Distinct Critical Scaling of Quantum Fisher Information in a Quantum Rabi Triangle System

TL;DR

The paper addresses critical quantum metrology in a three-mode quantum Rabi triangle, where multiple phases (NP, FSP, CSP) arise under an artificial magnetic field. It develops a circuit-QED realization and computes the quantum Fisher information (QFI) around phase boundaries, showing divergent scaling due to energy-gap closing. The authors find distinct critical exponents for QFI at various transitions and demonstrate Heisenberg-limit scaling, $I \sim \langle N \rangle^2 T^2$, when the hopping phase or scaled coupling is tuned to criticality with $\theta \neq 0$. They also propose a photon-number measurement that saturates the quantum Cramér-Rao bound in the normal phase, illustrating the metrological potential of criticality in the QRT platform.

Abstract

Critical properties of a quantum system are recognized as valuable resources for quantum metrology. In this work, we investigate the criticality-enhanced sensing in a quantum Rabi triangle system, which exhibits multiple phases. Around the phase boundary, enhanced parameter estimation precision can be achieved by tuning either the scaled coupling strength or the hopping phase controlled by an artificial magnetic field. We observe that the quantum Fisher information shows divergent scaling near different quantum phase transition points, characterized by distinct critical exponents. When the resource consumption is taken into account, we find that the divergent quantum Fisher information can reach the Heisenberg limit. Furthermore, we propose a measurement scheme of the average photon number and the quantum Cramér-Rao bound can be saturated.

Figures (9)

• Figure 1: Circuit diagram of a three-mode superconducting loop: each anharmonic LC oscillator (dashed boxes) consists of a Josephson junction ($L_{q1},\ L_{q2},\ L_{q3}$) and a capacitance ($C_{q1},\ C_{q2},\ C_{q3}$), which is coupled to the electromagnetic cavity mode of LC resonator ($L_{1},\ L_{2},\ L_{3},\ C_{1},\ C_{2},\ C_{3}$). The capacitance $C_J$ is added between cavities to modulate the hopping strength.
• Figure 2: (a)The phase diagram of the quantum Rabi triangle model. The red solid line stands for the second-order phase transition boundary $g_{1c}(q,\theta)$ between the normal phase (NP) and the superradiant phases. The blue dashed line $\pm\theta_c$ stands for the first-order phase transition boundary between the ferromagnetic superradiant phase (FSP) and the chiral superradiant phase (CSP). The two lines cross at the triple point (TP). The black dashed line stands for the special frustrated antiferromagnetic superradiant phase (FASP). (b)The excitation spectrum $\epsilon_q$ as a function of $g_1$ with $\theta=-\theta_c$ for $q={0,\pm2\pi/3}$.
• Figure 3: The QFI as a function of $g_1$ and $\theta$ normalized by $\pi$. (a) $I_{\mathrm{NP}}(g_1)$ in the NP. The black dot $(g_{1c}(-2\pi/3,\theta_0),\theta_0)$ is away from the phase boundary, and the red star $(g_{1c}(0,\theta^{\prime}),\theta^{\prime})$ is near the phase boundary. (b) $I_{\mathrm{FSP}}(g_1)$ in the FSP. (c) $I_{\mathrm{CSP}}(g_1)$ in the CSP. The inset presents a zoomed-in view of $I_{\mathrm{CSP}}(g_1)$ for $g_1$ in range of 0.49 to 0.51.
• Figure 4: The scaling of the QFI $I(g_1)$ as a function of $g_1$ normalized by $g_{1c}$ with different $\theta$. (a) Scaling behavior of $I(g_1)$ near the critical point $(g_{1c}(0,-2\pi/3),-2\pi/3)$ for the NP–FSP transition. (b) Scaling behavior of $I(g_1)$ near the triple point $(g_{1c}(0,-\theta_c),-\theta_c)$ for the NP-FSP transition. (c) Scaling behavior of $I(g_1)$ near the triple point $(g_{1c}(0,-\theta_c),-\theta_c)$ for the NP-CSP transition. The inset presents a zoomed-in view of $I_{\mathrm{NP}}(g_1)$. (d) Scaling behavior of $I(g_1)$ near the critical point $(g_{1c}(-2\pi/3,-\pi/3),-\pi/3)$ for the NP-CSP transition. The inset presents a zoomed-in view of $I_{\mathrm{NP}}(g_1)$. (e) Scaling behavior of $I(g_1)$ near the critical point $(g_{1c}(-2\pi/3,0),0)$ for the NP-FASP transition. The inset presents a zoomed-in view of $I_{\mathrm{NP}}(g_1)$.
• Figure 5: The scaling of the QFI $I(g_1)$ as a function of $\theta$ normalized by $\pi$ (or $\theta_c$) with different $g_1$. (a) The divergent feature of $I(g_1)$ around the triple points $(g_{1c}(0,-\theta_c),\pm\theta_c)$. (b) A zoomed-in view of $I(g_1)$ around the triple point $(g_{1c}(0,-\theta_c),-\theta_c)$, illustrating the scaling behavior of $I(g_1)$ for the FSP–CSP transition. (c) The peak feature of $I(g_1)$ around the phase transition points $(0.6,\pm\theta_c)$ and $(0.6,0)$ with $g_1$ far from the phase boundary $g_{1c}$.