128
R. Röhlsberger and J. Evers
3.6.1 Quantum Optics of the Empty Cavity
To illustrate the photonic environment in a cavity without nuclei, we restrict the
discussion to a single cavity mode and neglect the light polarization for the moment.
The probing x-rays have frequency ω and a wave vector k that defines the incidence
angle θ . The cavity modes are characterized by discrete wave number components
perpendicular to the cavity surface, but continuous wave number components along
the surface. The boundary conditions impose that the wave vector k C inside the cavity
has a component |k C | cos(θ ) along the cavity surface. The component transverse to
the cavity surface, however, is fixed by the guided mode standing wave condition to
|k| sin(θ 0 ), if θ 0 is the incidence angle under which this mode is driven resonantly
for incident wave number |k|. Thus, |k C | =
|k| 2 cos 2 (θ ) + |k| 2 sin
2
(θ 0 ), and a
detuning C = ω C − ω ≈ −ωθ 0 θ between the cavity resonance frequency and
the frequency of the incident light can be defined, which can be tuned via small
variations in the incidence angle θ = θ − θ 0 from the resonance condition. With
this detuning, the Heisenberg equation of motion for the cavity mode in the absence
of nuclear resonances characterized by annihilation [creation] operators a [a
† ] is
given by
d
dt
a = −(κ + i C )a +
2κ R a in ,
(3.40)
where κ is the overall damping rate of the cavity mode, κ R characterizes the evanescent coupling into and out of the cavity mode, and a in the applied x-ray field. In
practice, κ R can be adjusted, e.g., by choosing the thickness of the cavity top layer
through which the x-rays evanescently couple into the cavity mode. From the cavity
field operators, the empty cavity reflectance |R c |
2 can be obtained via the input-output
relations [77] a out = −a in +
√
2κ R a, using R c = =a out /a in . In the stationary state
(SS) ˙
a
(SS)
= 0, such that
a
(SS)
=
√
2κ R a in
κ + i C
(3.41)
and
R c =
2κ R
κ + i C
− 1.
(3.42)
At the so-called critical coupling condition 2κ R = κ, the reflectance |R c |
2 vanishes
on resonance C = 0, which can be interpreted as destructive interference between
light reflected from the outside of the cavity with that coupling out of the cavity
mode. If operated in this regime, the cavity can be employed to suppress a significant
part of the background photons, facilitating the detection.
R. Röhlsberger and J. Evers
3.6.1 Quantum Optics of the Empty Cavity
To illustrate the photonic environment in a cavity without nuclei, we restrict the
discussion to a single cavity mode and neglect the light polarization for the moment.
The probing x-rays have frequency ω and a wave vector k that defines the incidence
angle θ . The cavity modes are characterized by discrete wave number components
perpendicular to the cavity surface, but continuous wave number components along
the surface. The boundary conditions impose that the wave vector k C inside the cavity
has a component |k C | cos(θ ) along the cavity surface. The component transverse to
the cavity surface, however, is fixed by the guided mode standing wave condition to
|k| sin(θ 0 ), if θ 0 is the incidence angle under which this mode is driven resonantly
for incident wave number |k|. Thus, |k C | =
|k| 2 cos 2 (θ ) + |k| 2 sin
2
(θ 0 ), and a
detuning C = ω C − ω ≈ −ωθ 0 θ between the cavity resonance frequency and
the frequency of the incident light can be defined, which can be tuned via small
variations in the incidence angle θ = θ − θ 0 from the resonance condition. With
this detuning, the Heisenberg equation of motion for the cavity mode in the absence
of nuclear resonances characterized by annihilation [creation] operators a [a
† ] is
given by
d
dt
a = −(κ + i C )a +
2κ R a in ,
(3.40)
where κ is the overall damping rate of the cavity mode, κ R characterizes the evanescent coupling into and out of the cavity mode, and a in the applied x-ray field. In
practice, κ R can be adjusted, e.g., by choosing the thickness of the cavity top layer
through which the x-rays evanescently couple into the cavity mode. From the cavity
field operators, the empty cavity reflectance |R c |
2 can be obtained via the input-output
relations [77] a out = −a in +
√
2κ R a, using R c = =a out /a in . In the stationary state
(SS) ˙
a
(SS)
= 0, such that
a
(SS)
=
√
2κ R a in
κ + i C
(3.41)
and
R c =
2κ R
κ + i C
− 1.
(3.42)
At the so-called critical coupling condition 2κ R = κ, the reflectance |R c |
2 vanishes
on resonance C = 0, which can be interpreted as destructive interference between
light reflected from the outside of the cavity with that coupling out of the cavity
mode. If operated in this regime, the cavity can be employed to suppress a significant
part of the background photons, facilitating the detection.