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arXiv:2503.19198 · Advanced Quantum Technologies

Critical Quantum Metrology in a Stabilized Two‐Photon Rabi Model

Zu‐Jian Ying, Hang‐Hang Han, Bo‐Jian Li, Simone Felicetti, Daniel Braak

Published Sep 2, 2025• 10 citations
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Abstract

Abstract Applications of quantum systems with non‐linear light–matter interaction have to deal with a peculiar instability in the ultra‐ and deep strong coupling regime, the so‐called “spectral collapse.” To solve this problem, the present work investigates a generalized quantum Rabi model (QRM) with two‐ and four‐photon terms in view of applications for critical quantum metrology. In the introduced model, the spectral collapse occurring in the standard two‐photon QRM is stabilized by the presence of the quartic potential. The collapse is then transformed into a quantum phase transition, which occurs in the low‐frequency limit of the light mode, whose remnant at finite ratio between qubit and mode frequencies can be applied to critically enhanced quantum metrology. It is found that the four‐photon term entails a much higher measurement precision compared to the standard two‐photon QRM. The mechanism behind the higher precision can be traced to the different behavior of the ground state wave function as the system is tuned through the transition. As the standard two‐photon QRM, despite the absence of the spectral collapse, the proposed model allows for a finite preparation time for the probe state (PTPS).

DOI
10.1002/qute.202500263

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Abstract Applications of quantum systems with non‐linear light–matter interaction have to deal with a peculiar instability in the ultra‐ and deep strong coupling regime, the so‐called “spectral collapse.” To solve this problem, the present work investigates a generalized quantum Rabi model (QRM) with two‐ and four‐photon terms in view of applications for critical quantum metrology. In the introduced model, the spectral collapse occurring in the standard two‐photon QRM is stabilized by the presence of the quartic potential. The collapse is then transformed into a quantum phase transition, which occurs in the low‐frequency limit of the light mode, whose remnant at finite ratio between qubit and mode frequencies can be applied to critically enhanced quantum metrology. It is found that the four‐photon term entails a much higher measurement precision compared to the standard two‐photon QRM. The mechanism behind the higher precision can be traced to the different behavior of the ground state wave function as the system is tuned through the transition. As the standard two‐photon QRM, despite the absence of the spectral collapse, the proposed model allows for a finite preparation time for the probe state (PTPS).
The quantum Rabi model (QRM) with linear coupling between light mode and qubit exhibits the analog of a second-order phase transition for vanishing mode frequency which allows for criticality-enhanced quantum metrology in a few-body system. We show that the QRM including a nonlinear coupling term exhibits much higher measurement precisions due to its first-order-like phase transition at finite frequency, avoiding the detrimental slowing-down effect close to the critical point of the linear QRM. When a bias term is added to the Hamiltonian, the system can be used as a fluxmeter or magnetometer if implemented in circuit QED platforms.
Quantum Rabi model (QRM) is a fundamental model for light-matter interactions, the finite-component quantum phase transition (QPT) in the QRM has established a paradigmatic application for critical quantum metrology (CQM). However, such a paradigmatic application is restricted to a local regime of the QPT which has only a single critical point. In this work we propose a globalized CQM in the QRM by introducing an auxiliary nonlinear term which is realizable and can extend the critical point to a continuous critical regime. As a consequence, a high measurement precision is globally available over the entire coupling regime from the original critical point of the QRM down to the weak-coupling limit, as demonstrated by the globally accessible diverging quantum Fisher information in dynamics. We illustrate a measurement scheme by quadrature dynamics, with globally criticality-enhanced inverted variance as well as the scaling relation with respect to finite frequencies. In particular, we find that the globally high measurement precisions still survive in the presence of decoherence. Our proposal paves a way to break the local limitation of QPT of the QRM in CQM and enables a broader application, with implications of applicability in realistic situation.
Spectral collapse in the two-photon quantum Rabi model (tpQRM) has long been regarded as insufficient to establish quantum criticality because, under generic conditions, the lowest excitation gap remains finite. We show that, at a special qubit frequency, spectral collapse in the anisotropic tpQRM constitutes a continuous quantum phase transition governed by a single soft mode. The same-parity gap, set by the vanishing effective oscillator frequency, closes with exponent zν=1/2zν= 1/2zν=1/2, whereas the lower different-parity gap arises from symmetry-induced splitting. Parity symmetry excludes the different-parity excitation from the quantum Fisher information response and restricts Kibble-Zurek dynamics to excitations within the ground-state parity sector. Together, these results identify the same-parity excitation as the soft mode and its gap as the characteristic energy scale. The corresponding critical exponents, ν=1/4ν=1/4ν=1/4 and z=2z=2z=2, place the anisotropic tpQRM in the same universality class as the standard QRM. Our results establish spectral collapse as a potential route to experimentally accessible quantum criticality in a few-body system and show that universality is determined by the soft-mode structure rather than by the closing of the lowest excitation gap.
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 the preparation time of the 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 parameters 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.
Spectral collapse, the transition from a discrete to a continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model using optical phase space, and we find that, in the so-called degenerate qubit regime, the collapse is similar to the transition from a harmonic to an inverted oscillator with the free-particle potential as a critical transition point. In the degenerate qubit regime, we construct Dirac-normalizable eigenfunctions with well-defined parity for the model. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs. We numerically confirm the existence of an exceptional state at the critical coupling, and we argue its analytic provenance from both a Born-Oppenheimer and a variational approximation.
Abstract Squeezing and transition criticality are two main sensitivity resources for quantum metrology (QM), combination of them may yield an upgraded metrology protocol for higher upper bound of measurement precision (MP). The present work shows that such a combination is feasible in light–matter interactions by a realizable asymmetric non‐linear quantum Rabi model (QRM). Indeed, the non‐linear coupling possesses a squeezing resource for diverging MP while the non‐monotonous degeneracy lifting by the asymmetries induces an additional tunable transition which further enhances the MP by several orders, as demonstrated by the quantum Fisher information (QFI). The resource combination not only leads to cooperative boosts to the divergence of QFI but also brings divergence compensation with the merit of globally high MP. Moreover, the protocol is immune from the problem of diverging preparation time of probe state that may hinder the conventional linear QRM in application of QM. This work establishes a paradigmatic case of combining different sensitivity resources to manipulate QM and maximize MP.