3 Quantum Optical Phenomena in Nuclear Resonant Scattering
131
Inserting the steady state solution obtained from E| ˙
ρ|G = 0 into Eq. (3.44) finally
yields
R = R c − i
κ − i C
κ + i C
2κ R
κ
/2
( − L S ) +
i
2 (γ + )
,
(3.50)
with the empty cavity reflectance R c defined in Eq. (3.42). The nuclear response
therefore comprises a Lorentzian shifted with respect to the bare nuclear resonance
frequency by the collective Lamb shift LS , and with superradiant broadening of
the natural line width γ , as already found using a different formalism in Eq. (3.38).
3.7 Quantum Optical Effects in Cavities
3.7.1 Interferometric Phase Detection via Fano Resonance
Control
In Sect. 3.5 it was found that a resonantly driven cavity containing resonant twolevel nuclei features a Lorentzian spectral response, broadened by superradiance,
and shifted by the cooperative Lamb shift. Having the expression for the cavity
reflectance Eq. (3.50) at hand, we can start by exploring the cavity response offresonance with the cavity mode [79]. Close to the resonance, C = δ C (θ − θ min ),
such that the detuning between x-rays and cavity mode can experimentally be tuned
by varying the x-ray incidence angle θ around the resonance angle θ min .
To simplify the analysis, we specialize to strongly superradiant cavities (γ )
in critical coupling (κ = 2κ R ) and rewrite Eq. (3.50) using Eq. (3.48) to give
|R|
2
=
|ε + q|
2
1 + ε 2 σ 0 ,
(3.51)
where we defined the dimensionless energy ε = (( − LS )/((/2), the prefactor
σ 0 = [1 + κ
2
//
2
C ]
−1 , and the so-called q-factor q = κ// C . The cavity response
thus takes the form of a Fano resonance [80], which is an ubiquitous spectroscopic
signature in light-matter interactions [81, 82]. The Fano resonance arises, because
there are two interfering pathways for the light to propagate through the sample.
First, the spectrally broad cavity response, which is of relevance it the light does not
interact with the nuclei. Second, a spectrally narrow bound-state contribution arising
from the scattering on the nuclei. The relative phase between the two contributions
is given by φ = −arg(q − i), and it turns out that this phase determines the line
shape, which may range from Lorentzian absorption features via dispersion-like
asymmetric structures up inverted Lorentzian lines [83]. Conversely, external control
over this relative phase can be used to manipulate the lineshape [83, 84]. Since close
to resonance, q = κ/[δ C (θ − θ min )], we find that changing the incidence angle allows
131
Inserting the steady state solution obtained from E| ˙
ρ|G = 0 into Eq. (3.44) finally
yields
R = R c − i
κ − i C
κ + i C
2κ R
κ
/2
( − L S ) +
i
2 (γ + )
,
(3.50)
with the empty cavity reflectance R c defined in Eq. (3.42). The nuclear response
therefore comprises a Lorentzian shifted with respect to the bare nuclear resonance
frequency by the collective Lamb shift LS , and with superradiant broadening of
the natural line width γ , as already found using a different formalism in Eq. (3.38).
3.7 Quantum Optical Effects in Cavities
3.7.1 Interferometric Phase Detection via Fano Resonance
Control
In Sect. 3.5 it was found that a resonantly driven cavity containing resonant twolevel nuclei features a Lorentzian spectral response, broadened by superradiance,
and shifted by the cooperative Lamb shift. Having the expression for the cavity
reflectance Eq. (3.50) at hand, we can start by exploring the cavity response offresonance with the cavity mode [79]. Close to the resonance, C = δ C (θ − θ min ),
such that the detuning between x-rays and cavity mode can experimentally be tuned
by varying the x-ray incidence angle θ around the resonance angle θ min .
To simplify the analysis, we specialize to strongly superradiant cavities (γ )
in critical coupling (κ = 2κ R ) and rewrite Eq. (3.50) using Eq. (3.48) to give
|R|
2
=
|ε + q|
2
1 + ε 2 σ 0 ,
(3.51)
where we defined the dimensionless energy ε = (( − LS )/((/2), the prefactor
σ 0 = [1 + κ
2
//
2
C ]
−1 , and the so-called q-factor q = κ// C . The cavity response
thus takes the form of a Fano resonance [80], which is an ubiquitous spectroscopic
signature in light-matter interactions [81, 82]. The Fano resonance arises, because
there are two interfering pathways for the light to propagate through the sample.
First, the spectrally broad cavity response, which is of relevance it the light does not
interact with the nuclei. Second, a spectrally narrow bound-state contribution arising
from the scattering on the nuclei. The relative phase between the two contributions
is given by φ = −arg(q − i), and it turns out that this phase determines the line
shape, which may range from Lorentzian absorption features via dispersion-like
asymmetric structures up inverted Lorentzian lines [83]. Conversely, external control
over this relative phase can be used to manipulate the lineshape [83, 84]. Since close
to resonance, q = κ/[δ C (θ − θ min )], we find that changing the incidence angle allows