Collapse and revival of ultracold atoms in a microwave cavity and of photons in parametric down-conversion
To examine this question we made some simplifying assumptions on the behaviour of the condensate and reduced the Hamiltonian to a Hamiltonian that is trilinear in the annihilation operators a,b,c (a+ creates a photon, b+ creates an excited atom, and c+ creates an atom in the ground state):
This Hamiltonian is known to describe also parametric down-conversion in nonlinear crystals.
Using an elegant theoretical approach based on a first-order operator differential equation for the numer operator of excited atoms we predicted collapse and revivals for coherent initial states.
For number states we also examined the fluctuations of
the number of excited atoms. Interestingly these become very large:
The graphs are based on an exact analytical solution of
a 4th order nonlinear differential equation.