Heisenberg-limited spin squeezing in coupled spin systems

Long-Gang Huang, Xuanchen Zhang, Yanzhen Wang, Zhenxing Hua, Yuanjiang Tang, and Yong-Chun Liu

Phys. Rev. A 107, 042613 – Published 18 April 2023

Abstract

Spin squeezing plays a crucial role in quantum metrology and quantum information science. Its generation is the prerequisite for further applications but still faces an enormous challenge since the existing physical systems rarely contain the required squeezing interactions. Here we propose a universal scheme to generate spin squeezing in coupled spin models with collective spin-spin interactions, which commonly exist in various systems. Our scheme can transform the coupled spin interactions into squeezing interactions and reach the extreme squeezing with Heisenberg-limited measurement precision scaling as $1 / N$ for $N$ particles. Only constant and continuous driving fields are required, which is a requirement that is accessible to a series of current realistic experiments. This work greatly enriches the variety of systems that can generate the Heisenberg-limited spin squeezing, with broad applications in quantum precision measurement.

Received 4 February 2022
Accepted 13 March 2023

DOI:https://doi.org/10.1103/PhysRevA.107.042613

Physics Subject Headings (PhySH)

Quantum metrology Quantum state engineering Squeezing of quantum noise

Atomic, Molecular & Optical

Authors & Affiliations

Long-Gang Huang^1,2,*, Xuanchen Zhang^1,*, Yanzhen Wang ^1,*, Zhenxing Hua¹, Yuanjiang Tang ¹, and Yong-Chun Liu ^1,3,†

¹State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China
²China Fire and Rescue Institute, Beijing 102202, China
³Frontier Science Center for Quantum Information, Beijing 100084, People's Republic of China

^*These authors contributed equally to this work.
^†ycliu@tsinghua.edu.cn

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 107, Iss. 4 — April 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic diagram of coupled spin systems and spin squeezing via constant drivings. (a) Schematic diagram of the atom-light interaction system with Faraday magneto-optic rotation. (b) Schematic diagram of the spin-exchange interaction between two spins. (c) Schematic diagram of the dipole-dipole interaction between two spins. (d) Evolution of the quantum state for spin $S$ at the time instants denoted by the vertical gray dashed lines in panel (e), represented by the Husimi $Q$ function on the generalized Bloch spheres. (e) The blue, green, and orange dashed curves respectively present the free evolution of the squeezing parameters $ξ^{2}$ for spin $S$ under the Hamiltonians $H_{1}$ , $H_{2}$ , and $H_{3}$ as defined in the main text. The red solid balls denote the results of our scheme with $H_{2}$ under constant drivings, compared with the effective OAT interaction (5) (black solid curve). The vertical gray dashed lines mark time instants of $t = t_{\min} / 4$ , $t_{\min} / 2$ , $3 t_{\min} / 4$ , and $t_{\min}$ , with $t_{\min}$ being the optimal squeezing time of OAT. The parameters are $N_{s} = 50$ , $N_{j} = 50$ , and $Δ / g = 50$ . The initial state is the product of coherent spin states polarized along the $y$ axis for spin $S$ and along the $z$ axis for spin $J$ .
Reuse & Permissions
Figure 2
The effective OAT spin squeezing under Hamiltonian $H_{1}$ with constant DC driving fields. (a) Evolution of the spin squeezing parameter $ξ^{2}$ of our scheme with constant drivings (red solid balls), compared with that of the corresponding effective OAT Hamiltonian (5) (black solid curve). Panels (b) and (c) demonstrate the optimal spin squeezing parameter $ξ_{\min}^{2}$ and the corresponding squeezing time $t_{\min}$ as functions of the particle number $N_{s}$ (red solid balls). The blue solid lines are predicted by Eq. (6). The parameter $Ω = 50 N_{j} g$ , with particle numbers given in each subgraph.
Reuse & Permissions
Figure 3
The effective TAT spin squeezing under Hamiltonian $H_{2}$ with DC and AC driving fields. (a) Evolution of the spin squeezing parameter $ξ^{2}$ for coupled spin systems with drivings (red solid balls) and for the effective TAT Hamiltonian (8) (black solid curve), compared with the OAT spin squeezing (blue dashed curve) governed by Hamiltonian (5). (b) and (c) The optimal spin squeezing $ξ_{\min}^{2}$ and the optimal squeezing time $t_{\min}$ as functions of the particle number $N_{s}$ for the TAT scheme with drivings (red solid balls), compared with the effective TAT results (blue solid curves). The black dashed line in panel (b) corresponds to $ξ_{\min}^{2} = 1.8 / N_{s}$ . The parameter $Δ / g = 100$ , and the particle numbers are given in each subgraph.
Reuse & Permissions
Figure 4
The influence of parameter imperfections. The first row [panels (a1)–(a3)] shows the effective OAT spin squeezing under different deviations of $Ω$ , denoted as $ɛ$ . The second row [panels (b1)–(b3)] shows the results for the deviation of $Ω^{'}$ , denoted as $ɛ^{'}$ . The third [panels (c1)–(c3)] and fourth [panels (d1)–(d3)] rows are the corresponding results of the effective TAT spin squeezing under different $ɛ$ and $ɛ^{'}$ , respectively. The deviations $ɛ$ and $ɛ^{'}$ and relative error $δ$ are defined in the main text. For the first column [panels (a1)–(d1)], the dotted red, solid blue, and dashed green curves correspond to (a1) $ɛ = - 0.5$ , 0, and 0.5; (b1) $ɛ^{'} = - 1$ , 0, and 1; (c1) $ɛ = - 0.3$ , 0, and 0.3; and (d1) $ɛ^{'} = - 2 \times 10^{- 5}$ , 0, and $2 \times 10^{- 5}$ , respectively. The parameters are $Δ / g = 50$ for OAT and $Δ / g = 250$ for TAT spin squeezing, and the particle numbers are $N_{s} = N_{j} = 40$ .
Reuse & Permissions
Figure 5
Spin squeezing under different $Δ$ . The top row [panels (a1)–(a3)] demonstrates the evolution of the spin squeezing parameter $ξ^{2}$ under different $Δ$ , the optimal spin squeezing $ξ_{\min}^{2}$ , and the optimal squeezing time $t_{\min}$ as functions of $Δ$ , for the OAT spin squeezing. The bottom row [panels (b1)–(b3)] shows the results for the TAT spin squeezing. For the first column [panels (a1)–(b1)], the solid red, dashed orange, dotted blue, and dash-dotted green curves correspond to (a1) $Δ / g = 5$ , 10, 15, and 20 and (a2) $Δ / g = 50$ , 100, 150, and 200, respectively. The horizontal dashed lines denote the optimal spin squeezing achieved by a standard OAT squeezing in panels (a1) and (a2) and a standard TAT squeezing in panels (b1) and (b2). The particle numbers are $N_{s} = N_{j} = 40$ .
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information