3 Quantum Optical Phenomena in Nuclear Resonant Scattering
153
additional term describes the energy of the nuclei and their detuning from the
incoming beam.
We then continue by prediagonalizing the interaction between the cavity modes,
resulting in two cavity supermodes a ± , both coupling to the two nuclear ensembles
(see Methods in [39]). We then heuristically add driving terms for both of these cavity
modes, which are i
√
2κ R± (a in a
†
± + a
∗
in a ± ). The energy bandwidth of the empty cavities is very large, on the order of 100 eV, which renders them highly dissipative, i.e.,
their interactions and the temporal evolution of their modes take place on extremely
fast timescales, orders of magnitude faster than the evolution of the nuclear decay.
Therefore, the common practice is to adiabatically eliminate the cavity modes and
solely follow the temporal evolution of the three states |g 1 g 2 0 1 0 2 , |g 1 g 2 0 1 0 2 and
|g 1 e 2 0 1 0 2 , corresponding to states with no nuclear excitation or one delocalised,
collective excitation in layer 1 or 2, respectively, as depicted in Fig. 3.22d. Thus
we reduce the problem to that of two interacting low-excitation nuclear ensembles
similar to Ref. [79]. The effective Hamiltonian then reads (see Methods in [39]):
H =
⎛
⎝
0
1
2
∗
1 − δ 1 + iγ 1 g 12 − iγ 12
∗
2 g 12 − iγ 12 − δ 2 + iγ 2
⎞
⎠ ,
(3.60)
where the Hamiltonian is made up of a series of effective parameters given by:
(i) the effective driving strengths of both layers:
1 = g 1
√
N
√
2κ R+
κ + + i +
+
√
2κ R−
κ − + i −
a in ,
(3.61)
2 = g 2
√
N
√
2κ R+
κ + + i +
−
√
2κ R−
κ − + i −
a in ,
(3.62)
(ii) The effective decay widths, consisting of real and imaginary part, δ and γ ,
respectively:
δ i = −
g
2
i N
2
+
κ
2
+ +
2
+
+
−
κ
2
− +
2
−
,
(3.63)
γ i =
g
2
i N
2
κ +
κ
2
+ +
2
+
+
κ −
κ
2
− +
2
−
,
(3.64)
(iii) the effective cavity-mediated coupling strength (real and imaginary part):
g 12 = −
g 1 g 2 N
2
+
κ
2
+ +
2
+
−
−
κ
2
− +
2
−
,
(3.65)
γ 12 =
g 1 g 2 N
2
κ +
κ
2
+ +
2
+
−
κ −
κ
2
− +
2
−
.
(3.66)