Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors
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Semiconductor Laser 329
To include damping in the traditional way, we write
db
dt
=
$
−iω −
κ
2
%
b. (17.30)
The solution of this equation is b(t)=b(0) exp[−iω + κ/2)t], and it is a
simple exercise (see problem 17.1) to show that the commutator [b(t),b
†
(t)]
is not equal to one for all times. The damping term causes a decay of the
commutator. We can correct this inconsistency by adding a fluctuation
operator f(t) to the equation of motion
db
dt
=
−iω −
κ
2
b(t)+f(t) . (17.31)
Langevin equation
For simplicity, we always assume in this book that the fluctuation oper-
ators can be described as Markovian noise sources. This means that the
stochastic fluctuations at different times are uncorrelated (no memory)
f(t)f
†
(s) =2Dδ(t − s) . (17.32)
Markovian fluctuations
The average in Eq. (17.32) is taken over a reservoir to which the system
must be coupled in order to introduce irreversible behavior. D is called the
diffusion constant, which is related via the dissipation–fluctuation theorem
to the damping constant κ.
To obtain an explicit form for the fluctuation operator and the dissipa-
tion rate, let us discuss the coupling of the harmonic oscillator to a reservoir
(also called a bath), which consists of a large set of harmonic oscillators B
λ
with a continuous energy spectrum Ω
λ
. We assume that the bath oscilla-
tors are all in thermal equilibrium and are not disturbed by the coupling
to the harmonic oscillator which represents our system. In other words,
we assume that the bath is infinitely large in comparison to the system of
interest.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
330 Quantum Theory of the Optical and Electronic Properties of Semiconductors
The total system-bath Hamiltonian is
H
= ωb
†
b +
λ
Ω
λ
B
†
λ
B
λ
+
λ
g
λ
(bB
†
λ
+ B
λ
b
†
) . (17.33)
The equations of motion for the oscillators of the system and bath are
db
dt
= −iωb − i
λ
g
λ
B
λ
and
dB
λ
dt
= −iΩ
λ
B
λ
− ig
λ
b. (17.34)
Formally integrating the equation for the bath operator, we obtain
B
λ
(t)=B
λ
(0)e
−iΩ
λ
t
− ig
λ
t
0
dτ b(τ) e
−iΩ
λ
(t−τ )
. (17.35)
Inserting this result into the equation for the system operator yields
db
dt
= −iωb − i
λ
g
λ
B
λ
(0)e
−iΩ
λ
t
− ig
λ
t
0
dτ b(τ) e
−iΩ
λ
(t−τ )
, (17.36)
which we write in the form
db
dt
= −iωb + f(t) −
κ
2
b(t) . (17.37)
Now, we identify
κ
2
b(t)
λ
g
2
λ
b(0)e
−iΩ
λ
t
t
0
dτ e
i(Ω
λ
−ω)τ
λ
g
2
λ
πδ(Ω
λ
− ω)
b(t) ,
(17.38)
where we approximated b(t) b(0) exp(−iωt). In Eq. (17.38), we recog-
nize Fermi’s golden rule for the transition rate per unit time of an energy
quantum ω into the continuous spectrum of the bath. Using the density of
states ρ(ω) ρ, assuming g(ω) g, and integrating over all bath energies,
we get the damping constant
κ = g
2
2πρ . (17.39)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Semiconductor Laser 331
The fluctuation operator f(t) is given by the second term in Eq. (17.36) as
f(t)=−i
λ
g
λ
B
λ
(0)e
−iΩ
λ
t
. (17.40)
Obviously, if we average f (t) over the bath < ···>= tr ρ
B
...,with
ρ
B
=
e
−βH
B
tr e
−βH
B
, (17.41)
we find f(t) =0. For the second moments, we get
f
†
(t)f(s) =
λ
g
2
λ
B
†
λ
(0)B
λ
(0)e
iΩ
λ
(t−s)
g
2
g
0
(ω)ρ2πδ(t − s) ,
or
f
†
(t)f(s) = κg
0
(ω)δ(t − s) ,
f(t)f
†
(s) = κ
g
0
(ω)+1
δ(t − s) ,
f(t)f(s) =0and f
†
(t)f
†
(s) =0 . (17.42)
disipation-fluctuation theorem for harmonic oscillator
Here, g
0
(ω)=
exp(βω) − 1
−1
is the thermal Bose distribution for the
bath quanta. Eqs. (17.42) are called dissipation-fluctuation relations be-
cause they link the correlations of the fluctuations and the damping con-
stant κ, which describes the dissipation rate. We see that our simple model
yields, at least approximately, Markovian correlations for the fluctuations.
For the discussion of noise sources in the electron–hole system of the
semiconductor laser, one needs a more general formulation of the quan-
tum mechanical dissipation–fluctuation theorem. Following Lax (1966), we
therefore write the general Langevin equation for a set of quantum mechan-
ical variables O
µ
as
d
dt
O
µ
= A
µ
({O
µ
},t)+F
µ
({O
µ
},t) , (17.43)
where the dissipation rates A
µ
are calculated in second-order perturbation
theory in the interaction of the system with its reservoirs, as shown above
for the example of the harmonic oscillator. The Markov assumptions for
the fluctuations are then:
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332 Quantum Theory of the Optical and Electronic Properties of Semiconductors
F
µ
(t) is independent of all O
µ
(s) for t>s ;
F
µ
(t) =0 ;
F
µ
(t)F
ν
(s) =2D
µν
δ(t − s) . (17.44)
In order to determine the generalized diffusion coefficients D
µν
,weusethe
Langevin equation (17.43). We obtain O
µ
(t+∆t),where∆t is a small time
interval, from O
µ
(t) as
O
µ
(t +∆t) − O
µ
(t)=
t+∆t
t
ds A
µ
({O
µ
},s)+
t+∆t
t
ds F
µ
({O
µ
},s)
or
∆O
µ
A
µ
∆t +
t+∆t
t
ds F
µ
(s) . (17.45)
For the time derivative of the bilinear expression O
µ
(t)O
ν
(t), we find
d
dt
O
µ
(t)O
ν
(t)
1
∆t
O
µ
(t +∆t)O
ν
(t +∆t)−O
µ
(t)O
ν
(t)
1
∆t
(O
µ
(t)+∆O
µ
)(O
ν
(t)+∆O
ν
)−O
µ
(t)O
ν
(t)
1
∆t
O
µ
(t)∆O
ν
+ ∆O
µ
O
ν
(t) + ∆O
µ
∆O
ν
.
Using Eq. (17.45) we get
d
dt
O
µ
(t)O
ν
(t)O
µ
A
ν
+ A
µ
O
ν
+
1
∆t
O
µ
(t)
t+∆t
t
ds F
ν
(s) +
t+∆t
t
ds F
µ
(s)O
ν
(t)
+ A
µ
(t)
t+∆t
t
ds F
ν
(s) +
t+∆t
t
ds F
µ
(s)A
ν
(t)
+
1
∆t
t+∆t
t
ds F
µ
(s)
t+∆t
t
ds
F
ν
(s
) . (17.46)
The terms in the second line of Eq. (17.46) do not contribute because
O
µ
(t) is not correlated with the fluctuations F
ν
(s) for t<s<t+∆t
(see Eq. (17.44)). The terms in the third line are of third order in the
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Semiconductor Laser 333
system-bath interaction and are neglected. Inserting the Markov assump-
tion, Eq. (17.44), into the last line of Eq. (17.46), we get the result
2D
µν
=
d
dt
O
µ
O
ν
−A
µ
O
ν
−O
µ
A
ν
. (17.47)
disipation–fluctuation theorem
According to Eq. (17.47), one has to calculate the time-derivative of the
bath-averaged term O
µ
O
ν
and subtract the expressions which contain
the linear dissipation rate A
µ
in order to get the diffusion coefficients D
µν
.
As an application of Eq. (17.47), we now use the example of spontaneous
emission to calculate the diffusion coefficient for n
i,k
and P
k
. Note, that we
do not adiabatically eliminate the interband polarization dynamics in this
example. For more details, see Haug and Haken (1967). The interaction
Hamiltonian with the bath of photons is
H
I
=
d
cv
(a
†
c,k
a
v,k
B
λ
+ B
†
λ
a
†
v,k
a
c,k
) , (17.48)
where the index λ runs over the continuum of all photon modes. We assume
that the bath is in the vacuum state |0
λ
without photons, so that only
spontaneous emission and no absorption is possible. The change of the
population density in the state c, k calculated in second-order perturbation
theory is
d
dt
n
c,k
= −d
2
cv
λ
n
c,k
(1 − n
v,k
)πδ(
ck
−
vk
− Ω
λ
)=A
n
c,k
, (17.49)
in agreement with Fermi’s golden rule. In order to calculate the diffusion
coefficient D
n
c,k
,n
c,k
,wehavetoevaluate
d
dt
n
c,k
n
c,k
=
d
dt
n
c,k
= A
n
c,k
,
so that
2D
n
ck
,n
ck
= d
2
cv
λ
n
c,k
(1 − n
v,k
)πδ(
ck
−
vk
− Ω
λ
) , (17.50)
which describes the typical shot noise for population variables. In general,
shot noise consists of the sum of all transition rates in and out of the con-
sidered state. Because we assume that no photons are present in the bath,
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
334 Quantum Theory of the Optical and Electronic Properties of Semiconductors
only emission processes, i.e., transitions out of the state c, k are possible.
In local equilibrium, where n
ik
= f
ik
, we have to consider only the total
number of electrons in one band. Because the optical transition probabili-
ties at different k-values are not correlated, one gets for the fluctuations of
the total electron density N
c
the result
2D
N
c
,N
c
= d
2
cv
λ,k
f
ck
(1 − f
vk
) πδ(
ck
−
vk
− Ω
λ
) , (17.51)
which is just the total spontaneous transition rate. Similarly, one calculates
the auto-correlation of the fluctuations of the polarization P
k
2D
P
k
,P
†
k
+2D
P
†
k
,P
k
=2γ
k
f
ck
(1 − f
vk
)+f
vk
(1 − f
ck
)
. (17.52)
These polarization fluctuations determine mainly the coherence properties
of the semiconductor laser light.
The different dissipative contributions to the carrier rate equation (17.2)
can be modeled by a coupling to a suitable bath. The resulting noise sources
all have shot noise character (see e.g. Haug, 1967, 1969)
dN
dt
= r
p
− r
st
− r
sp
− r
nr
+ F
p
+ F
st
+ F
sp
+ F
nr
. (17.53)
The correlations of the fluctuations are discussed below.
Diode lasers are pumped by an injection current, which is driven by a
voltage source via a serial resistor R
s
. If the serial resistor is larger than the
differential resistance of the diode, R
s
suppresses the current fluctuations in
the diode. The resulting pump noise is, as Yamamoto and Machida (1987)
have shown, simply the Nyquist noise in the serial resistor
F
p
(t)F
p
(s) =
4k
B
T
e
2
L
3
R
s
δ(t − s) . (17.54)
Obviously, by increasing R
s
one can essentially suppress the pump noise.
For the stimulated emission, we obtain from the generalized dissipation
fluctuation theorem, Eq. (17.54),
F
st
(t)F
st
(s) =
|E
0
(t)|
2
2L
3
χ
a
(ω)+χ
e
(ω)
δ(t − s) , (17.55)
where we used Eqs. (17.1) and (17.6). In the evaluation of the various terms
in Eq. (17.55), we had to take the expression for χ before the expectation
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Semiconductor Laser 335
values were taken, i.e.,
f
e,k
→ a
†
c,k
a
c,k
(17.56)
and correspondingly for the other population terms. A close inspection of
Eq. (17.55) shows that F
st
(t)F
st
(s) is proportional to
f
e
f
h
+(1− f
e
)(1 − f
h
) ,
which describes the sum of the interband transition rates due to the laser
action.
The noise terms due to spontaneous emission have the correlation
F
sp
(t)F
sp
(s) =
r
sp
L
3
δ(t − s) , (17.57)
and the nonradiative transition noise yields
F
nr
(t)F
nr
(s) =
r
nr
L
3
δ(t − s) . (17.58)
17.4 Stochastic Laser Theory
The basic equations for our stochastic laser theory are (17.53) and (17.18)
with an added noise term. To get a feeling for the solutions of these equa-
tions, we analyze the situation, where we have small fluctuations around
the steady state values of Eqs. (17.26) – (17.28), which are now the mean
(bath averaged) values. Following Vahala and Yariv (1983), we express all
quantities in terms of their mean values and slowly varying amplitude and
phase perturbations, i.e.,
E
n
=[E
0
+ E(t)]e
−i
ω
m
t−φ(t)
,
N = N
0
+ n(t) ,
and
χ(N)=χ(N
0
)+
∂χ
∂N
0
n(t) , (17.59)
where the first-order Taylor coefficient is
∂χ
∂N
0
=
∂χ
∂N
N=N
0
.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
336 Quantum Theory of the Optical and Electronic Properties of Semiconductors
We insert (17.59) into Eq. (17.18) and linearize the resulting equation in
terms of the perturbations E(t), φ(t) and n(t). As described in prob-
lem (17.4), we obtain
i2ω
m
dE
dt
+ iE
0
dφ
dt
+ iω
m
E
0
8π
0
∂χ
∂N
0
dn
dt
+ ω
2
m
4π
0
∂χ
∂N
0
E
0
n
+
ω
2
m
− ω
2
n
+ iω
m
κc
n
b
+
4πω
2
m
0
χ(N
0
)
E
0
= F (t) . (17.60)
On the RHS of Eq. (17.60) we added the classical Markovian noise term
F (t)=F
(t)+iF
(t) with
F (t) =0 ,
F
(t)F
(t
) = F
(t)F
(t
) = Wδ(t − t
) ,
and
F
(t)F
(t) =0 , (17.61)
where W is given by the rate of spontaneous emission into the laser mode.
The expression for W can be determined from the quantum mechanical
theory of Sec. 17-3.
Eq. (17.60) describes the field and phase perturbations which are cou-
pled to the carrier density perturbations n(t). Introducing the expansion
(17.59) into the Langevin equation (17.53) for the carrier density yields
dn
dt
+
∂χ
∂N
0
1
2
E
2
0
n −
χ
(N
0
)E
0
E
+
n
τ
−
χ
(N
0
)
2
E
2
0
+
N
0
τ
− r
p
= F
n
,
(17.62)
where, as in Sec. 17.2, we again omitted the spontaneous emission and the
nonlinear nonradiative recombination term CN
3
. F
n
is the sum of the
relevant noise contributions.
The perturbations of the carrier density and the field amplitude and
phase have zero mean values. Therefore, taking the bath average of the
coupled Eqs. (17.60) and (17.62), yields the steady state laser equations
(17.22) – (17.24). Using these equations in Eqs. (17.60) and (17.62) gives
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Semiconductor Laser 337
three coupled equations which link the amplitude, phase and carrier-density
perturbations to the Langevin noise terms
dE
dt
+ E
0
4π
0
∂χ
∂N
0
dn
dt
+ E
0
ω
m
2π
0
∂χ
∂N
0
n =
F
ω
m
, (17.63)
dφ
dt
+
4π
0
∂χ
∂N
0
dn
dt
− ω
m
2π
0
∂χ
∂N
0
n = −
F
ω
m
E
0
, (17.64)
and
dn
dt
+
n
τ
r
−
ω
2
r
0
E
0
ω
m
2π∂χ
/∂N
0
E = F
n
. (17.65)
Here, we defined
1
τ
r
=
1
τ
+ E
2
0
∂χ
∂N
0
1
2
,
ω
2
r
= E
2
0
ω
m
2π
0
∂χ
∂N
0
χ
(N
0
)=−I
0
κ
4π
0
∂χ
∂N
0
, (17.66)
and
I
0
= E
2
0
cn
b
8π
. (17.67)
To appreciate these definitions, let us take the time derivative of the deter-
ministic part of Eq. (17.65):
d
2
n
dt
2
+
dn
dt
1
τ
r
−
ω
2
r
0
E
0
ω
m
2π∂χ
/∂N
0
dE
dt
=0 , (17.68)
and eliminate
dE
dt
using the deterministic part of Eq. (17.63). We obtain
d
2
n
dt
2
+
dn
dt
1
τ
r
+ ω
2
r
n =0 , (17.69)
where
1
τ
r
=
1
τ
r
+
2ω
2
r
ω
m
∂χ
/∂N
0
∂χ
/∂N
0
. (17.70)
Eq. (17.69) shows that the density exhibits damped oscillations, the so-
called relaxation oscillations. The parameter combinations ω
r
and τ
r
are
the frequency and damping time of these relaxation oscillations for ω
R
>>
1/τ
r
.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
338 Quantum Theory of the Optical and Electronic Properties of Semiconductors
Eqs. (17.63) – (17.65) are coupled first-order differential equations which
can be solved by Laplace transformation. From the resulting expressions
for E(t), φ(t),andn(t) we can compute the autocorrelation functions, such
as E(t + τ)E(t). We obtain for the phase autocorrelation function, see
problem (17.6),
φ(t + τ)φ(t) =
W
4ω
2
m
E
2
0
(1 + α
2
)t
+[B
1
cos(βτ)+B
2
sin(β|τ|)]e
−|τ |/2τ
r
, (17.71)
where B
1
and B
2
are combinations of the constants entering Eqs. (17.63)
– (17.65),
β
2
=
ω
2
r
−
1
4τ
2
r
and
α =
∂χ
/∂N
0
∂χ
/∂N
0
. (17.72)
is the so-called line-width enhancement factor.
The laser line width is obtained from
E
∗
(t + τ)E(t)E
2
0
e
iω
m
τ
ie
i
φ(t+τ )−φ(t)
E
2
0
e
iω
m
τ
e
−
1
2
φ(t+τ )−φ(t)
2
E
2
0
e
iω
m
τ
exp
,
−
W
8ω
2
m
E
2
0
(1 + α
2
)|τ|
+ e
−|τ |/τ
r
B
1
cos(βτ)+B
2
sin(β|τ|)
− B
1
-
. (17.73)
Assuming for the moment |τ| >> τ
r
, we obtain from Eq. (17.73)
E
∗
(t + τ)E(t)E
2
0
e
iω
m
τ
e
−
W
8ω
2
m
E
2
0
(1+α
2
)|τ |
. (17.74)
Fourier transformation of Eq. (17.74) yields the field spectrum