By studying a modified (unbiased) quantum multibaker map, we were able to obtain a finite asymptotic quantum current without a classical analog. This result suggests a general method for the design of purely quantum ratchets and sheds light on the investigation of the mechanisms leading to net transport generation by breaking symmetries of quantum systems. Moreover, we propose the multibaker map as a resource to study directed transport phenomena in chaotic systems without bias. In fact, this is a paradigmatic model in classical and quantum chaos, but also in statistical mechanics.