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Kluwer Academic Publishers eBooks

Spin Squeezing and Decoherence Limit in Ramsey Spectroscopy

Duger Ulam-Orgikh, Masahiro Kitagawa

Published Feb 24, 2006• 2 citations
Publisher

Abstract

It is known that partially entangled state gives improved sensitivity of Ramsey spectroscopy in the presence of decoherence, whereas maximally entangled state gives no improvement. However it has been an open question whether the absolute limit in the improvement is attainable by optimal entanglement alone or via measurement optimization. In this Paper, we answer this question by showing that even sub-optimal entanglement generated by the simplest spin squeezing can asymptotically attain the absolute improvement with increasing number of atoms.

DOI
10.1007/0-306-47114-0_72

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It is known that partially entangled state gives improved sensitivity of Ramsey spectroscopy in the presence of decoherence, whereas maximally entangled state gives no improvement. However it has been an open question whether the absolute limit in the improvement is attainable by optimal entanglement alone or via measurement optimization. In this Paper, we answer this question by showing that even sub-optimal entanglement generated by the simplest spin squeezing can asymptotically attain the absolute improvement with increasing number of atoms.
It is known that a partially entangled state gives improved sensitivity in generalized Ramsey spectroscopy in the presence of decoherence, whereas a maximally entangled state does not. However, it has been an open question whether the known decoherence limit in the improvement is achievable. We show that every spin-squeezed state possesses pairwise entanglement, thereby improving the spectroscopic sensitivity, and that even suboptimal entanglement in its easiest implementation suffices to asymptotically reach the decoherence limit without any measurement optimization.
The optimal precision of frequency measurements in the presence of decoherence is discussed. We analyze different preparations of nnn two-level systems as well as different measurement procedures. We show that standard Ramsey spectroscopy on uncorrelated atoms and optimal measurements on maximally entangled states provide the same resolution. The best resolution is achieved using partially entangled preparations with a high degree of symmetry.
The use of entanglement in sensing is often motivated by improved sensitivity arising from non-classical correlations among probes. For static signals, however, this quantum advantage in Ramsey spectroscopy with NNN probes is known to be highly susceptible to dephasing, which severely limits the possible gain from entanglement. In contrast, we consider sensing of signals that vary in time. Considering the simplest example of dephasing by classical white noise, we show that entanglement allows a large increase in bandwidth when estimating the frequency components of a time-dependent signal. Without sacrificing any sensitivity, entanglement allows for an increase in bandwidth by a factor of NNN. Alternatively, when interrogation times are limited by the signal bandwidth rather than by dephasing, the average estimation uncertainty over a fixed bandwidth is reduced by a factor of N\sqrt{N}N​ compared with separable states.
We investigate the presence of spin- and planar-squeezing in generalized superpositions of atomic (or spin) coherent states. Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise-limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases ϕ ∼ 0 . On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to enhanced metrological precision in measuring phases without the need for explicit prior knowledge of the phase value. In this paper, we show that the generalized superposition state can be parametrized to display both spin-squeezing along all orthogonal axes and planar-squeezing along all orthogonal planes for all values of J > 1 / 2 . We close with an application of the maximally spin- and planar-squeezed states to quantum metrology.
We consider optimally spin-squeezed states that maximize the sensitivity of the Ramsey spectroscopy, and for which the signal-to-noise ratio scales as the number of particles N. Using the variational principle we prove that these states are eigensolutions of the Hamiltonian $H(\ensuremath{\lambda})=\ensuremath{\lambda}{S}_{z}^{2}\ensuremath{-}{S}_{x},$ and that, for large N, the states become equivalent to the quadrature squeezed states of the harmonic oscillator. We present numerical results that illustrate the validity of the equivalence.
Techniques for making quantum-limited nonlinear measurements have attracted a great deal of recent attention because they allow a measurement sensitivity that improves as the number of measuring particles (e.g. photons) increases faster than an equivalent linear measurement. This super-Heisenberg scaling was recently demonstrated in a proof-of-principle experiment [1], in which a measurement Hamiltonian of the form studied in Ref. [2] was carefully engineered by exploiting nonlinearities in a Faraday rotation based measurement of atomic spins. However, despite the improved scaling, the absolute measurement sensitivity in this experiment remained significantly worse than a linear Faraday rotation measurement due to the weak atom-light coupling strength of the nonlinear Faraday rotation, and effect of higher-order nonlinearities at large photon number. This leaves open the question of whether it is possible to make a nonlinear measurement that both benefits from the improved scaling, and competes in absolute sensitivity with linear techniques.