  Recent numerical and experimental works have revealed a disorder-free
many-body localization (MBL) in an interacting system subjecting to a linear
potential, known as the Stark MBL. The conventional MBL, induced by disorder,
has been widely studied by using quantum simulations based on superconducting
circuits. Here, we consider the Stark MBL in two types of superconducting
circuits, i.e., the 1D array of superconducting qubits, and the circuit where
non-local interactions between qubits are mediated by a resonator bus. We
calculate the entanglement entropy and participate entropy of the
highly-excited eigenstates, and obtain the lower bound of the critical linear
potential $\gamma_{c}$, using the finite-size scaling collapse. Moreover, we
study the non-equilibrium properties of the Stark MBL. In particular, we
observe an anomalous relaxation of the imbalance, dominated by the power-law
decay $t^{-\xi}$. The exponent $\xi$ satisfies
$\xi\propto|\gamma-\gamma_{c}|^{\nu}$ when $\gamma<\gamma_{c}$, and vanishes
for $\gamma\geq \gamma_{c}$, which can be employed to estimate the
$\gamma_{c}$. Our work indicates that superconducting circuits are a promising
platform for investigating the critical properties of the Stark MBL transition.
