Coherent excitation of a two-level atom driven by a far-off-resonant classical field: Analytical solutions

We present an analytical treatment of coherent excitation of a two-level atom driven by a far-off-resonant classical field. A class of pulse envelope is obtained for which this problem is exactly solvable. The solutions are given in terms of the Heun function, which is a generalization of the hypergeometric function. Degeneracy of the Heun to a hypergeometric equation can give all the exactly solvable pulse shapes of Gauss hypergeometric form from the generalized pulse shape obtained here. We discuss the application of the results obtained to the generation of soft x-ray and ultraviolet radiations.

Figure 1

(a) Two-level atomic system: atomic transition frequency $\omega ={\omega }_a-{\omega }_b$, detuning $\Delta =\omega -\nu$, and Rabi frequency $\Omega (t)=\wp \mathcal{E}(t)/2\hslash$. (b) Classical electromagnetic field $E(t)=\text{sech}(\alpha t)\text{cos}(\nu t).$

Figure 2

Pulse shapes given by Eq. (18). (a) Pulse shapes with varying $\lambda$ and $c=2$, $q=-1$, and $\mathit{ab}=0$. (b) Pulse shapes with varying $c$ and $\lambda =2$, $q=-1$, and $\mathit{ab}=0.$

Figure 3

Pulse shapes given by Eq. (31). (a) Pulse shapes with varying $\lambda$ and $p=-q=1$. (b) Pulse shapes with varying $q$, $\lambda =2$, and $p=0.$

Figure 4

(a) Pulse shapes for different values of $\delta$. (b) Time dependence of the population in state $|a\rangle$ for a ${\Omega }_{\delta }(\tau )$ pulse for different values of $\delta >1$. For calculation we take $\alpha =0.08{\omega }_c$, varying $\delta$.

Figure 5

(a) Pulse shapes for ${\Omega }_{\pm }(t)={\Omega }_0\text{sech}\alpha t(\sqrt{1\pm \text{tanh}\alpha t})$. (b) Time dependence of the population in state $|a\rangle$ for the pulse shapes in (a). In calculations we take ${\Omega }_0=0.02{\omega }_c$, $\alpha =0.08{\omega }_c$, and $\Delta =0.2{\omega }_c$.

Figure 6

(a) Box pulse for $\delta -1={10}^{-9}$. (b) Time dependence of the population in state $|a\rangle$ for the box pulse ${\Omega }_{\delta }(\tau )$. In calculations we take ${\Omega }_0=0.02{\omega }_c$, $\alpha =0.08{\omega }_c$, and $\Delta =0.2{\omega }_c$.

Figure 7

Field configuration and level structure of $\mathrm{H}$ or He${}^{+}$. All population is initially in the ground state $|b\rangle$. First, a strong, short, far-of-resonant pulse with frequency ${\nu }_1$ is applied to the system to excite coherence between level $|b\rangle =|1s\rangle$ and level $|c\rangle =|2s\rangle$, and then a second pulse, with frequency ${\nu }_3$, which is close to the transition between level $2s$ and level $2p$, is applied to generate a XUV pulse with a higher frequency ${\nu }_4$.

Figure 8

Two-stage generation scheme for x-ray generation. (a) Applying a strong pulse allows one to excite an atomic system by transferring the population to electronic excited states. (b) Coherence is then induced by applying a resonant field.