Continuous weak measurement of the macroscopic quantum coherent oscillations

Abstract

The problem of continuous quantum measurement of coherent oscillations in an individual quantum two-state system is studied for a generic model of the measuring device. It is shown that for a symmetric detector, the signal-to-noise ratio of the measurement, defined as the ratio of the amplitude of the oscillation line in the output spectrum to background noise, is independent of the coupling strength between oscillations and the detector, and is equal to $(\hbar/ε)^2$, where $ε$ is the detector energy sensitivity. The fundamental quantum limit of 4 imposed by this result on the signal-to-noise ratio of the measurement with an ``ideal'' quantum-limited detector reflects the general tendency of a quantum measurement to localize the system in one of the eigenstates of the measured observable. These results are applied to specific measurements of the quantum oscillations of magnetic flux with a dc SQUID, and oscillations of charge measured with a Cooper-pair electrometer. They are also used to calculate the energy sensitivity of a quantum point contact as detector.