A parafermionic generalization of the Jaynes–Cummings model

We introduce a parafermionic version of the Jaynes–Cummings Hamiltonian, by coupling k Fock parafermions (nilpotent of order F) to a 1D harmonic oscillator, representing the interaction with a single mode of the electromagnetic field. We argue that for k = 1 and F ≤ 3 there is no difference between Fock parafermions and quantum spins $s=\frac{F-1}{2}$. We also derive a semiclassical approximation of the canonical partition function of the model by assuming hbar to be small in the regime of the large enough total number of excitations n, where the dimension of the Hilbert space of the problem becomes constant as a function of n. We observe in this case an interesting behaviour of the average of the bosonic number operator showing a single crossover between regimes with different integer values of this observable. These features persist when we generalize the parafermionic Hamiltonian by deforming the bosonic oscillator with a generic function Φ(x); the q-deformed bosonic oscillator corresponds to a specific choice of the deformation function Φ. In this particular case, we observe, at most k(F − 1), crossovers in the behaviour of the mean bosonic number operator, suggesting a phenomenology of superradiance similar to the k-atoms Jaynes–Cummings model.