  Quantum entanglement is an integral part of quantum optics and has been
exploited in areas such as computation, cryptography and metrology. The
entanglement between photons can be present in various degrees of freedom
(DOFs), and even the simplest bi-partite systems can occupy a large Hilbert
space. Therefore, it is desirable to exploit this multi-dimensional space for
various quantum applications by fully controlling the properties of the
entangled photons in multiple DOFs. While current entangled-photon sources are
capable of generating entanglement in one or more DOFs, there is currently a
lack of practical techniques that can shape and control the entanglement
properties in multiple DOFs. Here we show that cascading two or more
entangled-photon sources with tunable linear media in between allows us to
generate photon-pairs whose entanglement properties can be tailored and shaped
in the frequency and polarisation domains. We first develop a quantum
mechanical model to study the quantum state generated from the cascade
structure with special considerations paid to the effects of pump temporal
coherence, linear dispersion, and in-structure polarisation transformation
applied between the entangled-photon sources. We then experimentally generate
photon-pairs with tunable entanglement properties by manipulating the
dispersion and birefringence properties of the linear medium placed in between
two entangled-photon sources. This is done in an all-fibre, phase stable, and
alignment-free configuration. Our results show that the cascade structure
offers a great deal of flexibility in tuning the properties of entangled
photons in multiple DOFs, opening up a new avenue in engineering quantum light
sources.
