  We theoretically investigate the spectral property of a biphoton state from
multiplexed thermal atomic ensembles. This biphoton state originates from the
cascade emissions, which can be generated by two weak pump fields under
four-wave mixing condition. Under this condition, a signal photon from the
upper transition, chosen in a telecommunication bandwidth, can be generated
along with a correlated idler photon from the lower infrared transition. We can
spectrally shape the biphoton state by multiplexing the atomic ensembles with
frequency-shifted emissions, where the entropy of entanglement can be analyzed
via Schmidt decompositions. We find that this spectral entanglement increases
when more vapor cells are multiplexed with correlated or anti-correlated signal
and idler fields. The eigenvalues in Schmidt bases approach degenerate under
this multiplexing scheme, and corresponding Schmidt numbers can be larger than
the number of the multiplexed vapor cells, showing the enlarged entropy of
entanglement and excess correlated modes in continuous frequency spaces. We
also investigate the lowest entropy of entanglement allowed in the multiplexing
scheme, which is preferential for generating a pure single photon source. This
shows the potentiality to spectrally shape the biphoton source, where
high-capacity spectral modes can be applied in long-distance quantum
communication and multimode quantum information processing.
