Quantum Nondemolition Measurements

Abstract

Current attempts to detect gravitational radiation have to take into account the quantum uncertainties in the measurement process. Considering that the detectors are macroscopic objects in some cases as large as a 10-ton bar, the fact that quantum fluctuations in the detector must be taken into account seems surprising. However, as discussed in Chap. 8, gravitational waves interact so weakly with terrestrial detectors that a displacement of the order of 10⁻¹⁹cm is expected. To illustrate how the measurement process may introduce uncertainties which obscure the signal we consider the simple example of a free mass. A measurement of the position of a free mass with a precision Δx_i ≈ 10⁻¹⁹cm will disturb the momentum by an amount given by the uncertainty principles as Δp ≥ħ(2Δx_i)⁻¹. The period of the gravitational waves is expected to be about 10⁻³ s, hence a second measurement of the position should be made after this time. During this period, however, the position uncertainty will grow under free evolution by an amount Δx²(τ) = Δx²(0)+[Δp²(0)τ²/m²]. The following inequality then holds