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Relativistic Qutrit Quantum Rabi-type Model

S. Zarrinkamar

Published Oct 23, 2026• 0 citations
Publisher

Abstract

We formulate and solve an effective spin-one quantum Rabi model motivated by the physical spin-one sector of the ten-component Duffin--Kemmer--Petiau (DKP) theory. After the constrained DKP components are separated from the three physical spin projections, the internal sector is represented as a qutrit. Under a single-mode and magnetic-dipole approximation, its coupling to a quantized electromagnetic field has the spin-one Rabi form g(a+a†)Sx. We derive the regular spectrum of this model in the Bargmann representation. The exact parity symmetry reduces the three-component problem to independent scalar three-term recurrences, from which parity-resolved spectral functions are obtained as continued fractions. The pole energies generated by eliminating the outer spin components are treated separately, allowing exceptional finite-polynomial states to be identified without confusing them with regular zeros of the continued fractions. Numerical comparison with direct Fock-space diagonalization confirms the regular spectral construction to better than 5×10−13 for a representative parameter set. The result establishes an exact spectral framework for a qutrit generalization of the quantum Rabi model and states explicitly the assumptions required for its connection with spin-one DKP dynamics.

DOI
10.1364/opticaopen.34688829

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We formulate and solve an effective spin-one quantum Rabi model motivated by the physical spin-one sector of the ten-component Duffin--Kemmer--Petiau (DKP) theory. After the constrained DKP components are separated from the three physical spin projections, the internal sector is represented as a qutrit. Under a single-mode and magnetic-dipole approximation, its coupling to a quantized electromagnetic field has the spin-one Rabi form g(a+a†)Sx. We derive the regular spectrum of this model in the Bargmann representation. The exact parity symmetry reduces the three-component problem to independent scalar three-term recurrences, from which parity-resolved spectral functions are obtained as continued fractions. The pole energies generated by eliminating the outer spin components are treated separately, allowing exceptional finite-polynomial states to be identified without confusing them with regular zeros of the continued fractions. Numerical comparison with direct Fock-space diagonalization confirms the regular spectral construction to better than 5×10−13 for a representative parameter set. The result establishes an exact spectral framework for a qutrit generalization of the quantum Rabi model and states explicitly the assumptions required for its connection with spin-one DKP dynamics.
We formulate and solve an effective spin-one quantum Rabi model motivated by the physical spin-one sector of the ten-component Duffin--Kemmer--Petiau (DKP) theory. After the constrained DKP components are separated from the three physical spin projections, the internal sector is represented as a qutrit. Under a single-mode and magnetic-dipole approximation, its coupling to a quantized electromagnetic field has the spin-one Rabi form g(a+a†)Sx. We derive the regular spectrum of this model in the Bargmann representation. The exact parity symmetry reduces the three-component problem to independent scalar three-term recurrences, from which parity-resolved spectral functions are obtained as continued fractions. The pole energies generated by eliminating the outer spin components are treated separately, allowing exceptional finite-polynomial states to be identified without confusing them with regular zeros of the continued fractions. Numerical comparison with direct Fock-space diagonalization confirms the regular spectral construction to better than 5×10−13 for a representative parameter set. The result establishes an exact spectral framework for a qutrit generalization of the quantum Rabi model and states explicitly the assumptions required for its connection with spin-one DKP dynamics.
We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].
We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].
With the use of the general covariant matrix 10-dimensional Petiau-Duffin-Kemmer formalism in cylindrical coordinates exact solutions of the quantum-mechanical equation for a particle with spin 1 in the presence of an external homogeneous magnetic field are constructed. Three linearly independent types of solutions are separated; in each case the formula for the energy levels has been found.Within similar technique for the quantum-mechanical equation for a particle with spin 1 and additional intrinsic electromagnetic characteristics - polarizability, exact solutions are found in the presence of an external homogeneous magnetic field.
We investigate one- and two-mode variants of the Z 3 -symmetric quantum Rabi model, which describe the interaction of a qutrit with one or two bosonic modes and are directly relevant for circuit-QED and spin-qudit platforms. We find a canonical transformation that allows one to obtain the spectrum of the Z 3 Rabi model using perturbation theory in a magnetic field. We show that in a certain parameter regime (deep-strong coupling and a small magnetic field), the three lowest eigenstates become Z 3 qutrit-boson cat states. In order to characterize these states, we introduce a joint qutrit-boson Wigner function and derive its closed-form expression for the qutrit-boson cat states. Numerical calculations across a wide range of coupling strengths show that the proposed Wigner function is a useful tool that allows one to unambiguously identify the states of the Z 3 qutrit-boson cat states.