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arXiv:1605.00902 · Physical Review A

Bayesian parameter estimation by continuous homodyne detection

Alexander Holm Kiilerich, Klaus Mølmer

Published Sep 6, 2016• 75 citations
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Abstract

We simulate the process of continuous homodyne detection of the radiative emission from a quantum system, and we investigate how a Bayesian analysis can be employed to determine unknown parameters that govern the system evolution. Measurement backaction quenches the system dynamics at all times and we show that the ensuing transient evolution is more sensitive to system parameters than the steady state of the system. The parameter sensitivity can be quantified by the Fisher information, and we investigate numerically and analytically how the temporal noise correlations in the measurement signal contribute to the ultimate sensitivity limit of homodyne detection.

DOI
10.1103/physreva.94.032103

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We simulate the process of continuous homodyne detection of the radiative emission from a quantum system, and we investigate how a Bayesian analysis can be employed to determine unknown parameters that govern the system evolution. Measurement backaction quenches the system dynamics at all times and we show that the ensuing transient evolution is more sensitive to system parameters than the steady state of the system. The parameter sensitivity can be quantified by the Fisher information, and we investigate numerically and analytically how the temporal noise correlations in the measurement signal contribute to the ultimate sensitivity limit of homodyne detection.
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.
Continuous monitoring is a powerful tool for analyzing quantum systems and is increasingly used as a non-demolition technique in quantum metrology. In this context, noisy data acquired through continuous monitoring can be used to precisely pinpoint unknown parameters of the system. However, extending the theoretical framework that connects these data to the underlying parameters beyond Markovian dynamics is notoriously difficult, primarily because such dynamics cannot be expressed as completely positive divisible maps. To overcome this, we propose a method based on the reaction coordinate mapping to extend these parameter-estimation techniques beyond the Markovian regime. Our approach specifically targets linear systems with non-Markovian dynamics undergoing Gaussian continuous measurements, such as homodyne detection. Within this framework, we analyze Bayesian estimation and provide an analytical expression for the Fisher information, alongside the asymptotic scaling of the estimation precision for arbitrary parameters. Finally, we demonstrate the efficacy of our method through the example of thermometry of a bosonic bath.
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We derive an alternative formulation of the stochastic master equation describing the evolution of quantum systems continuously measured via homodyne detection in a squeezed-thermal environment. This formulation allows us to identify two distinct collapse operators, corresponding respectively to the monitored and unmonitored channels, and to express the quantum state evolution directly in terms of Kraus-like operators. As a consequence, it facilitates the application of advanced numerical methods that account for higher-order corrections and discrete-time monitoring, and it also enables the efficient numerical evaluation of Cramér-Rao bounds in parameter estimation problems. To demonstrate the latter, we evaluate the Fisher information of the continuous homodyne monitoring for qubit frequency estimation in the presence of a squeezed-thermal bath. Our results show that exponential scaling of the steady-state Fisher information rate with squeezing is retained even at non-zero temperature.
The deterioration of precision caused by quantum decoherence in dissipative environments is a longstanding problem in the development of quantum metrology. Therefore, it is significant to effectively analyze the evolution of quantum systems under dissipative environments and quantitatively estimate the parameters given a certain probe. Here, we propose a continuous-variable (CV) quantum metrology scheme within the Bayesian paradigm to estimate phase shift, loss coefficient and temperature of a dissipative environment. Through simulation analysis, we investigate the achievable precision of Bayesian quantum parameter estimation with Gaussian states and homodyne measurements, and the influence of different parameters on the estimation process. More interestingly, the scheme can be applied in more practical scenarios where not enough measurements are obtained and the frequentist analysis is not well defined.
Continuous monitoring of open quantum-optical systems offers a promising route towards quantum-enhanced sensing. In such sensing schemes, the ultimate precision limit is dictated by the quantum Cramér-Rao bound. However, reaching this fundamental bound with established continuous measurement techniques has remained an outstanding challenge. Here, we present an approach tailored to open quantum-optical sensors operating near dissipative critical points. The key insight of our approach is to exploit a synergy between the intrinsic critical dynamics and the unavoidable quantum measurement backaction of continuous homodyne detection. By carefully aligning the homodyne phase to the squeezing angle of the sensor state, quantum backaction can be evaded in the critical dynamics, thereby significantly boosting the attainable precision. We demonstrate the broad applicability of this approach across several sensor models and for both single- and multi-parameter estimation, showing that it consistently narrows the gap toward the ultimate precision bound.