  Ultra-short pulses propagating in nonlinear nanophotonic waveguides can
simultaneously leverage both temporal and spatial field confinement, promising
a route towards single-photon nonlinearities in an all-photonic platform. In
this multimode quantum regime, however, faithful numerical simulations of pulse
dynamics na\"ively require a representation of the state in an exponentially
large Hilbert space. Here, we employ a time-domain, matrix product state (MPS)
representation to enable efficient simulations by exploiting the entanglement
structure of the system. In order to extract physical insight from these
simulations, we develop an algorithm to unravel the MPS quantum state into
constituent temporal supermodes, enabling, e.g., access to the phase-space
portraits of arbitrary pulse waveforms. As a demonstration, we perform exact
numerical simulations of a Kerr soliton in the quantum regime. We observe the
development of non-classical Wigner-function negativity in the solitonic mode
as well as quantum corrections to the semiclassical dynamics of the pulse. A
similar analysis of $\chi^{(2)}$ simultons reveals a unique entanglement
structure between the fundamental and second harmonic. Our approach is also
readily compatible with quantum trajectory theory, allowing full quantum
treatment of propagation loss and decoherence. We expect this work to establish
the MPS technique as part of a unified engineering framework for the emerging
field of broadband quantum photonics.
