Stationary entanglement of a levitated oscillator with an optical field

Abstract

We report the generation of quantum entanglement between the center-of-mass motion of a levitated nanosphere, coupled by coherent scattering to an optical cavity mode, and the electromagnetic field. Using heterodyne detection, we reconstruct the full set of optical-mechanical correlations and observe a violation of separability bounds between the mechanical degrees of freedom and the propagating optical mode. Thus, we demonstrate the ability to distribute nonclassical correlations beyond the interaction region. Our results are obtained at room temperature and are robust over a broad range of detunings set by the cavity linewidth. These findings establish levitated optomechanical systems as a promising platform for macroscopic quantum optics and for future tests of fundamental physics.

Figures (4)

• Figure 1: Experimental configuration. The motion of a levitated nanoparticle is coupled to the field of an optical cavity via coherent scattering. The confining potential is obtained by overlapping in the optical tweezer two fields, $A$ and $B$, with different frequencies. This enables simultaneous driving of the cavity on opposite sides of its resonances, with detunings $\Delta_A$ (red) and $\Delta_B$ (blue): the former cools and stabilizes the motion, while the latter generates entanglement between the mechanical and optical subsystems. a) The detuning of the $A$ and $B$ fields is set with respect to different longitudinal modes separated by two free spectral ranges (FSR) at a value close to the mechanical bright mode frequency $\Omega_b$. b) This is achieved by phase locking (PL) both trapping lasers to an auxiliary laser, frequency stabilized to a cavity resonance using a Pound-Drever-Hall (PDH) locking scheme. The cavity output fields are measured in a heterodyne configuration using the same balanced detection (BHD). We use two different local oscillators frequencies, detuned by 1.4 MHz and 2.0 MHz with respect to the corresponding trapping fields by means of acousto-optical modulators (AOM). The heterodyne signals are recorded after a high-resolution analog-to-digital converter (ADC). The inset depicts the beam shape of the tweezer (polarized along Y) in the transverse plane. The label "b" ("d") refers to "bright" ("dark") and corresponds to the direction along (orthogonal to) the cavity axis. Measurements are performed in UHV environment, at a pressure of $\simeq3.5\times10^{-8}$ mbar.
• Figure 2: Optomechanical correlations. a) First row of the complex-valued spectral matrix $\mathbf{A}$, shown around the bright mode sidebands, along with the fitted theoretical model (dashed-red line). Each column in the figure represent the real (top) and imaginary (bottom) part of $\mathbf{A}$. b) 2D portrayal of the covariance matrix $\mathbf{V}$ fully characterizing the optical-mechanical Gaussian state, and corresponding numerical values. The estimated dynamical parameters for this dataset are $\Omega_b/2\pi=106\pm0.1$ kHz, $g_A/2\pi=11.7\pm0.2$ kHz, $g_B/2\pi=6.3\pm0.1$ kHz. The heating rates are $\Gamma_x/2\pi=3.00\pm0.05$ kHz and $\Gamma_y/2\pi=2.67\pm0.05$ kHz. All quoted errors indicate one SD of $\sim10$ independent samples.
• Figure 3: Optimization of the optical mode width. The smallest symplectic eigenvalue $\nu_{-}$ of the partially trasposed covariance matrix $\mathbf{V}$ is plotted as a function of mode width $\Gamma_\xi$. The data points with error bars (indicating one SD over about 10 independent data acquisitions) are obtained from $\mathbf{V}$ directly reconstructed from the experimental correlation matrix. The green shaded area derives from $\mathbf{V}$ computed with our optomechanical model, using the parameters that best fit the heterodyne spectra. The width of this band reflects systematic uncertainties in the model parameters (mainly the efficiency $\eta$), with negligible contribution from statistical uncertainty. The gray shaded region highlights the threshold above which the mechanical and optical subsystems become separable.
• Figure 4: Symplectic eigenvalue versus cavity detuning. Experimental data points shown by blue markers with error bars (SEM over about 10 independent data acquisitions) are obtained from the covariance matrix directly reconstructed from the measured correlations. Green markers are extracted instead from the same experimental data through the optomechanical model, using the parameters that best fit the heterodyne spectra. In this case the uncertainty is dominated by the systematic error on the detection efficiency $\eta$. In these measurements, we set $-\Delta_A\simeq\Delta_B=|\Delta|$. Intracavity entanglement is inferred from the symplectic eigenvalues associated to the covariance matrix calculated, within the same model and parameter set, for the joint state of the bright mode and the intracavity field (red points, with error bars again set by the systematic uncertainty on $\eta$). The green and red shaded regions are obtained by taking the extremal model predictions generated by the full set of parameters extracted from all data sets (at all detunings). These bands therefore illustrate the overall detuning dependence of the entanglement in our system.