Mukherjee Abhik
Abstract: we consider a single-mode microwave cavity coupled to low-capacitance Josephson junctions via the gauge-invariant Josephson phases. A recently proposed analytic tool: self-consistently ’rotating’ Holstein-Primakoff representation for the Cartesian components of the total spin, is used. We solve for the first time analytically the system of nonlinear semiclassical equations of motion of the coherent electromagnetic field (photonic condensate) bound to Cooper pair boxes (CPB) dipole moment for the Dicke model and solution is expressed via Jacobi elliptic functions of real time. This solution manifests emergence in the system of an intrinsic ’bound luminosity’ state that is characterized by periodic emission and re-absorption of the coherent electromagnetic radiation under the evolution of the collective state of CPB dipoles corresponding to coherent re-entrant population and de-population of the bare excited state by the two-level systems that represent e.g. Cooper pairs tunnelling in the Josephson junctions. The dynamical nature of this second quantized system in semiclassical limit may be compared with a classical phenomenon called " Dzhanibekov effect" which is also known as tennis racket theorem or intermediate axis theorem. It is named after Russian cosmonaut Vladimir Dzhanibekov who first noticed it in space in 1985. The theorem states that the rotation of a rigid body around its greatest and smallest moment of inertia principal axes is stable. Instability arises for rotation around its principal axis with intermediate moment of inertia. The stability pattern of our second quantized model in semiclassical limit has a deep connection with this classical phenomenon. We can show that stability is associated with the motion around p (photon momentum) and Sy (y-projection of spin) axes i.e, when these two quantities are constant, whereas there is an instability when Sz (z-projection of spin) is constant. Thus we show in this work that in thermodynamic limit, the second quantized Dicke model shows an instability pattern which is very similar to a purely classical phenomenon.