Asymmetric polaron picture for the quantum Rabi model

TL;DR

The paper introduces an asymmetric polaron picture (APP) for the quantum Rabi model to capture polaron–antipolaron asymmetries that are neglected by the conventional symmetric polaron approach. By constructing a variational wave function with explicit asymmetry parameters, the authors achieve higher accuracy in ground- and first-excited-state properties across coupling regimes and reveal new transitions such as asymmetry-imbalance reversals and polaron attraction/repulsion dynamics. The APP clarifies how displacement, frequency renormalization, and asymmetry jointly shape the energy landscape, leading to a richer phase diagram and sharper understanding of the finite-component quantum phase transition. They further demonstrate that APP improves quantum Fisher information predictions and critical coupling extraction, underscoring the relevance of polaron asymmetry as a resource in quantum metrology and its role in light-matter critical phenomena.

Abstract

The experimental access to ultra-strong couplings in light-matter interactions has made the quantum phase transition (QPT) in the quantum Rabi model practically relevant, while the physics of the QPT has not yet been fully explored. The polaron picture is a method capable of analyzing in the entire coupling regime and extracting the essential physics behind the QPT. However, the asymmetric deformation of polarons is missing in the current polaron picture. In the present work we propose an improved variational method in asymmetric polaron picture (APP). Our APP not only increases the method accuracy but also reveals more underlying physics concerning the QPT. We find that in the ground state both the polarons and antipolarons are asymmetrically deformed to a large extent, which leads to a richer phase diagram. We also analyze the first excited state in which we unveil an asymmetry direction reversal for the polarons and an attraction/replusion transition differently from the ground state. Finally, we apply the APP in quantum Fisher information analysis and critical coupling extraction, the improvements indicate that the polaron asymmetry makes a considerable contribution to the quantum resource in quantum metrology and plays an unnegligible role in the QPT. Our results and mechanism clarifications expose more subtle energy competitions and abundant physics, and the method potentially might have broader applications in light-matter interactions.

Figures (11)

• Figure 1: Schematic diagram for asymmetric polaron picture. (a) Polarons labeled by $\alpha$ (antipolaron labeled by $\beta$) with displacement $-\zeta_\alpha g ^\prime$ ($+\zeta_\beta g ^\prime$) renormalized from the potential displacement $-g ^\prime$ of $v_{+}$ (parabolic line). Symmetric (Asymmetric) polaron and antipolaron in dashed (solid) lines have an asymmetric factor $\delta_i=0$ ($\delta_i\neq 0$) where $i=\alpha,\beta$. (b) Enlarged overlap for asymmetric wave packets (solid) relative to symmetric ones (dashed). The overline over $\varphi$ denotes spin-down (- or $\downarrow$) component. (c) Four channels of tunneling between spin-up (+ or $\uparrow$) and spin-down components of polaron and antipolarons accommodated in effective potentials $v_{\pm}+\delta v_{\pm}$. (d) Different overlaps among farther asymmetric polarons and closer asymmetric antipolarons due to potential difference in (a).
• Figure 2: Accuracy improvements in the asymmetric polaron picture. The discrepancy $\Delta q= q_{var}-q_{\rm ED}$ of variational physical quantities $q_{var}$ from the results of exact diagonalization (ED) for $q$ being (a) the total energy $E$, (b) the photon number $\langle a^\dag a\rangle$, (c) the spin expectation $\langle \sigma_x \rangle$, (d) the coupling correlation $\langle \sigma_z (a^\dag +a)\rangle$. The circles (triangles) represent the symmetric (asymmetric) case with $\delta _i =0$ ($\delta _i \neq 0$), where $i=\alpha,\beta$, illustrated for the first excited state at $\omega=0.15\Omega$.
• Figure 3: Variational parameters and transitions in the ground state. Evolution of the variational parameters for (a) asymmetry factors $\delta_\alpha$ (dot-dashed), $\delta_\beta$ (dashed), $\delta_\alpha+\delta_\beta$ (solid), (b) weights of polaron ($\alpha$, solid) and antipolaron ($\beta$, dashed), (c) renormalization factors of polaron ($\xi_\alpha$, solid) and antipolaron ($\xi_\beta$, dashed), (d) displacement renormalization factors of polaron ($\zeta_\alpha$, solid) and antipolaron ($\zeta_\beta$, dashed). The dot-dashed line is the asymmetric polaron displacement $\zeta _{\alpha}+\zeta _{\alpha}^{\delta }$ from Eq. \ref{['Zata-Asymm']}. Here $\omega=0.15\Omega$.
• Figure 4: Ground-state phase diagram in the $\omega$-$g$ plane. The dark-yellow (light gray) solid line denotes the transition around $g_c$\ref{['Eq-gC']} as reflected by the maximum of $\xi_\beta$. The brown (dark gray) solid line around $g=1.2g_c$ (the dot-dashed line) labeled by Min${\delta _\beta}$ (Max${\delta _ \alpha}$) is the maximum asymmetry point of the antipolaron (polaron). The dotted boundary marks the reversal of asymmetry imbalance, with $\delta_\alpha +\delta_\beta >0$ inside the boundary and $\delta_\alpha +\delta_\beta >0$ outside. The dashed line separates the $\zeta_\alpha>0$ region above the line and the $\zeta_\alpha<0$ region below the line. The long-dashed slash is the boundary $\alpha =\beta$ for weight reversal with $\alpha >\beta$ above the boundary and $\alpha <\beta$ below.
• Figure 5: Variational parameters and transitions in the first excited state. Evolution of the variational parameters for (a) asymmetry factors $\delta_\alpha$ (solid) and $\delta_\beta$ (dashed), (b) weights of polaron ($\alpha$, solid) and antipolaron ($\beta$, dashed), (c) renormalization factors of polaron ($\xi_\alpha$, solid) and antipolaron ($\xi_\beta$, dashed), (d) displacement renormalization factors of polaron ($\zeta_\alpha$, solid) and antipolaron ($\zeta_\beta$, dashed). Here $\omega=0.5\Omega$.