# Quantum limit for precision estimation of classical forces

### Thursday, 17 Apr. 2014, 14:00 - 15:00

**Presenter:** Camille Latune Lombard - Instituto de Física, Universidade Federal do Rio de Janeiro

**Host:** M. Aspelmeyer

**Where:** Schrödingerroom, Boltzmanngasse 5, 4th floor, room 3425

In this talk I present a recently proposed solution for the problem of estimating a classical force through measurements made on a noisy quantum oscillator [1]. This problem may have applications to gravitational wave detection [2-4] and to the detection of weak forces in opto-mechanics [5,6], trapped ions [7], and atomic force microscopy.

Parameter estimation theory, first formalized by Fisher [8], allows one to analyze the best precision that can be reached in a parameter-estimation experiment, and to find out a precise protocol to reach it. The general framework considers a probe that interacts with a system characterized by the parameter to be estimated. The probe is prepared in a suitable initial state and measured after the interaction with the system. The value of the parameter is estimated from the measurement results, with a precision that depends on the initial state of the probe, the measurement procedure, and the interaction between the probe and the system.

Quantum probes introduce new features, which allow better precision, but also lead to new limitations: the precision may be spoiled by the back-action characteristics of quantum measurements, and also by the fact that outgoing states corresponding to different values of the parameter are not generally orthogonal, implying that they cannot be unambiguously distinguished. Quantum metrology [9,10] deals with these features, and seeks the determination of the best possible precision and the corresponding best initial state of the probe, as well as the best measurement procedure.

This talk introduces the main ideas of parameter estimation and the main issues of quantum metrology, and reviews a recently-proposed method [11] that allows, even for noisy systems, the derivation of lower bounds for the uncertainty in the estimation. This method leads to an exact analytical solution for the uncertainty in the estimation of a classical force applied to a noisy quantum harmonic oscillator. Applications to opto-mechanical systems are also discussed.

[1] C. L. Latune, B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Phys. Rev. A 88, 042112 (2013).

[2] K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, Phys. Rev. Lett. 40, 667 (1978).

[3] C. M. Caves, K. S. Thone, R. W. P. Drever, V. D. Sandberg, and M. Zimmerman, Rev. Mod. Phys. 52, 341 (1980).

[4] Schnabel, R. et al., Nat. Commun. 1:121 doi: 10.1038/ncomms1122 (2010).

[5] C. A. Regal, J. D. Teufel, and K. W. Lehnert, Nat. Phys. 4, 555 (2008).

[6] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim and C. Brukner, Nat. Phys. 8, 393-397 (2012).

[7] R. Maiwald, D. Leibfried, J. Britton, J. C. Bergquist, G. Leuchs, and D. J. Wineland, Nat. Phys. 5, 551 (2009).

[8] R. A. Fisher, Math. Proc. Cambridge Philos. Soc. 22, 700 (1925).

[9] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

[10] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).

[11] B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011)