Theory of polariton mediated Raman scattering in microcavities
L. M. León Hilario, A. Bruchhausen, A. M. Lobos, and A. A. Aligia
Centro Atómico Bariloche and Instituto Balseiro,
Comisión Nacional de Enerǵıa Atómica,
8400 S. C. de Bariloche, Argentina
(Dated: November 4, 2018)
Abstract
We calculate the intensity of the polariton mediated inelastic light scattering in semiconductor
microcavities. We treat the exciton-photon coupling nonperturbatively and incorporate lifetime
effects in both excitons and photons, and a coupling of the photons to the electron-hole continuum.
Taking the matrix elements as fitting parameters, the results are in excellent agreement with
measured Raman intensities due to optical phonons resonant with the upper polariton branches in
II-VI microcavities with embedded CdTe quantum wells.
PACS numbers: 71.36.+c, 78.30.Fs, 78.30.-j
http://arxiv.org/abs/0704.0198v1
Planar semiconductors microcavities (MC’s) have attracted much attention in the last
decade as they provide a novel means to study, enhance and control the interaction between
light and matter [1, 2, 3, 4, 5, 6, 7]. When the MC mode (cavity-photon) is tuned in near
resonance with the embedded quantum-well (QW) exciton transitions, and the damping
processes involved are weak in comparison to the photon-matter interaction, the eigenstates
of the system become mixed exciton-photon states, cavity-polaritons, which are in part light
and in part matter bosonic quasi-particles [1, 2, 3]. Examples of interesting new physics are
the recent evidence of a Bose-Einstein condensation of polaritons in CdTe MC’s [5, 6], and
the construction of devices which increase the interaction of sound and light, opening the
possibility of realizing a coherent monochromatic source of acoustic phonons [7].
Raman scattering due to longitudinal optical (LO) phonons, being a coherent process is
intrinsically connected with the cavity-polariton. The physics of strongly coupled photons
and excitons, the polariton–phonon interaction, and the polariton–external-photon coupling
are clearly displayed [8, 9, 10, 11, 12]. In particular resonant Raman scattering (RRS)
experiments, in which the wave-length of the incoming radiation is tuned in a way such that,
after the emission of a LO phonon, the energy of the outgoing radiation coincides with that of
the cavity polariton branches, have proven to be suited to sense the dynamics of the coupled
modes, and to obtain information about the dephasing of the resonant polaritonic state
[8, 9, 10, 11, 12]. Unfortunately, due to the large remaining luminesence, RRS experiments
in resonance with the lower polariton branch have not yet been achieved in intrinsic II-VI
MC’s [13]. Therefore all reported experiments in these kind of MC’s, with embedded CdTe
QW’s, consider the case of the scattered photons in outgoing resonance with the upper
polariton branch (for two branch-systems: coupling of one exciton mode and the cavity
photon) or with the middle polariton branch (for tree branch-systems: coupling of two
exciton modes and the cavity photon) [9, 10, 14]. The measured intensities in these two
systems were analyzed on the basis of a model in which one (two) exciton states |e〉 are
mixed with a photon state |f〉 with the same in-plane wave vector k, leading to a 2x2 (3x3)
matrix. Essentially, the Raman intensity is proportional to
I ∼ TiTs|〈Pi|H
′|Ps〉|
2, (1)
where Ti describe the probability of conversion of an incident photon |fi〉 into the polariton
state |Pi〉, Ts has an analogous meaning for the scattered polariton |Ps〉 and the outgoing
photon |fs〉, and H
′ is the interaction between electrons and the LO phonons. At the
conditions of resonance with the outgoing polariton, Ti is very weakly dependent on laser
energy or detuning (difference between photon and exciton energies), while Ts is proportional
to the photon strength of the scattered polariton |〈fs|Ps〉|
2. Similarly, in the simplest case
of only one exciton (2x2 matrix) one expects that the matrix element entering Eq. (1) is
proportional to the exciton part of the scattered polariton |〈Pi|H
′|Ps〉|
2 ∼ |〈es|Ps〉|
2. Thus
if the wave function is |Ps〉 = α|fs〉+ β|es〉, one has
I ∼ |〈fs|Ps〉|
2|〈es|Ps〉|
2 = |α|2|β|2. (2)
This model predicts a Raman intensity which is symmetric with detuning and is maximum
at zero detuning. In other words, the intensity is maximum for detunings such that the
scattered polariton is more easily coupled to the external photons, but at the same time
when the polariton is more easily coupled to the optical phonons, which requires a large
matter (exciton) component of the polariton. This result is in qualitative agreement with
experiment [10]. However, for positive detuning the experimental results fall bellow the
values predicted by Eq. (2) (see Fig. 1). This is ascribed to the effects of the electron-hole
continuum above the exciton energy, which are not included in the model [10].
For another sample in which two excitons are involved, the above analysis can be ex-
tended straightforwardly and the intensity depends on the amplitudes of a 3x3 matrix and
matrix element of H ′ involving both excitons [10]. However, comparison with experiments at
resonance with the middle polariton branch, shows a poorer agreement than in the previous
case (Fig. 6 of Ref. 10). In addition, a loss of coherence between the scattering of both
excitons with the LO phonon was assumed, which is hard to justify. Some improvement has
been obtained recently when damping effects are introduced phenomenologically as imagi-
nary parts of the photon and exciton energies, but still a complete loss of coherence resulted
from the fit [14, 15].
In this paper we include the states of the electron-hole continuum, and the damping
effects in a more rigorous way. Using some matrix elements as free parameters, we can
describe accurately the Raman intensities for both samples studied in Ref. 10.
In the experiments with CdTe QW’s inside II-VI MC’s, the light incides perpendicular to
the (x, y) plane of the QW’s and is collected in the same direction z. Therefore the in-plane
wave vector K = 0, the polarization of the electric field should lie in the (x, y) plane, and
the excitons which couple with the light should have the same symmetry as the electric
field (one of the two Γ5 states of heavy hole excitons [16]). Thus, to lighten the notation
we suppress wave vector and polarization indices. The basic ingredients of the theory are
two or three strongly coupled boson modes, one for the light MC eigenmode with boson
creation operator f †, another one for the 1s exciton (e1) and if is necessary, the 2s exciton
(e2) is also included. We assume that each of these boson states mixes with a continuum of
bosonic excitations which broadens its spectral density. In addition, we include the electron-
hole continuum above the exciton states, described by bosonic operators c
, ck, where k is
the difference between electron and hole momentum in the (x, y) plane (the sum is K = 0
because it is conserved).
The Hamiltonian reads:
H = Eff
iei +
if +H.c.) +
prp +
pf +H.c.)
iqdiq +
(Viqd
iqei +H.c.) +
f +H.c.). (3)
The first three terms describe the strong coupling between the MC photon and the exciton(s)
already included in previous approaches [10, 15]. The fourth and fifth terms describe a
continuum of radiative modes and its coupling to the MC light eigenmode. Their main effect
is to broaden the spectral density of the latter even in the absence of light-matter interaction.
The following two terms have a similar effect for the exciton mode(s). The detailed structure
of the states described by the d
iq operators is not important in what follows. They might
describe combined excitations due to scattering with acoustical phonons. The last two terms
correspond to the energy of the electron hole excitations and their coupling to the MC light
mode.
In Eq. (3) we are making the usual approximation of neglecting the internal fermionic
structure of the excitons and electron-hole operators and taking them as free bosons. This
is an excellent approximation for the conditions of the experiment. We also neglect terms
which do not conserve the number of bosons. Their effect is small for the energies of
interest [17]. These approximations allow us formally to diagonalize the Hamiltonian by a
Bogoliubov transformation. The diagonalized Hamiltonian has the form H =
νpν ,
where the boson operators p†ν correspond to generalized polariton operators and are linear
combinations of all creation operators entering Eq. (3). Denoting the latter for brevity as b
then p†ν =
j . In practice, instead of calculating the Eν and Aνj , it is more convenient
to work with retarded Green’s functions Gjl(ω) = 〈〈bj; b
l 〉〉ω and their equations of motion
ω〈〈bj; b
l 〉〉ω = δjl + 〈〈[bj, H ]; b
l 〉〉ω. (4)
As we show below, the RRS intensity can be expressed in terms of spectral densities derived
from these Green’s functions, which in turn can be calculated using Eq. (4).
Using Fermi’s golden rule, the probability per unit time for a transition from a polariton
state |i〉 = p
|0〉 to states |s〉 = p†νa
†|0〉, where a† creates a LO phonon is
|〈i|H ′|s〉|2ρ(ω), where ρ(ω) =
δ(ω − Eν) = −
ImGνν(ω + i0
+) (5)
is the density of final states. As argued above, we neglect the dependence of Ti on frequency
and take Ts = |Aνf |
2, the weight of photons in the scattered eigenstates. In addition, H ′
should be proportional to the matter (exciton) part of the scattered polariton states. Then,
the Raman intensity is proportional to:
WTs ∝ I = |Aνe1 + αAνe2|
2|Aνf |
2ρ(ω). (6)
Here we are neglecting the contribution of the electron-hole continuum to H ′, and α is the
ratio of matrix elements of the exciton-LO phonon interaction between 2s and 1s excitons.
If the 2s excitons are unimportant, α = 0.
Using Eqs. (4) it can be shown that
ρjl(ω) = −
[Gjl(ω + i0
+)−Gjl(ω − i0
−)] = AνjĀνlρ(ω). (7)
From here and
|Aνj|
2 = 1, it follows that ρ(ω) =
ρjj(ω). Replacing in Eq. (6) we
obtain:
I(ω) =
ρff [ρe1,e1 + |α|
2ρe2,e2 + 2Re(αρe2,e1)]∑
ρjj(ω)
. (8)
In practice, when ω is chosen such that the resonance condition for the outgoing polariton
is fulfilled, we can neglect the contribution of the continuum states in the denominator of
Eq. (8). In particular, if the contribution of the 2s exciton can be neglected (as in sample
A of Ref. 10)
I(ω) =
ρff (ω)ρe1,e1(ω)
ρff(ω) + ρe1,e1(ω)
. (9)
The Green’s functions are calculated from the equations of motion (4). In the final expres-
sions, the continuum states enter through the following sums:
Sf (ω) =
ω + i0+ − ǫp
, Si(ω) =
|Viq|
ω + i0+ − ǫiq
, S ′f(ω) =
ω + i0+ − ǫk
For the first two we assume that the results are imaginary constants that we take as param-
eters:
Sf(ω) = −iδf , Sj(ω) = −iδj (11)
This is the result expected for constant density of states and matrix elements. Our results
seem to indicate that this assumption is valid for the upper and middle polariton branches.
For the lower branch at small k it has been shown that the line width due to the interaction
of polaritons with acoustic phonons depends on detuning Ef −E1 and k‖, being smaller for
small wave vector [18, 19].
The electron-hole continuum begins at the energy of the gap and corresponds to vertical
transitions in which the light promotes a valence electron with 2D wave vector k to the
conduction band with the same wave vector. In the effective-mass approximation, the energy
is quadratic with k and this leads to a constant density of states beginning at the gap. The
matrix element Vk is proportional to Mk = 〈kv|pE |kc〉, where pE is the momentum operator
in the direction of the electric field, and |kv〉, |kc〉 are the wave functions for valence and
conduction electrons with wave vector k. Taking these wave functions as plane waves, one
has Mk ∼ kE , the wave vector in the direction of the electric field. Then |Vk|
2 ∼ k2E. Adding
the contributions of all directions of k one has |Vk|
2 ∼ |k|2 ∼ ǫk (linear with energy for small
energy). This leads to
S ′f = R(ω)− iA(ω −B)Θ(ω −B), (12)
where B is the bottom of the electron-hole continuum (the energy of the semiconductor gap)
and A is a dimensionless parameter that controls the magnitude of the interaction. The real
part R(ω) can be absorbed in a renormalization of the photon energy and is unimportant
in what follows. The imaginary part is a correction to the photon width for energies above
the bottom of the continuum.
Using the theory outlined above, we calculated the intensity of RRS corresponding to
the samples A and B measured in Refs. 10, 14 and compared them with the experimental
results.
Sample A corresponds to the simplest case. Two polariton branches are seen and therefore
only the 1s exciton plays a significant role. The binding energy of this exciton B−E1 is not
well known. The Rabi splitting 2V1 = 19 meV. The width of the Raman scan as a function
of frequency for zero detuning is of the order of w = 0.1 meV (see Fig. 3 of Ref. 10). This
implies the relation 2w2 = δ2f + δ
in our theory. In any case the results are weakly sensitive
to w. Therefore, we have three free parameters in our theory in addition to a multiplicative
constant: B −E1, the ratio of widths δf/δ1 and the slope A.
-15 -10 -5 0 5 10 15
Ef - E1  (meV)
FIG. 1: Raman intensity as a function of detuning for sample A. Solid squares: experimental
results [10]. Solid line: theory (Eq. 9) for B − E1 = 14 meV, δf = δ1 = 0.1 meV, and A = 0.031.
Dashed line: result for a 2x2 matrix (Eq. 2).
The comparison between the experimental and the theoretical intensities is shown in
Fig. 1 for a set of parameters that lead to a close agreement with experiment. The condition
of resonance is established choosing the energy ω for which the intensity given by Eq. (9) has
its second relative maximum (corresponding to the upper polariton branch). The dashed line
corresponds to the case in which only the first three terms (with i = 1) in the Hamiltonian,
Eq. (3) are included. In this case, the intensity is given in terms of the solution of a 2x2
matrix [Eq. (2)] and was used in Ref. 10 to interpret the data. This simple expression gives
a Raman intensity which is an even function of detuning. When the full model is considered,
the Raman intensity falls more rapidly for large detuning Ef − E1 as a consequence of the
hybridization of the photon with the electron-hole continuum. When the energy of the
polariton increases beyond B entering the electron-hole continuum (corresponding to the
kink in Fig. 1), the Raman peak broadens and loses intensity. The kink can be smoothed
if the effect of the infinite excitonic levels below the continuum is included in the model
(leading to an S ′f with continuous first derivative), but this is beyond the scope of this
work. If the ratio δf/δ1 is enlarged, the Raman intensity increases for negative detuning
with respect to its value for positive detuning.
In the experiments with sample B three polariton branches are observed [14] and the
2s exciton plays a role. Experimentally, it is known that the binding energy for the two
excitons are B − E1 =17 meV and B − E2 =2 meV. From the observed Rabi splitting one
has 2V1 = 13 meV, 2V2 = 2.5 meV. In comparison with the previous case, we have the
additional parameter α (the ratio of exciton-LO phonon matrix elements). In addition, to
be able to describe well the intensity for low energies of the middle polariton (left part of
the curve shown in Fig. 2), we need to assume a small linear dependence of E1 with the
position of the incident laser spot in the sample. This dependence is also inferred form the
observed luminescence spectrum [14]. In our model, this corresponds to a dependence of E1
with Ef :
E1 = E
+ z(Ef − E
) (13)
In Fig. 2 we show the intensity at the second maximum of I(ω) (corresponding to the
middle polariton branch) as a function of the energy of this maximum. We also show in the
figure experimental results taken at lower laser excitation and a slightly higher temperature
(4.5 K) than those reported in Ref. 10.
The slope which better describes the data is z = 0.14. This value is close to z = 0.155
which was obtained from a fit of the maxima of luminescence spectrum of the lower and
middle polaritons. We have taken the same value for A as in Fig. 1. The agreement between
theory and experiment is remarkable. As for sample A, the values of δi that result from the
fit are reasonable in comparison with calculated values [18].
1,645 1,650 1,655 1,660 1,665
Middle polariton energy (eV)
FIG. 2: Raman intensity as a function of the middle polariton energy. Solid squares: experimental
results [14]. Solid line: theory for δf = 0.2 meV, δ1 = 0.1 meV, δ2 = 0.12 meV, α = −0.45 and
z = 0.14. A is the same as in Fig. 1 .
In summary, we have proposed a theory to calculate Raman intensity for excitation
of longitudinal optical phonons in microcavities, in which different matrix elements are
incorporated as parameters of the model. The most important advance in comparison with
previous simplified theories [10, 15] is the inclusion of the strong coupling of the electron-
hole continuum with the microcavity photon. Inclusion of this coupling is essential when the
energy of the polariton is near the bottom of the conduction band (at the right of Fig. 1).
We also have included the effects of damping of excitons and photons, coupling them with
a continuum of bosonic excitations. Simpler approaches have included the spectral widths
δf and δi of photons and excitons as imaginary parts of the respective energies, leading to
non-hermitian matrices.
Taking some of the parameters of the model as free (δf/δi, B − E1 and A for sample A,
δf , δi, z and α for sample B), we obtain excellent fits of the observed Raman intensities. The
resulting values of the parameters agree with previous estimates, if they are available. We
are not aware of previous estimates for A and α. As an important improvement to previous
approaches [10, 15] for the case of sample B, we do not have to assume a partial loss of
coherence between 1s and 2s excitons in their scattering with the LO phonon.
Further progress in the understanding of the interaction of excitons with light and
phonons requires microscopic calculations of the parameters δf , δi, A and α, and the effects
of the temperature on them. However, taking into account the difficulties in calculating
these parameters accurately, the present results are encouraging and suggest that the main
physical ingredients are included in our model.
We thank A. Fainstein for useful discussions. This work was supported by PIP 5254 of
CONICET and PICT 03-13829 of ANPCyT.
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	References

  We calculate the intensity of the polariton mediated inelastic light
scattering in semiconductor microcavities. We treat the exciton-photon coupling
nonperturbatively and incorporate lifetime effects in both excitons and
photons, and a coupling of the photons to the electron-hole continuum. Taking
the matrix elements as fitting parameters, the results are in excellent
agreement with measured Raman intensities due to optical phonons resonant with
the upper polariton branches in II-VI microcavities with embedded CdTe quantum
wells.

Theory of polariton mediated Raman scattering in microcavities
L. M. León Hilario, A. Bruchhausen, A. M. Lobos, and A. A. Aligia
Centro Atómico Bariloche and Instituto Balseiro,
Comisión Nacional de Enerǵıa Atómica,
8400 S. C. de Bariloche, Argentina
(Dated: November 4, 2018)
Abstract
We calculate the intensity of the polariton mediated inelastic light scattering in semiconductor
microcavities. We treat the exciton-photon coupling nonperturbatively and incorporate lifetime
effects in both excitons and photons, and a coupling of the photons to the electron-hole continuum.
Taking the matrix elements as fitting parameters, the results are in excellent agreement with
measured Raman intensities due to optical phonons resonant with the upper polariton branches in
II-VI microcavities with embedded CdTe quantum wells.
PACS numbers: 71.36.+c, 78.30.Fs, 78.30.-j
http://arxiv.org/abs/0704.0198v1
Planar semiconductors microcavities (MC’s) have attracted much attention in the last
decade as they provide a novel means to study, enhance and control the interaction between
light and matter [1, 2, 3, 4, 5, 6, 7]. When the MC mode (cavity-photon) is tuned in near
resonance with the embedded quantum-well (QW) exciton transitions, and the damping
processes involved are weak in comparison to the photon-matter interaction, the eigenstates
of the system become mixed exciton-photon states, cavity-polaritons, which are in part light
and in part matter bosonic quasi-particles [1, 2, 3]. Examples of interesting new physics are
the recent evidence of a Bose-Einstein condensation of polaritons in CdTe MC’s [5, 6], and
the construction of devices which increase the interaction of sound and light, opening the
possibility of realizing a coherent monochromatic source of acoustic phonons [7].
Raman scattering due to longitudinal optical (LO) phonons, being a coherent process is
intrinsically connected with the cavity-polariton. The physics of strongly coupled photons
and excitons, the polariton–phonon interaction, and the polariton–external-photon coupling
are clearly displayed [8, 9, 10, 11, 12]. In particular resonant Raman scattering (RRS)
experiments, in which the wave-length of the incoming radiation is tuned in a way such that,
after the emission of a LO phonon, the energy of the outgoing radiation coincides with that of
the cavity polariton branches, have proven to be suited to sense the dynamics of the coupled
modes, and to obtain information about the dephasing of the resonant polaritonic state
[8, 9, 10, 11, 12]. Unfortunately, due to the large remaining luminesence, RRS experiments
in resonance with the lower polariton branch have not yet been achieved in intrinsic II-VI
MC’s [13]. Therefore all reported experiments in these kind of MC’s, with embedded CdTe
QW’s, consider the case of the scattered photons in outgoing resonance with the upper
polariton branch (for two branch-systems: coupling of one exciton mode and the cavity
photon) or with the middle polariton branch (for tree branch-systems: coupling of two
exciton modes and the cavity photon) [9, 10, 14]. The measured intensities in these two
systems were analyzed on the basis of a model in which one (two) exciton states |e〉 are
mixed with a photon state |f〉 with the same in-plane wave vector k, leading to a 2x2 (3x3)
matrix. Essentially, the Raman intensity is proportional to
I ∼ TiTs|〈Pi|H
′|Ps〉|
2, (1)
where Ti describe the probability of conversion of an incident photon |fi〉 into the polariton
state |Pi〉, Ts has an analogous meaning for the scattered polariton |Ps〉 and the outgoing
photon |fs〉, and H
′ is the interaction between electrons and the LO phonons. At the
conditions of resonance with the outgoing polariton, Ti is very weakly dependent on laser
energy or detuning (difference between photon and exciton energies), while Ts is proportional
to the photon strength of the scattered polariton |〈fs|Ps〉|
2. Similarly, in the simplest case
of only one exciton (2x2 matrix) one expects that the matrix element entering Eq. (1) is
proportional to the exciton part of the scattered polariton |〈Pi|H
′|Ps〉|
2 ∼ |〈es|Ps〉|
2. Thus
if the wave function is |Ps〉 = α|fs〉+ β|es〉, one has
I ∼ |〈fs|Ps〉|
2|〈es|Ps〉|
2 = |α|2|β|2. (2)
This model predicts a Raman intensity which is symmetric with detuning and is maximum
at zero detuning. In other words, the intensity is maximum for detunings such that the
scattered polariton is more easily coupled to the external photons, but at the same time
when the polariton is more easily coupled to the optical phonons, which requires a large
matter (exciton) component of the polariton. This result is in qualitative agreement with
experiment [10]. However, for positive detuning the experimental results fall bellow the
values predicted by Eq. (2) (see Fig. 1). This is ascribed to the effects of the electron-hole
continuum above the exciton energy, which are not included in the model [10].
For another sample in which two excitons are involved, the above analysis can be ex-
tended straightforwardly and the intensity depends on the amplitudes of a 3x3 matrix and
matrix element of H ′ involving both excitons [10]. However, comparison with experiments at
resonance with the middle polariton branch, shows a poorer agreement than in the previous
case (Fig. 6 of Ref. 10). In addition, a loss of coherence between the scattering of both
excitons with the LO phonon was assumed, which is hard to justify. Some improvement has
been obtained recently when damping effects are introduced phenomenologically as imagi-
nary parts of the photon and exciton energies, but still a complete loss of coherence resulted
from the fit [14, 15].
In this paper we include the states of the electron-hole continuum, and the damping
effects in a more rigorous way. Using some matrix elements as free parameters, we can
describe accurately the Raman intensities for both samples studied in Ref. 10.
In the experiments with CdTe QW’s inside II-VI MC’s, the light incides perpendicular to
the (x, y) plane of the QW’s and is collected in the same direction z. Therefore the in-plane
wave vector K = 0, the polarization of the electric field should lie in the (x, y) plane, and
the excitons which couple with the light should have the same symmetry as the electric
field (one of the two Γ5 states of heavy hole excitons [16]). Thus, to lighten the notation
we suppress wave vector and polarization indices. The basic ingredients of the theory are
two or three strongly coupled boson modes, one for the light MC eigenmode with boson
creation operator f †, another one for the 1s exciton (e1) and if is necessary, the 2s exciton
(e2) is also included. We assume that each of these boson states mixes with a continuum of
bosonic excitations which broadens its spectral density. In addition, we include the electron-
hole continuum above the exciton states, described by bosonic operators c
, ck, where k is
the difference between electron and hole momentum in the (x, y) plane (the sum is K = 0
because it is conserved).
The Hamiltonian reads:
H = Eff
iei +
if +H.c.) +
prp +
pf +H.c.)
iqdiq +
(Viqd
iqei +H.c.) +
f +H.c.). (3)
The first three terms describe the strong coupling between the MC photon and the exciton(s)
already included in previous approaches [10, 15]. The fourth and fifth terms describe a
continuum of radiative modes and its coupling to the MC light eigenmode. Their main effect
is to broaden the spectral density of the latter even in the absence of light-matter interaction.
The following two terms have a similar effect for the exciton mode(s). The detailed structure
of the states described by the d
iq operators is not important in what follows. They might
describe combined excitations due to scattering with acoustical phonons. The last two terms
correspond to the energy of the electron hole excitations and their coupling to the MC light
mode.
In Eq. (3) we are making the usual approximation of neglecting the internal fermionic
structure of the excitons and electron-hole operators and taking them as free bosons. This
is an excellent approximation for the conditions of the experiment. We also neglect terms
which do not conserve the number of bosons. Their effect is small for the energies of
interest [17]. These approximations allow us formally to diagonalize the Hamiltonian by a
Bogoliubov transformation. The diagonalized Hamiltonian has the form H =
νpν ,
where the boson operators p†ν correspond to generalized polariton operators and are linear
combinations of all creation operators entering Eq. (3). Denoting the latter for brevity as b
then p†ν =
j . In practice, instead of calculating the Eν and Aνj , it is more convenient
to work with retarded Green’s functions Gjl(ω) = 〈〈bj; b
l 〉〉ω and their equations of motion
ω〈〈bj; b
l 〉〉ω = δjl + 〈〈[bj, H ]; b
l 〉〉ω. (4)
As we show below, the RRS intensity can be expressed in terms of spectral densities derived
from these Green’s functions, which in turn can be calculated using Eq. (4).
Using Fermi’s golden rule, the probability per unit time for a transition from a polariton
state |i〉 = p
|0〉 to states |s〉 = p†νa
†|0〉, where a† creates a LO phonon is
|〈i|H ′|s〉|2ρ(ω), where ρ(ω) =
δ(ω − Eν) = −
ImGνν(ω + i0
+) (5)
is the density of final states. As argued above, we neglect the dependence of Ti on frequency
and take Ts = |Aνf |
2, the weight of photons in the scattered eigenstates. In addition, H ′
should be proportional to the matter (exciton) part of the scattered polariton states. Then,
the Raman intensity is proportional to:
WTs ∝ I = |Aνe1 + αAνe2|
2|Aνf |
2ρ(ω). (6)
Here we are neglecting the contribution of the electron-hole continuum to H ′, and α is the
ratio of matrix elements of the exciton-LO phonon interaction between 2s and 1s excitons.
If the 2s excitons are unimportant, α = 0.
Using Eqs. (4) it can be shown that
ρjl(ω) = −
[Gjl(ω + i0
+)−Gjl(ω − i0
−)] = AνjĀνlρ(ω). (7)
From here and
|Aνj|
2 = 1, it follows that ρ(ω) =
ρjj(ω). Replacing in Eq. (6) we
obtain:
I(ω) =
ρff [ρe1,e1 + |α|
2ρe2,e2 + 2Re(αρe2,e1)]∑
ρjj(ω)
. (8)
In practice, when ω is chosen such that the resonance condition for the outgoing polariton
is fulfilled, we can neglect the contribution of the continuum states in the denominator of
Eq. (8). In particular, if the contribution of the 2s exciton can be neglected (as in sample
A of Ref. 10)
I(ω) =
ρff (ω)ρe1,e1(ω)
ρff(ω) + ρe1,e1(ω)
. (9)
The Green’s functions are calculated from the equations of motion (4). In the final expres-
sions, the continuum states enter through the following sums:
Sf (ω) =
ω + i0+ − ǫp
, Si(ω) =
|Viq|
ω + i0+ − ǫiq
, S ′f(ω) =
ω + i0+ − ǫk
For the first two we assume that the results are imaginary constants that we take as param-
eters:
Sf(ω) = −iδf , Sj(ω) = −iδj (11)
This is the result expected for constant density of states and matrix elements. Our results
seem to indicate that this assumption is valid for the upper and middle polariton branches.
For the lower branch at small k it has been shown that the line width due to the interaction
of polaritons with acoustic phonons depends on detuning Ef −E1 and k‖, being smaller for
small wave vector [18, 19].
The electron-hole continuum begins at the energy of the gap and corresponds to vertical
transitions in which the light promotes a valence electron with 2D wave vector k to the
conduction band with the same wave vector. In the effective-mass approximation, the energy
is quadratic with k and this leads to a constant density of states beginning at the gap. The
matrix element Vk is proportional to Mk = 〈kv|pE |kc〉, where pE is the momentum operator
in the direction of the electric field, and |kv〉, |kc〉 are the wave functions for valence and
conduction electrons with wave vector k. Taking these wave functions as plane waves, one
has Mk ∼ kE , the wave vector in the direction of the electric field. Then |Vk|
2 ∼ k2E. Adding
the contributions of all directions of k one has |Vk|
2 ∼ |k|2 ∼ ǫk (linear with energy for small
energy). This leads to
S ′f = R(ω)− iA(ω −B)Θ(ω −B), (12)
where B is the bottom of the electron-hole continuum (the energy of the semiconductor gap)
and A is a dimensionless parameter that controls the magnitude of the interaction. The real
part R(ω) can be absorbed in a renormalization of the photon energy and is unimportant
in what follows. The imaginary part is a correction to the photon width for energies above
the bottom of the continuum.
Using the theory outlined above, we calculated the intensity of RRS corresponding to
the samples A and B measured in Refs. 10, 14 and compared them with the experimental
results.
Sample A corresponds to the simplest case. Two polariton branches are seen and therefore
only the 1s exciton plays a significant role. The binding energy of this exciton B−E1 is not
well known. The Rabi splitting 2V1 = 19 meV. The width of the Raman scan as a function
of frequency for zero detuning is of the order of w = 0.1 meV (see Fig. 3 of Ref. 10). This
implies the relation 2w2 = δ2f + δ
in our theory. In any case the results are weakly sensitive
to w. Therefore, we have three free parameters in our theory in addition to a multiplicative
constant: B −E1, the ratio of widths δf/δ1 and the slope A.
-15 -10 -5 0 5 10 15
Ef - E1  (meV)
FIG. 1: Raman intensity as a function of detuning for sample A. Solid squares: experimental
results [10]. Solid line: theory (Eq. 9) for B − E1 = 14 meV, δf = δ1 = 0.1 meV, and A = 0.031.
Dashed line: result for a 2x2 matrix (Eq. 2).
The comparison between the experimental and the theoretical intensities is shown in
Fig. 1 for a set of parameters that lead to a close agreement with experiment. The condition
of resonance is established choosing the energy ω for which the intensity given by Eq. (9) has
its second relative maximum (corresponding to the upper polariton branch). The dashed line
corresponds to the case in which only the first three terms (with i = 1) in the Hamiltonian,
Eq. (3) are included. In this case, the intensity is given in terms of the solution of a 2x2
matrix [Eq. (2)] and was used in Ref. 10 to interpret the data. This simple expression gives
a Raman intensity which is an even function of detuning. When the full model is considered,
the Raman intensity falls more rapidly for large detuning Ef − E1 as a consequence of the
hybridization of the photon with the electron-hole continuum. When the energy of the
polariton increases beyond B entering the electron-hole continuum (corresponding to the
kink in Fig. 1), the Raman peak broadens and loses intensity. The kink can be smoothed
if the effect of the infinite excitonic levels below the continuum is included in the model
(leading to an S ′f with continuous first derivative), but this is beyond the scope of this
work. If the ratio δf/δ1 is enlarged, the Raman intensity increases for negative detuning
with respect to its value for positive detuning.
In the experiments with sample B three polariton branches are observed [14] and the
2s exciton plays a role. Experimentally, it is known that the binding energy for the two
excitons are B − E1 =17 meV and B − E2 =2 meV. From the observed Rabi splitting one
has 2V1 = 13 meV, 2V2 = 2.5 meV. In comparison with the previous case, we have the
additional parameter α (the ratio of exciton-LO phonon matrix elements). In addition, to
be able to describe well the intensity for low energies of the middle polariton (left part of
the curve shown in Fig. 2), we need to assume a small linear dependence of E1 with the
position of the incident laser spot in the sample. This dependence is also inferred form the
observed luminescence spectrum [14]. In our model, this corresponds to a dependence of E1
with Ef :
E1 = E
+ z(Ef − E
) (13)
In Fig. 2 we show the intensity at the second maximum of I(ω) (corresponding to the
middle polariton branch) as a function of the energy of this maximum. We also show in the
figure experimental results taken at lower laser excitation and a slightly higher temperature
(4.5 K) than those reported in Ref. 10.
The slope which better describes the data is z = 0.14. This value is close to z = 0.155
which was obtained from a fit of the maxima of luminescence spectrum of the lower and
middle polaritons. We have taken the same value for A as in Fig. 1. The agreement between
theory and experiment is remarkable. As for sample A, the values of δi that result from the
fit are reasonable in comparison with calculated values [18].
1,645 1,650 1,655 1,660 1,665
Middle polariton energy (eV)
FIG. 2: Raman intensity as a function of the middle polariton energy. Solid squares: experimental
results [14]. Solid line: theory for δf = 0.2 meV, δ1 = 0.1 meV, δ2 = 0.12 meV, α = −0.45 and
z = 0.14. A is the same as in Fig. 1 .
In summary, we have proposed a theory to calculate Raman intensity for excitation
of longitudinal optical phonons in microcavities, in which different matrix elements are
incorporated as parameters of the model. The most important advance in comparison with
previous simplified theories [10, 15] is the inclusion of the strong coupling of the electron-
hole continuum with the microcavity photon. Inclusion of this coupling is essential when the
energy of the polariton is near the bottom of the conduction band (at the right of Fig. 1).
We also have included the effects of damping of excitons and photons, coupling them with
a continuum of bosonic excitations. Simpler approaches have included the spectral widths
δf and δi of photons and excitons as imaginary parts of the respective energies, leading to
non-hermitian matrices.
Taking some of the parameters of the model as free (δf/δi, B − E1 and A for sample A,
δf , δi, z and α for sample B), we obtain excellent fits of the observed Raman intensities. The
resulting values of the parameters agree with previous estimates, if they are available. We
are not aware of previous estimates for A and α. As an important improvement to previous
approaches [10, 15] for the case of sample B, we do not have to assume a partial loss of
coherence between 1s and 2s excitons in their scattering with the LO phonon.
Further progress in the understanding of the interaction of excitons with light and
phonons requires microscopic calculations of the parameters δf , δi, A and α, and the effects
of the temperature on them. However, taking into account the difficulties in calculating
these parameters accurately, the present results are encouraging and suggest that the main
physical ingredients are included in our model.
We thank A. Fainstein for useful discussions. This work was supported by PIP 5254 of
CONICET and PICT 03-13829 of ANPCyT.
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