  Synchronization has recently been explored deep in the quantum regime with
elementary few-level quantum oscillators such as qudits and weakly pumped
quantum Van der Pol oscillators. To engineer more complex quantum synchronizing
systems, it is practically relevant to study composite oscillators built up
from basic quantum units that are commonly available and offer high
controllability. Here, we consider a minimal model for a composite oscillator
consisting of two interacting qubits coupled to separate baths, and show that
this system exhibits a wide variety of synchronizing behaviors. We study the
phase response of the constituent qubits as well as the system as a whole, when
one of the qubits is weakly driven. We consider the thermal baths to have
positive as well as effective negative temperatures, and discover effects that
occur only when the temperatures of the baths for the two qubits are of
opposite signs. We propose and analyze a circuit quantum electrodynamics
implementation of this model, which exploits recent advances in dissipation
engineering to realize effective negative temperature baths. Our work
demonstrates the potential for assembling complex quantum synchronizing systems
from basic building units, which is of pragmatic importance for advancing the
field of quantum synchronization.
