8 Theory of enhanced second-harmonic generation
by the quadrupole-dipole hybrid exciton
Oleksiy Roslyak
Physics Department, The City College, CUNY
Convent Ave. at 138 St, New York, N.Y. 10031, USA
E-mail: avroslyak@gmail.com
Joseph L. Birman
Physics Department, The City College, CUNY
Convent Ave. at 138 St, New York, N.Y. 10031, USA
Abstract. We report calculated substantial enhancement of the second harmonic
generation (SHG) in cuprous oxide crystals resonantly hybridized with an appropriate
organic material (DCM2:CA:PS ’solid-state solvent’). The quadrupole origin of the
inorganic part of the quadrupole-dipole hybrid provides inversion symmetry breaking
and the organic part contributes to the oscillator strength of the hybrid. We show that
the enhancement of the SHG, compared to bulk cuprous oxide crystal, is proportional
to the ratio of the DCM2 dipole moment and the effective dipole moment of the
quadrupole transitions in the cuprous oxide. It is also inversely proportional to the
line-width of the hybrid and bulk excitons. The enhancement may be regulated by
adjusting the organic blend (mutual concentration of the DCM2 and CA part of the
solvent) and pumping conditions(varying the angle of incidence in case of optical
pumping or populating the minimum of the lower branch of the hybrid in case of
electrical pumping).
PACS numbers: 73.21.La, 73.22.Dj, 78.67.Hc
http://arxiv.org/abs/0704.1923v3
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton2
1. Introduction
Considerable attention has been paid to the relatively strong optical second-harmonic
generation (SHG) in thin films (D4h symmetry) and bulk (Oh symmetry) of
cuprous oxide crystals. Which was first addressed in the pioneering work of Shen
[Shen et al. (1996)]. This effect is attributed to the electric-quadrupole h̄ω1S = 2.05 eV
exciton effect. The quadrupole exciton has very small oscillator strength but it possess
rather narrow line-width h̄γ1S. So the effect is well pronounced when the exciting laser
energy is close to one h̄ω1S − h̄ω ≪ h̄γ1S or two photon resonance h̄ω1S − 2h̄ω ≪ h̄γ1S.
In the dipole approximation this effect disappears [Atanasov et al. (1994)].
We propose to amplify the SHG characteristic of the 1S quadrupole Wannier exciton
(WE) in cuprous oxide by making a hybrid with an organic Frenkel exciton (FE)
(See next section for more details). The idea of resonant enhancement of some non-
linear properties generic to semiconductor dipole-allowed Wannier-Mott (WE) excitons
was presented in pioneering work of [Agranovich et al. (1998)] for the layered organic-
inorganic heterostructures. It was also developed for quantum wires and dots embedded
into organic shell [Engelmann et al. (1998)],[Gao et al. (2004)] or attached to dendrimer
structure [Huong and Birman (2000)],[Huong and Birman (2003)].
In our previous work [Roslyak and Birman (2007a)] we demonstrated considerable
enhancement of another non-linear effect in cuprous oxide, photo-thermal bi-stability
[Dasbach (2004)]. We demonstrated a considerable enhancement in the hysteresis-like
region size (from µeV for bulk cuprous oxide to meV for the hybrid). The enhancement
was attributed to the large oscillator strength of the hybrid exciton inherited from
the organic part and still rather narrow line-width of the same order as the coupling.
Analogous enhancement can be expected for the SHG, which is the subject of this paper.
In Section 2 we propose a pump-prob experiment to reveal the SHG enhancement
due to the resonant dynamical hybridization and briefly discuss relevant quadrupole
hybrid exciton properties. In the next Section 3 we address the question how this
resonant‡ enhancement depends on such parameters of the system as oscillator strengths
and damping of the FE and WE constituting the hybrid. Using a classical model of
nonlinear coupled oscillators, we demonstrate that while the big ratio of the hybrid
oscillator strength suggests many orders of the enhancement magnitude it is actually
somehow reduced by the rather small coupling parameter and density of the FE.
Because the FE is dynamically brought into resonance with the WE there is an
important hybridization time τh parameter. Hence, in the Section 4, we develop more
sophisticated quantum mechanical model to address the dynamics of the hybrid SHG.
Namely we show that the signal enhancement drastically depends on the either the one
probes the system before or after the hybridization occurred.
‡ The resonance occurs between the FE and WE
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton3
2. Proposed experimental set-up for the SHG
In this work we adopt the concept of a layered organic-inorganic heterostructure. The
inorganic component of the hybrid is a thin layer of Cu2O (quantum well, latter in
the text referred to as QW) grown upon a film of the organic composite (See Fig.1).
Due to the small radius of both the WE and FE exciton part of the hybrid one can
neglect the effect of confinement. In this case one can not tune the two types of excitons
in resonance by adjusting the confinement (Lw > a
B ≈ to the cuprous oxide unit cell
a = 4.6 Å). The QW confinement just assures the WE propagate along the interface and
is the subjected to the electric field gradient of the FE propagating along the adjacent
chain of the DCM2 molecules.
x, kx, Eg
143 meV
2.17 eV
≫ 2.17 eV
Cu2O PS :DCM2: CA
Lw ≈ 4.6 Å
probe
Figure 1. Schematic representation and the energy offset of a possible experimental
set-up to observe the enhanced SHG by the quadrupole-dipole exciton. Here the
inorganic Cu2O quantum well provides the 1S quadrupole WE. The DCM2 part of
the organic ’solid state solute’ provides dipole allowed FE (set of small arrows); the
PS host prevents wave function overlapping between organic and inorganic excitons;
CA under proper concentration allows tuning of the excitons into the resonance.
To provide resonance between WE in cuprous oxide and FE in the organic, we pro-
posed utilization of ’solid state solvation’ (SSS) of the DCM2 § molecules in transpar-
ent polystyrene (PS) host doped with camorphic anhydride (CA)[Bulovic et al. (1999)].
The SSS is a type of solvatochromism manifesting itself as some change in the spec-
tral position of the absorption/luminescence band due to change in the polarity of the
medium. The Förster dipole-dipole non-resonant interaction between DCM2 and CA
modifies the energy structure of the involved molecules.
§ [2-methyl-6-2-(2,3,6,7-tetrahydro-1H, 5H - benzo[i,j] - quinolizin - 9 - yl) - ethenyl] - 4H - pyran - 4 -
ylidene] propane dinitritle.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton4
During the ’slow’ phase (τs ≈ 3.3ns) the energy of the FE ‖ experiences a red shift
linear with the CA concentration due to non-resonant dipole-dipole interaction with the
CA molecules. Not that our model capitalizes on the fact that DCM2 molecules form
a 2D layer rather than been diluted in the PS:CA solvent which is the case of currently
manufactured optical light emitting devices (OLED). This allows us to neglect rather
complicated problem of the inhomogeneous broadening of the FE energy by utilizing a
mean field approximation ¶. For the mean field approximation the red spectral shift
of the FE energy in resonance with the quadrupole WE can be accomplished with
ρCA ≈ 22% CA concentration.
To avoid complicated problems of time dependent hybridization and stay within
the analytical model framework, we assume that the FE and WE are in exact resonance
once the DCM2 energy is in close proximity to the WE energy i.e. h̄ωDCM2 −
h̄ω1S ≤ Γk. We introduced the quadrupole-dipole coupling parameter Γk ≤ 4 µeV
[Roslyak and Birman (2007b)] (See also Appendix (A.1)). This resonant coupling gives
rise to the upper and lower branches of the quadrupole-dipole hybrid (QDH) dispersion
+: h̄ωu,l = h̄ω1S ± Γk. To populate both of the branches one needs a second pumping
photon tuned into resonance with the 1S transition.
The radiation field interacts through both dipole and quadrupole part of the hybrid.
The dipole interaction can be utilized to produce linear response signal due to the
pumping [Roslyak and Birman (2007a)]. By using the non-linear response to the prob
signal, the second harmonic can be generated through the quadrupole part of the hybrid.
Different SHG regimes can be achieved by changing the timing between pumping and
prob signals (See section 4 for more details).
According to the selection rules for the quadrupole-dipole hybrid the pumping
signal, running along the organic-inorganic interface of the heterostructure, induces
the linear polarization in the z direction [Roslyak and Birman (2007b)]. The prob
signal induces the second order non-linear response in the cuprous oxide. Which is
perpendicular to the interface, and defined by the second order polarization along the
x direction (See Fig.1). The net polarization is given by a second rank tensor through
the following expression:
P (1)z + P
l=ĵ×x̂
i,zEi + iχ
l,i,j,xkxEiEj (1)
Here Ei, Ej are the electric field of the pumping and prob lasers correspondingly. The x
component of the prob signal wave vector is taken to be close to zero to avoid possible
interference in momentum conservation. For the sake of simplicity we are going to omit,
x and l indexes of the tensor keeping in mind that the wave vector of the pump signal
‖ in our case we define the FE as DCM2 excitation
¶ Indeed, in our simplified model the DCM2 molecules are not randomly situated but rather form
a uniform (homogeneous) thin layer near the interface. Also the experimentally observed FE energy
relaxation shows no significant energy fluctuations. This experiments are performed at MIT by the Dr.
Bulovic. Although this results are not officially published yet, but reported in the MIT proceedings.
+ See the eigenvalues of the linearized system (4) or the Hamiltonian (8)
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton5
has only x component and the SH signal is perpendicular to it and the prob signal
polarization: iχ
l,i,j,xkx = χ
i,j .
In this paper we develop both classical and quantum mechanical models, which can
be used to find a specific form of the hybrid second order nonlinear susceptibility. In
section 3 we demonstrate that the second order non linearity (generic to the cuprous
oxide and introduced through a small parameter λ) is enhanced due to the resonant
quadrupole-dipole hybridization with the organic (See (5)). In section 4 we develop
quantum theory of the enhanced SHG. It allows investigation of different regimes of the
process defined by the time ordering between the prob pulse and the time when the FE
and WE energies are close enough to form the hybrid. We generalize the concept of the
double-sided Feynman diagrams [Mukamel (1995)] to include non radiative processes of
the energy exchange between DCM2 and CA as well as resonant QDH between DCM2
and cuprous oxide.
3. Anharmonic coupled oscillators model
As a first step, we will use the simplest classical model neglecting the non-local effects
of the linear χ
i,z and non-linear susceptibility χ
l,i,j,x to describe the hybrid SHG.
Namely, we adopt an extension of the anharmonic oscillator model [Bloembergen (1965),
Mukamel (1995)] generalized for the case of resonant coupling between two distinct sets
of oscillators. This simplified picture only covers the case when the pumping field is
polarized along ẑ ‖ [001] axis (Ei = Ez) and we prob the hybrid system (ω1S = ωF )
with a signal perpendicular to the interface and polarized along ŷ ‖ [010] direction
(Ej = Ey).
We consider the WE in cuprous oxide as an assembly of the oscillators with
the oscillator strength per unit cell given by fxz,k ∝ kx (See for example reference
[Moskalenko and Liberman (2002)]). The second set of the oscillators with the oscillator
strength given by fF corresponds to the FE in the organic.
Treating the wave vector k as just another parameter∗, the polarization PW , P F
due to WE and FE can be written in terms of the effective electron-hole displacements
X, Y as:
aWB S
fxz,keX (2)
P F =
ρDCM2N
fFeY (3)
Here S is the area of the interface and aWB , a
B are the WE and FE Bohr radius, e is the
electron charge. The surface density of the WE and FE excitons are Nfxz,k/ (aBS) and
ρDCM2Nf
F/ (aBS) correspondingly and N is the total number of the oscillators. Here
we also took into account the low density (ρDCM2 = 0.05%) of the DCM2 molecules in
the organic to avoid the aggregation effect [Madigan and Bulovic (2003)].
∗ In the text we are going to omit index k unless we put an emphasis on it
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton6
In the time frame of the hybridization τs−τh < t < τs, the WE and FE energies are
at perfect resonance. Hence, the system of equations governing the oscillators dynamics
can be written in the form:
Ẍ + ω21SX + γẊ −
2ω1SΓk
Y − ω21SλX2 = 0
Ÿ + ω21SY + γẎ −
2ω1SΓk
The nonlinear factor ω21Sλ appears due to the prob signal. It is defined such that λ
has dimensions of reciprocal length and is considered to be small in a sense that it is
much less than the reciprocal of the maximum displacement of FE (Y ) and WE (X)
oscillator. The exact value of λ can be obtained either from an experiment or from the
microscopic quantum theory (See next section for more details).
The terms proportional to γ describe the QDH damping. The terms proportional
to 2ω1SΓk/h̄ describe the quadrupole-dipole coupling. Hence, the eigenvalues of the
linearized system of equations (4) give both branches of the QDH.
The system is driven dominantly by the light-dipole interaction in the organic and
the quadrupole-light interaction is neglected (m is the electron mass).
Using standard perturbation theory with respect to the small parameter λ in zero
order (neglecting the quadratic term) and combining equations (1,2,4) one has the linear
response of the hybrid and bulk cuprous oxide given by the following expressions:
Hy (ω) =
ρDCM2N
fFe2/m (ω21S − ω2 + iωγ)
(ω21S − ω2 + iωγ)
2 − (2ω1SΓk/h̄)2
Cu2O (ω) =
aWB S
fxz,ke
ω21S − ω2 + iγ
Including the nonlinear term as a source for the SHG to first order in the perturbation
parameter, there is a displacement at 2ω. The SHG response is given by a solution of
the following system:
Ẍ + ω21SX + γẊ −
2ω1SΓk
Y − ω21SλX2λ=0 = 0
Ÿ + ω21SY + γẎ −
2ω1SΓk
X = 0
Using the definitions (1) and (2) one gets the following non-linear second order response
function for the hybrid and bulk cuprous oxide correspondingly:
Hy (2ω;ω, ω) =
ρDCM2N
fFe3/m2ω21Sλ (2ω1SΓk/h̄)
ω21S − (2ω)
+ i2ωγ
− (2ω1SΓk/h̄)2
(ω21S − ω2 + iωγ)
(ω21S − ω2 + iωγ)
2 − (2ω1SΓk/h̄)2
Cu2O (2ω;ω, ω) =
aWB S
fxz,ke
3/m2ω21Sλ
ω21S − (2ω)
+ i2γ
(ω21S − ω2 + iγ)
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton7
Straightforward comparison of the expressions above evinces the resonant rise of the
second order nonlinearity owing to hybridization. There are several competing factors
involved. The enhancement by means of big oscillator strength ratio fF/fxz,k is reduced
by rather small coupling parameter ΓK and small DCM2 density ρDCM2 (see more
numerical details in Section 5).
4. Quantum theory of SHG due to the QDH
Although the system of non-linear susceptibilities (5) in principle solves the problem of
SHG due to the hybrid it does not clarify the origin of the nonlinearity λ. Also, such an
important parameter as the hybridization time τh is left out of the classical description.
Hence, in this section we propose a unified quantum theory of the hybrid SHG.
The linear response of the hybrid is due to dipole transitions from the ground |g〉
state♯ to the FE |F 〉 in the organic and due to quadrupole transitions to the WE |1S〉
in the cuprous oxide. The non-linearities are the result of some intermediate inter-band
transitions in the cuprous oxide [Mukamel (1995)].
In cuprous oxide the nearest state in energy to the quadrupole ortho-exciton
h̄ω1S
is the h̄ω2P
dipole allowed excitonic band |2P 〉, Eg > h̄ω2P > h̄ωF >
h̄ω1S. Hence it plays the main role in formation of the non-linear response and can be
excited by the properly tuned prob signal. We neglect all the rest of inter-band and
intra-band†† transitions. Therefore, the states above form a complete basis for the SHG
problem:
|g〉 , |1S〉 , |F 〉 , |2P 〉 (6)
Inversion symmetry of the DCM2 is also broken by the CA induced local field and the
interface effect. Therefore, unlike in classical model, the contribution from the organic
to the SHG has to be consider as well. But due to the smallness of the symmetry
breaking local field it contributes a little to the SHG enhancement.
Using the basis above let us introduce creation operators for the FE and the 1S
and the 2P WE exciton b† = |F 〉 〈g|, B†1S = |1S〉 〈g|, B
2P = |2P 〉 〈g| respectively. The
commutation algebra of the operators is presented in the Appendix (B.1).
The net polarization of the sample is defined as [Mukamel (1995)]:
P = µi1S,k
1S +B1S
+ µi2P
2P +B2P
b† + b
1S,2P
1SB2P +B1SB
Here µi1S,k = î · ẑ kxQx,z = 3 · 10−5(kx/k0,x) D is an effective dipole moment
[Moskalenko and Liberman (2002), Roslyak and Birman (2007b)] due to the quadrupole
transitions associated with the oscillator strength; k0 is the resonant wave vector for bulk
♯ when no excitations are present in the system
††due to small radius of the quadrupole WE
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton8
cuprous oxide( Appendix (A.2)). The dipole moment of the transitions from |1S〉 to
〈2P | is defined by [Artoni et al. (2002), Elliott (1961)]:
1S,2P
Ne2h̄2f2P
SaWB 2m
ĵ × x̂
= 6 · 10−3 D2
Finally, the DCM2 dipole moment of the transition from |g〉 to 〈F | per unit area of the
interface is given by [Madigan and Bulovic (2004)]:
ρDCM2Ne
2h̄2fF
SaFB2m
∗h̄ω1S
= 0.2 D2
Using equation (7) and the rotating wave approximation for the resonant wave
vector k, the hybrid Hamiltonian can be written as:
H = h̄ωF b
†b+ h̄ω1SB
1SB1S + E2PB
2PB2P + Γk
1Sb+B1Sb
+ (8)
i + bEi
+ µi1S,k
i +B1SEi
+ µi2P
i +B2PEi
1S,2P
1SB2PE
j +B1SB
The linear response from both branches of the hybrid may be observed by pumping the
hybrid with two signals Ei||ẑ ∝ eiωt. The first photon h̄ω = EDCM2 excites DCM2
molecules. During the time period τs − τh the system relaxes to the FE exciton energy
close to h̄ω1S thus providing resonance between WE and FE. Then the second pumping
photon h̄ω = h̄ω1S enters and excites quadrupole WE so that both QDH branches are
populated. The QDH exciton lives for τh nano-seconds and then both branches of the
hybrid relaxes to the ground state emitting photons of the energy h̄ω1S ± Γk.
Generalizing conventional double-sided Feynman diagrams [Mukamel (1995)] to
include the non-radiative processes, the linear response from the QDH can be represented
by the following diagram:
|g〉 〈g|
〈1S ⊕ F 〉
h̄ω1S−Γk
dd d$
d$ h̄ω1S+Γk
8x8x8x
8x8x8x
h̄ω1S
〈DCM2| //
h̄ω1S
EDCM2
τs−τh
i (ω, k) = µ
1S,0 +B1S,0
+ µiF
0 + b0
(µiF )
(h̄ω − h̄ω1S + ih̄γ) + µiFµi1SΓk
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
+ c.c. (10)
On the diagram the wavy lines represent the incoming and outgoing photons; the straight
lines stand for the non-radiative transitions. The diagram shows energy exchange
between photon-exciton and exciton-exciton as well as the time separation between
two pumping signals. Time increases from bottom to the top of the diagram as for the
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton9
conventional Feynman diagram. The hybrid life time is denoted as τh = 1/γ and the
hybridization between FE and WE is denoted as ⊕.
In the derivation of the linear response χ
i (ω, k) we used equation (7) along with
solutions of the Heisenberg equations of motion presented in the Appendix II (B.2).
Formally the linear response can be written in terms of the hybrid Green’s functions as:
i (ω, k) =
a,b={g,1S,F}
µiabµ
baIab (ω)
I1S,g = IF,g =
h̄ω − h̄ω1S + ih̄γ
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
I1S,F =
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
Iab = I
Here the dipole matrix elements in the corresponding basis (6) are given by:
0 µ1S µF 0
µ1S 0
µ1SµF 0
µFµ1S 0 0
0 0 0 0
Note that we neglected the non-resonant term associated with ground state dipole
moment of the organic µg.
The SHG is due to second order response Ej⊥Ei||z and given by the last term in
the equation (7) and the solutions of the equations of motion (B.2,B.3). The first type
of the SHG is formed when the branches of the hybrid interacts with the |2P 〉 level
excited by the prob signal. Using all the diagram conventions we adopted above, the
diagram for this non linear process is given below:
|g〉 〈g|
|2P 〉 oo
µ1S,2P
// 〈1S ⊕ F 〉
h̄ω1S−Γkd$
h̄ω1S+Γk8x8x
888x8x8x
h̄ω2P
h̄ω1S
〈DCM2| //
h̄ω1S
τ2P>τs−τh
〈g| 〈g|
EDCM2
τs−τh
ij (2ω;ω, ω) = µ1S,2P
1S,0B2P,1 + c.c.
= (11)
µi2Pµ
1S,2P (µ
1S (h̄ω − h̄ω1S + ih̄γ) + µiFΓk)
(2h̄ω − h̄ω2P )
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
) + c.c. (12)
Here the prob signal comes after the hybrid is formed: τ2P > τs − τh.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton10
Another second order non linear response can be formed if the prob signal is coming
before the hybridization τ2P < τs − τh. It can be represented by the following diagram:
|g〉 |g〉 〈g|
|g〉 h̄ω1S///o/o/o 〈1S ⊕ F 〉
1S±Γk
1S∓Γk
h̄ω2P
µ1S,2P
|2P 〉 〈DCM2|
// |CA〉
τs−τh
τ2P<τs−τh
h̄ω2P
EDCM2
ij (2ω;ω, ω) = µ1S,2P
1S,1B2P,0 + c.c.
µi2Pµ
1S,2P (µ
1S (2h̄ω − h̄ω1S + ih̄γ) + µiFΓk)
(h̄ω − h̄ω2P )
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
The Green’s function representation of the SHG due to the second order response is
given by the following expression:
ij (2ω;ω, ω) = µ
1S,2P
a={g,1S,F,2P}
µia,1Sµ
2P,a ×
× [Ia,1S (ω) Ia,2P (2ω) + I1S,a (2ω) I2P,a (ω)]
I2P,g =
h̄ω − h̄ω2P
The dipole matrix elements on the basis (6) are given by:
0 µ1S µF µ2P
µ1S 0
µ1SµF µ1S,2P
µFµ1S 0 0
µ2P µ1S,2P 0 0
According to the last term in the equation (7), the signal at 2h̄ω = h̄ω1S ± Γk may
generate the signal at h̄ω = h̄ω1S ± Γk:
ij (ω; 2ω,−ω) =
1S,2P
(2h̄ω − h̄ω1S + ih̄γ)
(h̄ω − h̄ω2P )2
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
) + c.c.
This type of signal has been experimentally detected [Shen et al. (1996)] in bulk cuprous
oxide (Γk = 0) when the pumping signal was tuned to the wave length between 12285 Å
and 12195 Å. A strong SH signal was detected at 6096 Å which has to be attributed not
only to the narrow line-width of the quadrupole exciton but to the fact that µ1S,2P ≫ µ1S
as well. From the last expression it follows that in this case no increment in the outgoing
signal can be expected due to the hybridization effect.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton11
The third order nonlinearity is responsible as well for some small contribution to
the SHG due to the non-zero ground state dipole moment of the DCM2 molecules
[Kishida et al. (1994)]. In the local electric field created by the polar CA molecules
on the interface Eloc (0) ,the SH signal is due to the third order susceptibility
ij (2ω;ω, ω, 0). The exact expression in terms of the corresponding Green’s functions
is too lengthy to be listed here [Mukamel (1995)], therefore we provide numerical
calculations of the total SHG including the above correction in the next section.
5. Results and discussion
In order to make a numerical comparison of the hybrid and bulk SHG the life-
time of the hybrid plays a major role. Considering the bi-stability effect in the
hybrid [Roslyak and Birman (2007a)] we assumed that the cuprous oxide has purity
of 99.99% with the reported line-width of h̄γ1S = 0.1 meV (pico-second lifetime)
[Shen et al. (1996)]. Therefore the hybrid life-time is dominated by its inorganic part
h̄γ ≈ h̄γ1S. To compensate for such big line-width we also assumed that the DCM2 is
presented as a thin film embedded into PS host close to the interface with the cuprous
oxide. For the non-linear absorption experiment this assumption can be justified as
it makes the absorption length of the hybrid equal to the narrow region around the
interface, of the size of the hybrid itself. But there is a drawback in that model due to
possible aggregation of the DCM2.
Hence in this article we adopted the picture of disordered organic and higher
purity of the inorganic crystal. This will bring the line-width and the coupling
parameter to the same order. For pure cuprous oxide crystal the life-time of the
quadrupole 1S exciton is reported to be τ1S = 1.7 . . . 3.0 ns (h̄γ = 1 . . . 0.5 µeV )
[Dasbach (2004), Frohlich et al. (2005), Elliott (1961)]. Such crystals and thin films
are widely used in searching for BEC of excitons.
In this case the life-time of the 1S quadrupole exciton is mainly determined by
the ortho-para exciton conversion. The life-time of the organic part of the hybrid
is determined by the time the excited DCM2 molecule reaches an equilibrium with
the bath of polar CA molecules. The life-time for the given concentration of the CA
is reported to be 3.3 ns [Madigan and Bulovic (2003), Madigan and Bulovic (2004)].
Because these processes are of the same order, the effective life-time of the hybrid is a
non-trivial combination of the effects described above and will be reported elsewhere.
Here we assume the simplest case of non-coherent life-time of the hybrid h̄γ = 0.29 µeV
[Roslyak and Birman (2007b)].
The intensity of the second-harmonic is proportional to
∣χ(2)kx
(See for example
[Haueisen and Mahr (1973)]). Therefore an important measurable quantity is a relative
value of nonlinear susceptibility
∣χ(2)kx
∣ presented in Fig.2.
The SHG signal is split according to the response from the lower and upper branch
of the hybrid. Asymmetry between this two branches is a result of quantum effects
and not present in the classical anharmonic oscillator picture. We also included the
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton12
−3 −2 −1 0 1 2 3
(2ω − ω1S) /2γ
−1 −0.5 0 0.5 1
(2ω − ω1S) /2Γk
Figure 2. (Color on-line) Relative value of the nonlinear susceptibility in case of bulk
cuprous oxide (dotted curves) and the quadrupole-dipole hybrid (solid curves). The
density of the disordered DCM2 is taken ρDCM2 = 0.005% while the CA density is
ρCA = 22%. The Fig.2a represents moderate coupling Γk = h̄γ1S = 0.29 µeV and
Fig.2b corresponds to strong coupling regime Γk = 3.5 µeV . In the last case the
enhancement is evident and indicated by the different scales for the bare cuprous oxide
(left) and hybrid (right) SHG
corrections due to interface effect in the organic in our numerical simulation.
For the sake of simplicity let us consider two distinct cases. First, the pump laser is
perpendicular to the interface. The states up to ka = k0a are populated thermally. No
hybridization occurs and it is equivalent to the bulk case SHG (See Fig.2 dotted curve).
The maximum power generated by the second-harmonic is proportional to the square
of the following expression:
ij,max (2h̄ω = h̄ω1S)
µ2Pµ1S,2P
h̄ω − h̄ω2P
µ1S,kkx
h̄γ1S
The small relative value of the SHG is due to the narrowness of the cuprous oxide
quantum well.
Second, the pump laser incidence angle is reduced to acquire the wave vector
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton13
k0a ≪ ka. The maximum power generated by the second-harmonic is proportional
to the square of the following expression:
ij,max (2h̄ω = h̄ω1S ± Γk)
µ2Pµ1S,2P
h̄ω − h̄ω2P
αkh̄γ
Here the incidence angle dependent coefficient αk =
5 (See Fig.2a) for ka = 0.13
(Γk = h̄γ = 0.29 µeV ) and αk = 2 for maximum value of the coupling Γk = 3.5 µeV at
ka = 1.57 (See Fig.2b). Finally, comparing the last expression (14) to the bulk cuprous
oxide (13) the second order response of the hybrid is amplified by the factor:
h̄γ1S
Therefore the amplification can be adjusted by manipulating the organic composition
(DCM2 and CA densities) or changing the pump laser incidence angle.
Finally, we would like to note that there is another merit in using the hybrid
structure for the SHG. Namely the fact that the optical pumping can be replaced by
an electrical pumping. For this the hybrid sample has to be placed between Alq3 and
a-NPD [Madigan and Bulovic (2004)] semiconductor plates. The bond structure and
offset of these materials provide electrons and holes to form the hybrid exciton on the
interface. Although, in this case one can expect the SHG only from the lower branch
of the hybrid as the excitons are accumulated in the minimum of the hybrid dispersion
[Roslyak and Birman (2007b)].
6. Conclusion
In this paper we addressed possibility to enhance SHG signal χ(2) generic to a cuprous
oxide bulk crystal as the lowest excitation in this material has quadrupole origin. To
demonstrate the concept we proposed to consider a pump-prob experiment performed
on cuprous oxide sandwiched between the organic composite. An intense pump signal
excites one part of the organic known as DCM2 (FE). Non-resonant (Förster) energy
transfer in the organic layer (”solid state solvation” effect) provides dynamical red shift
of the FE. When the FE energy is close enough to the quadrupole allowed 1S exciton
in the adjacent cuprous oxide the quadrupole-dipole hybridization occurs due to the
FE induced gradient of the electric field penetrating into the inorganic layer. The prob
signal is designed to reveal the SHG signal.
The resonant enhancement of the χ(2) occurs for the hybrid exciton shares properties
of both quadrupole WE (long radiative lifetime) and organic FE (big oscillator strength).
I’t quadrupole part allows χ(2) to be non-vanishing. While it’s FE part provides the
enhancement of the SHG signal compared to the bare cuprous oxide crystal due to
more efficient absorption of the pump signal by means of large oscillator strength of the
hybrid.
However, as we demonstrated in the classical coupled oscillator model
framework,the enhancement is determined not only by the big ratio of the corresponding
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton14
organic/inorganic oscillator strengths but somehow quenched by the small coupling
parameter and low DCM2 density. By varying those parameters one the hybrid SHG
signal may be enhanced by orders of magnitude compared with the generic (cuprous
oxide) one.
To reveal the enhancement dependence on such an important parameter of the
hybrid as the hybridization time τh we proposed more sophisticated quantum theory.
It suggests that there is substantial difference in the hybrid SHG signal provided one
probes the system before or after the hybridization occurred. In the first case the SH is
generated at ω2P/2 frequency. Hence it is vastly suppressed by the short life time of the
dipole-allowed 2P WE in cuprous oxide. Nevertheless, if one probes the system after
the hybridization had happened, the SH is generated at ω1S/2 and is heightened by the
big oscillator strength of the hybrid and it’s small damping coefficient.
7. Acknowledgments
We would like to thank Ms. Upali Aparajita for helpful discussion and comments. The
project was supported in part by PCS-CUNY.
References
[Shen et al. (1996)] M. Shen, S. Koyama, M. Saito, T. Goto, and N. Kuroda, Phys. Rev. B 53, 13477
(1996).
[Atanasov et al. (1994)] R. Atanasov, F. Bassani, and V. Agranovich, Phys. Rev. B 50, 7809 (1994).
[Agranovich et al. (1998)] V. Agranovich, D. Basko, G. L. Rocca, and F. Bassani, J. Phys.: Condens.
Matter 13, 9369 (1998).
[Engelmann et al. (1998)] A. Engelmann, V. I. Yudson, and P. Reineker, Phys. Rev. B 57, 1784 (1998).
[Gao et al. (2004)] Y. Gao, J. Birman, N. Huong, and M. Potasek, Journal of applied physics 96, 1
(2004).
[Huong and Birman (2000)] N. Q. Huong and J. L. Birman, Phys. Rev. B 61, 13131 (2000).
[Huong and Birman (2003)] N. Q. Huong and J. L. Birman, Phys. Rev. B 67, 075313 (2003).
[Roslyak and Birman (2007a)] O. Roslyak and J. Birman, Solid State Communications 143, 487
(2007a).
[Dasbach (2004)] G. Dasbach, Phys. Rev. B 70, 121202 (2004).
[Bulovic et al. (1999)] V. Bulovic, R. Deshpande, and S. Forrest, Chem. Phys. Lett. 308, 317 (1999).
[Roslyak and Birman (2007b)] O. Roslyak and J. L. Birman, Physical Review B (Condensed Matter
and Materials Physics) 75, 245309 (pages 11) (2007b).
[Mukamel (1995)] S. Mukamel, Principles of nonlinear optical spectroscopy (Oxford Press, New York,
1995).
[Bloembergen (1965)] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965).
[Moskalenko and Liberman (2002)] S. Moskalenko and M. Liberman, Phys. Rev. B 65, 064303 (2002).
[Madigan and Bulovic (2003)] C. Madigan and V. Bulovic, Physical review letters 91, 247403 (2003).
[Artoni et al. (2002)] M. Artoni, I. Carusotto, G. La Rocca, and F. Bassani, J. Opt. B.: Quantum
Semiclass. Opt. 4, S345 (2002).
[Elliott (1961)] R. J. Elliott, Phys. Rev. 124, 340 (1961).
[Madigan and Bulovic (2004)] C. Madigan and V. Bulovic (Bringing Materials Research Together,
2004).
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton15
[Kishida et al. (1994)] H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, Y. Tokura, H. Tachibana,
M. Matsumoto, S. Wada, T. Lay, and H. Tashiro, Phys. Rev. B 50, 7786 (1994).
[Frohlich et al. (2005)] D. Frohlich, G. Dasbach, G. B. Hogersthal, M. Bayer, R. Kliebera, D. Sutera,
and H. Stolzb, Solid State Communications 134, 139 (2005).
[Haueisen and Mahr (1973)] D. Haueisen and H. Mahr, Phys. Rev. B 8, 734 (1973).
Appendix A.
Explicit expression for the quadrupole-dipole coupling is given below:
(ε+ ε̃)Lw
ke−kz
(A.1)
Here aFB, a
B are the Bohr radius of the FE and WE exciton; ε̃ and ε are the
corresponding dielectric constants, z′ is the distance to the DCM2 layer, Lw is the
quantum well width. The quadrupole transition matrix element Qxz may be estimated
from the corresponding oscillator strength per unit cell through the following identity
[Moskalenko and Liberman (2002)] and depends on polarization of the pumping laser
field:
fxz,k0 =
4πmEg
3e2h̄2
(z · k0,x ·Qx,z)2 (A.2)
fxz,k0||[1,1,0] = 3.9× 10−9
fxz,k0||[1,1,1] =
3.9× 10−9
Here the energy gap of cuprous oxide is denoted as Eg = 2.173 eV ; k0 = 2.62×105 cm−1
is the resonant wave vector; a is the unit cell size; the unit vector in the pumping field
polarization is z.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton16
Appendix B.
The non-zero commutator relations for the organic and inorganic parts of the hybrid
yield [Mukamel (1995)]:
1S, B1S
= −1 +B†2PB2P + b†b;
1S, B2P
1SB2P ;
2P , B1S
2PB1S (B.1)
2P , B2P
= −1 +B†1SB1S + b†b;
b†, B1S
= b†B1S;
b†, B2P
= b†B2P
b†, b
= −1 +B†2PB2P +B
1SB1S;
1S, b
2P , b
In the TDHF approximate factorization for the averages, the corresponding Heisenberg
equations up to the second order in the creation and annihilation operators are:
= h̄ω1SB
1S + Γkb
† − µFEiB†1Sb+ µ1S,kEi
1− B†2PB2P − b†b
µ2PEiB
1SB2P + µ1S,2PEjB
= −h̄ω1SB1S − Γkb+ µFE⋆i b†B1S − µ1S,kE⋆i
1− B†2PB2P − b†b
2PB1S − µ1S,2PE⋆jB2P
= h̄ω2PB
2P − µFEiB
2P b− µ1S,kEiB
2PB1S + µ2PEi
1− B†1SB1S − b†b
+µ1S,2PEjB
= −h̄ω2PB2P + µFE⋆i b†B2P + µ1S,kE⋆i B
1SB2P − µ2PE⋆i
1−B†1SB1S − b†b
−µ1S,2PE⋆jB1S
= EF b† + ΓkB
1S + µ
1−B†1SB1S −B
2PB2P
− µ1S,kEib†B1S − µ2PEib†B2P
= −EF b− ΓkB1S − µFE⋆i
1−B†1SB1S −B
2PB2P
+ µ1S,kE
1Sb+ µ2PE
Here we omitted the average brackets for the shorter notation. In the exact resonance
between FE and WE excitons h̄ω1S = h̄ωF the linear approximation is straightforward.
The creation operators are proportional to ∝ eiωt and the system above is reduced to:
1S,0 = (h̄ω1S − ih̄γ)B
1S,0 + Γkb
0 + µ1S,kEi − h̄ωB1S,0
= − (h̄ω1S + ih̄γ)B1S,0 − Γkb0 − µ1S,kE⋆i
2P,0 = h̄ω2PB
2P,0 + µ2PEi − h̄ωB2P,0
= −h̄ω2PB2P,0 − µ2PE⋆i
0 = (h̄ω1S − ih̄γ) b
0 + ΓkB
1S,0 + µ
FEi − h̄ωb0
= − (h̄ω1S + ih̄γ) b0 − ΓkB1S,0 − µFE⋆i
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton17
The system above has a solution:
2P,0 =
µ2PEi
h̄ω − h̄ω2P
(B.2)
1S,0 =
µ1SEi (h̄ω − h̄ω1S + ih̄γ) + µFΓkEi
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
µFEi (h̄ω − h̄ω1S + ih̄γ) + µ1SΓkEi
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
The SHG is due to response to induced polarization and is proportional to ∝ ei2ωt:
2h̄ωB
1S,1 = (h̄ω1S − ih̄γ)B
1S,1 + Γkb
1 + µ1S,2PEjB
2h̄ωB1S,1 = (h̄ω1S + ih̄γ)B1S,1 + Γkb1 + µ1S,2PE
2h̄ωB
2P,1 = h̄ω2PB
2P,1 + µ1S,2PEjB
2h̄ωB2P,1 = h̄ω2PB2P,1 + µ1S,2PE
2h̄ωb
1 = (h̄ω1S − ih̄γ) b
1 + ΓkB
2h̄ωb1 = (h̄ω1S + ih̄γ) b1 + ΓkB1S,1
The system has a solution:
2P,1 =
µ1S,2PEjB
2h̄ω − h̄ω2P
(B.3)
1S,1 =
µ1S,2PEj (2h̄ω − h̄ω1S + ih̄γ)B†2P,0
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
µ1S,2PEjΓkB
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
This solutions are implemented in the main text to calculate the linear and nonlinear
responses of the hybrid.
	Introduction
	Proposed experimental set-up for the SHG
	 Anharmonic coupled oscillators model
	 Quantum theory of SHG due to the QDH
	Results and discussion
	Conclusion
	Acknowledgments

  We report calculated substantial enhancement of the second harmonic
generation (SHG) in cuprous oxide crystals resonantly hybridized with an
appropriate organic material (DCM2:CA:PS 'solid-state solvent'). The quadrupole
origin of the inorganic part of the quadrupole-dipole hybrid provides inversion
symmetry breaking and the organic part contributes to the oscillator strength
of the hybrid. We show that the enhancement of the SHG, compared to bulk
cuprous oxide crystal, is proportional to the ratio of the DCM2 dipole moment
and the effective dipole moment of the quadrupole transitions in the cuprous
oxide. It is also inversely proportional to the line-width of the hybrid and
bulk excitons. The enhancement may be regulated by adjusting the organic blend
(mutual concentration of the DCM2 and CA part of the solvent) and pumping
conditions(varying the angle of incidence in case of optical pumping or
populating the minimum of the lower branch of the hybrid in case of electrical
pumping).

Introduction
Considerable attention has been paid to the relatively strong optical second-harmonic
generation (SHG) in thin films (D4h symmetry) and bulk (Oh symmetry) of
cuprous oxide crystals. Which was first addressed in the pioneering work of Shen
[Shen et al. (1996)]. This effect is attributed to the electric-quadrupole h̄ω1S = 2.05 eV
exciton effect. The quadrupole exciton has very small oscillator strength but it possess
rather narrow line-width h̄γ1S. So the effect is well pronounced when the exciting laser
energy is close to one h̄ω1S − h̄ω ≪ h̄γ1S or two photon resonance h̄ω1S − 2h̄ω ≪ h̄γ1S.
In the dipole approximation this effect disappears [Atanasov et al. (1994)].
We propose to amplify the SHG characteristic of the 1S quadrupole Wannier exciton
(WE) in cuprous oxide by making a hybrid with an organic Frenkel exciton (FE)
(See next section for more details). The idea of resonant enhancement of some non-
linear properties generic to semiconductor dipole-allowed Wannier-Mott (WE) excitons
was presented in pioneering work of [Agranovich et al. (1998)] for the layered organic-
inorganic heterostructures. It was also developed for quantum wires and dots embedded
into organic shell [Engelmann et al. (1998)],[Gao et al. (2004)] or attached to dendrimer
structure [Huong and Birman (2000)],[Huong and Birman (2003)].
In our previous work [Roslyak and Birman (2007a)] we demonstrated considerable
enhancement of another non-linear effect in cuprous oxide, photo-thermal bi-stability
[Dasbach (2004)]. We demonstrated a considerable enhancement in the hysteresis-like
region size (from µeV for bulk cuprous oxide to meV for the hybrid). The enhancement
was attributed to the large oscillator strength of the hybrid exciton inherited from
the organic part and still rather narrow line-width of the same order as the coupling.
Analogous enhancement can be expected for the SHG, which is the subject of this paper.
In Section 2 we propose a pump-prob experiment to reveal the SHG enhancement
due to the resonant dynamical hybridization and briefly discuss relevant quadrupole
hybrid exciton properties. In the next Section 3 we address the question how this
resonant‡ enhancement depends on such parameters of the system as oscillator strengths
and damping of the FE and WE constituting the hybrid. Using a classical model of
nonlinear coupled oscillators, we demonstrate that while the big ratio of the hybrid
oscillator strength suggests many orders of the enhancement magnitude it is actually
somehow reduced by the rather small coupling parameter and density of the FE.
Because the FE is dynamically brought into resonance with the WE there is an
important hybridization time τh parameter. Hence, in the Section 4, we develop more
sophisticated quantum mechanical model to address the dynamics of the hybrid SHG.
Namely we show that the signal enhancement drastically depends on the either the one
probes the system before or after the hybridization occurred.
‡ The resonance occurs between the FE and WE
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton3
2. Proposed experimental set-up for the SHG
In this work we adopt the concept of a layered organic-inorganic heterostructure. The
inorganic component of the hybrid is a thin layer of Cu2O (quantum well, latter in
the text referred to as QW) grown upon a film of the organic composite (See Fig.1).
Due to the small radius of both the WE and FE exciton part of the hybrid one can
neglect the effect of confinement. In this case one can not tune the two types of excitons
in resonance by adjusting the confinement (Lw > a
B ≈ to the cuprous oxide unit cell
a = 4.6 Å). The QW confinement just assures the WE propagate along the interface and
is the subjected to the electric field gradient of the FE propagating along the adjacent
chain of the DCM2 molecules.
x, kx, Eg
143 meV
2.17 eV
≫ 2.17 eV
Cu2O PS :DCM2: CA
Lw ≈ 4.6 Å
probe
Figure 1. Schematic representation and the energy offset of a possible experimental
set-up to observe the enhanced SHG by the quadrupole-dipole exciton. Here the
inorganic Cu2O quantum well provides the 1S quadrupole WE. The DCM2 part of
the organic ’solid state solute’ provides dipole allowed FE (set of small arrows); the
PS host prevents wave function overlapping between organic and inorganic excitons;
CA under proper concentration allows tuning of the excitons into the resonance.
To provide resonance between WE in cuprous oxide and FE in the organic, we pro-
posed utilization of ’solid state solvation’ (SSS) of the DCM2 § molecules in transpar-
ent polystyrene (PS) host doped with camorphic anhydride (CA)[Bulovic et al. (1999)].
The SSS is a type of solvatochromism manifesting itself as some change in the spec-
tral position of the absorption/luminescence band due to change in the polarity of the
medium. The Förster dipole-dipole non-resonant interaction between DCM2 and CA
modifies the energy structure of the involved molecules.
§ [2-methyl-6-2-(2,3,6,7-tetrahydro-1H, 5H - benzo[i,j] - quinolizin - 9 - yl) - ethenyl] - 4H - pyran - 4 -
ylidene] propane dinitritle.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton4
During the ’slow’ phase (τs ≈ 3.3ns) the energy of the FE ‖ experiences a red shift
linear with the CA concentration due to non-resonant dipole-dipole interaction with the
CA molecules. Not that our model capitalizes on the fact that DCM2 molecules form
a 2D layer rather than been diluted in the PS:CA solvent which is the case of currently
manufactured optical light emitting devices (OLED). This allows us to neglect rather
complicated problem of the inhomogeneous broadening of the FE energy by utilizing a
mean field approximation ¶. For the mean field approximation the red spectral shift
of the FE energy in resonance with the quadrupole WE can be accomplished with
ρCA ≈ 22% CA concentration.
To avoid complicated problems of time dependent hybridization and stay within
the analytical model framework, we assume that the FE and WE are in exact resonance
once the DCM2 energy is in close proximity to the WE energy i.e. h̄ωDCM2 −
h̄ω1S ≤ Γk. We introduced the quadrupole-dipole coupling parameter Γk ≤ 4 µeV
[Roslyak and Birman (2007b)] (See also Appendix (A.1)). This resonant coupling gives
rise to the upper and lower branches of the quadrupole-dipole hybrid (QDH) dispersion
+: h̄ωu,l = h̄ω1S ± Γk. To populate both of the branches one needs a second pumping
photon tuned into resonance with the 1S transition.
The radiation field interacts through both dipole and quadrupole part of the hybrid.
The dipole interaction can be utilized to produce linear response signal due to the
pumping [Roslyak and Birman (2007a)]. By using the non-linear response to the prob
signal, the second harmonic can be generated through the quadrupole part of the hybrid.
Different SHG regimes can be achieved by changing the timing between pumping and
prob signals (See section 4 for more details).
According to the selection rules for the quadrupole-dipole hybrid the pumping
signal, running along the organic-inorganic interface of the heterostructure, induces
the linear polarization in the z direction [Roslyak and Birman (2007b)]. The prob
signal induces the second order non-linear response in the cuprous oxide. Which is
perpendicular to the interface, and defined by the second order polarization along the
x direction (See Fig.1). The net polarization is given by a second rank tensor through
the following expression:
P (1)z + P
l=ĵ×x̂
i,zEi + iχ
l,i,j,xkxEiEj (1)
Here Ei, Ej are the electric field of the pumping and prob lasers correspondingly. The x
component of the prob signal wave vector is taken to be close to zero to avoid possible
interference in momentum conservation. For the sake of simplicity we are going to omit,
x and l indexes of the tensor keeping in mind that the wave vector of the pump signal
‖ in our case we define the FE as DCM2 excitation
¶ Indeed, in our simplified model the DCM2 molecules are not randomly situated but rather form
a uniform (homogeneous) thin layer near the interface. Also the experimentally observed FE energy
relaxation shows no significant energy fluctuations. This experiments are performed at MIT by the Dr.
Bulovic. Although this results are not officially published yet, but reported in the MIT proceedings.
+ See the eigenvalues of the linearized system (4) or the Hamiltonian (8)
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton5
has only x component and the SH signal is perpendicular to it and the prob signal
polarization: iχ
l,i,j,xkx = χ
i,j .
In this paper we develop both classical and quantum mechanical models, which can
be used to find a specific form of the hybrid second order nonlinear susceptibility. In
section 3 we demonstrate that the second order non linearity (generic to the cuprous
oxide and introduced through a small parameter λ) is enhanced due to the resonant
quadrupole-dipole hybridization with the organic (See (5)). In section 4 we develop
quantum theory of the enhanced SHG. It allows investigation of different regimes of the
process defined by the time ordering between the prob pulse and the time when the FE
and WE energies are close enough to form the hybrid. We generalize the concept of the
double-sided Feynman diagrams [Mukamel (1995)] to include non radiative processes of
the energy exchange between DCM2 and CA as well as resonant QDH between DCM2
and cuprous oxide.
3. Anharmonic coupled oscillators model
As a first step, we will use the simplest classical model neglecting the non-local effects
of the linear χ
i,z and non-linear susceptibility χ
l,i,j,x to describe the hybrid SHG.
Namely, we adopt an extension of the anharmonic oscillator model [Bloembergen (1965),
Mukamel (1995)] generalized for the case of resonant coupling between two distinct sets
of oscillators. This simplified picture only covers the case when the pumping field is
polarized along ẑ ‖ [001] axis (Ei = Ez) and we prob the hybrid system (ω1S = ωF )
with a signal perpendicular to the interface and polarized along ŷ ‖ [010] direction
(Ej = Ey).
We consider the WE in cuprous oxide as an assembly of the oscillators with
the oscillator strength per unit cell given by fxz,k ∝ kx (See for example reference
[Moskalenko and Liberman (2002)]). The second set of the oscillators with the oscillator
strength given by fF corresponds to the FE in the organic.
Treating the wave vector k as just another parameter∗, the polarization PW , P F
due to WE and FE can be written in terms of the effective electron-hole displacements
X, Y as:
aWB S
fxz,keX (2)
P F =
ρDCM2N
fFeY (3)
Here S is the area of the interface and aWB , a
B are the WE and FE Bohr radius, e is the
electron charge. The surface density of the WE and FE excitons are Nfxz,k/ (aBS) and
ρDCM2Nf
F/ (aBS) correspondingly and N is the total number of the oscillators. Here
we also took into account the low density (ρDCM2 = 0.05%) of the DCM2 molecules in
the organic to avoid the aggregation effect [Madigan and Bulovic (2003)].
∗ In the text we are going to omit index k unless we put an emphasis on it
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton6
In the time frame of the hybridization τs−τh < t < τs, the WE and FE energies are
at perfect resonance. Hence, the system of equations governing the oscillators dynamics
can be written in the form:
Ẍ + ω21SX + γẊ −
2ω1SΓk
Y − ω21SλX2 = 0
Ÿ + ω21SY + γẎ −
2ω1SΓk
The nonlinear factor ω21Sλ appears due to the prob signal. It is defined such that λ
has dimensions of reciprocal length and is considered to be small in a sense that it is
much less than the reciprocal of the maximum displacement of FE (Y ) and WE (X)
oscillator. The exact value of λ can be obtained either from an experiment or from the
microscopic quantum theory (See next section for more details).
The terms proportional to γ describe the QDH damping. The terms proportional
to 2ω1SΓk/h̄ describe the quadrupole-dipole coupling. Hence, the eigenvalues of the
linearized system of equations (4) give both branches of the QDH.
The system is driven dominantly by the light-dipole interaction in the organic and
the quadrupole-light interaction is neglected (m is the electron mass).
Using standard perturbation theory with respect to the small parameter λ in zero
order (neglecting the quadratic term) and combining equations (1,2,4) one has the linear
response of the hybrid and bulk cuprous oxide given by the following expressions:
Hy (ω) =
ρDCM2N
fFe2/m (ω21S − ω2 + iωγ)
(ω21S − ω2 + iωγ)
2 − (2ω1SΓk/h̄)2
Cu2O (ω) =
aWB S
fxz,ke
ω21S − ω2 + iγ
Including the nonlinear term as a source for the SHG to first order in the perturbation
parameter, there is a displacement at 2ω. The SHG response is given by a solution of
the following system:
Ẍ + ω21SX + γẊ −
2ω1SΓk
Y − ω21SλX2λ=0 = 0
Ÿ + ω21SY + γẎ −
2ω1SΓk
X = 0
Using the definitions (1) and (2) one gets the following non-linear second order response
function for the hybrid and bulk cuprous oxide correspondingly:
Hy (2ω;ω, ω) =
ρDCM2N
fFe3/m2ω21Sλ (2ω1SΓk/h̄)
ω21S − (2ω)
+ i2ωγ
− (2ω1SΓk/h̄)2
(ω21S − ω2 + iωγ)
(ω21S − ω2 + iωγ)
2 − (2ω1SΓk/h̄)2
Cu2O (2ω;ω, ω) =
aWB S
fxz,ke
3/m2ω21Sλ
ω21S − (2ω)
+ i2γ
(ω21S − ω2 + iγ)
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton7
Straightforward comparison of the expressions above evinces the resonant rise of the
second order nonlinearity owing to hybridization. There are several competing factors
involved. The enhancement by means of big oscillator strength ratio fF/fxz,k is reduced
by rather small coupling parameter ΓK and small DCM2 density ρDCM2 (see more
numerical details in Section 5).
4. Quantum theory of SHG due to the QDH
Although the system of non-linear susceptibilities (5) in principle solves the problem of
SHG due to the hybrid it does not clarify the origin of the nonlinearity λ. Also, such an
important parameter as the hybridization time τh is left out of the classical description.
Hence, in this section we propose a unified quantum theory of the hybrid SHG.
The linear response of the hybrid is due to dipole transitions from the ground |g〉
state♯ to the FE |F 〉 in the organic and due to quadrupole transitions to the WE |1S〉
in the cuprous oxide. The non-linearities are the result of some intermediate inter-band
transitions in the cuprous oxide [Mukamel (1995)].
In cuprous oxide the nearest state in energy to the quadrupole ortho-exciton
h̄ω1S
is the h̄ω2P
dipole allowed excitonic band |2P 〉, Eg > h̄ω2P > h̄ωF >
h̄ω1S. Hence it plays the main role in formation of the non-linear response and can be
excited by the properly tuned prob signal. We neglect all the rest of inter-band and
intra-band†† transitions. Therefore, the states above form a complete basis for the SHG
problem:
|g〉 , |1S〉 , |F 〉 , |2P 〉 (6)
Inversion symmetry of the DCM2 is also broken by the CA induced local field and the
interface effect. Therefore, unlike in classical model, the contribution from the organic
to the SHG has to be consider as well. But due to the smallness of the symmetry
breaking local field it contributes a little to the SHG enhancement.
Using the basis above let us introduce creation operators for the FE and the 1S
and the 2P WE exciton b† = |F 〉 〈g|, B†1S = |1S〉 〈g|, B
2P = |2P 〉 〈g| respectively. The
commutation algebra of the operators is presented in the Appendix (B.1).
The net polarization of the sample is defined as [Mukamel (1995)]:
P = µi1S,k
1S +B1S
+ µi2P
2P +B2P
b† + b
1S,2P
1SB2P +B1SB
Here µi1S,k = î · ẑ kxQx,z = 3 · 10−5(kx/k0,x) D is an effective dipole moment
[Moskalenko and Liberman (2002), Roslyak and Birman (2007b)] due to the quadrupole
transitions associated with the oscillator strength; k0 is the resonant wave vector for bulk
♯ when no excitations are present in the system
††due to small radius of the quadrupole WE
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton8
cuprous oxide( Appendix (A.2)). The dipole moment of the transitions from |1S〉 to
〈2P | is defined by [Artoni et al. (2002), Elliott (1961)]:
1S,2P
Ne2h̄2f2P
SaWB 2m
ĵ × x̂
= 6 · 10−3 D2
Finally, the DCM2 dipole moment of the transition from |g〉 to 〈F | per unit area of the
interface is given by [Madigan and Bulovic (2004)]:
ρDCM2Ne
2h̄2fF
SaFB2m
∗h̄ω1S
= 0.2 D2
Using equation (7) and the rotating wave approximation for the resonant wave
vector k, the hybrid Hamiltonian can be written as:
H = h̄ωF b
†b+ h̄ω1SB
1SB1S + E2PB
2PB2P + Γk
1Sb+B1Sb
+ (8)
i + bEi
+ µi1S,k
i +B1SEi
+ µi2P
i +B2PEi
1S,2P
1SB2PE
j +B1SB
The linear response from both branches of the hybrid may be observed by pumping the
hybrid with two signals Ei||ẑ ∝ eiωt. The first photon h̄ω = EDCM2 excites DCM2
molecules. During the time period τs − τh the system relaxes to the FE exciton energy
close to h̄ω1S thus providing resonance between WE and FE. Then the second pumping
photon h̄ω = h̄ω1S enters and excites quadrupole WE so that both QDH branches are
populated. The QDH exciton lives for τh nano-seconds and then both branches of the
hybrid relaxes to the ground state emitting photons of the energy h̄ω1S ± Γk.
Generalizing conventional double-sided Feynman diagrams [Mukamel (1995)] to
include the non-radiative processes, the linear response from the QDH can be represented
by the following diagram:
|g〉 〈g|
〈1S ⊕ F 〉
h̄ω1S−Γk
dd d$
d$ h̄ω1S+Γk
8x8x8x
8x8x8x
h̄ω1S
〈DCM2| //
h̄ω1S
EDCM2
τs−τh
i (ω, k) = µ
1S,0 +B1S,0
+ µiF
0 + b0
(µiF )
(h̄ω − h̄ω1S + ih̄γ) + µiFµi1SΓk
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
+ c.c. (10)
On the diagram the wavy lines represent the incoming and outgoing photons; the straight
lines stand for the non-radiative transitions. The diagram shows energy exchange
between photon-exciton and exciton-exciton as well as the time separation between
two pumping signals. Time increases from bottom to the top of the diagram as for the
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton9
conventional Feynman diagram. The hybrid life time is denoted as τh = 1/γ and the
hybridization between FE and WE is denoted as ⊕.
In the derivation of the linear response χ
i (ω, k) we used equation (7) along with
solutions of the Heisenberg equations of motion presented in the Appendix II (B.2).
Formally the linear response can be written in terms of the hybrid Green’s functions as:
i (ω, k) =
a,b={g,1S,F}
µiabµ
baIab (ω)
I1S,g = IF,g =
h̄ω − h̄ω1S + ih̄γ
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
I1S,F =
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
Iab = I
Here the dipole matrix elements in the corresponding basis (6) are given by:
0 µ1S µF 0
µ1S 0
µ1SµF 0
µFµ1S 0 0
0 0 0 0
Note that we neglected the non-resonant term associated with ground state dipole
moment of the organic µg.
The SHG is due to second order response Ej⊥Ei||z and given by the last term in
the equation (7) and the solutions of the equations of motion (B.2,B.3). The first type
of the SHG is formed when the branches of the hybrid interacts with the |2P 〉 level
excited by the prob signal. Using all the diagram conventions we adopted above, the
diagram for this non linear process is given below:
|g〉 〈g|
|2P 〉 oo
µ1S,2P
// 〈1S ⊕ F 〉
h̄ω1S−Γkd$
h̄ω1S+Γk8x8x
888x8x8x
h̄ω2P
h̄ω1S
〈DCM2| //
h̄ω1S
τ2P>τs−τh
〈g| 〈g|
EDCM2
τs−τh
ij (2ω;ω, ω) = µ1S,2P
1S,0B2P,1 + c.c.
= (11)
µi2Pµ
1S,2P (µ
1S (h̄ω − h̄ω1S + ih̄γ) + µiFΓk)
(2h̄ω − h̄ω2P )
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
) + c.c. (12)
Here the prob signal comes after the hybrid is formed: τ2P > τs − τh.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton10
Another second order non linear response can be formed if the prob signal is coming
before the hybridization τ2P < τs − τh. It can be represented by the following diagram:
|g〉 |g〉 〈g|
|g〉 h̄ω1S///o/o/o 〈1S ⊕ F 〉
1S±Γk
1S∓Γk
h̄ω2P
µ1S,2P
|2P 〉 〈DCM2|
// |CA〉
τs−τh
τ2P<τs−τh
h̄ω2P
EDCM2
ij (2ω;ω, ω) = µ1S,2P
1S,1B2P,0 + c.c.
µi2Pµ
1S,2P (µ
1S (2h̄ω − h̄ω1S + ih̄γ) + µiFΓk)
(h̄ω − h̄ω2P )
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
The Green’s function representation of the SHG due to the second order response is
given by the following expression:
ij (2ω;ω, ω) = µ
1S,2P
a={g,1S,F,2P}
µia,1Sµ
2P,a ×
× [Ia,1S (ω) Ia,2P (2ω) + I1S,a (2ω) I2P,a (ω)]
I2P,g =
h̄ω − h̄ω2P
The dipole matrix elements on the basis (6) are given by:
0 µ1S µF µ2P
µ1S 0
µ1SµF µ1S,2P
µFµ1S 0 0
µ2P µ1S,2P 0 0
According to the last term in the equation (7), the signal at 2h̄ω = h̄ω1S ± Γk may
generate the signal at h̄ω = h̄ω1S ± Γk:
ij (ω; 2ω,−ω) =
1S,2P
(2h̄ω − h̄ω1S + ih̄γ)
(h̄ω − h̄ω2P )2
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
) + c.c.
This type of signal has been experimentally detected [Shen et al. (1996)] in bulk cuprous
oxide (Γk = 0) when the pumping signal was tuned to the wave length between 12285 Å
and 12195 Å. A strong SH signal was detected at 6096 Å which has to be attributed not
only to the narrow line-width of the quadrupole exciton but to the fact that µ1S,2P ≫ µ1S
as well. From the last expression it follows that in this case no increment in the outgoing
signal can be expected due to the hybridization effect.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton11
The third order nonlinearity is responsible as well for some small contribution to
the SHG due to the non-zero ground state dipole moment of the DCM2 molecules
[Kishida et al. (1994)]. In the local electric field created by the polar CA molecules
on the interface Eloc (0) ,the SH signal is due to the third order susceptibility
ij (2ω;ω, ω, 0). The exact expression in terms of the corresponding Green’s functions
is too lengthy to be listed here [Mukamel (1995)], therefore we provide numerical
calculations of the total SHG including the above correction in the next section.
5. Results and discussion
In order to make a numerical comparison of the hybrid and bulk SHG the life-
time of the hybrid plays a major role. Considering the bi-stability effect in the
hybrid [Roslyak and Birman (2007a)] we assumed that the cuprous oxide has purity
of 99.99% with the reported line-width of h̄γ1S = 0.1 meV (pico-second lifetime)
[Shen et al. (1996)]. Therefore the hybrid life-time is dominated by its inorganic part
h̄γ ≈ h̄γ1S. To compensate for such big line-width we also assumed that the DCM2 is
presented as a thin film embedded into PS host close to the interface with the cuprous
oxide. For the non-linear absorption experiment this assumption can be justified as
it makes the absorption length of the hybrid equal to the narrow region around the
interface, of the size of the hybrid itself. But there is a drawback in that model due to
possible aggregation of the DCM2.
Hence in this article we adopted the picture of disordered organic and higher
purity of the inorganic crystal. This will bring the line-width and the coupling
parameter to the same order. For pure cuprous oxide crystal the life-time of the
quadrupole 1S exciton is reported to be τ1S = 1.7 . . . 3.0 ns (h̄γ = 1 . . . 0.5 µeV )
[Dasbach (2004), Frohlich et al. (2005), Elliott (1961)]. Such crystals and thin films
are widely used in searching for BEC of excitons.
In this case the life-time of the 1S quadrupole exciton is mainly determined by
the ortho-para exciton conversion. The life-time of the organic part of the hybrid
is determined by the time the excited DCM2 molecule reaches an equilibrium with
the bath of polar CA molecules. The life-time for the given concentration of the CA
is reported to be 3.3 ns [Madigan and Bulovic (2003), Madigan and Bulovic (2004)].
Because these processes are of the same order, the effective life-time of the hybrid is a
non-trivial combination of the effects described above and will be reported elsewhere.
Here we assume the simplest case of non-coherent life-time of the hybrid h̄γ = 0.29 µeV
[Roslyak and Birman (2007b)].
The intensity of the second-harmonic is proportional to
∣χ(2)kx
(See for example
[Haueisen and Mahr (1973)]). Therefore an important measurable quantity is a relative
value of nonlinear susceptibility
∣χ(2)kx
∣ presented in Fig.2.
The SHG signal is split according to the response from the lower and upper branch
of the hybrid. Asymmetry between this two branches is a result of quantum effects
and not present in the classical anharmonic oscillator picture. We also included the
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton12
−3 −2 −1 0 1 2 3
(2ω − ω1S) /2γ
−1 −0.5 0 0.5 1
(2ω − ω1S) /2Γk
Figure 2. (Color on-line) Relative value of the nonlinear susceptibility in case of bulk
cuprous oxide (dotted curves) and the quadrupole-dipole hybrid (solid curves). The
density of the disordered DCM2 is taken ρDCM2 = 0.005% while the CA density is
ρCA = 22%. The Fig.2a represents moderate coupling Γk = h̄γ1S = 0.29 µeV and
Fig.2b corresponds to strong coupling regime Γk = 3.5 µeV . In the last case the
enhancement is evident and indicated by the different scales for the bare cuprous oxide
(left) and hybrid (right) SHG
corrections due to interface effect in the organic in our numerical simulation.
For the sake of simplicity let us consider two distinct cases. First, the pump laser is
perpendicular to the interface. The states up to ka = k0a are populated thermally. No
hybridization occurs and it is equivalent to the bulk case SHG (See Fig.2 dotted curve).
The maximum power generated by the second-harmonic is proportional to the square
of the following expression:
ij,max (2h̄ω = h̄ω1S)
µ2Pµ1S,2P
h̄ω − h̄ω2P
µ1S,kkx
h̄γ1S
The small relative value of the SHG is due to the narrowness of the cuprous oxide
quantum well.
Second, the pump laser incidence angle is reduced to acquire the wave vector
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton13
k0a ≪ ka. The maximum power generated by the second-harmonic is proportional
to the square of the following expression:
ij,max (2h̄ω = h̄ω1S ± Γk)
µ2Pµ1S,2P
h̄ω − h̄ω2P
αkh̄γ
Here the incidence angle dependent coefficient αk =
5 (See Fig.2a) for ka = 0.13
(Γk = h̄γ = 0.29 µeV ) and αk = 2 for maximum value of the coupling Γk = 3.5 µeV at
ka = 1.57 (See Fig.2b). Finally, comparing the last expression (14) to the bulk cuprous
oxide (13) the second order response of the hybrid is amplified by the factor:
h̄γ1S
Therefore the amplification can be adjusted by manipulating the organic composition
(DCM2 and CA densities) or changing the pump laser incidence angle.
Finally, we would like to note that there is another merit in using the hybrid
structure for the SHG. Namely the fact that the optical pumping can be replaced by
an electrical pumping. For this the hybrid sample has to be placed between Alq3 and
a-NPD [Madigan and Bulovic (2004)] semiconductor plates. The bond structure and
offset of these materials provide electrons and holes to form the hybrid exciton on the
interface. Although, in this case one can expect the SHG only from the lower branch
of the hybrid as the excitons are accumulated in the minimum of the hybrid dispersion
[Roslyak and Birman (2007b)].
6. Conclusion
In this paper we addressed possibility to enhance SHG signal χ(2) generic to a cuprous
oxide bulk crystal as the lowest excitation in this material has quadrupole origin. To
demonstrate the concept we proposed to consider a pump-prob experiment performed
on cuprous oxide sandwiched between the organic composite. An intense pump signal
excites one part of the organic known as DCM2 (FE). Non-resonant (Förster) energy
transfer in the organic layer (”solid state solvation” effect) provides dynamical red shift
of the FE. When the FE energy is close enough to the quadrupole allowed 1S exciton
in the adjacent cuprous oxide the quadrupole-dipole hybridization occurs due to the
FE induced gradient of the electric field penetrating into the inorganic layer. The prob
signal is designed to reveal the SHG signal.
The resonant enhancement of the χ(2) occurs for the hybrid exciton shares properties
of both quadrupole WE (long radiative lifetime) and organic FE (big oscillator strength).
I’t quadrupole part allows χ(2) to be non-vanishing. While it’s FE part provides the
enhancement of the SHG signal compared to the bare cuprous oxide crystal due to
more efficient absorption of the pump signal by means of large oscillator strength of the
hybrid.
However, as we demonstrated in the classical coupled oscillator model
framework,the enhancement is determined not only by the big ratio of the corresponding
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton14
organic/inorganic oscillator strengths but somehow quenched by the small coupling
parameter and low DCM2 density. By varying those parameters one the hybrid SHG
signal may be enhanced by orders of magnitude compared with the generic (cuprous
oxide) one.
To reveal the enhancement dependence on such an important parameter of the
hybrid as the hybridization time τh we proposed more sophisticated quantum theory.
It suggests that there is substantial difference in the hybrid SHG signal provided one
probes the system before or after the hybridization occurred. In the first case the SH is
generated at ω2P/2 frequency. Hence it is vastly suppressed by the short life time of the
dipole-allowed 2P WE in cuprous oxide. Nevertheless, if one probes the system after
the hybridization had happened, the SH is generated at ω1S/2 and is heightened by the
big oscillator strength of the hybrid and it’s small damping coefficient.
7. Acknowledgments
We would like to thank Ms. Upali Aparajita for helpful discussion and comments. The
project was supported in part by PCS-CUNY.
References
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(1996).
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[Huong and Birman (2000)] N. Q. Huong and J. L. Birman, Phys. Rev. B 61, 13131 (2000).
[Huong and Birman (2003)] N. Q. Huong and J. L. Birman, Phys. Rev. B 67, 075313 (2003).
[Roslyak and Birman (2007a)] O. Roslyak and J. Birman, Solid State Communications 143, 487
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and Materials Physics) 75, 245309 (pages 11) (2007b).
[Mukamel (1995)] S. Mukamel, Principles of nonlinear optical spectroscopy (Oxford Press, New York,
1995).
[Bloembergen (1965)] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965).
[Moskalenko and Liberman (2002)] S. Moskalenko and M. Liberman, Phys. Rev. B 65, 064303 (2002).
[Madigan and Bulovic (2003)] C. Madigan and V. Bulovic, Physical review letters 91, 247403 (2003).
[Artoni et al. (2002)] M. Artoni, I. Carusotto, G. La Rocca, and F. Bassani, J. Opt. B.: Quantum
Semiclass. Opt. 4, S345 (2002).
[Elliott (1961)] R. J. Elliott, Phys. Rev. 124, 340 (1961).
[Madigan and Bulovic (2004)] C. Madigan and V. Bulovic (Bringing Materials Research Together,
2004).
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton15
[Kishida et al. (1994)] H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, Y. Tokura, H. Tachibana,
M. Matsumoto, S. Wada, T. Lay, and H. Tashiro, Phys. Rev. B 50, 7786 (1994).
[Frohlich et al. (2005)] D. Frohlich, G. Dasbach, G. B. Hogersthal, M. Bayer, R. Kliebera, D. Sutera,
and H. Stolzb, Solid State Communications 134, 139 (2005).
[Haueisen and Mahr (1973)] D. Haueisen and H. Mahr, Phys. Rev. B 8, 734 (1973).
Appendix A.
Explicit expression for the quadrupole-dipole coupling is given below:
(ε+ ε̃)Lw
ke−kz
(A.1)
Here aFB, a
B are the Bohr radius of the FE and WE exciton; ε̃ and ε are the
corresponding dielectric constants, z′ is the distance to the DCM2 layer, Lw is the
quantum well width. The quadrupole transition matrix element Qxz may be estimated
from the corresponding oscillator strength per unit cell through the following identity
[Moskalenko and Liberman (2002)] and depends on polarization of the pumping laser
field:
fxz,k0 =
4πmEg
3e2h̄2
(z · k0,x ·Qx,z)2 (A.2)
fxz,k0||[1,1,0] = 3.9× 10−9
fxz,k0||[1,1,1] =
3.9× 10−9
Here the energy gap of cuprous oxide is denoted as Eg = 2.173 eV ; k0 = 2.62×105 cm−1
is the resonant wave vector; a is the unit cell size; the unit vector in the pumping field
polarization is z.
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton16
Appendix B.
The non-zero commutator relations for the organic and inorganic parts of the hybrid
yield [Mukamel (1995)]:
1S, B1S
= −1 +B†2PB2P + b†b;
1S, B2P
1SB2P ;
2P , B1S
2PB1S (B.1)
2P , B2P
= −1 +B†1SB1S + b†b;
b†, B1S
= b†B1S;
b†, B2P
= b†B2P
b†, b
= −1 +B†2PB2P +B
1SB1S;
1S, b
2P , b
In the TDHF approximate factorization for the averages, the corresponding Heisenberg
equations up to the second order in the creation and annihilation operators are:
= h̄ω1SB
1S + Γkb
† − µFEiB†1Sb+ µ1S,kEi
1− B†2PB2P − b†b
µ2PEiB
1SB2P + µ1S,2PEjB
= −h̄ω1SB1S − Γkb+ µFE⋆i b†B1S − µ1S,kE⋆i
1− B†2PB2P − b†b
2PB1S − µ1S,2PE⋆jB2P
= h̄ω2PB
2P − µFEiB
2P b− µ1S,kEiB
2PB1S + µ2PEi
1− B†1SB1S − b†b
+µ1S,2PEjB
= −h̄ω2PB2P + µFE⋆i b†B2P + µ1S,kE⋆i B
1SB2P − µ2PE⋆i
1−B†1SB1S − b†b
−µ1S,2PE⋆jB1S
= EF b† + ΓkB
1S + µ
1−B†1SB1S −B
2PB2P
− µ1S,kEib†B1S − µ2PEib†B2P
= −EF b− ΓkB1S − µFE⋆i
1−B†1SB1S −B
2PB2P
+ µ1S,kE
1Sb+ µ2PE
Here we omitted the average brackets for the shorter notation. In the exact resonance
between FE and WE excitons h̄ω1S = h̄ωF the linear approximation is straightforward.
The creation operators are proportional to ∝ eiωt and the system above is reduced to:
1S,0 = (h̄ω1S − ih̄γ)B
1S,0 + Γkb
0 + µ1S,kEi − h̄ωB1S,0
= − (h̄ω1S + ih̄γ)B1S,0 − Γkb0 − µ1S,kE⋆i
2P,0 = h̄ω2PB
2P,0 + µ2PEi − h̄ωB2P,0
= −h̄ω2PB2P,0 − µ2PE⋆i
0 = (h̄ω1S − ih̄γ) b
0 + ΓkB
1S,0 + µ
FEi − h̄ωb0
= − (h̄ω1S + ih̄γ) b0 − ΓkB1S,0 − µFE⋆i
Theory of enhanced second-harmonic generationby the quadrupole-dipole hybrid exciton17
The system above has a solution:
2P,0 =
µ2PEi
h̄ω − h̄ω2P
(B.2)
1S,0 =
µ1SEi (h̄ω − h̄ω1S + ih̄γ) + µFΓkEi
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
µFEi (h̄ω − h̄ω1S + ih̄γ) + µ1SΓkEi
(h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
The SHG is due to response to induced polarization and is proportional to ∝ ei2ωt:
2h̄ωB
1S,1 = (h̄ω1S − ih̄γ)B
1S,1 + Γkb
1 + µ1S,2PEjB
2h̄ωB1S,1 = (h̄ω1S + ih̄γ)B1S,1 + Γkb1 + µ1S,2PE
2h̄ωB
2P,1 = h̄ω2PB
2P,1 + µ1S,2PEjB
2h̄ωB2P,1 = h̄ω2PB2P,1 + µ1S,2PE
2h̄ωb
1 = (h̄ω1S − ih̄γ) b
1 + ΓkB
2h̄ωb1 = (h̄ω1S + ih̄γ) b1 + ΓkB1S,1
The system has a solution:
2P,1 =
µ1S,2PEjB
2h̄ω − h̄ω2P
(B.3)
1S,1 =
µ1S,2PEj (2h̄ω − h̄ω1S + ih̄γ)B†2P,0
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
µ1S,2PEjΓkB
(2h̄ω − h̄ω1S + ih̄γ)2 − Γ2k
This solutions are implemented in the main text to calculate the linear and nonlinear
responses of the hybrid.
	Introduction
	Proposed experimental set-up for the SHG
	 Anharmonic coupled oscillators model
	 Quantum theory of SHG due to the QDH
	Results and discussion
	Conclusion
	Acknowledgments
