Nonreciprocal electrical transport in linear systems with balanced gain and loss in the bulk

We investigate electrical transport in a quantum wire of $N$ sites connected to an equal number $(N_i/2)$ of sources and drains of charges in bulk. Each source and drain injects and extracts charges at the same rate, respectively. We show that the linear-response electrical current is nonreciprocal in such a system when the arrangement of sources and drains breaks the system's parity. We prove that inelastic scattering is essential for nonreciprocity in this system. For this, we invoke a master equation description of classical charge transport in a similar system. The nonreciprocal current in quantum wire matches that in the classical model for $N_i/N \sim 1$, generating a finite scattering length much smaller than the length of the wire. The nonreciprocity in the quantum wire oscillates with wire length when $N_i/N \ll 1$, and it can vanish at specific lengths.