Quantum Nondemolition Measurements of Harmonic Oscillators

Abstract

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 ${X}_{1}$ and ${X}_{2}$ simultaneously ($\ensuremath{\Delta}{X}_{1}\ensuremath{\Delta}{X}_{2}>~\frac{\ensuremath{\hbar}}{2m\ensuremath{\omega}}$), ${X}_{1}$ 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.