We report the discovery of a faint H \alpha pulsar wind nebula ( PWN ) powered by the radio pulsar B0740 - 28 . The characteristic bow-shock morphology of the PWN implies a direction of motion consistent with the previously measured velocity vector for the pulsar . The PWN has a flux density more than an order of magnitude lower than for the PWNe seen around other pulsars , but , for a distance 2 kpc , it is consistent with propagation through a medium of atomic density n _ { H } \sim 0.25 cm ^ { -3 } , and neutral fraction of 1 % . The morphology of the PWN in the area close to the pulsar is distinct from that in downstream regions , as is also seen for the PWN powered by PSR B2224 + 65 . In particular , the PWN associated with PSR B0740–28 appears to close at its rear , suggesting that the pulsar has recently passed through a transition from low density to high density ambient gas . The faintness of this source underscores that deep searches are needed to find further examples of optical pulsar nebulae .