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. CavesCenter for Quantum Information and Control, University of New Mexico

http://info.phys.unm.edu/~caves

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].

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 densitiesZero-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, 133601 (2010).

Cable BeachWestern Australia

QCRB for waveform estimation Classical waveform Prior 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=qto 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.

Optomechanical force detector

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. 647.

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, 022002 (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).