Assisted metrology and preparation of macroscopic superpositions with split spin-squeezed states

Jiajie Guo, Fengxiao Sun, Qiongyi He, and Matteo Fadel

Phys. Rev. A 108, 053327 – Published 30 November 2023

Abstract

We analyze the conditional states in which one part of a split spin-squeezed state is left, upon performing a collective spin measurement on the other part. For appropriate measurement directions and outcomes, we see the possibility of obtaining states with high quantum Fisher information, even reaching the Heisenberg limit. This allows us to propose a metrological protocol that can outperform standard approaches, for example, in a situation where the number of particles in the probe is bounded. The robustness of this protocol is investigated by considering realistic forms of noise present in cold-atom experiments, such as particle number fluctuations and imperfect detection. Ultimately, we show how this measurement-based state-preparation approach can allow for the conditional (i.e., heralded) preparation of spin Schrödinger's cat states even when the initial state before splitting is only mildly squeezed.

Received 5 September 2023
Accepted 30 October 2023

DOI:https://doi.org/10.1103/PhysRevA.108.053327

Physics Subject Headings (PhySH)

Entanglement in quantum gases Quantum metrology

Bose-Einstein condensates

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Jiajie Guo ¹, Fengxiao Sun ¹, Qiongyi He ^1,2,3,*, and Matteo Fadel ^4,†

¹State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
²Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
³Peking University Yangtze Delta Institute of Optoelectronics, Nantong 226010, Jiangsu, China
⁴Department of Physics, ETH Zürich, 8093 Zürich, Switzerland

^*qiongyihe@pku.edu.cn
^†fadelm@phys.eth.ch

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Vol. 108, Iss. 5 — November 2023

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Images

Figure 1
State preparation and metrology with split atomic ensembles. (a) To prepare nonclassical states of many-body systems, traditional approaches rely on implementing a nonlinear dynamic in a trapped ensemble. (b) Instead, our assisted protocol is based on first spatially splitting a mildly entangled ensemble to then measure one of the two parts. Because of the shared correlations, this results in projecting the other half into a multipartite state that can have strong quantum correlations.
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Figure 2
Measurements on the $xy$ plane. Properties of the conditional states obtained from a split spin-squeezed state with $N = 100, N_{A} = N_{B} = N / 2$ . (a) For $l_{A} = N_{A} / 2$ , $F_{Q} / N_{B}$ as a function of the measurement direction $θ_{A}$ and selected Wigner functions showing squeezed and cat-like states. Fixing $θ_{A}$ such that the measurement direction is either $y^{'}$ or $z^{'}$ , we show the probability of measuring $l_{A}$ , panels (b), (c) respectively, and the $F_{Q} / N_{B}$ of the associated states, panels (d), (e). For $l_{A} = N_{A} / 2$ , we show in (f) a comparison between $F_{Q} / N_{B}$ of conditional states and of OAT states, as a function of the squeezing $μ$ . Here, the comparison is for a fixed number of particles $N_{B}$ in the probe state.
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Figure 3
Measurements along the $x$ direction. Properties of the conditional states obtained from a split spin-squeezed state with $N = 100, N_{A} = N_{B} = N / 2$ . (a) Probability of measuring $l_{A}$ for different levels of squeezing and (b) the $F_{Q} / N_{B}$ of the associated conditional states. (c) Selected Wigner functions showing cat-like states of different size. (d) Comparison between $F_{Q} / N_{B}$ of conditional states and of OAT states, as a function of the squeezing $μ$ . Here, the comparison is for a fixed number of particles $N_{B}$ in the probe state.
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Figure 4
Robustness to detection noise. Sensitivity of the conditional states obtained from a split spin-squeezed state with $N = 100, N_{A} = N_{B} = N / 2$ . Panels (a), (b) consider conditional states with $l_{A}^{*} = N_{A} / 2$ resulting from measurements along $y^{'}$ and $z^{'}$ , while panels (c), (d) along $x$ . Panels (a), (c) show $F_{Q} / N_{B}$ as a function of the detection noise $σ$ , while (b), (d) as a function of the squeezing $μ$ . Dashed lines are for a level of noise $σ_{1} = 0.49$ , while dotted lines for $σ_{2} = 1.37$ , which correspond to a $10 %$ probability of having $l_{A} = l_{A}^{*} \pm 1$ or $l_{A} = l_{A}^{*} \pm 2$ , respectively.
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Figure 5
Heralded generation of cat states. Nonclassical features of the conditional states obtained from a split spin-squeezed state with $N = 100, N_{A} = N_{B} = N / 2$ , after measuring $S_{x}^{A}$ . We show, as a function of the level of detection noise $σ$ . (a) Wigner functions of conditional states associated to different $l_{A}^{*}$ and (b) Wigner function negativity as defined in Eq. (9).
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