Quantum phase estimation in noisy systems

The precision in estimation of unknown parameters can go beyond the classical limits by using nonclassical properties of quantum mechanics. However, nonclassical properties are very sensitive to the interaction with environment. This interaction could result in the loss of photons and low visibility. The purpose of this thesis is to find states and measurement schemes which give enhancement in estimation accuracy over the classical methods in the presence of such effects. We begin with estimation of an unknown phase in a Mach-Zehnder interferometer in the presence of photon loss. We propose a scheme to produce states which are loss-resistant and perform very close to optimal states. We propose sequences of such states combined with single photon states to obtain an unambiguous phase estimate with better accuracy than the standard quantum limit. We then consider the case that the loss is due to the interaction of the beam with an ensemble of atoms. Traditionally, the transition frequencies of atoms are measured via absorption, but there is also information available in the phase shift. We numerically find states that give maximum information about the transition frequency, obtained from both the absorption and the phase and quantified by the Fisher information. We also consider phase estimation in a Ramsey interferometer using an NV centre to measure an unknown time-independent magnetic field. The low visibility in Ramsey measurements requires different measurement schemes than optical interferometers. We find an optimised adaptive scheme which reaches the limit analogous to the Heisenberg-like scaling in the estimation of the magnetic field, and outperforms the optimal non-adaptive scheme. Finally, we consider continuous measurement on a single spatial mode field with a time varying phase. We consider a phase which varies in time with power law spectral density. We show that by using squeezed coherent states in an adaptive homodyne measurement scheme we can estimate the phase with accuracy scaling at the Heisenberg limit.