ece 407 –spring 2009 –farhan rana –cornell university1 ece 407 –spring 2009 –farhan rana...

1

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Chapter 7: Quantum States of Light

In this lecture you will learn:

• Coherent States• Quadrature Operators and Quadrature Fluctuations• Squeezed States• Two-Photon Coherent States• Number and Phase Noise Operators• Number-Phase Uncertainty• Thermal States


Radiation in a Cavity

Cavity

mmmm

momo

mmmm

mo

n

mmmm

mom

rUtatatrH

rUtataitrE

rUtatatrA

)()(ˆ)(ˆ2

1),(ˆ

)()(ˆ)(ˆ2

),(ˆ

)()(ˆ)(ˆ2

),(ˆ

Field operators are:

We consider only one mode for simplicity:

)()(ˆ)(ˆ2

1),(ˆ

)()(ˆ)(ˆ2

),(ˆ

)()(ˆ)(ˆ2

),(ˆ

rUtatatrH

rUtataitrE

rUtatatrA

ooo

o

o

oo

21

ˆˆˆ aaH o

2


Photon Number States

nnnaann

n

an

n

ˆˆˆ

0!

)(

Photon number states are defined as:

These are eigenstates of the photon number operator

nnnH o

21ˆ

11ˆ

1ˆ

nnna

nnna

0|ˆ||ˆ| nannan

Since:

0),(ˆ),(ˆ ntrHnntrEn

Average values of fields in number states are zero!


What Quantum States are Generated by Oscillating Currents?

Cavity

Maxwell’s Equations:

t

trHtrE

ttrE

rtrJtrH

o

o

,,

,,,

Hamiltonian Description:

rAtrJrdaatH o

ˆ.,ˆˆˆ 3

trStrJ ocos,

Schrodinger Picture

)(ˆˆ2

)(ˆ rUaarAoo

taaJaatH ooo cosˆˆˆˆˆ )(.2

3 rUrSrdJoo

o

Initial State:

00 t

3



Cavity

taaJaatH ooo cosˆˆˆˆˆ

Start from the Schrodinger equation:

ttHt

ti

ˆ

aaH oo ˆˆˆ

Let:

t

tietH

t

ti

tet

tHi

o

tHi

o

o

ˆ

ˆ

ˆ



Cavity

0

ˆˆ2

cosˆˆ

cosˆˆ

cosˆˆˆˆ

ˆ

ˆˆ

ˆˆ

ˆ

* atat

iio

otiti

o

tHi

oo

tHi

ooo

tHi

o

et

teaeaJ

t

ti

tteaeaJt

ti

tetaaJet

ti

ttaaJHt

tietH

t

ti

ttHt

ti

oo

oo

o

We get:

teJ

it io 2

Ignore non-resonant terms

4


0

0

0

0

0

ˆˆ

ˆˆ

ˆˆ

ˆˆˆˆ

ˆ

ˆˆˆ

ˆ

*

*

*

*

*

atat

eateat

tattat

tHi

tHi

atattHi

atattHi

tHi

e

e

e

eeee

ee

tet

toitoi

ooo

o

o


Cavity

The state of the radiation is then:

tio oteJ

it2

What is this state??


Coherent States of Radiation

Define a displacement operator:

aaeD ˆ*ˆ)(ˆ ie||

A coherent state is defined as:

0)(ˆ D

Recall the relation:

0ˆ,ˆ,ˆ0ˆ,ˆ,ˆprovidedˆˆ2

]ˆ,ˆ[ˆˆ

BABBAAeeee BA

BABA

aa

aa

eeeD

eeeD

ˆˆ*2||

ˆ*ˆ2||

2

2

)(ˆ

)(ˆ

Therefore:

5


Properties of Coherent States

)(ˆ)(ˆ

1)(ˆ)(ˆ

1

DD

DD

i) The displacement operator is unitary:

ii)

aDaD ˆ)(ˆ)(ˆ

1ˆ,ˆifˆˆ ˆˆ BAAeAe BB

Recall that:

aeae aaaa ˆˆ ˆ*ˆˆ*ˆ

Therefore:

00)(ˆ ˆ*ˆ aaeD

Similarly:

*ˆ)(ˆˆ)(ˆ aDaD

iii) Coherent states are properly normalized:

10|00)(ˆ)(ˆ0| DD


Properties of Coherent Statesiv) Coherent states are linear superposition of photon number states:

v) If a photon number measurement is performed on a coherent state, the probability of finding n photons in a coherent state is:

nn

n

a

n

na

a

aaD

n

n

n

nn

n

n

0

2

0

2

0

2

2

2

!2||

exp

0!

)ˆ(

!2||

exp

0!

)ˆ(2||

exp

0)ˆexp(2||

exp

0)ˆ*(exp)ˆ(exp2||

exp0)(ˆ

22

2

!2exp|

nnnP

n

)exp(

!2

2

n

nPoisson Statistics!!

6



vi) Coherent states are eigenstates of the destruction operator:

aProof:

aDaD ˆ)(ˆˆ)(ˆSince:

Therefore:

0)(ˆ

0ˆ)(ˆ0)(ˆˆ)(ˆ)(ˆˆ

D

aDDaDDa

vii) Mean photon number is:

*ˆ aSimilarly:

22ˆˆˆ aann

0ˆˆ * atatet

Recall the state generated by the antenna inside a cavity:

teJ

it io 2

Cavity



viii) Variance in the photon number is equal to the mean:

2

42

2

ˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆ,ˆˆˆˆˆ

nn

aaaaaa

aaaaaa

aaaaaaaaaan

nnnn ˆˆˆˆ 222

ix) Coherent states are not orthogonal:

?0)(ˆ)(ˆ0| DD

*ˆ*ˆ*ˆˆ22

ˆ*ˆ2ˆ*ˆ2ˆ*ˆˆ*ˆ

22

22

)(ˆ)(ˆ

eeeeee

eeeeeeeeDD

aaaa

aaaaaaaa

]ˆ,ˆ[ˆˆˆˆ BAABBA eeeee

7


Properties of Coherent Statesix) Coherent states are not orthogonal:

*22

*22

22

22

|

0)(ˆ)(ˆ0

e

eDD

000)(ˆ)(ˆ0 *ˆ*ˆ*ˆˆ22

22

eeeeeeDD aaaa

x) Coherent states form a complete set:

11

ir dd ir i



xi) Mean values of field operators are non-zero for coherent states:

*ˆ

ˆ

a

a

)(ˆˆ2

),(ˆ rUeaeatrA titi

o

oo

)(*2

),(ˆ rUeetrA titi

o

oo

Similarly:

)(*2

),(ˆ rUeeitrE titio oo

)(*2

1),(ˆ rUeetrH titi

oo

oo

8


Signal Quadratures

Recall that one can write a narrowband signal as:y()

tietxty )(Re)(

)()()( 21 txitxtx

x(t)

x2(t)

x1(t) x1

x2


Quadrature Operators and Quadrature Fluctuations

We define quadrature operators as:

titi eaeatrA ˆˆ21

),(

Note that:

titi etxetxty )()(21

)( *

21 ˆˆˆ xixa

2

ˆ,ˆ

2

ˆˆˆ

2

ˆˆˆ

ˆˆˆ

21

21

21

ixx

iaa

xaa

x

xixa

If follows that:

161

ˆˆ 22

21 xx

The commutators imply the uncertainty relation:

Hermitian

9


Mean Quadrature Values and Fluctuations for Coherent States

Consider: ie

We get:

1*

41

1*2*41

ˆ

1*41

1*2*41

ˆ

sinIm2

*ˆ

cosRe2

*ˆ

22222

22221

2

1

x

x

ix

x

41

ˆ

ˆˆˆ

21

21

21

21

x

xxx

The quadrature fluctuations are:

41

ˆ 22 x

Similarly: 161

ˆˆ 22

21 xx


Vacuum Quadrature Fluctuations

The vacuum state is also a coherent state with = 0

41

0ˆ0

41

0ˆ0

22

21

x

x

00ˆ0

00ˆ0

2

1

x

x

10


Generalized Quadratures

Recall that one can write a narrowband signal as:y()

tietxty )(Re)(

iii etxitxetxetxtx 22

2 )()()(

x(t) x+/ 2(t)

x(t)

x1

x2


Generalized Quadrature Operators

ieaea

x

eaeax

exixexexa

exixexexa

ii

ii

iii

iii

2

ˆˆˆ

2

ˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

2

22

2

22

2

2

ˆ,ˆ 2i

xx

161

ˆˆ 22

2 xx

We define generalized quadrature operators as:

It follows that:

11


Mean Quadrature Values and Fluctuations for Coherent States

Consider: ie

sinIm2

*ˆ

cosRe2

*ˆ

2 iee

x

eex

ii

iiWe get:

For the fluctuations we get:

41

ˆ

41

ˆ

2

2

x

x


Error Diagram for Coherent States

x1

A coherent statex2

1x

2x

41

ˆ

41

ˆ

22

2

x

x

41

ˆ 22 x

ie

41

ˆ 21 x

But fluctuations are the same in any direction:

2ˆ x

2ˆ x

A coherent state

Radius=1/2

Radius=1/2

a21 ˆˆˆ xixa

iexixa 2ˆˆˆ

1

1

ˆ cos

ˆ sin

x

x

12


Coherent States as Displaced Vacuum States

Consider the field operator:

)(ˆˆ2

))(ˆ

)(ˆ rUaarUq

rAo

0)(ˆ)(ˆ nrAnrA

)(*2

)(ˆ rUrAo

What if we consider the eigenstates of the operator:

qqqq ˆ

q

1

qqdq

1

ˆ ˆ2ˆ2

2ˆ

o

o

a aq

x


ntrAqqndqnrAn ),(ˆ)(ˆ


Consider the expectation value again:

)()(

)()()(

2

*

rUqqdq

qrUq

qdq

n

nn

Hermite-Gaussian functions

o

q

oeqq

2

1)(0 2

02

q0

Vacuum state

21 )(q

20 )(q

22 )(q

23 )(q

13


Coherent States as Displaced Vacuum StatesConsider the following state:

qqo0

)(' 00 oqqq

o

oqq

oeq

2)(2

01

)('

)()(')(ˆ 20

rUqrUqqdqrA o

The average of the field operator will be non-zero:

3

33

2

22

!3!21

q

q

q

q

qqe oo

oq

qoRecall that:

)(

)('

0

/241

0

2

qe

eq

qq

qq

o

o

o

o



21

4 2 /0 0| ' ( ) ( )

oo

o

q qq

qoq q e e q

We need a state such that:

)(0ˆ 0 qqi

pq

Remember that:

ipq ˆ,ˆ

0

0)()('|

ˆ

ˆ

00

piq

piq

qq

o

o

e

eqqeqq

So we can write:

But:

iaa

p o

2

ˆ

Therefore:

0ˆ0

00

*

222

De

ee

aa

aqaqi

aaiqo

2* oq

14



qqo0

Vacuum state

Coherent state

Displacement operator

20 )(q

20 )(' q

p

qx

o

o

ˆ2

1x

ˆ2

ˆ

2

1

x1

A coherent statex2

1x

2x

Vacuum state

412

cos2

12

1ˆ

21

1

2

x

x ex

412

sin2

12

2ˆ

22

2

2

x

x ex

Note that:

One can construct eigenstates of the quadrature operators:

412

sin

21

2

2ˆ

412

cos21

2

ˆ

22

2

2

2

2

x

x

x

x

ex

ex

D

2* oq


Coherent States: Time Dependence

0

0ˆ

0ˆ

0)(

)(ˆ)(ˆ

ˆˆˆ

ˆ

ˆˆ

tata

tHi

tHi

tHi

tHi

tHi

tHi

e

eeDe

De

e(tet

Suppose:

00ˆ0 ˆˆ aaeDt

At time t:

ti

ti

o

o

eata

eata

ˆ)(ˆ

ˆ)(ˆ

0)( ˆ)(ˆ)( atatet

ti

ti

o

o

et

et

)(tt

1ˆ ˆ ˆ2oH a a

15


Time Dependence of the Quadrature Operators

For real signals:

tietxty Re txitxtx 21

)(ˆ)(ˆ21

),(ˆ tatatrA

For field operator:

titititi oooo eetaeetatrA ˆˆ21

,

)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ)(ˆ

21

21

txitxeta

txitxetati

ti

o

o

So at time t:

ti

ti

o

o

eata

eata

ˆ)(ˆ

ˆ)(ˆ

This implies:

ietaeta

tx

etaetatx

titi

titi

oo

oo

2)(ˆ)(ˆ

)(ˆ

2)(ˆ)(ˆ

)(ˆ

2

1



ieetaeeta

tx

eetaeetatx

tiitii

tiitii

oo

oo

2)(ˆ)(ˆ

)(ˆ

2)(ˆ)(ˆ

)(ˆ

2

Similarly:

DO NOT USE:

Htxdt

txdi ˆ,ˆ

ˆ

t

Hit

Hi

exetx ˆˆ

ˆˆ

Averages (e.g. ) in the Schrodinger and Heisenberg Pictures:

0t

Compute t

0)(ˆ0 ttxt

Then use:

Compute tata ˆ,ˆ

Find )(ˆ tx

Then use: teeaeeat

tiitii oo

2

ˆˆ

ˆ ( )x tGiven

16



Example:

ttt 0

2

2)(ˆ)(ˆ

)(ˆ

ii

tiitii

ee

eetaeetatx

At later time, using Heisenberg picture:

2

2

ˆˆ

2

ˆˆ

ii

tiitii

tiitii

ee

teeaeea

t

teeaeea

t

oo

oo

At later time, now using Schrodinger picture:

ti oet

ti

ti

o

o

eata

eata

ˆ)(ˆ

ˆ)(ˆ


Quantum States Generated by Parametric Down Conversion

CavityHamiltonian Description:

3 ˆ ˆˆ ˆ ˆ , .o TH t a a d r P r t E r

)(ˆˆ2

)(ˆ

)(ˆˆ2

)(ˆ

rUaairE

rUaarA

o

o

oo

op 2

o

221 ,ˆ,ˆ,ˆ trErEtrErEtrP popo

o

ignore

tipp

tipp

po

pp

oo erUerUitrE

2**2 )()(2

),(

trErEtrP po ,ˆ2,ˆ 2

Only relevant term in the polarization will be:

ipp e

V

erU

rki

p

.

17


Cavity

op 2

o o


trErErErdaatH poo ,ˆˆ2ˆˆˆ 32

)()()(

222

232 rUrUrUrd p

po

p

o

oo

2222* ˆˆ*

2ˆˆˆ aeae

iaatH ti

pti

pooo

o

o

op 2

o

o

op 2

Down conversionUp conversion



Cavity

op 2

o o

Suppose: 00 t

Then:

0ˆ

0

022

2222

ˆ2

ˆ2

*

ˆˆ2

tS

e

et

at

at

aeaet toiitoii

What is this state?

2222 ˆˆ

2ˆˆˆ aeaeiaatH itiiti

ooo

tii ote 2

ip e

2222* ˆˆ*

2ˆˆˆ aeae

iaatH ti

pti

pooo

Answer: squeezed vacuum state

18


Squeezed States of Light

Squeezed States are obtained by first squeezing the vacuum state with the squeezing operator :

00)(ˆ22 )ˆ(

2ˆ

2

aa

eS

)(ˆ S

And then displacing the resulting state with the displacement operator:

0)(ˆ0)(ˆ)(ˆ, ˆ* SeSD aa

2ier

ie

0)(ˆ,0 S

The state obtained by just squeezing the vacuum is called the squeezed vacuum:


Properties of the Squeezing Operator

i) The squeezing operator is unitary:

ii) Action on the creation and destruction operators:

)(ˆ)(ˆ)(ˆ1)(ˆ)(ˆ 1 SSSSS

rearaSaS

rearaSaSi

i

sinhˆcoshˆ)(ˆˆ)(ˆ

sinhˆcoshˆ)(ˆˆ)(ˆ

2

2

2ier

The proof follows from using the identity and collecting terms:

BAABABABA ˆ,ˆ,ˆ!2

1ˆ,ˆˆ)ˆ(expˆ)ˆ(exp

10)(ˆ)(ˆ0,0,0 SS

19


Properties of the Squeezed States

x1

x2

Vacuum state

2ier

)(ˆ S

x1

x2

Squeezed vacuum state

x1

x2


x1

x2 Squeezedstate

1x

2x

)(ˆ D

ie

Action of the squeezing operator:

Action of the dispalcement operator:

ˆˆ, ( ) ( ) 0D S



i) Averages of creation of destruction operators:

,ˆ,

0)(ˆˆ)(ˆ0

0)(ˆ)(ˆˆ)(ˆ)(ˆ0,ˆ,

a

SaS

SDaDSa

0,0ˆ,0

00)(ˆˆ)(ˆ0,0ˆ,0

a

SaSa

For squeezed vacuum states:

ii) The photon number operator average is:

22

2

sinh

0ˆˆˆˆˆˆ0

0ˆˆˆˆ0

0ˆˆˆˆˆˆˆˆ0,ˆˆ,,ˆ,

r

SaSSaS

SaaS

SDaDDaDSaan

Lots photons of in the squeezed vacuum state (=0)

20



iii) Average of the quadrature operators:

2

,2

ˆˆ,,ˆ,

ii

ii

ee

eaeax

iee

xii

2,ˆ,

2

iv) Variances of the quadrature operators:

re

SaSSaS

SaaS

SaS

SDaDDaDS

SDaDSa

i sinhrcosh

0)(ˆˆ)(ˆ)(ˆˆ)(ˆ0

0)(ˆˆ2ˆ)(ˆ0

0)(ˆ)ˆ()(ˆ0

0)(ˆ)(ˆˆ)(ˆ)(ˆˆ)(ˆ)(ˆ0

0)(ˆ)(ˆˆ)(ˆ)(ˆ0,ˆ,

22

2

22

2

22

rea i sinhrcosh,)ˆ(, 222



iv) Variances of the quadrature operators (contd…):

41

sinh

41

cosh4

sinhcosh

4sinhcosh

,ˆˆˆˆˆ,41

,ˆ,

22

222

22

222

222 22

r

re

rre

erre

aaaaeaeax

ii

ii

ii

x1

x2 Squeezedstate

1x

2x

r

r

errx

errx

2222

22

41

sinhcosh41

,ˆ,

41

sinhcosh41

,ˆ,2

Choose :

21



iv) Variances of the quadrature operators (contd…):

x1

x2 Squeezedstate

1x

2x

161

,ˆ,,ˆ, 222

2222

iiii rexrerexre

Squeezed states are minimum uncertainty states

Squeezed states, compared to coherent states, have more fluctuations in one quadrature to the orthogonal quadrature – while satisfying the uncertainty relation


Two-Photon Coherent States

Squeezed states are:

0)(ˆ)(ˆ, DSP

0)(ˆ)(ˆ, SDS

What about the states:

First squeeze and then displace

First displace and then squeeze

These are called two-photon coherent states and are more common than squeezed states

We know that coherent states are eigenstates of the destruction operator:

Turns out that two-photon coherent states are also eigenstates of a certain “destruction” operator:

a

0)(ˆ)(ˆ0)(ˆ)(ˆˆ DSDSb

)(ˆˆ)(ˆˆ SaSb

Where:

22


Two-Photon Coherent States

0)(ˆ)(ˆ

)(ˆ

ˆ)(ˆ

0)(ˆˆ)(ˆ

)(ˆ)(ˆ)(ˆˆ)(ˆ0)(ˆ)(ˆˆ

DS

S

aS

DaS

oDSSaSDSb

rearab

SaSbi sinhˆcoshˆˆ

)(ˆˆ)(ˆˆ

2

rareab

SaSbi coshˆsinhˆˆ

)(ˆˆ)(ˆˆ

2

Define two new creation and destruction operators:

1

ˆˆ,ˆˆˆ,ˆ

SaSSaSbb

Find the action of the destruction operator on the two-photon squeezed state:

00)(ˆˆ Sb


Two-Photon Coherent States and Squeezed States

The squeezed vacuum state is the “ground” state of the operator :

00)(ˆˆ Sb

b

Suppose there is an eigenstate of the operator such that:b

b

00ˆ aCompare with:

aCompare with:

0)(ˆˆ*ˆ Se bb

It is easy to construct:

Compare with:

0ˆ*ˆ aae b

0)(ˆ)(ˆ0)(ˆ)(ˆˆ DSDSb

But we know:

Therefore:

0)(ˆ0)(ˆ ˆ*ˆˆ*ˆ aabb eSSe

23


Two-Photon Coherent States and Squeezed States

0)(ˆ0)(ˆ)(ˆ)(ˆ

0)(ˆ0)(ˆ

ˆ*ˆˆ*ˆ

ˆ*ˆˆ*ˆ

aabb

aabb

eSSeSS

eSSe

rareab

rearabi

i

coshˆsinhˆˆ

sinhˆcoshˆˆ

2

2

We had:

Recall that:

S

i

arrearer

bb

rer

Se

SeSSii

,sinhcosh

0)(ˆ

0)(ˆ)(ˆ)(ˆ

2*

ˆcoshsinhˆsinhcosh

ˆˆ

*22*

*

LHS:

RHS:

0)(ˆ, ˆ*ˆ aaP eS

s

ip rer ,sinhcosh, 2

This implies:


Generation of Two-Photon Coherent States

Cavity

op 2

o o

If: 00 t

Then:

0ˆ

0

022

2222

ˆ2

ˆ2

*

ˆˆ2

tS

e

et

at

at

aeaet toiitoii

Suppose: Dt ˆ0

Then:

P

at

at

aeaet

tt

tDtS

tDe

tDet

toiitoii

,

0ˆˆ

0ˆ

0ˆ

22

2222

ˆ2

ˆ2

*

ˆˆ2


Two-photon coherent state

24


Squeezed Vacuum States

0)(ˆ,0 S

Consider the squeezed vacuum state:

It can be expressed in terms of the photon number states as follows:

mm

mre

r mm

mim 2!2

!2tanh1

cosh

1,0

0

2

Only even photon number states are included in the summation


Time Dependence of Squeezed States

Suppose:

0ˆˆ,0 SDt

tt

tStD

eeSeeDe

SDe

e(tet

tHi

tHi

tHi

tHi

tHi

tHi

tHi

tHi

,

0ˆˆ

0ˆˆ

0ˆˆ

,0)(

ˆˆˆˆˆ

ˆ

ˆˆ

aaH o ˆˆ

Then:

ti

ti

o

o

et

et

2

25


Phase in Quantum Optics

t

The “phase of a photon” is an ill-defined concept

The “phase of electromagnetic field” makes more sense

• There is no such thing as “absolute” phase and absolute phase is not an observable

• There is no universally accepted quantum mechanical Hermitian operator for the phase of electromagnetic field

• Phase can only be measured in a relative way ………. but relative to what??

E



)()(ˆ)(ˆ2

),(ˆ rUtataitrE

Consider the field operator:

Suppose:

nt 0 00),(ˆ0 ttrEt

Obviously the field has no well-defined phase

Now consider:

10 1

2it n e n ˆ ( , ) sin ( )

2o

oE r t n t U r

Therefore, states for which the phase could be well defined must not be states of definite photon number

Question: Are the zeroes of the field in time well defined???

26



t

E

10 1

2it n e n ˆ ( , ) sin ( )

2o

oE r t n t U r

2ˆ ˆ ˆ ˆ( , ). ( , ) ( , ) . ( , ) 2 sin ( ). ( )2

oE r t E r t E r t E r t n t U r U r

t

Histogram obtained for E-field measurements at different times:



1

0

10

N in

nt e n

N

Now consider the following state:

1

1

2ˆ ( , ) sin ( )2

No

on

E r t n t U rN

Average field value:

Variance:

1 1

0 0

21

0

ˆ ˆ ˆ ˆ( , ). ( , ) ( , ) . ( , )

2 1 2cos 2 2 1

2( ). ( )

2 1 21 os 2 2

2

N N

n no

N

n

E r t E r t E r t E r t

n t n nN N

U r U r

n c tN

for large N values, field variance goes to zero (almost) when cos(2t – 2) is unity!

And this happens exactly at the zero crossings of the average field value

Maximum photon number superposition

27



t

N=3000

1

1

2ˆ ( , ) sin ( )2

No

on

E r t n t U rN

Average field value:

Variance:

1 1

0 0

21

0

ˆ ˆ ˆ ˆ( , ). ( , ) ( , ) . ( , )

2 1 2cos 2 2 1

2( ). ( )

2 1 21 os 2 2

2

N N

n no

N

n

E r t E r t E r t E r t

n t n nN N

U r U r

n c tN

Histogram obtained for E-field measurements at different times:



States for which the phase could be well defined must not be states of definite photon number

The above points to a phase-photon number uncertainty relation (assuming a Hermitian phase operator exists):

One would be tempted to write down a Fourier Transform relation:

Which in turn would imply (assuming completeness of phase states):

ˆˆ,n i

2

inen

2 2

0 0 2

inen d n d

0 0 2

in

n n

en n n

Problem is that the spectrum of photon number operator is limited to non-negative integers only!! So the Fourier transform relation does not follow from the commutator!

• Not eigenstates of the phase operator!• Not orthogonal! Form an overcomplete set• Cannot construct a unitary phase operator using these states

28


Phase in Quantum OpticsIf there were a Hermitian phase operator in quantum optics then one would expect a decomposition of the destruction operator of the form:

ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆ ˆ

i ia e n a n e

a a n

Then operator would have been unitary:îe

ˆ ˆ

ˆ ˆ ˆ ˆ

1

1

i i

i i i i

e e

e e e e

And the average phase, say , of any quantum state of the field could be obtained as follows:

î ie e

• No Hermitian phase operator of the form exists, and no unitary operator of the form exists in the full Hilbert space! • However, there are several approximate ways of handling phase in quantum optics!

îe

or perhaps as ˆ



Lets look at a coherent state:

0

2!

0)(ˆ

2

n

nn

neD

ie

)(sin2

2),(ˆ rUttrEo

x1

A coherent statex2

|| cos

|| sin

sin)(ˆ

cos)(ˆ

2

1

tx

tx

21

22

1ˆ ( )

41

ˆ ( )4

x t

x t

29



x1

An arbitrary state of radiation

x2

A cos

A sin

Consider an arbitrary quantum state:

oo iti eAeta )(ˆ

o

o

Atx

AAtx

sin)(ˆ

real is cos)(ˆ

2

1

The phase o is well defined as long as 1A

A

Then we can write:

txitxeta

txitxtxitxtxitxeta

ti

ti

o

o

21

212121

ˆˆˆ

ˆˆˆˆˆˆˆ

iititi etxietxetaeta oo2ˆˆˆˆ

Or

(not too helpful)

(not too helpful either)


Phase Fluctuation Operator in Quantum Optics

x1


x2

A cos

A sin

A

2

2

ˆ ˆ ˆ ˆ

ˆ ˆ

o o o oo o

oo o

i t i t i i

i

a t e a t e x t e i x t e

A x t i x t e

oo iti eAeta )(ˆ

One can define a Hermitian phase fluctuation operator as follows:

A

txt o

)(ˆ)(ˆ

2

Example: Consider a coherent state: ˆ( ) o oi t ia t e e

oie tn

txt

txt

o

o

ˆ41

4

1)(ˆ)(ˆ

0)(ˆ

)(ˆ

222

22

2

(helpful!)

x1

x2

A

2ox

ox

30


Photon Number Fluctuation Operator in Quantum Optics

ontatatn )(ˆˆ)(ˆ

Suppose:

ˆ( ) o oi t ia t e Ae 1A

txAA

txitxAtxitxAtata

tntntn

o

oooo

ˆ2

ˆˆˆˆˆˆ

ˆˆˆ

2

22

Then:

txntxAtnoo o ˆ2ˆ2ˆ

2ˆ Antn o

x1

x2

A


A cos

A sin

2ox

ox

2

2

ˆ ˆ ˆ ˆ

ˆ ˆ

o o o oo o

oo o

i t i t i i

i

a t e a t e x t e i x t e

A x t i x t e


Photon Number Fluctuation Operator in Quantum Optics

Example: Consider a coherent state: oiti eeta )(ˆ

tntxtn

txtn

o

o

ˆ)(ˆ4)(

0)(ˆ2)(

2222

2ˆ tn

txtno ˆ2ˆ

x1

x2

A


A cos

A sin

2ox

ox

31


Photon Number Fluctuation and Phase Fluctuation Operators

Suppose:

ˆ( ) o oi t ioa t e n e

2ˆ ˆ ˆ

ˆˆ

2

o oo o

o

i t io

io o

o

a t e n x t i x t e

n tn i n t e

n

1on

Photon Number Fluctuation Operator:

txntn

txnntntntn

o

o

o

oo

ˆ2ˆ

ˆ2ˆˆˆ

Phase Fluctuation Operator:

on

txt o

)(ˆ)(ˆ

2

x1

x2

A


A cos

A sin

2ox

ox


Photon Number and Phase Uncertainty Relation

itxtxttnoo

)(ˆ,)(ˆ2)(ˆ,ˆ 2

txntnoo ˆ2ˆ

on

txt o

)(ˆ)(ˆ

2

41

)(ˆ)(ˆ

)(ˆ),(ˆ

22

ttn

ittn

Therefore, the photon number and the phase (with respect to a reference) of a radiation field cannot be measured simultaneously with high accuracy

If a quantum state of radiation has a well defined value for the phase of the field then this quantum state cannot have a well defined number of photons and it must be a superposition of different photon number states.

On the other hand, if a quantum state has a well defined number of photons then it cannot have a well defined value for the phase of the field.

o

o

nt

ntn

41

4

1)(ˆ

)(ˆ

22

22

x1

x2

A


A cos

A sin

2ox

ox

32


Photon Number and Phase Squeezed States

x1

A squeezed state with increased phase fluctuations and squeezed photon number fluctuations

x2

|| cos

|| sin

x1

A squeezed state with increased photon number fluctuations and squeezed phase fluctuations

x2

|| cos

|| sin


Photon Number States: Another New Look

21

11 2x

nn exHx

0!

)ˆ(

0!

)ˆ(

111 n

axnxx

n

an

n

n

n

x1

x2

In the limit n → ∞ :

21

11 ~xn

n exx

-20 -10 0 10 20x

1

Maximum around: nx ~1

n=8

For large n:

33


Thermal Radiation: A Statistical Mixture

nnnPn

0

Cavity

We assume that in thermal equilibrium:

From statistical physics:

TK

n

B

o

enP

21

We must have:

10

nnP

TKTKn BoBo eenP 1

1

1ˆˆTraceˆ

0

TK

n BoenPnnn

The average number of photons in the mode is:

Bose-Einstein Distribution

Bose-Einstein factor

T

2ˆˆ o

o aaH


Thermal Radiation: A Statistical Mixture

Cavity n

TKTKn

n

n

neenP BoBo

ˆ1

ˆ

ˆ11

1

More general way of writing the thermal distribution:

The fluctuations in the photon number is:

nnnnn ˆ1ˆˆˆˆ 222 Larger variance compared to Poisson distribution

34


ece 407 –spring 2009 –farhan rana –cornell university1 ece 407 –spring 2009 –farhan rana...

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