động học ngưng tụ của polariton trong khuôn khổ phương trình boltzmann master

W
n
g
2
(0)
g
2
(0) P/P
th
W
n
0.9P
th
2.5P
th
W
n
P = 3P
th
g
(2)
(0, t)
g
2
(0)
g
2

(0)

k
λ
2
= L
c
ω
c
0
=
ck
z
√
ǫ
0
=
πc
L
c
√
ǫ
0
.
ω
c,k
=


ω
c
0
2
+ ω
2

=
c
√
ǫ
0

k
2
z
+ k
2

=
c
√
ǫ
0




π

2
L
2
c
+ k
2

= ω
c,q

.
E(q, t) = −
1
c
∂
∂t
A(q, t),
A(q, t) A(r, t)

∂
2
∂t
2
− c
2
∇
2

A(r, t) = 0.
A(r, t) = u

q
(r)e
−iω
q
t
ω
q
=
c | q |
√
ǫ
0
, u
q
(r) =
1
L
3/2
e
iqr
ǫ
0
u
k
(r)
A(r, t) =

k

2πc

2
¯h
ω
k

b
k
u
k
(r) + b
†
k
u
∗
k
(r)

.
A(r, t)
u
q
(r) = χ
q
(z)
1
√
S
e
iq


r

r = (z, x, y) = (z, r

) q = (q
z
, q
x
, q
y
) = q(q
z
, q

) A(r, t)
A(r, t) =

k

2πc
2
¯h
ω
k

b
k
χ
k
(z)

1
√
S
e
ik

r

+ b
†
k
χ
∗
k
(z)
1
√
S
e
−ik

r


ω
k
A(r, t)
A(q

, z, t) =

1
S

dr


k

2πc
2
¯h
ω
k

b
k
χ
k
(z)
e
ik

r

√
S
+ b
†
k
χ

∗
k
(z)
e
−ik

r

√
S

e
−iqr

=

q
z

2πc
2
¯h
ω
k

b
q
z
,q


χ
q
z
,q

(z) + b
†
q
z
,−q

χ
∗
q
z
,−q

(z)

.
E(q

, z, t) =

q
z
i

2π¯hω
k


b
q
z
,q

χ
q
z
,q

(z) − b
†
q
z
,−q

χ
∗
q
z
,−q

(z)

.
ψ
n,m
(r) = −





1
πa
2
0
(n + 1/2)
3
(n− | m |)!
[(n+ | m |)!]
3
ρ
|m|
e
−ρ/2
L
2|m|
n+|m|
(ρ)e
imφ
ρ =
2r
(n +
1
2
)a
0
.
1s n = 1, m = 0

f
1,0
(r) = 2a
−3/2
0
e
−r/a
0
,
Ψ
2d
(r, R) = ξ(z
e
)ξ(z
h
)
e
ikR
√
S
ψ
2d
1s
(r)
R =
r
e
m
e
+ r

h
m
h
m
e
+ m
h
, r = r
e
− r
h
ψ
2d
1s
(r) =

2
π
αe
αr
=

2
π
1
a
2d
0
e
αr

,
ξ(z) =

2
π
cos(
πz
L
z
).
H
I
=

d
3
r
ˆ
ψ
†
(r, t)[er]E(r, t)
ˆ
ψ(r, t).
ˆ
ψ
s
(r, t) =

λ,k


a
λ,k

,s
(t)ψ
λ
(k

, r)
ψ
λ
(r, k

) =

k

ξ
λ
(z)
e
ik

r

√
S
ω
λ
(k


, r)
ψ
∗
λ
(r, k

) =

k

ξ
∗
λ
(z)
e
−ik

r

√
S
ω
∗
λ
(k

, r)
ω
λ

(k

, r) ω
∗
λ
(k

, r)
H
I
= −

k

,k
′

,q


λ,λ
′
a
†
k

,λ
a
k
′


,λ
′
1
S

d
3
rξ
∗
λ
(z)ξ
λ
′
(z)e
i(k
′

−k

+q

)r

ω
∗
λ
(0, r)erω
λ
′

(0, r)E(q

, z, t),
E(r, t) =

q

E(z, q

, t)e
iq

r

.
r r
u
+ r
L
ξ
λ
(z) E(r, t)
H
I
= −

k

,k
′


,q


λ,λ
′
a
†
k

,λ
a
k
′

,λ
′
N

j=1
e
i(k
′

−k

+q

)r
L


S
ξ
∗
λ
(z)ξ
λ
′
(z)E(q

, z, t)
×

d
3
r
u
ω
∗
λ
(0, r
u
)e(r
u
+ r
L
)ω
λ
′
(0, r

u
)e
i(k
′

−k

+q

)r
u

= −

k

,k
′

,q


λ,λ
′
a
†
k

,λ
a

k
′

,λ
′

dzd
2
r
L

e
i(k
′

−k

+q

)r
L

S
ξ
∗
λ
(z)ξ
λ
′
(z)

×E(q

, z, t)
1
l
3

d
3
r
u
ω
∗
λ
(0, r
u
)er
u
ω
λ
′
(0, r
u
)e
i(k
′

−k

+q


)r
u

= −

k

,k
′

,q


λ,λ
′
a
†
k

,λ
a
k
′

,λ
′

dzξ
∗

λ
(z)ξ
λ
′
(z)E(q

, z, t)
×
1
l
3

d
3
r
u
ω
∗
λ
(0, r
u
)er
u
ω
λ
′
(0, r
u
)δ
k


,k
′

+q

.
η(z) = ξ
∗
λ
(z)ξ
λ
′
(z)
d
λ,λ
′
=
1
l
3

d
3
r
u
ω
∗
λ
(0, r

u
)er
u
ω
λ
′
(0, r
u
)
H
I
= −i

k

,k
′

,q


λ,λ
′

dzη(z)
√
2π¯hω
k
√
S

d
λ,λ
′
a
†
k

,λ
a
k

−q

,λ
′

b
q
z
,q

χ
q
z
,q

(z) − b
†
q
z

,−q

χ
∗
q
z
,−q

(z)

.
H
I
= −i

k

,k
′

,q


λ,λ
′

dzη(z)χ
q
z
,q


(z)
√
2π¯hω
k
√
S
d
λ,λ
′
×

a
†
k

,c
a
k

−q

,v
+ a
†
k

,v
a
k


−q

,c

b
q
z
,q

− b
†
q
z
,−q


U(q

, q
z
) =

dzη( z)χ
q
z
,q

(z)
¯hg

ν
(q

, q
z
) = d
cv
ψ
2d
ν
(r

= 0)
√
2πω
q
U(q

, q
z
)
a
†
k

,λ
, a
k

,λ

B
†
ν,K
, B
ν,K
H
I
= −i

ν,q
z
,q

¯hg
ν
(q

, q
z
)

B
†
ν,q

+ B
ν,−q


b

q
z
,q

− b
†
q
z
,−q


.
H
I
= −i

q

¯hg(q

)(B
†
ν,q

+ B
ν,−q

)(b
q
z

,q

− b
†
q
z
,−q

).
H
tot
= H
ex
+ H
ph
+ H
I
= ¯h

q

ω
x,q

B
†
q

B
q


+

q

¯hω
c,q

b
†
q

b
q

− i

q

¯hg(q

)(B
†
ν,q

+ B
ν,−q

)(b
q

z
,q

− b
†
q
z
,−q

)
=

q

¯h

ω
x,q

B
†
q

B
q

+ ω
c,q

b

†
q

b
q

− ig(q

)(B
†
ν,q

+ B
ν,−q

)(b
q
z
,q

− b
†
q
z
,−q

)

,
ω

x,q

= ω
x
0
+
¯h
2
k
2
2M
ω
c,q

a
†
q

, a
q

H
tot
H =

q

¯hΩ
q


a
†
q

a
q

a
q

= x
q

B
q

+ c
q

b
q

Ω
q

,±
=
1
2
(ω

x,q

+ ω
c,q

) ±
1
2

(ω
x,q

− ω
c,q

)
2
+ 4g
2
,
x
q

,j
=




Ω

q

,j
− ω
c,q

2Ω
q

,j
− ω
c,q

− ω
x,q

ic
q

,j
=




Ω
q

,j
− ω

x,q

2Ω
q

,j
− ω
c,q

− ω
x,q

.
H
tot
= H
polariton
+ H
phonon
+ H
polariton−phonon
+ H
polariton−polariton
U(
e
,
h
) = a
e
∆(

e
) + a
h
∆(
h
)
a
e
, a
h
∆(r
e
), ∆(r
h
)
∆( ) =

i

¯h| |
2V ̺u
[b
+ b
†
−
]e
i
b , b
†
−

; V, ̺, u
H
def
=

e h

k
′

B
†
k
′

ψ
∗
k
′

(
e
,
h
)

a
e

i


¯h| |
2V ̺u
(b
+ b
†
−
)e
i
e
+a
h

i

¯h| |
2V ̺u
(b
+ b
†
−
)e
i
h


k

B
k


ψ
k

(
e
,
h
)
=

k
′

,k

,

e h
i

¯h| |
2V ̺u
ψ
∗
k
′

(
e

,
h
)ψ
k

(
e
,
h
)
×

a
e
e
i
e
+ a
h
e
i
h

B
†
k
′

B
k


(b + b
†
−
).
e/h
→ (r
e/h
, z
e/h
),
→ (q, q
z
),
ψ
k
(r
e
, z
e
, r
h
, z
h
) = ξ(z
e
)ξ(z
h
)
e

ikR
√
S
ψ
2d
1s
(r
e
− r
h
)
ξ(z) =

2
L
z
cos(
πz
L
z
), R =
r
e
m
e
+ r
h
m
h
m

e
+ m
h
, ψ
2d
1s
(r) =

2
π
e
r/a
2d
0
a
2d
0
k, k
′
k

, k
′

(x, y)
G(q, q
z
, k, k
′
) =


e h
i

¯h| |
2V ̺u
ψ
∗
k
′
(
e
,
h
)ψ
k
(
e
,
h
)

a
e
e
i
e
+ a
h
e

i
h

= i

¯h| |
2V ̺u
(a
e
I
e
+ a
h
I
h
),
I
e/h
=

e h
ψ
∗
k
′
(
e
,
h
)ψ

k
(
e
,
h
)e
i
e/h
.
I
e
=

dr
e
dr
h
dz
e
dz
h

2
L
z
cos(
πz
e
L
z

)

2
L
z
cos(
πz
h
L
z
)
e
−ik
′
r
e
m
e
+r
h
m
h
m
e
+m
h
√
S

2

π
e
r
e
−r
h
a
2d
0
a
2d
0
×
×

2
L
z
cos(
πz
e
L
z
)

2
L
z
cos(
πz

h
L
z
)
e
ik
r
e
m
e
+r
h
m
h
m
e
+m
h
√
S

2
π
e
r
e
−r
h
a
2d

0
a
2d
0
e
iq
z
z
e
e
iqr
e
=

dz
e
2
L
z
cos
2
(
πz
e
L
z
)e
iq
z
z

e

dz
h
2
L
z
cos
2
(
πz
h
L
z
)

dr
e
dr
h
e
−ik
′
r
e
m
e
+r
h
m

h
m
e
+m
h
√
S
×
×

2
π
e
r
e
−r
h
a
2d
0
a
2d
0
e
ik
r
e
m
e
+r

h
m
h
m
e
+m
h
√
S

2
π
e
r
e
−r
h
a
2d
0
a
2d
0
e
iqr
e
=

L
z

/2
−L
z
/2
dz
e
2
L
z
cos
2
(
πz
e
L
z
)e
iq
z
z
e

L
z
/2
−L
z
/2
dz
h

2
L
z
cos
2
(
πz
h
L
z
) ×
×

+∞
−∞
dr
e
dr
h
e
−i(k
′
−k)
r
e
m
e
+r
h
m

h
m
e
+m
h
√
S
2
π
e
2
r
e
−r
h
a
2d
0
a
2d
0
e
iqr
e
=
8π
2
q
z
L

z
(4π
2
− q
2
z
L
2
z
)
sin(
q
z
L
z
2
)

+∞
−∞
dr
e
dr
h
e
−i(k
′
−k)
r
e

m
e
+r
h
m
h
m
e
+m
h
+iqr
e
√
S
2
π
e
2
r
e
−r
h
a
2d
0
a
2d
0
=
8π

2
q
z
L
z
(4π
2
− q
2
z
L
2
z
)
sin(
q
z
L
z
2
)

1 + (
a
b
q
e
2
)
2


−3/2
δ(k
′
− k + q)
q
e
=
m
e
m
e
+ m
h
(k
′
− k) + q, b
e/h
=
m
e/h
m
e
+ m
h
qa
b
I
h
G(q, q

z
, k) =

k
′
G(q, q
z
, k, k
′
)
= i

¯h| |
2V ̺u
8π
2
q
z
L
z
(4π
2
− q
2
z
L
2
z
)
sin(

q
z
L
z
2
)

a
e
[1 +
b
2
e
4
]
−3/2
+ a
h
[1 +
b
2
h
4
]
−3/2

.
H
def
=


q,q
z
,k
G(q, q
z
, k)B
†
k
B
k−q
(b
q,q
z
+ b
†
−q,q
z
).
a, a
†
B
k
=

k
x
k
a
k

x
k
H
def
= H
p−ph
=

q,q
z
,k
G(q, q
z
, k)x
∗
k
x
k−q
a
†
k
a
k−q
(b
q,q
z
+ b
†
−q,q
z

)
=

q,q
z
,k
ˆ
G(q, q
z
, k)a
†
k
a
k−q
(b
q,q
z
+ b
†
−q,q
z
).
H
p−p
=
1
4S

k,k
′

,q
V (k, k
′
, q)a
†
k+q
a
†
k
′
−q
a
k
′
a
k
.
V (k, k
′
, q) = 6E
B
a
2
B
x
k+q
x
k
′
−q

x
k
′
x
k
x
k
E
B
a
B
H
tot
=

k
e
k
a
†
k
a
k
+

q,q
z
ω
q,q
z

b
†
q,q
z
b
q,q
z
+

q,q
z
,k
G(q, q
z
, k)a
†
k
a
k−q
(b
q,q
z
+ b
†
−q,q
z
) +
+
1
4S


k,k
′
,q
V (k, k
′
, q)a
†
k+q
a
†
k
′
−q
a
k
′
a
k
= H
p
+ H
ph
+ H
p−ph
+ H
p−p
.
f
k

= a
†
k
a
k
d
dt
f
k
=
i
¯h
[H
tot
, f
k
] =
i
¯h
[H
p
+ H
ph
+ H
p−ph
+ H
p−p
, f
k
]

=
i
¯h


k
′
e

a
†
k
′
a
k
′
, a
†
k
a
k

+

q,q
z
ω
q,q
z


b
†
q,q
z
b
q,q
z
, a
†
k
a
k

+
+

q,q
z
,k
G(q, q
z
, k
′
)

a
†
k
′
a

k
′
−q
(b
q,q
z
+ b
†
−q,q
z
), a
†
k
a
k

+
+
1
4S

k,k
′
,q
V (k, k
′
, q)

a
†

k+q
a
†
k
′
−q
a
k
′
a
k
, a
†
k
a
k

.
[a
†
k
′
a
k
′
, a
†
k
a
k

] = δ(k − k
′
)(a
k
′
a
†
k
− a
k
a
†
k
′
)
[a
†
k
′
a
k
′
−q
, a
†
k
a
k
] = −δ
k,k

′
a
†
k
a
k
′
−q
+ δ
k,k
′
−q
a
†
k
′
a
k
[H
p−ph
, f
k
] =


q,q
z
G(q, q
z
, k + q)a

†
k+q
a
k
−

q,q
z
G(q, q
z
, k)a
†
k
a
k−q

(b
q,q
z
+ b
†
−q,q
z
)
=


q,q
z
G(q, q

z
, k + q)

a
†
k+q
a
k
b
q,q
z
+ a
†
k+q
a
k
b
†
−q,q
z

+
−

q,q
z
G(q, q
z
, k)


a
†
k
a
k−q
b
q,q
z
+ a
†
k
a
k−q
b
†
−q,q
z

=

q,q
z

G(q, q
z
, k + q)

a
†
k+q

a
k
b
q,q
z
+ a
†
k+q
a
k
b
†
−q,q
z

+
−G(q, q
z
, k)

a
†
k
a
k−q
b
q,q
z
+ a
†

k
a
k−q
b
†
−q,q
z

.
d
dt
a
†
k+q
a
k
b
q,q
z
=
i
¯h
[H, a
†
k+q
a
k
b
q,q
z

]
=
i
¯h


k
′
e
k
′

a
†
k
′
a
k
′
, a
†
k+q
a
k
b
q,q
z

+


q
′
,q
′
z
ω
q
′
,q
′
z

b
†
q
′
,q
′
z
b
q
′
,q
′
z
, a
†
k+q
a
k

b
q,q
z

+

q
′
,q
′
z
,k
′
G(q
′
, q
′
z
, k
′
)[a
†
k
′
a
k
′
−q
′
(b

q
′
,q
′
z
+ b
†
−q
′
,q
′
z
), a
†
k+q
a
k
b
q,q
z
]

=
i
¯h
(I + J + K).
I =

k
′

e
k
′

a
†
k
′
a
k
′
, a
†
k+q
a
k
b
q,q
z

=

k
′
e
k
′
(δ
k
′

,k+q
a
†
k
′
a
k
− δ
kk
′
a
†
k+q
a
k
′
)b
q,q
z
= (e
k+q
− e
k
)a
†
k+q
a
k
b
q,q

z
.
J =

q
′
,q
′
z
ω
q
′
,q
′
z

b
†
q
′
,q
′
z
b
q
′
,q
′
z
, a

†
k+q
a
k
b
q,q
z

=

q
′
,q
′
z
ω
q
′
,q
′
z
(−δ
qq
′
a
†
k+q
a
k
b

q
′
)
= −ω
q
a
†
k+q
a
k
b
q,q
z
.
K =

q
′
,q
′
z
,k
′
G(q
′
, q
′
z
, k
′

)

a
†
k
′
a
k
′
−q
′
b
†
−q
′
,q
′
z
a
†
k+q
a
k
b
q,q
z
− a
†
k+q
a

k
b
q,q
z
a
†
k
′
a
k
′
−q
′
b
†
−q
′
,q
′
z

=

q
′
,q
′
z
,k
′

G(q
′
, q
′
z
, k
′
)

a
†
k
′
a
k
a
†
k+q
a
k
′
−q
′
− a
†
k
′
a
k
δ

jj
′
δ
k+q,k
′
−q
′

b
†
−q
′
,q
′
z
b
q,q
z
+
−

a
†
k+q
a
k
′
−q
′
a

†
k
′
a
k
− a
†
k+q
a
k
′
−q
′
δ
k,k
′

b
q,q
z
b
†
−q
′
,q
′
z

= G(k, −q, q
z

)(a
†
k
a
k
a
†
k+q
a
k−q
− a
†
k
a
k
)b
†
−q,q
z
b
q,q
z
+
−G(k, −q, q
z
)(a
†
k+q
a
k−q

a
†
k
a
k
− a
†
k+q
a
k−q
)b
q,q
z
b
†
−q,q
z
.
K = G(k, −q, q
z
)

f
k
f
k+q
N
q
− f
k+q

(f
k
+ 1)(N
q
+ 1)

,
N
q
= b
†
−q,q
z
b
q,q
z
 = (e
ω
q
/k
B
T
− 1)
−1
d
dt
a
†
k+q
a

k
b
q,q
z
 = i(e
k+q
− e
k
− ω
q,q
z
)a
†
k+q
a
k
b
q,q
z
 +
+
i
¯h
G(k, −q, q
z
)

f
k
f

k+q
N
q
− f
k+q
(f
k
+ 1)(N
q
+ 1)

.
d
dt
a
†
k+q
a
k
b
†
−q,q
z
=
i
¯h
[H, a
†
k+q
a

k
b
†
q,q
z
]
=
i
¯h


k
′
e
k
′

a
†
k
′
a
k
′
, a
†
k+q
a
k
b

†
q,q
z

+
+

q
′
,q
′
z
ω
q
′
,q
′
z

b
†
q
′
,q
′
z
b
q
′
,q

′
z
, a
†
k+q
a
k
b
†
q,q
z

+

q
′
,q
′
z
,k
′
G(q
′
, q
′
z
, k
′
)
×


a
†
k
′
a
k
′
−q
′
(b
q
′
,q
′
z
+ b
†
−q
′
,q
′
z
), a
†
k+q
a
k
b
†

q,q
z

= i(e
k+q
− e
k
− ω
q,q
z
)a
†
k+q
a
k
b
q,q
z
+

q
′
,q
′
z
,k
′
G(q
′
, q

′
z
, k
′
)
×

a
†
k
′
a
k
a
†
k+q
a
k
′
−q
′
+ a
†
k
′
a
k
δ
k+q,k
′

−q
′

b
q,q
z
b
†
−q
′
,q
′
z
+
−

a
†
k+q
a
k
′
−q
′
a
†
k
′
a
k

+ a
†
k+q
a
k
′
−q
′
δ
k,k
′

b
†
−q
′
,q
′
z
b
q,q
z

.
d
dt
a
†
k+q
a

k
b
†
−q,q
z
 = i(e
k+q
− e
k
+ ω
q,q
z
)a
†
k+q
a
k
b
†
−q,q
z
 +
i
¯h
G(k, −q, q
z
) ×
×

f

k
(f
k+q
+ 1)(N
q
+ 1) − f
k+q
(f
k
+ 1)N
q

.
d
dt
a
†
k
a
k−q
b
q,q
z
 = i(e
k
− e
k−q
− ω
q,q
z

)a
†
k
a
k−q
b
q,q
z
 +
i
¯h
G(k − q, −q, q
z
) ×
×

f
k−q
(f
k
+ 1)N
q
− f
k
(f
k−q
+ 1)(N
q
+ 1)


d
dt
a
†
k
a
k−q
b
†
−q,q
z
 = i(e
k
− e
k−q
+ ω
q,q
z
)a
†
k
a
k−q
b
†
−q,q
z
 +
i
¯h

G(k − q, −q, q
z
) ×
×

f
k−q
(f
k
+ 1)(N
q
+ 1) − f
k
(f
k−q
+ 1)N
q

.
˙
A = ieA + Γ(t)
A(t) = A(t
0
)e
ie(t−t
0
)
+

t

t
0
dτe
ie(t−τ)
Γ(τ).
A(t
0
) = 0
a
†
k+q
a
k
b
q,q
z
 =

t
t
0
dτe
i(e
k+q
−e
k
−ω
q,q
z
)(t−τ)

×
×
i
¯h
G(k, −q, q
z
)

f
k
f
k+q
N
q
− f
k+q
(f
k
+ 1)(N
q
+ 1)

.
a
k
(τ) = a
k
(t)e
ie
k

(t−τ)
a
†
k
(τ) = a
†
k
(t)e
−ie
k
(t−τ)
f
k
(τ) = f
k
(t) G(k, −q, q
z
) τ
t
0
→ −∞
a
†
k+q
a
k
b
q,q
z
 =

i
¯h
G(k, −q, q
z
)

f
k
f
k+q
N
q
− f
k+q
(f
k
+ 1)(N
q
+ 1)

×
×

t
−∞
dτe
i(e
k+q
−e
k

−ω
q,q
z
+iδ)(t−τ)
.

t
−∞
dτe
i(e
k+q
−e
k
−ω
q,q
z
)(t−τ)
= −

0
∞
dt
′
e
i(e
k+q
−e
k
−ω
q,q

z
+iδ)t
′
=
i
e
k+q
− e
k
− ω
q,q
z
+ iδ
= D(e
k+q
− e
k
− ω
q,q
z
)
= P (
1
e
k+q
− e
k
− ω
q,q
z

) − iπδ(e
k+q
− e
k
− ω
q,q
z
).
a
†
k+q
a
k
b
q,q
z
 =
i
¯h
G(k, −q, q
z
)D(e
k+q
− e
k
− ω
q,q
z
)
×


f
k
f
k+q
N
q
− f
k+q
(f
k
+ 1)(N
q
+ 1)

a
†
k+q
a
k
b
†
−q,q
z
 =
i
¯h
G(k, −q, q
z
)D(e

k+q
− e
k
+ ω
q,q
z
)
×

f
k
(f
k+q
+ 1)(N
q
+ 1) − f
k+q
(f
k
+ 1)N
q

a
†
k
a
k−q
b
q,q
z

 =
i
¯h
G(k −q, −q, q
z
)D(e
k
− e
k−q
− ω
q,q
z
)
×

f
k−q
(f
k
+ 1)N
q
− f
k
(f
k−q
+ 1)(N
q
+ 1)

a

†
k
a
k−q
b
†
−q,q
z
 =
i
¯h
G(k −q, −q, q
z
)D(e
k
− e
k−q
+ ω
q,q
z
)
×

f
k−q
(f
k
+ 1)(N
q
+ 1) − f

k
(f
k−q
+ 1)N
q

df
k
/dt
i
¯h
[H
ph
, f
k
] =
i
¯h

q,q
z
G(k + q, q, q
z
)
i
¯h
G(k, −q, q
z
)D(e
k+q

− e
k
− ω
q,q
z
)
×

f
k
f
k+q
N
q
− f
k+q
(f
k
+ 1)(N
q
+ 1) +

+
i
¯h

q,q
z
G(k + q, q, q
z

)
i
¯h
G(k, −q, q
z
)D(e
k+q
− e
k
+ ω
q,q
z
)
×

f
k
(f
k+q
+ 1)(N
q
+ 1) − f
k+q
(f
k
+ 1)N
q

+
−

i
¯h

q,q
z
G(k, q, q
z
)
i
¯h
G(k −q, −q, q
z
)D(e
k
− e
k−q
− ω
q,q
z
)
×

f
k−q
(f
k
+ 1)N
q
− f
k

(f
k−q
+ 1)(N
q
+ 1)

+
−
i
¯h

q,q
z
G(k, q, q
z
)
i
¯h
G(k −q, −q, q
z
)D(e
k
− e
k−q
+ ω
q,q
z
)
×


f
k−q
(f
k
+ 1)(N
q
+ 1) − f
k
(f
k−q
+ 1)N
q

.
q → −q
D(e)
D
(2.32)
= D(e
k
− e
k−q
− ω
q,q
z
)
(q → −q) : D
(2.32)
= D(e
k

− e
k+q
− ω
−q,q
z
) = D(−[e
k+q
− e
k
+ ω
q,q
z
])
= −P
i
e
k+q
− e
k
+ ω
q,q
z
+ πδ(e
k+q
− e
k
+ ω
q,q
z
)

D
(2.31)
= P
i
e
k+q
− e
k
+ ω
q,q
z
+ πδ(e
k+q
− e
k
+ ω
q,q
z
).
δ
P
i
¯h
[H
ph−p
, f
k
] = −
2π
¯h

2

q,q
z
,j
G(k + q, q, q
z
)G(k, −q, q
z
)δ(e
k+q
− e
k
− ω
q,q
z
) ×
×

f
k
(f
k+q
+ 1)N
q
− f
k+q
(f
k
+ 1)(N

q
+ 1)

+
−
2π
¯h
2

q,q
z
,j
G(k + q, q, q
z
)G(k, −q, q
z
)δ(e
k+q
− e
k
+ ω
q,q
z
) ×
×

f
k
(f
k+q

+ 1)(N
q
+ 1) − f
k+q
(f
k
+ 1)N
q

.
˙
f
k




p−ph
= −
2π
¯h
2

q,q
z
G(k + q, q, q
z
)G(k, −q, q
z
)

×

f
k
(f
k+q
+ 1)

N
q
δ(e
k+q
− e
k
− ω
q,q
z
) + (N
q
+ 1)δ(e
k+q
− e
k
+ ω
q,q
z
)

−f
k+q

(f
k
+ 1)

(N
q
+ 1)δ(e
k+q
− e
k
− ω
q,q
z
) + N
q
δ(e
k+q
− e
k
+ ω
q,q
z
)

= −

k
′
=k+q


W
kk
′
f
k
(f
k
′
+ 1) − W
k
′
k
f
k
′
(f
k
+ 1)

W
kk
′
=
2π
¯h
2

q,q
z
G(k + q, q, q

z
)G(k, −q, q
z
)

N
q
δ(e
k+q
− e
k
− ω
q,q
z
)
+(N
q
+ 1)δ(e
k+q
− e
k
+ ω
q,q
z
)

W
k
′
k

=
2π
¯h
2

q,q
z
G(k + q, q, q
z
)G(k, −q, q
z
)

(N
q
+ 1)δ(e
k+q
− e
k
− ω
q,q
z
)
+N
q
δ(e
k+q
− e
k
+ ω

q,q
z
)

.
N
ǫ
q
N
ǫ
q
=



N
q
: ǫ = 1
N
q
+ 1 : ǫ = −1
N
ǫ
q
= N
q
−
1
2
(ǫ −1)

W
kk
′
=
2π
¯h
2

q,q
z
,ǫ=±1
G(k + q, q, q
z
)G(k, −q, q
z
)N
ǫ
q
z
,|k
′
−k|
δ(e
k+q
− e
k
− ǫω
q,q
z
)

W
k
′
k
=
2π
¯h
2

q,q
z
,ǫ=±1
G(k + q, q, q
z
)G(k, −q, q
z
)N
ǫ
q
z
,|k
′
−k|
δ(e
k+q
− e
k
+ ǫω
q,q
z

).
W
kk
′
=
2π
¯h
2

q,q
z
,ǫ=±1
G(k + q, q, q
z
)G(k, −q, q
z
)N
ǫ
q
z
,|k
′
−k|
δ(e
k+q
− e
k
− ǫω
q,q
z

)
=
2π
¯h
2

q,q
z
,ǫ=±1

x
∗
k
x
k
′
i

¯h
2V ̺u
8π
2
q
z
L
z
(4π
2
− q
2

z
L
2
z
)
sin(
q
z
L
z
2
)

×

x
∗
k
′
x
k
i

¯h
2V ̺u
8π
2
q
z
L

z
(4π
2
− q
2
z
L
2
z
)
sin(
q
z
L
z
2
)

×

a
e
[1 + b
2
e
/4]
−3/2
+ a
h
[1 + b

2
h
/4]
−3/2

2
δ(e
k+q
− e
k
− ǫω
q,q
z
).
A(q
z
) =
8π
2
q
z
L
z
(4π
2
− q
2
z
L
2

z
)
sin(
q
z
L
z
2
)
D(q) = a
e
[1 + b
2
e
/4]
−3/2
+ a
h
[1 + b
2
h
/4]
−3/2
K
kk
′
=
e
k
′

− e
k
u
, ω
q,q
z
= uq, q
z
|
| =

q
2
z
+ | k
′
− k |
2
.
⇒ W
kk
′
= −
2π
¯h
2

q,q
z
,ǫ=±1

(x
∗
k
x
k
′
)
2
¯h|
|
2V ̺u
A
2
(q
z
)D
2
(q)δ(e
k+q
− e
k
− ǫω
q,q
z
).

k
′′
,k
′

,q
V (k
′′
, k
′
, q)[a
†
k
′′
+q
a
†
k
′
−q
a
k
′
a
k
′′
, a
†
k
a
k
]
= −2

k

′
,q
V (k − q, k
′
, q)a
†
k
′
−q
a
†
k
a
k
′
a
k−q
+ 2

k
′
,q
V (k
′
, k, q)a
†
k
′
+q
a

†
k−q
a
k
′
a
k
.
p = a
†
k
′
−q
a
†
k
a
k
′
a
k−q
a
†
k
′
+q
a
†
k−q
a

k
′
a
k
−i¯h
d
dt
a
†
k
′
−q
a
†
k
a
k
′
a
k−q
= [H
tot
, a
†
k
′
−q
a
†
k

a
k
′
a
k−q
] = [H
tot
, p],
H
tot
= H
p
+ H
ph
+ H
p−ph
+ H
p−p
[H
p
, p] = (−e
k
′
−q
+ e
k
+ e
k
′
− e

k−q
)p.
[H
p−p
, p] ∼

k
1
,k
′′
,q
′
V (k
1
, k
′′
, q
′
)[a
†
k
1
+q
′
a
k
′′
a
†
k

′′
−q
′
a
k
1
, a
†
k
′
−q
a
†
k
a
k
′
a
k−q
]
˙
f
k




p−p
= −
2π

¯h

k
′
,q
X
k+q,k
k
′
−q,k
′

f
k
f
k
′
(1 + f
k+q
)(1 + f
k
′
−q
) − f
k+q
f
k
′
−q
(1 + f

k
)(1 + f
k
′
)

.
k
p
≃ 1.7 × 10
−2
−1
P
c
(k, t) = P
0
e
(
E(k)−E(k
p
)
Γ
)
2
tanh(t/t
0
)
t
0
P

p
(k, t) = P
0
e
[
E(k)−E(k
p
)
Γ
]
2
e
(
t−t
p
∆t
)
2
t
p
∆t
τ
x
τ
c
1/τ
p
= x
2
/τ

x
+ c
2
/τ
c
.
f
k
˙
f
k
= P
c/p
(k, t) −
f
k
τ
p
+
˙
f
k




p−ph
+
˙
f

k




p−p
11 −2

k = 0
n
0
= f
k=0
/S = f
0
/S
∂
∂t
f
k
= P
k
−
f
k
τ
k
+
∂
∂t

f
k




p−ph
+
∂
∂t
f
k




p−p
∂
∂t
n
0
= −
n
0
τ
0
+
∂
∂t
n

0




p−ph
+
∂
∂t
n
0




p−p
∂
∂t
f
k




p−ph
= −

q,σ=±
W
p−ph

k,q,σ

f
k
(1 + f
k+q
)N
q,σ
− f
k+q
(1 + f
k
)N
q,−σ
