Solution of the two-qubit quantum Rabi model and its exceptional eigenstates

Abstract

We have studied the two-qubit quantum Rabi model in the asymmetric case and its generalizations with dipole and Heisenberg-type qubit–qubit interactions. The solutions are obtained analytically with eigenstates given in terms of the extended coherent states or photon number states. For identical qubit–photon couplings, a novel type of quasi-exact solution which exists for all coupling values with constant eigenenergy is found, leading to level crossings within the same parity subspace even for non-identical qubits. In contrast to the quasi-exact eigenstates of the single-qubit model, these exceptional eigenstates are formed by just a few Fock states (photon number bounded from above at one or three), and the condition for them depends only on a fine-tuning of the qubit level splittings with respect to the photon energy, but not on the coupling to the photon field if the qubit–qubit interaction is not taken into consideration. This makes them excellent candidates for direct experimental observation within circuit quantum electrodynamics (QED) and application in single photon experiments. Besides, exceptional solutions with finite photon numbers N are also found.

Two-photon Rabi model: analytic solutions and spectral collapse

“…This state will also not collapse as g = is reached. Because several eigenstates for each parity are located below the first pole for small g according to 18, and correspond to zeros of the G-function in a pole-free region, none of them is constrained by the argument above and may be separated from the continuum at the critical coupling, if they do not cross the zeroth baseline for some g < g c .…”

“…G R (E) was then recovered within the extended coherent states approach, which avoids the mapping into the Bargmann space of analytic functions [7]. These results have stimulated extensive research in the QRM and related models [8,9,10,11,12,13,14,15,16,17,18,19,20].…”

Algebraic structure of the two-qubit quantum Rabi model and its solvability using Bogoliubov operators

“…which is independent of g, coinciding with [19]. So for ∆ 1 − ∆ 2 = 1 = ω and ∆ 1 − ∆ 2 = −1 = − ω, we obtain two quasi-exact solutions…”

“…Now, we have demonstrated all the exceptional eigenstates of the two-qubit quantum Rabi model with finite photon numbers presented in [19] by finding its algebraic structure in the photon number space. The special eigenstates |ψ g1 , |ψ g2 and |ψ e originate from the permutation symmetry of the qubit-photon coupling terms, and we may conjecture there are similar solutions for similar models.…”

“…which is the famous "dark state" [19,39], where the spin singlet is decoupled from the photon field. If ∆ 1 = ∆ 2 , considering the coefficient of {|M − 1, g, g , |M − 1, e, e } must be 0, it is required that E = M, which contradicts with E = N, so that the only possible choice is M = 0.…”