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arXiv:2412.02918 · Physical Review Research

Analytical study of the non-Hermitian semiclassical Rabi model

Yibo Liu, Liwei Duan, Qing‐Hu Chen

Published Apr 2, 2025• 4 citations
arXivHTMLPDF

Abstract

The PT -symmetric semiclassical Rabi model explores the fundamental interaction between a two-level atom and a classical field, revealing novel phenomena in open systems through the inclusion of non-Hermitian terms. We propose a single similarity transformation that yields an effective Hamiltonian in rotating-wave approximation, enabling an analytical solution. The phase boundary of the PT -broken phase, derived from the analytical eigenvalues, closely matches the numerically exact one over a wide range of atomic frequencies, demonstrating the effectiveness of the analytical approach, especially at the main resonance. The Floquet parity operator is also introduced, providing a deeper physical understanding of the emergence of the PT -broken phase. Furthermore, by analyzing the dynamics of excited-state population, we observe several stable oscillations in the Fourier spectrum, demonstrating the applicability of the analytical method beyond the single-photon resonance region. The Bloch-Siegert shift is also discussed, and surprisingly, resembles its Hermitian counterpart, except for the higher-order terms in the coupling strength. The present analytical treatment provides a concise and accurate description of the main physics of this non-Hermitian atom-field interaction system.

DOI
10.1103/physrevresearch.7.023004

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The PT -symmetric semiclassical Rabi model explores the fundamental interaction between a two-level atom and a classical field, revealing novel phenomena in open systems through the inclusion of non-Hermitian terms. We propose a single similarity transformation that yields an effective Hamiltonian in rotating-wave approximation, enabling an analytical solution. The phase boundary of the PT -broken phase, derived from the analytical eigenvalues, closely matches the numerically exact one over a wide range of atomic frequencies, demonstrating the effectiveness of the analytical approach, especially at the main resonance. The Floquet parity operator is also introduced, providing a deeper physical understanding of the emergence of the PT -broken phase. Furthermore, by analyzing the dynamics of excited-state population, we observe several stable oscillations in the Fourier spectrum, demonstrating the applicability of the analytical method beyond the single-photon resonance region. The Bloch-Siegert shift is also discussed, and surprisingly, resembles its Hermitian counterpart, except for the higher-order terms in the coupling strength. The present analytical treatment provides a concise and accurate description of the main physics of this non-Hermitian atom-field interaction system.
We study a non-Hermitian version of the Rabi model, where a two-level system is periodically driven with an imaginary-valued drive strength, leading to alternating gain and loss. In the Floquet picture, the model exhibits PT\mathcal{PT}PT symmetry, which can be broken when the drive is sufficiently strong. We derive the boundaries of the PT\mathcal{PT}PT phase diagram for the different resonances by doing perturbation theory beyond the rotating-wave approximation. For the main resonance, we show that the non-Hermitian analog of the Bloch-Siegert shift corresponds to maximal PT\mathcal{PT}PT breaking. For the higher-order resonances, we capture the boundaries to lowest order. We also solve the regime of high frequency by mapping to the Wannier-Stark ladder. Our model can be experimentally realized in waveguides with spatially modulated loss or in atoms with time-modulated spontaneous decay.
A generalized quantum Rabi Hamiltonian with both one- and two-photon terms has emerged in the circuit quantum electrodynamics system for a decade. The usual parity symmetry is broken naturally in the simultaneous presence of both couplings, which complicates analytical treatments, even in the rotating-wave approximations. In this paper, we propose an adiabatic approximation to this generic model by using Bogoliubov operators and obtain a very concise analytical solution for both eigenvalues and eigenstates. Although the adiabatic approximation is only exact in the vanishing limit of the qubit frequency, the results for some physical observables nevertheless agree well with the numerical ones in a wide parameter regime. In the rotating-wave approximations, we also derive an analytical eigensolution. Two dominant Rabi frequencies are found in the Rabi oscillations of this generalized model. We also apply the present analytical theory to the vacuum Rabi splitting. It is found that some different phenomena emerge just because of the presence of the additional two-photon coupling term.
We considered the semiclassical Rabi model, that is, a two-level system interacting with a single-mode, classical field. In resonance and for an arbitrary initial state of the system, we obtained a simple, approximate, and analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/2 (1/4) of the angular frequency of the field and 1 (10) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and allows one to describe the evolution of the Bloch vector of the system in terms of a \textit{slow-precessional} and a \textit{fast-nutational} motion similar to that of a symmetric top with one point fixed. Finally, the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA.
The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number NNN cross the negative branch lines for N+2N+2N+2, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.
In this work, we explore the PT\mathcal{PT}PT-symmetric quantum Rabi model, which describes a PT\mathcal{PT}PT-symmetric qubit coupled to a quantized light field. By employing the adiabatic approximation (AA), we are able to solve this model analytically in the parameter regime of interest and analyze various physical aspects. We investigate the static and dynamic properties of the model, using both the AA and numerical diagonalization. Our analysis reveals a multitude of exceptional points (EPs) that are closely connected with the exactly solvable points in the Hermitian counterpart of the model. Intriguingly, these EPs vanish and revive depending on the light-matter coupling strength. Furthermore, we discuss the time evolution of physical observables under the non-Hermitian Hamiltonian. Rich and exotic behaviors are observed in both strong and ultrastrong coupling regimes. Our work extends the theory of PT\mathcal{PT}PT symmetry into the full quantum light-matter interaction regime and provides insights that can be readily enlarged to a broad class of quantum optical systems.