5 Coherent Nonlinear Processes in Metal-Semiconductor …
107
or the SPP oscillator emits a photon to surrounding via spontaneous emission (loss
of phase information), which is absorbed by the other oscillator. This incoherent
exchange affects the damping of the coupled modes by an amount that is given by
√
exc SPP [35, 36]. Such co-operative damping effects have been observed in various systems like trapped ions and molecular aggregates. Though the coupled mode
formation in metal-semiconductor hybrid nanostructures has been investigated extensively, the co-operative damping and the dynamics of the coupled modes has been
challenging.
Before discussing the experimental results, we briefly discuss the framework
of a phenomenological coupled oscillator model to describe the optical response
of the metal-J-aggregate hybrid nanostructures [8, 9, 14, 22, 23, 25, 26]. It is a
straightforward method similar to that used to describe coupled mechanical harmonic oscillators, yet it can account for co-operative damping phenomena and it is
also possible to add some of the other quantum mechanical effects. It is effective
to rationalize the observations of linear optical response but needs to be extended
to more advanced quantum mechanical Bloch equation formalism for applying to
nonlinear response [8, 25–27]. Here, we assume that both emitter and resonator are
two-level systems, i.e. each one can either be in the ground state or the excited state.
The states corresponding to both of them in the ground state or the excited state are
forbidden. Due to the presence of damping, the energy finally leaks out of the system.
Thus, the non-hermitian Hamiltonian representing the dipole light-matter interaction
can in the form of a 2 × 2 matrix as
H exc-SPP =
ω exc R
∗
R ω SPP
− i
exc R
R SPP
.
(5.1)
Here, the first matrix of the hamiltonian,
H exc-SPP represents the coherent periodic energy exchange governed by R , whereas the second matrix represents the
incoherent transfer of energy to the surrounding or radiative losses via spontaneous
emission [8, 25–27]. The term R represents the co-operative damping given by
√
exc SPP . Since excitons in an ensemble exhibit considerable non-radiative damping,
exc = exc +
nr
exc denotes the sum of the radiative and non-radiative decay
rates of the excitons and SPP denotes the radiative damping of the SPP. It has been
shown that the major contribution to damping in metal nanostructures is because of
radiative damping [37, 38]. Hence, it assumed that there are no non-radiative losses
within the resonator (no light is absorbed by the metal nanostructure). The complex
eigenfrequencies of the UP and LP modes, ω U P,L P are then given by the eigenvalues of
H exc-SPP as ω U P,L P =
1
2
ω exc + ω SPP − i((
exc + SPP ) ± A 0
, where A 0 =
2 + 4 A 1 A 2 and = ω exc − ω SPP − i((
exc − SPP ). Here A 1 = R − i R , and
A 2 =
∗
R − i R . The UP and LP polariton wavefunctions are then obtained as
the normalized eigenvectors of
H exc-SPP : |U P, L P =
((±A 0 )|10+2 A 2 |01
B U P,L P
with a normalization constant B U P,L P =
| ± A 0 | 2 + 4|A 2 | 2 . Since we are considering the
interaction in linear regime, the terms corresponding to both the exciton and the
SPP in ground state or in the excited state are omitted. Using this coupled oscillator
107
or the SPP oscillator emits a photon to surrounding via spontaneous emission (loss
of phase information), which is absorbed by the other oscillator. This incoherent
exchange affects the damping of the coupled modes by an amount that is given by
√
exc SPP [35, 36]. Such co-operative damping effects have been observed in various systems like trapped ions and molecular aggregates. Though the coupled mode
formation in metal-semiconductor hybrid nanostructures has been investigated extensively, the co-operative damping and the dynamics of the coupled modes has been
challenging.
Before discussing the experimental results, we briefly discuss the framework
of a phenomenological coupled oscillator model to describe the optical response
of the metal-J-aggregate hybrid nanostructures [8, 9, 14, 22, 23, 25, 26]. It is a
straightforward method similar to that used to describe coupled mechanical harmonic oscillators, yet it can account for co-operative damping phenomena and it is
also possible to add some of the other quantum mechanical effects. It is effective
to rationalize the observations of linear optical response but needs to be extended
to more advanced quantum mechanical Bloch equation formalism for applying to
nonlinear response [8, 25–27]. Here, we assume that both emitter and resonator are
two-level systems, i.e. each one can either be in the ground state or the excited state.
The states corresponding to both of them in the ground state or the excited state are
forbidden. Due to the presence of damping, the energy finally leaks out of the system.
Thus, the non-hermitian Hamiltonian representing the dipole light-matter interaction
can in the form of a 2 × 2 matrix as
H exc-SPP =
ω exc R
∗
R ω SPP
− i
exc R
R SPP
.
(5.1)
Here, the first matrix of the hamiltonian,
H exc-SPP represents the coherent periodic energy exchange governed by R , whereas the second matrix represents the
incoherent transfer of energy to the surrounding or radiative losses via spontaneous
emission [8, 25–27]. The term R represents the co-operative damping given by
√
exc SPP . Since excitons in an ensemble exhibit considerable non-radiative damping,
exc = exc +
nr
exc denotes the sum of the radiative and non-radiative decay
rates of the excitons and SPP denotes the radiative damping of the SPP. It has been
shown that the major contribution to damping in metal nanostructures is because of
radiative damping [37, 38]. Hence, it assumed that there are no non-radiative losses
within the resonator (no light is absorbed by the metal nanostructure). The complex
eigenfrequencies of the UP and LP modes, ω U P,L P are then given by the eigenvalues of
H exc-SPP as ω U P,L P =
1
2
ω exc + ω SPP − i((
exc + SPP ) ± A 0
, where A 0 =
2 + 4 A 1 A 2 and = ω exc − ω SPP − i((
exc − SPP ). Here A 1 = R − i R , and
A 2 =
∗
R − i R . The UP and LP polariton wavefunctions are then obtained as
the normalized eigenvectors of
H exc-SPP : |U P, L P =
((±A 0 )|10+2 A 2 |01
B U P,L P
with a normalization constant B U P,L P =
| ± A 0 | 2 + 4|A 2 | 2 . Since we are considering the
interaction in linear regime, the terms corresponding to both the exciton and the
SPP in ground state or in the excited state are omitted. Using this coupled oscillator