Feedback stabilization of a nanoparticle at the intensity minimum of an optical double-well potential

TL;DR

This work develops a practical adaptive feedback strategy to stabilize a levitated nanoparticle at the intensity minimum of an optical double-well, addressing measurement nonlinearities and slow drifts. By linearizing the nonlinear potential near the apex and augmenting the state with an unknown apex drift, the authors implement a stochastic LQG controller on FPGA hardware, including a projected Kalman filter to maintain apex-tracking within bounds. Simulation and experimental results show that the adaptive 2D controller (including z-axis information) significantly improves confinement and reduces residual variance compared to simpler variants, while remaining robust to drift and nonlinear detection. The approach advances dark-trap technologies with potential applicability to quantum-state preparation and fundamental tests of quantum physics at the mesoscopic scale, leveraging the natural extension of LQG to quantum filtering and fast FPGA-based control.

Abstract

In this work, we develop and analyze adaptive feedback control strategies to stabilize and confine a nanoparticle at the unstable intensity minimum of an optical double-well potential. The resulting stochastic optimal control problem for a noise-driven mechanical particle in a nonlinear optical potential must account for unavoidable experimental imperfections such as measurement nonlinearities and slow drifts of the optical setup. To address these issues, we simplify the model in the vicinity of the unstable equilibrium and employ indirect adaptive control techniques to dynamically follow changes in the potential landscape. Our approach leads to a simple and efficient Linear Quadratic Gaussian (LQG) controller that can be implemented on fast and cost-effective FPGAs, ensuring accessibility and reproducibility. We demonstrate that this strategy successfully tracks the intensity minimum and significantly reduces the nanoparticle's residual state variance, effectively lowering its center-of-mass temperature. While conventional optical traps rely on confining optical forces in the light field at the intensity maxima, trapping at intensity minima mitigates absorption heating, which is crucial for advanced quantum experiments. Since LQG control naturally extends into the quantum regime, our results provide a promising pathway for future experiments on quantum state preparation beyond the current absorption heating limitation, like matter-wave interference and tests of the quantum-gravity interface.

Figures (9)

• Figure 1: Simplified layout of the experimental setup. A TEM00 and a TEM01 beam are superimposed to create a one-dimensional double-well potential. Optical detection in the TEM00 beam lets one measure the particle's position in the $x$ and $z$ directions. The unstable $x$ direction is equipped with electrodes to act on the charged particle through electrostatic forces. The misalignment of the electrodes is exaggerated for illustration purposes.
• Figure 2: Potential landscape and nonlinear detection. Top: Changes of the potential due to the relative drift $\Delta_{1}$ of TEM01 to TEM00 (i.e., for $\Delta_0 = 0$ defining the reference frame of the optical detection). The black line depicts the potential for aligned beams, changing towards red for $\Delta_{1} =60nm$. Blue crosses mark the apex (local maximum) of the optical potential. Bottom: Normalized nonlinear detection sensitivity (blue) and resulting effective potential shapes (dashed) for aligned (black) and misaligned (red) cases.
• Figure 3: Behavior of the apex position $\Delta_{apex}(\mathbf{p})$ and the corresponding local stiffness $k_{apex}(\mathbf{p})$ for relative drift $\Delta_{1}$ of the two beams up to $\pm 30nm$.
• Figure 4: Qualitative comparison of the controller performance in simulation: All scenarios are initialized with identical noise realizations and initial conditions. As expected, including the estimate of $\Delta_{apex}$ into the Kalman Filter allows the controller to follow the drift of the local maximum. However, omitting the $z$-axis (Adaptive 1D controller) leads to a significantly worse confinement of the particle (i.e., cooling). As illustrated in the detailed plot of the 8 ms mark on the right-hand side, this is due to the apex estimator wrongfully attributing cross-talk of $z$-axis motion in the $x$-detection signal.
• Figure 5: Power spectral densities of the two measurements for different simulated control algorithms: Non-adaptive 1D (blue), Adaptive 1D (green), and Adaptive 2D (red) for $x$-detection (top) and $z$-detection (bottom). The $\Omega_{well}$ peak in the non-adaptive controller indicates that the particle is exploring one of the potential wells.