Resonance fluorescence in ultrafast and intense x-ray free-electron-laser pulses

The spectrum of resonance fluorescence is calculated for a two-level system excited by an intense, ultrashort x-ray pulse made available, for instance, by free-electron lasers such as the Linac Coherent Light Source. We allow for inner-shell hole decay widths and destruction of the system by further photoionization. This two-level description is employed to model neon cations strongly driven by x rays tuned to the $1s2p^{-1}\to 1s^{-1}2p$ transition at $848\mathrm{eV}$; the x rays induce Rabi oscillations which are so fast that they compete with Ne $1s$-hole decay. We predict resonance fluorescence spectra for two different scenarios: first, chaotic pulses based on the self-amplified spontaneous emission principle, like those presently generated at x-ray free-electron-laser facilities and, second, Gaussian pulses which will become available in the foreseeable future with self-seeding techniques. As an example of the exciting opportunities derived from the use of seeding methods, we predict, in spite of the above obstacles, the possibility to distinguish at x-ray frequencies a clear signature of Rabi flopping in the spectrum of resonance fluorescence.

Figure 1

An atom ensemble (red) is driven by x rays (blue) tuned to a resonance. The emitted photons (green) are measured perpendicularly to the propagation direction of the x rays.

Figure 2

Two-level model used to describe the coherent interaction between Ne${}^{+}$ and the external driving field tuned to the $1s2p^{-1}\to 1s^{-1}2p$ transition at $848\mathrm{eV}$ [52]. The ground state $1s2p^{-1}$ is given by $|1\rangle =|L=1,M_L=0\rangle$ and the core-excited state $1s^{-1}2p$ is written as $|2\rangle =|L=0,M_L=0\rangle$. The external field is linearly polarized along the $z$ direction and induces Rabi flopping between these states. Spontaneous decay, however, also allows the core-excited state to decay to valence-ionized states with $M_L=\pm 1$.

Figure 3

Time evolution of the population of a two-level system driven by a Gaussian x-ray pulse (36) of peak intensity $I_{\mathrm{G}}=2.6\times {10}^{17}\mathrm{W}/{\mathrm{cm}}^2$ and FWHM duration ${\tau }_{\mathrm{G}}=5\mathrm{fs}$, corresponding to a pulse area of $Q_{\mathrm{G}}=14\pi$. The dashed lines show the total population of the two-level system ${\rho }_{11}\left(t\right)+{\rho }_{22}\left(t\right)$ [Eq. (9)] in the absence (black line) and presence (red line) of Auger decay. The solid lines show the corresponding occupation of the excited state ${\rho }_{22}\left(t\right)$.

Figure 4

Energy spectrum of resonance fluorescence for the Gaussian pulse used in Fig. 3 in the absence (black, dashed line) and presence (red, solid line) of Auger decay. As indicated by the arrows, the scale on the left refers to the black, dashed curve, whereas the scale on the right refers to the red, solid curve.

Figure 5

Time evolution of the population of the excited state ${\rho }_{22}\left(t\right)$ for a two-level system driven by Gaussian x-ray pulses [Eq. (36)] of different peak intensities (shown in the legend) and a FWHM duration ${\tau }_{\mathrm{G}}=2\mathrm{fs}$. In panel (a) pulse areas $Q_{\mathrm{G}}=2\pi (n-1/2)$, for $n=1$ (red, dotted line), $n=2$ (black, dashed line) and $n=3$ (green, solid line) are used. In panel (b) pulse areas $Q_{\mathrm{G}}=2\pi n$, for $n=1$ (red, dotted line), $n=2$ (black, dashed line), and $n=3$ (green, solid line) are used.

Figure 6

Spectrum of resonance fluorescence of a two-level system driven by Gaussian x-ray pulses of different peak intensities (shown in the legend) and a FWHM duration ${\tau }_{\mathrm{G}}=2\mathrm{fs}$. Line styles of panels (a) and (b) as in Fig. 5.

Figure 7

(a) Total emitted energy $\mathcal{E}={\int }_{-\infty }^{+\infty }{S}_z(\omega ,\Omega )d\omega$ and (b) peak value of the spectrum $S\left({\omega }_{\mathrm{X}}\right)$ multiplied by $\pi {\Gamma }_{\mathrm{A}}/2$ as functions of the normalized pulse area $Q_{\mathrm{G}}/\left(2\pi \right)$ [Eq. (37)] for ${\tau }_{\mathrm{G}}=2\mathrm{fs}$ (red, dashed line), ${\tau }_{\mathrm{G}}=5\mathrm{fs}$ (black, dotted line), and ${\tau }_{\mathrm{G}}=10\mathrm{fs}$ (green, solid line).

Figure 8

Average resonance fluorescence spectrum for 500 different realizations of the Gaussian driving pulses [Eq. (36)]. The mean duration of the pulses is ${\overline{\tau }}_{\mathrm{G}}=7\mathrm{fs}$, and the mean peak intensity is ${\overline{I}}_{\mathrm{G}}=7\times {10}^{17}\mathrm{W}/{\mathrm{cm}}^2$. Here ${\tau }_{\mathrm{G}}$ and $I_{\mathrm{G}}$ are Gaussian random variables independently chosen for each realization with a variance equal to the $20\%$ of the respective mean values.

Figure 9

(a) The Rabi frequency ${\Omega }_{\mathrm{R}}\left(t\right)$ induced by the amplitude of a SASE pulse and (b) the phase ${\varphi }_{\mathrm{X}}\left(t\right)$ of the SASE pulse (red, solid lines) and their mean value (black, dashed lines). The mean pulse has a duration ${\tau }_{\mathrm{env}}=6.5\mathrm{fs}$ and a peak intensity $I=3.8\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$. Its bandwidth is $\Delta {\omega }_{\mathrm{SASE}}=6\mathrm{eV}$.

Figure 10

Time evolution of a two-level system driven by a SASE pulse with the time-dependent Rabi frequency of Fig. 9a. The phase is assumed to be (a) constant, ${\varphi }_{\mathrm{X}}\left(t\right)=0$, or (b) to be equal to the phase of Fig. 9b. The red, dashed line shows the evolution of the total population ${\rho }_{11}\left(t\right)+{\rho }_{22}\left(t\right)$ [Eq. (9)]; the black, solid line represents the occupation of the excited state ${\rho }_{22}\left(t\right)$.

Figure 11

Time evolution of a two-level system driven by SASE pulses generated with the PCM described in Appendix . In both cases the pulses have a mean duration ${\tau }_{\mathrm{env}}=6.5\mathrm{fs}$ and a bandwidth $\Delta {\omega }_{\mathrm{SASE}}=6\mathrm{eV}$. The peak intensity is (a) $I=3.8\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $I=8.8\times {10}^{17}\mathrm{W}/{\mathrm{cm}}^2$. The red, dashed line shows the evolution of the total population ${\rho }_{11}\left(t\right)+{\rho }_{22}\left(t\right)$ [Eq. (9)]; the black, solid line represents the occupation of the excited state ${\rho }_{22}\left(t\right)$.

Figure 12

Resonance fluorescence spectrum for SASE pulses. The black, dotted line shows the spectrum from a Gaussian pulse with FWHM duration ${\tau }_{\mathrm{G}}=6.5\mathrm{fs}$ and peak intensity $I_{\mathrm{G}}=3.8\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$. The red, solid line is the arithmetic mean over 1000 SASE pulses with average peak intensity $I_{\mathrm{G}}$, FWHM duration ${\tau }_{\mathrm{G}}$, and a bandwidth of $\Delta {\omega }_{\mathrm{SASE}}=6\mathrm{eV}$. The green, dashed line is for the pulse in Fig. 9.

Figure 13

Average resonance fluorescence spectrum over SASE pulses. The pulses have an average peak intensity of $I=1.6\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$ and bandwidth $\Delta {\omega }_{\mathrm{SASE}}=6\mathrm{eV}$. The red, solid line shows the average over SASE pulses with average FWHM duration of ${\tau }_{\mathrm{env}}=6.5\mathrm{fs}$; the black, dotted line is associated with pulses of average FWHM duration of ${\tau }_{\mathrm{env}}=2\mathrm{fs}$.