Two-level-atom excitation probability for single- and $N$-photon wave packets

We study how the transient excitation probability of a two-level atom by a quantized field depends on the temporal profile of the incident pulse, in the presence of external losses, for both coherent and Fock states, and in two complementary limits: when the pulse contains only one photon (on average), and when the number of photons $N$ is large. For the latter case we derive analytical expressions for the scaling of the excitation probability with $N$ that can be easily evaluated for any pulse shape.

Figure 1

Optimized excitation probability for various single-photon pulse shapes, as a function of the ratio of coupling ${\mathrm{\Gamma }}_P$ to coupling plus losses, $\mathrm{\Gamma }={\mathrm{\Gamma }}_P+{\mathrm{\Gamma }}_B$. From top to bottom, the lines are for a rising exponential pulse, a Gaussian pulse modified by interacting with an atom in a single-sided cavity, a “square” pulse, an ordinary Gaussian pulse, a pulse emitted by an (initially excited) atom in a cavity, and a decaying exponential pulse.

Figure 2

Optimized excitation probability for various pulse shapes, as a function of the ratio ${\mathrm{\Gamma }}_P/({\mathrm{\Gamma }}_P+{\mathrm{\Gamma }}_B)$, for coherent states with $\overline{n}=1$. From top to bottom, the curves are for a rising exponential pulse, a “square” pulse, a Gaussian pulse, and a decaying exponential pulse.

Figure 3

Optimized excitation probability for multiphoton coherent states, for various pulse shapes, as a function of the average photon number $\overline{n}$, in the absence of external losses (${\mathrm{\Gamma }}_B=0$). From top to bottom, the curves are for a rising exponential pulse, a “square” pulse, a Gaussian pulse (these two are virtually indistinguishable on this scale), and a decaying exponential pulse. The solid lines show the analytical approximation, and the dashed lines the results of numerical calculations.

Figure 4

Optimized excitation probability for multiphoton Fock states, for various pulse shapes, as a function of the photon number $N$, in the absence of external losses (${\mathrm{\Gamma }}_B=0$). From top to bottom, the curves are for a rising exponential pulse, a “square” pulse, a Gaussian pulse (these two are virtually indistinguishable on this scale), and a decaying exponential pulse. The solid lines show the analytical approximation, and the dashed lines the results of numerical calculations.