Quantum Optics SeminarTalya 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 zkkq

AH

zktqAtzE

jj j

jjy

jj

jjx

,...3,2,1

/

j

Ljk j

0

22Vm

A jjjLcjj / constm j

Canonical momentum of the jth mode

j j

jjjjyxV m

pqmHEd

2222

02

0 21)(

21

jjj qmp

0,,

,

jjjj

jjjj

ppqq

ipq

)(2

1

)(2

1

jjjjjj

tij

jjjjjj

tij

ipqmm

ea

ipqmm

ea

j

j

0,,

,

jjjj

jjjj

aaaa

aa

jjjj aa

21

Annihilation (absorption) operator

Creation operator

cceav

ktrH

cceatrE

k

rkitikk

k

k

k

rkitikkk

k

k

.ˆ1),(

.ˆ),(

,,

)(

0

,,

)(

),,( zyx kkkk

Ln

k ii

2

2/1

02

Vk

k

Fock states Single mode of frequency

nEnaan n )21(

nn

n

n

EE

nna

naEna

1

1

)(

Eigenstate

Eigenvalue

21

21

00

0)(0

0

0

nE

E

a

aEa

n

Naa

Normalization

Complete set

Multi mode fields

0!)(

11

1

nan

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

na

na

nannn

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|

||

cnvc

cnvc

ncvnnc

ncv

n

n

nn

nn

nn

nn

2/0

00

2

|!

|

v

n

n

ec

nnvcv

))(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

nv

evnnpn

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

212

2/)(2/

2/)2(1*22/2/

1*22/2/

12

!

)!()!(

vv

vvvvvv

vvvv

n

nvv

m n

mnvv

evv

ee

envv

ee

mnmn

vveevv

)(~ 212

221 vve vv

110

2

n

nnvdvv

A resolution of the identity operator 1 in terms of coherent state projectors:

vdvv 21

Over- complete set

Displacement operator0|0|

!)ˆ(| ˆ2/

0

2/ 22

avv

n

nnv ee

navev

BABBAA

BABABAˆ,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*

avave 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(2/12

2/12 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

(( ˆˆˆˆˆˆ

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(21exp)(ˆ 22* azazzS

parameter squeeze

rrez i

aa

reara

azzazaza

zSazSzA

i

ˆˆsinhˆcoshˆ

...!3ˆ

!2ˆ

ˆˆ

)(ˆˆ)(ˆ)(ˆ22

re

ri sinh

cosh

aa

zSazSzAˆˆ

)(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

vzAAAAeAAeAAvzvzQvzii

ii

Coherent representation of Thermal states

Density operator

Fock state representation- exponent

Coherent state representation- Gaussian distribution (Distribution function)

)(ˆ

ˆ

ˆ

H

H

eTre

kkn

n

n

nkk

n nnn

nnnee

kk

k

k

kkk

1)1(

)1(ˆ 1

1

ken

vdvvvvP 2* ),(̂

nnenn

n

nkk

nn

n

kk

k 11exp1)1(

ˆ 2

1

2

nven

vvP21),( *

22

* **22

ˆ),( deeevvP 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