Results and Analysis

The resonant layer in the cavity consisted of ⁵⁷Fe enriched steel; the cladding was made out of Palladium, and the core of Carbon. The design parameters intended for the cavity to be perfectly symmetric, i.e. both cavities were supposed to have the same core sizes. But we have not managed to find a fit to the reflectivity measurements that confirms this. Therefore we analyze the data purely with the quantum mechanical model developed in ref. [58]. In Fig. 39 we show the reflectivity curve of the multilayer around the region of interest along with a quantum optical fit of the cavity parameters, which are given in Tab. V.5. The angular range over which we analyze the data is very small, and we will analyze only one spectrum, so we ignore modifications of the reflected intensity by beam size effects and the envelope of the reflectivity. To include the effects of the hyperfine magnetic field distribution and the possibility of isomer shift distributions etc., we have assumed that the nuclear resonance has a width of 4Γ0. In Fig. 40 we show the energy spectrum measured at 0.176^◦, along with a fit taking into account the time-gating detection. The figure also shows the calculated reflectivity, without taking into account the detection process.

The splitting is obvious, and has a width of about 24Γ0. The parameters used for the fit are given in Tab. V.6 . In Fig. 41 we show the measured temporal decay pattern of the cavity. The beating indicative of Rabi oscillations is clearly visible. We compare the result to the Fourier

Figure 39:Reflectivity of the double cavity around the first two supermodes along with a quantum optical simulation.

We have assumed a very slight divergence of8×10⁻⁶. The second supermode seems to overestimate the quality of the cavity, but this is possible; note that the resolution of the curve is not too good. Using these parameters in the fits of the spectra to come yields good results.

Figure 40:(a) Spectrum of the reflectivity at0.176^◦, along with a simulation taking into account the7ns time gating window. The agreement is good. (b) simulation of the bare spectrum without any time gating. The splitting is about24Γ0, meaningΩR≈12Γ0.

transformation of the bare spectrum in Fig. 40 and a standard exponentially damped cosine with the frequencyΩR. Both models adequately predict the periodicity of the beating, although there are some shortcomings in the description of the intensities. In the case of the damped cosine, it

4. Rabi Oscillations in a double cavity system 95 Mode 1 Mode 2

g₁[10⁵Γ0] 5.508 5.508 g2[10⁵Γ0] 6.05 6.05

Table V.6:Coupling strength parameters for the supermodes.

clearly does not take into account the additional dephasing that stems from the hyperfine magnetic field distributions, therefore the calculated interference fringes are much more visible. The full quantum optical model mimics these distribution effects via the interactions strength parameters, and therefore does a much better job of predicting the intensities from about 20 ns after the excitation on; but it fails to predict the intensity at the beginning of the spectrum. Part of that might be just due to the leakage associated with the Fourier transform; but it is also conceivable that this is a systematic error that results from discrepancies between the energy spectrum and the simulation. Note that in Fig. 40 the dip in the spectrum at zero detuning is not accurately reproduced. In the bare spectrum, this manifests itself as a peak, which should be higher. This could be interpreted as a kind of superradiance that comes from interference between the two normal modes; the superradiant part at the beginning of the temporal decay pattern would be larger if this peak were larger.

We now examine whether the splitting and the decay oscillations we have observed is due to the collective Lamb shifts or the interaction, and whether the strong coupling condition is met (i.e.

whether the splitting is larger than the superradiant decay enhancement of the eigenmodes). In Fig. 42 we plot the superradiant decay enhancements, Lamb shifts, and interaction strengths.

The results are clear. At the angular position where we have measured the spectrum (indicated by the black line), the Rabi frequency is almost exclusively a result of the interaction strength between the layers, since the Lamb shifts are both almost zero there. Hence, the splitting is a genuine splitting due to the strong coupling of two ensembles mediated by several cavity modes andΩR=_Ωc=_12Γ0. Furthermore, as can be seen in Fig. 42, this is larger than the superradiant decay widths. We therefore expect a strong signal showing Rabi oscillations, which are due to the exchange of the excitation populations between the two layers. We conclude that we have achieved strong coupling between two nuclear ensembles, and observed the coherent exchange of population between them in the form of Rabi oscillations. The splitting has a magnitude of 24Γ0≈160 MHz. It is interesting that this is almost in the range where microwave cavities become available commercially. This offers the tantalizing opportunity to further manipulate this system with microwave photons, and perform quantum optics by interfacing x-rays and microwaves via

Figure 41:Rabi oscillations in the double cavity system.ΩR=_Ωc=_12Γ0. The cosine is from the calculation of the Rabi oscillations; The quantum optical simulation is a Fourier transform of the energy spectrum calculated from the quantum optical model. Deviations are probably due to the hyperfine fields of the layers not being properly implemented.

nuclear ensembles.

A more immediately accessible benefit is that this system offers the opportunity of designing and tuning three-level systems in the hard x-ray range. Suppose, for instance, that one ensemble was coupled to the cavity more strongly then the other. This can be achieved almost effortlessly through layer placement, or by making one layer thinner, reducing the collective coupling strength.

Upon tuning the angle, one layer then experiences a greater Lamb shift then the other, and also a greater superradiant decay enhancement. Right at the center between two cavity minima, the Lamb shifts would cancel each other anyway, in what would appear as an anti - crossing, because of the interaction strength. Closer to the cavity minima, one branch would appear very far detuned and superradiant, with the other barely detuned and with a normal decay time, the interaction between them remaining. The whole setup would effectively turn into an artificial three-levelΛ - atom with tunable level energies, decay times and interactions between the upper levels. And of course there remains the opportunity to add more degrees of freedom by using the multiple hyperfine levels available inα−⁵⁷Feand adding more cavities. This brings closer to creating the

4. Rabi Oscillations in a double cavity system 97 foundations for some of the more complicated proposals advanced by theorists in recent years for the coherent control of x-rays. Naturally, the principle can be extended to the electronic resonances we have discussed in Chapter V.2, see for reference Fig. 43.

Figure 42:(a) Interaction strengths between the layers vs. superradiant decay enhancements. The vertical black line marks the position at which the data were taken. The interaction between the nuclear ensembles has a frequency of about24Γ0, obviously much stronger than the decay at the position where the reflectivity spectrum was taken. In (b) it is obvious that the collective Lamb shifts are minuscule at that angle as well, as laid out in the theory section. In (c) finally we compare the Rabi frequency that is due to the interaction strength to that due to the detunings. Obviously the interaction strength contributes the lion’s share to the Rabi frequency.

4. Rabi Oscillations in a double cavity system 99

Figure 43:(a) Parratt-simulated reflectivity of a double cavity with the dimensions 2 nm Pt/20 nm C/1 nm Ta/20 nm C/1 nm Pt/20 nm C/1 nm Ta/20 nm C/20 nm Pt. In the center between the two supermodes, marked by the black line, the matter-like parts of the excitation spectra are split by the effective interaction. The detailed spectrum is shown in (b).