  We study the coupled atom-molecular quantized ring vortices of 87Rb
Bose-Einstein Condensates (BEC) trapped in a rotating three dimensional (3D)
anisotropic cylindrical trap both in time independent and time-dependent
Gross-Pitaevskii approaches. For atom to molecular conversion and vice-versa
two-photon Raman photoassociation scheme has been used. Atomic and molecular
stationary state solutions show that the different number of nodes and crests
formed in the density profile (as a function of r and z) for different
combinations of radial (n) and axial (nz) quantum numbers at a fixed azimuthal
quantum number l=2, give rise to different structure around the 3D ring vortex
centered at r=0. We have considered both spontaneous and induced decays and
compared the results with those without considering the decays. The out of
phase oscillation of atomic and molecular numbers in two vortex states both in
presence and absence of external decays is the signature of coherence due to
the atom-molecular coupling. This coherence is also implemented in the
evolution of coupled atomic and molecular vortices. Intensity of molecular ring
vortices grows with progress in time in expense of that of atomic ring vortices
and vice versa. It is found that the intensity of the coupled atomic and
molecular ring vortices starts oscillation out of phase during evolution.
Dependence of the atom-molecular conversion efficiency and the lifetime of the
system on the laser intensity of photoassociation lasers and the total number
of atoms in two different vortex states reveals that formation of
atom-molecular coupled vortices and the efficiency of formation can be
controlled by varying these parameters. Linear stability analysis of vortex
states as a function of different system parameters shows that the atomic
vortices are more stable than molecular vortices.

Quantum Gases;Quantum Physics