Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing. II. Applications to atomic magnetometry and Hardy’s paradox

Abstract

The quantum smoothing theory [Tsang, Phys. Rev. Lett. 102, 250403 (2009); Phys. Rev. A, in press (e-print arXiv:0906.4133)] is extended to account for discrete jumps in the classical random process to be estimated, discrete variables in the quantum system, such as spin, angular momentum, and photon number, and Poissonian measurements, such as photon counting. The extended theory is used to model atomic magnetometers and study Hardy's paradox in phase space. In the phasespace picture of Hardy's proposed experiment, the negativity of the predictive Wigner distribution is identified as the culprit of the disagreement between classical reasoning and quantum mechanics.

Quantum metrology with open dynamical systems

“…The waveform of interest x(t) can then be the mirror position, momentum or the force. The phase shift φ(t) is given by (49), which is a linear function of x(t) with impulse-response function g(t), and measured in the experiment by a homodyne phase-locked loop, followed by smoothing of the data [29,30,45,[55][56][57][58][59]. The force, for example, is a realization of the Ornstein-Uhlenbeck process, which has a prior power spectral density in the form of…”

Past Quantum States of a Monitored System

“…As demonstrated in Refs. [26][27][28][29], a similar formalism enables the estimation of unknown classical perturbations, applied to a probed quantum system. These works generalize the socalled smoothing procedure applied in the classical probability theory of hidden Markov models (HMMs) [30] to hybrid quantum classical systems.…”

Frequency tracking and parameter estimation for robust quantum state estimation

“…(8)-(10) together as the hybrid master equation, since it involves both quantum and classical states-of-knowledge. Unlike the smoothing filters developed by Tsang [13][14][15], the hybrid master equation propagates forwards in time only.…”

“…There has been work on parameter estimation by Gambetta and Wiseman * jfralph@liverpool.ac.uk † kurt.jacobs@umb.edu ‡ cdhill@unimelb.edu.au [9], Verstraete et al [10], Stockton et al [11], Chase and Geremia [12], and Tsang [13][14][15]. Gambetta and Wiseman have examined a situation similar to the one we consider here, and derived the full Bayesian estimation equations.…”

“…In tracking the classical parameter it was necessary to track the state of the system, and thus the authors also derived the full Bayesian estimation equations. More recently, Chase and Geremia have developed a quantum analog of a classical tracking filter called a particle filter [17], which estimates the evolution of continuous probability density function for a parameter by considering a finite set of weighted sample points [12], and Tsang has developed a generalisation of classical smoothing filters [13][14][15], where measurements are processed forward and backward in time to improve state estimates and the probability distributions for classical parameters. In addition, related work has been done by Yamamoto [18], who considered the problem of robust state estimation for linear quantum systems, using results from classical control theory.…”