Bacteria such as Escherichia coli exhibit biased motion if kept in a spatially nonuniform chemical environment. Here, we bring out unique time-dependent characteristics of bacterial chemotaxis, in response to a diffusing spatial step ligand profile. Experimentally obtained temporal characteristics of the drift velocity are compared with the theoretical and Monte Carlo simulation-based estimates, and excellent agreements can be obtained. These results bring insights to the time-responsive facets of bacterial drift, bearing far-reaching implications in understanding their migratory dynamics in the quest of finding foods by swimming toward the highest concentration of food molecules, or for fleeing from poisons, as well as toward the better understanding of therapeutic response characteristics for certain infectious diseases.