||The light wave propagating in a dispersive medium can be formularized as electromagnetic (EM) fields or quantized as a pair of Hermitian operators. The fascinating optical phenomena display classical and nonclassical properties, depending on whether they can be described by the classical EM field theory. A cold atomic ensemble with neligible inhomogenous Doppler broadening is ideal to explore the interaction between light and matter. A typical problem in classical pulse propagation is optical transients, which induced by rapid response of dispersive medium to the sudden change of the driving radiation fields. With cold atoms confined in the two-dimensional magneto-optical trap, whose optical depth is controllable in our system, optical precursor and free-induction decay both classified as optical transients are observed in a single experiment with different conditions. Moreover, the information velocity limit is verified in superluminal medium, with the observation and analysis of precursors. Characterized with narrow linewidth and long coherence time, paired photons generated from cold atomic ensemble in four-wave mixing process possess irreplaceable advantage in quantum optics area. Temporal modulation of narrow-band paired photons can be achieved by directly modulating the generated photons or by modulating the classical control fields. After successfully reshaping the temporal waveform of the paired photons, heralded single photon source is well prepared for studying the propagation of nonclassical single photon in dispersive medium. The speed limit imposed by Einstein Causality principle is firstly experimentally verified on single photon level.