Enhancing collective spin squeezing via one-axis twisting echo control of individual atoms

Abstract

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.

Figures (2)

• Figure 1: Dependence of the enhancement factor on the coupling strength for (a) integer $f$ and (b) half-integer $f$. (c) Schematic of the internal-state twisting echo protocol for enhancing inter-atom entanglement, depicted on a Bloch sphere. Initial CSS (i) evolves under $\hat{U}_{\rm{OAT}}$ into a GHZ state $\ket{\rm{GHZ_+}}$ (ii). The QND interaction drives a transition from $\ket{\rm{GHZ_+}}$ to $\ket{\rm{GHZ_-}}$ (iii), while both states exhibit the same probability distribution on the Bloch sphere. Finally, the reversed OAT evolution $\hat{U}_{\rm{OAT}}^\dag$ transforms $\ket{\rm{GHZ_-}}$ ($\ket{\rm{GHZ_+}}$) back into the magnetic sublevel $\ket{-f}$ ($\ket{f}$) (iv). (d) Schematic of collective squeezing. The first OAT evolution transforms the collective state from the reference state $\ket{f}^{\otimes N}$ to $\ket{\rm{GHZ}_+}^{\otimes N}$. The subsequent QND measurement then coherently transfers an even number of atoms to $\ket{\rm{GHZ}_-}$, generating inter-atom entanglement encoded in the two orthogonal $\rm{GHZ}_\pm$ states. This entanglement is finally mapped onto the magnetic sublevels $\ket{-f}$ and $\ket{f}$ via inverse OAT evolution.
• Figure 2: Spin squeezing versus internal OAT coupling strength for (a) weak and (b) strong QND measurement. Dashed curves: the cooperative squeezing scheme; solid curves: the present scheme; dot-dashed lines: the spin squeezing achieved for the internal state in CSS. (c) Schematics for implementing internal OAT and collective QND interactions (middle). The linearly polarized probe pulse propagating along the $x$-axis drives a $\Delta m_f=2$ transition (left), realizing the internal $\hat{U}_{\rm{OAT}}$ evolution (blue detuning) and the $\hat{U}_{\rm{OAT}}^{\dag}$ evolution (red detuning). The pulse propagating along the $y$-axis is used to drive the $\Delta m_f=1$ transition (right) and responsible for the collective QND measurement. (d) Pulse sequence. Atoms are first prepared in the CSS by optical pumping, followed by the first $x$-probe, which spreads the quantum uncertainty along the $y$-direction. The subsequent $y$-probe slightly squeezes $\hat{F}_y$. The second $x$-probe untwists the spin to restore the variance of $\hat{F}_y$ to the CSS level. Finally, the rf-fields convert the atomic oscillator squeezing into spin squeezing, thereby reducing the uncertainty in $\hat{F}_y$.