Zeptonewton force sensing with nanospheres in an optical lattice

Optically trapped nanospheres in high vacuum experience little friction and hence are promising for ultrasensitive force detection. Here we demonstrate measurement times exceeding ${10}^5$ s and zeptonewton force sensitivity with laser-cooled silica nanospheres trapped in an optical lattice. The sensitivity achieved exceeds that of conventional room-temperature solid-state force sensors by over an order of magnitude, and enables a variety of applications including electric-field sensing, inertial sensing, and gravimetry. The particle is confined at the antinodes of the optical standing wave, and by studying the motion of a particle which has been moved to an adjacent trapping site, the known spacing of the antinodes can be used to calibrate the displacement spectrum of the particle. Finally, we study the dependence of the trap stability and lifetime on the laser intensity and gas pressure, and examine the heating rate of the particle in vacuum without feedback cooling.

Figure 1

(a) Standing-wave trap for 300 nm beads is formed using counterpropagating 1064 nm laser beams focused at nearly the same spatial location. Active feedback cooling is performed using 780 nm lasers (shown as green) in three dimensions. (b) Calculated optical force along the $z\mathrm{axis}$ assuming total power of $2.2$ W, waist of $8\mu \mathrm{m}$, and a $0.2\%$ intensity modulation due to interference from the counterpropagating beam, corresponding well with the measured trap frequencies. (c) Time trace for 1 s of particle motion in the axial direction at $P=2$ Torr. When subject to an applied sinusoidal optical force, the particle hops to an adjacent trapping site as a result of the perturbation. Dotted lines indicate expected antinode spacing.

Figure 2

Measured force on a bead as a function of averaging time at 2 Torr and $5\times {10}^{-6}$ Torr (HV) for charged and uncharged beads, while driving with a sinusoidally varying electric field of 1 kV/m. (Inset) Measured $x$ displacement spectrum of a 300 nm sphere at 2 Torr and HV with feedback cooling applied. Lorentzian fits indicate cooling to 460 mK at HV.

Figure 3

(a) Mean pressure at which beads are lost for various laser trapping intensities, with no feedback cooling applied. Statistics are shown for 30 beads of each size. (b) Calculated trap depth for 300 nm and $3\mu \mathrm{m}$ beads. (c) Trap lifetime at high-vacuum vs laser intensity for higher intensity trapping, along with linear fit. (d) Expected internal temperature rise in a 300 nm sphere vs time for laser intensities of ${10}^{10}$ and $2\times {10}^{10}$ W/${\mathrm{m}}^2$, respectively. The shaded bands are determined by varying ${\epsilon }_2=2.5\times {10}^{-7}$ to ${\epsilon }_2=1.0\times {10}^6$.

Figure 4

Measured $y$-displacement signal of the bead as optical feedback cooling is turned off under high vacuum conditions. “FB off” indicates the time of blocking feedback lasers. Arrows represent bead hopping to (a) and returning from (b) and then returning to (c) an adjacent lattice site (in the $z$ direction). (Inset) Variance signal with averaged moving window of 0.12 s and fit to exponential growth before bead loss.