X-ray-frequency modulation via periodic switching of an external magnetic field

Single x-ray photons can be resonantly scattered and stored with the help of suitable transitions in the atomic nucleus. Here, we investigate theoretically means of mechanical-free modulation for the frequency spectra of such x-ray photons via periodic switching of an external magnetic field. We show that periodically switching on and off an external magnetic field generating hyperfine splitting of the nuclear transition leads to the generation of equidistant narrow sidebands of the resonantly scattered response. This frequency-comb-like structure depends on the magnitude and orientation of the applied magnetic field and on the switching period. An analytical approach for the characterization of the comblike frequency spectrum is presented. The feasibility of the external control on the frequency modulation of the x-ray response is discussed in view of possible applications in high-resolution spectroscopy or quantum technology.

Figure 1

NFS setup. The x ray (orange line and arrow) propagating along the $z$ direction impinges on the Mössbauer target. The probe experiences a magnetic field $\mathbf{B}$ that can be switched off, retrieved, and can be rotated by $\pi$, inverting the quantization axis. The inset shows the level scheme of the ground $g$ and first excited $e$ state of the ${}^{57}\mathrm{Fe}$ sample including the magnetic sublevels.

Figure 2

(a) Time spectrum for a constant magnetic field and optical thickness $\xi =0.5$ (blue solid line) and in the absence of any hyperfine magnetic field with $\xi =0.25$ (orange dashed line). The peak at $t=10\mathrm{ns}$ shows the incoming Gaussian pulse. The magnetic field introduces an absolute-cosine-like oscillation known as quantum beat. (b) Corresponding frequency spectrum. In the absence of the magnetic field, the spectrum consists of a single peak at the nuclear transition frequency ${\omega }_0$. The magnetic field introduces the splitting into the two transition frequencies ${\omega }_{41}$ and ${\omega }_{32}$ separated by ${\mathrm{\Delta }}_{\mathcal{F}}$.

Figure 3

(a) Time and (b) frequency spectra for ${\tau }_{\mathrm{on}}={\tau }_{\mathrm{off}}=\frac{T_{QB}}{2}$ (thick red line). The periodically manipulated scattered field intensity is presented together with the case of a constant magnetic field $B_0=34.4\mathrm{T}$ (blue line) and of an approximate exponential decay in the absence of the magnetic field (dashed orange line). For the computations an optical thickness of $\xi =0.5$ was used.

Figure 4

Frequency spectra for different ${\tau }_{\mathrm{off}}$ values and constant ${\tau }_{\mathrm{on}}=T_{QB}/2$. With increasing storage time, the peaks multiply and narrow towards ${\omega }_0$, while the overall intensity drops. For illustration purposes, the spectrum for ${\tau }_{\mathrm{off}}=T_{QB}/2$ was manually downscaled by 40%.

Figure 5

(a) $\mathbf{E}$-field amplitude time spectrum for the inverting sequence at ${\tau }_{\mathrm{on}}={\tau }_{\mathrm{off}}=T_{QB}/2$ (thick red line). The magnetic-field orientation is inverted after every storage. (b) Comparison of the frequency spectra of the inverting and regular stepwise switching for the same choice of ${\tau }_{\mathrm{on}}$ and ${\tau }_{\mathrm{off}}$.

Figure 6

Frequency spectra in the case of the inverted stepwise switching scheme for several storage times ${\tau }_{\mathrm{off}}$ and constant ${\tau }_{\mathrm{on}}=T_{QB}/2$. For illustration purposes, the spectrum for ${\tau }_{\mathrm{off}}=T_{QB}/2$ was manually downscaled by 40%.

Figure 7

Basic components of the analytical model. The combination of the sinusoidal signal $sin\left({\omega }_{QB}t\right)$ given via the quantum beat and the periodic sequence $f$ determined by the $B$-field switching leads to the periodic function $P$.

Figure 8

(a) Peak distance ${\mathrm{\Delta }}_{\mathcal{F}}$ as a function of the storage time ${\tau }_{\mathrm{off}}$. Numeric data points are shown together with the analytic solution given by Eq. (13). (b) Frequency spectra for regular stepwise switching schemes with random variations in ${\tau }_{\mathrm{off}}$. For both graphs ${\tau }_{\mathrm{on}}=T_{QB}/2$ and $\xi =0.5$ have been considered.

Figure 9

(a) Magnetic-field modulations: stepwise (orange solid line), smoothed stepwise (green dash-dotted line) with $\varepsilon =0.5$, and sinusoidal (red dashed line). (b) Examples of smoothed stepwise functions as in Eq. (14) with $\varepsilon =0.01$ and $\varepsilon =0.1$.

Figure 10

(a) Time and corresponding frequency spectra (b) for abrupt (orange dashed line), smoothed (green dotted line), and sinusoidal (red dashed-dotted line) magnetic switching sequences in comparison to the case of a constant magnetic field (blue solid line). The smoothing radius for the hyperbolic function smoothed step was set to $\varepsilon =0.5$. For all cases we considered ${\tau }_{\mathrm{on}}={\tau }_{\mathrm{off}}=T_{QB}/2$ and $\xi =0.5$.