Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum

We describe the implementation of laser-cooled silica microspheres as force sensors in a dual-beam optical dipole trap in high vacuum. Using this system we have demonstrated trap lifetimes exceeding several days, attonewton force detection capability, and wide tunability in trapping and cooling parameters. Measurements have been performed with charged and neutral beads to calibrate the sensitivity of the detector. This work establishes the suitability of dual-beam optical dipole traps for precision force measurement in high vacuum with long averaging times, and enables future applications including the study of gravitational inverse square law violations at short range, Casimir forces, acceleration sensing, and quantum optomechanics.

Figure 1

The optical dipole trap is created by focusing two orthogonally polarized laser beams using 50 mm focal length inside a vacuum chamber. A 780 nm laser provides active feedback cooling to stabilize the trapped beads in vacuum and provide optical damping. Inset: A set of calibration electrodes is used to conduct force measurements with a known applied electric field at the location of the trap.

Figure 2

Optical force along the laser axis with coincident foci (left) and offset foci (right) for various power imbalance between the $S$- and $P$-polarized beams. Power imbalance is defined so that, e.g., $1\%$ indicates $51\%$ of the power is in the $P$-polarized beam. The total laser power is $2.2$ W and the waist is $9\mu \mathrm{m}$.

Figure 3

(a) Mean pressure at which beads are lost from the trap for various laser trapping intensities, with no laser-feedback cooling applied. Statistics are shown for 30 beads. (b) Calculated internal temperature (right) and photothermal force (left) assuming ${\epsilon }_2={10}^{-6}$ and $1\%$ internal temperature gradient across the bead. (c) Measured damping rate along $x$ versus gas pressure for two different beads (dashed line). Calculated damping rate for a bead of diameter $3.0\mu \mathrm{m}$ in ${\mathrm{N}}_2$ gas. (d) Measured center-of-mass temperature in the $x$ direction versus pressure. For data shown in (b)–(d), $I_{\mathrm{trap}}=2\times {10}^9$ W/${\mathrm{m}}^2$.

Figure 4

(a) Typical position spectrum of a bead held at low vacuum of $1.7$ Torr with no feedback cooling applied (red), and at high vacuum of $5\times {10}^{-6}$ Torr with feedback cooling applied (blue). Also shown (light blue) is a Lorentzian fit to the peaks in the high vacuum data, with fit parameters as discussed in the text. (b) $x$ spectrum of a bead at $5\times {10}^{-6}$ Torr with varying feedback cooling rates.

Figure 5

(a) On-resonance horizontal ($x$) force on the bead at $5\times {10}^{-6}$ Torr, versus averaging time, with no applied forces. (b) Force on a charged bead ($2e^{-}$) and neutral bead ($0e^{-}$) versus averaging time for varying applied driving voltage $V_{\text{ac}}$. (c) Force on beads of varying charge (neutral, $2e^{-}, 18\pm 2e^{-}$, and $41\pm 5e^{-}$) versus applied driving voltage $V_{\text{ac}}$ and corresponding electric field with 100 s averaging time. Data are shown at $V_{\text{ac}}=2,4,8,12,20,24,28$ V.