Control and measurement of electric dipole moments in levitated optomechanics

Levitated optomechanical systems are rapidly becoming leading tools for precision sensing, enabling a high level of control over the sensor's center-of-mass motion, rotation, and electric charge state. Higher-order multipole moments in the charge distribution, however, remain a major source of backgrounds. By applying controlled precessive torques to the dipole moment of a levitated microsphere in vacuum, we demonstrate cancellation of dipole-induced backgrounds by 2 orders of magnitude. We measure the dipole moments of nanogram-mass spheres and determine their scaling with sphere size, finding that the dominant torques arise from induced dipole moments related to dielectric-loss properties of the ${\mathrm{SiO}}_2$ spheres. Control of multipole moments in the charge distribution of levitated sensors is a key requirement to sufficiently reduce background sources in future applications.

Figure 1

Experimental setup and dipole control. (a) Ten-, 15-, and 20-$\mu \mathrm{m}$-diameter ${\mathrm{SiO}}_2$ microspheres are optically levitated between a pair of parallel electrodes, one of which is segmented to enable application of arbitrary electric fields and gradients. The spheres can be spun optically, using a circularly polarized trapping beam, or electrically by coupling their electric dipole moment to a rotating electric field. (b) Once the sphere reaches high angular frequencies a DC electric field can be applied to precess it, coupling to the remaining component of the dipole moment. One example is performing a sequence of “$\pi$ pulses,” inverting the direction of the dipole vector and canceling dipole-related backgrounds. The pulse length is given by $t_{\pi }=\pi /{\mathrm{\Omega }}_p$ [Eq. (1)].

Figure 2

Cancellation of backgrounds arising from dipole coupling to an externally applied electric field gradient. (a) Precession causes sinusoidal oscillations (blue circles, sine fit in the solid red line) in the angular acceleration of the sphere, as the axis of ${\mathrm{\Omega }}_s$ flips with respect to the constant optical torque. (b) The phase delay of the response of the sphere to a $2\pi \times 99$ Hz oscillating $\partial E_x/\partial z$ gradient (blue squares) exhibits a flip from in-phase to a $\pi$-phase lag, synchronous with the precession angle $\theta$ obtained from residual signal in the $y$ center-of-mass sensor (red circles). (c), (d) The force acting on the sphere in the $x$ direction, sampled at two opposing points in the precession [purple (dark) and yellow (light), corresponding to the dash-dotted lines in panel (b)], bandpass-filtered to 10 Hz around the drive frequency. Over the 100-ms window, the sum of the two [magenta line surrounded by dash-dotted rectangle, zoomed-in in panel (d)] demonstrates cancellation by a factor of $\sim 120$.

Figure 3

Measurement of the dipole moment using the precession and libration methods of Eqs. (1) and (2). The sphere's rotation is measured in response to an electric field of varying amplitude (color bar), rotating at a frequency of ${\mathrm{\Omega }}_s=2\pi \times 10$ kHz. (a) Libration is observed as sidebands around the second harmonic $2{\mathrm{\Omega }}_s$. (b) Precession is observed at lower frequencies. (c) Data from panels (a) and (b) are rescaled according to Eq. (2) (blue circles) and Eq. (1) (red squares), respectively. Yellow diamonds represent high-pressure lock-loss measurements rescaled according to Eq. (3) with $\beta =6.97\pm 0.07$. The inset shows the anticorrelated $1\sigma$ and $2\sigma$ confidence interval contours.

Figure 4

Scaling of the dipole moment with sphere mass. Ten-, 15-, and 20-$\mu \mathrm{m}$-diameter spheres are tested to compare the permanent (a) and induced (b) components of the total dipole. The dashed black line in panel (b) indicates the expected magnitude of $\alpha$ for a perfectly conducting sphere.