X-ray frequency combs from optically controlled resonance fluorescence

An x-ray pulse-shaping scheme is put forward for imprinting an optical frequency comb onto the radiation emitted on a driven x-ray transition, thus producing an x-ray frequency comb. A four-level system is used to describe the level structure of $N$ ions driven by narrow-bandwidth x rays, an optical auxiliary laser, and an optical frequency comb. By including many-particle enhancement of the emitted resonance fluorescence, a spectrum is predicted consisting of equally spaced narrow lines which are centered on an x-ray transition energy and separated by the same tooth spacing as the driving optical frequency comb. Given an x-ray reference frequency, our comb could be employed to determine an unknown x-ray frequency. While relying on the quality of the light fields used to drive the ensemble of ions, the model has validity at energies from the 100 eV to the keV range.

Figure 1

(a) An ensemble of ions is driven by narrow-bandwidth x rays (${\mathit{k}}_{\mathrm{X}}$, brown), an auxiliary optical laser (${\mathit{k}}_{\mathrm{L}}$, red), both linearly polarized along the $z$ direction, and an optical frequency comb (${\mathit{k}}_{\mathrm{C}}$, green), linearly polarized along the $x$ direction. All fields propagate in the $y$ direction. The resonance fluorescence spectrum (${\mathit{k}}_{\mathrm{F}}$, blue) exhibits an induced x-ray frequency-comb structure. (b) Four-level scheme of $\mathrm{He}$-like ions interacting with the three light fields.

Figure 2

Time evolution of the periodic density operator ${\widehat{\varrho }}^{\mathrm{eq}}\left(t\right)$ and spectrum of resonance fluorescence for ${\mathrm{Be}}^{2+}$ ions. Present-day parameters are used to model the optical frequency comb [Eq. (5)], $T_{\mathrm{FWHM}}=120\mathrm{fs}$, $T_{\mathrm{p}}=1\mathrm{ns}$, $1/T_{\mathrm{p}}=1\mathrm{GHz}$ [11, 36], i.e., $2\pi /T_{\mathrm{p}}=4.1\times {10}^{-6}\mathrm{eV}$. The ion sample has $N={10}^6$ particles over a length of $L=1\mathrm{cm}$ and area $1{\mathrm{mm}}^2$. The driving fields have intensities $I_{\mathrm{X}}=1.5\times {10}^4\mathrm{W}/{\mathrm{cm}}^2$, $I_{\mathrm{L}}=1.7\times {10}^8\mathrm{W}/{\mathrm{cm}}^2$, and $I_{\mathrm{C},\max }=3.0\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$, associated with $2\pi$ optical-frequency-comb pulses. The periodic solutions are (a) ${\varrho }_{4_{+}1}^{\mathrm{eq}}\left(t\right)$ for $n{T}_{\mathrm{p}}<t<n{T}_{\mathrm{p}}+T_{\mathrm{d}}$, with $T_{\mathrm{d}}=\pi T_{\mathrm{FWHM}}/\left[2\arccos \left(\sqrt[4]{1/2}\right)\right]$, and (b) ${\varrho }_{31}^{\mathrm{eq}}\left(t\right)$ for $n{T}_{\mathrm{p}}<t<(n+1)T_{\mathrm{p}}$. The power $P_m$ of each peak in the spectrum of Eq. (40) is displayed (c) for the whole comb, centered on ${\omega }_{41}=123.7\mathrm{eV}$, and (d) around the maximum. In panel (d), $a_1={10}^5{\mathrm{nW}}^{-1}$, $a_2=1.86\mathrm{nW}$.