  Integrated entangled photon pair source is an essential resource for both
fundamental investigations and practical applications of quantum information
science. Currently there have been several types of entanglement, among which
the transverse-mode entanglement is becoming attractive because of its unique
advantages. Here, we report an on-chip transverse-mode entangled photon pair
source via the spontaneous four-wave mixing processes in a multimode silicon
waveguide. Transverse-mode photon pairs are verified over multiple frequency
channels within a bandwidth of $\sim$2~THz, and a maximally entangled Bell
state is also produced with a net fidelity of $0.96\pm0.01$. Our entangled
photon pair source is the key element for quantum photonics based on
transverse-mode, and also has the possibility to extend to higher-dimensional
Hilbert space. Furthermore, the transverse-mode entanglement can be converted
coherently to path and polarization entanglement, which paves the way to
realizing highly complex quantum photonic circuits with multiple degrees of
freedom.
