Quantum Rabi Model: N-State Extensions

• Quantum Rabi Model is a framework that generalizes two-level light–matter interactions to N-state systems using advanced group-theoretical methods and generalized spin matrices.
• It employs novel analytical techniques such as the Symmetric Generalized Rotating Wave Approximation to capture strong coupling effects and symmetry-induced phenomena like ground-state parity inversion.
• The model finds practical applications in atom-cavity QED, molecular aggregates, and dissipative spin–boson systems, offering robust insights into quantum dynamics.

The Quantum Rabi Model (QRM) generalizes the paradigmatic two-level light–matter quantum Hamiltonian to accommodate a multistate (“N-state”) quantum system interacting with a single-mode bosonic field. The model systematically extends the two-state symmetry structure, introduces generalized conservation laws, and enables tractable analysis—both analytic and numeric—of diverse physical scenarios, including atom-cavity, molecular, and dissipative environments. This framework provides foundational tools for studying the quantum dynamics, spectral properties, and symmetry-induced phenomena in complex atomic and molecular systems.

1. Extension to N-State Systems via Generalized Spin Matrices

The standard two-state (two-level) QRM describes a bosonic mode (annihilation/creation operators $b$, $b^\dagger$; mode frequency $\omega$) coupled to a two-level system (Pauli matrices $\sigma_x, \sigma_z$; tunneling $J$):

$$H_2 = \omega\,b^\dagger b + (b + b^\dagger)\sigma_z + J\sigma_x$$

The N-state generalization considers a quantum system with a basis $\{|n\rangle\}$, $n = 0,\ldots,N-1$, and introduces generalized spin matrices $S_{j,k}$:

$$S_{j,k} = \sum_{n=0}^{N-1} e^{i(2\pi/N)nj}\,|n\rangle\langle n+k|$$

The N-state QRM Hamiltonian is constructed as

$$H = \omega\,b^\dagger b + \left( b S_{1,0} + b^\dagger S_{1,0}^\dagger \right) + J S_{0,N/2} + \sum_{k=1}^\kappa J_k (S_{0,k} + S_{0,k}^\dagger)$$

where $J_k$ dictates tunneling amplitudes between system states. The interaction term couples the field’s coherent displacement to the system’s internal degrees of freedom with explicitly N-dependent phase weights, embedding a generalized $N$-fold symmetry.

This formalism treats atomic, molecular, or periodic N-level units consistently, allowing rich internal connectivity (e.g., ring, chain, or cascade topologies) to be parameterized via $J_k$.

2. Group-Theoretical Structure and Symmetry Sectors

A central technical feature of the N-state QRM is its extended symmetry, operationally described via the commutation of $H$ with operators of the form

$\mathcal{O}_n = \mathcal{R}_n S_{0,n}$

$$\mathcal{R}_n = \exp\left[ i\frac{2\pi}{N} n b^\dagger b \right]$$

for $n=0,\ldots,N-1$. These commuting operators encode an $N$-fold generalized parity symmetry—reducing to the familiar two-level parity for $N=2$—and partition the Hilbert space into $N$ independent infinite-dimensional boson chains. For each value of $N$, additional quantum numbers (e.g., “cascade” or $\delta$ for $N=4,3$) arise, enabling the labelling and classification of eigenstates according to conserved symmetry sectors.

A group-theoretical (unitary) transformation $U$ exploits this symmetry to block-diagonalize the Hamiltonian, significantly simplifying both analytic (via symmetry-adapted bases) and numerical calculations (allowing truncation within symmetry-constrained chains).

3. Strong Coupling and Ground State Parity Inversion in N=4

In the four-state ($N=4$) scenario, the symmetric topological structure (double-Λ, tripod, or diamond, depending on $J, K$ parameters) enables new qualitative phenomena. A notable prediction is the “parity switch” of the ground state: as the system’s coupling parameter grows large, the lowest-energy state crosses between different symmetry sectors. This abrupt change in parity as a function of interaction strength is a property absent in the $N=2$ model and directly follows from the enhanced Hilbert-space decomposition afforded by the group-theoretical construction. Such ground-state parity inversion leads to distinct spectroscopic and dynamical signatures, most directly observable in strongly coupled four-level atom-cavity experiments.

4. Symmetric Generalized Rotating Wave Approximation (S-GRWA)

Accurate analytical treatment in the strong- and ultrastrong-coupling regimes requires approximations beyond the standard rotating wave approximation (RWA). In the N-state model, the S-GRWA is derived to respect the symmetry of $H$ and fully include the counter-rotating contributions:

• The S-GRWA yields analytical energies, obtained in the decoupled boson chain representation, which are in excellent agreement with numerical results even for coupling strengths $\lambda \sim \omega$ where traditional approximations fail.
• The approximation is constructed such that the conserved quantum numbers (from the symmetry operators) are preserved, ensuring physically correct assignment of energy levels and eigenstates.

As a result, S-GRWA becomes a powerful tool for both analytic insight and quantitative estimation throughout the parameter space of N-state models.

5. Dissipative N-State Spin-Boson Generalization

The model admits a natural extension to the dissipative (infinite-mode) environment:

$$H \to \sum_l [\omega_l b_l^\dagger b_l + (b_l S_{1,0} + b_l^\dagger S_{1,0}^\dagger)] + J S_{0,N/2} + \cdots$$

Here, the single mode $b$ is replaced by a set $\{b_l\}$, each with its own frequency $\omega_l$, representing a bosonic bath. This yields a periodic generalization of the standard spin-boson (Leggett) model incorporating full $N$-fold symmetry. This construction is directly applicable to the dynamics of excitation energy transfer in periodic molecular aggregates—such as molecular rings and chains—enabling studies of geometric effects, symmetry-induced interference, and decoherence in complex excitonic systems.

6. Practical Significance and Analytical Formulation

The N-state QRM provides:

• A compact, tractable representation for multi-level systems coupled to quantized fields, with flexibility in coupling geometry and symmetry.
• Access to new dynamical and spectral phenomena (ground-state parity switches, extra conservation laws, configuration-dependent transitions) not present in two-level models.
• Analytical formulas for critical components such as the generalized spin matrices,

$$S_{j,k} = \sum_{n=0}^{N-1} e^{i (2\pi/N) n j} |n\rangle\langle n+k|$$

the full Hamiltonian,

$$H = \omega\,b^\dagger b + (b S_{1,0} + b^\dagger S_{1,0}^\dagger) + J S_{0,N/2} + \sum_{k=1}^\kappa J_k (S_{0,k} + S_{0,k}^\dagger)$$

and the symmetry operations,

$$\mathcal{O}_n = \mathcal{R}_n S_{0,n},\quad \mathcal{R}_n = \exp\left[ i (2\pi/N) n b^\dagger b \right]$$

consistent with group theory and facilitating practical computation.

7. Application Domains and Outlook

This framework enables quantitative treatment of:

• Multi-level atom–cavity QED systems, including experiments exceeding the two-level approximation,
• Molecular ring and chain aggregates relevant for photosynthetic energy transfer, emphasizing the importance of spatial and rotational symmetry,
• Theoretical exploration of symmetry effects on quantum phase transitions, energy transfer efficiency, and decoherence suppression,
• Extensions to open quantum systems and the analysis of energy relaxation pathways, by mapping to periodic dissipative spin–boson models.

By systematically extending the symmetry structure and providing robust analytical and numerical handling, the N-state QRM advances both fundamental understanding and experimental exploration of quantum light–matter systems far beyond the two-level case (Albert, 2011).