3 Quantum Optical Phenomena in Nuclear Resonant Scattering
155
parameters in the effective Hamiltonian can be derived from g 1(2)
N 1(2) , κ ± and
± , and κ R± . While the latter three can be extracted from an off-resonant reflectivity
curve, the quantity g i
√
N i must be fitted from the experimental data. Note that
these new, effective parameters can be tuned via the angle of incidence θ through a
wide range of values, as they all depend on the detuning, which is given by ± =
(sin(θ ± )/ sin(θ ) − 1) ω 0 . Here, θ ± is the angle for which one cavity supermode is
exactly at resonance at the nuclear transition energy ω 0 .
We can essentially regard the new, effective Hamiltonian as a three-level system
(Fig. 3.22d). The first level, corresponding to the ground state (no excited ensemble)
serves only to probe the properties of the other two through the (very weak) driving
terms. The energies of these two states, in turn, can be tuned via the collective Lamb
shifts, as can their superradiant decay terms and their mutual interaction. These
upper two levels form a two-level system with tunable energies, decay constants and
a tunable mutual interaction. Just as for a regular two-level system interacting with
a cavity mode, strong coupling and Rabi oscillations between the two levels can be
achieved if the interaction term is larger than the two respective decay terms, in this
case g 12 > γ 1 , γ 2 . If this can be reached by appropriate tuning, the two upper states
will exchange a photon multiple times before the whole system decays, giving rise to
Rabi oscillations [129]. This is an equivalent of the standard strong-coupling set-up,
where the coupling strength between an atom and a cavity must exceed the latter’s
decay constants [130–133]. This effective three-level system may also be regarded
as the implementation of an artificial Autler-Townes set-up, which also consists of
two upper levels whose interaction is probed from a lower ground state. A crucial
difference is that the interaction here is cavity-induced, whereas in the Autler-Townes
case, a laser beam tuned to the energy difference between the two levels couples them.
The equivalent of the classical Rabi frequency in our case is given by
R =
g
2
12 + γ
2
12 + (δ 1 − δ 2 ) 2 ,
(3.67)
which includes the cavity-mediated coupling strength g 12 between the two ensembles/states, as well as their relative energy detuning δ 1 − δ 2 . The behaviour of R
in comparison with the superradiant decay rates, and the collective Lamb shifts as a
function of the incident angle, are shown in Figs. 3.23b,c. Any change in the incident
angle θ leads to modifications of all parameters δ i , γ i and g 12 , which makes it difficult to reach the desired behaviour of the coupling and the decay properties of the
system. This raises the question of whether we can find an angle θ at which the strong
coupling conditions (g 12 > γ i ) are fulfilled. This is indeed the case: In between both
supermodes, at large detunings from them, the superradiant decay widths γ 1 , γ 2 are
insignificant compared to the interaction strength between the two layers. This can
be explained because the real part g 12 of the interaction strength is suppressed by
1//
−1
± at angles sufficiently far away from the cavity angle of zero detuning, while
the imaginary part γ 12 describing the superradiant decay is suppressed by 1//
−2
± .
However, the collective Lamb shifts do not go to zero as quickly, and, according to
Eq. (3.67), they are also involved in determining the oscillation rate R . To observe
155
parameters in the effective Hamiltonian can be derived from g 1(2)
N 1(2) , κ ± and
± , and κ R± . While the latter three can be extracted from an off-resonant reflectivity
curve, the quantity g i
√
N i must be fitted from the experimental data. Note that
these new, effective parameters can be tuned via the angle of incidence θ through a
wide range of values, as they all depend on the detuning, which is given by ± =
(sin(θ ± )/ sin(θ ) − 1) ω 0 . Here, θ ± is the angle for which one cavity supermode is
exactly at resonance at the nuclear transition energy ω 0 .
We can essentially regard the new, effective Hamiltonian as a three-level system
(Fig. 3.22d). The first level, corresponding to the ground state (no excited ensemble)
serves only to probe the properties of the other two through the (very weak) driving
terms. The energies of these two states, in turn, can be tuned via the collective Lamb
shifts, as can their superradiant decay terms and their mutual interaction. These
upper two levels form a two-level system with tunable energies, decay constants and
a tunable mutual interaction. Just as for a regular two-level system interacting with
a cavity mode, strong coupling and Rabi oscillations between the two levels can be
achieved if the interaction term is larger than the two respective decay terms, in this
case g 12 > γ 1 , γ 2 . If this can be reached by appropriate tuning, the two upper states
will exchange a photon multiple times before the whole system decays, giving rise to
Rabi oscillations [129]. This is an equivalent of the standard strong-coupling set-up,
where the coupling strength between an atom and a cavity must exceed the latter’s
decay constants [130–133]. This effective three-level system may also be regarded
as the implementation of an artificial Autler-Townes set-up, which also consists of
two upper levels whose interaction is probed from a lower ground state. A crucial
difference is that the interaction here is cavity-induced, whereas in the Autler-Townes
case, a laser beam tuned to the energy difference between the two levels couples them.
The equivalent of the classical Rabi frequency in our case is given by
R =
g
2
12 + γ
2
12 + (δ 1 − δ 2 ) 2 ,
(3.67)
which includes the cavity-mediated coupling strength g 12 between the two ensembles/states, as well as their relative energy detuning δ 1 − δ 2 . The behaviour of R
in comparison with the superradiant decay rates, and the collective Lamb shifts as a
function of the incident angle, are shown in Figs. 3.23b,c. Any change in the incident
angle θ leads to modifications of all parameters δ i , γ i and g 12 , which makes it difficult to reach the desired behaviour of the coupling and the decay properties of the
system. This raises the question of whether we can find an angle θ at which the strong
coupling conditions (g 12 > γ i ) are fulfilled. This is indeed the case: In between both
supermodes, at large detunings from them, the superradiant decay widths γ 1 , γ 2 are
insignificant compared to the interaction strength between the two layers. This can
be explained because the real part g 12 of the interaction strength is suppressed by
1//
−1
± at angles sufficiently far away from the cavity angle of zero detuning, while
the imaginary part γ 12 describing the superradiant decay is suppressed by 1//
−2
± .
However, the collective Lamb shifts do not go to zero as quickly, and, according to
Eq. (3.67), they are also involved in determining the oscillation rate R . To observe