Download Quantum Imaging

MURI Kick-Off Meeting
Rochester, June 9-10, 2005
Quantum Imaging
- Entangled state and thermal light
- Foundamental and applications
Optical Projection (Chinese shadow, x-ray, …)
Momentum (p1)
 Momentum (p )
No image plane is defined.
2
Optical Imaging:
Point (object Plane)
Position (x1)
1
1
1


S0 Si
f
 Point (image plane)
 Position (x )
and (x1  x2 )  0
2
{
}
Geometric optics
Image lens:  
Spatial Resolution
Imaging lens: finite size
Image Plane
Point (object plane)
function


Spot (image plane)
somb-function
(x1  x2 )  comb( ) 
2J1 ( )

D1
X2
X2
C.C
D2
S.C
Two-Photon Imaging
S.C
X2
X2
“Ghost” Imaging with entangled photon pairs
x1  
x 2    x1  x 2  0
k1  
k2    k1  k2  0

Point x (image plane)

1
1
1
Point x1 (object plane)
S0
2

Si

f
So
Si
1 1 1
 
S0 Si f
 Interference
“Ghost” Image and “Ghost”
EPR Experiment in momentum-position
PRL, 74, 3600 (1995); PRA, 52, R3429 (1995).
Classical: never!
- classical statistical measurements
(x1  x 2 )  (x1 ) 2  (x 2 ) 2  Max(x1, x 2 )
( p1  p2 )  (p1 ) 2  (p2 ) 2  Max(p1, p2 )


* H  H1  H2; Hinteraction  0
* Space-like separated measurement events.
(1) No interaction between two distant quanta;
(2) No action-at-a-distance between individual
measurements.
To EPR: the two quanta are independent as well as
the measurements, so that
(x1  x 2 )  (x1 ) 2  (x 2 ) 2  Max(x1, x 2 )
( p1  p2 )  (p1 ) 2  (p2 ) 2  Max(p1, p2 )
Classically correlated systems: one may consider
building an ensemble of particle-pairs to force each
pair with p1  p2  p0 and p1  0, p2  0, so that
(p1  p2 )  0 . In this case, however, (x1  x2 ) ~ 




Quantum: yes!
- EPR: if the two quanta are entangled
(x1  x 2 )  0
( p1  p2 )  0
Although


x1  , x 2  
{ p  ,
1
p2  
}
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 an surprising phenomenon: the momentum
(position) for neither subsystem is determinate; however, if

one particle is measured to have a certain momentum (position), the momentum (position) of its “twin” is determined
with certainty, despite the distance between them!
The apparent contradiction deeply troubled Einstein.
While one sees the measurement on (p1+p2) and
(x1-x2) of two individual particles satisfy the EPR
-function and believes the classical inequality,
one might easily be trapped into considering either
there is a violation of the uncertainty principle or
there exists action-at-a-distance.

Violation of the uncertainty principle ?
 p1  p2   0  x1  x 2   0
Simultaneously !
(p1+p2) and (x1-x2) are not
conjugate variables !!!!
1
(x1, x 2 ) 
2
1

2

ip1 x1 /
ip2 (x 2 x 0 )/
dp
dp

(
p

p
)
e
e
 1 2 1 2
 d( p  p )( p  p ) e
1
2
1
i( p1  p 2 )(x1 x 2' )/ 2
2
 d( p1 p2 ) /2 e i( p1  p2 )(x1 x2 )/ 2
'
 1  (x1  x 2  x 0 )

( p1, p2 ) 
1
2
1

2

 dx dx (x
1
ip1 x1 /
ip2 (x 2 x 0 )/

x

x
)
e
e
1
2
0
2
 d(x  x ) e
1
,
2
i( p1  p 2 )(x1 x ,2 )/ 2
 d(x  x ) /2 (x
1
  ( p1  p2 ) 1
,
2
1
 x )e
,
2
i( p1  p 2 )(x1 x ,2 )/ 2
Conjugate Variables:
(x1  x 2 )  ( p1  p2 )
(x1  x 2 )  ( p1  p2 )


(x1  x 2 )    ( p1  p2 )  0
(x1  x 2 )  0  ( p1  p2 )  

EPR -function:
-- perfect entangled system
(x1  x2 )  0, (p1  p2 )  0.
Although: x1  , x2  , p1  , p2  .


EPR Inequality:
-- non-perfect entangled system
(x1  x 2 )  min( x1, x 2 )
( p1  p2 )  min( p1, p2 )
Then, why Einstein … ?
Observation:
(x1  x2 )  0, (p1  p2 )  0
Believing:
(x1  x 2 )  Max(x1, x 2 )

( p1  p2 )  Max(p1, p2 )
Conclusion:

x1  0, p1  0
x 2  0, p2  0
(Violation of the …)
The interpretation ?
Quantum entanglement
Two-photon is not two photons !
2  1 1

Classical:
Two Wavepackets
Entanglement:
A non-factorable
2-D Wavepacket
Biphoton State: Spontaneous Parametric Down Conversion
Two-photon Pure State
The signal (idler) photon can
have any energy (momentum),
however, if one of the
photons is measured at
certain energy (momentum) its
twin must be at a certain
energy (momentum).
   ( s   i   p )  (k s  k i  k p ) aˆ s aˆ i 0
s,i
Operational approach:
G (x1, x2 )  E E E E
(2)
()
1
()
2
()
2
()
1
Pure state:
G(2) (x1, x 2 )

  E1() (x1 )E 2() (x 2 )E 2() (x 2 )E1() (x1 ) 
 0 E (x 2 )E (x1) 
()
2

()
1
2
SPDC
(t1,t2 )  0 E (t2 )E (x1) 
()
2
()
1
A biphoton
Effective Two-photon
wavefunction
(t1,t 2 )  Ft1 t2 { f ( p   p0 )} Ft1 t2 {g( s   s0 )} ei s 0 t1 ei i 0 t2

Two-photon imaging
Field Operators:
E ()
j (r j ,t j ) 
 dkT
 d a(,kT ) e
it j
g j (,kT ;z j , j )
g j (,kT ;z j , j ) : Green’s function (optical transfer function).
determined by the experimental setup.


G
(2)
 (1, 2 )
2
The calculation of G(2) is lengthy but straightforward:


(1  2 )EPR ~ 0
It is the two-photon coherent superposition made it possible!
Although questions regarding fundamental
issues of quantum theory still exist, quantum
entanglement has indeed brought up a novel
concept or technology in nonlocal positioning and timing measurements with high
accuracy, even beyond the classical limit.
Question:
Can “ghost” image be simulated classically ?

Image but not projection!!!

Yes
Experimentally
Thermal Light Imaging
Magic Mirror and Ghost Imaging
M = 2.15 (Mtheory = 2.16); V = 12 % (Vtheory =16.5%)
Experimental Result: Ghost image of a double-slit.
A. Valencia, G. Scarcelli, M. D'Angelo, and Y.H. Shih, Phys. Rev. Lett. 94, 063601 (2005).
Measurement on the image plan.
Two-photon thermal light Imaging:
Incoherent imaging: G(2)  Gk(2)   [ G11(1)G22(1)  G12(1)G21(1) ]
k
k
Magic Mirror ?
Measurement on the mirror plan.
It is useful !
A “Ghost” Camera in Space
(Nonlocal)
A “Magic Mirror” for X-ray 3-D Imaging
It is fundamentally interesting !!
50% momentum-momentum, position-position EPR correlation
Where it comes from ?
Remember: thermal light is chaotic !
It comes from Hanbeury Brown - Twiss … ???
It comes from “photon bunching” … ???
We are not satisfied !
The physics behind ???
x
“Two-photon” film
Slit A

Fourier transform
function (or ) ?
Slit B
(2)
G (x1, x 2 )
f
Different
Input State

Fourier Transform
Plane
x1, x 2
Correlated
Lasers

f
G(2) = G(2) (x1)  G(2) (x 2 )
A product of two independent first-order-pattern.

x1, x 2
SPDC
2
0
+

=
0
f
  [as ai  bsbi ei ] 0 ;  const.
G(2)  sinc 2[
af
bf
af
bf
(x1  x 2 )] cos2[
(x1  x 2 )]  sinc 2 (
x) cos2
x


 /2
 /2
x1
Thermal
2
0
2
2
+
+
+
=
0
  ak ak 0 ;
  bkbk 0 ;
 
1  
(ak bk   bk ak) 0
2
ˆ        

2
2
G(2)  1 + sinc 2[
2
af
bf
(x1  x 2 )] cos2[
(x1  x 2 )]


x2
Quantum lithography
(ultra-resolution: beyond classical limit)
Optical Lithography
Fourier Transform
One
Fourier Transform
Two
Optical Lithography
Laser
Fourier Transform
One
Fourier Transform
Two
Optical Lithography
SPDC
Fourier Transform
One
Fourier Transform
Two
Optical Lithography
Thermal
Fourier Transform
One
Fourier Transform
Two
Two-photon diffraction and quantum lithography
M2
PBS
P
s and i
(916 nm)
D1
F
D2
M1
BBO
Pump
(458 nm)

Slit
Coincidence
Circuit
M F
Rc
Experiment: M. D’Angelo, et al, PRL, 87, 013602 (2001).
Theory: A.N. Boto, et al. PRL 85, 2733 (2000).
CC in 100 sec
Experimental Data
250
SPDC two-photon at
200
s  i  916nm
150
100
50
-6
-4
-2
0
2
4
6

Counts per sec
R c ( )  sinc 2[(2a/)  ]  cos 2[(2
b/)  ]
16000
Classical laser light at
  916nm
12000
8000
4000
-6
-4
-2
0
(mrad)
2
4
6
I()  sinc 2[(a/)  ]  cos2[(b/
) ]
It is the result of two-photon coherent superposition.
It measures the second-order correlation between the
object plane and the image plane, defined by the
Gaussian thin lens equation.
The published measurement was on the Fourier transform plane (far-field).
PRL, 87, 013602 (2001).
Super-resolution:
x  px  h
Classical diffraction

Diffraction of a pair
x  px  h


Double (super)
Spatial Resolution
on the Image Plane
“Ghost” Shadow (Projection)
ˆ cl 

 dk  dk
1
2
2)
P(k1)(k1  k 2 ) 1(k 1 )  (k
2

Bennink et al. PRL 89, 113601 (2002)