  We study an one-dimensional non-Hermitian lattice with complex hopping rates,
which can be realized by a quasi-one-dimensional sawtooth-type Hermitian
lattice after adiabatic elimination with proper conditions. By means of
synthetic magnetic fluxes, the imaginary parts of the complex hopping rates can
be modulated by additional phase, thus a non-reciprocal structure arises. With
this lattice, one can realize robust unidirectional transport for both
single-site and Gaussian excitations, which is immune to defects and
backscattering. Furthermore, we proposed a sandwich structure based on the
non-Hermitian lattice, which can be used for realizing controllable photon
storage and reversal. The storage time and range can be artificially controlled
within limits, and the storage efficiency can be increased via a finite gain
compensation. The proposal of controllable photon transport in this paper opens
up a new path for unidirectional photon transport and provides a promising
platform for optical control and manipulation.
