quantupm optics

8/12/2019 quantupm optics

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3, Coherent and Squeezed States

1. Coherent states

2. Squeezed states

3. Field Correlation Functions

4. Hanbury Brown and Twiss experiment

5. Photon Antibunching6. Quantum Phenomena in Simple Nonlinear Optics

Ref:

Ch. 2, 4, 16in Quantum Optics,by M. Scully and M. Zubairy.

Ch. 3, 4in Mesoscopic Quantum Optics,by Y. Yamamoto and A. Imamoglu.

Ch. 6in The Quantum Theory of Light,by R. Loudon.

Ch. 5, 7in Introductory Quantum Optics,by C. Gerry and P. Knight.

Ch. 5, 8in Quantum Optics,by D. Wall and G. Milburn.

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Uncertainty relation

Non-commuting observable do not admit common eigenvectors.

Non-commuting observables can not have definite values simultaneously.

Simultaneous measurement of non-commuting observables to an arbitrary degreeof accuracy is thus incompatible.

variance: A2 =|(A A)2|=|A2| |A|2.

A2B2 14 [F2 + C2],

where

[A, B] =iC, and F = AB+ BA 2AB.

Take the operators A= q (position) and B = p(momentum) for a free particle,

[q, p] =i q2p2 2

4 .

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ifRe() = 0, A + i

B is a normal operator, which have orthonormal eigenstates.

the variances,

A2 = i

2

[F + iC], B2 = i

2

[F iC],

set =r + ii,

A2 = 1

2[i

F

+ r

C

], B2 =

1

||2A2, i

C

r

F

= 0.

if||= 1, thenA2 = B2, equal variance minimum uncertainty states.

if

|

|= 1along withi = 0, thenA

2 = B2 and

F

= 0, uncorrelatedequal

variance minimum uncertainty states.

ifr= 0, thenF= ir C, A2 =

||22r

C, B2 = 12r

C.If Cis a positive operator then the minimum uncertainty states exist only if r >0.

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Minimum Uncertainty State

(q q)|=i(p p)|

if we define = e2r , then

(er q+ ier p)|= (erq + ierp)|,

the minimum uncertainty state is defined as an eigenstateof a non-Hermitian

operatorer q+ ier pwith a c-number eigenvalueerq + ierp.

the variances of q and pare

q2= 2

e2r , p2= 2

e2r .

herer is referred as the squeezing parameter.

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Phase diagram for EM waves

Electromagnetic waves can be represented by

E(t) =E0[ X1sin(t) X2cos(t)]

where

X1 = amplitude quadrature

X2 = phase quadrature

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Quadrature operators

the electric and magnetic fields become,

Ex(z, t) =Xj

(j

0V)1/2[aje

ijt + ajeijt] sin(kjz),

=Xj c

j [a1jcos j t + a2jsin jt]uj(r),

note thataand a are not hermitian operators, but(a) = a.

a1 = 12 (a + a)and a2 = 12i (a a)are two Hermitian (quadrature) operators.the commutation relation foraanda is[a, a] = 1,

the commutation relation foraanda is[a1, a2] = i2 ,

anda21a22 116 .

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Minimum Uncertainty State

(a1 a1)|=i(a2 a2)|if we define = e2r , then(er a1+ ier a2)|= (era1 + iera2)|,

the minimum uncertainty state is defined as an eigenstateof a non-Hermitian

operatorera1+ iera2 with a c-number eigenvalueer

a1

+ ier

a2

.

the variances ofa1 anda2 are

a21= 1

4e2r, a22=

1

4e2r .

herer is referred as the squeezing parameter.

whenr = 0, the two quadrature amplitudes have identical variances,

a21=a22= 14 ,

in this case, the non-Hermitian operator,er a1+ ier a2 = a1+ ia2 = a, and this

minimum uncertainty state is termed a coherent stateof the electromagnetic field, aneigenstate of the annihilation operator,a|=|.

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Coherent States

in this case, the non-Hermitian operator,e

r

a1+ ier

a2 = a1+ ia2 = a, and thisminimum uncertainty state is termed a coherent stateof the electromagnetic field, an

eigenstate of the annihilation operator,

a

|

=

|

.

expand the coherent states in the basis of number states,

|

= X

n |n

n

|

= X

n |n

0

|

an

n! |

= X

n

n

n! 0

|

|n

,

imposing the normalization condition, |= 1, we obtain,

1 =|=Xn

Xm

m|n ()mnm!

n!

=e||2 |0||2,

we have

|= e 12 ||2 Xn=0

nn!

|n,

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Poisson distribution

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Photon number statistics

For photons are independent of each other, the probability of occurrence ofn

photons, or photoelectrons in a time intervalT is random. DivideT intoN

intervals, the probability to find one photon per interval is,p = n/N,

the probability to find no photon per interval is,1

p,

the probability to findn photons per interval is,

P(n) = N!

n!(N

n)!

pn(1 p)Nn,

which is a binomial distribution.

whenN ,P(n) =

nnexp(n)n!

,

this is the Poisson distributionand the characteristics of coherent light.

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Real life Poisson distribution

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Displacement operator

the coherent state is the displaced form of the harmonic oscillator ground state,

|= D()|0=eaa|0,

where D()is the displacement operator, which is physically realized by a classical

oscillating current,

the displacement operator D()is a unitary operator, i.e.

D() = D(

) = [D()]1,

D()acts as a displacement operator upon the amplitudesaanda, i.e.

D1()aD() = a + ,

D1()aD() = a+ ,

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Radiation from a classical current

the Hamiltonian (p

A) that describes the interaction between the field and the

current is given by

V=

Z J(r, t) A(r, t)d3r,

whereJ(r, t)is the classical current and A(r, t)is quantized vector potential,

A(r, t) =iXk

1

kEkake

ikt+ikr +H.c.,

the interaction picture Schrdinger equation obeys,

d

dt|(t)= i

V|(t),

the solution is |(t)= Qkexp[ka kak]|0k, wherek =

1k

EkRt0

dtR

drJ(r, t)eitikr ,

this state of radiation field is called a coherent state,

|=(a a)|0.IPT5340 Fall 06 . 16/


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Properties of Coherent States

therefore, any coherent state can be expanded using other coherent state,

|= 1

Z d2||= 1

Z d2e

1

2||2 |,

this means that a coherent state forms an overcompleteset,

the simultaneous measurement ofa1 anda2, represented by the projection

operator|

|, is not an exact measurement but instead an approximate

measurement with a finite measurement error.

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q-representation of the coherent state

coherent state is defined as the eigenstate of the annihilation operator,

a|= |,

wherea= 12

(q+ ip),

theq-representation of the coherent state is,

(q +

q )q|= 2q|,

with the solution,

q|= (

)1/4exp[ 2

(q q)2 + i p

q+ i],

where is an arbitrary real phase,

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Phase diagram for coherent states

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Generation of Coherent States

In classical mechanics we can excite a SHO into motion by, e.g. stretching the

spring to a new equilibrium position,

H = p2

2m+

1

2kx2 eE0x,

= p22m

+12

k(x eE0k

)2 12

( eE0k

)2,

upon turning off the dc field, i.e. E0 = 0, we will have a coherent state

|

which

oscillates without changing its shape,

applying the dc field to the SHO is mathematically equivalent to applying the

displacement operator to the state|0.

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Attenuation of Coherent States

Glauber showed that a classical oscillating current in free space produces a

multimode coherent state of light.

The quantum noise of a laser operating at far above threshold is close to that of a

coherent state.

A coherent state does not change its quantum noise properties if it is attenuated,

a beam splitter with inputs combined by a coherent state and a vacuum state|0,

HI = (ab + ab), interaction Hamiltonian

where is a coupling constant between two modes,

the output state is, with=

T and=

1 T ,

|out= U|a|0b =|a|b, with U=exp[i(ab + ab)t],

The reservoirs consisting of ground state harmonic oscillators inject the vacuum

fluctuation and partially replace the original quantum noise of the coherent state.

Since the vacuum state is also a coherent state, the overall noise is unchanged.

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C h t d S d St t


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Coherent and Squeezed States

Uncertainty Principle: X1X2 1.

1. Coherent states: X1 = X2 = 1,

2. Amplitude squeezed states: X1


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Squeezed States and SHO

Suppose we again apply a dc field to SHO but with a wallwhich limits the SHO to a

finite region,

in such a case, it would be expected that the wave packet would be deformed or

squeezedwhen it is pushed against the barrier.

Similarly the quadratic displacement potential would be expected to produce a

squeezed wave packet,

H= p2

2m

+1

2

kx2

eE0(ax

bx2),

where theax term will displace the oscillator and thebx2 is added in order to give

us a barrier,

H= p2

2m

+1

2

(k+ 2ebE0)x2

eaE0x,

We again have a displaced ground state, but with the larger effective spring

constantk =k+ 2ebE0.

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Squeezed Operator


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Squeezed Operator

To generate squeezed state, we need quadratic terms inx, i.e. terms of the form

(a + a)2,

for the degenerate parametric process, i.e. two-photon, its Hamiltonian is

H=i

(ga2

ga2

),

whereg is a coupling constant.

the state of the field generated by this Hamiltonian is

|(t)= exp[(ga2 ga2)t]|0,

define the unitary squeeze operator

S() =exp[1

2a2 1

2a2]

where =rexp(i)is an arbitrary complex number.

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Properties of Squeezed Operator


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Properties of Squeezed Operator

define the unitary squeeze operator

S() =exp[1

2a2 1

2a2]


squeeze operator is unitary, S() = S1() = S(),and the unitarytransformation of the squeeze operator,

S()aS() = a cosh r

aei sinh r,

S()aS() = acosh r aei sinh r,

with the formulaeABeA = B+ [A, B] + 12!

[A, [A, B]], . . .

A squeezed coherent state|, is obtained by first acting with the displacementoperator D()on the vacuum followed by the squeezed operator S(), i.e.

|, = S()D()|0,

with =||exp(i).

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ifRe() = 0, A + iB is a normal operator, which have orthonormal eigenstates.

the variances,

A2 = i2

[F + iC], B2 = i2

[F iC],

set =r + ii,

A2 = 1

2

[i

F

+ r

C

], B2 =

1

||2

A2, i

C

r

F

= 0.

if||= 1, thenA2 = B2, equal variance minimum uncertainty states.

if

|

|= 1along withi = 0, thenA

2 = B2 and

F

= 0, uncorrelatedequal

variance minimum uncertainty states.

ifr= 0, thenF= ir C, A2 =

||22r

C, B2 = 12r

C.If Cis a positive operator then the minimum uncertainty states exist only if r >0.

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Squeezed State


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Squeezed State

define the squeezed state as

|s= S()|,

where the unitary squeeze operator

S() =exp[1

2a2 1

2a2]


squeeze operator is unitary, S() = S1() = S(),and the unitarytransformation of the squeeze operator,

S()aS() = a cosh r

aei sinh r,

S()aS() = acosh r aei sinh r,

for| is the vacuum state|0, the|sstate is the squeezed vacuum,

|= S()|0,

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Squeezed Vacuum State


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Squeezed Vacuum State

for

|

is the vacuum state

|0

, the

|s

state is the squeezed vacuum,

|= S()|0,

the variances for squeezed vacuum are

a21 = 1

4[cosh2 r+ sinh2 r 2 sinh r cosh r cos ],

a22 = 1

4[cosh2 r+ sinh2 r+ 2 sinh r cosh r cos ],

for = 0, we have

a21 = 1

4

e2r, and a22 = 1

4

e+2r,

and squeezing exists in the a1 quadrature.

for =, the squeezing will appear in the a2 quadrature.

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Quadrature Operators


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Quadrature Operators

define a rotated complex amplitude at an angle/2

Y1+ iY2 = (a1+ ia2)ei/2 = aei/2,

where

0@ Y1

Y2

1A =

0@ cos /2 sin /2

sin /2 cos /2

1A0@ a1

a2

1A

then S()(Y1+ iY2)S() = Y1er + iY2er,

the quadrature variance

Y2

1 =

1

4e2r , Y2

2 =

1

4e+2r, and Y

1Y

2=

1

4,

in the complex amplitude plane the coherent state error circle is squeezed into an

error ellipseof the same area,

the degree of squeezing is determined byr =||which is called the squeezedparameter.

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Vacuum, Coherent, and Squeezed states


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Vacuum, Coherent, and Squeezed states

vacuum coherent squeezed-vacuum

amp-squeezed phase-squeezed quad-squeezed

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Squeezed Coherent State


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Squeezed Coherent State

A squeezed coherent state

|,

is obtained by first acting with the displacement

operator D()on the vacuum followed by the squeezed operator S(), i.e.

|, = D()S()|0,

where S() = exp[12 a

2

12 a

2

],

for = 0, we obtain just a coherent state.

the expectation values,

, |a|, =, a2= 2 ei sinh r cosh r, and aa=||2 + sinh2 r,

with helps of D()aD() = a + and D()aD() = a+ ,

forr0we have coherent state, and 0 we have squeezed vacuum.furthermore

, |Y1+ iY2|, =ei/2, Y21= 1

4e2r, and Y22=

1

4e+2r ,

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Squeezed State


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Squee ed State

from the vacuum statea

|0

= 0, we have

S()aS()S()|0= 0, or S()aS()|= 0,

since S()aS() = a cosh r+ aei sinh r

a + a, we have,

(a + a)|= 0,

the squeezed vacuum state is an eigenstate of the operator a + a with

eigenvalue zero.

similarly,

D()S()aS()D()D()|= 0,

with the relation D()aD() = a

, we have

(a + a)|, = ( cosh r+ sinh r)|, |, ,

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Squeezed State and Minimum Uncertainty State


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q y

write the eigenvalue problem for the squeezed state

(a + a)|, = ( cosh r+ sinh r)|, |, ,

in terms of in terms ofa= (Y1+ iY2)ei/2 we have

(Y1+ ie2rY2)|, =1|, ,

where

1 =er

ei/2

=Y1 + i

Y2e

2r

,

in terms ofa1 and a2 we have

(a1+ ia2)

|,

=2

|,

,

where

= +

, and 2 =

+ ,

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Squeezed State in the basis of Number states


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q

consider squeezed vacuum state first,

|=Xn=0

Cn|n,

with the operator of(a + a)|= 0, we have

Cn+1 =

( n

n + 1)1/2Cn1,

only the even photon states have the solutions,

C2m = (1)m(ei tanh r)m[ (2m 1)!!(2m)!!

]1/2C0,

whereC0 can be determined from the normalization, i.e. C0 =

cosh r,

the squeezed vacuum state is

|= 1cosh r

Xm=0

(1)mp(2m)!2mm!

eim tanhm r|2m,

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Squeezed State in the basis of Number states


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q

the squeezed vacuum state is

|= 1cosh r

Xm=0

(1)mp

(2m)!

2mm! eim tanhm r|2m,

the probability of detecting2mphotons in the field is

P2m =|2m||2 = (2m)!22m(m!)2

tanh2m r

cosh r,

for detecting2m + 1 statesP2m+1 = 0,

the photon probability distribution for a squeezed vacuum state is oscillatory,

vanishing for all odd photon numbers,

the shape of the squeezed vacuum state resembles that of thermal radiation.

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Number distribution of the Squeezed State


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n

Pn

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

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Number distribution of the Squeezed Coherent State


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For a squeezed coherent state,

Pn =|n|, |2 =( 12

tanh r)n

n! cosh r exp[||2 1

2(2ei+2ei) tanh r]H2n((e

i sinh(2r))1/

30 40 50 60 70 80

0.02

0.04

0.06

0.08

0.1

Ref:

Ch. 5, 7in Introductory Quantum Optics,by C. Gerry and P. Knight.

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Number distribution of the Squeezed Coherent State


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A squeezed coherent state

|,

is obtained by first acting with the displacement

operator D()on the vacuum followed by the squeezed operator S(), i.e.

|, = D()S()|0,

the expectation values,

aa=||2 + sinh2 r,

50 100 150 200

0.02

0.04

0.06

0.08

50 100 150 200

0.002

0.004

0.006

0.008

0.01

0.012

||2 = 50, = 0, r = 0.5 ||2 = 50, = 0, r = 4.0

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Generations of Squeezed States


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Generation of quadrature squeezed light are based on some sort of parametric

processutilizing various types of nonlinear optical devices.

for degenerate parametric down-conversion, the nonlinear medium is pumped by a

field of frequencyp and that field are converted into pairs of identical photons, of

frequency =p/2each,

H= aa + pbb + i(2)(a2b a2b),

whereb is the pump mode anda is the signal mode.

assume that the field is in a coherent state|eiptand approximate theoperators bandb by classical amplitudeeipt andeipt, respectively,

we have the interaction Hamiltonian for degenerate parametric down-conversion,

HI =i(a2 a2),

where =(2).

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Generation and Detection ofSqueezed Vacuum


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1. Balanced Sagnac Loop (to cancel the mean field),

2. Homodyne Detection.

M. Rosenbluh and R. M. Shelby,Phys. Rev. Lett. 66, 153(1991).

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Homodyne detection


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the detectors measure the intensitiesIc =

cc

andId =

dd

, and the

difference in these intensities is,

Ic Id =ncd=c c dd=iab ab,

assuming theb mode to be in the coherent state|eit, where=||ei , wehave

ncd=||{aeitei + aeitei},

where =+ /2,

assume thata mode light is also of frequency (in practice both thea andb

modes derive from the same laser), i.e. a= a0eit, we have

ncd

= 2

|

|X()

,

where X() = 12

(a0ei + a0ei)is the field quadrature operator at the angle ,

by changing the phase of the local oscillator, we can measure an arbitrary

quadrature of the signal field.

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Squeezed States in Quantum Optics


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Generation of squeezed states:

nonlinear optics: (2) or(3) processes,

cavity-QED,

photon-atom interaction,

photonic crystals,

semiconductor, photon-electron/exciton/polariton interaction,

Applications of squeezed states:

Gravitational Waves Detection,

Quantum Non-Demolition Measurement (QND),

Super-Resolved Images (Quantum Images),

Generation of EPR Pairs,

Quantum Informatio Processing, teleportation, cryptography, computing,

IPT5340 Fall 06 . 50/

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