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arXiv:2006.08974 · Physical Review Letters

Noisy Quantum Metrology Enhanced by Continuous Nondemolition Measurement

Matteo A. C. Rossi, Francesco Albarelli, Dario Tamascelli, Marco G. Genoni

Published Nov 12, 2020• 68 citations
arXivHTMLPDF

Abstract

We show that continuous quantum nondemolition (QND) measurement of an atomic ensemble is able to improve the precision of frequency estimation even in the presence of independent dephasing acting on each atom. We numerically simulate the dynamics of an ensemble with up to N=150 atoms initially prepared in a (classical) spin coherent state, and we show that, thanks to the spin squeezing dynamically generated by the measurement, the information obtainable from the continuous photocurrent scales superclassically with respect to the number of atoms N. We provide evidence that such superclassical scaling holds for different values of dephasing and monitoring efficiency. We moreover calculate the extra information obtainable via a final strong measurement on the conditional states generated during the dynamics and show that the corresponding ultimate limit is nearly achieved via a projective measurement of the spin-squeezed collective spin operator. We also briefly discuss the difference between our protocol and standard estimation schemes, where the state preparation time is neglected.

DOI
10.1103/physrevlett.125.200505

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We show that continuous quantum nondemolition (QND) measurement of an atomic ensemble is able to improve the precision of frequency estimation even in the presence of independent dephasing acting on each atom. We numerically simulate the dynamics of an ensemble with up to N=150 atoms initially prepared in a (classical) spin coherent state, and we show that, thanks to the spin squeezing dynamically generated by the measurement, the information obtainable from the continuous photocurrent scales superclassically with respect to the number of atoms N. We provide evidence that such superclassical scaling holds for different values of dephasing and monitoring efficiency. We moreover calculate the extra information obtainable via a final strong measurement on the conditional states generated during the dynamics and show that the corresponding ultimate limit is nearly achieved via a projective measurement of the spin-squeezed collective spin operator. We also briefly discuss the difference between our protocol and standard estimation schemes, where the state preparation time is neglected.
Ultimate quantum fluctuations which limit the fundamental precision in metrology can be only reduced by entanglement of particles. In particular, squeezing of the fluctuations called the spin projection noise by means of generation of entanglement in an ensemble of atoms can improve the precision of spectroscopy and of atomic clocks. A powerful tool for generation of multiparticle entanglement is a quantum nondemolition measurement (QND). We will describe the fundamentals of QND measurement of atomic collective state with light and present a recent experiment [1] where this method has been used to reduce the projection noise on the clock transition in an ensemble of 105 cold and trapped Cesium atoms. Another method for generation of spin squeezing via electron-nucleus entanglement applicable only for atoms with angular momentum higher than ½ will be also discussed [2].
We study the generation of planar quantum squeezed (PQS) states by quantum nondemolition (QND) measurement of an ensemble of ^{87}Rb atoms with a Poisson distributed atom number. Precise calibration of the QND measurement allows us to infer the conditional covariance matrix describing the F_{y} and F_{z} components of the PQS states, revealing the dual squeezing characteristic of PQS states. PQS states have been proposed for single-shot phase estimation without prior knowledge of the likely values of the phase. We show that for an arbitrary phase, the generated PQS states can give a metrological advantage of at least 3.1 dB relative to classical states. The PQS state also beats, for most phase angles, single-component-squeezed states generated by QND measurement with the same resources and atom number statistics. Using spin squeezing inequalities, we show that spin-spin entanglement is responsible for the metrological advantage.
We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.
Sensing a magnetic field with an atomic magnetometer operated in real time presents significant challenges, primarily due to sensor nonlinearity, the presence of noise, and the need for one-shot estimation. To address these challenges, we propose a comprehensive approach that integrates measurement, estimation, and control strategies. Specifically, this involves implementing a quantum nondemolition measurement based on continuous light probing of the atomic ensemble. The resulting photocurrent is then directed into an extended Kalman filter to produce instantaneous estimates of the system’s dynamical parameters. These estimates, in turn, are utilized by a linear quadratic regulator, whose output is applied back to the system through a feedback loop. This procedure automatically steers the atomic ensemble into a spin-squeezed state, yielding a quantum enhancement in precision. Furthermore, thanks to the feedback proposed, the atoms exhibit entanglement even when the measurement data are discarded. To prove that our approach constitutes the optimal strategy in realistic scenarios, we derive ultimate bounds on the estimation error applicable in the presence of both local and collective decoherence, and show that these are indeed attained. Additionally, we demonstrate that, for large ensembles, the extended Kalman filter not only reliably predicts its own estimation error in real time, but also accurately estimates spin squeezing at short timescales.
We use the vacuum Rabi splitting to perform quantum nondemolition measurements that prepare a conditionally spin squeezed state of a collective atomic psuedospin. We infer a 3.4(6) dB improvement in quantum phase estimation relative to the standard quantum limit for a coherent spin state composed of uncorrelated atoms. The measured collective spin is composed of the two-level clock states of nearly 10(6) (87)Rb atoms confined inside a low finesse F=710 optical cavity. This technique may improve atomic sensor precision and/or bandwidth, and may lead to more precise tests of fundamental physics.
We review some recent results regarding the use of time-continuous measurements for quantum-enhanced metrology. First, we present the underlying quantum estimation framework and elucidate the correct figures of merit to employ. We then report results from two previous works where the system of interest is an ensemble of two-level atoms (qubits) and the quantity to estimate is a magnetic field along a known direction (a frequency). In the first case, we show that, by continuously monitoring the collective spin observable transversal to the encoding Hamiltonian, we get Heisenberg scaling for the achievable precision (i.e., 1 / N for N atoms); this is obtained for an uncorrelated initial state. In the second case, we consider independent noises acting separately on each qubit and we show that the continuous monitoring of all the environmental modes responsible for the noise allows us to restore the Heisenberg scaling of the precision, given an initially entangled GHZ state.