150
R. Röhlsberger and J. Evers
nuclear resonance. The splitting is then 4g
√
N f j (qρ), where f 1 (qρ) = cos(qρ/2)
and f 2 (qρ) = sin(qρ/2). Depending on whether ρ is smaller or larger than 0.25a, the
bands marked by j = 1 or j = 2 form the inner bands. We assume that ρ is smaller,
giving a splitting of 4g
√
N cos(qρ/2) ≈ 8 0 ≈ 57.3 MHz. Again, this is the result
for 30 layers and should not uncritically be equated with the formula resulting from
the infinite model. Nevertheless, the agreement between transfer matrix model simulations and measurements is excellent. An exception is the almost dispersionless gap
visible in Fig. 3.18a which could not be resolved experimentally since the energetic
width of the analyzer foil smears out sharp spectral features.
To summarize, we have simulated and measured the energy-resolved reflectivities
of an isotopic 1.64 nm
56 Fe/1.14 nm
57 Fe multilayer around the nuclear resonant
Bragg peak. The results were explained in terms of the polaritonic propagation of
light and excitations of nuclear resonant matter. Within a quantum optical model, we
could connect the dispersion to the coupling of x-rays with nuclear excitons and have
given a lower bound to the collective coupling strength. Dissipation and the interface
roughness that grow with the number of layers thwart any attempt to observe fullyformed band-gaps, and therefore precise values cannot be determined. However,
the distinctly observable dispersion and splitting of bands is the first unambiguous
evidence of collective strong coupling in the hard x-ray energy range.
Nuclear optical lattices display several unique features absent in other systems,
such as extremely high number densities (on the order of 10
28 m
−3 ) and stability over
a wide temperature range. This work could be extended to other isotopes with higher
energies and less electronic interaction, such as
119 Sn (23.9 keV) or
61 Ni (67.4 keV).
We therefore anticipate that nuclear resonant periodic multilayers will stimulate xray quantum optics research and bring it closer to coherent control of the x-ray-matter
interaction. Even further, the concept of polaritons in bichromatic optical lattices or
other periodic systems itself is a subject that holds great interest far beyond the x-ray
range.
To put our results into perspective we briefly summarize previous work in similar
systems, as given in [118]. To the best of our knowledge, two physical systems have
yielded phenomena and observations similar to those described here: genuine optical
lattices [104, 106, 107] and gratings of excitonic quantum wells. In excitonic quantum wells, a semiconductor is doped periodically; that way, the background refractive
index is identical throughout, but there are periodically spaced regions where quantum well excitations are possible [119]. This medium is particular interesting, since
it suffers from a similar drawback as ours: too few layers result in an unclear or
incomplete formation of band gaps. Since the early 1990s, the results obtained from
excitonic quantum wells have been described in a different framework. Instead of
assuming an infinite structure, researchers calculated the eigenmodes of these systems for a small number (∼10) of layers [120, 121]. In that case, the eigenmodes
are one superradiant Bragg mode, which reflects the radiation in a band much wider
than the exciton resonance, and a number of dark modes. In a sense, this is the
incipient Bragg band gap. However, experiments [108, 122] showed that with an
increasing number of layers, dips in the superradiant mode and a saturation of its
width appeared; researchers explained this later in terms of band gaps and standing
R. Röhlsberger and J. Evers
nuclear resonance. The splitting is then 4g
√
N f j (qρ), where f 1 (qρ) = cos(qρ/2)
and f 2 (qρ) = sin(qρ/2). Depending on whether ρ is smaller or larger than 0.25a, the
bands marked by j = 1 or j = 2 form the inner bands. We assume that ρ is smaller,
giving a splitting of 4g
√
N cos(qρ/2) ≈ 8 0 ≈ 57.3 MHz. Again, this is the result
for 30 layers and should not uncritically be equated with the formula resulting from
the infinite model. Nevertheless, the agreement between transfer matrix model simulations and measurements is excellent. An exception is the almost dispersionless gap
visible in Fig. 3.18a which could not be resolved experimentally since the energetic
width of the analyzer foil smears out sharp spectral features.
To summarize, we have simulated and measured the energy-resolved reflectivities
of an isotopic 1.64 nm
56 Fe/1.14 nm
57 Fe multilayer around the nuclear resonant
Bragg peak. The results were explained in terms of the polaritonic propagation of
light and excitations of nuclear resonant matter. Within a quantum optical model, we
could connect the dispersion to the coupling of x-rays with nuclear excitons and have
given a lower bound to the collective coupling strength. Dissipation and the interface
roughness that grow with the number of layers thwart any attempt to observe fullyformed band-gaps, and therefore precise values cannot be determined. However,
the distinctly observable dispersion and splitting of bands is the first unambiguous
evidence of collective strong coupling in the hard x-ray energy range.
Nuclear optical lattices display several unique features absent in other systems,
such as extremely high number densities (on the order of 10
28 m
−3 ) and stability over
a wide temperature range. This work could be extended to other isotopes with higher
energies and less electronic interaction, such as
119 Sn (23.9 keV) or
61 Ni (67.4 keV).
We therefore anticipate that nuclear resonant periodic multilayers will stimulate xray quantum optics research and bring it closer to coherent control of the x-ray-matter
interaction. Even further, the concept of polaritons in bichromatic optical lattices or
other periodic systems itself is a subject that holds great interest far beyond the x-ray
range.
To put our results into perspective we briefly summarize previous work in similar
systems, as given in [118]. To the best of our knowledge, two physical systems have
yielded phenomena and observations similar to those described here: genuine optical
lattices [104, 106, 107] and gratings of excitonic quantum wells. In excitonic quantum wells, a semiconductor is doped periodically; that way, the background refractive
index is identical throughout, but there are periodically spaced regions where quantum well excitations are possible [119]. This medium is particular interesting, since
it suffers from a similar drawback as ours: too few layers result in an unclear or
incomplete formation of band gaps. Since the early 1990s, the results obtained from
excitonic quantum wells have been described in a different framework. Instead of
assuming an infinite structure, researchers calculated the eigenmodes of these systems for a small number (∼10) of layers [120, 121]. In that case, the eigenmodes
are one superradiant Bragg mode, which reflects the radiation in a band much wider
than the exciton resonance, and a number of dark modes. In a sense, this is the
incipient Bragg band gap. However, experiments [108, 122] showed that with an
increasing number of layers, dips in the superradiant mode and a saturation of its
width appeared; researchers explained this later in terms of band gaps and standing