Perturbative Model of the Rabi System

• Perturbative model of the Rabi system is an analytic framework that defines eigenstates, spectral properties, and observables in light–matter interactions.
• It employs systematic perturbative expansions—such as the Bloch–Siegert transformation in pUSC and effective Hamiltonians in pDSC—to capture level splittings and state dynamics.
• The model delineates clear energetic boundaries for perturbative validity, distinguishing between controlled analytic predictions and inherently non-perturbative dynamics.

The perturbative model of the Rabi system provides a systematic analytic framework for describing the eigenstates, spectrum, and observables of the @@@@1@@@@ in parameter regimes where either the light-matter coupling or the two-level system splitting can be regarded as small. This approach is crucial for delineating the boundaries of validity where controlled, closed-form expansions apply, for distinguishing between perturbatively accessible and essentially non-perturbative dynamics, and for elucidating static and dynamical observables in all coupling regimes.

1. Quantum Rabi Model: Hamiltonian Structure and Regimes

The quantum Rabi Hamiltonian (QRM) captures the minimal model for a two-level system (qubit) interacting with a single mode of the quantized electromagnetic field: $$H_{\mathrm{R}} = \omega\,a^\dagger a + \frac{\Omega}{2}\,\sigma_z + g\,\sigma_x (a + a^\dagger)$$ where $\omega$ is the oscillator (cavity) frequency, $\Omega$ the qubit energy splitting, $g$ the light-matter coupling, and $\sigma_{x,z}$ the Pauli operators. The model features solvable limits and supports several qualitatively distinct spectral regimes, classified as follows:

• Perturbative Ultrastrong Coupling (pUSC): $g/(\omega+\Omega) \ll 1$, counter-rotating terms treated perturbatively;
• Perturbative Deep-Strong Coupling (pDSC): $\Omega/\omega \ll 1$, the bare qubit splitting is a perturbation to the dominant interaction;
• Non-perturbative (intermediate) regime: Neither $g$ nor $\Omega$ is a small parameter, precluding simple expansion (Rossatto et al., 2016).

These are sharply separated by “Juddian crossings,” i.e., spectral level crossings set by the ratios of $g$, $\omega$, $\Omega$, and the system energy—which define the maximal excitation number accessed.

2. Perturbative Expansions: Methods and Effective Hamiltonians

2.1 pUSC: Expansion in $g/(\omega + \Omega)$

In the pUSC regime, the Rabi Hamiltonian is partitioned as

$H_R = H_{JC} + V_{CR}$

where $H_{JC}$ is the Jaynes–Cummings Hamiltonian (rotating-wave approximation), and $V_{CR} = g (a\sigma_- + a^\dagger \sigma_+)$ contributes counter-rotating corrections. An explicit Bloch–Siegert (BS) transformation eliminates $V_{CR}$ to second order in $g/(\omega + \Omega)$, yielding

$$U^\dagger H_R U \approx H_{JC} + \omega_{BS} \sigma_z (a^\dagger a + \tfrac12) - \frac{\omega_{BS}}{2}$$

with $\omega_{BS} = g^2/(\omega+\Omega)$ the BS shift. The resulting spectrum contains JC-level avoided crossings, modified by a state-dependent frequency renormalization and leading-order level splittings (Rossatto et al., 2016, Gartner et al., 2021).

2.2 pDSC: Expansion in $\Omega/\omega$

In the deep strong coupling limit ($g/\omega \gg 1$), $H_R = H_0 + V$ with $$H_0 = \omega\,a^\dagger a + g\,\sigma_x(a+a^\dagger)$$ and $V = (\Omega/2)\sigma_z$. Diagonalizing $H_0$ gives displaced Fock states $$|\pm, n\rangle = |\pm\rangle \otimes D(\mp \alpha)|n\rangle$$ where $\alpha = g/\omega$, $D$ is the displacement operator, and $|\pm\rangle$ are $\sigma_x$ eigenstates. The splitting within each doublet emerges at first order in $\Omega$: $$E_{n,\pm} = n\omega - \alpha^2 \omega \pm \frac{\Omega}{2} e^{-2\alpha^2} L_n(4\alpha^2)$$ with $L_n$ the $n$-th Laguerre polynomial. The effective Hamiltonian in this basis succinctly captures all leading corrections (Rossatto et al., 2016).

2.3 Spectral Convergence and Series Uniformity

Rigorous justification of such expansions relies on bounds such as those from Rellich’s theory, which guarantee the convergence of finite families of eigenvalues and eigenvectors within a finite domain of the parameter $\epsilon$ provided non-degeneracy of first-order splittings and avoidance of accidental resonances (e.g., zeros of Laguerre polynomials in $g/\omega$) (Malagutti et al., 25 Jan 2026).

3. Domain of Validity, Crossovers, and Spectral Classification

The perturbative approach specifies sharp energetic boundaries for its applicability:

Regime	Expansion parameter	Validity condition
pUSC	$\Lambda=g/(\omega+\Omega)$	$g/\omega \lesssim 1/\sqrt{2(2n+1)}$ (for max. occupied $n$)
pDSC	$\Omega/\omega$	$g/\omega$ beyond last Juddian crossing, splitting $|E_{n,+} - E_{n,-}|/\omega \lesssim 0.1$

For fixed mean energy $E\simeq n\omega$, these boundaries translate to explicit lines in the $(g/\omega, n)$ plane. Between them, non-perturbative effects dominate and only numerical or analytic non-perturbative methods are reliable (Rossatto et al., 2016, Malagutti et al., 25 Jan 2026).

4. Static and Dynamical Observables: Perturbative Predictions

A key virtue of the perturbative model is transparent predictions for observables:

Static Quantities

• Total excitation number ($$n_e = \langle a^\dagger a + \sigma_+ \sigma_-\rangle$$): exactly conserved in pUSC ($[H_{BS}, n_e]=0$), weakly violated in pDSC.
• Photon statistics (Fano–Mandel parameter $Q$): sub-Poissonian ($Q<0$) in pUSC, super-Poissonian ($Q>0$) in pDSC, oscillatory in the intermediate regime.
• Qubit–cavity entanglement (von Neumann entropy $S$): $S\lesssim 1$ in pUSC, $S\approx 1 - O(e^{-4(g/\omega)^2})$ in pDSC, showing non-trivial minima between Juddian points in intermediate regions.

Dynamical Features

• Population dynamics: in pUSC, coherent Rabi oscillations as governed by the JC Hamiltonian and corrected by the Bloch–Siegert shift; in pDSC, collapse and revival patterns with characteristic timescales set by photon wave packet evolution in displaced oscillator chains; in non-perturbative regions, irregular oscillations lacking simple structure (Rossatto et al., 2016).

5. Mathematical Structure and Spectral Theorems

Mathematically, perturbative expansions in the Rabi model are justified via analytic perturbation theory for self-adjoint operators. Upon suitable unitary transformation, the Hamiltonian can be written as $A+\epsilon B$, where $A$ is the unperturbed harmonic oscillator (possibly matrix-valued for multilevel generalizations), and $B$ encodes spin interactions and mode displacements.

Rellich’s theorems ensure for fixed finite oscillator levels ($n\leq n_{\text{max}}$) the eigenvalues and eigenstates of $A+\epsilon B$ are analytic in $\epsilon$ within some radius, provided non-resonant conditions hold. This yields explicit expansions for each dressed doublet. Notably, verification of Braak’s conjecture for the Rabi model within this framework confirms the presence of at most two eigenvalues per unit spectral interval at small splitting, and the nonadjacency of empty/doubly occupied blocks in the spectrum (Malagutti et al., 25 Jan 2026).

6. Extensions: Dissipation, Drive, and Generalizations

Dissipation and Weak Drive

Incorporation of dissipation and external drive requires extension of the perturbative treatment to Lindblad master-equation frameworks. For weak drive,

• Perturbation in the drive amplitude yields analytic corrections to energies, eigenstates, and the damping-basis structure of the master equation;
• Decoherence features, such as doubled oscillation frequency for highly-inverted systems and analytical corrections to vacuum Rabi splitting, emerge in the weak-driving regime (Yu et al., 2016).

Multilevel and Multimode Extensions

The perturbative machinery generalizes naturally to $N$-level atoms and multiple cavity modes. The leading spectral asymptotics (Weyl law) is determined by the phase-space volume of $N$ harmonic oscillators, with the principal correction from the coupling encoded in the trace of the perturbation on the energy surface (Malagutti et al., 25 Jan 2026).

7. Physical Interpretation and Limitations

The perturbative model of the Rabi system provides a framework for understanding light–matter interactions in the experimentally relevant pUSC and pDSC regimes, with clearly defined boundaries and analytic predictions for a range of observables. The expansions break down precisely at Juddian crossings, corresponding to the emergence of exact spectral degeneracies, and fail in broad non-perturbative regions, where one must resort to non-perturbative or numerical treatments.

The validity of the perturbative treatment is fundamentally restricted by the ratio of microscopic parameters ($g/\omega$, $\Omega/\omega$) and by the system energy or excitation manifold accessed during dynamics. The analytic structure—rooted in operator perturbation theory—guarantees series convergence for any finite energy segment provided resonance points (zeros of Laguerre polynomials in $g/\omega$) are avoided.

Experimental and theoretical access to both pUSC and pDSC regimes has enabled direct tests of the perturbative predictions, revealing not only energy spectra but also qualitative features in photon statistics, excitation conservation, and entanglement signatures characteristic of each regime (Rossatto et al., 2016, Gartner et al., 2021, Malagutti et al., 25 Jan 2026).