126
R. Röhlsberger and J. Evers
photon field have already be demonstrated [69]. Even more interesting it is to consider
a 0-dimensional confinement of the atoms, e.g., in a 3D cavity. This will be practically
impossible for x-rays, but has been demonstrated with microcavities in the optical
regime [70].
The cavity is an ideal laboratory to study features of cooperative emission. In the
following we exploit this to study the dependence of the CLS on the size of the sample.
For that purpose we increased the thickness of the resonant layer while keeping the
areal density of the resonant atoms constant. Calculations of the cavity reflectivity
for an extended ensemble of atoms distributed over the standing wave within a 3rd
order guided mode are shown in Fig. 3.8a. A close inspection reveals two prominent
features: First, one observes a sharp dip in the reflectivity spectrum at the exact
resonance energy ( = 0). This structure is very reminiscent of the transparency dips
that appear in the phenomenon of electromagnetically induced transparency (EIT)
in quantum optics [71, 72]. As will be discussed in Sect. 3.7.2, there is indeed a
mechanism which leads to EIT in the case of nuclear resonant scattering from a cavity
that contains resonant atoms. Second, the CLS (determined from the center of gravity
of the curve), is a non-monotonous function of the thickness of the atomic ensemble
within the cavity. This behavior can be studied particularly well if higher-order
modes are employed where the resonant atoms can be distributed over a large range
of k R values within the cavity, see Fig. 3.8b. The results are displayed in Fig. 3.8c.
For comparison, we have used the function a + b(sin 2k R)/(2k R) (dashed lines) to
pinpoint the functional dependence of the oscillations in the CLS with increasing
sample size as it was predicted first in [31] and recently experimentally verified [45].
A rigorous theory to describe this behaviour on the basis of the cavity geometry used
here, however, still has to be developed.
3.6 Quantum Optics of Mössbauer Nuclei in X-Ray Cavities
The reflectance and spectral response of an x-ray cavity can be calculated using
different techniques (see Fig. 2 in [40]). One approach is Parratt’s formalism [73],
in which all possible scattering pathways arising from the material boundaries are
summed up. A generalization of this technique which enables one to include resonant multipole scattering with its polarization dependence has been formulated in a
transfer-matrix formalism (for an overview, see [6]). A numerical implementation
of this formalism is provided via the CONUSS software package [74]. An alternative approach involves the direct numerical integration of Maxwell’s equations, via
a finite-difference time-domain method [75]. The analysis so far, however, focused
to the case of linear light-matter interaction with classical light fields. Moreover,
these methods do not enable one to interpret the obtained spectra in terms of the
underlying nuclear dynamics. In the following, we therefore focus on a recently
developed quantum optical framework for the description of ensembles of nuclei
in x-ray cavities [35, 37]. The key idea of this approach is to relate the entire system comprising the x-ray cavity and the large ensemble of multi-level nuclei to that
R. Röhlsberger and J. Evers
photon field have already be demonstrated [69]. Even more interesting it is to consider
a 0-dimensional confinement of the atoms, e.g., in a 3D cavity. This will be practically
impossible for x-rays, but has been demonstrated with microcavities in the optical
regime [70].
The cavity is an ideal laboratory to study features of cooperative emission. In the
following we exploit this to study the dependence of the CLS on the size of the sample.
For that purpose we increased the thickness of the resonant layer while keeping the
areal density of the resonant atoms constant. Calculations of the cavity reflectivity
for an extended ensemble of atoms distributed over the standing wave within a 3rd
order guided mode are shown in Fig. 3.8a. A close inspection reveals two prominent
features: First, one observes a sharp dip in the reflectivity spectrum at the exact
resonance energy ( = 0). This structure is very reminiscent of the transparency dips
that appear in the phenomenon of electromagnetically induced transparency (EIT)
in quantum optics [71, 72]. As will be discussed in Sect. 3.7.2, there is indeed a
mechanism which leads to EIT in the case of nuclear resonant scattering from a cavity
that contains resonant atoms. Second, the CLS (determined from the center of gravity
of the curve), is a non-monotonous function of the thickness of the atomic ensemble
within the cavity. This behavior can be studied particularly well if higher-order
modes are employed where the resonant atoms can be distributed over a large range
of k R values within the cavity, see Fig. 3.8b. The results are displayed in Fig. 3.8c.
For comparison, we have used the function a + b(sin 2k R)/(2k R) (dashed lines) to
pinpoint the functional dependence of the oscillations in the CLS with increasing
sample size as it was predicted first in [31] and recently experimentally verified [45].
A rigorous theory to describe this behaviour on the basis of the cavity geometry used
here, however, still has to be developed.
3.6 Quantum Optics of Mössbauer Nuclei in X-Ray Cavities
The reflectance and spectral response of an x-ray cavity can be calculated using
different techniques (see Fig. 2 in [40]). One approach is Parratt’s formalism [73],
in which all possible scattering pathways arising from the material boundaries are
summed up. A generalization of this technique which enables one to include resonant multipole scattering with its polarization dependence has been formulated in a
transfer-matrix formalism (for an overview, see [6]). A numerical implementation
of this formalism is provided via the CONUSS software package [74]. An alternative approach involves the direct numerical integration of Maxwell’s equations, via
a finite-difference time-domain method [75]. The analysis so far, however, focused
to the case of linear light-matter interaction with classical light fields. Moreover,
these methods do not enable one to interpret the obtained spectra in terms of the
underlying nuclear dynamics. In the following, we therefore focus on a recently
developed quantum optical framework for the description of ensembles of nuclei
in x-ray cavities [35, 37]. The key idea of this approach is to relate the entire system comprising the x-ray cavity and the large ensemble of multi-level nuclei to that