Braginsky, Gorodetsky, Khalili, Matsko, Thorne and Vyatchanin (Phys. Rev. D 67, 082001 (2003)) proved that in interferometric gravitational-wave detectors and other linear meters of classical forces, photon shot noise and radiation-pressure back-action are the only quantum noises: with appropriate output filtering, the test masses' initial quantum state and the measurement-induced state reduction contribute nothing to the force-monitoring error. The authors explicitly left open the strongly nonlinear regime, posing the concrete question whether a mechanical oscillator's initial quantum state influences the accuracy with which a classical force acting on its mass can be monitored by a highly nonlinear meter — their example being the null-detector scheme in which the oscillating mass is the end mirror of a Fabry–Perot cavity and the oscillator phase is read out through the cavity's momentary transmissivity at resonance crossings, with both the mirror displacements and the zero-point oscillation amplitude comparable to or larger than the cavity linearity scale λ/F\lambda/F.

Background

Interferometric gravitational-wave (GW) detectors are, at bottom, meters of a classical time-varying force — the tidal force of a passing GW — acting on macroscopic test masses (the suspended cavity mirrors). Their quantum noise comes from the probing light: shot noise (photon-counting fluctuations at the output photodetector) and radiation-pressure back-action (momentum kicks delivered to the mirrors by fluctuating light amplitude). The trade-off between the two yields the standard quantum limit (SQL), the familiar sensitivity floor of a conventional interferometer. Historically the SQL could be derived in two seemingly independent ways — from the quantum mechanics of the test masses (the nonvanishing commutator of the test-mass position at different times) or from the quantum fluctuations of the light — and it was unclear whether test-mass quantization, and the state reduction (collapse of the wavefunction) that accompanies any measurement, constituted an additional, independent noise source on top of the light's noise. Braginsky, Gorodetsky, Khalili, Matsko, Thorne and Vyatchanin (2003) resolved this question for linear measurements. A linear measurement, in their sense, is one in which (i) the meter output is the sum of the measured test-object variable and an additive meter noise that does not depend on the test object's initial state, and (ii) the measurement-induced perturbations (back action) of the relevant test-object observables are linear and independent of the initial state — a condition guaranteed when the commutators of the involved observables are c-numbers (ordinary numbers rather than operators), as for a free mass or a harmonic oscillator. For such measurements they showed that the interferometer output (the photon number flux at the photodetector) commutes with itself at different times and is therefore a classical record; the contribution of the test-mass initial position and momentum operators x^o\hat{x}_o and p^o\hat{p}_o to the output commutator is exactly canceled by the commutation of the light's shot noise and back-action noise; and the initial-state influence appears only at the test-mass pendular swinging frequency, so that ordinary filtering (e.g. high-passing below ~10 Hz) removes it entirely. Hence no additional noise arises from test-mass quantization, from the test masses' initial quantum state, or from state reduction. The authors stressed, however, that strictly speaking all real meters are nonlinear, and that their conclusions rely on linearity. A LIGO-like meter is linear only because the mirror displacements are far smaller than the width of a cavity resonance, λ/F\lambda/F, where λ\lambda is the light wavelength and FF is the cavity finesse (a measure of how narrow the cavity's transmission resonances are). If the displacements are comparable to or larger than λ/F\lambda/F, the readout becomes strongly nonlinear. A concrete scheme of this kind is the null-detector technique (Vyatchanin 1979; Braginsky, Khalili and Kulaga 1995; Kulaga 1995) for measuring the phase of a mechanical oscillator: the oscillating mass is the end mirror of a Fabry–Perot cavity, and the times at which the mirror passes through cavity-resonant positions are registered with high accuracy through the cavity's momentary transmissivity (a null detector is one arranged so that its output sits at a null — near-zero transmission — except when the signal condition is met). In this scheme the mirror displacements and even the oscillator's zero-point oscillation amplitude δxzp\delta x_{\mathrm{zp}} are large compared to λ/F\lambda/F, and state reduction plays an essential constructive role: it drives the oscillator into a squeezed-phase state (a state with reduced phase uncertainty at the price of enlarged conjugate fluctuations), thereby enabling high-precision phase monitoring. The source paper notes that in nonlinear measurements genuinely alien effects — for example the quantum Zeno effect (the freezing of a system's evolution by sufficiently frequent measurements) — can arise, and that the linear-theory tools (self-commuting outputs, c-number commutators, commutator cancellation, filtering) have no obvious analogue there. The unresolved question is therefore whether the central conclusion of the linear theory — that a classical force can be monitored with accuracy limited only by the meter's own quantum noise, with no influence from the measured object's initial quantum state — survives in the strongly nonlinear regime exemplified by this null detector.

Problem Statement

Analyze the use of the highly nonlinear null-detector meter as a monitor of a classical force. In this meter, proposed for quantum-nondemolition (QND) measurement — a measurement whose back action does not degrade the measured observable — of a mechanical oscillator's phase, the oscillating mass is the end mirror of a Fabry–Perot cavity, and the times at which the mirror passes through cavity-resonant positions are registered with high accuracy through the cavity's momentary transmissivity; the measurement is strongly nonlinear because both the mirror displacements and the oscillator's zero-point oscillation amplitude δxzp\delta x_{\mathrm{zp}} are comparable to or larger than the cavity's linearity scale λ/F\lambda/F (λ\lambda the light wavelength, FF the cavity finesse), and state reduction in this scheme drives the oscillator into a squeezed-phase state. The concrete question posed by the source authors is: when this meter is used to monitor a classical force acting on the oscillator's mass, does the oscillator's initial quantum state influence the accuracy of the monitoring? Determine whether the initial quantum state of the oscillator contributes irreducibly — that is, not removably from the force estimate by any processing of the output — to the achievable force-estimation error, quantifying any such contribution (e.g. its dependence on initial-state parameters such as squeezing, temperature, or phase uncertainty) and, where it is removable, exhibiting the data processing that removes it in analogy with the filtering that removes x^o\hat{x}_o and p^o\hat{p}_o influence in linear measurements; this requires a quantum model of the null detector operating as a force monitor (nonlinear readout observable, explicit role of state reduction). The objective is to establish whether the principle that a classical force can be monitored with accuracy limited only by the meter's own quantum noise — with no contribution from the measured object's initial quantum state — extends from linear meters to strongly nonlinear ones. The associated general question, in what conditions initial-state influence can and cannot be eliminated in nonlinear measurement dynamics, is the natural generalization of the same objective, and is recorded here as broader context rather than as a required deliverable.

References

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