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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.