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arXiv:gr-qc/9310038 · Physical Review A

Quantum limits for measurements on macroscopic bodies: A decoherence analysis

Carlos O. Escobar, Léa Ferreira dos Santos, Paulo C. Marques F.

Published Aug 1, 1994• 3 citations
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Abstract

We consider in this paper the quantum limits for measurements on macroscopic bodies which are obtained by employing the concept of decoherence coming from an analysis of the quantum mechanics of dissipative systems. Two cases are analyzed, the free mass and the harmonic oscillator, and for both system we compare our approach with previous treatments of such limits.

DOI
10.1103/physreva.50.1913

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We consider in this paper the quantum limits for measurements on macroscopic bodies which are obtained by employing the concept of decoherence coming from an analysis of the quantum mechanics of dissipative systems. Two cases are analyzed, the free mass and the harmonic oscillator, and for both system we compare our approach with previous treatments of such limits.
Recent achievements in the isolation of macroscopic mechanical objects from a heat bath make it possible to implement quantum measurements with such systems. In this case, either a free mass or an oscillator can be used as a test object. The advantage of the first variant is in significantly longer relaxation times achieved for free masses. The advantage of the second variant is in the absence of restrictions on the limiting measurement accuracy associated with internal losses in the meter. This restriction can be bypassed, retaining a long relaxation time typical of free masses, if a test oscillator with a ponderomotive electromagnetic rigidity is used. Estimates show that the potential sensitivity of this test body for the modern level of technology may be considerably higher than the standard quantum limit.
Careful monitoring of harmonically bound (or as a limiting case, free) masses is the basis of current and future gravitational wave detectors, and of nanomechanical devices designed to access the quantum regime. We analyze the effects of stochastic localization models for state vector reduction, and of related models for environmental decoherence, on such systems, focusing our analysis on the non-dissipative forced harmonic oscillator, and its free mass limit. We derive an explicit formula for the time evolution of the expectation of a general operator in the presence of stochastic reduction or environmentally induced decoherence, for both the non-dissipative harmonic oscillator and the free mass. In the case of the oscillator, we also give a formula for the time evolution of the matrix element of the stochastic expectation density matrix between general coherent states. We show that the stochastic expectation of the variance of a Hermitian operator in any unraveling of the stochastic process is bounded by the variance computed from the stochastic expectation of the density matrix, and we develop a formal perturbation theory for calculating expectation values of operators within any unraveling. Applying our results to current gravitational wave interferometer detectors and nanomechanical systems, we conclude that the deviations from quantum mechanics predicted by the continuous spontaneous localization (CSL) model of state vector reduction are at least five orders of magnitude below the relevant standard quantum limits for these experiments. The proposed LISA gravitational wave detector will be two orders of magnitude away from the capability of observing an effect.
We discuss limitations in precision measurements of a weak classical force coupled to quantum mechanical systems, the so-called standard quantum limit (SQL). Among the several contexts exploiting the measurement of classical signals, gravitational wave (GW) detection is of paramount importance. In this framework, we analyze the quantum limited sensitivity of a free test mass, a quantum mechanical harmonic oscillator, two harmonic oscillators with equal masses and different resonance frequencies, and finally two mechanical oscillators with different masses and resonating at the same frequency. The sensitivity analysis of the latter two cases illustrates the potentialities of back-action reduction and classical impedance matching schemes, respectively. By examining coupled quantum oscillators as detectors of classical signals, we found a viable path to approach the SQL for planned or operating GW detectors, such as DUAL and AURIGA.
The standard quantum limits on several macroscopic quantities and the conditions for attaining these limits are evaluated. Possibilities for exceeding them by means of quantum nondemolition measurements are examined. The best frequency stability in self-excited oscillators, the smallest mechanical displacements, and the highest Q values of mechanical and electromagnetic resonators which have been achieved are reviewed. The outlook for improving the sensitivity in several gravitational experiments is evaluated.