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Fundamental Quantum Limit to Waveform Estimation

Mankei Tsang, University of New Mexico

(Session 5 : Friday from 4:45pm-5:15pm)

Abstract. Quantum noise is now routinely observed in atomic, optical, electrical, and mechanical systems and will impose severe limits to the precision of sensors in the near future. Atomic magnetometers limited by quantum noise have been demonstrated, while optomechanical force sensors, including the gravitational wave detector in LIGO, are expected to reach the so-called standard quantum limit (SQL) within the next few years. Such technological advances have motivated renewed interest in the theoretical quantum limits to sensing. Recent work on such quantum limits focuses on the relatively simple problem of parameter estimation, but in realistic applications, the unknown signal of interest, such as a gravitational wave or a magnetic field, is rarely a parameter constant in time but a continuously varying waveform, in which case no rigorous quantum limit has yet been established. Here we present a rigorous limit to the error of waveform estimation in quantum sensing in the form of a quantum Cramer-Rao bound (QCRB). Unlike the one first derived by Helstrom for parameter estimation, our QCRB shows how the prior information crucial for waveform estimation can be taken into account and is more directly applicable to force sensing and magnetometry applications. As an important example, we calculate the QCRB for optomechanical force sensing, show that the QCRB is in general below the SQL, and demonstrate that the QCRB can be achieved by applying a quantum estimation technique called quantum smoothing to the observations and a quantum noise cancellation technique to the experimental setup, thus proving the optimality of these techniques. Being a rigorously proven and demonstrably achievable limit, our QCRB supercedes the heuristic SQL as the fundamental quantum limit to force sensing. We are thus able to relate the modern theoretical program of quantum metrology, which has so far relied on toy parameter-estimation models, to the classic but more practical studies of continuous quantum measurement theory initiated by Braginsky and others.