Optimal State Estimation for Cavity Optomechanical Systems

We demonstrate optimal state estimation for a cavity optomechanical system through Kalman filtering. By taking into account nontrivial experimental noise sources, such as colored laser noise and spurious mechanical modes, we implement a realistic state-space model. This allows us to obtain the conditional system state, i.e., conditioned on previous measurements, with a minimal least-squares estimation error. We apply this method to estimate the mechanical state, as well as optomechanical correlations both in the weak and strong coupling regime. The application of the Kalman filter is an important next step for achieving real-time optimal (classical and quantum) control of cavity optomechanical systems.

Figure 1

Kalman filter for cavity optomechanical systems. (a) Working principle of the Kalman filter: (i) The conditional state is depicted by the green phase-space ellipse, which (ii) evolves in time according to the system dynamics. (iii) After a time $dt$, a Bayesian update is applied based on the measurement outcome to find the new conditional state. This procedure minimizes the mean-square estimation error, which makes the Kalman filter optimal for real-time state estimation. (b) Schematic of the experiment: The optomechanical cavity is driven by two laser beams each of which carry amplitude and phase noise. The mechanical motion is typically driven by Brownian noise. After their interaction with the cavity, the optical fields are detected by two independent homodyne measurements (signals $z_d$ and $z_r$), which themselves are subject to optical losses and noise. Building an accurate Kalman filter requires appropriate modeling of all relevant noise sources.

Figure 2

Measurement signals and Kalman-filter predictions. Shown are the measurement signals of the two homodyne detections of the detuned and resonant beam, $z_d(t)$ (red dots) and $z_r(t)$ (blue dots), respectively, and their Kalman-filter predictions ${\widehat{z}}_i(t)$ (gray line) both for the weak (left) and the strong coupling regime (right). Error bars of the prediction ($\pm 2\sigma$) are indicated by the width of the gray line. Kalman-filter innovations ${\nu }_i(t)=z_i(t)-{\widehat{z}}_i(t)$ are plotted below each data set and demonstrate the accuracy of the implemented filter. To assess the performance of the filter, we calculate the fraction of normalized innovations that are contained in a two-sided 95% confidence region ($\pm 2\sigma$ indicated by the lines) of a zero-mean Gaussian distribution (beside each plot). The experiment was performed at room temperature with a micromechanical oscillator of ${\omega }_m=2\pi \times 1.278\mathrm{MHz}$, ${\gamma }_m=2\pi \times 265\mathrm{Hz}$, and optomechanical parameters $\kappa =0.34{\omega }_m$, $g_0=2\pi \times 7.7\mathrm{Hz}$ [39]. For both coupling strengths of the detuned beam (${\mathrm{\Delta }}_d={\omega }_m$), we use ${\phi }_d\approx 0$, ${\mathrm{\Delta }}_r=0$, $g_r=0.2\kappa$, and ${\phi }_r=\pi /2$. Note that the fast oscillation of $z_r(t)$, which is due to the 20 MHz Pound-Drever-Hall phase modulation for frequency locking, is taken into account by the Kalman filter.

Figure 3

Estimating optomechanical quadratures. (a) Shown are Kalman-filter real-time estimates for the optical amplitude (${\widehat{x}}_i$, straight line) and phase (${\widehat{y}}_i$, dashed line) quadrature of the detuned (red, top) and resonant (blue, middle) beam along with the mechanical position ($\widehat{q}$, straight green line) and momentum quadrature ($\widehat{p}$, dashed green line) of the optomechanical system for the weak and strong coupling regime. The mechanical [(b), green line] and optomechanical [(c), gray line] phase-space trajectories are estimated over a period of $100\mu \mathrm{s}$. A histogram along each quadrature is shown as a side panel and estimated over 10 ms. The uncertainty ellipse of the unconditional (conditional) state is shown as dashed (straight) line. Note that the length of the shown trajectory is not sufficient to adequately represent the state’s statistics. All units are given in terms of quadrature zero-point fluctuations (zpf). For our experimental parameters $q_{\mathrm{zpf}}=2.73\times 10^{-16}\mathrm{m}$, $p_{\mathrm{zpf}}=3.87\times 10^{-19}\mathrm{kg}\mathrm{m}/\mathrm{s}$.