# The population of the upper state versus the area of the pulse and the frequency spectrum of the pulse

**Figure 6.** The population of the upper state versus the area of the pulse and the frequency spectrum of the pulse. The pulse width parameter *T* is 0.7 *T*_{0} for (a) and (b), and is 0.2 *T*_{0} for (c) and (d). The area of the pulse is 2.38 π for (b) and 0.8 π for (d).

**Abstract**

The interactions of sub-cycle and single-cycle pulses with two- or three-level quantum systems are studied, respectively. For the two-level quantum system, two cases in which the carrier frequency of pulses is in resonance and far from resonance with the atom are analysed. The ultrafast complete population transfer can be obtained. The sub-cycle pulse with a far off-resonant carrier frequency is found to be more suitable for the population transfer. The relation between the area of pulses and population transfer is clarified in the sub-cycle and single-cycle domain. For the Lambda-type three-level quantum system, more than 90% of population transfer can be achieved from one level to another. The scheme is insensitive to the variation of the laser parameters such as the peak Rabi frequency and the carrier frequency for both two- and three-level quantum systems.