Quantum Rabi
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 in a truncated bosonic Fock basis and compares the full excited-state trace against the Jaynes-Cummings reference. The sliders vary the boson cutoff and coupling . At weak coupling and modest times the two curves nearly coincide; as grows, the counter-rotating terms drive the expected departure from the simple reference.
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 in a truncated bosonic Fock basis and compares the full excited-state trace against the Jaynes-Cummings reference. The sliders vary the boson cutoff and coupling . At weak coupling and modest times the two curves nearly coincide; as grows, the counter-rotating terms drive the expected departure from the simple reference.
We use the resonant Hamiltonian
with and bosonic truncation
Writing the state as
and removing the trivial free phases, the amplitudes satisfy nearest-neighbor ladder equations,
The first term in each equation is the familiar rotating-wave channel, which moves amplitude inside the usual one-excitation ladder. The second term is the counter-rotating channel. Even though it oscillates rapidly, it is exactly the piece that the Jaynes-Cummings truncation discards, so it is the only source of disagreement between the two plotted traces.
The comparison curve keeps only the rotating-wave terms,
which for gives
For this initial condition, Jaynes-Cummings never needs more than the two-state subspace . The full Rabi model does not respect that simplification. Repeated counter-rotating excursions can feed amplitude into higher Fock sectors and then back again, so a trace that still looks smooth can already be sampling a larger truncated Hilbert space than the analytic reference would suggest.
Numerically, the demo integrates the truncated ODE system with fourth-order Runge-Kutta over a fixed visible time grid . Each displayed sample is internally subdivided into smaller micro-steps chosen to keep the rapidly oscillating counter-rotating phase resolved as and increase. That means the cutoff slider and the coupling slider stress different parts of the computation: one enlarges the state space, while the other increases the stiffness of the oscillatory coupling.
The most useful way to read the figure is therefore sequentially. First, pick a coupling and compare the blue and dashed orange curves. Then increase until the blue curve stops moving appreciably. If the trace stabilizes while still differing from , that difference is genuine Rabi physics. If it keeps drifting with the cutoff, the plot is telling you that the chosen bosonic truncation is still too aggressive for that coupling.
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.