X-ray quantum optics with Mössbauer nuclei embedded in thin-film cavities

A promising platform for the emerging field of x-ray quantum optics consists of Mössbauer nuclei embedded in thin-film cavities probed by near-resonant x-ray light, as used in a number of recent experiments. Here we develop a quantum optical framework for the description of experimentally relevant settings involving nuclei embedded in x-ray waveguides. We apply our formalism to two settings of current experimental interest based on the archetype Mössbauer isotope ${}^{57}$Fe. For the present experimental conditions, we derive compact analytical expressions and show that the alignment of medium magnetization, as well as incident and detection polarization, enable the engineering of advanced quantum optical level schemes. The model encompasses nonlinear and quantum effects which could become accessible in future experiments.

Figure 1

Schematic of the considered setup. The cavity contains a layer of resonant nuclei, as indicated in the inset. It is probed by hard x rays (red lines, $a_{\mathrm{in}}$) with propagation direction $\widehat{\mathbf{k}}$. The angle of incidence $\phi$ is of the order of a few mrad. The incident polarization in the (${\widehat{\mathbf{a}}}_1,{\widehat{\mathbf{a}}}_2$) plane (blue) together with the alignment of the magnetization ${\mathbf{B}}_{\mathrm{hf}}$ of the nuclei (green) sensitively determine the properties of the scattered light. Both light reflected from the cavity ($a_{\mathrm{out}}$) at output angle $\phi$ and light exiting the cavity on the front side ($b_{\mathrm{out}}$) are considered.

Figure 2

(a) Electronic reflectivity curve of a thin-film cavity, showing the reflectance as a function of the x-ray incidence angle $\phi$. The dips correspond to resonant excitations of guided modes of the waveguide. The cavity consists of a 2.6-nm Pt top layer, followed by 7.9 nm C, 1.5 nm ${}^{57}$Fe, 9.3 nm C, and a thick bottom layer of Pt. (b) Field intensity distributions inside the waveguide (without Fe layer for simplicity) for probing fields resonant to the first three guided modes, respectively. The three panels differ only in the incidence angle of the field and show a range of 1 mm in horizontal direction and of 50 nm in vertical direction. The white horizontal lines indicate the layer boundaries, and above the layer system, the standing wave formed by incident and reflected field is visible.

Figure 3

The Mössbauer transition in ${}^{57}$Fe. In the presence of a magnetic hyperfine field the two levels split up and six $M1$ transitions can be driven.

Figure 4

Reflectance of the cavity containing an unmagnetized ${}^{57}$Fe layer. The top panel shows ${\left|R\right|}^2$ as a function of the grazing incidence angle $\phi$. The nuclear part is strongly detuned such that only the electronic reflectivity curve is visible. Parameters are as in the main text and ${\phi }_0=2.96$ mrad. The dashed line corresponds to the reflectivity curve from Fig. 2 calculated by conuss. The bottom left panel shows the reflectance for fixed $\phi ={\phi }_0$. The narrow nuclear resonance is located in the center of the broad cavity resonance, where it appears as a sharp spike. The bottom right panel shows a magnification of the bottom left panel around the nuclear resonance. The nuclear spectrum is a Lorentzian which is significantly broadened due to superradiance.

Figure 5

Engineering of nuclear level schemes. Depending on the choice of the input polarization and the nuclear magnetization axes, different level schemes are obtained. The four rows show the cases (a) ${\widehat{\mathbf{a}}}_{\mathrm{in}}\parallel {\widehat{\mathbf{a}}}_{\mathrm{out}}\parallel {\widehat{\mathbf{B}}}_{\mathrm{hf}}$; (b) ${\widehat{\mathbf{a}}}_{\mathrm{in}}\parallel {\widehat{\mathbf{a}}}_{\mathrm{out}}\perp {\widehat{\mathbf{B}}}_{\mathrm{hf}}$; (c) ${\widehat{\mathbf{a}}}_{\mathrm{in}}\parallel {\widehat{\mathbf{a}}}_1-{\widehat{\mathbf{a}}}_2$, ${\widehat{\mathbf{a}}}_{\mathrm{out}}\parallel {\widehat{\mathbf{a}}}_1+{\widehat{\mathbf{a}}}_2$, ${\widehat{\mathbf{B}}}_{\mathrm{hf}}\parallel {\widehat{\mathbf{a}}}_2$; and (d) ${\widehat{\mathbf{a}}}_{\mathrm{in}}\parallel {\widehat{\mathbf{a}}}_1$, ${\widehat{\mathbf{a}}}_{\mathrm{out}}\parallel {\widehat{\mathbf{a}}}_2$, ${\widehat{\mathbf{B}}}_{\mathrm{hf}}\parallel {\widehat{\mathbf{a}}}_2+\widehat{\mathbf{k}}$. The left column shows the obtained level scheme, and the right column shows the corresponding reflectance. The excited states $|E_{\mu }^{(+)}\rangle$ are mutually coupled due to $H_{\mathrm{LS}}$ and ${\mathcal{L}}_{\mathrm{cav}}$ (red wavy arrows) and coherently probed by $H_{\Omega }$ (blue). Spontaneous decay channels and Lamb shifts are not shown in the level diagram for clarity. The two geometries in (c),(d) correspond to situations considered in [2], where only the nuclear part of $R$ contributes. The vertical lines in the reflectance plots indicate the resonance frequencies of the six transitions. Other parameters are as in Fig. 4.

Figure 6

The effective level system obtained if the linearly polarized transitions are driven by the probing field in the presence of a magnetic splitting. Collective Lamb shifts are not considered in the figure for clarity. (a) The collective ground state $|G\rangle$ is coherently coupled to the two possible excited states (solid blue arrows). Both states decay superradiantly (singly headed red wavy arrows) and are coupled via cross-decay terms (double-headed wavy arrow). (b) After a basis transition, only the symmetric state $|+\rangle$ is probed by the incident field. It is coupled to the antisymmetric state $|-\rangle$, which is metastable on the superradiantly accelerated decay time scale of $|+\rangle$ since it decays at the single-nucleus incoherent decay rate.