Quantum limits on estimating a waveform What’s the problem?

1. Quantum limits on estimating a waveform • What’s the problem? • 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 • http://info.phys.unm.edu/~caves Collaborators: M. Tsang, UNM postdoc H. M. Wiseman, Griffith University M. Tsang and C. M. Caves, “Coherent quantum-noise cancellation for optomechanical sensors,” PRL 105,123601 (2010). M. Tsang, H. M. Wiseman, and C. M. Caves, “Fundamental quantum limit to waveform estimation,” arXiv:1006.5407 [quant-ph].

2. SQL Back-action force measurement noise

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

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

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

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

7. Achieving the force-estimation QCRB Oscillator and negative-mass oscillator paired sidebands paired collective spins W. Wasilewski , K. Jensen, H. Krauter, J. J. Renema, M. V. Balbas, and E. S. Polzik, PRL 104, 133601 (2010). Quantum noise cancellation (QNC) Collective and relative coördinates

8. Cable Beach Western Australia

9. QCRB for waveform estimation Classical waveform Prior information Measurements Estimator Bias

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

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

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

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

14. Optomechanical force detector Optomechanical force detector Flowchart of signal and noise. Simplified flowchart.

15. 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. 647. M. T. Jaekel and S. Reynaud, Europhys. Lett. 13, 301 (1990). S. P. Vyatchanin and A. B. Matsko, JETP 77, 218 (1993). H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, S. P. Vyatchanin, PRD 65, 022002 (2002).

16. QNC II QNC by introduction of anti-noise path. Simplified flowchart. Detailed flowchart of ponderomotive coupling and intra-cavity matched squeezing. Implementation of matched squeezing scheme.

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