3 Quantum Optical Phenomena in Nuclear Resonant Scattering
135
γ 2 = γ 0
γ 3 = γ 0 [1 + d 2 f 0 E 2−− ]
(3.53)
2
c = d 1 d 2 ( f 0 γ 0 )
2 E 2−+ E 1+−
This result prompts an obvious interpretation, supported by the illustration in
Fig. 3.11b, c: The two ensembles of nuclei in the node and the antinode of the standing
wave field experience two significantly different photonic densities of states, leading
to two different collective decay rates γ 2 and γ 3 . This effectively converts the nuclei
in the cavity into three-level systems with two degenerate upper levels represented
by the states |2 and |3, as illustrated in the level scheme of Fig. 3.11c. The expression for
2
C is proportional to the two transfer matrix elements E 2−+ and E 1+− that
describe the transition amplitudes between the two counterpropagating fields in the
cavity at the position of the two resonant layers. This indicates that the coupling field
arises from the radiative coupling of the two resonant layers via the cavity field: An
excited atom in the antinode |3 decays back to the ground state |1 and releases
a photon into the cavity. This photon can promote an atom in the node from the
ground state into state |2 that eventually decays and again releases a photon into the
cavity, and so on. As a result, the two excited states |2 and |3 are coupled through
their common ground state |1 via the vacuum field of the cavity, which effectively
establishes a control field between the two upper states, represented by the horizontal arrow in Fig. 3.11c. The resulting arrangement of levels in Fig. 3.11c and their
coupling resembles closely a -type level scheme as in Fig. 3.11a. It should be noted
that the control field Rabi frequency of Eq. (3.53) enters Eq. (3.52) as a complexvalued quantity
2
C rather than a real number | C |
2 in the usual expression for an
EIT susceptibility. A closer inspection reveals that the imaginary part of
2
C is small
compared to its real part for the cavity configurations employed here. It remains to
be investigated in which way the imaginary part of
2
C affects EIT in these systems.
We want to emphasize that cooperative emission is critical to EIT in this system. While one of the atomic ensembles undergoes single-photon superradiant
enhancement leading to a decay width of C = 2γ 3 = d 2 f 0 Re[E 2−− ] 0 and a
collective Lamb shift of L C = −d 2 f 0 Im[E 2−− ] 0 /2, the decay width 2γ 2 of
the other’subradiant’ ensemble is given by just the natural line width 0 , so that
γ 3 ≈ 50 γ 2 in the example shown in Fig. 3.10. Thus, in the presence of a strong
superradiant enhancement of state |3, the state |2 is relatively long-lived and thus
can be considered as metastable. This is an important condition for a pronounced EIT
effect. The superradiantly broadened transition of the nuclei in the antinode provides
the continuum of states relative to which the Fano interference in this system takes
place. The collective Lamb shift of this level introduces an asymmetry that leads to
the characteristic Fano profile of the transparency window. For vanishing CLS, the
profile would simply be a Lorentzian line [79, 82, 83].
For an experimental verification of EIT in the x-ray regime, we have prepared
an x-ray cavity, shown in Fig. 3.11b, that consists of a Pt(3 nm)/C(38 nm)/Pt(10nm)
sandwich structure containing two 3 nm
57 Fe layers that occupy a node and an
antinode of the cavity field. These two layers represent the subradiant state |2 and
135
γ 2 = γ 0
γ 3 = γ 0 [1 + d 2 f 0 E 2−− ]
(3.53)
2
c = d 1 d 2 ( f 0 γ 0 )
2 E 2−+ E 1+−
This result prompts an obvious interpretation, supported by the illustration in
Fig. 3.11b, c: The two ensembles of nuclei in the node and the antinode of the standing
wave field experience two significantly different photonic densities of states, leading
to two different collective decay rates γ 2 and γ 3 . This effectively converts the nuclei
in the cavity into three-level systems with two degenerate upper levels represented
by the states |2 and |3, as illustrated in the level scheme of Fig. 3.11c. The expression for
2
C is proportional to the two transfer matrix elements E 2−+ and E 1+− that
describe the transition amplitudes between the two counterpropagating fields in the
cavity at the position of the two resonant layers. This indicates that the coupling field
arises from the radiative coupling of the two resonant layers via the cavity field: An
excited atom in the antinode |3 decays back to the ground state |1 and releases
a photon into the cavity. This photon can promote an atom in the node from the
ground state into state |2 that eventually decays and again releases a photon into the
cavity, and so on. As a result, the two excited states |2 and |3 are coupled through
their common ground state |1 via the vacuum field of the cavity, which effectively
establishes a control field between the two upper states, represented by the horizontal arrow in Fig. 3.11c. The resulting arrangement of levels in Fig. 3.11c and their
coupling resembles closely a -type level scheme as in Fig. 3.11a. It should be noted
that the control field Rabi frequency of Eq. (3.53) enters Eq. (3.52) as a complexvalued quantity
2
C rather than a real number | C |
2 in the usual expression for an
EIT susceptibility. A closer inspection reveals that the imaginary part of
2
C is small
compared to its real part for the cavity configurations employed here. It remains to
be investigated in which way the imaginary part of
2
C affects EIT in these systems.
We want to emphasize that cooperative emission is critical to EIT in this system. While one of the atomic ensembles undergoes single-photon superradiant
enhancement leading to a decay width of C = 2γ 3 = d 2 f 0 Re[E 2−− ] 0 and a
collective Lamb shift of L C = −d 2 f 0 Im[E 2−− ] 0 /2, the decay width 2γ 2 of
the other’subradiant’ ensemble is given by just the natural line width 0 , so that
γ 3 ≈ 50 γ 2 in the example shown in Fig. 3.10. Thus, in the presence of a strong
superradiant enhancement of state |3, the state |2 is relatively long-lived and thus
can be considered as metastable. This is an important condition for a pronounced EIT
effect. The superradiantly broadened transition of the nuclei in the antinode provides
the continuum of states relative to which the Fano interference in this system takes
place. The collective Lamb shift of this level introduces an asymmetry that leads to
the characteristic Fano profile of the transparency window. For vanishing CLS, the
profile would simply be a Lorentzian line [79, 82, 83].
For an experimental verification of EIT in the x-ray regime, we have prepared
an x-ray cavity, shown in Fig. 3.11b, that consists of a Pt(3 nm)/C(38 nm)/Pt(10nm)
sandwich structure containing two 3 nm
57 Fe layers that occupy a node and an
antinode of the cavity field. These two layers represent the subradiant state |2 and