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Physical Review A

Quantum nondemolition measurements via quadratic coupling

Gerard J Milburn, Daniel F. Walls

Published Oct 1, 1983• 149 citations
Publisher

Abstract

We consider making quantum nondemolition measurements on a harmonic oscillator by coupling it to an oscillator readout system using a four-wave-mixing interaction. The quantum nondemolition variable of interest appears as a nonlinear function of the number operator. It has the form ${\stackrel{^}{D}}_{1}(t)=cos(\ensuremath{\chi}{a}^{\ifmmode\dagger\else\textdagger\fi{}}at)$. A sequence of measurements of this operator may be made with completely predictable results. The possible use of this system to detect gravitational radiation is suggested.

DOI
10.1103/physreva.28.2065

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We consider making quantum nondemolition measurements on a harmonic oscillator by coupling it to an oscillator readout system using a four-wave-mixing interaction. The quantum nondemolition variable of interest appears as a nonlinear function of the number operator. It has the form {\stackrel{^}{D}}_{1}(t)=cos(\ensuremath{\chi}{a}^{\ifmmode\dagger\else\textdagger\fi{}}at). A sequence of measurements of this operator may be made with completely predictable results. The possible use of this system to detect gravitational radiation is suggested.
In the measurements on an oscillator to detect short bursts of gravitational radiation, one important demand is the ability to use short integration times in order to decrease the contribution of thermal noise. This requires a strong coupling of the antenna to the measuring apparatus, i.e., a high value of the so-called electromechanical coupling parameter ${\mathrm{\ensuremath{\beta}}}_{0}$. A well-known solution is in the use of a resonant transducer, which entails the observation of a system of two harmonic oscillators. We extend to this system the back-action evasion technique by showing that one can perform a quantum nondemolition measurement on a double harmonic oscillator with essentially the same configuration as for the single one.
The complex amplitude ${X}_{1}+i{X}_{2}\ensuremath{\equiv}(x+\frac{\mathrm{ip}}{m\ensuremath{\omega}}){e}^{i\ensuremath{\omega}t}$ of a harmonic oscillator is constant in the absence of driving forces. Although the uncertainty principle forbids precise measurements of X1{X}_{1}X1​ and X2{X}_{2}X2​ simultaneously ($\ensuremath{\Delta}{X}_{1}\ensuremath{\Delta}{X}_{2}>~\frac{\ensuremath{\hbar}}{2m\ensuremath{\omega}}$), X1{X}_{1}X1​ alone can be measured precisely and continuously ("quantum nondemolition measurement"). Examples are given of measuring systems that do this job. Such systems might play a crucial role in gravitational-wave detection and elsewhere.
The current generation of Weber gravitational wave antennae is a multimode system which makes use of a transducer resonant with the antenna and is expected to reach the standard quantum limit of sensitivity and possibly, by using quantum nondemolition measurement strategies, to surpass it. Quantum nondemolition measurement strategies have been studied until now only for a single harmonic oscillator. In this paper we generalize the stroboscopic quantum nondemolition strategy to coupled harmonic oscillators. We calculate the commutator of the relative displacement at different times for two and three coupled harmonic oscillators. It is shown that the commutator is time dependent in such a way as to retain values close to zero for a duration of the order of the uncoupled period of oscillations. The stability of this result with respect to frequency detunings is studied. The implications of this result for adapting the proposed quasistroboscopic quantum nondemolition schemes to coupled oscillators are also discussed.