130
R. Röhlsberger and J. Evers
addition of the scattering channels, the responses of the different transitions within
this few-level system may interfere, providing access to a rich variety of quantum
optical phenomena.
3.6.3 Nuclear Dynamics in the Cavity
It remains to determine the nuclear dynamics, i.e., its evolution under the action of
an x-ray pulse, in order to determine the density matrix ρ entering the reflectance
in Eqs. (3.44) and (3.45). Here we want to illustrate this for the simplest case of
two-level nuclei and a single cavity mode a. The original Hamiltonian is of JaynesCummings-type [2, 29], and contains interaction terms of the form S
(n)
+ a, describing
the annihilation of a cavity photon (a) together with an excitation of nucleus n
(S
(n)
+ ), as well as the reverse process. The problem can be simplified considerably be
exploiting that the fastest timescale in the problem typically is given by the cavity
lifetime 1/κ. In this “bad cavity” limit characterized by short photon trapping times,
the cavity modes adiabatically follow the much slower evolution of the nuclei. As
a consequence, the cavity operators can approximately be replaced by their steadystate values Eq. (3.43), which results in an effective Hamiltonian for the nuclei alone.
In the radiative eigenmode basis it is given by [35, 37]
H = −|EE| + (( eff |EG| + H.c.) + L S |EE|.
(3.46)
The interpretation of this Hamiltonian is straightforward. It is equivalent to the Rabi
model for a driven two-level system [2]. However, the effective Rabi coupling constant eff = g
√
N a
(SS) in H is not given by the bare nucleus-cavity coupling g,
but modified by cooperative effects as indicated by the superradiant enhancement
factor
√
N , as well as by the cavity field as indicated by the presence of the steadystate value of the field operator a
(SS) . Furthermore, the usual detuning between
x-ray frequency and bare nuclear transition frequency is augmented by an additional
contribution
LS = |g|
2 N Im[(κ + i C )
−1
],
(3.47)
which arises due to the radiative coupling between the nuclei and which can be
interpreted as the cooperative Lamb shift. Similar to the Hamiltonian parts, also the
incoherent spontaneous emission of the individual nuclei γ is modified by
= 2|g|
2 N Re[(κ + i C )
−1
].
(3.48)
In linear response, the desired nuclear polarization E|ρ|G is governed by
E| ˙
ρ|G = −i eff + i
( − L S ) +
i
2
(γ + )
E|ρ|G.
(3.49)
R. Röhlsberger and J. Evers
addition of the scattering channels, the responses of the different transitions within
this few-level system may interfere, providing access to a rich variety of quantum
optical phenomena.
3.6.3 Nuclear Dynamics in the Cavity
It remains to determine the nuclear dynamics, i.e., its evolution under the action of
an x-ray pulse, in order to determine the density matrix ρ entering the reflectance
in Eqs. (3.44) and (3.45). Here we want to illustrate this for the simplest case of
two-level nuclei and a single cavity mode a. The original Hamiltonian is of JaynesCummings-type [2, 29], and contains interaction terms of the form S
(n)
+ a, describing
the annihilation of a cavity photon (a) together with an excitation of nucleus n
(S
(n)
+ ), as well as the reverse process. The problem can be simplified considerably be
exploiting that the fastest timescale in the problem typically is given by the cavity
lifetime 1/κ. In this “bad cavity” limit characterized by short photon trapping times,
the cavity modes adiabatically follow the much slower evolution of the nuclei. As
a consequence, the cavity operators can approximately be replaced by their steadystate values Eq. (3.43), which results in an effective Hamiltonian for the nuclei alone.
In the radiative eigenmode basis it is given by [35, 37]
H = −|EE| + (( eff |EG| + H.c.) + L S |EE|.
(3.46)
The interpretation of this Hamiltonian is straightforward. It is equivalent to the Rabi
model for a driven two-level system [2]. However, the effective Rabi coupling constant eff = g
√
N a
(SS) in H is not given by the bare nucleus-cavity coupling g,
but modified by cooperative effects as indicated by the superradiant enhancement
factor
√
N , as well as by the cavity field as indicated by the presence of the steadystate value of the field operator a
(SS) . Furthermore, the usual detuning between
x-ray frequency and bare nuclear transition frequency is augmented by an additional
contribution
LS = |g|
2 N Im[(κ + i C )
−1
],
(3.47)
which arises due to the radiative coupling between the nuclei and which can be
interpreted as the cooperative Lamb shift. Similar to the Hamiltonian parts, also the
incoherent spontaneous emission of the individual nuclei γ is modified by
= 2|g|
2 N Re[(κ + i C )
−1
].
(3.48)
In linear response, the desired nuclear polarization E|ρ|G is governed by
E| ˙
ρ|G = −i eff + i
( − L S ) +
i
2
(γ + )
E|ρ|G.
(3.49)