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The quantum Rabi model is the minimal fully quantized light-matter Hamiltonian, while the Jaynes-Cummings model is its rotating-wave approximation [1, 2]. This demo propagates the initial state $|e,0\rangle$ in a truncated bosonic Fock basis and compares the full excited-state trace against the Jaynes-Cummings reference.

The sliders vary the boson cutoff $n_{\max}$ and coupling $g$ . At weak coupling and modest times the two curves nearly coincide; as $g$ grows, the counter-rotating terms drive the expected departure from the simple $\cos^2(gt)$ reference.

Method

We use the resonant Hamiltonian

H=\omega a^\dagger a+\frac{\omega_0}{2}\sigma_z+g(a+a^\dagger)(\sigma_++\sigma_-),

with $\omega=\omega_0=1$ and bosonic truncation

n=0,\dots,n_{\max}.

After removing the trivial free phases, the amplitudes satisfy nearest-neighbor ladder equations with explicit $e^{\pm i(\omega+\omega_0)t}$ counter-rotating factors. The demo integrates that truncated system with RK4, so increasing $n_{\max}$ is the direct numerical check on how much bosonic support is needed at the selected coupling.

The comparison curve keeps only the rotating-wave terms,

H_{\mathrm{JC}}=\omega a^\dagger a+\frac{\omega_0}{2}\sigma_z+g(a\sigma_++a^\dagger\sigma_-),

which for $|e,0\rangle$ gives

P_e^{\mathrm{JC}}(t)=\cos^2(gt).

So the plotted difference is a direct visual measure of how much the neglected counter-rotating sector matters at the chosen $(g,n_{\max})$ .

References

[1] E. T. Jaynes and F. W. Cummings. “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proceedings of the IEEE 51 (1), 89--109 (1963). DOI: 10.1109/PROC.1963.1664.
[2] D. Braak. “Integrability of the Rabi model,” Physical Review Letters 107 (10), 100401 (2011). DOI: 10.1103/PhysRevLett.107.100401.