# Simultaneous retrodiction of multi-mode optomechanical systems using matched filters

## TL;DR

This paper develops a method for retrodicting the initial states of multiple optomechanical oscillators using matched filters, achieving quantum-limited sensitivity and demonstrating the approach experimentally for two modes.

## Contribution

It introduces a linear matched filter approach for simultaneous multi-mode state retrodiction in optomechanical systems, reaching the Standard Quantum Limit.

## Key findings

- Optimal estimators are derived from a detailed model of the homodyne signal.
- High-cooperativity measurements enable retrodiction at the SQL.
- Experimental two-mode retrodiction demonstrates practical feasibility.

## Abstract

Generation and manipulation of many-body entangled states is of considerable interest, for applications in quantum simulation or sensing, for example. Measurement and verification of the resulting many-body state presents a formidable challenge, however, which can be simplified by multiplexed readout using shared measurement resources. In this work, we analyze and demonstrate state retrodiction for a system of optomechanical oscillators coupled to a single-mode optical cavity. Coupling to the shared cavity field facilitates simultaneous optical measurement of the oscillators' transient dynamics at distinct frequencies. Optimal estimators for the oscillators' initial state can be defined as a set of linear matched filters, derived from a detailed model for the detected homodyne signal. We find that the optimal state estimate for optomechanical retrodiction is obtained from high-cooperativity measurements, reaching estimate sensitivity at the Standard Quantum Limit (SQL). Simultaneous estimation of the state of multiple oscillators places additional limits on the estimate precision, due to the diffusive noise each oscillator adds to the optomechanical signal. However, we show that the sensitivity of simultaneous multi-mode state retrodiction reaches the SQL for sufficiently well-resolved oscillators. Finally, an experimental demonstration of two-mode retrodiction is presented, which requires further accounting for technical fluctuations of the oscillator frequency.