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Quantum Optics II
Cozumel, Mexico, December 6-9 2004
Correlated imaging, quantum
and classical aspects
INFM, Università dell’Insubria, Como, Italy
Theory:
Lab. I:
Lab. II:
Alessandra Gatti,
Enrico Brambilla,
Morten Bache and
Luigi Lugiato
Ottavia Jederkievicz,
Yunkun Jiang
Paolo Di Trapani
Fabio Ferri,
Davide Magatti
INTRODUCTION
Spatial aspects of quantum optical fluctuations
New potential applications exploiting the quantum properties of the light for
image processing or multi-channel operation
quantum imaging
•The quantum laser pointer
•Quantum lithography
•Entangled two-photon imaging
(ghost imaging)
•Quantum superresolution
•Quantum teleportation of images
•Noiseless image amplification
Parametric down-conversion process (PDC) in a (2) nonlinear crystal
(2)
SIGNAL

PUMP
2
PUMP
2
IDLER

Large emission bandwidth in the spatial frequency domain
OUTLINE OF THE TALK
I -First experimental observation of spatial correlation at the
quantum level in the macroscopic regime of parametric downconversion
II-Ghost imaging tecnique: optical imaging by means of the
spatial correlation (spatial entanglement) of two beams
•Comparison between ghost imaging with entangled beams and
classically correlated beam from a thermal source
 Results which combine in a surprising way quantum and classical
optics bringing together the two communities to a common discussion.
Twin-photons generated by parametric down-conversion
Microscopic generation of twin photons: at the origin of spatial
correlation of signal and idler beams at the crystal output (near field)
Signal/ idler twin photons are are always created at the same position  the
intensity distributions of signal and idler beams are spatially correlated
NEAR FIELD
pump
•Finite crystal length--> uncertainty in the relative
position of the twin photons due to diffraction spread
(2)
lcoh ~ lc 
•
uncertainty in the position
of photon 1 from a measurement of the position of
photon 2
lc=5mm
SIGNAL
IDLER
•Perfect spatial intensity correlation for detection areas
2
broader than l coh
Phase matching: at the origin of far-field spatial correlation of PDC photons
Plane-wave
pump
q=0,
signal
idler
Perfect intensity correlation in symmetric far field positions of the two beams
FAR FIELD
SIGNAL
pump
(2)
IDLER
Finite size of the pump waist wP --> uncertainty in the propagation directions of twin photons
 1 / wP
 Perfect intensity correlation only for detection areas broader than a “coherence area”  1 / w P2
Brambilla, Gatti, Bache, Lugiato, Phys Rev A 69, 023802 (2004); quant-ph/0306116 (2003)
Detection of sub-shot-noise Spatial Correlation in the
high-gain regime of PDC
Experiment performed at Como Lab. (Ottavia Jedrkiewicz, Yunkun Jiang,
Paolo Di Trapani)
•Literature in the low gain regime: single photon pairs resolved in time by
photodetectors  coincidence measurements
•In the high-gain regime: large number of photons emitted into each mode
detection in single shot by means of a high Q.E. CCD
GOALS of THE EXPERIMENT
•Investigate the single-shot spatial intensity correlation in the far field,
between the signal and idler beams.
•Check if the far-field signal and idler intensity distributions “coincide”
within the shot noise
Experimental set-up
Pump pulses
@352 nm
The nonlinear crystal: BBO (L=4mm) =49.05°, =0° type
II; degenerate s,i @ 704 nm
Pump pulses @352 nm, 3rd harmonic of Nd:Glass laser, 1.5ps,
Rep. rate 2 Hz, Ep ~ 0.1mJ – 0.5 mJ, 1 mm waist
Spatial filter
+200 mm
teflon pnh
Gain varying between 10 and 103
Selection of a portion of PDC
fluorescence around collinear
direction
type II BBO
rectangular
aperture
M
3
M
5
M
2
h ~ 89% @704nm
Polarizing
Beamsplitter
M
3
CCD
M
1
M
4
M 3 ,M 4 ,M 5 (HR@352,HT@704)
Low-band pass
filter
htot ~ 75%
No Interference filter during
measurements
Far field image of the selected portion of PDC fluorescence
Boxes correspond to a
20x8 mrad angular
bandwidth around
collinear direction and
<10 nm bandwidth
around degeneracy
SPATIAL statistics
performed inside boxes
(4000 pix) for each single
laser pulse
Zoomed signal
Zoomed idler
evident spatial correlation between the two images
Photocounts (signal-idler) difference noise statistics
3.0
s2s i  ns  ni 2  ns  ni
2
s2s-i/<ns+ni>
2.5
Jedrkiewicz, Jiang, Gatti,
Brambilla,Bache, Di Trapani,
and Lugiato, PRL in press,
Quant-ph/0407211
2.0
1.5
SNL
1.0
0.5
noise reduction limit
0.0
10
<ns+ni> (pe)
100
Sub-shot-noise correlation up to gains characterized by ns  ni 15-18 corresponding to
100 pe per mode (transverse size of the coherence areas in that regime about 2-4 pixels)
Transition from the quantum to classical regime: attributed to a
broadening of the far field coherence area with increasing gain
Pump intensity I~ 5 GW/cm2
Pump intensity I~ 50 GW/cm2


The down-converted fields map the gain profile G( r )  sinh 2 s I ( r )L
On increasing the pump intensity, the gain profile gets narrow despite of the
fixed pump waist  the far-field coherence area broadens
 Detection areas (single pixels) become smaller than the coherence area
In summary: twin beam effect over several phase conjugate
signal and idler modes
Perspectives:
IMAGING OF A FAINT OBJECT (WEAK ABSORBTION) WITH A
SENSITIVITY BEYOND STANDARD QUANTUM LIMIT
I1
PDC
crystal
Signal
Noise
Signal
Noise
I2
RATIO LOW FOR I1
RATIO HIGH FOR I1-I2
GHOST IMAGING TECHNIQUE
Optical imaging by means of the spatial correlation (spatial entanglement)
of two beams
Flexible way of performing coherent imaging with incoherent light
IN THIS TALK:
Comparison between ghost imaging with entangled beams and classically
correlated beam from a thermal source
Results which combine in a surprising way quantum and classical optics
bringing together the two communities to a common discussion.
Ghost imaging by means of two-photon quantum entanglement
•Pittman, Shih , Strekalov and Sergienko, PRA 52, R3429 (1995)
•Ribeiro, Padua, Machado da Silva, Barbosa, PRA. 49, 4176, (1994)
•Strekalov, Sergienko, Klyshko and Shih, PRL 74, 3600 (1995)
GHOST IMAGE
GHOST
DIFFRACTION
x1
Photon-pair created
by PDC in the ultralow gain regime
Pump
POINT-LIKE DETECTOR,
FIXED POSITION
OR BUCKET DETECTOR
1
Coincidence
counts as a
function of x2
(2)
2
ARRAY
OF
DETECTORS
x2
The imaging information is extracted from the coincidence
counts as a function of the position of the reference photon 2
Generalization to the regime of many photon pairs: signal-idler
intensity correlation function [Gatti, Brambilla, Lugiato, PRL 90, 133603 (2003)]
POINT-LIKE DETECTOR,
FIXED POSITION
1
Correlation function
of intensities

x1
Pump


I 2  x2 I1  x1 
(2)
2
ARRAY
OF
DETECTORS

x2
THE IMAGING INFORMATION IS CONTAINED IN THE CORRELATION FUNCTION
OF INTENSITY FLUCTUATIONS




I 2  x2 I1 x1   I 2  x2  I1 x1 
no information,
background

I i  I i  I i
.


I 2 x2 I1x1 
Imaging
information
Is entanglement really necessary to perform ghost imaging?
Yes:
•Abouraddy, Saleh, Sergienko, Teich, Phys. Rev. Lett. 87, 123602 (2002); Josa B 19,1174 (2002)
“the distributed quantum-imaging scheme truly requires entanglement in the
source and cannot be achieved by using a classical source with correlations but
without entanglement”
No,but..
Ghost image experiment by using laser pulses with classical angular correlation.
•Bennink, Bentley and Boyd, PRL 89, 11389 (2002)  Although the result of any
single ghost imaging experiment can be reproduced by classical sources,
“a classical source cannot mimic a quantum source in a pure state for all test and
reference systems unless that state is nonentangled.”
Theory in arbitrary gain regime
•Gatti, Brambilla and Lugiato, PRL 90, 11389 (2003)  The results of each single
experiment can be reproduced by a classical source.
But...
f-f scheme:ghost diffraction
DOUBLE SLIT
2f-2f scheme:ghost image
DOUBLE SLIT
test beam 1
f
f
f
(2)
f
x
f
x
2f
2f
reference beam 2


I 2 ( x2 )I1( x1 ) numerics


I 2 ( x2 )I1( x1 ) numerics
600
250
200
150
10000 SHOTS
100
50
correlation function (a.u.)
correlation function (a.u.)
f
(2)
reference beam 2
0
test beam 1
500
400
300
100
0
-8
-6
-4
-2
0
2
x2/x0
4
6
8
10000 SHOTS
200
-15
-10
-5
0
5
x2/lcoh
10
15
By only operating on the optical set-up in the path of beam 2 (which never went through
the object), one is able to pass from the interference pattern to the image of the object.
Key point: simultaneous presence of spatial correlation both in the near and in the farfield of the PDC beams. Feature that distinguishes the entangled from the classical source ?
Intensity operators in the far field and in the near field of each beam

 

 
Iˆi q   bi  q bi q  ,
Iˆi x   bi  x bi x 
ARE NON COMMUTING OBSERVABLES
Replace the pure EPR state with a statistical
mixture that exactly preserves the far-field
spatial correlation  the near field spatial
correlation is completely lost
f-f scheme: diffraction pattern of the
object
Replace the pure EPR state with a statistical
mixture that exactly preserves the nearfield spatial correlation  the far-field
spatial correlation is completely lost
2f-2f scheme: image of the object
BUT no information about the image in
the 2f-2f scheme
BUT no information about the diffraction
pattern in the f-f scheme
Gatti, Brambilla, Lugiato, Phys. Rev. Lett . 90, 133603 (2003)
Simultaneous presence of “perfect” spatial correlation in the near and in the far-field
of the PDC beams. [Brambilla, Gatti, Bache, Lugiato, PRA 69, 023802 (2004)}
q
(2)
q’
-q’
-q
SIGNAL
SIGNAL
IDLER
IDLER
wP=160mm
FAR FIELD INTENSITY
CORRELATION
Directions of propagation of twin photons
are correlated because of phase matching
NEAR-FIELD INTENSITY CORRELATION
Twin photons are generated at the same position
inside the cristal
Position x of signal photon determined from a
measurement of the position of the idler photon
Momentum q of signal photon determined
from a measurement of the momentum -q of
the idler photon
EPR-like inequality for the conditional variancies of position and momentum
of two photons
x q  1
satisfied only by entangled (nonseparable) states
Bennink, Bentley, Boyd, PRL 92 033601 (2004) ; see also D’Angelo Kim Kulik Shih
PRL 92, 233601 (2004)
Claim: this inequality limits the resolution capabilities of ghost imaging
with classically correlated beams. High-resolution ghost image and ghost
diffraction are possible only with an entangled source of photons
Is that true?
Nothing prevents two classical beams from being spatially correlated both in the near
and in the far field up to an imperfect degree
Beam in a thermal-like
state
(i.e. classically, or at shot noise)
N1
N   N1  N 2
50:50 BS
N 2  N1  N 2
SHOT - NOISE LEVEL
N2
vacuum
C
N1N 2
N12
N 22
 1
N1
N12
0 C 1
coherent state
(no correlation)
Cauchy-Schwartz
(perfect correlation)
Coherent beam
N12  N1
Thermal beam N12  N1
Intense thermal beam
 N1
C 0
 1  C  0
N12  N1  C  1
HIGH LEVEL OF CORRELATION
BUT STILL CLASSICAL!
“Twin” speckle pattern generated by impinging a laser beam
on a ground glass and then splitting simmetrically.
Fabio Ferri and Davide
Magatti lab in Como
TO CCD
LASER
ROTATING
GROUND GLASS
BS
Moreover, the correlation is preserved from the near-field to the far-field, provided the
source cross-section is much larger than the coherence length  the classically
correlated thermal beams can be used for ghost imaging exactly in the same way
as the entagled beams from PDC
beam in a thermal (or
pseudo-thermal) state
50:50 BS
vacuum
x


I1 x1 I 2  x2 
Correlated imaging : parallel between the use of
(a) ENTANGLED PDC BEAMS
and (b) CLASSICALLY CORRELATED BEAMS
BY SPLITTING THERMAL RADIATION
Correlation function of intensity
fluctuations at the detection planes


I1  x1 I 2  x2  


 
 


 dx1'  dx2' h1x1 , x1' h2 x2 , x2'  b1x1' b2 x2' 
Signal-idler field cross-correlation
(two-photon propagator)


b1  x' b2  x' 
correlation length = coherence
length of PDC beams1/q
Correlation length in the far field:
inversely proportional to the
pump beam-waist
Correlation function of intensity
fluctuations at the detection planes
2


I1  x1 I 2  x2   1 4


 
 

 







d
x
'
d
x
'
h
*
x
,
x
'
h
x
,
x
'
a
x
'
a
x
1
2
1
1
1
2
2
2
1
2' 
 
Second order correlation of the
thermal radiation
 
a   x ax' 
correlation length = coherence
length of thermal radiation1/q
Correlation length in the far field:
inversely proportional to the crosssection of the thermal source
Gatti et al. quant-phys/0307187 (2003), PRL 93, 093602 (2004), Phys. Rev. A 70, 013802 (2004)
2
RELEVANT DIFFERENCE: VISIBILITY OF THE INFORMATION
RETRIEVED VIA CORRELATION MEASUREMENTS




I 2  x2 I1 x1   I 2  x2  I1 x1 
no information,
background



I 2 x2 I1x1 
Imaging
information
a) ENTANGLED PDC BEAMS
(b) CORRELATEDTHERMAL BEAMS
I 2 x2 I1x1  scales as 1  n


I 2  x2 I1  x1  scales as n


2
I 2  x2  I1  x1  scales as n


2
I 2 x2  I1x1  scales as n


n
2
Background term is negligible in the
coincidence count regime n  1
Background term is never negligible
The entangled configuration, in the regime of coincidence
counts, offers a better visibility of the information
Experimental evidence of high resolution ghost image and ghost diffraction
with classically correlated beams from a pseudo thermal source
Ferri, Magatti,Gatti, Bache, Brambilla, Lugiato, quant-ph/0408021 (2004), submitted to PRL
near-field
plane
He-Ne LASER
OBJECT
400 mm
F
CCD
F
D=3mm
GROUND
GLASS
TURBID
MEDIUM
BS
p2
F'
q2
1  1  1
q2 p2 Feff
Feff focal of the two lens system
IMAGES OF A DOUBLE SLIT (190 mm needle inside a 690 mm aperture)
OBTAINED BY CROSS-CORRELATING THE REFERENCE ARM INTENSITY
DISTRIBUTION WITH THE TOTAL LIGHT IN THE OBJECT ARM
-3
x10
1,0
correlation
laser illumination
(a.u.)
0,8
0,6
0,4
0,2
0,0
-750
5000 FRAMES
30000 FRAMES
-500
-250
0
x mm
250
SECTION
IMAGE OBTAINED BY
SHINING LASER LIGHT
500
750
BY SIMPLY REMOVING THE LENS F’ IN THE REFERENCE ARM:
DIFFRACTION PATTERN OF THE DOUBLE SLIT
50
a.u.
40
correlation function
G(x2-x1)
30
laser illumination
20
10
0
-200 -150 -100 -50
FRINGES OBTAINED BY
CROSS CORRELATION (500 FRAMES)
FRINGES OBTAINED BY
SHINING LASER LIGHT
0
50
x mm
100 150 200
SECTION
INTENSITY DISTRIBUTION IN
THE OBJECT ARM
RESOLUTION OF GHOST IMAGING WITH CORRELATED THERMAL
BEAMS
The resolution of the ghost imaging and ghost diffraction schemes are determined by
the widths of the near- field and far-field auto-correlation functions xn and xf.
2,0
far-field correlation
Gaussian fit s=7.8 mm
near-field correlation
Gaussian fit s=14.3 mm
1,8
1,6
1,4
1,2
We find xn = 34.3 mm
xf =15.6 mm 
1,0
0,8
0
10
20
30
40
50
60
70
|x -x'| mm
80
90 100
q  2 x f  1.93  10 3 mm 1
f
xn q = 0.066 < 1
The product of xn q we obtain is much smaller than the value 1, which was
suggested as a lower bound for the resolution of classically correlated beams.
SUMMARY AND CONCLUSIONS
First experimental investigation of quantum spatial correlation in the
high-gain regime of PDC: sub-shot noise intensity correlations of signal
and idler far fields
Ghost Imaging: results that question the role of entanglement
-
Experimental evidence of high resolution ghost imaging and ghost diffraction with a
pseudo thermal source .
-
Information processed by only operating on the reference beam.
-
The suggested lower bound for the product in the resolutions (near and far field) does
not exist.
The only difference from an entangled source is a lower visibility of the
information.
Entanglement can be advantageous in high sensitivity measurements (e.g.
imaging of a faint object) or in quantum information (e.g. cryptographic)
schemes, no evident practical advantage in imaging macroscopic classical
object
a) ENTANGLED PDC BEAMS
(b) CORRELATED THERMAL BEAMS
f-f scheme: diffraction pattern


f
1
2

x  x  1  nq   nq 



  2   
Pobj q
f
q   x2
 2 
I 2 ( x2
2
I 2 ( x2 )I1( x1 ) 


mean photon
number per mode

I 2
   
x2
I1
x1

Tobj(  x2

)
2
1  nq   nq 
spatial resolution determined
by the near-field PDC
coherence length
)I1( x1 )


2
mean photon
number per mode
spatial resolution determined by the farfield coherence length , inverse of the pump
waist
2f-2f scheme: image
f-f scheme: diffraction pattern
f



f
q   x2
Pobj  q 
x2  x1  nq   2 

 2
  2   
f
q  x1
 2 
spatial resolution determined by the far
field coherence length , inverse of the source
cross-section
2f-2f scheme: image
I 2  x2 I1  x1   Tobj (  x2 ) nq 


 2

f
q  x1
 2 
2
spatial resolution determined by the
near-field thermal coherence length
Gatti et al. quant-phys/0307187 (2003), PRL 93, 093602 (2004), Phys. Rev. A 70, 013802 (2004)
1D NUMERICAL SIMULATION FOR THE RECOSTRUCTION OF THE INTERFERENCE


I 2 ( x2 )I1( x1 )
FRINGES VIA
IN THE f-f SCHEME


I 2  x2 I1  x1  0
1000 shots
1000 shots
300
entangled
thermal
analytic (a.u)
100
0
-2
-1
0
1
2
x2/x0
350
<I2(x2)I1(x1=0)>
<I2(x2)I1(x1=0)>
entangled
thermal
5000
200
<I2(x2)I1(x1=0)>
<I2(x2)I1(x1=0)>
400


I 2  x2 I1  x1  0 




I 2  x2  I1  x1  0  I 2  x2 I1  x1  0
4000
3000
2000
1000
entangled
thermal
analytic (a.u.)
10000 shots
300
250
200
0
-6
-4
-2
0
x2/x0
2
4
6
FRINGE VISIBILITY  5
IN BOTH CASES
150
100
50
0
-2
-1
0
x2/x0
1
2
HOWEVER, EFFICIENT
RECONSTRUCTION AFTER
A REASONABLE NUMBER
OF PUMP SHOTS
Precise formal analogy between the use of classically correlated beams
from a thermal source and entangled beams from PDC  all the features
of ghost imaging can be reproduced without entanglement !
Gatti et al. quant-phys/0307187 (2003), PRL 93, 093602 (2004), Phys. Rev. A 70, 013802 (2004)
ONLY RELEVANT DIFFERENCE: VISIBILITY OF THE INFORMATION




I 2  x2 I1 x1   I 2  x2  I1 x1 
no information,
background
a) ENTANGLED PDC BEAMS
I 2 x2 I1x1  scales as 1  n


I 2 x2  I1x1  scales as n


n
2
Background term is negligible in the
coincidence count regime n  1



I 2 x2 I1x1 
Imaging
information
(b) CORRELATED THERMAL BEAMS
I 2  x2 I1  x1  scales as n


2
I 2  x2  I1  x1  scales as n


2
Background term is never negligible
The entangled configuration, in the regime of coincidence counts, offers
a better visibility of the information