Quantum nondemolition tracking of a resonator's phase

Resonators driven to self-oscillation by active feedback play an important role in technology. The laser is an example of such a device. Generally, the feedback that sustains self-oscillation is contaminated with amplifier noise, which leads to phase diffusion. Here we show that under suitable conditions phase noise from the feedback amplifier can be evaded, allowing one to track the resonator's phase in a quantum nondemolition manner. In particular, we consider a resonator with a Kerr nonlinearity. The output of the resonator is fed into a linear amplifier, phase shifted, fed through an amplitude limiter, and then fed back into the resonator. Stable operation is achieved, even on portions of the resonance curve for which the resonator would exhibit optical bistability under open-loop conditions. By operating the system at a point on the resonance curve for which the slope of the phase versus frequency is infinite, the long-term phase stability of the oscillator becomes insensitive to the amplifier's output noise. In addition, the amplifier's input-port noise can be evaded by making the coupling between the resonator and the amplifier sufficiently weak. We used an analog system to test the operating principles of this device. The resonator consisted of a vibrating silicon beam that was rigidly anchored at each end. Tension built up along the beam as it was deflected gave rise to a nonlinear restoring force, providing an analogue for the Kerr nonlinearity. When the resonator was placed in a magnetic field, a current path along the beam provided a means of driving the beam through the Lorentz force and a means of detecting the resonator's motion through the electromotive force developed along the conductor. Amplification, phase shifting, and amplitude limiting were all done electronically. A 10-dB reduction in phase diffusion due to amplifier noise was demonstrated.