Free expansion of a charged nanoparticle via electrostatic compensation

Abstract

Coherent wavepacket expansion is a key component of recent proposals aiming to create non-classical states of a levitated dielectric nanoparticle. Free evolution, i.e., releasing the particle from its harmonic trapping potential and allowing its position variance to grow, is a simple but effective expansion scheme, but requires accurate force compensation to avoid significant mean displacements of the wavepacket during the free evolution. Here, using an optical trap-release-recapture sequence, we demonstrate an electrostatic compensation technique that enables free evolution without significant mean displacement in 3D, successfully compensating both gravity and stray electric fields. To achieve 100 microsecond free evolution times with charged particles, we developed methods to map and correct for force cross-talk, as well as to control the environmental charge state. Combined with a low decoherence environment, our approach enables the preparation of largely delocalized states without the need for long freefall trajectories or a low-gravity environment.

Figures (5)

• Figure 1: The electrostatic compensation procedure for the axial direction. Left: The three panels show the experimental sequence, with only the axial electrode pair illustrated: cooling (trap + feedback + DC bias), free evolution (DC bias) and recapture (trap + DC bias). The corresponding phase space distribution at each step is also illustrated. Right: A measurement showing the increase in mean energy over the sequence as a function of the applied DC bias in the axial ($z$) direction, with five repetions per voltage. A parabolic fit (black line) indicates that the optimal compensation voltage is 0.04 V in this example. Insets are sample timetraces at the various biases, with blue indicating the time before free evolution, red the time after recapture, and gray the free evolution plus a buffer to account for the detector response.
• Figure 2: The effect of force cross-talk on axial compensation scans. We compute power spectral densities using 5 ms of data before and after the 10 µs free evolutions to isolate the response of the individual motional modes. The $z$ mode shows the expected parabolic behavior (blue), while the $x$ mode shows a clear dependence on $V_z$ due to cross-talk (red dashed line). When the voltages are instead scanned in a proportion given by the third column of $C^{-1}$ (solid lines), the cross-talk into $x$ is significantly reduced. Each point contains five repetitions and error bars are standard errors of the mean.
• Figure 3: Spectrogram of a single 200 µs (a) and a single 300 µs (b) free evolution followed by recapture as measured in the back-scattered homodyne detection. The trap and feedback are switched off at $t=0$ and re-enabled simultaneously after the free expansion. Excitation and re-cooling of the motional modes $\Omega_{x,y,z}/(2\pi) = [302, 268, 92]$ kHz is visible upon recapture. In the 300 µs spectrogram, a several kHz frequency drop and slow recovery is also visible, likely due to imperfect radial compensation and a resulting exploration of trap nonlinearites (see Supplementary Material).
• Figure 4: The relative mean energy increase of the axial mode as a function of $\tau$ at pressures below $8\cdot10^{-8}$ mbar. For force-free evolution, this should evolve as $1+\Omega^2\tau^2/2$ (blue line). The data is consistent with a $2\cdot 10^{-18}$ N residual force (blue dashed line, corresponding to a 2.5 nm mean displacement over 100 µs) but strongly excludes larger forces. For example, the gray line corresponds to a force which would yield mean displacements of 10 nm over 100 µs. The mean energy is extracted from the squared amplitudes of a sine fit, with each data point corresponding to 150 repetitions and error bars the standard error of the mean.
• Figure 5: Growth of the position standard deviation of the axial mode as a function of $\tau$ for the same dataset as Fig. \ref{['fig:meanE']}. The state variance evolution is extracted by analyzing the statistics of the sine-fitted trajectories upon recapture. We plot the maximum of the standard deviation as a function of $\tau$, which grows as $\Omega\tau$ for large $\tau$ (blue dashed line), with the full theory the blue solid line (the effect of gas and recoil is negligible). The occupancy $\bar{n}=117$ used in the theory curve is estimated by fitting to $z_{\text{zp}}^2(2\bar{n}+1)(1+\Omega^2\tau^2)$, which agrees to within 10% with the occupancy estimated via integration of the calibrated spectrum. Each point contains 150 repetitions, with error bars estimated via binning (see Supplementary Material).