Phonon-induced two-axis spin squeezing with decoherence reduction in hybrid spin-optomechanical system

TL;DR

This work presents a hybrid spin-optomechanical platform to realize Heisenberg-limited spin squeezing by engineering a Bogoliubov-type spin-spin interaction through dispersive elimination of the optical mode and a squeezing transformation of the phonon mode. By tuning a dissipation-related parameter $\Gamma$, the scheme interpolates between OAT, TAT, and weighted two-axis squeezing, enabling flexible control of squeezing dynamics. In the presence of spin dephasing and phonon dissipation, the authors derive both analytical and numerical bounds showing that the minimum achievable squeezing saturates to a finite limit in the large-$N$ limit, but parameter optimization recovers improved scaling with $N$ (e.g., $\xi_R^2 \propto N^{-0.6}$ for OAT and $\propto N^{-0.73}$ for TAT under mild thermal occupancy), along with shortened preparation times. These insights illuminate the impact of dissipation on spin squeezing and point to optimized hybrid platforms as promising avenues for high-precision quantum metrology in many-body systems.

Abstract

We propose a scheme to implement Heisenberg-limited spin squeezing in a hybrid cavity optomechanical-spin system. In our system, $N$ two-level systems are coupled via Tavis-Cummings interactions to a mechanical resonator (MR) in a standard optomechanical setup. Within the dispersive coupling regime, adiabatic elimination of the optical mode induces a squeezing effect on the MR, which, in the squeezed representation, effectively transforms the collective spin operators into a Bogoliubov form. Under large detuning conditions, the phonon mode mediates interactions among the Bogoliubov collective spins, thereby enabling a two-axis twisting squeezing protocol through appropriate parameter tuning. Both theoretical analysis and numerical simulations show that in the presence of dephasing and phonon dissipation, the maximum squeezing degree asymptotically converges to a constant as $N$ increases, which implies the metrological precision asymptotically approaches the standard quantum limit without parameter optimization. Nevertheless, in parameter optimization we extract a scaling relation of the optimal squeezing which surpasses existing schemes in the literature. Moreover, the optimization also leads to a considerable reduction of the preparation time for the optimal squeezing. Our work may provide insights into dissipation effects in spin squeezing and offer a potential route for high-precision quantum metrology in many-body systems.

Figures (7)

• Figure 1: Scheme of the hybrid optomechanical platform for Heisenberg-limited spin squeezing. (a) A mechanical resonator (MR) with phonon mode $b$ is coupled to an optical cavity mode $a$ with single-photon optomechanical coupling $g_{0}$, and simultaneously interacts with an atomic ensemble $S$ (spins in square) with coupling strength $g$. The cavity is driven by an external laser of frequency $\omega_{p}$ and amplitude $\Omega_{p}$. (b) Phonon dissipation at rate $\gamma$ induces effective dissipative channels on both the optical and spin degrees of freedom.
• Figure 2: The Husimi $Q$ function at different moments of time $t$ under the one-axis twisting (OAT) [(a)-(c)] and two-axis twisting (TAT) [(d)-(f)] schemes. Here $\theta$ and $\phi$ denote the polar and azimuthal angles. Panels (a) and (d) correspond to the initial coherent spin state (CSS) $|\mathrm{\pi/2,0}\rangle$ at $t=0$. Panels (b) at $t=0.105\chi^{-1}$ and (e) at $t=0.051\chi^{-1}$ show the quasiprobability distributions at the maximum-squeezing times. Panels (c) at $t=0.3\chi^{-1}$ and (f) at $t=0.1\chi^{-1}$ depict the oversqueezing.
• Figure 3: Time evolution of the squeezing parameter [$\xi _R ^2\left( \mathrm{dB} \right) =10\log_{10}\xi _R ^2$] under the four schemes in Eq.\ref{['con:8']} in the absence of dissipation, with $N=100$. The system parameters are chosen as $g/(2\pi) = 1\,\mathrm{kHz}$, $\omega_b/(2\pi) = 1\,\mathrm{GHz}$, and $\tilde{\Omega} = 0$. The blue (red) solid line represents the evolution under the OAT (TAT) scheme, while the purple dashed and dotted-dashed lines correspond to schemes with $\Gamma = -0.05\omega_b$ and $\Gamma = -2\omega_b$, respectively.
• Figure 4: Time evolution under OAT (dotted-dashed line) and TAT (solid line) schemes in the presence of dissipation. The dark blue, blue, and light blue lines indicate the evolutions at different thermal phonon occupations $\bar{n}_{th}= 1$, 10, and 100 for two schemes. Here we set $N=100$, $\omega _r/(2\pi)=200$kHz, $Q_m=10^6$, $g/2\pi = 1\,\mathrm{kHz}$ and a spin dephasing time $T_2=0.01\,\mathrm{s}$.
• Figure 5: Asymptotic behavior of the squeezing parameter in dissipation. (a) Comparison between numerical fitting and analytical results. The red solid line corresponds to the analytical expression given by Eq.\ref{['con:16']}, while the black dashed line represents the fitted curve based on Eq.\ref{['con:19']}. The blue dashed line indicates the asymptotic squeezing bound obtained from both approaches. Parameters are $\omega_r/(2\pi) = 53$ kHz and $\bar{n}_{th} = 20$. (b) Dependence of the squeezing bound on $\bar{n}_{th}$. From bottom to top, the curves correspond to the unitary case and $\bar{n}_{th}= 10, 30, 50$, respectively, with fitting according to Eq.\ref{['con:19']}. Here, $\omega_r /(2\pi)= 80$kHz. The remaining parameters in each panel are $g/2\pi =1\mathrm{kHz}, Q_m=10^6, T_2=0.01\mathrm{s}$. (c) Lower bound $\xi ^2_{lb}$ versus number of $\bar{n}_{th}$. The black solid line represents the result from Eq.\ref{['con:18']}, while the red crosses correspond to the fitting based on Eq.\ref{['con:19']}. The parameters used are the same as those in panel (b).