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 th e local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, suc h a measurement introduces noise into the phase estimate, with a varia nce scaling as N-1, where N is the maximum photon number. This represe nts the shot-noise limit or standard quantum limit (SQL). Recently, on e of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a wa y to improve upon this: a real-time feedback loop can control the loca l oscillator phase to be equal to the estimated system phase plus pi/2 , so that the phase quadrature of the system is measured preferentiall y. The phase estimate used in the feedback loop at time t is a functio nal of the photocurrent from time 0 up to time t in the single-shot me asurement. In this paper we consider a very simple feedback scheme inv olving only linear electronic elements. Approaching the problem from s emiclassical detection theory, we obtain analytical results for asympt otically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as N-3/ 2. This is much less than the SQL variance, but still much greater tha n the minimum intrinsic phase variance which scales as N-2. We briefly discuss the effect of detector inefficiencies and delays in the feedb ack loop.