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Quantum-Atom Optics Downunder (QAO)

Quantum Parameter Estimation With Cold Atomic Spins

Anthony Miller, John K. Stockton, Magnus T. L. Hsu, Orion Crisafulli, Gopal Sarma, Luc Bouten, Andrew Silberfarb, Hideo Mabuchi

Published Jan 1, 2007• 0 citations
Publisher

Abstract

Optical Faraday rotation of a detuned probe laser can be used to monitor the dynamic response of atomic hyperfine spins to time-varying magnetic fields, with minimal backaction. We present a modern formulation of broadband atomic magnetometry as an example of quantum parameter estimation, establishing important connections between abstract quantum measurement theory and concrete physical models. We discuss experimental challenges and progress in this area that sits at the interface of quantum information science and precision measurement.

DOI
10.1364/qao.2007.qwa1

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•Jan 1, 2006•0 citations
Optical Faraday rotation of a detuned probe laser can be used to monitor the dynamic response of atomic hyperfine spins to time-varying magnetic fields, with minimal backaction. We present a modern formulation of broadband atomic magnetometry as an example of quantum parameter estimation, establishing important connections between abstract quantum measurement theory and concrete physical models. We discuss experimental challenges and progress in this area that sits at the interface of quantum information science and precision measurement.
We present an experimental demonstration of closed-loop quantum parameter estimation in which real-time feedback is used to achieve robustness to modeling uncertainty. By performing broadband estimation of a magnetic field acting on hyperfine spins in a cold atom ensemble, we show that accuracy is not compromised by fluctuations in total atom number even though the measured signal in our canonical configuration depends only on the product of the field and atom number. This methodology could be essential for efforts to utilize conditional squeezing in spin-resonance measurements.
We measure the sensitivity of a broadband atomic magnetometer using quantum non-demolition spin measurements. A cold, dipole-trapped sample of rubidium atoms provides a long-lived spin system in a non-magnetic environment, and is probed non-destructively by paramagnetic Faraday rotation. The calibration procedure employs a known reference state, the maximum-entropy or 'thermal' spin state and quantitative imaging-based atom counting to identify electronic, quantum, and technical noise in both the probe and spin system. The measurement achieves sensitivity 2.8 dB better than the projection noise level (6dB better if optical noise is suppressed) and will enable squeezing-enhanced broadband magnetometry [Geremia, et al. PRL 94, 203002 (2005)].
We have implemented a continuous measurement of the mean magnetic moment of an ensemble of atoms trapped in a far-off-resonance optical lattice, by detecting the Faraday rotation of one of the lattice beams after it has passed through the atom cloud. In a first demonstration experiment we have observed Larmor precession with high signal-to-noise ratio, and compared the performance of the measurement with a simple theory. Faraday spectroscopy offers an ideal method to monitor the atomic dynamics and will be applied to the study of quantum chaos in magneto-optical lattices. In principle the measurement sensitivity can be increased to the point where quantum backaction becomes significant, thereby opening the door to studies of quantum feedback, spin squeezing and the role played by quantum measurement in quantum/classical correspondence.
We study the problem of estimating a time-dependent magnetic field by continuous optical probing of an atomic ensemble. The magnetic field is assumed to follow a stochastic Ornstein-Uhlenbeck process and it induces Larmor precession of the atomic ground-state spin, which is read out by the Faraday polarization rotation of a laser field probe. The interactions and the measurement scheme are compatible with a hybrid quantum-classical Gaussian description of the unknown magnetic field, and the atomic and field variables. This casts the joint conditional quantum dynamics and classical parameter estimation problem in the form of update formulas for the first and second moments of the classical and quantum degrees of freedom. Our hybrid quantum-classical theory is equivalent to the classical theory of Kalman filtering and to the quantum theory of Gaussian states. By reference to the classical theory of smoothing and with the quantum theory of past quantum states, we show how optical probing after time ttt improves our estimate of the value of the magnetic field at time ttt, and we present numerical simulations that analyze and explain the improvement over the conventional filtering approach.
The linear Faraday effect is used to implement a continuous measurement of the spin of a sample of laser-cooled atoms trapped in an optical lattice. One of the optical lattice beams serves also as a probe beam, thereby allowing one to monitor the atomic dynamics in real time and with minimal perturbation. A simple theory is developed to predict the measurement sensitivity and associated cost in terms of decoherence caused by the scattering of probe photons. Calculated signal-to-noise ratios in measurements of Larmor precession are found to agree with experimental data for a wide range of lattice intensity and detuning. Finally, quantum back-action is estimated by comparing the measurement sensitivity to spin projection noise, and shown to be insignificant in the current experiment. A continuous quantum measurement based on Faraday spectroscopy in optical lattices may open up new possibilities for the study of quantum feedback and classically chaotic quantum systems.
Quantum measurement is one of the most important features of quantum theory. Although mathematical predictions have been verified in great detail, practical implementation has lagged behind. Only recently have people begun to take advantage of quantum measurement properties to produce new technologies. This research helps fill that technological gap by experimental examination of a continuous, optical measurement for an ensemble of cold atomic spins. The essential physics reduces to the interaction between an atomic ensemble and a weak optical field, which has many well known results. While this work demonstrates many novel applications of the interaction, it also shows that the whole can be more than the sum of the individual parts. Starting with basic characterization of the measurement system using laser-cooled cæsium atoms, the mean value of a spin component is obtained in real time. In essence, the angular momentum of the atomic spins creates a Faraday-like rotation in the polarization of a laser probe beam. With slight modifications, additional spin components are also observed, and are shown to be in good agreement with predictions. In measuring spin dynamics, it is important to account for effects of the probe on the spin states as well. Capitalizing on this as a resource, the probe-induced ac-Stark shift is used to transform a quasi-classical spin-coherent state into a highly quantum Schrödinger cat type of superposition between two spin states. Finally, this work combines all the previous results to demonstrate how a continuous measurement of the spin with a carefully crafted evolution created in part by the probe, allows for nearly real-time determination of the complete spin density matrix. In a single 1.5 millisecond run, a spin density matrix is determined with fidelities ranging from about 85% to 90% across a wide spectrum of test states.