Quantum Rabi Model forN-State Atoms

Abstract

A tractable N -state Rabi Hamiltonian is introduced by extending the parity symmetry of the two-state model. The single-mode case provides a few-parameter description of a novel class of periodic systems, predicting that the ground state of certain four-state atom-cavity systems will undergo parity change at strong coupling. A group-theoretical treatment provides physical insight into dynamics and a modified rotating wave approximation obtains accurate analytical energies. The dissipative case can be applied to study excitation energy transfer in molecular rings or chains. Interactions between spin systems and harmonic oscillators (boson modes) have been studied for over 70 years [1][2][3]. One of the most well-known, the quantum Rabi model [1], is a phenomenological Hamiltonian describing interactions between a two-level system and a cavity mode. The model has also formed a basis of understanding for exciton-phonon interactions [4] and, along with its multi-mode extension, has numerous established applications in chemistry and physics (see [5][6][7] and refs. therein). The Jaynes-Cummings (J-C) [3] model is obtained by taking the Rabi model in the rotating wave approximation (RWA), where the "counter-rotating" terms are ignored (see e.g. [8]). While the J-C model is sufficient to study small atom-field coupling, the RWA breaks down at large coupling [9,10] Many-site spin-boson interaction, e.g. multi-state atom-cavity interaction [14] or excitation energy transfer in multi-chromophoric systems [15], continues to be a subject of significant interest, dictating a need for extensions of the two-state model. Extensions of the J-C model have been studied extensively [16][17][18][19], but are no longer applicable in the strong-coupling regime. Exciton-phonon generalizations which extend the parity/reflection symmetry of the Rabi system [20] are neither tractable nor applicable to atom-cavity systems. Most importantly, the Rabi model is the single-mode version of a dissipative (infinite-mode) spin-boson model [21], signifying that light-matter interaction is a simplified manifestation of a more fundamental interaction between a two-state system and a dissipative environment. Previous dissipative [22][23][24][25] generalizations have neither extended the symmetry nor preserved this correspondence. Motivated by these properties, this Letter presents a symmetry-preserving N -state extension of the Rabi model. The extension includes counter-rotating terms in a rigorous, intuitive, and mathematically manageable way, using a minimal number of parameters and paving the way for applications to multi-level atom-cavity experiments at both weak and strong coupling. A grouptheoretical approach [2] provides numerical advantages and physical insight into dynamics of the single-mode case. The symmetric generalized RWA [7] is applied to obtain accurate analytical energies/eigenstates valid for strong coupling. The above procedures are significantly simplified via the generalized spin matrices [26], providing a new tool for the treat...

Initial states of qubit–environment models leading to conserved quantities

“…As a second example, we consider the k-photon Rabi model [25] for which H Q = ωσ z and H E = νa † a read exactly as in the James-Cummings model (1) but the interaction is given by…”

Haydock’s recursive solution of self-adjoint problems. Discrete spectrum

“…was found to reduce to a TTRR [9,10,[12][13][14][15]28,29] φ n+1 + a n φ n + b n φ n−1 = 0 (n ≥ 1), (11) and a two-term condition on φ 0 and φ 1 [13,14], [9,10,14,15,[28][29][30], {φ n } ∞ n=0 are the sought expansion coefficients of a one-dimensional irreducible component of in the spin subspace. Keeping the two-term condition (12) separate from the truly TTRR (11) is important for the analysis of recurrence solutions [13,14].…”

Full spectrum of the two-photon and the two-mode quantum Rabi models

“…[2][3][4][5][6] Indeed, the Rabi model has been the subject of many generalizations including a multi-photon Rabi model, 7,8 two-mode Rabi model, 9 multi-atom Rabi model, 10 and a N-state Rabi model. 11 All these models have been investigated with various approaches leading however to quasi-exact resolutions. The multi-photon Rabi models describing the interaction of a two-level atom (a qubit) with quantized fields (oscillators) are of the form…”