Entanglement-Enhanced Phase Estimation without Prior Phase Information

Abstract

We study the generation of planar quantum squeezed (PQS) states by quantum non-demolition (QND) measurement of a cold ensemble of 87 Rb atoms. Precise calibration of the QND measurement allows us to infer the conditional covariance matrix describing the Fy and Fz components of the PQS, revealing the dual squeezing characteristic of PQS. PQS states have been proposed for singleshot phase estimation without prior knowledge of the likely values of the phase. We show that for an arbitrary phase, the generated PQS gives a metrological advantage of at least 3.1 dB relative to classical states. The PQS also beats traditional squeezed states generated with the same QND resources, except for a narrow range of phase values. Using spin squeezing inequalities, we show that spin-spin entanglement is responsible for the metrological advantage. Estimation of interferometric phases is at the heart of precision sensing, and is ultimately limited by quantum statistical effects [1]. Entangled states can improve sensitivity beyond the "classical limits" that restrict sensing with independent particles, and a diversity of entangled states have been demonstrated for this task, including photonic squeezed states [2,3] and spin-squeezed states [4]. These give improved sensitivity for a narrow range of phases, but worsened sensitivity for most phases. Optical "NOON" states [5] give improved sensitivity over the whole phase range, but introduce additional phase ambiguity that increases with the size, and thus sensitivity advantage, of the NOON state. Recent proposals [6-8] suggest using planar quantum squeezed (PQS) states to obtain an entanglement-derived advantage for all phase angles, with no additional phase ambiguity. A natural application is in high-bandwidth atomic sensing [9][10][11], in which the precession angle may not be predictable in advance. PQS states may also be valuable for ab initio phase estimation using feedback [12][13][14].Discussion of such states under the name "intelligent spin states" [15] predates modern squeezing terminology, and analogous states have been studied with optical polarization [16][17][18]. Generation of PQS states in material systems has been proposed using two-well BoseEinstein condensates with tunable and attractive interactions [7,8], and using quantum non-demolition (QND) measurements [19]. Here we take the latter approach, using Faraday rotation QND measurements [20,21] applied to an ensemble of cold atomic spins with f = 1. As the ensemble spin precesses about the x axis in an external magnetic field [22][23][24], we measure the y and z spin components to generate measurement-induced squeezing in these two components, creating a PQS state. The resulting state has enhanced sensitivity to precession angle, i.e., to Zeeman-shift induced phase. The demonstrated PQS state beats the best possible classical state at any precession angle, and beats traditional spin-squeezed states when averaged over the possible angles. Spin-squeezing inequalities [7,8,25] detect spin entanglement in the PQS state...

Longitudinal short-distance constraints for the hadronic light-by-light contribution to (g − 2)μ with large-Nc Regge models

“…[52], which involve the model-dependent concept of an off-shell pion. This becomes particularly important when combined with other intermediate states, ensuring that the pseudoscalar poles are consistent with, for instance, the dispersive definition of two-pion intermediate states [34,35], which in turn are determined by the corresponding on-shell quantities, in this case the helicity amplitudes for γ * γ * → ππ [53][54][55][56][57][58]. However, in contrast to HVP there is no closed formula that resums all possible intermediate states (in terms of the cross section for e + e − → hadrons [59,60]), in such a way that the consideration of exclusive channels will break down eventually, irrespective of the complications when extending the dispersive formalism to higher-multiplicity intermediate states.…”

“…In fact, as we will show below, with modern input for the TFFs the corresponding increase would become even larger. While there is no doubt that the SDC is important-it is, in fact, one of the few constraints on the mixed-energy regions in which one photon virtuality remains small-modifying the expression for the pseudoscalar poles in this way is not compatible with the dispersive description of the four-point HLbL tensor [31][32][33][34][35] and spoils consistency with other intermediate states in the same framework.…”

“…Since the HVP contribution can be systematically calculated with a data-driven dispersive approach [5-9], lattice QCD [10][11][12][13][14][15][16], and potentially be accessed independently by the proposed MUonE experiment [17,18], which aims to measure the space-like finestructure constant α(t) in elastic electron-muon scattering, the HLbL contribution may end up dominating the theoretical error. 1 Apart from lattice QCD [27][28][29], recent data-driven approaches towards HLbL scattering are again rooted in dispersion theory, either for the HLbL tensor [30][31][32][33][34][35], the Pauli 1 Note that higher-order insertions of HVP [5,19,20] and HLbL [21] are already under sufficient control, as are hadronic corrections in the anomalous magnetic moment of the electron, where recently a 2.5 σ tension between the direct measurement [22] and the SM prediction [23] using the fine-structure constant from Cs interferometry [24] emerged [25,26].…”

Dispersive construction of two-loop P→πππ(P=K,η) amplitudes

“…The determination of the subtraction constants arises from matching [64,65] with the NLO results of chiral perturbation theory. A more recent analysis [39] within this framework uses instead information from the chiral perturbation theory calculation at NNLO of Ref. [30].…”

Lattice calculation of the hadronic leading order contribution to the muon g − 2

“…In the low-energy regime one relies on experimental data for the R-ratio R had (s), which implies that the SM prediction is subject to experimental uncertainties. While a dispersive framework to address a hlbl µ is being developed [9][10][11][12][13][14][15][16][17][18][19], the current SM prediction is still largely based on hadronic models [20].…”