118
R. Röhlsberger and J. Evers
In a synchrotron experiment with broadband excitation, we observe the decay of the
excitation probability that is given by
I (t) = = e (t)| e (t) =
m,n
a
∗
n a m e
i(ω
∗
n −ω m )t
n | m
(3.20)
Since the normal modes | m are transpose orthogonal rather than Hermitian orthogonal, we have in general n | m = 0. This gives rise to dynamical beats between
the modes in the temporal evolution I (t) of the decay. In the following subsections we discuss the most frequently encountered cases, i.e., forward scattering and
Bragg scattering with particular emphasis on cooperative effects encountered in these
geometries.
3.4.2 Forward Scattering
The state vector in Eq. (3.8) corresponds to the small sample limit, also called the
simple Dicke limit. The time evolution of the decay of this state to the ground state
is strictly exponential, but due to the lack of spatial phasing there is no directionality
involved. On the other hand, for extended samples (k R 1) the spatial phasing in
Eq. (3.9) leads to directional emission that is the situation most frequently encountered in experiments, especially in the regime of hard x-rays.
The exciton | e (k 0 ) created by the synchrotron pulse can be considered a Bloch
wave given by Eq. (3.9). However, the Bloch waves are generally not the true radiative normal modes in a crystal. In general, the Bloch state | e (k 0 ) is a superposition
of radiative eigenmodes, which exhibit a distribution of eigenfrequencies and decay
rates. In all cases, the initial decay is always superradiant but the decay at delayed
times is drastically different, depending on whether the exciton | e (k 0 ) is an eigenmode or not.
1 In the case of an eigenmode, the scattered signal I (t) exhibits a pure
exponential decay with an enhanced decay rate. On the other hand, if | e (k 0 ) is a
superposition of eigenmodes, the superradiant components die out quickly, leaving
a superposition of slowly decaying components with a distribution of eigenmode
frequencies. This leads to a slowly decaying beating signal at delayed times, referred
to as dynamical beats or propagation quantum beats, as illustrated in Fig. 3.5. They
have been observed not only for nuclear resonant scattering [50], but also for coherent forward scattering from excitons in the optical domain [60]. Quantitatively, the
response function of the sample, characterizing the amplitude of the scattered light
for an incident field δ(t), is given by
A(t) = δ(t) − e
− 0 t/2 C
J 1 (
√
4 C t/)
√
C t/
,
(3.21)
1 In a great part of the literature about the collective Lamb shift the values given are valid only for
the initial phase of the temporal evolution where the decay can be considered superradiant.
R. Röhlsberger and J. Evers
In a synchrotron experiment with broadband excitation, we observe the decay of the
excitation probability that is given by
I (t) = = e (t)| e (t) =
m,n
a
∗
n a m e
i(ω
∗
n −ω m )t
n | m
(3.20)
Since the normal modes | m are transpose orthogonal rather than Hermitian orthogonal, we have in general n | m = 0. This gives rise to dynamical beats between
the modes in the temporal evolution I (t) of the decay. In the following subsections we discuss the most frequently encountered cases, i.e., forward scattering and
Bragg scattering with particular emphasis on cooperative effects encountered in these
geometries.
3.4.2 Forward Scattering
The state vector in Eq. (3.8) corresponds to the small sample limit, also called the
simple Dicke limit. The time evolution of the decay of this state to the ground state
is strictly exponential, but due to the lack of spatial phasing there is no directionality
involved. On the other hand, for extended samples (k R 1) the spatial phasing in
Eq. (3.9) leads to directional emission that is the situation most frequently encountered in experiments, especially in the regime of hard x-rays.
The exciton | e (k 0 ) created by the synchrotron pulse can be considered a Bloch
wave given by Eq. (3.9). However, the Bloch waves are generally not the true radiative normal modes in a crystal. In general, the Bloch state | e (k 0 ) is a superposition
of radiative eigenmodes, which exhibit a distribution of eigenfrequencies and decay
rates. In all cases, the initial decay is always superradiant but the decay at delayed
times is drastically different, depending on whether the exciton | e (k 0 ) is an eigenmode or not.
1 In the case of an eigenmode, the scattered signal I (t) exhibits a pure
exponential decay with an enhanced decay rate. On the other hand, if | e (k 0 ) is a
superposition of eigenmodes, the superradiant components die out quickly, leaving
a superposition of slowly decaying components with a distribution of eigenmode
frequencies. This leads to a slowly decaying beating signal at delayed times, referred
to as dynamical beats or propagation quantum beats, as illustrated in Fig. 3.5. They
have been observed not only for nuclear resonant scattering [50], but also for coherent forward scattering from excitons in the optical domain [60]. Quantitatively, the
response function of the sample, characterizing the amplitude of the scattered light
for an incident field δ(t), is given by
A(t) = δ(t) − e
− 0 t/2 C
J 1 (
√
4 C t/)
√
C t/
,
(3.21)
1 In a great part of the literature about the collective Lamb shift the values given are valid only for
the initial phase of the temporal evolution where the decay can be considered superradiant.