Adaptive single-shot phase measurements: A semiclassical approach

The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as [Formula Presented], where [Formula Presented] is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus [Formula Presented], so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time [Formula Presented] is a functional of the photocurrent from time [Formula Presented] up to time [Formula Presented] in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as [Formula Presented]. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as [Formula Presented]. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.