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Transcript
Quantum Imaging
Yanhua Shih, Morton H. Rubin, and Fow-Sen Choa
University of Maryland Baltimore County
Baltimore, MD 21250
Quantum Imaging - UMBC
Objective
• Study the physics of multi-photon imaging for
entangled state, coherent state and chaotic
thermal state: distinguish their quantum and
classical nature, in particular, the necessary
and/or unnecessary role of quantum entanglement in quantum imaging and lithography.
• “Magic” mirror for “ghost” imaging.
• Muti-photon sources and measurement devices.
Approach
• Using entangled two-photon and three-photon
states created via optical nonlinear interaction in
spontaneous and stimulated modes for multiphoton spatial correlation study and imaging;
• Using chaotic pseudothermal source, coherent
source for two-photon spatial correlation study
and ghost imaging;
• Using photon counting and classical currentcurrent correlation circuit to explore the nature of
two-photon correlation.
Experimentally observed triphoton wavepacket.
Accomplishments
* The physics of quantum lithography: “Is entanglement dispensable in quantum lithography?”
(submitted for publication).
* The physics of two-photon imaging of chaotic
thermal light: PRL, 96, 063602 (2006).
* The source: Generation of true triphoton state:
immediately applicable for three-photon imaging
and lithography (submitted for publication).
* The detector: Fabrication of 2-D APD arrays
with improved photon counting characteristics.
Quantum Imaging - UMBC
Part I
The physics of quantum imaging and lithography
- Is entanglement dispensable in quantum lithography?
Classical Lithography
Optical lithography is a printing method in which light is used to
etch a substrate (a reduced-size image of complicated patterns is
reproduced onto a microchip).
Geometric Optics: POINT-to-POINT
relationship between the OBJECT and
the IMAGE planes.
t(
E(i,zi ,ti ) 
 d d g(,;   ,z  z ) e
i (t i to )
i
o
i
o
I( i ) 
E o (,)
 d t( )  (   /m)
o
o
o
2
i
If light always follows the law of geometrical optics, the image plane
and the object plane would have a “point-to-point” relationship which
means an unlimited ability of making demagnified image.
Classical Lithography
Optical lithography is a printing method in which light is used to
etch a substrate (a reduced-size image of complicated patterns is
reproduced onto a microchip).
DIFFRACTION: POINT-to-SPOT
relationship between the OBJECT
and the IMAGE planes
t(
E(i,zi ,ti ) 
i (t i to )
d

d

g(

,

;



,z

z
)
e
E o (,)

i
o i
o
I( i ) 
 do t(o )
2J1 (R /so 2 /  o  i /m )
R /so 2 /  o  i /m
2
Unfortunately, light is wave. The finite size of the spot, defined by the
“point-spread function”, determines the spatial resolution of the
imaging setup and
 limits the ability to produce demagnified images !!
Classical Rayleigh Limit
t
1/so+1/si = 1/f
m =si / so
R: lens radius
| ∫obj dρo t(ρo) somb (R/so 2/ |ρo + ρi/m|) |2 Coherent
I(ρi) =
∫obj dρo |t(ρo)|2 |somb (R/so 2/ |ρo + ρi/m|)|2 Incoherent
somb(x) = 2J1(x) / x  “point-spread function” (resolution)
The resolution is constrained by the Rayleigh diffraction limit: /2 !!!
Quantum Lithography: beyond classical limit
1/so+1/si = 1/f
m =si / so
R: lens radius
()
()
()
()
2
ˆ
ˆ
ˆ
ˆ
G ( 1,t1; 2 ,t 2 )   E1 E 2 E 2 E1   (1,t1; 2,t 2 )
(2)
somb( /2)
R 2
G ( 1,2 )   do t (o ) somb(
o  i /m )
so  /2
(2)
2
2
The resolution is improved by a factor of 
2 (as if one used a classical
source with wavelength λ/2) !!
What is so special about
entangled two-photon state?
Can quantum mechanical physical reality be
considered complete?
Einstein, Poldosky, Rosen, Phys. Rev. 47, 777 (1935).
(1) Proposed the entangled two-particle state according to
the principle of quantum superposition:
x1, x 2  

 p1, p2  
dp  p x 2  u p x1   (x1  x 2  x 0 )

dx  x x 2 v x x1    ( p1  p2 )
(2) Pointed out a surprising phenomenon: the momentum
(position) for neither subsystem is known; however, if one

particle is measured to have a certain momentum (position),
the momentum (position) of its “twin” is known with
certainty, despite the distance between them!
What is so special about entangled two-photon states?
   ( s   i   p )  (k s  k i  k p ) aˆ s aˆ i 0
s,i
EPR correlation:
(s  i )  (s  i )
()
()
()
()
2
ˆ
ˆ
ˆ
ˆ
G ( 1,t1; 2 ,t 2 )   E1 E 2 E 2 E1   (1,t1; 2 ,t 2 )
(2)

(s, i )  (s  i )
On the output
plane of the
source
In EPR’s language, the signal and the idler may come out
from any point of the object plane; however, if the signal
(idler)
is found in a certain position, the idler (signal) must
be found in the same position, with 100% certainty.
What is so special about entangled two-photon states?
(s  i )
ks
ki
(s  i )
somb( /2)
The entangled photon pair comes out from a point of the object
plane, undergoes
thus, results in twice
 two-photon diffraction,
narrower point spread function on the image plane.
E(, z,t) 
it ˜
d

d

g(

,

;

,
z)
e
E (,)

g(,;  o,z  zo )

Green’s function, or optical transfer
function (Fourier optics)
( )
( )
( )
( )
( )
( )
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
G (1, 2 )   E1 E 2 E 2 E1   0 E 2 E1 
(2)
(1, 2 )  0 Eˆ 2 Eˆ 1 
()
()
2
2
Biphoton

Coherent
G(2) ( ) 
R 2
o  /m )
so  /2
R 2
G ( 1,2 )   do t( o ) somb(
o  1 /m )
so 
2
R 2
o  2 /m )
so 
2
(2)

Chaotic

 do t 2 (o ) somb(
2
 do t(o ) somb(
2 2

R 2
G(2) ( )   do t( o ) 2 somb(
o  /m ) 


s

o


Two-photon diffraction: proof of principle of
Quantum Lithography
Boto et al., PRL 85, 2733 (2000)  M. D’Angelo, M.V. Checkova,
and Y.H. Shih, PRL, 87, 013602 (2001).
The measurement was on the Fourier transform plane instead of the
image plane: twice narrower interference/diffraction pattern!!
Unfolded version
Two-photon diffraction and quantum
lithography
M2
PBS
P
s and i
(916 nm)
D1
F
D2
M1
BBO
Pump
(458 nm)

Slit
M F
Coincidence
Circuit
M. D’Angelo, M.V.
Checkova, and Y.H. Shih,
PRL, 87, 013602 (2001).
Rc
 Degenerate Collinear type-II SPDC
 Double-slit VERY close to the crystal   b D    (asai  bsbi ) 0
 The measurement is on the Fourier transform plane.

Experimental Data
SPDC: 916nm
R c ( )  sinc 2[(2a/)  ]  cos 2[(2b/)  ]
Classical Laser light:
916nm
I()  sinc 2[(a/)  ]  cos2[(b/)  ]
After 2nd Fourier transform, on the image plane, the spatial
resolution gains a factor of 2, beyond the classical limit!
Classical Diffraction
Diffraction of an
entangled pair

Double
Spatial Resolution
on the Image Plane
Is Entanglement Dispensable in Quantum Lithography?
The unique EPR correlation of (x1 - x2) & (p1 + p2) made entangled state
very special: the pair comes out from a point on the object plane, under goes
two-photon diffraction, and stops at a point on the image plane. The twophoton diffraction provide us sub-wavelength spatial resolution (/).
“Is Entanglement Dispensable in Quantum Lithography?” to be
published, 2006, G. Scarcelli, M. D’Angelo and Y.H. Shih.
Quantum Imaging - UMBC
Part II
The physics of second-order correlation of chaotic light
- “ghost” imaging of thermal light
Hanbury Brown and Twiss 1956
Counts
D1
a
b
Position of Detector
Counts
D2
Position of Detector
Coincidences
Joint
Detection
Position of Detector
(2)  I1 I2 ~ I02{1 sinc 2[(x1  x2 )/ ]}
It isinterpretation:
a trivial wrong idea:
An
G(2) = constant for laser light.
Laser Beam
Copies of “Speckles”

More Problems
Coincidences
Entangled Photon
Source
Joint
Detection
I1I2  I11I2 I2 I1 II12I2 I1I2
1
0
Position of Detector
I1  0
I

0
sinc
[



(x

x
)
/

]
~ sinc [~1
 (x

x
)
/

]
2
1
2
1
2
2
2
???
Classical Statistical Correlation of Intensity Fluctuations does
not work for Entangled States
The physics ?
“Can Two-photon Correlation of Chaotic Light Be Considered as
Correlation of Intensity Fluctuations?” PRL, 96, 063602 (2006)
(G. Scarcelli,V. Berardi, and Y.H. Shih).
Two-Photon Imaging
S.C
D2
X2
X2
C.C
D1
S.C
X2
X2
Ghost Image with chaotic light
Pseudothermal
source
Martienssen-Spiller
(1964)
Correlator
Photon Counting
Correlation
Measurement
Near-field
  10 / d
Correlation
Results
True Image ?
1 1 1
 
S0 Si f

Correlation
Measurement
so  db  d a
si  d
'
b
Results
Question 1
What is the similarity and difference
between this experiment and HBT?
Ghost Imaging and HBT
k-k
Momentum-Momentum
Correlation
IMAGE
x-x
Position-Position Correlation
  10 / d
Question 2
Why the statistical correlation between
intensity fluctuations is not valid for
this experiment ?
(1)Statistical Correlation of Intensity Fluctuations?
No !
Unlike HBT, it is not in “far-field”…every point of the image
plane is “hit” by many k’s of the radiation. For chaotic light, all
modes are chaotically independent!
Source
BS
Image Plane
Object Plane
(2) Statistical Correlation of Intensity Fluctuations?
No !
The correlations are washed out by the “bucket detection”
Source
BS
Image Plane
Object Plane
Bucket
Detector
Question 3
Alternative Interpretation ?
Two-photon interference
Glauber theory
G(2) (x1, x2 )  E () (x1)E () (x2 )E () (x2 )E () (x1)
Source
BS
Image Plane

x1


 
E1(  ) ( x1 )   g1 ( x1 ; q )aˆ q

q
E
Object Plane

x2
   1q ,1q ' 1q ,1q '
 
q ,q '
()
2

 
( x2 )   g 2 ( x2 ; q )aˆ q

q
Two-photon Chaotic light
Two-photon interference
 
() 
() 
G ( x2 ; x1 )   0 E ( x2 ) E ( x1 ) 1q ,1q '
( 2)
2
 
q ,q '
 
 
 
  2
  g 2 ( x2 , q ) g1 ( x1 , q ' )  g 2 ( x2 , q ' ) g1 ( x1 , q )
 
q ,q '
Indistinguishable two-photon probability amplitudes
result in a “click-click” joint detection event.
Source

q'
D1

q'

q

q
D2
Two-photon interference
 
G ( x1 , x2 ) 
( 2)
  2
  2
 g1 ( x1 , q )  g 2 ( x2 , q ) 

q

q
 
 
 g ( x2 , q ) g1 ( x1 , q )

q
*
2
IMAGE
G (x1, x1 )G (x 2 , x 2 )  G (x1, x 2 )G (x 2, x1 )
(1)
11
(1)
22
(1)
12
(1)
21
Back to the “standard result” in terms of
first-order correlation functions
2
A possible application: “ghost” camera.
The ground camera is pointed to a “magic mirror” in space. A point like photo-detector
D1 collects all photons coming from the object and records the photon registration time.
By studying the time correlation between the photon registration times of D1 and each of
the CCD element, a “ghost” image is obtained in “coincidences”.
Quantum Imaging - UMBC
Part III
Multi-photon sources
- A true entangled three-photon state
1
p

1’
3

p
2’
Feynman diagram:
three-photon generation.
Keller, Rubin, Shih, Wu,
PRA 57, 2076 (1998).

2
 
 (2
 

ˆ
ˆ
ˆ






)

(2k

G

k

k

k
)
a
a
a
p
1
2
3
p
1
2
3
k1 k 2 k 3 0
k 1 ,k 2 ,k 3
The triphoton state
2D photonic crystal: hexagonally poled LiTaO3
Schematic setup of the three-photon correlation measurement
G(3) measurement
Experimental data &
Numerical simulation