A correspondence between the Rabi model and an Ising model with long-range interactions

By means of Trotter’s formula, we show that transition amplitudes between a class of generalized coherent states in the Rabi model can be understood in terms of a certain Ising model featuring long-range interactions beyond nearest neighbors in its thermodynamic limit. Specifically, we relate the transition amplitudes in the Rabi model to a sum over binary variables of the form of a partition function of an Ising model with a number of spin sites equal to the number of steps in Trotter’s formula applied to the real-time evolution of the Rabi model. From this, we show that a perturbative expansion in the energy splitting of the two-level subsystem in the Rabi model is equivalent to an expansion in the number of spin domains in the Ising model. We conclude by discussing how calculations in one model give nontrivial information about the other model, and vice versa, as well as applications and generalizations this correspondence may find.

💡 Research Summary

The paper establishes a novel correspondence between the quantum‑optical Rabi model and a one‑dimensional Ising model with long‑range (non‑nearest‑neighbour) interactions. By introducing a spin‑dependent bosonic operator (b\equiv\sigma_x a) (and its Hermitian conjugate) the original Rabi Hamiltonian
(H=\omega_0\sigma_z+\omega a^\dagger a+g\sigma_x(a+a^\dagger))
is rewritten as
(H=-P\omega_0 e^{i\pi b^\dagger b}+\omega b^\dagger b+g(b^\dagger+b)),
where (P=\pm1) is the parity eigenvalue that commutes with the Hamiltonian. This reformulation isolates the parity dynamics from the bosonic degrees of freedom, allowing a clean application of the Trotter product formula.

The evolution operator is split into three non‑commuting pieces (H_1, H_2, H_3) and approximated as
(U(t)=\lim_{n\to\infty}\big(e^{-iH_1t/n}e^{-iH_2t/n}e^{-iH_3t/n}\big)^n).
Each Trotter step flips the sign of the eigenvalue of (b); representing these flips by binary variables (s_i=\pm1) yields a sum over (2^n) terms. The authors recognize that this sum has exactly the structure of an Ising partition function:

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