  A photonic system with a tunable bath environment provides an extra degree of
freedom for quantum simulators. Such a system can be realized by parametrically
modulating the coupling between the system and bath. In this letter, by
coupling a transmon qubit to a lossy Superconducting Nonlinear Asymmetric
Inductive eLement (SNAIL) mode, we experimentally create a tunable chemical
potential for the qubit mode. We show that the qubit can be thermalized to
equilibrium with different population distributions under different parametric
pumping conditions. We further extend our method to the third level of the
transmon, showing its practical use beyond the simple two-level case. Our
results provide a useful tool that can be readily integrated with quantum
simulators that would benefit from a non-trivial photon population
distribution.
