  Parametric down-conversion (PDC) is the established standard for the
practical generation of a multiplicity of quantum optical states. These include
two-mode squeezed vacuum, heralded non-Gaussian states and entangled photon
pairs. Detailed theoretical studies provide insight into the time-frequency
(TF) structure of PDC, which are governed by the complex-valued joint spectral
amplitude (JSA) function. However in experiments, the TF structure of PDC is
mostly characterised by intensity measurementsthat forbid access to the
important phase of the JSA. In this paper, we present an amplitude-sensitive
quantum process tomography technique that combines methods from ultrafast
optics and classical three-wave mixing. Our approach facilitates a direct and
phase-sensitive time-frequency tomography of PDC with high spectral resolution
and excellent signal-to-noise ratio. This is important for all quantum optical
applications, which rely on engineered parametric processes and base on minute
knowledge of the quantum wave-function for the generation of tailored photonic
quantum states.
