Multilevel Quantum Rabi Model

• Multilevel Quantum Rabi Model is a generalization of the two-level model that replaces a simple atom with multiple ground and excited states, enabling complex light–matter coupling.
• It employs techniques such as singular value decomposition to distinguish bright and dark sectors, effectively renormalizing coupling strengths and broadening the parameter space.
• The model underpins applications in circuit QED, quantum thermometry, and multimode interactions, paving the way for robust ultrastrong coupling and innovative quantum control.

Searching arXiv for the specified papers and closely related multilevel quantum Rabi model work. The multilevel quantum Rabi model denotes a family of generalizations of the quantum Rabi model in which the idealized two-level atom is replaced by a multilevel structure, while retaining nonperturbative light–matter coupling to one or more bosonic modes. In its most direct formulation, the atomic subsystem consists of $m$ ground states and $n$ excited states coupled to a single field mode, often under the assumption that the two manifolds are well separated while the splittings within each manifold remain small. Under these conditions, the model preserves key Rabi features such as parity symmetry, but acquires additional structure—multiple effective couplings, bright and dark sectors, manifold-dependent renormalizations, and qualitatively new dynamical and thermometric regimes (Doicin et al., 8 Apr 2025). In realistic platforms, especially circuit QED, the multilevel formulation is not merely a refinement: once the coupling becomes large, the bare multilevel structure of the artificial atom cannot be ignored, even when the device is strongly anharmonic (Manucharyan et al., 2016).

1. Conceptual scope and relation to the standard Rabi model

The standard quantum Rabi model is built around a single bosonic mode interacting with a perfect two-level system. Multilevel extensions relax that atomic idealization in several distinct ways. One important class replaces the two-level atom by two near-degenerate manifolds, with $m$ ground and $n$ excited states all coupled to the same field mode (Doicin et al., 8 Apr 2025). A second class arises in superconducting circuits, where nominal qubits such as transmons, fluxonium devices, or Cooper-pair boxes are intrinsically multilevel and must be treated beyond a two-state truncation under strong driving or ultrastrong coupling (Pietikäinen et al., 2017). A third class incorporates additional transitions, modes, or ancillary states, as in mixed Rabi–Jaynes–Cummings models and multi-$\Lambda$ cavity systems (Torosov et al., 2015).

Across these formulations, the multilevel structure changes both the spectral organization and the effective coupling strengths. Some linear combinations of levels hybridize strongly with the field, while orthogonal combinations can become dark. In the simplest near-degenerate-manifold setting, the model reduces approximately to a direct sum of ordinary Rabi Hamiltonians with different couplings, so the multilevel problem retains a recognizable QRM backbone while substantially enlarging the accessible parameter space (Doicin et al., 8 Apr 2025).

A related but distinct line of work considers classically driven multilevel manifolds rather than a quantized field mode. There the relevant phenomenon is not a decomposition into independent quantum Rabi blocks, but the emergence of anomalous Rabi oscillations whose frequency is fixed by an internal level splitting rather than by the pulse area. This is not the same Hamiltonian problem as the single-mode MQRM, but it extends multilevel Rabi physics in a way that is structurally analogous: a larger manifold qualitatively alters the effective two-state dynamics (Chang et al., 2017).

2. Canonical Hamiltonians and symmetry structure

For $m$ ground states $\{|g_j\rangle\}$, $n$ excited states $\{|e_i\rangle\}$, and a single bosonic mode of frequency $\omega$, one representative MQRM Hamiltonian is

$n$0

with $n$1 and $n$2 (Doicin et al., 8 Apr 2025). The parameters $n$3 and $n$4 encode small intramanifold splittings, with $n$5 in the near-degenerate regime. The interaction contains only ground-to-excited transitions; direct transitions within each manifold are assumed negligible.

This model retains the usual parity symmetry,

$n$6

so parity remains a conserved quantum number even though the atomic subsystem is no longer two dimensional (Doicin et al., 8 Apr 2025). That conservation law is one of the principal continuities with the ordinary QRM.

A closely related thermometric formulation uses atomic manifolds of sizes $n$7 and $n$8, bare splitting $n$9, bosonic frequency $m$0, and a complex coupling matrix $m$1: $m$2 where $m$3 (Doicin et al., 13 Feb 2026). The two formulations differ in notation, but both assume two well-separated atomic bands with weak internal splittings and a general multichannel light–matter coupling.

In driven circuit-QED variants, the Hamiltonian becomes explicitly time dependent. For a multilevel transmon coupled to a cavity, one writes

$m$4

with

$m$5

supplemented by cavity drive and probe terms proportional to $m$6 (Pietikäinen et al., 2017). Here the multilevel structure is encoded directly in the device spectrum $m$7 and matrix elements $m$8.

3. Bright and dark sectors, SVD reduction, and coupling enhancement

A central structural result is that the coupling matrix $m$9 can be singular-value decomposed, and the atomic basis can then be rotated into collective states adapted to the radiation field. In the notation of the near-degenerate-manifold MQRM, one writes

$n$0

with singular values ordered as $n$1, and defines radiation-basis states

$n$2

In this basis the interaction becomes

$n$3

so for $n$4 the full Hamiltonian decomposes into a direct sum of $n$5 independent two-level quantum Rabi Hamiltonians, each with its own coupling $n$6 (Doicin et al., 8 Apr 2025).

The thermometric MQRM expresses the same structure in the language of bright doublets and dark manifolds. After an SVD,

$n$7

the interacting atomic states form bright pairs $n$8, each coupled with strength $n$9, while the remaining states form a completely decoupled dark manifold of dimension $\Lambda$0 (Doicin et al., 13 Feb 2026). The bright/dark distinction is not merely algebraic: it governs which sectors hybridize with the cavity mode, which sectors remain spectrally inert, and which transitions dominate thermal sensitivity.

The simplest coupling pattern is the uniform case $\Lambda$1 for all $\Lambda$2 with $\Lambda$3. Then $\Lambda$4, the only nonzero singular value is

$\Lambda$5

and all other $\Lambda$6 vanish. The multilevel problem therefore collapses onto a single QRM block with dressed coupling

$\Lambda$7

(Doicin et al., 8 Apr 2025). More generally, any factorisable coupling matrix $\Lambda$8 has a single nonzero singular value $\Lambda$9.

For random couplings, the largest singular value is governed by Wishart statistics. If $m$0 are IID complex Gaussian variables, the largest coupling satisfies

$m$1

in the large-$m$2 limit, and the variance vanishes as $m$3 (Doicin et al., 8 Apr 2025). The paper interprets the resulting coupling boost as analogous to superradiant enhancement in the Dicke model: collective superpositions couple strongly, while orthogonal combinations become dark.

4. Dynamical regimes: anomalous oscillations, Floquet response, and breakdown of two-level truncations

In classically driven multilevel manifolds, the dressed-state picture produces a qualitatively different oscillation law from the ordinary two-level Rabi formula. For a two-level system with detuning $m$4,

$m$5

By contrast, in a multilevel manifold one can isolate a two-level dressed substructure (TLDS) consisting of two dressed eigenvalues $m$6 and $m$7. At low field, $m$8, one has $m$9, which yields normal, area-controlled Rabi oscillations. At high field, $\{|g_j\rangle\}$0, one instead has $\{|g_j\rangle\}$1, where $\{|g_j\rangle\}$2 is the characteristic internal splitting. Defining the generalized pulse area

$\{|g_j\rangle\}$3

the strong-field limit gives

$\{|g_j\rangle\}$4

so the oscillation frequency is pinned to the internal splitting rather than to the pulse area (Chang et al., 2017). The regime of validity is succinctly stated as

$\{|g_j\rangle\}$5

The same work reports that for $\{|g_j\rangle\}$6 and $\{|g_j\rangle\}$7, one finds $\{|g_j\rangle\}$8 almost independent of area, and that inversion fidelity remains above $\{|g_j\rangle\}$9 even when $n$0 fluctuates by $n$1 (Chang et al., 2017).

A different dynamical problem appears in periodically driven multilevel cavity QED, where the relevant tool is Floquet theory. For the driven multilevel generalized Rabi model, one seeks solutions

$n$2

and diagonalizes the Floquet operator $n$3 in a basis $n$4 of transmon levels, cavity Fock states, and Fourier harmonics (Pietikäinen et al., 2017). Linear response to a weak probe then produces a susceptibility whose poles occur at transitions between Floquet states, and the resonance shift is extracted from the resulting probe spectrum.

The numerical results in that setting quantify how quickly multilevel physics invalidates the qubit approximation. At very low power, $n$5, the ground transmon state dominates and an $n$6 truncation is adequate. As the cavity occupation rises through $n$7, excited transmon levels become appreciably populated and the two-level truncation visibly fails. The crossover cavity occupation is $n$8, nearly independent of drive–cavity detuning, and the two-level approximation is stated to be accurate only up to $n$9 (Pietikäinen et al., 2017). If the resonance frequency is required to be accurate within $\{|e_i\rangle\}$0, the paper gives breakdown thresholds at $\{|e_i\rangle\}$1 for the two-level model, $\{|e_i\rangle\}$2 for the three-level model, and $\{|e_i\rangle\}$3 for the five-level model.

5. Circuit-QED realizations and multilevel renormalization of Rabi physics

In circuit QED, the MQRM arises directly from circuit quantization rather than from a phenomenological enlargement of a two-state atom. Two paradigmatic examples are fluxonium and the Cooper-pair box (CPB), each coupled to a resonator mode. In both cases, one obtains a Hamiltonian of the form

$\{|e_i\rangle\}$4

or its charge-gauge analogue involving $\{|e_i\rangle\}$5, after expressing the microscopic circuit variables in the eigenbasis of the bare artificial atom (Manucharyan et al., 2016). At the charge- and flux-degeneracy points, projection onto the two lowest levels recovers the canonical Rabi Hamiltonian, but that reduction ceases to be reliable once the coupling becomes large.

The principal conclusion is that the hallmark low-energy structure of the QRM survives, but in renormalized form. When $\{|e_i\rangle\}$6, both fluxonium and CPB develop nearly degenerate vacuum doublets $\{|e_i\rangle\}$7 whose splitting rapidly decreases with increasing coupling (Manucharyan et al., 2016). In the ideal two-level QRM,

$\{|e_i\rangle\}$8

For fluxonium, however, the low-energy splitting is instead found to decay algebraically,

$\{|e_i\rangle\}$9

when $\omega$0. Higher anticrossings are also strongly renormalized by off-resonant couplings to many bare levels.

Despite this renormalization, the ground-state entanglement spectrum remains essentially two dimensional. For $\omega$1, the reduced atomic density matrix has eigenvalues

$\omega$2

closely resembling the catlike vacuum of the two-level QRM even though many bare atomic levels participate in the full state (Manucharyan et al., 2016). The paper attributes the resulting two-fold near-degeneracy of the vacuum to environmental suppression of flux or charge tunneling by virtual resonator photons: in fluxonium this is interpreted as shunting by the resonator capacitance, while in the CPB it is described as dynamical Coulomb blockade.

The near-degenerate-manifold MQRM proposed for generic light–matter systems points to a different experimental direction. Systems with many near-degenerate levels, including colloidal quantum dots coupled to plasmonic nanoresonators, are identified as candidates for exploiting collective coupling enhancement in order to reach ultrastrong or deep-strong coupling without requiring unphysically large dipoles (Doicin et al., 8 Apr 2025).

6. Multi-mode, multi-$\omega$3, and controllable effective Rabi models

A three-level atom interacting with two quantized bosonic modes provides a mixed Rabi–Jaynes–Cummings realization of multilevel Rabi physics. In that model, the $\omega$4 transition is ultrastrongly coupled to one mode and treated in full Rabi form, while the $\omega$5 transition is weakly coupled to a second mode and treated under the rotating-wave approximation (Torosov et al., 2015). Fixing the second-mode photon number at $\omega$6 produces an Autler–Townes doublet

$\omega$7

and adiabatic elimination of the off-resonant branch yields an effective two-level QRM with

$\omega$8

When $\omega$9, the resulting effective Hamiltonian is the degenerate-level QRM, and the dynamics becomes exactly periodic with period

$n$00

even though $n$01 (Torosov et al., 2015). The Hilbert space then splits into invariant parity chains, as in the ordinary QRM.

An even more elaborate extension is the multi-photon quantum Rabi model with center-of-mass motion for multi-$n$02 atoms in a cavity (Hartmann et al., 7 Jul 2025). Starting from a fully second-quantized description of bosonic atomic fields $n$03, two quantized cavity modes $n$04 and $n$05, and a dipole interaction $n$06, the authors apply Hamiltonian averaging theory to obtain an effective Hamiltonian

$n$07

Here $n$08 contains a $n$09 “Rabi” block with self-couplings $n$10 and Raman couplings $n$11, while $n$12 describes particle–particle interactions generated by virtual ancillary-state processes. The effective description includes AC-Stark shifts, Bloch–Siegert shifts, and counter-rotating corrections from all involved ancillary states.

Under plane-wave modes and large single-photon detuning, the same framework yields Raman Rabi oscillations. Applied to a single-atom state, the propagator gives transitions such as

$n$13

so the populations oscillate sinusoidally (Hartmann et al., 7 Jul 2025). For a specific two-particle input state and $n$14, the same Raman configuration produces both atomic and optical Hong–Ou–Mandel effects. This places MQRM physics within a broader framework of multimode cavity QED, matter-wave interferometry, and fully quantized effective interactions.

7. Thermometric regimes and large-scale approximations

The MQRM has also been developed as an equilibrium quantum thermometer. In the thermometric model, one first separates bright and dark sectors by SVD, then works in the adiabatic or “large-displacement” regime

$n$15

(Doicin et al., 13 Feb 2026). For $n$16 and $n$17, the reduced Hamiltonian is

$n$18

and each bright doublet is diagonalized in a displaced-oscillator basis

$n$19

with zeroth-order energies

$n$20

After perturbatively reintroducing $n$21 and the intraband detunings, the $n$22-th bright ladder acquires energies

$n$23

while dark states remain diagonal (Doicin et al., 13 Feb 2026).

The thermometric figure of merit is the thermal quantum Fisher information

$n$24

Within the adiabatic approximation, the paper derives a closed-form expression for $n$25 in terms of the bright and dark partition-function contributions $n$26 and $n$27 (Doicin et al., 13 Feb 2026).

Two complementary thermometric limits are then identified. In dark-manifold saturation, $n$28, a large dark band lies above a few bright doublets, and the dominant thermal sensitivity comes from bright–dark population transfer. As $n$29, the QFI peak approaches the “ideal thermometer” bound

$n$30

provided the bright–dark gap is optimally tuned (Doicin et al., 13 Feb 2026). The optimal coupling $n$31 is stated to lie in the intermediate coupling regime.

In bright-manifold saturation, $n$32, there are $n$33 bright doublets and no dark states. For random couplings from the complex Ginibre ensemble, the bright ladders spread over a broad set of displacements, and as $n$34 the total QFI profile becomes broad and smooth, sample-to-sample fluctuations diminish through self-averaging, and the sensitivity remains above that of any single two-level reference across a wide temperature band (Doicin et al., 13 Feb 2026). This establishes the MQRM as a setting in which degeneracy can either sharpen thermometric response through a bright–dark gap or broaden it through a dense bright-manifold spectrum.

Taken together, these developments show that the multilevel quantum Rabi model is not a single special-purpose extension of the QRM but a general framework for organizing collective couplings, dark-state structure, strong-driving effects, multilevel renormalizations, and high-dimensional spectral engineering. A plausible implication is that its main significance lies precisely in this flexibility: the same multilevel enlargement that complicates the textbook two-level picture also creates new routes to ultrastrong coupling, robust state preparation, multimode interference, and equilibrium sensing (Doicin et al., 8 Apr 2025).