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arXiv:1703.02298 · Physical Review Letters

Entanglement-Enhanced Phase Estimation without Prior Phase Information

Giorgio Colangelo, Ferran Martin Ciurana, Graciana Puentes, Morgan W. Mitchell, R. J. Sewell

Published Jun 9, 2017• 21 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.118.233603

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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.
Abstract Planar quantum squeezed (PQS) states, i.e. quantum states which are squeezed in two orthogonal spin components on a plane, have recently attracted attention due to their applications in atomic interferometry and quantum information [1, 2]. In this paper we present an application of the framework described by Puentes et al [3] for planar quantum squeezing via quantum nondemolition (QND) measurement, for the particular case of spin-1/2 systems and nonzero covariance between orthogonal spin components. Our regime, consisting of spin-1/2 entangled planar-squeezed (EPQS) states, is of interest as it can present higher precision for quantum parameter estimation and quantum magnetometry at finite intervals. We show that entangled planar-squeezed states (EPQS) can be used to reconstruct a specific quantum parameter, such as a phase or a magnetic field, within a finite interval with higher precision than PQS states, and without iterative procedures. EPQS is of interest in cases where limited prior knowledge about the specific subinterval location for a phase is available, but it does not require knowledge of the exact phase a priori , as in the case of squeezing on a single spin component.
Planar squeezed states, i.e. quantum states which are squeezed in two orthogonal spin components, have recently attracted attention due to their applications in atomic interferometry and quantum information (He et al 2012 New J. Phys. 14 093012). While canonical variables such as quadratures of the radiation field can be squeezed in at most one component, simultaneous squeezing in two orthogonal spin components can be achieved due to the angular momentum commutation relations. We present a novel scheme for planar squeezing via quantum non-demolition (QND) measurements in spin-1 systems. The QND measurement is achieved via near-resonant paramagnetic Faraday rotation probing, and the planar squeezing is obtained by sequential QND measurement of two orthogonal spin components. We compute the achievable squeezing for a variety of optical depths, initial conditions and probing strategies. The planar squeezed states generated in this way contain entanglement detectable by spin-squeezing inequalities and give an advantage relative to non-squeezed states for any precession phase angle, a benefit for single-shot and high-bandwidth magnetometry.
Reduction of quantum noise in one spin component is a significant tool for enhancing the sensitivities of interferometers and atomic clocks. It has been recently implemented for ultra-cold atomic Bose-Einstein condensate (BEC) interferometer. This type of quantum noise reduction reduces the measurement noise near some predetermined phase. However, if the phase is completely unknown prior to measurement, then it is not known which phase quadrature should be in a squeezed state. We introduce a novel planar squeezing uncertainty relation for spin variance in a plane, and analyze how to obtain such a planar quantum squeezed (PQS) state by using a double-well single component BEC, through the use of local nonlinear S-wave scattering interaction between trapped atoms. Here, we consider the PQS that is generated by using two hyperfine states in a two components BEC system, which is useful for quantum metrology. By comparison with the case of two spatial wells, the Hamiltonian parameters can be controlled in a more efficient way. The spin component can be measured by detecting the occupation number difference between the two internal modes, while one needs to observe a spatial interference pattern in the double well BEC case. This is the major difference between the internal and external cases. Another difference is that one can use the Rabi frequency Ω instead of the Josephson parameters to switch the Hamiltonian parameters through using a diabatic technique. Therefore the coupling could be switched off or on to study the different evolutions. PQS simultaneously reduces the quantum noises of two orthogonal spin projections below the standard quantum limit, while increases the noise in the third dimension. This allows the improvement in phase measurement at any phase-angle. PQS states that reductions of fluctuations everywhere in a plane have potential utility in "one-shot" phase measurement, where iterative or repeated measurement strategies cannot be utilized. The improved interferometric phase measurements and planar uncertainty relations are useful for detecting the entanglement in mesoscopic system between two distinguished modes regardless of the third component.
We introduce an entanglement-depth criterion optimized for planar quantum-squeezed (PQS) states. It is connected with the sensitivity of such states for estimating a phase generated by rotations about an axis orthogonal to its polarization. We compare numerically our criterion with the well-known extreme spin-squeezing condition of S\o{}rensen and M\o{}lmer [Phys. Rev. Lett. 86, 4431 (2001)] and show that our condition detects a higher depth of entanglement when both planar spin variances are squeezed below the standard quantum limit. We employ our theory to monitor the entanglement dynamics in a PQS state produced via quantum nondemolition measurements using data from a recent experiment [Phys. Rev. Lett. 118, 233603 (2017)].
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.