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arXiv:2609.36861

Exact Trotterization in the Degenerate Quantum Rabi Model

I. Lizuain, A. Rodriguez-Prieto

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
v1 · Sep 29, 2026v2 · Oct 5, 2026
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Abstract

Digital quantum simulation of the quantum Rabi model in the deep strong coupling regime motivates the need for an exact characterization of Trotterization errors, so far known only through qualitative bounds. Here we derive such a characterization in the exactly solvable degenerate limit, valid for arbitrary coupling strength. The first-order Trotter operator admits a closed expression, showing that the generated phase-space trajectory lies exactly on a circle that is rescaled and rigidly rotated with respect to the exact one, rather than merely approximating it. This geometric renormalization yields a closed form for the state fidelity, structurally identical to the physical survival probability governing revivals in the deep strong coupling regime: both are the same geometric question -- the distinguishability of a displaced Fock state -- applied to two different pairs of points in phase space. We further show that this geometric error is exactly correctable by promoting the Trotter-step coupling to a complex amplitude, at no additional circuit cost. These results, obtained in the exactly solvable degenerate limit, provide a benchmark for Trotterization errors in the general Rabi model.

Comments
9 pages, 4 figures

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Digital quantum simulation of the quantum Rabi model in the deep strong coupling regime motivates the need for an exact characterization of Trotterization errors, so far known only through qualitative bounds. Here we derive such a characterization in the exactly solvable degenerate limit, valid for arbitrary coupling strength. The first-order Trotter operator admits a closed expression, showing that the generated phase-space trajectory lies exactly on a circle that is rescaled and rigidly rotated with respect to the exact one, rather than merely approximating it. This geometric renormalization yields a closed form for the state fidelity, structurally identical to the physical survival probability governing revivals in the deep strong coupling regime: both are the same geometric question -- the distinguishability of a displaced Fock state -- applied to two different pairs of points in phase space. We further show that this geometric error is exactly correctable by promoting the Trotter-step coupling to a complex amplitude, at no additional circuit cost. These results, obtained in the exactly solvable degenerate limit, provide a benchmark for Trotterization errors in the general Rabi model.
We derive a quantum master equation in Lindblad form for the open quantum Rabi model in the perturbative deep strong coupling regime, where the spectrum is close to degenerate and harmonic. The Lindblad form is obtained by combining secular and singular-coupling approximations, working in a Schrieffer-Wolff transformed frame. We benchmark our equation against a Bloch-Redfield one, analyzing the time-evolution and steady state properties of several observables and find excellent agreement. Analytical results are obtained in limiting cases.
The Quantum Rabi model is a crucial theoretical framework for understanding the interplay between light and matter. This paper focuses on a transformative digital quantum simulation that utilizes Trotterization with an increased number of steps to analyze a complex unitary Hamiltonian. This method demonstrates the advantages of digital techniques in capturing the dynamic behaviors of Quantum Rabi models in the deep strong coupling regime. Moreover, this proposal highlights the universality, flexibility, and scalability of digital methods, particularly in coherent measurements during time evolution with varying photon counts.
The quantum Rabi model is in the scientific spotlight due to the recent theoretical and experimental progress. Nevertheless, a full-fledged classification of its coupling regimes remains as a relevant open question. We propose a spectral classification dividing the coupling regimes into three regions based on the validity of perturbative criteria on the quantum Rabi model, which allows us the use of exactly solvable effective Hamiltonians. These coupling regimes are (i) the perturbative ultrastrong coupling regime which comprises the Jaynes-Cummings model, (ii) a region where nonperturbative ultrastrong and nonperturbative deep strong coupling regimes coexist, and (iii) the perturbative deep strong coupling regime. We show that this spectral classification depends not only on the ratio between the coupling strength and the natural frequencies of the unperturbed parts, but also on the energy to which the system can access. These regimes additionally discriminate the completely different behaviors of several static physical properties, namely the total number of excitations, the photon statistics of the field, and the cavity-qubit entanglement. Finally, we explain the dynamical properties which are traditionally associated with the deep strong coupling regime, such as the collapses and revivals of the state population, in the frame of the proposed spectral classification.
The quantum Rabi model describing the fundamental interaction between light and matter is a cornerstone of quantum physics. It predicts exotic phenomena like quantum phase transitions and ground-state entanglement in ultrastrong and deep-strong coupling regimes, where coupling strengths are comparable to or larger than subsystem energies. Demonstrating dynamics remains an outstanding challenge, the few experiments reaching these regimes being limited to spectroscopy. Here, we employ a circuit quantum electrodynamics chip with moderate coupling between a resonator and transmon qubit to realise accurate digital quantum simulation of deep-strong coupling dynamics. We advance the state of the art in solid-state digital quantum simulation by using up to 90 second-order Trotter steps and probing both subsystems in a combined Hilbert space dimension of ∼80, demonstrating characteristic Schrödinger-cat-like entanglement and large photon build-up. Our approach will enable exploration of extreme coupling regimes and quantum phase transitions, and demonstrates a clear first step towards larger complexities such as in the Dicke model.
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We revisit the analytic solution of the two-qubit quantum Rabi model with inhomogeneous coupling and transition frequencies using a displaced oscillator basis. This approach enables us to apply the same truncation rules and techniques adopted in the Rabi model to the two qubits system. The derived analytical spectra match perfectly with the numerical solutions in the parameter regime where the qubits’ transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultrastrong coupling regime. We further explore the dynamical behavior of the two qubits as well as the evolution of entanglement. The analytical methods provide unexpectedly accurate results in describing the dynamics of the two qubits in the present experimentally accessible coupling regime. The time evolutions of the probability for the qubits show that the collapse-revival phenomena emerge, survive and finally disappear when one coupling strength increases from weak to strong coupling regimes and the other coupling strength is well into the ultrastrong coupling regime. The inhomogeneous coupling system exhibits new dynamics, which are different from the homogeneous coupling case.