Globalized Nonlinear Critical Quantum Metrology by Two-photon Rabi-Stark model

TL;DR

The paper introduces the two-photon Rabi-Stark model to overcome the local nature of criticality in nonlinear quantum metrology. By adding a Stark coupling, the critical point becomes tunable, enabling a global divergence of the quantum Fisher information (QFI) over a wide range of couplings, with a universal critical exponent $\gamma=2$. It identifies squeezing as the dominant resource driving the QFI near criticality and shows that a globally strong squeezing persists across parameters, while the probe-state preparation time remains finite. The approach is supported by exact diagonalization and polaron theory, with Wigner-function analysis confirming robust squeezing, and the framework is positioned as experimentally feasible in superconducting circuits and trapped ions.

Abstract

Squeezing as a quantum resource for quantum metrology is robust against decoherence and dissipation, while the conventional nonlinear two-photon quantum Rabi model (QRM) provides a squeezing resource immune to the divergence problem of preparation time of probe state (PTPS). However the critical point of the two-photon QRM is locally restricted to one single point, which hinders a wider application. In the present work we propose to combine the Stark coupling with the two-photon QRM to realize a tunable critical point so that the nonlinear critical quantum metrology can be globalized. As demonstrated by the diverging quantum Fisher information (QFI) the protocol enables us to acquire a high measurement precision in a wide range of coupling parameter rather than locally at a single critical point. Moreover, We find that the QFI not only manifests criticality but also exhibits universality. As a particular merit of our protocol, a strong squeezing can be globally retained as the leading quantum resource, while at the same time the PTPS remains in a finite order.

Figures (8)

• Figure 1: Tunable critical point by Stark coupling $\chi$. (a) Effective potential $v_{\pm}$ at $\chi =0$. (b) Wider potential $\tilde{v}_{-}$ and narrower $\tilde{v}_{+}$ at $\chi \neq 0$. (c) Potential frequency at $\chi =0$ ($\varpi _{\pm}$, broken lines) and $\chi \neq 0$ ($\tilde{\varpi} _{\pm}$, solid lines). (d) Phase diagram of $\tilde{m} _{-}$ and $\tilde{\varpi} _{-}$ in the $\chi$-$g_2$ plane. The dotted and dashed lines in (a,b) represent the effective spin-dependent single-particle energy. In (a-c) $\chi=0.8$ is illustrated for $\chi \neq 0$ case.
• Figure 2: Criticality both at zero and finite $\chi$. (a) Spin expectation $\langle\hat{\sigma}_{x,z}\rangle$. (b) Rotated spin expectation $\langle\hat{\tilde{\sigma}}_{x,z}\rangle$. (c) $\langle\hat{x}^2\rangle$. (d) $\langle\hat{p}^2\rangle$. Here broken (solid) lines denote the $\chi =0$ ($\chi \neq 0$) case in all panels. Here the results are obtained by exact diagonalization (ED).
• Figure 3: Diverging quantum Fisher information (QFI) $F_Q$ by ED. (a) $F_q$ (in natural logarithm) at different values of $\chi$ at fixed $\Omega=1.0\omega$. (b) 3D plot for $F_Q$ vs $g_2$ and $\chi$ at fixed $\Omega=\omega$. (c) Equi-$\tilde{\Omega} _c$ lines in the $\chi$-$\Omega$ plane. (d) $F_Q$ at different values of $\chi$ with fixed $\tilde{\Omega} _c=0.2\omega$. We set $\omega=1$ as the unit throughout all figures.
• Figure 4: Evolution of wave function with respect to $g_2$. (a,b) $\psi _{\pm}(x)$ at $\chi=0$. (c,d) $\psi _{\pm}(x)$ on unrotated spin basis at $\chi=0.8$. (e,f) $\tilde{\psi} _{\pm}(x)$ on rotated spin basis at $\chi=0.8$. $\tilde{\Omega}_c=0.1\omega$ in all panels ($\Omega=1.0\omega$ for $\chi=0.0$ and $\Omega=0.97\omega$ for $\chi=0.8$).
• Figure 5: Analysis of sensitivity resources: (a) $F_Q$ by polaron picture (PP, green solid) in agreement with exact diagonalization (ED, blue dots). (b) Contributions of squeezing ($F^{\xi}$, blue solid), basis weight ($F_Q^{\rho}$, orange solid), spin basis rotation ($F_Q^{\sigma}$, yellow solid) and mixed term ($F_Q^{\rm mixed}$, gray dotted) to the total $F_Q$ (green dashed). $F_Q^{\rm mixed}$ is negative before the sharp dip and $ln |F_Q^{\rm mixed}|$ is plotted. (c) Evolutions of effective mass $\tilde{m} _{\pm}$ and basis frequency $\tilde{\xi} _{\pm}$. (d) Evolutions of spin basis weight $\tilde{C} _{\pm}$ and spin basis variation products $\langle\tilde{\Uparrow}'|\tilde{\Uparrow}'\rangle$ ($= \langle\tilde{\Downarrow}'|\tilde{\Downarrow}'\rangle$) and $\langle\tilde{\Uparrow}|\tilde{\Downarrow}'\rangle$ ($= -\langle\tilde{\Downarrow}|\tilde{\Uparrow}'\rangle$). In all panels $\chi=0.8$ and $\tilde{\Omega} _c=0.2$ ($\Omega =1.13\omega$).