Revealing the Velocity Uncertainties of a Levitated Particle in the Quantum Ground State

We demonstrate time-of-flight measurements for an ultracold levitated nanoparticle and reveal its velocity for the translational motion brought to the quantum ground state. We discover that the velocity distributions obtained with repeated release-and-recapture measurements are significantly broadened via librational motions of the nanoparticle. Under feedback cooling on all the librational motions, we recover the velocity distributions in reasonable agreement with an expectation from the occupation number, with approximately twice the width of the quantum limit. The strong impact of librational motions on the translational motions is understood as a result of the deviation between the libration center and the center of mass, induced by the asymmetry of the nanoparticle. Our results elucidate the importance of the control over librational motions and establish the basis for exploring quantum mechanical properties of levitated nanoparticles in terms of their velocity.

Figure 1

Overview of the experiments. (a) A levitated nanoparticle, whose motional degree of freedom is cooled to the ground state, is released from a harmonic trap and recaptured to the same trap. From the amplitude of the oscillation, the displacement during the TOF is derived. In the presence of librational motions in an optical trap, the nanoparticle can rotate during the TOF. (b) Schematic of the experimental setup. A nearly spherical nanoparticle is trapped in an optical lattice. Three translational motions are feedback-cooled via optical cold damping, while three librational motions are electrically feedback-cooled. To turn off the light for the TOF measurements, an acousto-optic modulator (AOM) is used. Optical cold damping is realized with an electro-optic modulator (EOM) for the $z$ direction and with an AOM for the $x$ and $y$ directions. In this Letter, we explore the motions along the $z$ direction with the oscilloscope.

Figure 2

TOF measurements. (a) Time sequence of the TOF measurements. Feedback cooling for translational and librational motions is turned off during the TOF measurements. (b) A typical oscillation signal after the nanoparticle is recaptured. In the inset, an expanded view is shown. The signal is obtained through a high-pass filter such that the motions in $x$ and $y$ directions are excluded from the signal. (c) Velocity distribution for $n_z=0.80$ with LC. The solid line is a Gaussian fit. (d) Velocity distribution for $n_z=0.87$ without LC. The width of the distribution is significantly broadened by the presence of librational motions in the trap. The solid line is a Gaussian fit.

Figure 3

Measured velocity width with respect to the occupation number. The vertical error bars reflect both statistical errors in fitting the distributions and systematic errors in calibrating the displacement, while the horizontal error bars indicate systematic errors in temperature measurements. The blue solid line shows calculations with Eq. (2) with $\mathrm{\Delta }\omega =0$, where the uncertainty in calculations due to the error in $m$ is shown by shaded area. With LC, the observed velocity widths are in reasonable agreement with the calculations. Without LC, the velocity widths are significantly broader than the calculation. The red solid line shows a fit on the results without LC via Eq. (2). The dashed line shows the quantum limited velocity uncertainty of $\sqrt{\hslash {\mathrm{\Omega }}_z/m}$.

Figure 4

Comparison between two models to explain the observed broadening via librational motions. (a) Definition of coordinates for the first model at $t=0$. (b) Definition of coordinates for the second model at $t=0$. (c) Numerically obtained histogram for the first model. The parameter ${ϵ}_1$ is set to 6.7 nm to reproduce the observed broadening. The solid line is a Gaussian fit. (d) Numerically obtained histogram for the second model. The parameter ${ϵ}_2$ is set to 0.29 nm to reproduce the observed broadening. The solid line is a Gaussian fit.

Figure 5

Deviation of the libration center from the COM. (a) Definition of coordinates for an asymmetric nanoparticle to derive the libration center. We consider an asymmetric nanoparticle obtained by attaching a semisphere (lower half; radius of $a$) and a prolate semispheroid (upper half; major semiaxis $c$ and minor semiaxis $a$) at the $r=-r_0$ plane. The COM lies at the origin of the $pqr$ coordinate. The $xyz$ coordinates are defined by the optical trap (lab frame) and are related to the $pqr$ coordinates by $(p,q,r)=\{-z\cos \psi +(x-{ϵ}_2)\sin \psi ,y,z\sin \psi +{ϵ}_2+(x-{ϵ}_2)\cos \psi \}$. The nanoparticle is rotated around an axis parallel to the $y$ axis, which passes through the point $(x,y,z)=({ϵ}_2,0,0)$, by $\psi$. Because of the rotation, the COM does not lie at the origin of the $xyz$ coordinate. The coordinates $(p,q,r)$ are used for integrating the optical potential within the nanoparticle. The libration center lies at $(p,q,r)=(0,0,{ϵ}_2)$. (b) Calculated ${ϵ}_2$ as a function of the asymmetry of the nanoparticle $c/a$. The deviation between the libration center and the COM is calculated for the geometry shown in (a). The radius $a$ is assumed to be 174 nm. The COM motional frequencies are set to ${\mathrm{\Omega }}_x/2\pi =62\mathrm{kHz}$ and ${\mathrm{\Omega }}_z/2\pi =209\mathrm{kHz}$. These values are required for calculating ${ϵ}_2$ via Eq. (7) in the Supplemental Material [27].