A pedagogical derivation of the first-order effective Hamiltonian for the two-mode Jaynes-Cummings model

TL;DR

This work presents a clear, self-contained derivation of the first-order effective Hamiltonian for the two-mode Jaynes–Cummings model in the dispersive regime, showing how a perturbative small-rotation transformation removes nonresonant atom-field terms and reveals an atom-mediated beam-splitter coupling between modes. The authors diagonalize the resulting quadratic Hamiltonian via a geometric SU(2) rotation, obtaining atom-state–dependent normal modes with frequencies $\Omega_{A,B}^{(\pm)}$ and a clear interpretation of the mixing angle $\theta_{\pm}$, including an explicit diagonalization condition $\tan(2\theta_{\pm}) = \pm 2J/(\tilde{\omega}_{a}^{(\pm)} - \tilde{\omega}_{b}^{(\pm)})$. Time evolution reduces to slow, mode-mixing dynamics governed by the effective parameters $\chi_{a,b} = 2 g_{a,b}^{2}/\Delta_{a,b}$ and $J = 2 g_{a} g_{b}/\Delta_{ab}$, leading to observable beating patterns and conditional dynamics depending on the atomic state. The paper provides worked examples for Fock and coherent inputs, highlighting the pedagogy of effective-Hamiltonian methods and their relevance to contemporary platforms in cavity QED, trapped ions, and circuit QED.

Abstract

This work presents a pedagogical and self-contained derivation of the first-order effective Hamiltonian for the two-mode Jaynes-Cummings model in the dispersive regime. A perturbative unitary transformation removes nonresonant atom-field terms, revealing dispersive frequency shifts leading to an atom-induced effective beam-splitter interaction between the field modes. The resulting Hamiltonian is diagonalized through a simple geometric rotation in the two-mode bosonic space, providing a transparent interpretation of the underlying dynamics. The exposition emphasized clarity and physical insight, making effective Hamiltonian methods accessible for teaching and learning in multimode light-matter interactions.

Figures (2)

• Figure 1: Branches of the rotation angle $\theta_\pm$ obtained from the diagonalization condition in Eq. \ref{['eq:theta_s']}. The two branches correspond to the atomic eigenstates $\ket{\pm}$ and represent the angles required to rotate the original fieldmodes $(\hat{a},\hat{b})$ into the normal modes $(\hat{A},\hat{B})$ that diagonalize the effective Hamiltonian in each atomic subspace. The vertical asymptotes occur when the effective detuning $\tilde{\omega}_a^{(s)}-\tilde{\omega}_b^{(s)}$ approaches zero, signaling near-degeneracy of the dispersively shifted modes. In this regime, a small change in parameters produces a rapid variation of $\theta_s$, reflecting strong atom-mediated mode hybridization. Away from these points, $\theta_s$ varies smoothly, and the normal modes remain predominantly aligned with one of the original field modes. This figure provides a geometric interpretation of the atom’s role as a tunable media of mode mixing: the atomic state selects a specific rotation branch, thereby controlling how strongly the two modes are hybridized in the dispersive regime.
• Figure 2: Time evolution of the mean photon numbers $\langle\hat{n}_a(t)\rangle$ and $\langle\hat{n}_b(t)\rangle$ for representative initial states and parameters, computed using the first-order effective Hamiltonian. Distinct curves correspond to the two atomic eigenstates $\ket{+}$, illustrating how the atomic state conditionally controls the field dynamics even though no real atom--field excitation exchange occurs in the dispersive regime. The oscillatory exchange of population between the modes is governed by the effective beam-splitter interaction induced by virtual atomic transitions. The characteristic oscillation period is set by the normal-mode splitting $|\Omega_{A}^{(\pm)} - \Omega_{B}^{(\pm)}|$, which scales as $g_{a} g_{b}/\Delta$ and is therefore much longer than the bare Rabi period $1/g_{k}$. This separation of time scales explains the slow dynamics observed (horizontal tick labels) against the normal time dynamics of the full system (tilded tick labels) in the figure, and illustrates how the effective-Hamiltonian treatment rescales the relevant time axis by eliminating fast atom--field oscillations. Overall, the figure demonstrates how the effective Hamiltonian reorganizes the dynamics into slow, physically transparent mode-exchange processes, whose frequency and amplitude are directly linked to the rotation angle $\theta_s$ and to the atomic state.