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arXiv:1403.1192 · Physical Review A

Estimation of atomic interaction parameters by photon counting

Alexander Holm Kiilerich, Klaus Mølmer

Published May 12, 2014• 57 citations
arXivHTMLPDF

Abstract

Detection of radiation signals is at the heart of precision metrology and sensing. In this article we show how the fluctuations in photon-counting signals can be exploited to optimally extract information about the physical parameters that govern the dynamics of the emitter. For a simple two-level emitter subject to photon counting, we show that the Fisher information and the Cram\'er-Rao sensitivity bound based on the full detection record can be evaluated from the waiting-time distribution in the fluorescence signal which can, in turn, be calculated for both perfect and imperfect detectors by a quantum trajectory analysis. We provide an optimal estimator achieving that bound.

DOI
10.1103/physreva.89.052110

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The physical parameters governing the dynamics of a light-emitting quantum system can be estimated from the photon counting signal. The information available in the full detection record can be analyzed by means of the distribution of waiting times between detection events. Our theory allows calculation of the asymptotic, long-time behavior of the sensitivity limit, and it applies to emission processes with branching towards different final states accompanied by the emission of distinguishable photons. We illustrate the theory by application to a laser-driven $\ensuremath{\Lambda}$-type atom.
Precision measurements with quantum systems rely on our ability to trace the differences between experimental signals to variations in unknown physical parameters. In this Letter we derive the Fisher information and the ensuing Cramér-Rao sensitivity limit for parameter estimation by continuous measurements on an open quantum system. We illustrate our theory by its application to resonance fluorescence from a laser-driven two-state atom; we show that photon counting and homodyne detection records yield different sensitivity to the atomic parameters, while none of them exceed our general result.
The quantum Cramer-Rao bound relates estimation sensitivity in quantum precision measurements to the distinguishability of different quantum states. We show that the theoretical sensitivity limit to parameters that govern the dynamics of an open quantum system coupled to a fully quantized environment can be evaluated from the reduced system master equation. We provide the bound for a laser driven two-state atom, where photon counting and homodyne detection of the fluorescence signal yield different sensitivity to the atomic and field parameters, while none of them exceed our general sensitivity limit.
In this chapter we consider the physical limits to information extraction. This is an important aspect of optical quantum information processing in that many high-precision experiments (such as gravitational wave detection) are implemented in optical systems, i.e., interferometers. It is not surprising that just as in computation and communication, the use of quintessentially quantum mechanical properties allows us to improve the sensitivity in interferometry. We start this chapter with a derivation of the Fisher information and the Cramér-Rao bound, which tell us how much information we can extract about a parameter in a set of measurements. In Section 13.2 we introduce the statistical distance between two probability distributions. This can in turn be used to determine how many times the system needs to be queried before we can determine which probability distribution governs the system. In addition, we make a connection between the statistical distance and the angle between states in Hilbert space. In Sections 13.3 and 13.4 we derive bounds on how fast quantum states evolve to orthogonal states, and how entangled states can be used to improve parameter estimation. Finally, in Section 13.5 we present a number of approaches for implementing quantum metrology in optical systems, most importantly in optical interferometers. Parameter estimation and Fisher information In the theory of computation, discrete variables have the benefit that a practically perfect readout is often possible.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.