Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors
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Coulomb Quantum Kinetics 419
D.F. DuBois, in Lectures in Theoretical Physics, eds. W.E. Brittin, and
A.O. Barut, and M. Guenin, Gordon and Breach, New York (1967)
For applications of the nonequilibrium Green’s functions to ultrafast semi-
conductor spectroscopy see:
H. Haug and A.P. Jauho, Quantum Kinetics in Transport and Optics of
Semiconductors, Springer, Berlin (1966)
M. Bonitz (ed), Progress in Non-Equilibrium Green’s Functions, World Sci-
entific, Singapore (2000)
W. Schäfer and M. Wegener, Semiconductor Optics, Springer, Berlin (2002)
R. Binder and S.W. Koch, Prog. Quant. Electr. 19, 307 (1995)
F. Jahnke, M. Kira, and S.W. Koch, Z. Phys. B104, 559 (1997)
For applications of the second Born results to compute absorption and gain
spectra see:
F. Jahnke, M. Kira, S.W. Koch, G. Khitrova, E. Lindmark, T. Nelson, D.
Wick, J. Berger, O. Lyngnes, H.M. Gibbs, and K. Thai, Phys. Rev. Lett.
77, 5257 (1996)
J. Hader, D. Bossert, J. Stohs, W.W. Chow, S.W. Koch, and J. Moloney,
Appl. Phys. Lett. 74, 2277 (1999)
W.W. Chow and S.W. Koch, Semiconductor-Laser Fundamentals, Springer,
Berlin, 1999
For the treatment of the time-dependent screening in the density matrix
theory see:
H.W. Wyld and B.D. Fried, Annals of Physics 23, 374 (1963)
T. Woltering, V.M. Axt, and T. Kuhn, Phys. Rev. B67, 115311 (2003)
For the treatment of the time-dependent screening in terms of nonequilib-
rium Green’s functions and the experimental measurement of the build-up
of screening see:
L. Banyai, Q.T. Vu, B. Mieck, and H. Haug, Phys. Rev. Lett. 81, 882
(1998)
Q.T. Vu and H. Haug, Phys. Rev. B62, 7179 (2000)
P. Gartner, L. Banyai, and H. Haug, Phys. Rev. B62, 7116 (2000)
W.A. Hügel, M.F. Heinrich, M. Wegener, Q.T. Vu, L. Banyai, and H. Haug,
Phys.Rev.Lett.83, 3313 (1999)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
420 Quantum Theory of the Optical and Electronic Properties of Semiconductors
Q.T.Vu, H. Haug, W.A. Hügel, S. Chatterjee, and M. Wegener, Phys. Rev.
Lett. 85, 3508 (2000)
R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A.
Leitenstorfer, Nature 414, 286 (2001)
H. Haug, Nature 414, 261 (2001)
PROBLEMS
Problem 21.1: Derive the generalized Kadanoff–Baym ansatz
(a) for free particles
(b) for particles governed by a mean-field Hamiltonian.
Problem 21.2: Show that the ladder integral equation of Fig 21.3 yields
after the first iteration the vertex diagram of Fig. 21.2.
Problem 21.3: Proof the optical theorem (21.7) for the “greater” function
explicitly.
Problem 21.4: Give the two by two matrix of the mean-field Hamilto-
nian (21.26) with coupling to the light field and Hartree–Fock Coulomb
corrections explicitly.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Appendix A
Field Quantization
In this appendix, we review the general quantization procedure for a given
classical field. As a result, we obtain a formulation of quantum mechanics
in terms of generation and annihilation operators, which is well suited to
describe many-body systems. In this language, the transition from state l
to state m is described as annihilation of the respective quasi-particle in
state l and simultaneous creation of that quasi-particle in state m.
Formally, one can also apply the quantization procedure to the one-
particle Schrödinger field, which is the origin of the name second quantiza-
tion, which is often used instead of field quantization. In this appendix, we
discuss the general formalism of field quantization and illustrate it with
three examples: i) the electromagnetic field, ii) the displacement field,
iii) the Schrödinger field. In the process, we introduce the appropriate
quanta, namely, photons for the electromagnetic field, phonons for the dis-
placement field, and electrons for the Schrödinger field. Anticipating the
introduction of these quanta, we use their names as headings for our sub-
sections. For the electromagnetic and displacement fields, we use the or-
dinary commutator (with a minus sign), while for the electrons we use
anti-commutators. These simple choices automatically yield the correct
statistics, namely Bose-Einstein statistics for the electromagnetic and dis-
placement fields and Fermi-Dirac statistics for the electrons.
A.1 Lagrange Functional
We denote by φ
j
(r,t) the general field variable, which may be the vector
potential A
j
, the displacement field ξ
j
, or any other field. Generalizing
Lagrangian and Hamiltonian point mechanics to the continuum case, we
421
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422 Quantum Theory of the Optical and Electronic Properties of Semiconductors
introduce a Lagrange functional which depends on fields according to
L[φ
j
]=
d
3
rL(φ
j
,∂φ
j
/∂t, ∂φ
j
/∂x
i
, r,t) . (A.1)
The Euler–Lagrange equations (field equations) are obtained from Hamil-
ton’s principle
δ
t
1
t
0
Ldt=0=δ
t
1
t
0
dt
d
3
rL(φ
j
,∂φ
j
/∂t, ∂φ
j
/∂x
i
, r,t) . (A.2)
In Eq. (A.2), the variation δ acts on the fields (not on the coordinates)
δL =
j
δL
δφ
j
δφ
j
+
δL
δ
∂φ
j
∂t
δ
˙
φ
j
+
i
δL
δφ
i
j
δφ
i
j
, (A.3)
where we introduced the abbreviations
˙
φ = ∂φ/δt, φ
i
j
= ∂φ
j
/∂x
i
. δL/δφ
j
denotes the functional derivative of L with respect to the j-th component
of the vector field φ. Assuming as usual that variations and derivatives
commute, we can use partial integration. Since δφ(t
0
)=δφ(t
1
)=0and
since δφ also vanishes at the boundaries of the system, we obtain
δ
t
1
t
0
dtL =
t
1
t
0
dt
d
3
r
δL
δφ
j
−
∂
∂t
δL
δ
∂φ
j
∂t
−
i
∂
∂x
i
δL
δ
∂φ
j
∂x
i
δφ
j
=0 . (A.4)
The variations δφ
j
are arbitrary, so that Eq. (A.4) can be satisfied only if
δL
δφ
j
−
∂
∂t
δL
δ
∂φ
j
∂t
−
i
∂
∂x
i
δL
δ
∂φ
j
∂x
i
=0 . (A.5)
These equations are the Euler–Lagrange or field equations.
i) photons
Photons are the quanta of the electromagnetic field. To keep things as
simple as possible, we restrict our discussion to the electromagnetic field in
vacuum. For this, we have
dt L =
dt
d
3
r
1
8π
(E
2
− H
2
) , (A.6)
where
H = ∇×A =curlA , (A.7)
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Appendix A: Field Quantization 423
A is the vector potential,
E = −∇φ −
1
c
∂A
∂t
, (A.8)
and φ is the scalar potential. Using Eqs. (A.7) and (A.8) in Eq. (A.6), we
write the Lagrangian density as
L =
1
8π
∇φ +
1
c
∂A
∂t
2
− (curlA)
2
. (A.9)
The components A
i
of the vector potential and the scalar potential φ are
the elements of the general field variable φ
i
(r,t) discussed above, i.e.,
φ
i
= A
i
for i =1, 2, 3 and φ
4
= φ. (A.10)
To obtain the field equations, we need the results
δL
δA
j
=
δL
δφ
=
δL
δ
∂φ
∂t
=0, (A.11)
δL
δ
∂A
j
∂t
=
1
4πc
∂φ
∂x
j
+
1
c
∂A
j
∂t
, (A.12)
δL
δ
∂φ
∂x
i
=
1
4π
∂φ
∂x
i
+
1
c
∂A
i
∂t
, (A.13)
and
δL
δ
∂A
j
∂x
l
= −
δ
δ
∂A
j
∂x
l
imk
∂A
k
∂x
m
iqn
∂A
n
∂x
q
1
8π
= −
1
8π
ilj
iqn
∂A
n
∂x
q
−
1
8π
imk
∂A
k
∂x
m
ilj
= −
1
4π
ilj
iqn
∂A
n
∂x
q
= −
1
4π
∂A
j
∂x
l
−
∂A
l
∂x
j
. (A.14)
In Eq. (A.14), we have used the so-called Levi–Civita tensor
ijk
, which has
the following properties
123
=1,
213
= −1 ,
iij
=0 , (A.15)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
424 Quantum Theory of the Optical and Electronic Properties of Semiconductors
where cyclic permutations of the indices do not change the result, i.e.,
123
=
312
=
231
, etc. For the product of the tensors, we have the rule
i
ijl
imn
= δ
jm
δ
ln
− δ
jn
δ
lm
. (A.16)
The Levi–Civita tensor is very useful when evaluating vector products, curls
and the like. For example, the curl of a vector is
curlA =
e
i
∂
∂x
j
A
l
ijl
. (A.17)
Combining Eqs. (A.9) – (A.14) with the Euler–Lagrange equations, we get
1
4π
∂
∂x
i
∂A
j
∂x
i
−
∂A
i
∂x
j
−
1
c
∂
∂t
∂φ
∂x
j
+
1
c
∂A
j
∂t
=0 (A.18)
or, using Eqs. (A.7) and (A.8)
curl H =
1
c
∂
∂t
E , (A.19)
which is one of Maxwell’s equations. Analogously we obtain
∂
∂x
i
∂φ
∂x
i
+
1
c
∂A
i
∂t
=0 , (A.20)
which is nothing but
div E =0. (A.21)
The other two Maxwell equations are just the definitions of the potentials.
Equation (A.8) yields
curl E = −
1
c
∂H
∂t
, (A.22)
since
∇×E= ∇×
−∇φ −
1
c
∇A
∂t
= −
1
c
∂
∂t
∇×A (A.23)
and from Eq. (A.7) we have
div H =0 (A.24)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Appendix A: Field Quantization 425
since
∇·(∇×A)=0 . (A.25)
Equations (A.19), (A.21), (A.22) and (A.24) are the complete Maxwell
equations in vacuum, which occur here as our field equations.
ii) phonons
As second example, we treat waves in an elastic medium. The quanta of
these waves are the phonons. We denote by ξ
i
(r,t) the i-th component of
the displacement field at point r and time t. The displacement field satisfies
the classical wave equation
∂
2
∂t
2
ξ
i
− c
2
∇
2
ξ
i
=0 . (A.26)
The corresponding Lagrangian is
L =
1
2
ρ
i
∂ξ
i
∂t
2
−
ij
c
2
∂ξ
i
∂x
j
2
, (A.27)
where we have the density ρ as a prefactor so that L has the correct unit
of an energy density. We arrived at Eq. (A.27) by an educated guess and
by verifying that the field equations are indeed just Eq. (A.26).
iii) electrons
As we see below, the electrons appear through the quantization of the
Schrödinger equation. This is the origin of the name second quantization,
although the name field quantization is generally more appropriate. Here,
we consider the single-particle Schrödinger equation as our classical wave
equation
i
∂ψ
∂t
−H
sch
ψ =0 , (A.28)
where
H
sch
= −
2
∇
2
2m
+ V (r) (A.29)
and ψ is the complex wave function. We treat ψ and ψ
∗
as the quan-
tities that have to be varied independently. Our educated guess for the
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
426 Quantum Theory of the Optical and Electronic Properties of Semiconductors
Lagrangian is
L = ψ
∗
(i
˙
ψ −H
sch
ψ) , (A.30)
which can be verified by evaluating Eq. (A.5). The variation with respect
to ψ
∗
yields
δL
δψ
∗
−
∂
∂t
δL
δ
∂ψ
∗
∂t
−
∂
∂x
i
δL
δ
∂ψ
∗
∂x
i
=0 . (A.31)
The last two terms on the LHS of this equation are identically zero and
δL
δψ
∗
=0=i
∂ψ
∂t
−H
sch
ψ, (A.32)
which is the Schrödinger equation (A.28). The variation with respect to
ψ is more lengthy, but it can be shown to yield the conjugate complex
Schrödinger equation
−i
∂ψ
∗
∂t
−H
sch
ψ
∗
=0 . (A.33)
A.2 Canonical Momentum and Hamilton Function
The next step in our general procedure is to introduce the canonical mo-
mentum and the Hamilton functional. As in point mechanics, the canonical
momentum Π
i
(r,t) for φ
i
(r,t) is defined as
Π
i
=
δL
δ
∂φ
i
∂t
(A.34)
and the Hamilton density is defined as
h =
i
Π
i
∂φ
i
∂t
−L , (A.35)
where all variables have to be expressed in terms of Π and φ.
i) photons
Using the Lagrangian (A.9) in Eq. (A.34), we obtain
Π
i
=
δL
δ
∂A
i
∂t
=
1
4πc
∂φ
∂x
i
+
1
c
∂A
i
∂t
, (A.36)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Appendix A: Field Quantization 427
so that Π
i
= −E/4πc. Solving (A.36) for ∂A
i
/∂t yields
∂A
i
∂t
=4πc
2
Π
i
− c
∂φ
∂x
i
. (A.37)
The momentum density canonical to the scalar potential φ vanishes iden-
tically because L does not depend on ∂φ/∂t. Therefore, we cannot treat
φ in the same way as A. Actually, as we will show below, the Hamilton
density is independent of φ. Therefore, we can completely eliminate φ from
the discussion by choosing the gauge φ =0.
The Hamilton density is
h =
Π
i
4πc
2
Π
i
− c
∂φ
∂x
i
−
1
8π
[(4πcΠ)
2
− (curl A)
2
]
=2πc
2
Π
2
+
1
8π
(curl A)
2
, (A.38)
since the term cΠ·∇φ vanishes. To prove this, we make a partial integration
of the total Hamiltonian. This transforms cΠ ·∇φ into −cφ∇·Π.Wecan
drop this term if we limit the treatment to solutions for which ∇·Π=
∇·E=0. To see that this is possible, we consider the Hamilton equation
for Π,
∂Π
∂t
= −
1
4π
∇×∇×A =0 . (A.39)
This equation shows that ∇·Π always remains zero if we choose it to be
zero initially.
ii) phonons
Using the Lagrangian (A.27) in Eq. (A.34), we obtain the canonical mo-
mentum
Π
i
=
δL
δ
∂ξ
i
∂t
= ρ
∂ξ
i
∂t
. (A.40)
This result is the continuum version of the well-known relation from point
mechanics, where p = m ˙x. The Hamiltonian is computed as
h =
Π
2
2ρ
+
ρc
2
2
ik
∂ξ
i
∂x
k
2
, (A.41)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
428 Quantum Theory of the Optical and Electronic Properties of Semiconductors
where the first term on the RHS is the continuum version of p
2
/2m and
the second term is the potential.
iii) electrons
Here we use the Lagrangian (A.30) to obtain
Π=iψ
∗
(A.42)
and
h =Π
˙
ψ −
1
i
Πi
˙
ψ +
1
i
Π H
sch
ψ
=
1
i
Π H
sch
ψ. (A.43)
A.3 Quantization of the Fields
Now, we have prepared everything to come to the crucial step of field quan-
tization. First a reminder: the transition from classical mechanics to quan-
tum mechanics can be done formally by replacing the classical variables r
and p by operators
ˆ
r and
ˆ
p that fulfill the commutation relations
[ˆr
l
, ˆp
j
]=iδ
lj
,
[ˆr
l
, ˆr
j
]=0 ,
and
[ˆp
l
, ˆp
j
]=0 , (A.44)
where here and in the following a commutator without a subscript always
denotes the minus commutator
[
ˆ
A,
ˆ
B] ≡ [
ˆ
A,
ˆ
B]
−
=
ˆ
A
ˆ
B −
ˆ
B
ˆ
A . (A.45)
In this appendix, we use hats on top of the variables to denote operators.
The field quantization is now done by replacing the fields φ
i
→
ˆ
φ
i
,
Π
i
→
ˆ
Π
i
and the Hamiltonian H→
ˆ
H. For the field operators, we demand
the commutation relations
[
ˆ
φ
l
(r,t),
ˆ
Π
j
(r
,t)]
±
= iδ
lj
δ(r − r
) ,
[
ˆ
φ
l
(r,t),
ˆ
φ
j
(r
,t)]
±
=0 ,