  We consider an open Dicke model comprising a single infinite-component vector
spin and a single-mode harmonic oscillator which are connected by
Jaynes--Cummings-type interaction between them. This open quantum model is
referred to as the OISD (Open Infinite-component Spin Dicke) model. The
algebraic structure of the OISD Liouvillian is studied in terms of
superoperators acting on the space of density matrices. An explicit invertible
superoperator (precisely, a completely positive trace-preserving map) is
obtained that transforms the OISD Liouvillian into a sum of two independent
Liouvillians, one generated by a dressed spin only, the other generated by a
dressed harmonic oscillator only. The time evolution generated by the OISD
Liouvillian is shown to be asymptotically equivalent to that generated by an
adjusted decoupled Liouvillian with some synchronized frequencies of the spin
and the harmonic oscillator. This asymptotic equivalence implies that the time
evolution of the OISD model dissipates completely in the presence of any (tiny)
dissipation.
