Transient nuclear inversion by x-ray free electron laser in a tapered x-ray waveguide

The enhancement of x-ray-matter interaction by guiding and focusing radiation from x-ray free electron lasers is investigated theoretically. We show that elliptical waveguides using a cladding material with a high atomic number, such as platinum, can maintain an x-ray intensity of up to three orders of magnitude larger than in free space. This feature can be used to place a nuclear sample in the waveguide focal area and drive nuclear Mössbauer transitions up to transient nuclear population inversion. The latter is a long-standing goal related to gamma-ray lasers or nuclear state population control for energy storage. We show that inverted nuclei numbers of up to approximately $2\times {10}^5$ are achievable in the realistic region of longitudinal x-ray-free-electron-laser coherence time $\le 10$ fs. Our results anticipate the important role of tapered x-ray waveguides and strategically embedded samples in the field of x-ray quantum optics.

Figure 1

(a) A nuclear sample (yellow cylinder) is placed at the WG focal point (sketch not to scale). The intrawaveguide x-ray intensity is depicted below. (b) A bottle-like tapered WG with input radius $r_i$, output radius $r_o$ and focusing length $L_f$. (c) ${}^{57}\mathrm{Fe}$ nuclear level scheme. A linearly polarized XFEL pulse (blue arrows) drives the two $\mathrm{\Delta }m=0$ transitions. We label the two ground states with $|1\rangle$, $|2\rangle$ and the relevant excited states with $|3\rangle$ and $|4\rangle$, respectively.

Figure 2

(a) Intrawaveguide x-ray concentration ratio $R_c(r=0,z)$ (left vertical axis) and transmission $T$ (right vertical axis) of the CWG (blue empty squares and blue dashed line) and EWG (red dots and red dashed-dotted line) with common parameters ($r_i$, $L_f$, $r_o$, $\sigma$) = (150 nm, 160 $\mu \mathrm{m}$, 15 nm, 120 nm). (b) Contour plots of $R_c(r,z)$ for a Pt-cladding EWG. The red-upward arrow pinpoints the EWG focal point. White dashed lines indicate the platinum cladding and vacuum interface. The inset zooms in on the WG focal point. The white horizontal (vertical) arrow indicates the intrawaveguide Rayleigh length (focal spot size).

Figure 3

(a) Axial x-ray $R_c(0,z,t)$ for 14.4 keV x-ray propagation in a Pt-cladding EWG. (b) Spatial distribution of $I_v(r,z)$ driven by a propagating XFEL with ($n_p$, $\sigma$, $\tau$) = ($2.88\times {10}^{11}$, 120 nm, 42.5 fs). The maximum $I_v$ for ${}^{133}\mathrm{Ba}$ (unfilled red square), ${}^{57}\mathrm{Fe}$ (filled blue triangle), ${}^{187}\mathrm{Os}$ (filled yellow circle), and ${}^{169}\mathrm{Tm}$ (unfilled green diamond) is depicted as a function of the photon number per pulse in a (c) platinum- (d) silicon-cladding EWG. The solid lines are calculated using Eq. (8) with steady-state $R_c$. ($r_i$, $L_f$, $r_o$) = (150 nm, 160 $\mu \mathrm{m}$, 15 nm) are used for all graphs.

Figure 4

(a) $A_p$-dependent number of inverted ${}^{187}\mathrm{Os}$ (red solid line), ${}^{169}\mathrm{Tm}$ (green dashed line), ${}^{57}\mathrm{Fe}$ (blue dotted line), and ${}^{133}\mathrm{Ba}$ (orange dashed-dotted line) nuclei. The shaded regions depict the error bars averaged over 100 realizations. (b), (c) ${\tau }_c$-dependent number of inverted nuclei. The XFEL FWHM is 100 fs and $A_p=0$ for all cases. The shaded regions depict the error bars from 5000 SASE realizations, for illustration purposes scaled by a factor of 0.2. An EWG with ($r_i$, $L_f$, $r_o$) = (150 nm, 160 $\mu \mathrm{m}$, 15 nm) is used for both panels.