3 Quantum Optical Phenomena in Nuclear Resonant Scattering
115
3.4 The Nuclear Exciton, Radiative Eigenstates and
Single-Photon Superradiance
Soon after the discovery of the Mössbauer effect it became clear that ensembles of
nuclei collectively excited by single photons bear a number of fascinating properties. These states have been called ‘nuclear excitons’ and the physics of them was
explored theoretically by Afanas’ev and Kagan [23] as well as Hannon and Trammell
[24, 25]; for an extensive review see [26]. With the advent of high-brilliance synchrotron radiation it became possible to prepare such states and study their properties
systematically. Due to the small number of photons per mode of the radiation field
at these sources, however, there is in most cases only one photon interacting with
the resonant ensemble at a time. In the following we investigate collectively excited
atomic (nuclear) states that have been created by short-pulse excitation containing
one photon at most. Since we do not know which nucleus is excited, all possible
Fock states |b 1 b 2 . . . a j . . . b N containing one excited nucleus (a j ) while the others
(b i ) are in the ground state, contribute with equal weight to the state vector of the
whole system. In this sense, the superradiant excitonic states are those of maximum
delocalization of the excitation energy.
For the case that the sample extension R is much smaller than the wavelength of
the radiation, k 0 R 1, the exciton state is written as
| e =
1
√
N
|b 1 b 2 . . . a j . . . b N .
(3.8)
This state is fully symmetric with respect to exchange of any two atoms, therefore
it is often called the symmetric Dicke state. In most cases of optical physics up into
the x-ray regime, however, the opposite limit is encountered where k 0 R 1, so that
the spatial position of the atoms within the ensemble has to be taken into account:
| e (k 0 ) =
1
√
N
j
e
i k 0 ·R j |b 1 b 2 . . . a j . . . b N ,
(3.9)
where k 0 is the wave vector of the incident photon and R j denotes the position of the
jth atom. This state was introduced to describe coherent nuclear resonant scattering
as ‘nuclear exciton’ [23, 26] or more recently as ‘timed Dicke state’ [41], because
atoms at various locations within the extended sample are excited at different times.
3.4.1 Radiative Normal Modes
The collective spectral response of a given ensemble of emitters and the temporal evolution of its decay can be obtained by determination of the radiative normal
modes. The scattering of an external wave proceeds via virtual excitation of these
115
3.4 The Nuclear Exciton, Radiative Eigenstates and
Single-Photon Superradiance
Soon after the discovery of the Mössbauer effect it became clear that ensembles of
nuclei collectively excited by single photons bear a number of fascinating properties. These states have been called ‘nuclear excitons’ and the physics of them was
explored theoretically by Afanas’ev and Kagan [23] as well as Hannon and Trammell
[24, 25]; for an extensive review see [26]. With the advent of high-brilliance synchrotron radiation it became possible to prepare such states and study their properties
systematically. Due to the small number of photons per mode of the radiation field
at these sources, however, there is in most cases only one photon interacting with
the resonant ensemble at a time. In the following we investigate collectively excited
atomic (nuclear) states that have been created by short-pulse excitation containing
one photon at most. Since we do not know which nucleus is excited, all possible
Fock states |b 1 b 2 . . . a j . . . b N containing one excited nucleus (a j ) while the others
(b i ) are in the ground state, contribute with equal weight to the state vector of the
whole system. In this sense, the superradiant excitonic states are those of maximum
delocalization of the excitation energy.
For the case that the sample extension R is much smaller than the wavelength of
the radiation, k 0 R 1, the exciton state is written as
| e =
1
√
N
|b 1 b 2 . . . a j . . . b N .
(3.8)
This state is fully symmetric with respect to exchange of any two atoms, therefore
it is often called the symmetric Dicke state. In most cases of optical physics up into
the x-ray regime, however, the opposite limit is encountered where k 0 R 1, so that
the spatial position of the atoms within the ensemble has to be taken into account:
| e (k 0 ) =
1
√
N
j
e
i k 0 ·R j |b 1 b 2 . . . a j . . . b N ,
(3.9)
where k 0 is the wave vector of the incident photon and R j denotes the position of the
jth atom. This state was introduced to describe coherent nuclear resonant scattering
as ‘nuclear exciton’ [23, 26] or more recently as ‘timed Dicke state’ [41], because
atoms at various locations within the extended sample are excited at different times.
3.4.1 Radiative Normal Modes
The collective spectral response of a given ensemble of emitters and the temporal evolution of its decay can be obtained by determination of the radiative normal
modes. The scattering of an external wave proceeds via virtual excitation of these