  In systems where interactions couple a central degree of freedom and a bath,
one would expect signatures of the bath's phase to be reflected in the dynamics
of the central degree of freedom. This has been recently explored in connection
with many-body localized baths coupled with a central qubit or a single cavity
mode -- systems with growing experimental relevance in various platforms. Such
models also have an interesting connection with Floquet many-body localization
via quantizing the external drive, although this has been relatively
unexplored. Here we adapt the multilayer multiconfigurational time-dependent
Hartree (ML-MCTDH) method, a well-known tree tensor network algorithm, to
numerically simulate the dynamics of a central degree of freedom, represented
by a $d$-level system (qudit), coupled to a disordered interacting 1D spin
bath. ML-MCTDH allows us to reach $\approx 10^2$ lattice sites, a far larger
system size than what is feasible with exact diagonalization or kernel
polynomial methods. From the intermediate time dynamics, we find a well-defined
thermodynamic limit for the qudit dynamics upon appropriate rescaling of the
system-bath coupling. The spin system shows similar scaling collapse in the
Edward-Anderson spin glass order parameter or entanglement entropy at
relatively short times. At longer time scales, we see slow growth of the
entanglement, which may arise from dephasing mechanisms in the localized system
or long-range interactions mediated by the central degree of freedom. Similar
signs of localization are shown to appear as well with unscaled system-bath
coupling.
