  Hybrid interfaces between distinct quantum systems play a major role in the
implementation of quantum networks. Quantum states have to be stored in
memories to synchronize the photon arrival times for entanglement swapping by
projective measurements in quantum repeaters or for entanglement purification.
Here, we analyze the distortion of a single photon wave packet propagating
through a dispersive and absorptive medium with high spectral resolution.
Single photons are generated from a single In(Ga)As quantum dot with its
excitonic transition precisely set relative to the Cesium D1 transition. The
delay of spectral components of the single photon wave packet with almost
Fourier-limited width is investigated in detail with a 200 MHz narrow-band
monolithic Fabry-P\'erot resonator. Reflecting the excited state hyperfine
structure of Cesium, "slow light" and "fast light" behavior is observed. As a
step towards room-temperature alkali vapor memories, quantum dot photons are
delayed for 5 ns by strong dispersion between the two 1.17 GHz hyperfine-split
excited state transitions. Based on optical pumping on the hyperfine-split
ground states, we propose a simple, all-optically controllable delay for
synchronization of heralded narrow-band photons in a quantum network.
