Effects of free-electron-laser field fluctuations on the frequency response of driven atomic resonances

We study the effects of field fluctuations on the total yields of Auger electrons, obtained in the excitation of neutral atoms to a core-excited state by means of short-wavelength free-electron-laser pulses. Beginning with a self-contained analysis of the statistical properties of fluctuating free-electron-laser pulses, we analyze separately and in detail the cases of single and double Auger resonances, focusing on fundamental phenomena such as power broadening and ac Stark (Autler-Townes) splitting. In certain cases, field fluctuations are shown to influence dramatically the frequency response of the resonances, whereas in other cases the signal obtained may convey information about the bandwidth of the radiation as well as the dipole moment between Auger states.

Figure 1

(a), (c) The dotted curves show two of the profiles $f_s\left(t\right)$ used in our simulations. (b), (d) A sample of two random spiky pulses, typically produced in a single realization of the algorithm discussed in Sec. 2, by superimposing Gaussian-correlated noise (${\sigma }_{\omega }^{-1}=7.2$ fs) with the deterministic profiles of (a) and (c). The solid curves in (a) and (c) show the average intensity $\langle I_s\left(t\right)\rangle /I_s^{\left(0\right)}$ on a sample of 1000 random spiky pulses.

Figure 2

Convergence of the algorithm discussed in Sec. 2. The ratio $\langle I_s{\left(t\right)}^r\rangle /{\langle I_s\left(t\right)\rangle }^r$ is plotted as a function of time, with the average taken over two different numbers of pulses (realizations). The thick grey (horizontal) lines refer to $r!$, which is the theoretically expected value of the ratio. Other parameters: ${\tau }_s=10$ fs, ${\sigma }_{\omega }^{-1}=2$ fs.

Figure 3

An atomic transition pertaining to a subvalence electron, driven by chaotic SASE-FEL pulses. The resulting inner-shell vacancy in the excited atom decays with rate ${\Gamma }_2$.

Figure 4

Single resonance driven by a Fourier-limited pulse. The Auger signal $Q_2$ is plotted as a function of the detuning ${\Delta }_s$. (a) Various values of the ratio ${\Omega }_s^{\left(0\right)}/{\Gamma }_2$ and for ${\tau }_s=3{\Gamma }_2^{-1}$. (b) Various values of ${\Gamma }_2{\tau }_s$ for fixed ${\Omega }_s^{\left(0\right)}=2{\Gamma }_2$. A close-up of (a) and (b) around the peak is shown in (c) and (d), respectively. The dashed curves show the best Lorentzian fits to the numerical data for (b) ${\Gamma }_2{\tau }_s=4$ and ${\Gamma }_2{\tau }_s=8$; (c) ${\Omega }_s^{\left(0\right)}=0.5{\Gamma }_2$ and ${\Omega }_s^{\left(0\right)}=4.0{\Gamma }_2$. Other parameters: Gaussian pulse profile, $T_f=32{\Gamma }_2^{-1}$.

Figure 5

Single resonance driven by stochastic pulses. Stochastic Auger signal averaged over 5000 stochastic pulses as a function of the detuning ${\Delta }_s$, for various values of the ratio ${\Omega }_s^{\left(0\right)}/{\Gamma }_2$ and (a) $\chi =1.67$; (b) $\chi =2.5$; (c) $\chi =5.0$; (d) $\chi =10.0$. For the sake of comparison, the signal for a Fourier-limited Gaussian pulse with intensity profile the same as $\langle I_s\left(t\right)\rangle$ is also shown for ${\Omega }_s^{\left(0\right)}=0.5{\Gamma }_2$ and ${\Omega }_s^{\left(0\right)}=2{\Gamma }_2$ (dashed curves). Other parameters: Gaussian profile $\langle I_s\left(t\right)\rangle$ with ${\tau }_s=3{\Gamma }_2^{-1}$, $T_f=32{\Gamma }_2^{-1}$, Gaussian-correlated noise.

Figure 6

Single resonance driven by stochastic pulses. Stochastic Auger signal averaged over 1000 stochastic pulses as a function of the detuning ${\Delta }_s$ for (a) Gaussian-correlated noise and various combinations of pulse durations and coherence times; (b) various types of fluctuations and ${\tau }_s=3{\Gamma }_2^{-1}$, ${\sigma }_{\omega }=10{\tau }_s^{-1}$. The curves of (a) correspond to the combinations (from 1 to 4) ${\tau }_s=3{\Gamma }_2^{-1}$ and ${\sigma }_{\omega }=10{\tau }_s^{-1}$; ${\tau }_s=4.5{\Gamma }_2^{-1}$ and ${\sigma }_{\omega }=10{\tau }_s^{-1}$; ${\tau }_s=3{\Gamma }_2^{-1}$ and ${\sigma }_{\omega }=20{\tau }_s^{-1}$; ${\tau }_s=4.5{\Gamma }_2^{-1}$ and ${\sigma }_{\omega }=20{\tau }_s^{-1}$, respectively. The degrees of first-order coherence for the fluctuations chosen in (b) are shown in Table . For the sake of comparison, we also plot the signal corresponding to a phase-diffusion model (PDM) [4, 5] with $\gamma =2\sqrt{2\ln \left(2\right)}{\sigma }_{\omega }$. Other parameters: Gaussian profile for $\langle I_s\left(t\right)\rangle$, ${\Omega }_s^{\left(0\right)}=2{\Gamma }_2$, $T_f=32{\Gamma }_2^{-1}$.

Figure 7

Double Auger resonance driven by a short-wavelength SASE-FEL pulse (lower transition), and a Fourier-limited pulse (upper transition). The subvalence vacancy in the excited atom decays via two distinct Auger channels with rates ${\Gamma }_2$ and ${\Gamma }_3$.

Figure 8

DR arrangement I with stochastic probe and Fourier-limited pump. The average stochastic signal is plotted as a function of the probe detuning ${\Delta }_s$ for increasing values of $\chi$, and for (a) resonant (${\Delta }_d=0$) and (b) off-resonant (${\Delta }_d=2{\Gamma }_2$) pump. The thin dashed curves correspond to a Fourier-limited probe (i.e., $\chi \ll 1$). Synchronized Gaussian profiles have been chosen for the average probe intensity $\langle I_s\left(t\right)\rangle$ as well as for the pump. Other parameters: ${\tau }_s=4.5{\Gamma }_2^{-1}$, ${\tau }_d=6{\Gamma }_2^{-1}$, ${\Omega }_s^{\left(0\right)}=0.1{\Gamma }_2$, ${\Omega }_d^{\left(0\right)}=10{\Gamma }_2$, ${\Gamma }_3={\Gamma }_2$, $T_f=32{\Gamma }_2^{-1}$, 5000 realizations, Gaussian-correlated noise.

Figure 9

DR arrangement I with stochastic probe and Fourier-limited pump. (a)–(c) The separation of the peaks in the average stochastic signal $\langle Q_2\rangle$, normalized to the separation in the signal for Fourier-limited probe of the same shape (approximately equal to $19.2{\Gamma }_2$), as a function of ${\chi }^{-1}$ for various noises. (d)–(f) The depth of the doublet in the average stochastic signal [given by Eq. (28)], as a function of ${\chi }^{-1}$ for various noises. Other parameters as in Fig. 8.

Figure 10

DR arrangement I with stochastic probe and Fourier-limited pump. (a) The FWHM of the two peaks in the average stochastic signal $\langle Q_2\left({\Delta }_s\right)\rangle$ is plotted as a function of $\chi$ for various types of noises. Other parameters as in Fig. 8. (b) As in (a), but for Gaussian-correlated noise only, and various combinations of atomic and field parameters. Circle: Gaussian profile for $\langle I_s\left(t\right)\rangle$, ${\tau }_s=4.5{\Gamma }_2^{-1}$, ${\Gamma }_3={\Gamma }_2$; square: Gaussian profile for $\langle I_s\left(t\right)\rangle$, ${\tau }_s=4.5{\Gamma }_2^{-1}$, ${\Gamma }_3=0.5{\Gamma }_2$; star: Gaussian profile for $\langle I_s\left(t\right)\rangle$, ${\tau }_s=3{\Gamma }_2^{-1}$, ${\Gamma }_3={\Gamma }_2$; triangle: profile 2 for $\langle I_s\left(t\right)\rangle$, ${\tau }_s=4.5{\Gamma }_2^{-1}$, ${\Gamma }_3={\Gamma }_2$. Other parameters: Gaussian pump pulse, ${\tau }_d=6{\Gamma }_2^{-1}$, ${\Omega }_s^{\left(0\right)}=0.1{\Gamma }_2$, ${\Omega }_d^{\left(0\right)}=10{\Gamma }_2$, $T_f=32{\Gamma }_2^{-1}$, 5000 realizations.

Figure 11

DR arrangement II with stochastic pump and Fourier-limited probe. The average stochastic signal is plotted as a function of the probe detuning ${\Delta }_d$, for decreasing values of ${\chi }^{-1}$, and for pump detunings (a) ${\Delta }_s=0$; (b) ${\Delta }_s=0.1{\Gamma }_2$; (c) ${\Delta }_s=2{\Gamma }_2$. The thin dashed curves correspond to a Fourier-limited pump (i.e., $\chi \ll 1$). Synchronized Gaussian profiles have been chosen for the average pump intensity $\langle I_s\left(t\right)\rangle$ as well as for the probe. Other parameters: ${\tau }_s=6{\Gamma }_2^{-1}$, ${\tau }_d=3{\Gamma }_2^{-1}$, ${\Omega }_s^{\left(0\right)}=10{\Gamma }_2$, ${\Omega }_d^{\left(0\right)}=0.1{\Gamma }_2$, ${\Gamma }_3={\Gamma }_2$, $T_f=32{\Gamma }_2^{-1}$, Gaussian-correlated noise.

Figure 12

DR arrangement I with Fourier-limited pump and probe. The Auger signal $Q_2$ is plotted as a function of the detuning ${\Delta }_s$, for two different ratios of ${\Omega }_d^{\left(0\right)}/{\Gamma }_2$ and pump detunings (a) ${\Delta }_d=0$; (b) ${\Delta }_d=2{\Gamma }_2$. The dashed curves correspond to synchronized Gaussian probe and pump pulses with durations ${\tau }_s=4.5{\Gamma }_2^{-1}$ and ${\tau }_d=6{\Gamma }_2^{-1}$, respectively. The solid curves correspond to constant pump and probe fields lasting for time equal to $T_f$. For the sake of comparison, the signal has been normalized to its maximum value. Other parameters: ${\Omega }_s^{\left(0\right)}=0.1{\Gamma }_2$, ${\Gamma }_3={\Gamma }_2$, $T_f=32{\Gamma }_2^{-1}$.

Figure 13

DR arrangement I with Fourier-limited pump and probe. (a)–(c) Signal $Q_2$ at the end of the sequence of probe and pump pulses shown in the upper panel, as a function of the detuning ${\Delta }_s$. The probe follows the profile 2, discussed in Sec. 2, and its FWHM is approximately $7.5{\Gamma }_2^{-1}$. The pump pulse is taken Gaussian with duration (a) ${\tau }_d=3{\Gamma }_2^{-1}$; (b), (c) ${\tau }_d=6{\Gamma }_2^{-1}$. For (a) and (b) the probe is delayed relative to the pump by approximately $0.6{\Gamma }_2^{-1}$, whereas for (c) the probe precedes by approximately $4{\Gamma }_2^{-1}$. Other parameters: ${\Omega }_s^{\left(0\right)}=0.1{\Gamma }_2$, ${\Omega }_d^{\left(0\right)}=10{\Gamma }_2$, ${\Gamma }_3={\Gamma }_2$, ${\Delta }_d=0$, $T_f=32{\Gamma }_2^{-1}$.

Figure 14

DR arrangement II with Fourier-limited pump and probe. The Auger signal $Q_3$ is plotted as a function of the detuning ${\Delta }_d$, for two different ratios of ${\Omega }_s^{\left(0\right)}/{\Gamma }_2$, and for pump detunings (a) ${\Delta }_s=0$; (b) ${\Delta }_s=0.1{\Gamma }_2$. The dashed curves correspond to Gaussian probe and pump pulses with durations ${\tau }_d=3{\Gamma }_2^{-1}$ and ${\tau }_s=6{\Gamma }_2^{-1}$, respectively. The solid curves correspond to constant pump and probe fields lasting for time equal to $T_f$. For the sake of comparison, the signal has been normalized to its maximum value. Other parameters: ${\Omega }_d^{\left(0\right)}=0.1{\Gamma }_2$, ${\Gamma }_3={\Gamma }_2$, $T_f=32{\Gamma }_2^{-1}$.