quantum optics seminar talya vaknin. quantization of the electromagnetic field fock states and fock...
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Quantum Optics Seminar
Talya Vaknin
Quantization of the electromagnetic field Fock states and Fock spaceCoherent states Squeezed statesCoherent representation of Thermal states
Quantization of the free electromagnetic field
Electromagnetic field contained in a very large cube of side L
Periodic boundary conditions Electric field linearly polarized in the x- direction
)cos(
)sin()(),(
0 zkk
qAH
zktqAtzE
jj j
jjy
jj
jjx
,...3,2,1
/
j
Ljk j
0
22
V
mA jj
jLcjj / constm j
Canonical momentum of the jth mode
j j
jjjjyxV m
pqmHEd
2222
02
0 2
1)(
2
1
jjj qmp
0,,
,
jjjj
jjjj
ppqq
ipq
)(2
1
)(2
1
jjjj
jj
tij
jjjj
jj
tij
ipqmm
ea
ipqmm
ea
j
j
0,,
,
jjjj
jjjj
aaaa
aa
jjjj aa
2
1
Annihilation (absorption) operator
Creation operator
cceav
ktrH
cceatrE
k
rkiti
kkk
k
k
rkiti
kkk
k
k
.ˆ1
),(
.ˆ),(
,,
)(
0
,,
)(
),,( zyx kkkk
L
nk ii
2
2/1
02
Vk
k
Fock states
Single mode of frequency
nEnaan n )2
1(
nn
n
n
EE
nna
naEna
1
1
)(
Eigenstate
Eigenvalue
2
12
1
00
0)(0
0
0
nE
E
a
aEa
n
Naa
Normalization
Complete set
Multi mode fields
0!
)(
11
1
n
an
nnna
nnna
n
nn
n
nc
nn
10
,...1,...,1,...,...,
,...1,...,,...,...,
0!
)(...
!
)(
!
)(,...,
2121
2121
1
2
22
1
11
21
llll
llll
m
mkkk
ml
kkkkkkkk
kkkkkkkk
k
nk
k
nk
k
nk
kkk
nnnnnnna
nnnnnnna
n
a
n
a
n
annn
Coherent states
Eigen states of the annihilation operator Poisson distribution of Fock states States of minimum uncertainty product A product of the displacement operator on
the vacuum state.
Fock representation of the coherent state
|ˆ|
||ˆ* vvav
vvva
0
1
01
0
!
|1|
||
cn
vc
cn
vc
ncvnnc
ncv
n
n
nn
nn
nn
nn
2/0
00
2
|!
|
v
n
n
ec
nn
vcv
))(ˆ)(ˆ(2
)(ˆ tatatq
2ˆ 2
vqv
2ˆ 2
vpv
2)ˆ)(ˆ( 22 pq
))(ˆ)(ˆ(2
)(ˆ tataitp
Minimum uncertainty
The photon distribution p(n) for a coherent state
1.02 v
12 v
102 v
!|)(
22 2
n
vevnnp
n
v
Probability that n photons will be found in the coherent state
0
2)(
n
vnnp
Mean number of photons
222
22)(
vvvaaaav
vNvvNvnVar
Variance
Complete set
221
*121
*2
221
1*2
21
22
22
21
22
21
2
12
2/)(2/
2/)2(1*22/2/
1*22/2/
12
!
)!()!(
vv
vvvvvv
vvvv
n
nvv
m n
mnvv
evv
ee
en
vvee
mnmn
vveevv
)(~ 212
221 vve vv
11
0
2
n
nnvdvv
A resolution of the identity operator 1 in terms of coherent state projectors:
vdvv 21
Over- complete set
Displacement operator
0|0|!
)ˆ(| ˆ2/
0
2/22
avv
n
nnv ee
n
avev
BABBAA
BABABA
ˆ,ˆ,ˆ0ˆ,ˆ,ˆ
)2/ˆ,ˆexp(ˆexpˆexp)ˆˆexp(
avBavA ˆˆ,ˆˆ *
avavevD ˆˆ *
)(ˆ
0|| ˆˆ2/ *2avavv eeev
0|0|...!2
)ˆ(ˆ10|
2**ˆ*
av
ave av
0|| ˆˆ *avavev
)(ˆ)(ˆ1)(ˆ)(ˆ vDvDvDvD )(ˆ)(ˆ vDvD
Squeezed states Squeezing a single mode field
iPQ
aaiP
aaQ
2ˆ,ˆ
)ˆˆ(ˆ
ˆˆˆ
1)ˆ()ˆ(2/1
22/1
2 PQ
12ˆ,ˆˆˆ2ˆˆ
ˆˆˆˆˆˆˆ
ˆˆˆ
ˆˆ)ˆ(
22*222
222
*
222
vvvvaaaaaav
vaaaaaavQ
vvvaavQ
AAA
1)ˆ(
1)ˆ(
2
2
P
Q
)]sin(ˆ)cos(ˆ[)(),(ˆ
]ˆˆ[)(),(ˆ )()(
vtrkPvtrkQvltrE
eaeavltrE vtrkivtrki
(( ˆˆˆ
ˆˆˆ
ii
ii
eaeaP
eaeaQ
)]sin(ˆ)cos(ˆ[)(
),(ˆ
vtrkPvtrkQvl
trE
1)ˆ(1)ˆ( 22 PQ
Vacuum state
Coherent state
Squeezed state with reduced phase uncertainty
Squeezed state with reduced amplitude uncertainty
The unitary squeeze operator
)ˆˆ(2
1exp)(ˆ 22* azazzS
parameter squeeze
r
rez i
aa
reara
azzazaza
zSazSzA
i
ˆˆ
sinhˆcoshˆ
...!3
ˆ
!2
ˆˆˆ
)(ˆˆ)(ˆ)(ˆ
22
re
ri sinh
cosh
aa
zSazSzA
ˆˆ
)(ˆˆ)(ˆ)(ˆ
*
122
1)(ˆ),(ˆ zAzA
Two photon coherent state vzSvz )(ˆ,
vzvzAvz
vzvvzSvvazSvzSzSazSvzzA
,)(ˆ,
,)(ˆˆ)(ˆ)(ˆ)(ˆˆ)(ˆ,)(ˆ
*
ii
iiii
evvevv
vzeAAeAAvzvzeaeavzvzQvz
)()(
,)ˆˆ()ˆˆ(,,ˆˆ,,ˆ,***
**
)2cos(2sinh2cosh)(12ˆ 22*22 rreeQ ii
2
1,)ˆ(,
2sinh2cosh,)ˆ(,
2
22
vzQvz
errvzQvz r revzPvz 22 ,)ˆ(,
2/14 )1( re
122])[(])[(
,1)ˆˆ)(ˆˆ(2)ˆˆ()ˆˆ(,,ˆ,
22**22*2*2**
**2222*2
vvevvevv
vzAAAAeAAeAAvzvzQvz
ii
ii
Coherent representation of Thermal states
Density operator
Fock state representation- exponent
Coherent state representation- Gaussian distribution (Distribution function)
)(ˆ
ˆ
ˆ
H
H
eTr
e
kkn
n
n
nkk
n nnn
nnnee
k
k
k
k
kkk
1)1()1(ˆ
1
1
ke
n
vdvvvvP 2* ),(̂
nn
enn
n
nkk
nn
n
k
k
k 11exp
1)1(ˆ 2
1
2
nven
vvP21
),( *
22
* **2
2
ˆ),( deee
vvP vavv
Bibliography
Leonard Mandel and Emil Wolf, Optical coherence and quantum optics, chap 10, 11 and 21 (Cambridge University press, Cambridge 1995)
Marlan O. Scully and M. Suhail Zubairy, Quantum Optics, chap 1 and 2 (Cambridge University press, Cambridge 1997)
Minimum uncertainty
)argcos(2
)(2
ˆ
))(ˆ)(ˆ(2
)(ˆ
*
vtv
evvevqv
tatatq
titi
)argsin(2
)(2
ˆ
))(ˆ)(ˆ(2
)(ˆ
*
vtv
evveivpv
tataitp
titi
2)ˆ)(ˆ( 22 pq
)12(2
ˆ
)1)(ˆ)(ˆ2)(ˆ)(ˆ(2
))(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ(2
ˆ
*22*222
22
222
vvevevvqv
tatatata
tatatatatataq
titi
2ˆˆˆ
222 vqvvqvvqv
2ˆ 2
vpv