocorrelation function (VACF) of the par- ticle is (14) y(t) �� e���t=2tp cos w1t ��� sin w1t 2w1tp ��2�� In the simplified scheme of our optical trap and vacuum chamber (Fig. 1), the trap is formed inside a vacuum chamber by two counterpropa- gating laser beams focused to the same point by two identical aspheric lenses with focal lengths of 3.1 mm and numerical apertures of 0.68 (15). The two 1064-nm-wavelength laser beams are orthogonally polarized, and their frequencies dif- fer by 160 MHz to avoid interference. The scat- tering forces exerted on the bead by the two beams cancel, and the gradient forces near the center of the focus create a three-dimensional harmonic potential for the bead. When the bead deviates from the center of the trap, it deflects both trapping beams. The position of the bead is monitored by measuring the deflection of one of the beams, which is split by a mirror with a sharp edge. The difference between the two halves is measured by a fast balanced detector (7, 16). The lifetime of a bead in our trap in air is much longer than our measurement times over a wide range of pressures and trap strengths. We have tested it by trapping a 4.7-mm bead in air con- tinuously for 46 hours, during which the power of both laser beams was repeatedly changed from 5 mW to 2.0 W. The trap becomes less stable in vacuum. The lowest pressure at which we have trapped a bead without extra stabilization is about 0.1 Pa. For studying the Brownian motion of a trapped bead, unless otherwise stated, the powers of the two laser beams were 10.7 and 14.1 mW (15), the diameter of the bead was 3 mm, the temperature of the system was 297 K, and the air pressure was 99.8 or 2.75 kPa. The trapping was stable and the heating due to laser absorption was negligible un- der these conditions. In typical samples of position and velocity traces of a trapped bead (Fig. 2), the position traces of the bead at these two pressures appear to be very similar. On the other hand, the velocity traces are clearly different. The instanta- neous velocity of the bead at 99.8 kPa changes more frequently than that at 2.75 kPa, because the momentum relaxation time is shorter at higher pressure. Figure 3 shows the MSDs of a 3-mm silica bead as a function of time. The measured MSDs fit with Eq. 1 over three decades of time for both pressures. The calibration factor a = position/ voltage of the detection system is the only fit- ting parameter of Eq. 1 for each pressure. tp and w0 are obtained from the measured normalized VACF. The two values of a obtained for these two pressures differ by 10.8%. This is because the vacuum chamber is distorted slightly when the pressure is decreased from 99.8 to 2.75 kPa. The measured MSDs are completely different from those predicted by Einstein���s theory of Brownian motion in a diffusive regime. The Fig. 3. (A) The MSDs of a 3-mm silica bead trapped in air at 99.8 kPa (red square) and 2.75 kPa (black circle). They are calculated from 40 mil- lion position measure- ments for each pressure. The ���noise��� signal (blue triangle) is recorded when there is no particle in the optical trap. The solid lines are theoretical predictions of Eq. 1. The prediction of Einstein���s theory of free Brownian motion in the diffusive regime is shown in dashed lines for com- parison. (B)MSDs at short time scales are shown in detail. The dash-dotted line indicates ballistic Brownian motion of a free particle. A B Fig. 2. One-dimensional trajectories of a 3-mm-diameter silica bead trapped in air at 99.8 kPa (A) and 2.75 kPa (B). The instantaneous velocities of the bead corresponding to these trajectories are shown in (C) and (D). 25 JUNE 2010 VOL 328 SCIENCE www.sciencemag.org 1674 REPORTS on August 12, 2010 www.sciencemag.org Downloaded from

slopes of measured MSD curves at short time scales are double those of the MSD curves of diffusive Brownian motion in the log-log plot (Fig. 3A). This is because the MSD is propor- tional to t 2 for ballistic Brownian motion, whereas it is proportional to t for diffusive Brownian mo- tion. In addition, the MSD curves are indepen- dent of air pressure at short time scales, which is predicted by �����Dx��t����2��� �� ��kBT=m��t2 for bal- listic Brownian motion, whereas the MSD in the diffusive regime does depend on the air pressure. At long time scales, the MSD saturates at a con- stant value because of the optical trap. Figure 3B displays 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 instanta- neous velocities (Fig. 4A) agree very well with the Maxwell-Boltzmann distribution. The mea- sured rms velocities are vrms = 0.422 mm/s at 99.8 kPa and vrms = 0.425 mm/s at 2.75 kPa. These values are very close to the prediction of the energy equipartition theorem, vrms �� kBT/m, p��������������������������������������� which is 0.429 mm/s. As expected, the velocity distribution is independent of pressure. The rms value of the noise signal is 0.021 mm/s, which means we have 1.0 �� spatial resolution in 5 ms. This measurement noise is about 4.8% of the rms velocity. Figure 4A represents direct verification of the Maxwell-Boltzmann distribution of veloc- ities and the equipartition theorem of energy for Brownian motion. For a Brownian particle in liquid, the inertial effects of the liquid become im- portant. The measured rms velocity of the particle will be vrms �� kBT/m* p��������������������������������������������� in the ballistic regime, where the effective mass m* is the sum of the mass of the particle and half of the mass of the displaced fluid (17). This is different from the equipar- tition theorem. To measure the true instantaneous velocity in liquid as predicted by the equiparti- tion theorem, the temporal resolution must be much shorter than the time scale of acoustic damping, which is about 1 ns for a 1-mm particle in liquid (17). Figure 4B shows the normalized VACF of the bead at two different pressures. At 2.75 kPa, one can see the oscillations due to the optical trap. Equation 2 is independent of the calibration factor a of the detection system. The only in- dependent variable is time t, which we can mea- sure with high precision. Thus the normalized VACF provides an accurate method to measure tp and w0. By fitting the normalized VACF with Eq. 2, we obtained tp = 48.5 T 0.1 ms, w0 = 2p �� (3064 T 4) Hz at 99.8 kPa and tp = 147.3 T 0.1 ms, w0 = 2p �� (3168 T 0.5) Hz at 2.75 kPa. The trapping frequency changed by 3% because of the distortion of the vacuum chamber at dif- ferent pressures. For a particle at a certain pres- sure and temperature, tp should be independent of the trapping frequency. We verified this by changing the total power of the two laser beams from 25 to 220 mW. The measured tp changed less than 1.3% for both pressures, thus proving that the fitting method is accurate, and the heat- ing due to the laser beams (which would change the viscosity and affect tp) is negligible. We can also calculate the diameter of the silica bead from the tp value at 99.8 kPa (18). The obtained diameter is 2.79 mm. This is within the uncer- tainty range given by the supplier of the 3.0-mm silica beads. We used this value in the calcu- lation of MSD and normalized VACF. The ability to measure the instantaneous ve- locity of a Brownian particle will be invaluable in studying nonequilibrium statistical mechanics (19, 20) and can be used to cool Brownian mo- tion by applying a feedback force with a direction opposite to the velocity (21, 22). In a vacuum, our optically trapped particle should be an ideal system for investigating quantum effects in a mechanical system (16, 23���25) because of its near-perfect isolation from the thermal environ- ment. Combining feedback cooling and cavity cooling, we expect to cool the Brownian motion of a bead starting from room temperature to the quantum regime, as predicted by recent theoret- ical calculations (24, 25). We have directly ver- ified the energy equipartition theorem of Brownian motion. However, we also expect to observe de- viation from this theorem when the bead is cooled to the quantum regime. The kinetic energy of the bead will not approach zero even at 0 K because of its zero-point energy. The rotational energy of the bead should also become quantized. References and Notes 1. A. Einstein, Zeit. f. Elektrochemie 13, 41 (1907). 2. A. Einstein, Investigations on the Theory of the Brownian Movement, R. F��rth, Ed., A. D. Cowper, Transl. (Methuen, London, 1926), pp. 63���67. 3. F. M. Exner, Ann. Phys. 2, 843 (1900). 4. M. Kerker, J. Chem. Educ. 51, 764 (1974). 5. B. Luki�� et al., Phys. Rev. Lett. 95, 160601 (2005). 6. Y. Han et al., Science 314, 626 (2006). 7. I. Chavez, R. Huang, K. Henderson, E.-L. Florin, M. G. Raizen, Rev. Sci. Instrum. 79, 105104 (2008). 8. P. D. Fedele, Y. W. Kim, Phys. Rev. Lett. 44, 691 (1980). 9. J. Blum et al., Phys. Rev. Lett. 97, 230601 (2006). 10. D. R. Burnham, P. J. Reece, D. McGloin, Brownian dynamics of optically trapped liquid aerosols. In press preprint available at http://arxiv.org/abs/0907.4582. 11. A. Einstein, Ann. Phys. 17, 549 (1905). 12. P. Langevin, C. R. Acad. Sci. (Paris) 146, 530 (1908). 13. G. E. Uhlenbeck, L. S. Ornstein, Phys. Rev. 36, 823 (1930). 14. M. C. Wang, G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945). 15. Materials and methods are available as supporting material on Science online. 16. K. G. Libbrecht, E. D. Black, Phys. Lett. A 321, 99 (2004). 17. R. Zwanzig, M. Bixon, J. Fluid Mech. 69, 21 (1975). 18. A. Moshfegh, M. Shams, G. Ahmadi, R. Ebrahimi, Colloids Surf. A Physicochem. Eng. Asp. 345, 112 (2009). 19. R. Kubo, Science 233, 330 (1986). 20. G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, D. J. Evans, Phys. Rev. Lett. 89, 050601 (2002). 21. A. Hopkins, K. Jacobs, S. Habib, K. Schwab, Phys. Rev. B 68, 235328 (2003). 22. D. Kleckner, D. Bouwmeester, Nature 444, 75 (2006). 23. A. Ashkin, J. M. Dziedzic, Appl. Phys. Lett. 28, 333 (1976). 24. D. E. Chang et al., Proc. Natl. Acad. Sci. U.S.A. 107, 1005 (2010). 25. O. Romero-Isart, M. L. Juan, R. Quidant, J. Ignacio Cirac, N. J. Phys. 12, 033015 (2010). 26. M.G.R. acknowledges support from the Sid W. Richardson Foundation and the R. A. Welch Foundation grant number F-1258. D.M. acknowledges support from El Consejo Nacional de Ciencia y Tecnolog��a (CONACYT) for his graduate fellowship (206429). The authors would also like to thank E.-L. Florin and Z. Yin for helpful discussions and I. Popov for his help with the experiment. Supporting Online Material www.sciencemag.org/cgi/content/full/science.1189403/DC1 Materials and Methods 10 March 2010 accepted 10 May 2010 Published online 20 May 2010 10.1126/science.1189403 Include this information when citing this paper. Fig. 4. (A) The distribu- tion of the measured in- stantaneous velocities of a 3-mm silica bead. The statistics at each pressure is calculated from 4 mil- lion instantaneous veloc- ities. The solid lines are Maxwell-Boltzmann dis- tributions. We obtained vrms = 0.422 mm/s at 99.8 kPa (red square) and vrms = 0.425 mm/s at 2.75 kPa (black circle) from the measurements. The rms value of the noise (blue triangle) is 0.021 mm/s. (B) The normalized velocity autocorrelation functions of the 3-mm bead at two different pressures. The solid lines are fittings with Eq. 2. 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