Quantum limits on estimating a waveform I.What’s the problem? II.Quantum Cramér-Rao bound (QCRB) for classical force on a linear system Carlton M. Caves Center for Quantum Information and Control, University of New Mexico M. Tsang and C. M. Caves, “Coherent quantum-noise cancellation for optomechanical sensors,” PRL 105, (2010). M. Tsang, H. M. Wiseman, and C. M. Caves, “Fundamental quantum limit to waveform estimation,” arXiv: [quant-ph]. Collaborators: M. Tsang, UNM postdoc H. M. Wiseman, Griffith University

measurement noise Back-action force SQL

Langevin force When can the Langevin force be neglected? Narrowband, on- resonance detection Wideband detection

The right story. But it’s still wrong. SQL for force detection Use the tools of quantum information theory to formulate a general framework for addressing quantum limits on waveform estimation.

Noise-power spectral densities Zero-mean, time-stationary random process u(t) Noise-power spectral density of u

QCRB: spectral uncertainty principle Prior-information term At frequencies where there is little prior information, Quantum-limited noise No hint of SQL, but can the bound be achieved? 1.Back-action evasion. Monitor a quantum nondemolition (QND) observable. 2.Quantum noise cancellation (QNC). Add an auxiliary negative-mass oscillator on which the back-action force pulls instead of pushes.

Achieving the force-estimation QCRB Oscillator and negative-mass oscillator paired sidebands paired collective spins Collective and relative coördinates Quantum noise cancellation (QNC) W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balbas, and E. S. Polzik, PRL 104, (2010).

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QCRB for waveform estimation Classical waveformPrior information Measurements Estimator Bias

Handling the measurements Hamiltonian evolution of pure states, but all ancillae subject to measurements

QCRB for waveform estimation Gives a classical Fisher information for the prior information. Apply the Schwarz inequality! Gives a quantum Fisher information involving the generators.

QCRB for waveform estimation Two-point correlation function of generator h(t)

QCRB for force on linear system Force f(t) coupled h=q to position Gaussian priorClassical prior Fisher information is the inverse of the two-time correlation matrix of f(t). Time-stationaryTwo-time correlation matrices are processesdiagonal in the frequency domain. Diagonal elements are spectral densities. QCRB becomes a spectral uncertainty principle.

Optomechanical force detector (a)Flowchart of signal and noise. (b)Simplified flowchart.

QNC I (a) QNC and frequency- dependent input squeezing. (b) QNC by output optics (variational measurement) and squeezing. W. G. Unruh, in Quantum Optics, Experimental Gravitation, and Measurement Theory, edited by P. Meystre and M. O. Scully (Plenum, New York, 1983), p M. T. Jaekel and S. Reynaud, Europhys. Lett. 13, 301 (1990). H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, S. P. Vyatchanin, PRD 65, (2002). S. P. Vyatchanin and A. B. Matsko, JETP 77, 218 (1993).

QNC II (a)QNC by introduction of anti-noise path. (b)Simplified flowchart. (c)Detailed flowchart of ponderomotive coupling and intra- cavity matched squeezing. (d)Implementation of matched squeezing scheme.

QNC III Input (a) and output (b) matched squeezing schemes and associated flowcharts (c) and (d).