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arXiv:0906.0962 · Physical Review A

Quantum-limited metrology and Bose-Einstein condensates

Sergio Boixo, Animesh Datta, Matthew J. Davis, Anil Shaji, Alexandre B. Tacla, Carlton M. Caves

Published Sep 4, 2009• 72 citations
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Abstract

We discuss a quantum-metrology protocol designed to estimate a physical parameter in a Bose-Einstein condensate of $N$ atoms, and we show that the measurement uncertainty can decrease faster than $1/N$. The $1/N$ scaling is usually thought to be the best possible in any measurement scheme. From the perspective of quantum information theory, we outline the main idea that leads to a measurement uncertainty that scales better than $1/N$. We examine in detail some potential problems and challenges that arise in implementing such a measurement protocol using a Bose-Einstein condensate. We discuss how some of these issues can be dealt with by using lower-dimensional condensates trapped in nonharmonic potentials.

DOI
10.1103/physreva.80.032103

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We study the performance of initial product states of nnn-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear kkk-body $(k\ensuremath{\ll}n)$ Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as 1/nk1/{n}^{k}1/nk, and for initial product states, it scales as $1/{n}^{k\ensuremath{-}1/2}$. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian (k=2)(k=2)(k=2), implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O({n}^{\ensuremath{-}3/2})$ scaling for k=2k=2k=2; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the $O({n}^{\ensuremath{-}3/2})$ sensitivity scaling for initial product states.
Questions about quantum limits on measurement precision were once viewed from the perspective of how to reduce or avoid the effects of quantum noise. With the advent of quantum information science came a paradigm shift to proving rigorous bounds on measurement precision. These bounds have been interpreted as saying, first, that the best achievable sensitivity scales as 1/n, where n is the number of particles one has available for a measurement and, second, that the only way to achieve this Heisenberg-limited sensitivity is to use quantum entanglement. We review these results and show that using quadratic couplings of n particles to a parameter to be estimated, one can achieve sensitivities that scale as 1/n^2 if one uses entanglement, but even in the absence of any entanglement at any time during the measurement protocol, one can achieve a super-Heisenberg scaling of 1/n^(3/2)