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arXiv:1712.00874 · Physical Review A

Smoothing of Gaussian quantum dynamics for force detection

Zhishen Huang, Mohan Sarovar

Published Apr 10, 2018• 12 citations
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Abstract

Building on recent work by Gammelmark et al. [Phys. Rev. Lett. 111, 160401 (2013)] we develop a formalism for prediction and retrodiction of Gaussian quantum systems undergoing continuous measurements. We apply the resulting formalism to study the advantage of incorporating a full measurement record and retrodiction for impulselike force detection and accelerometry. We find that using retrodiction can only increase accuracy in a limited parameter regime, but that the reduction in estimation noise that it yields results in better detection of impulselike forces.

DOI
10.1103/physreva.97.042106

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Building on recent work by Gammelmark et al. [Phys. Rev. Lett. 111, 160401 (2013)] we develop a formalism for prediction and retrodiction of Gaussian quantum systems undergoing continuous measurements. We apply the resulting formalism to study the advantage of incorporating a full measurement record and retrodiction for impulselike force detection and accelerometry. We find that using retrodiction can only increase accuracy in a limited parameter regime, but that the reduction in estimation noise that it yields results in better detection of impulselike forces.
Gaussian states of quantum oscillators are fully characterized by the mean values and the covariance matrix of their quadrature observables. We consider the dynamics of a system of oscillators subject to interactions, damping, and continuous probing which maintain their Gaussian state property. Such dynamics is found in many physical systems that can therefore be efficiently described by the ensuing effective representation of the density matrix $\ensuremath{\rho}(t)$. Our probabilistic knowledge about the outcome of measurements on a quantum system at time ttt is not only governed by $\ensuremath{\rho}(t)$ conditioned on the evolution and measurement outcomes obtained until time ttt but is also modified by any information acquired after ttt. It was shown [S. Gammelmark, B. Julsgaard, and K. M\o{}lmer, Phys. Rev. Lett. 111, 160401 (2013)] that this information is represented by a supplementary matrix, E(t)E(t)E(t). We show here that the restriction of the dynamics of $\ensuremath{\rho}(t)$ to Gaussian states implies that the matrix E(t)E(t)E(t) is also fully characterized by a vector of mean values and a covariance matrix. We derive the dynamical equations for these quantities and we illustrate their use in the retrodiction of measurements on Gaussian systems.
What knowledge can be obtained from the record of a continuous measurement about the quantum state of the measured system at the beginning of the measurement? The task of quantum state retrodiction, the inverse of the more common state prediction, is rigorously addressed in quantum measurement theory through retrodictive positive operator-valued measures (POVMs). This introduction to this general framework presents its practical formulation for retrodicting Gaussian quantum states using continuous-time homodyne measurements and applies it to optomechanical systems. We identify and characterize achievable retrodictive POVMs in common optomechanical operating modes with resonant or off-resonant driving fields and specific choices of local oscillator frequencies in homodyne detection. In particular, we demonstrate the possibility of a near-ideal measurement of the quadrature of the mechanical oscillator, giving direct access to the position or momentum distribution of the oscillator at a given time. This forms the basis for complete quantum state tomography, albeit in a destructive manner.
Continuous measurements can prepare macroscopic mechanical oscillators in conditional quantum states, but their covariance is difficult to verify. The conventional retrodictive estimator assumes a forward--backward covariance symmetry and can be biased, because physical dynamics such as feedback damping reduces the observability of the state from future records. Here, we derive an exact linear-Gaussian estimator from causal, retrodictive, and smoothed trajectories. For a milligram-scale mirror, it agrees with a Riccati prediction based on parameters fixed independently, while the conventional estimate exhibits a covariance-space bias of dM≃3.5d_M\simeq3.5dM​≃3.5. Our method paves the way toward unbiased testing of macroscopic entanglement within a calibrated linear-Gaussian model, applicable to both tabletop mirrors and kg-scale gravitational-wave test masses.
We analyze the problem of estimating past quantum states of a monitored system from a mathematical perspective in order to ensure self-consistency with the principle of quantum nondemolition. Despite several claims of “measuring noncommuting observables” in the physics literature, we show that we are always measuring commuting processes. Our main interest is in the notion of quantum smoothing, or retrodiction. In particular, we examine proposals to estimate the result of an external measurement made on an open quantum systems during a period where it is also undergoing continuous monitoring. A full analysis shows that the nondemolition principle is in place, and so a well-posed statistical inference problem can be formulated. We extend the formalism to consider multiple independent external measurements made on the system over the course of a continual period of monitoring.
Estimating the state of an open quantum system monitored over time requires incorporating information from past measurements (filtering) and, for improved accuracy, also from future measurements (smoothing). While classical smoothing is well understood within a Bayesian framework, its quantum generalization has been challenging, leading to distinct and seemingly incompatible approaches. In this work, we demonstrate that quantum state smoothing hinges on a uniquely quantum feature: the fundamental dependence of retrodiction on prior correlations. We introduce auxiliary systems into the prior to represent specified correlations formed during preparation and evolution, and formulate quantum state smoothing as extended Bayesian retrodiction. This framework identifies previous approaches as different choices of the extended prior, and naturally extends it to other choices that have not been considered before. We also give an information-theoretic characterization of the choices of prior, in terms of the average entropy of the smoothed states. Our results establish quantum state smoothing as a fundamentally retrodictive process just like classical smoothing, with proper quantum features clearly identified.
Smoothing is a technique for estimating the state of an imperfectly monitored open system by combining both prior and posterior measurement information. In the quantum regime, current approaches to smoothing either give unphysical outcomes, due to the non-commutativity of the measurements at different times, or require assumptions about how the environment is measuring the system, which with current technology is unverifiable. We propose a novel definition of the smoothed quantum state based on quantum Bayesian retrodiction, which mirrors the classical retrodictive approach to smoothing. This approach always yields physical results and does not require any assumption on the environment. We show that this smoothed state has, on average, greater purity than the state reconstructed using just the prior information. Finally, we make a connection with the smoothing theory of Guevara and Wiseman in a well-studied regime, and describe from a purely quantum perspective how it conditions on the posterior information.