Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors
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Excitonic Optical Stark Effect 259
∂
∂t
{N +1,M +1} =
∂c
†
(ϕ
N+1
)
∂t
{N,M}c(ψ
M+1
)
+ c
†
(ϕ
N+1
)
∂{N,M}
∂t
c(ψ
M+1
)
+ c
†
(ϕ
N+1
) {N,M}
∂c(ψ
M+1
)
∂t
. (13.89)
To analyze to which correlations functions {N,M} are coupled, one can
evaluate the equations of motion for the operators, i.e.,
∂c
†
(ϕ
i
)
∂t
and
∂c(ϕ
j
)
∂t
.
Using Heisenberg’s equation, one finds that {1, 0} ({0, 1}) is coupled to
itself by the single-particle (band structure) part, coupled to {0, 1} ({1, 0})
by the light-matter interaction, and coupled to {2, 1} ({1, 2})bythemany-
body Coulomb interaction.
Inserting these results into Eq. (13.89) and restoring normal ordering,
i.e., commuting all creation operators to the left of the destruction op-
erators, one can see that the light-matter interaction couples {N,M} to
{N −1,M+1}, {N +1,M−1}, {N −2,M},and{N, M −2}, since it is given
by pairs of creation and destruction operators. The many-body Coulomb
interaction, however, couples {N, M} to {N,M} and {N +1,M +1}.
Therefore, the four-operator part of the Hamiltonian generates the cou-
pling to products that contain more operators, i.e., the many-body hierar-
chy. Clearly, for both parts of the Hamiltonian, operators with N −M being
odd are only coupled to other operators where N −M is odd, and operators
with N −M being even are only coupled to other operators where N −M
is even. Thus, if we start from the ground state as the initial condition, i.e.,
{0, 0} =1and all other expectation values are zero, the operators {N, M}
with N − M being odd vanish at all times, since they contain no sources.
These considerations are visualized in Fig. 13.6. This, furthermore, clearly
shows that in any finite order in the external field, only a finite number of
expectation values contribute to the optical response. To find the lowest
order in the light field of any term {N, M}, one only needs to evaluate the
minimum number of dotted lines in Fig. 13.6, which are needed to connect
it to {0, 0}.
As can be seen from Fig. 13.6 and can be proven rigorously, see Lindberg
et al. (1994), the minimum order in the field in which {N,M} is finite is
(N + M)/2 if N and M are both even and (N + M )/2+1 if N and M are
both odd, i.e.,
c
†
(ϕ
2N
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
2M
) = O(E
N+M
) , (13.90)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
260 Quantum Theory of the Optical and Electronic Properties of Semiconductors
and
c
†
(ϕ
2N+1
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
2M+1
) = O(E
N+M+2
) . (13.91)
If one starts initially from the ground state of the semiconductor and the
dynamics is fully coherent, one can factorize expectation values containing
a mix of creation and destruction operators to lowest order into products
of expectation values which contain either only creation or only destruction
operators. For our case, we can write
c
†
(ϕ
2N
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
2M
) =
c
†
(ϕ
2N
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
2M
) + O(E
N+M+2
) , (13.92)
and
c
†
(ϕ
2N+1
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
2M+1
) =
c
†
(ϕ
2N+1
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
2M+1
) + O(E
N+M+4
) . (13.93)
To prove Eqs. (13.92) and (13.93), it is important to note that in the Heisen-
berg picture, where the operators are time-dependent, all expectation values
are taken with respect to the initial state, which is the ground state |0 of
the semiconductor, i.e.,
c
†
(ϕ
N
) c
†
(ϕ
N−1
) ... c
†
(ϕ
1
) c(ψ
1
) ... c(ψ
M
) =
0 |c
†
H
(ϕ
N
,t) ... c
†
H
(ϕ
1
,t) c
H
(ψ
1
,t) ... c
H
(ψ
M
,t) |0 . (13.94)
Now, one can define an interaction picture by decomposing the Hamil-
tonian into H = H
0
+ H
I
,whereH
0
contains the free-particle and the
Coulomb interaction parts. As is well known, in the interaction picture an
operator O becomes time-dependent according to
O
I
(t)=e
iH
0
t
Oe
−iH
0
t
. (13.95)
Inserting the interaction representation of the identity operator in be-
tween the creation and destruction operators of a ground state expectation
value of Heisenberg operators, i.e., in between c
†
H
(ϕ
1
,t) and c
H
(ψ
1
,t) of
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Excitonic Optical Stark Effect 261
Eq. (13.94), one finds by iterating the exponentials e
±iH
0
t
of Eq. (13.95)
0 |c
†
H
(ϕ
N
,t) ... c
†
H
(ϕ
1
,t) c
H
(ψ
1
,t) ... c
H
(ψ
M
,t) |0 =
0 |c
†
H
(ϕ
N
,t) ... c
†
H
(ϕ
1
,t) |0 0 |c
H
(ψ
1
,t) ... c
H
(ψ
M
,t) |0
+
δ
1
0 |c
†
H
(ϕ
N
,t) ... c
†
H
(ϕ
1
,t) c
†
I
(δ
1
,t) |0
×0 |c
I
(δ
1
,t) c
H
(ψ
1
,t) ... c
H
(ψ
M
,t) |0
+
1
2
δ
1
,δ
2
0 |c
†
H
(ϕ
N
,t) ... c
†
H
(ϕ
1
,t) c
†
I
(δ
2
,t) c
†
I
(δ
1
,t) |0
×0 |c
I
(δ
1
,t) c
I
(δ
2
,t) c
H
(ψ
1
,t) ... c
H
(ψ
M
,t) |0
+ ··· . (13.96)
This general result can be used to expand all correlation functions into their
contributions up to any defined order in the field.
We now apply this scheme to obtain the equations of motion which
describe the optical semiconductor response in the coherent χ
(3)
-limit. In
order to be able to distinguish between the uncorrelated Hartree-Fock part
and the correlation contributions, it is convenient to define a pure four-
particle correlation function via
¯
B
eh
e
h
k,k
,k
,k
=
3
α
†
e,k
β
†
h
,k
α
†
e
,k
β
†
h,k
4
−
3
α
†
e,k
β
†
h
,k
43
α
†
e
,k
β
†
e,k
4
−
3
α
†
e,k
β
†
h,k
43
α
†
e
,k
β
†
h
,k
4
= B
eh
e
h
k,k
,k
,k
−
3
α
†
e,k
β
†
h
,k
43
α
†
e
,k
β
†
e,k
4
−
3
α
†
e,k
β
†
h,k
43
α
†
e
,k
β
†
h
,k
4
.
(13.97)
Using the above introduced expansions up to third order in the field,
the polarization equation can be written as
∂
∂t
P
eh
k
=
∂
∂t
P
eh
k
|
hom
+
3
n=1
∂
∂t
P
eh
k
|
inhom,n
, (13.98)
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262 Quantum Theory of the Optical and Electronic Properties of Semiconductors
with
i
∂
∂t
P
eh
k
|
hom
= −
&
E
e
k
+ E
h
k
'
P
eh
k
+
q
V
q
P
eh
k−q
, (13.99)
i
∂
∂t
P
eh
k
|
inhom,1
=
d
eh
−
e
,h
(P
eh
k
)
∗
P
e
h
k
d
e
h
−
h
,e
d
eh
e
(P
e
h
k
)
∗
P
e
h
k
·E , (13.100)
i
∂
∂t
P
eh
k
|
inhom,2
= −
q,e
,h
V
q
P
eh
k
$
P
e
h
k
%
∗
P
e
h
k−q
− P
eh
k+q
$
P
e
h
k+q
%
∗
P
e
h
k
+ P
eh
k+q
$
P
e
h
k
%
∗
P
e
h
k
− P
eh
k
$
P
e
h
k−q
%
∗
P
e
h
k−q
, (13.101)
i
∂
∂t
P
eh
k
|
inhom,3
=
q,k
,e
,h
V
q
$
P
e
h
k
%
∗
$
¯
B
eh
e
h
k,k
,k
−q,k−q
−
¯
B
eh
e
h
k+q,k
,k
−q,k
+
¯
B
eh
e
h
k+q,k
+q,k
,k
−
¯
B
eh
e
h
k,k
+q,k
,k−q
%
. (13.102)
The homogeneous part of Eq. (13.98) contains the kinetic energies of elec-
trons and holes plus their Coulomb attraction. This term is diagonal in
the basis of exciton eigenstates. The contributions denoted with the sub-
script inhom in Eq. (13.98) are the different inhomogeneous driving terms,
i.e., they are the sources for the microscopic polarizations. The direct cou-
pling of the carrier system to the electromagnetic field is represented by
the terms proportional to d · E, see Eq. (13.100). These terms include
the linear optical coupling (d · E) and the phase-space filling contributions
(d · E P
∗
P ).
As a consequence of the many-body Coulomb interaction, Eq. (13.98)
contains further optical nonlinearities. The contributions that are of first-
order in the Coulomb interaction, which are proportional to VPP
∗
P ,are
given by Eq. (13.101). Those, together with the phase-space filling terms,
correspond to the Hartree–Fock limit of Eq. (13.98). The correlation contri-
butions to the polarization equation, see Eq. (13.102), consist of four terms
of the structure VP
∗
¯
B,where
¯
B is the genuine four-particle correlation
function defined in Eq. (13.97). Hence, as a consequence of the many-body
Coulomb interaction, the two-particle electron–hole amplitude P is coupled
to higher-order correlation functions
¯
B.
If the calculation of the coherent nonlinear optical response is limited to
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Excitonic Optical Stark Effect 263
a finite order in the applied field, the many-particle hierarchy is truncated
automatically and one is left with a finite number of correlation functions.
This truncation occurs since one considers here purely optical excitation
where only the optical field exists as linear source which in first-order in-
duces a linear polarization, in second order leads to carrier occupations,
and so on. In the more general situation, where incoherent populations
may be present, as, e.g., by pumping via carrier injection or in the presence
of relaxed occupations, a classification of the nonlinear optical response in
terms of powers of a field is no longer meaningful. For such cases, one
can use a cluster expansion or nonequilibrium Green’s functions to obtain
systematic approximation schemes to the many-body hierarchy.
Equation (13.98) has been derived within the coherent χ
(3)
-limit, i.e., by
keeping all contributions up to third-order in the electromagnetic field. If
this condition is fulfilled, the nonlinear optical response is fully determined
by the dynamics of single and two electron–hole–pair excitations, P and
¯
B,
respectively. The equation for
¯
B’s is obtained as
∂
∂t
¯
B
eh
e
h
k,k
,k
,k
=
∂
∂t
¯
B
eh
e
h
k,k
,k
,k
|
hom
+
∂
∂t
¯
B
eh
e
h
k,k
,k
,k
|
inhom
, (13.103)
with
i
∂
∂t
¯
B
eh
e
h
k,k
,k
,k
|
hom
= −
$
E
e
k
+ E
h
k
+ E
e
k
+ E
h
k
%
¯
B
eh
e
h
k,k
,k
,k
+
q
V
q
$
¯
B
eh
e
h
k+q
,k
+q
,k
,k
−
¯
B
eh
e
h
k+q
,k
,k
−q
,k
+
¯
B
eh
e
h
k+q
,k
,k
,k
+q
+
¯
B
eh
e
h
k,k
+q
,k
+q
,k
−
¯
B
eh
e
h
k,k
+q
,k
,k
−q
+
¯
B
eh
e
h
k,k
,k
+q
,k
+q
%
,(13.104)
i
∂
∂t
¯
B
eh
e
h
k,k
,k
,k
|
inhom
= −V
|k−k
|
&
P
eh
k
− P
eh
k
'
$
P
e
h
k
− P
e
h
k
%
+ V
|k−k
|
$
P
eh
k
− P
eh
k
%$
P
e
h
k
− P
e
h
k
%
. (13.105)
The homogeneous part of the equation for
¯
B contains the kinetic energies as
well as the attractive and repulsive interactions between two electrons and
two holes, i.e., the biexciton problem. The sources of Eq. (13.103) consist
of the Coulomb interaction potential times nonlinear polarization terms,
VPP.
Since the equations of motion in the coherent χ
(3)
-limit are quite
lengthy, in order to see the structure of the many-body couplings, it is
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
264 Quantum Theory of the Optical and Electronic Properties of Semiconductors
useful to reduce the notation and to consider a schematical set of equa-
tions of motion. These can be obtained from Eqs. (13.98) and (13.103) by
neglecting all indices and superscripts,
i
∂
∂t
P = −E
P
P +(d − bP
∗
P ) E − V
HF
P
∗
PP+ V
corr
P
∗
¯
B
(13.106)
and
i
∂
∂t
¯
B = −E
B
¯
B −
¯
V
corr
PP . (13.107)
Here, E
P
and E
B
are the energies of single- and biexciton states, and b
denotes the Pauli blocking, V
HF
the first-order (Hartree–Fock) Coulomb
terms, and V
corr
as well as
¯
V
corr
Coulomb correlation contributions.
Formal integration of Eq. (13.107) yields
¯
B(t)=
i
V
t
−∞
dt
e
iE(t−t
)/
P (t
)P (t
) . (13.108)
Inserting this expression for
¯
B(t) into Eq. (13.106) shows that P (t) depends
on contributions P (t
) with t
<t, i.e., on terms with earlier time argu-
ments. Such Coulombic memory effects would be neglected if one solved
the coupled equations adiabatically within the Markov approximation.
As an application, we use the coherent χ
(3)
-equations to compute differ-
ential absorption spectra for different circular polarizations of the pump and
probe pulses assuming excitation of the semiconductor energetically below
the exciton resonance. Numerical results for the polarization-dependent
excitonic optical Stark effect, are displayed in Fig. 13.7. In the upper
right panel in Fig. 13.7, we see the computed differential absorption spec-
trum for co-circularly polarized pulses. This spectrum corresponds to the
usual blueshift of the exciton resonance discussed in the previous sections
of this chapter. The detailed analysis reveals that the phase-space filling
nonlinearity and the first-order Coulomb terms, i.e., the Hartree–Fock con-
tributions induce a blueshift, whereas the correlations alone would yield
aredshift. However,fortheσ
+
σ
+
configuration the magnitude of the
correlation-induced contribution is relatively small and in turn, the total
signal is dominated by the blueshift of the Hartree–Fock terms.
The situation is, however, very different if opposite circularly polarized
(σ
+
σ
−
) pump and probe pulses are considered, see the lower right panel
in Fig. 13.7. For this configuration, the Hartree–Fock contributions for
heavy-hole transitions vanish and the response is completely determined by
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Excitonic Optical Stark Effect 265
-0.5
0.0
0.5
s+s+
1.490 1.491 1.492 1.493 1.494 1.495
-0.1
0.0
0.1
s+s-
-0.5
0.0
0.5
s+s+
1.490 1.491 1.492 1.493 1.494 1.495
-0.1
0.0
0.1
s+s-
Energy (eV) Energy (eV)
Differential Absorption
Fig. 13.7 Experimental (left column) and computed (right column) differential absorp-
tion spectra for a InGaAs/GaAs quantum-well sample. The pump pulses were detuned
to 4.5 meV below the exciton resonance and the shown spectra were obtained at zero
time delay. The usual blueshift is obtained for equal polarization (top row) and a redshift
is obtained for opposite circular polarizations of the pump and probe pulses. [After Sieh
et al. (1999).]
correlations. These correlations yield a redshift which is not compensated
by other terms and thus survives. As Sieh et al. (1999) have shown, the
occurrence of this redshift is not directly related to the existence of a bound
biexciton that can be excited with σ
+
σ
−
polarized pulses. This can be seen
already from the analysis of the σ
+
σ
+
excitation configuration, where no
bound biexciton can be excited but still the correlation term alone amounts
to a redshift of the exciton resonance.
Experimental measurements of the excitonic optical Stark effect for dif-
ferent pump–probe polarization configurations are displayed in the left col-
umn of Fig. 13.7. These measurements nicely confirm the predictions of
the third-order theory.
REFERENCES
The first reports on the observation of the nonresonant optical Stark effect
are:
A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W.T. Masselink, and
H. Morkoc, Phys. Rev. Lett. 55, 1335 (1985)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
266 Quantum Theory of the Optical and Electronic Properties of Semiconductors
A. von Lehmen, J.E. Zucker, J.P. Heritage, and D.S. Chemla, Optics Lett.
11, 609 (1986)
The Stark effect theory presented in this chapter has been developed in:
S. Schmitt-Rink, D.S. Chemla and H. Haug, Phys. Rev. B37, 941 (1988)
S.W. Koch, N. Peyghambarian, and M. Lindberg, J. Phys. C21, 5229
(1988)
C. Ell, J.F. Müller, K. El Sayed, and H. Haug, Phys. Rev. Lett. 62, 306
(1989)
The polarization dependence and memory effects in the excitonic optical
Stark effect have been presented in:
C. Sieh, T. Meier, A. Knorr, S.W. Koch, P. Brick, M. Hübner, C. Ell, J.
Prineas, G. Khitrova, and H.M. Gibbs, Phys. Rev. Lett. 82, 3112 (1999)
The systematic truncation of the equation hierarchy under coherent exci-
tation conditions has been derived in:
V.M.AxtandA.Stahl,Z.Phys.B93, 195 and 205 (1994)
M. Lindberg, Y.Z. Hu, R. Binder, and S. W. Koch, Phys. Rev. B50,
18060 (1994)
see also:
W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phe-
nomena, Springer Verlag, Berlin (2002)
The expansion of optical nonlinearities in terms of nonlinear susceptibilities
is summarized in:
N. Bloembergen, Nonlinear Optics, Benjamin Inc., New York (1965)
PROBLEMS
Problem 13.1: Show for a two-level system that (1 − 2n)
2
+4|P |
2
is a
conserved quantity.
Problem 13.2: Use Eqs. (13.2) – (13.5) to derive the optical Stark shift
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Excitonic Optical Stark Effect 267
results of Sec. 2-3 for the two-level atom.
Problem 13.3: Show that δp
+
k
>> δp
−
k
and derive the linearized
Eq. (13.19).
Problem 13.4: Calculate the enhancement factors (13.40) using the exci-
ton wave functions in 2D and 3D.
Problem 13.5: Use the Heisenberg equation for the four-operator terms
to derive Eq. (13.103) in the coherent χ
(3)
limit.
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