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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
What Quantum States are Generated by Oscillating Currents?
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
What Quantum States are Generated by Oscillating Currents?
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
What Quantum States are Generated by Oscillating Currents?
Cavity
The state of the radiation is then:
tio oteJ
it2
What is this state??
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of Coherent States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of Coherent States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of Coherent States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Signal Quadratures
Recall that one can write a narrowband signal as:y()
tietxty )(Re)(
)()()( 21 txitxtx
x(t)
x2(t)
x1(t) x1
x2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
ntrAqqndqnrAn ),(ˆ)(ˆ
Coherent States as Displaced Vacuum States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
o
o
o
o
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Coherent States as Displaced Vacuum States
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
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Coherent States as Displaced Vacuum States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Time Dependence of the Quadrature Operators
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Time Dependence of the Quadrature Operators
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Cavity
op 2
o o
Quantum States Generated by Parametric Down Conversion
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Quantum States Generated by Parametric Down 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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of the Squeezed States
x1
x2
Vacuum state
2ier
)(ˆ S
x1
x2
Squeezed vacuum state
x1
x2
Squeezed vacuum state
x1
x2 Squeezedstate
1x
2x
)(ˆ D
ie
Action of the squeezing operator:
Action of the dispalcement operator:
ˆˆ, ( ) ( ) 0D S
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of the Squeezed States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of the Squeezed States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of the Squeezed States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Properties of the Squeezed States
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
Squeezed vacuum state
Two-photon coherent state
24
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
)()(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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
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:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
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:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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:ˆie
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:
ˆi 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!
ˆie
or perhaps as ˆ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase in Quantum Optics
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)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Phase Fluctuation Operator in Quantum Optics
x1
An arbitrary state of radiation
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
An arbitrary state of radiation
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
An arbitrary state of radiation
A cos
A sin
2ox
ox
31
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
An arbitrary state of radiation
A cos
A sin
2ox
ox
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
An arbitrary state of radiation
A cos
A sin
2ox
ox
32
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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 University