Stochastic Mean-field Theory for Conditional Spin Squeezing by Homodyne Probing of Atom-Cavity Photon Dressed States

TL;DR

The paper develops a stochastic variant of cumulant mean-field theory to tackle conditional spin squeezing under homodyne probing in large, multi-level atom-cavity systems. It closes the hierarchy using a second-order cumulant expansion and identical-atom symmetry, enabling efficient simulations with thousands of atoms and strong light-mavity coupling. Benchmarking against exact stochastic density-matrix methods confirms accuracy for standard two-level scenarios, and application to a realistic $N=10^4$-atom setup demonstrates detailed spin-squeezing dynamics, including AC Stark rotations and dissipation-driven scaling between SQL and Heisenberg limits. The framework supports realistic experimental protocols and offers pathways to more exotic quantum-measurement effects and feedback-enabled squeezing in large quantum ensembles.

Abstract

Projective measurements of collective observables can be employed to herald the preparation of entangled states of quantum systems, and the resulting conditional dynamics is usually handled by stochastic master equation (SME) for small systems, and by an approximate Gaussian-state formalism for large systems. In this work, we present an alternative technique by developing a stochastic variant of cumulant mean-field theory, benchmark it against an exact stochastic collective density matrix approach by the simulations of hundreds of identical two-level atoms. More importantly, we demonstrate its full power by studying the conditional spin squeezing of thousands of three-level atoms coupled strongly with an optical cavity subject to individual decay and dephasing, and by simulating the experimental protocol to reveal formation and detection of the spin squeezed state. The proposed technique might be further extended to study more exotic quantum-measurement effects of large quantum systems, such as deterministic spin squeezing with quantum feedback, spin squeezing of optical clock transitions, and retrodictive spin squeezing by posterior measurements, and so on.

Figures (12)

• Figure 1: Benchmark of the results based on the proposed stochastic mean-field approach (solid line) and the exact stochastic collective density matrix approach (dots). (a) shows the spin squeezing parameter $\xi_z^2$ for a system with $N=100$ atoms evolving from a coherent spin state (CSS, left inset) to the spin squeezed state (SSS, right inset), which can be well fitted with a function BLMadsen$\xi_z^2(t)=1/(1+kt)$ with a spin squeezing rate $k$. (b) shows $k$ as function of the number of atoms $N$.
• Figure 2: System and energy diagram. (a) shows $N$ rubidium-87 atoms inside an optical cavity, and the balanced homodyne detection of the field formed by mixing the probe field at frequency $\omega_{p}$ with the field transmitted through the cavity. (b) shows the simplified energy diagram of the atoms with two hyper-fine ground states, represented as the up $|2\rangle$ and down $|1\rangle$ state of a spin-1/2 particle, and an electronic excited state $|3\rangle$, coupling strongly with the optical cavity of frequency $\omega_c$ , which leads to two atoms-photon dressed states of transition frequencies $\omega_{\pm}$ (horizontal dashed lines). (c,d) show the intra-cavity field amplitude $\langle \hat{a} \rangle$ as function of $\omega_p$ relative to the atomic transition frequency $\omega_{32}$ (c) for the atomic ensemble prepared in an equal superposition of the two hyper-fine ground states, and the imbalance of the two ground state populations $J_z/N$ for $\omega_p =\omega_{32} + \sqrt{N/2}g$ (d). For more details see text.
• Figure 3: Conditional spin squeezing dynamics. (a) shows the evolution of uncertainty of the collective spin components $\Delta J_x$,$\Delta J_y$,$\Delta J_z$ for an ideal system without the individual atomic dissipation, where $\Delta J_z$ is amplified tenfold for visual clarity, and all quantities are normalized to the atomic number $N$. (b-e) show the spin squeezing parameter $\xi_z^2$ as function of time for different probe amplitudes $\Omega_p$, atomic decay rates $\gamma$, atomic dephasing rates $\chi$ and numbers of atoms $N$, respectively. (f) shows the $\xi_z^2=1$ (the standard quantum limit, SQL) for the systems in the CSS (black line), and the reduction of the minimal $\xi_z^2$ with increasing $N$ for the systems in the absence (red dots and line) and presence (blue dots and line) of the individual atomic dissipations, which can be fitted with $\xi_z^2 \sim 1/N$ (i.e. Heisenberg limit, HL) and $\xi_z^2 \sim 1/N^{0.6}$ (between the SQL and the HL), respectively.
• Figure 4: Simulation of experimental protocol. (a) shows the spin squeezing parameter $\xi_z^2$ following the protocol shown on the top. (b) shows the dynamic of the photo-current in the homodyne detection, where the current during the four laser pulses are integrated to form the quantities $n_i$ ($i=1,2,3,4$), and the inset shows $J_{z,2}=n_4 -n_3$ versus $J_{z,1}=n_1 -n_2$ during the preparation and verification of the conditional spin squeezing for one hundred simulations. For more details see the text.
• Figure A5: Julia codes to construct (a) and solve (b) the stochastic mean-field equations (\ref{['A1']}-\ref{['A7']}) derived from the standard SME, and the calculation of the collective spin components and spin squeezing parameter $\xi_z^2$.