Imaging Correlations in Heterodyne Spectra for Quantum Displacement Sensing

The extraordinary sensitivity of the output field of an optical cavity to small quantum-scale displacements has led to breakthroughs such as the first detection of gravitational waves and of the motions of quantum ground-state cooled mechanical oscillators. While heterodyne detection of the output optical field of an optomechanical system exhibits asymmetries which provide a key signature that the mechanical oscillator has attained the quantum regime, important quantum correlations are lost. In turn, homodyning can detect quantum squeezing in an optical quadrature but loses the important sideband asymmetries. Here we introduce and experimentally demonstrate a new technique, subjecting the autocorrelators of the output current to filter functions, which restores the lost heterodyne correlations (whether classical or quantum), drastically augmenting the useful information accessible. The filtering even adjusts for moderate errors in the locking phase of the local oscillator. Hence we demonstrate the single-shot measurement of hundreds of different field quadratures allowing the rapid imaging of detailed features from a simple heterodyne trace. We also obtain a spectrum of hybrid homodyne-heterodyne character, with motional sidebands of combined amplitudes comparable to homodyne. Although investigated here in a thermal regime, the method’s robustness and generality represents a promising new approach to sensing of quantum-scale displacements.

Figure 1

(a) Experimental scheme: a membrane-in-the-middle setup within a cavity with a single optical mode coupled to multiple mechanical modes of the membrane, simultaneously cooling and detecting them. The cavity output signal is beat with a reference oscillator $a_{\mathrm{LO}}\propto e^{i(\mathrm{\Omega }t+\theta )}$ of beat frequency $\mathrm{\Omega }$ and phase $\theta$. Balanced detection produces an output current $i_h(t)$; then $\mathrm{\Omega }=0$ corresponds to homodyne detection, while $\mathrm{\Omega }\ne 0$ corresponds to heterodyne. The output field from an optical cavity provides exquisitely sensitive detection of even quantum-scale displacements via the power spectral density (PSD) of $i_h(t)$. (b) Schematic illustration of homodyne vs heterodyne features: While heterodyning detects quantum regimes via the dependence of the motional sidebands on phonon occupancy $\overline{n}$, their combined height is only half of the homodyne amplitude. For homodyne detection, the presence of field noise correlations yields squeezing, evidenced by PSD values below the quantum imprecision noise floor (the region between white lines of the map). In contrast, the PSD of the heterodyne spectra shows neither squeezing nor any $\theta$ dependence. (c) Outline of the $r$-heterodyne method relative to the standard heterodyne PSD. (d) Example spectra using ${\mathcal{F}}_{-1}$ (upper panel) and ${\mathcal{F}}_0$ (lower panel). Spectra are shown normalized to the heterodyne peak amplitudes (orange peaks). For the upper panel, the correlations are isolated from the main peaks; for the lower panel, the restored correlations interfere with the heterodyne PSDs, doubling the sideband height for $\theta =\pi /2$.

Figure 2

(a) Mapping correlations: In contrast to the usual featureless ($\theta$-independent) heterodyne spectra, the correlations are extracted efficiently, even in an experiment where $\theta$ is not actively stabilized. The figure compares $r$-heterodyne experimental spectra, subjected to different filters, with analytically calculated spectra. (i) distills isolated correlations near $\omega ={\omega }_M$. (ii) obtains correlations plus heterodyne peaks. (iii) distills correlations only, but displaced to the position of the heterodyne peaks. Agreement between the experiment and theory is excellent, even if some distortion is evident in (iii), which we attribute to experimental noises (frequency noise, the drift in $\theta$). Data are in the thermal regime, and all spectra are normalized to the heterodyne peak amplitude of the 0.38 MHz mode. Maps are on a linear color scale where $\text{white}=1$ and $\text{dark}\text{blue}=-1$. (b) Two spectra, i.e., “cuts” from (iii), and compared with heterodyne (orange) showing that, despite distortion, the strongest $\theta =\pi /2$ peak amplitudes remain close to the expected amplitude ($2/\pi$ of the heterodyne amplitude). (c) Inset of (i) thermal vs pure quantum backaction spectra (analytical) illustrating future possible signatures of quantum regimes facilitated by a high-resolution mapping of correlations: The zero contour (white) rotates away from the vertical as the quantum backaction contribution becomes dominant.

Figure 3

(a) A spectrum of hybrid homodyne-heterodyne character. Map for a ${\mathcal{F}}_0\otimes \mathcal{A}(t_0)$ filter that most closely mimics homodyne behavior, since the correlations interfere with the static heterodyne peaks. In future experiments, mapping will allow the efficient one-shot detection of quantum behavior in the mechanical oscillator (via sideband asymmetry) and in the cavity mode (via quantum squeezing). (b) Data for two mechanical modes: The $r$-heterodyne peaks have nearly double the heterodyne amplitudes; hence, their combined amplitudes become comparable to the corresponding homodyne values allowing sensitive displacement detection [white cut in (b) indicates maxima $\theta =\pi /2$]. Peaks are normalized to a heterodyne amplitude of the 0.38 MHz mode. The white contour indicates where the heterodyne amplitude is exceeded. Map colors on a linear scale ranging from $\mathrm{white}=2$ to $\text{dark}\text{blue}=0$.