Low-intensity light detection necessitates high-responsivity photodetectors. To achieve this, we report ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}/\mathrm{InAs}/{\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$ quantum well (InAs QW) photo-field-effect-transistors (photo-FETs) integrated on a Si substrate using direct wafer bonding. Structure of the InAs QW channel was carefully designed to achieve higher effective mobility and a narrower bandgap compared with a bulk ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}$, while suppressing the generation of defects due to lattice relaxations. High-performance 2.6 nm InAs QW photo-FETs were successfully demonstrated with a high on/off ratio of ${10}^5$ and a high effective mobility of $2370{\mathrm{cm}}^2/(\mathrm{V}·\mathrm{s})$. The outstanding transport characteristics in the InAs QW channel result in an optical responsivity 1.8 times greater than InGaAs photo-FETs and the fast rising/falling times. Further, we experimentally confirmed that the InAs QW photo-FET can detect light in the short-wavelength infrared (SWIR; 1.0–2.5 μm) near 2 μm thanks to bandgap engineering through InAs QW structures. Our result suggests that the InAs QW photo-FET is promising for high-responsivity and extended-range SWIR photodetector applications.