Dipole spectrum structure of nonresonant nonpertubative driven two-level atoms

We analyze the dipole spectrum of a two-level atom excited by a nonresonant intense monochromatic field under the electric dipole approximation and beyond the rotating wave approximation. We show that the apparently complex spectral structure can be completely described by two families: harmonic frequencies of the driving field and field-induced nonlinear fluorescence. Our formulation of the problem provides quantitative laws for the most relevant spectral features: harmonic ratios and phases, nonperturbative Stark shift, and frequency limits of the harmonic plateau. In particular, we demonstrate the locking of the harmonic phases at the wings of the plateau opening the possibility of ultrashort pulse generation through harmonic filtering.

Figure 1

Dipole spectrum, $|u(\omega )|{}^2$, of the driven two-level atom. (a) Spectrum at low field amplitude ($\left|\chi \right|/{\omega }_0=0.005$ and $\omega /{\omega }_0=0.1$); (b) spectrum at higher field amplitude ($\left|\chi \right|/{\omega }_0=1.5$ and $\omega /{\omega }_0=0.2$). (Inset) Detail of the dipole spectrum in (b) showing the structure of satellite peaks around the harmonics.

Figure 2

Dipole spectrum, $|u(\omega )|{}^2$, plotted for negative and positive frequencies in two different cases, (a) $\omega /{\omega }_0=0.1$, $\left|\chi \right|/{\omega }_0=0.5$, and (b) $\omega /{\omega }_0=0.2$, $\left|\chi \right|/{\omega }_0=1.5$. This latter one corresponds to the same case as in Fig. 1b, including the inset. The two families composing the spectrum have been labeled with solid black triangles (harmonic family) and solid green boxes (fluorescence family). The complex conjugate of this later is labeled with solid red circles.

Figure 3

(a) Plot of formula (16) for $\omega /{\omega }_0=0.1$ versus the Rabi frequency, resulting from the truncation of $M$ to $1\times 1$. ${\alpha }_{0,\pm }$ refer to the frequencies of the central peak in each of the two families composing the dipole spectrum. Since ${\alpha }_{0,-}$ corresponds to the harmonic family, it is practically insensitive to the intensity field. (b) Test of the analytical solution ${\alpha }_{0,+}$ against the exact numerical results from Eqs. (1) and (2) for $\omega /{\omega }_0=0.1$. The numerical data shows a good agreement with the analytical formula for the Stark shift although for large intensities small deviations appear. Parts (c) and (d) of this figure show the results for ${\alpha }_{0,\pm }$ resulting from the truncation of $M$ to $3\times 3$. The results are obtained numerically from the equation $det(M_{3\times 3})=0$ and are compared with (c) the exact numerical result also shown in part (b) of this figure and (d) with the results for ${\alpha }_{0,-}$ from the $1\times 1$ truncation shown in part (a) of this figure.

Figure 4

Dipole spectral intensities (a) and phases (b) for the strongly driven case with $\left|\chi \right|/{\omega }_0=4$ and $\omega /{\omega }_0=0.2$. The harmonic peak family is highlighted with open circles. The shadowed box encloses the harmonic plateau structure. The limits of this box have been defined using the expressions (22) and (24) for the plateau onset and cut-off frequencies. The arrow points to the plateau’s onset frequency in the weak driving limit, equal to ${\omega }_0$.

Figure 5

(Black-filled curve) Detail of the inverse Fourier transform of the higher frequency part of the dipole spectrum shown in Fig. 4. Frequencies below $34\omega$ are filtered out. Only the field envelope is represented. (Red line) Sketch of the amplitude of the driving field at the same time interval.

Figure 6

Frequencies for the plateau’s onset (red line) and cutoff (green line) derived from Eqs. (22) and (24) for the case $\omega /{\omega }_0=0.1$. The visual estimation from the numerically computed spectra are plotted with black and blue points, we have added an error bar of $\pm 2\omega$ as an error estimation of the method.

Figure 7

Scheme of the instantaneous eigenenergies of the two level atom during one driving field period. In the adiabatic limit, the maximum (a) and minimum (b) level differences define the energies of the plateau onset and cutoff, respectively.