  Multi-photon quantum interference is the underlying principle for optical
quantum information processing protocols. Indistinguishability is the key to
quantum interference. Therefore, the success of many protocols in optical
quantum information processing relies on the availability of photon states with
a well-defined spatial and temporal mode. Photons in single spatial mode can be
obtained from nonlinear processes in single-mode waveguides. For the temporal
mode, the common approach is to engineer the nonlinear processes. But it is
complicated because the spectral properties and the nonlinear interaction are
often intertwined through phase matching condition. In this paper, we study a
different approach which is based on an SU(1,1) nonlinear interferometer with a
pulsed pump and a controllable linear spectral phase shift for precise
engineering. We systematically analyze the important figures of merit such as
modal purity and heralding efficiency to investigate the feasibility of this
approach. Specifically, we analyze in detail the requirement on the spectral
phase engineering to optimize the figures of merit and apply numerical
simulations to a fiber system. Both modal purity and efficiency are improved
simultaneously. Furthermore, a novel multi-stage nonlinear interferometer is
proposed and shown to achieve more precise state engineering for near-ideal
single-mode operation and near-unity efficiency. We also extend the study to
the case of high gain in the four-wave mixing process for the spectral
engineering of quantum entanglement in continuous variables. Our investigation
provides a new approach for precisely tailoring the spectral property of
quantum light sources, especially, photon pairs can be engineered to
simultaneously possess the features of high purity, high collection efficiency,
high brightness, and high flexibility in wavelength and bandwidth selection.
