Quantum and Classical Coincidence Imaging and

InterferenceShiyao Zhu and Yangjian Cai

Physics DepartmentHong Kong Baptist University

2. Ghost Imaging and Interference with Laser beam --- Coherent Gaussian Beam

(Classically Correlated Light)

1. Ghost Imaging and Interference with Entangled Photon Pairs

3. Ghost Image with Blackbody Radiation

4.Comparison and Conclusion

1. Ghost Imaging and Interference with entangled photon pairs

Ghost Imaging and Interference

with entangled photon pairs

Y. H. Shih ， Phys. Rev. Lett. 74, 3600 (1995)

1

22

1

22 cossinzz

dx

zz

axcRc

1

22

1

22 cossinzz

dx

zz

axcRc

For a thin planar crystal under quasi-monochromatic

conditions, the biphoton state is written as:

The positive-frequency portions of the signal and idler electric field are expressed as:

0ˆˆ0| ''' xaxaxxxEdxdx isp

dxxaxxhxE ss ,ˆ111

dxxaxxhxE ii ,ˆ222

hs(i) are the response functions of the signal and idler

paths B.E.A.Saleh, Physics Review A,043816, 2000Y. H. Shih ， Phys. Rev. Lett. 74, 3600 (1995)

The fourth-order correlation function

dxxxhxxhxExx isp ,,, 2121

2

21

2

1122

11222211

1122221121)2(

,

)(ˆ)(ˆ0

)(ˆ)(ˆ00)(ˆ)(ˆ

)(ˆ)(ˆ)(ˆ)(ˆ),(

xx

xExE

xExExExE

xExExExExxG

are the response functions of the signal and idler (the first and second) paths.

)(ish

Y. H. Shih et al ， Phys. Rev. A 52, R3429 (1995)

fSS

111

21

z

fSS

111

21

What is the Ghost Image?

1. The object in one path, while the image in another path in coincident counting.

2. The position and magnification of the image determined by both paths.

2. Ghost Imaging and Interference with Classically

correlated light(Coherent Gaussian Beam)

Laser beam

Phys. Rev. Lett. 89, 113601 (2002)

Ghost Imaging and Interference with classically correlated light

Boyd et al. PRL 92, 033601 (2004) No image equation

Phys. Rev. Lett. 92, 033601 (2004).

Did not give the fringe equation

The angle between two peaks:

mradm

nm

d 4.1

450

633

Experimental observation of ghost imaging with thermal light

A.Valencia, G. Scarcelli, M D’Angelo and Y. Shih,

quant-ph/0408001 v1 30 Jul 2004

The Double Slits

where N is the number of distinct features in the object plane. In experiment, detector one is point-like detector

and is fixed at x1=0, the visibility for a single slit is about 26%, for a double slit is about 12%

Assume 21 tt

• D. Magatti, F. Ferri, A. Gatti, M. Bache, E. Brambilla, and L.A. Lugiato, quant-ph/0408021, 3 Aug 2004

'2

111

FFF

2212

111

Fqdp

1212 , dxxxGxG

Detector 1 is a bucket detector,

Analytical Derivation for the Ghost Imaging and Interference with Coherent Gaussian Beams

Ghost image with coherent Gaussian beam

In the input plane

sinsinexp

sinexp 0

2

0

0 zxikw

zxxE

,,,

)()(

,

2

22222

2

11111

21

2211

122121)2(

dxuxhxEdxuxhxE

uIuI

uEuEuEuE

uEuEuEuEuuG

fzzzz11111

21 21 zz 21 aa

2

0

01

2

1

22

0

01

1

121

121)2(

2)2(

sin

2exp

sin

exp1

),(,

w

za

u

dvw

za

v

vHa

duuuGuG

)sin()sin( 02

22

10 za

uazH

Point by point projection

fzzz 221 mmw 01.00

A point-to-point image. Is it a real Ghost Imaging?

0

0

),(2

12

)2(

02

)2(

duGuG av

fzzzz11111

21

fzzz

mmExp

2

05.0:

21

0

Small waist width: Small spots on the object and on the detector 2.

For large waist width, the spots can become small through focusing.

fzzzz11111

21

Different amplification factor

1/ 1 ZZ

10/ 1 ZZ

50/ 1 ZZ

Ghost interference with coherent Gaussian beam

2

121

3

21

1

120

2

20

01

11

120

2

2

220

2

22

120

0

2

220

20

220

23132

)2(

22exp

2

14

sinsin2

exp

2

1

2

14

sinsin2

exp

2

1

sin4exp

1,

dvvz

ikv

z

ik

z

ik

w

ikw

zv

z

ik

vH

z

ik

w

b

ika

w

ub

ikik

w

z

b

ika

ww

z

bzzuG

at 0 1 u

.sinsin

cossin

sin1

, 22212

32

)2(

fu

hhc

fzuG

For we have 0w

mmwfz 2, 02

For all angles we have

0

0

),(2

12

)2(

02

)2(

duGuG av

This gives the interference fringes.

fz 2

The angle between two peaks:

mradm

nm

d 4.1

450

633

fzz

111

2

2

121

3

21

1

120

2

20

01

11

120

2

0

01

2

20

220

23122

)2(

22exp

21

4

sinsin2

exp

21

sin

2expsin2

exp1

,

dvvz

ikv

z

ik

zik

w

ikw

zv

zik

vH

zik

w

w

zau

w

z

azzuG

For small 0w

By refocusing

Clear fringes can be obtained

fz 2

z2 varies from satisfy to not satisfy image condition

Comparison

• Similarity:

Ghost image and interference can be

generated with entangled photon pairs and

coherent Gaussian beams both with high

visibility and good quality.

• Difference:

ImageThe Function of Lens

Quantum case: ImagingCoherent case: Focusing (both paths)

InterferenceQuantum case: No Lens needed.

Classical Case: Lens needs for focusing.

(a)Equation for image formation:

Coherent Gaussian beams: basically determined by

path one where the object is. First order-correlation

Entangled photon pairs: both paths. Second order

(b) Correlation:

Coherent Gaussian beams: Probability correlation,

(intensity-intensity (particle-particle) correlation).

Entangled photon pairs: Probability amplitude

correlation (electric field amplitude correlation).

• In above, we see that classical coherent field could not form the real ghost image.

• Can classical light produce a real ghost image?

• YES, Blackbody Radiation

3. Blackbody Radiation

• Blackbody Radiation is also a Classical source.

Can it produce a real ghost image?

• YES.

The Ghost image and Interference with Blackbody Radiation

2

221

2

22

21

2*

1 2

)(exp

4exp)(

gIg

xxxxIxExE

.),( )()(

,)()()()(

,

2

21

2

2*

1)2(

2

2*

121

122121)2(

uuuEuEG

uEuEuIuI

uEuEuEuEuuG

Ghost Interference with blackbody radiation

Ghost Imaging and Interference

with entangled photon pairs

Y. H. Shih ， Phys. Rev. Lett. 74, 3600 (1995)

1

22

1

22 cossinzz

dx

zz

axcRc

1

22

1

22 cossinzz

dx

zz

axcRc

I mm2Sufficient Large

flzl

111

211

mmfmmzl 10,2011

Ghost Image with Blackbody Radiation

, 111

211 flzl

2

1212

2

21

1 ),( auH

azuu

Ghost Image with Blackbody Radiation

Y. H. Shih et al ， Phys. Rev. A 52, R3429 (1995)

fSS

111

21

z

,exp1/exp22

1 3122

02

*1 kdi

k

kk

Tkh

hEE ji

ijB

ji rrkrr

For sufficient high temperature T in one-dimension

212

22

21

0

2

221

2

22

21

2*

1

4exp

2

)(exp

4exp)(

xxxx

I

xxxxIxExE

I

gIg

g High temperature, small

,),(.),( )()(

,,,,

,

2

21

2

2121

43214321232222141111

122121)2(

uuconstuuuIuI

dxdxdxdxxExExExEuxhuxhuxhuxh

uEuEuEuEuuG

Fourth-order correlation function

,22

and 2222

,22

,0

2222 and

2222,1

)(ad

vad

vadad

v

adv

adadv

ad

vH

,),(,2

2121)2( uuuuG

The image at different positions

mmI 2

Image for different surface sizes

mmg 00001.0 mmg 0005.0 mmg 003.0 (a)

(b)

(c) with mmI 5

.lity visibi,quality , ture tempera, g

Quality and visibility of image

(a) (b) with mmI 10,1mmI mmg 001.0

.lity visibi,quality , I

1

22

1

22 cossinzz

dx

zz

axcRc

1

22

1

22 cossinzz

dx

zz

axcRc

Difference

Interference

entangled photon

blackbody radiation

Image

flzl

111

211

entangled photon

flzl

111

211

blackbody radiation

Visibility Entangled photon pairs (high) Blackbody radiation (low)

Comparison between entangled photon pairs and blackbody

Wave function of entangled photon pares 

2121 0,0, xExExx

Fourth order correlation

2

2121)2( ,, xxxxG

Blackbody radiation

2*

121, xExExx

2

212121)2( ),( )()( , xxxIxIxxG

Fourth order correlation

Conclusion• Ghost image and interference can be generated with

blackbody radiation.

• Difference between Ghost image with entangled photons

and blackbody radiation:

(a) Equation for imaging: -z1 replaced with z1,

(b) Low visibility for blackbody radiation.

• The quality and visibility of the image with blackbody

radiation is determined by the surface size and

temperature (correlation length) of the blackbody.