Position Measurement of a Levitated Nanoparticle via Interference with Its Mirror Image

Interferometric methods for detecting the motion of a levitated nanoparticle provide a route to the quantum ground state, but such methods are currently limited by mode mismatch between the reference beam and the dipolar field scattered by the particle. Here we demonstrate a self-interference method to detect the particle’s motion that solves this problem. A Paul trap confines a charged dielectric nanoparticle in high vacuum, and a mirror retro-reflects the scattered light. We measure the particle’s motion with a sensitivity of $1.7\times {10}^{-12}\mathrm{m}/\sqrt{\mathrm{Hz}}$, corresponding to a detection efficiency of 2.1%, with a numerical aperture of 0.18. As an application of this method, we cool the particle, via feedback, to temperatures below those achieved in the same setup using a standard position measurement.

Figure 1

Basic principles of the self-homodyne technique. (a) The particle is illuminated by a laser beam and two confocal lenses collect the scattered light. The image created by one lens is reflected back to the particle by a distant flat mirror. The reflected light interferes with the directly scattered light, and the other lens images the two fields on a detector. The phase of the interferometric signal measured by the detector depends on the mutual particle-mirror distance along the $q$ axis. Inset shows a camera image of the nanoparticle and its mirror image. Here, for visualization purposes, the images are displaced from each other. (b) Typical detector output (blue) when the mirror is linearly displaced in time with a piezo translation stage. The driving ramp voltage of the mirror (red) occurs with a subhertz period. Insets show camera images obtained at maximum and minimum of the interferometric fringe, respectively. (c) Time trace of the particle’s displacement $q$ for the case of a fixed mirror position. Particle oscillations occur at frequencies of a few kilohertz.

Figure 2

Simultaneous self-homodyne and forward detection of a levitated nanoparticle. (a) Schematic overview of the setup. Lasers for the two detection methods are decoupled using polarization optics. For self-homodyne detection, the particle is illuminated with a weakly focused beam, and the particle’s motion along the detection axis $q$ is detected by an avalanche photodiode (APD). The APD signal is sent to feedback electronics consisting of a low-pass filter and a proportional–integral–derivative controller (PID) that stabilize the mirror position. For forward detection, light propagating along $\widehat{q}$ is focused onto the particle and then guided, together with the forward-scattered light from the particle, to a set of balanced photodiodes. The light scattered by the particle from the two lasers is collected by a pair of aspheric lenses ($\mathrm{NA}=0.18$), which also focus and collimate the 987 nm beam. The inset shows the Paul trap axes. (b) Power spectral density (PSD) of the radial motion of one particle acquired simultaneously with the forward-detection method (purple) and with the self-homodyne method (red). Dashed gray lines indicate the trap radial frequencies ${\nu }_x$ and ${\nu }_y$, the Paul trap’s drive frequency ${\nu }_{\text{drive}}$, and the first two harmonics of ${\nu }_{\text{drive}}$. The measurement imprecisions $S_{\mathrm{imp}}^{\mathrm{self}}$ and $S_{\mathrm{imp}}^{\mathrm{fw}}$ are indicated.

Figure 3

Motional spectra acquired with self-homodyne detection for different values of the total scattered power $P$ and with the same trapped particle. (a) Spectra over a frequency range that includes all detected mechanical resonances. (b) Closer view of the first radial resonance. (c) Closer view of the noise floor separation. Lower values of the noise floor are reached for higher values of the scattered power. Gain-dependent excess noise is subtracted. (d) Imprecision $S_{\mathrm{imp}}$ of self-homodyne detection as a function of the scattered power $P$ of the particle, plotted as triangles. Error bars correspond to the uncertainty in the NA estimation. The black solid line indicates the expected imprecision for ideal detection [19] and the purple solid line indicates the imprecision for the forward detection inferred from our measurements. The gray dashed line and area represent the theoretically expected imprecision and its uncertainty, respectively. The distance between the black and gray lines is the detection efficiency ${\eta }_{\det }$.

Figure 4

Feedback cooling of the particle’s radial motion. (a) Motional spectra acquired with self-homodyne detection for different electronic feedback gains. Black lines are fits to Lorentzians, used to determine the temperature and cooling rate for each of the two modes. (b) Temperature of the higher-frequency radial mode as a function of the cooling rate ${\gamma }_{\mathrm{fb}}$ obtained with the self-homodyne (blue diamonds) and forward-detection methods (purple circles). In the self-homodyne case, temperatures are fitted with $T_y=A/{\gamma }_{\mathrm{fb}}$, where $A=112(7)\mathrm{rad}\mathrm{K}$ (blue line). In the forward-detection case, the temperature fit function includes the effect of measurement imprecision in the feedback loop [6].