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Quantum limits for stationary force sensing

Interferometric force and position sensors have reached the sensitivity where the quantum noise of the meter, e.g., light, is significant or even dominant. In particular, this sensitivity is close to the Standard Quantum Limit (SQL), which directly follows from the Heisenberg uncertainty relation and corresponds to balancing the measurement imprecision and the perturbation of the probe by the quantum back action of the meter. This limit is not a truly fundamental one and several methods for its overcoming have been proposed and demonstrated. At the same time, two other more fundamental quantum sensitivity constraints exist beyond the SQL. They also originate from the Heisenberg uncertainty relation, but in more subtle ways. The first limit arises from the finiteness of the probing strength (in the case of optical interferometers - of the circulating optical power) and is known as the Energetic Quantum Limit or, in a more general context, as the Quantum Cram\'{e}r-Rao Bound (QCRB). The second limit arises from the dissipative dynamics of the probe, which prevents full efficacy of the quantum back action evasion techniques developed for overcoming the SQL. No particular name has been assigned to this limit; we propose the term Dissipative Quantum Limit (DQL) for it. Here we develop a unified theory of these two fundamental limits by deriving the general sensitivity constraint from which they follow as particular cases. Our analysis reveals a phase transition occurring at the boundary between the QCRB-dominated and the DQL regimes, manifested by the discontinuous derivatives of the optimal spectral densities of the meter field quantum noise. We show that the DQL originates from the non-autocommutativity of the internal thermal noise of the probe and that it can be overcome in non-stationary measurements.