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arXiv:2410.22032 · Quantum

From spin squeezing to fast state discrimination

Michael R. Geller

Published May 20, 2026• 0 citations

Abstract

There is great interest in generating and controlling entanglement in Bose-Einstein condensates and similar ensembles for use in quantum computation, simulation, and sensing. One class of entangled states useful for enhanced metrology are spin-squeezed states of N two-level atoms. After preparing a spin coherent state of width 1 / N centered at coordinates ( θ , ϕ ) on the Bloch sphere, atomic interactions generate a nonlinear evolution that shears the state's probability density, stretching it to an ellipse and causing squeezing in a direction perpendicular to the major axis. Here we consider the same setup but in the N → ∞ limit . This shrinks the initial coherent state to zero area. Large N also suppresses two-particle entanglement and squeezing, as required by a monogamy bound. The torsion (1-axis twist) is still present, however, and the center of the large N coherent state evolves as a qubit governed by a two-state Gross-Pitaevskii equation. The resulting nonlinearity is known to be a powerful resource in quantum computation. It can be used to implement single-input quantum state discrimination, an impossibility within linear one-particle quantum mechanics. We obtain a solution to the discrimination problem in terms of a Viviani curve on the Bloch sphere. We also consider an open-system variant containing both Bloch sphere torsion and dissipation. In this case it should be possible to generate two basins of attraction within the Bloch ball, having a shared boundary that can be used for a type of autonomous state discrimination. We explore these and other connections between spin squeezing in the large N limit and nonlinear quantum gates, and argue that a two-component condensate is a promising platform for realizing a nonlinear qubit.

DOI: 10.22331/q-2026-05-20-2108

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{10}^{4}

particles with ${\ensuremath{\xi}}_{\mathrm{S}}^{2}=\ensuremath{-}2.8(4)$ dB.

A major challenge in quantum metrology is the generation of entangled states with a macroscopic atom number. Here, we demonstrate experimentally that atomic squeezing generated via nonlinear dynamics in Bose-Einstein condensates, combined with suitable trap geometries, allows scaling to large ensemble sizes. We achieve a suppression of fluctuations by 5.3(5) dB for 12,300 particles, from which we infer that similar squeezing can be obtained for more than 10(7) atoms. With this resource, we demonstrate quantum-enhanced magnetometry by swapping the squeezed state to magnetically sensitive hyperfine levels that have negligible nonlinearity. We find a quantum-enhanced single-shot sensitivity of 310(47) pT for static magnetic fields in a probe volume as small as 90 μm3.

Squeezed states, a special kind of entangled states, are known as a useful resource for quantum metrology. In interferometric sensors they allow to overcome the "classical" projection noise limit stemming from the independent nature of the individual photons or atoms within the interferometer. Motivated by the potential impact on metrology as wells as by fundamental questions in the context of entanglement, a lot of theoretical and experimental effort has been made to study squeezed states. The first squeezed states useful for quantum enhanced metrology have been proposed and generated in quantum optics, where the squeezed variables are the coherences of the light field. In this tutorial we focus on spin squeezing in atomic systems. We give an introduction to its concepts and discuss its generation in Bose-Einstein condensates. We discuss in detail the experimental requirements necessary for the generation and direct detection of coherent spin squeezing. Two exemplary experiments demonstrating adiabatically prepared spin squeezing based on motional degrees of freedom and diabatically realized spin squeezing based on internal hyperfine degrees of freedom are discussed.

Spin squeezing generated via inter-atom entanglement in multilevel atomic ensembles provides a powerful resource for quantum-enhanced metrology. Existing schemes that harness internal atomic degrees of freedom to boost squeezing typically encode the collective squeezing in complex superpositions of magnetic sublevels, which complicates state control and limits practical applications. Here, we propose a coherent control scheme that simultaneously enhances collective spin squeezing and maps the resulting atom-atom entanglement onto two well-defined magnetic sublevels suitable for subsequent metrology experiments. Our protocol sandwiches a quantum non-demolition measurement between two internal one-axis-twisting interactions arranged in an echo sequence. We show that this approach can optimally leverage internal states to boost the inter-atom entanglement and, at the same time, encode it in two magnetic sublevels, which is readily convertible into metrologically useful spin squeezing. Our results offer a straightforward and efficient strategy for generating highly entangled yet readily accessible quantum states in multilevel atomic systems.

Squeezed states, special kinds of entangled states, are known as a useful resource for quantum metrology. In interferometric sensors, they allow us to overcome the ‘classical’ projection noise limit stemming from the independent nature of the individual photons or atoms within the interferometer. Motivated by the potential impact on metrology as well as by fundamental questions in the context of entanglement, a lot of theoretical and experimental effort has been made to study squeezed states. The first squeezed states useful for quantum-enhanced metrology have been proposed and generated in quantum optics, where the squeezed variables are the coherences of the light field. In this tutorial, we focus on spin squeezing in atomic systems. We give an introduction to its concepts and discuss its generation in Bose–Einstein condensates. We discuss in detail the experimental requirements necessary for the generation and direct detection of coherent spin squeezing. Two exemplary experiments demonstrating adiabatically prepared spin squeezing based on motional degrees of freedom and diabatically realized spin squeezing based on internal hyperfine degrees of freedom are discussed.

The production of large-scale entangled states is one of the main goals of next-generation quantum technologies, with an immediate potential for applications in the context of entanglement-assisted quantum sensing. A very promising platform to achieve this goal is offered by ultracold spinor gases, made of atoms with a large internal spin sensitive to magnetic fields. Here we show that the native spin-changing collisions in a spinor Bose-Einstein condensate, combined with an arbitrary quadratic Zeeman shift, can generate scalable spin squeezing in the collective spin of the ensemble, following the universal paradigm of the celebrated one-axis-twisting model. Squeezing dynamics is driven by the quadratic Zeeman shift when this shift is small; and by the spin-changing collisions for large shifts, in the form of stroboscopic squeezing. Turning off the Zeeman shift freezes out the collective-spin dynamics, so that the ensuing collective spin dynamics can be uniquely governed by an external field to be sensed. Our theoretical results pave the way for the use of spinor Bose gases with a large spin in fundamental studies of entanglement, as well as in advanced metrological applications.