A parafermionic generalization of the Jaynes–Cummings model

Abstract

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 . We also derive a semiclassical approximation of the canonical partition function of the model by assuming ℏ 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.