We introduce a driven-dissipative Bose-Hubbard chain describing coupled lossy photonic modes, in which time-reversal symmetry is broken by a coherent drive with a uniform phase gradient. We investigate this model by means of a Gaussian variational ansatz and numerically prove that the steady-state solution is stabilized by an inhomogeneous profile of the driving amplitude, which damps out boundary effects. Our calculations unveil a nonequilibrium phase diagram showing low- and high-density phases for photons separated by a phase coexistence region in which the system exhibits the phenomenon of topological amplification and is characterized by a finite non-Hermitian winding number. Our Letter shows the emergence of non-Hermitian topological phases in an interacting model that can be naturally implemented with superconducting circuits.

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• Open quantum systems
• Quantum optics
• Quantum phase transitions
• Topological effects in photonic systems
• Non-Hermitian systems
• Superconducting devices