Correlated imaging, quantum and classical aspects

INFM, Università dell’Insubria, Como, Italy

Quantum Optics II

Cozumel, Mexico, December 6-9 2004

Theory:

Alessandra Gatti, Enrico Brambilla,

Morten Bache and Luigi Lugiato

Lab. I:

Ottavia Jederkievicz, Yunkun Jiang Paolo Di Trapani

Lab. II:

Fabio Ferri, Davide Magatti

INTRODUCTION

Large emission bandwidth in the spatial frequency domain

Parametric down-conversion process (PDC) in a (2) nonlinear crystal

PUMP2

(2)

SIGNAL

IDLER

PUMP2

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

•Entangled two-photon imaging (ghost imaging)

•Noiseless image amplification

•Quantum lithography

•Quantum superresolution

•Quantum teleportation of images

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.

OUTLINE OF THE TALK

I -First experimental observation of spatial correlation at the quantum level in the macroscopic regime of parametric down-conversion

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)

•Finite crystal length--> uncertainty in the relative

position of the twin photons due to diffraction spread

• uncertainty in the position

of photon 1 from a measurement of the position of

photon 2

•Perfect spatial intensity correlation for detection areas

broader than

ccoh l~l

2cohl

lc=5mm

SIGNAL IDLER

(2)

NEAR FIELD

pump

Signal/ idler twin photons are are always created at the same position the intensity distributions of signal and idler beams are spatially correlated

SIGNAL

IDLER

(2)

FAR FIELD

pump

Finite size of the pump waist wP --> uncertainty in the propagation directions of twin photons

Perfect intensity correlation only for detection areas broader than a “coherence area”

Brambilla, Gatti, Bache, Lugiato, Phys Rev A 69, 023802 (2004); quant-ph/0306116 (2003)

Pw/1

21 Pw/

Perfect intensity correlation in symmetric far field positions of the two beams

q=0,

-q

q signal

idler-

Phase matching: at the origin of far-field spatial correlation of PDC photons

Plane-wave pump

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

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

Spatial filter+200 mteflon pnh

M3,M

4,M

5(HR@352,HT@704)

M5

M4

Low-band pass filter

M3

M2

M1

PolarizingBeamsplitter

M3

type II BBOrectangularaperture

CCD

~ 89% @704nm

Selection of a portion of PDC fluorescence around collinear direction

No Interference filter during measurements

tot ~ 75%

Experimental set-up

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

Gain varying between 10 and 103

Pump pulses @352 nm

Zoomed signal Zoomed idler

evident spatial correlation between the two images

Far field image of the selected portion of PDC fluorescence

SPATIAL statistics performed inside boxes (4000 pix) for each single laser pulse

Boxes correspond to a 20x8 mrad angular bandwidth around collinear direction and <10 nm bandwidth around degeneracy

Sub-shot-noise correlation up to gains characterized by 15-18 corresponding to 100 pe per mode (transverse size of the coherence areas in that regime about 2-4 pixels)

is nn

Photocounts (signal-idler) difference noise statistics

SNL

10 1000.0

0.5

1.0

1.5

2.0

2.5

3.0

2 s-i/<

n s+n i>

<ns+n

i> (pe)

noise reduction limit

Jedrkiewicz, Jiang, Gatti, Brambilla,Bache, Di Trapani, and Lugiato, PRL in press, Quant-ph/0407211

222isisis nnnn

The down-converted fields map the gain profile

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

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

LG 2 )r(Isinh)r(

In summary: twin beam effect over several phase conjugate signal and idler modes

PDC crystal

I1

I2

RATIO LOW FOR I1

RATIO HIGH FOR I1-I2

NoiseSignal

NoiseSignal

Perspectives: IMAGING OF A FAINT OBJECT (WEAK ABSORBTION) WITH A

SENSITIVITY BEYOND STANDARD QUANTUM LIMIT

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

Photon-pair created by PDC in the ultra-low gain regime

POINT-LIKE DETECTOR, FIXED POSITIONOR BUCKET DETECTOR

2

1

(2)Pump

ARRAYOF

DETECTORS

h2(x

2 ,x2’)

h 1(x 1

,x 1’)

Coincidence counts as a function of x2

OBJECT

x2

x1

The imaging information is extracted from the coincidence counts as a function of the position of the reference photon 2

•Pittman, Shih , Strekalov and Sergienko, PRA 52, R3429 (1995) GHOST IMAGE

•Ribeiro, Padua, Machado da Silva, Barbosa, PRA. 49, 4176, (1994)

•Strekalov, Sergienko, Klyshko and Shih, PRL 74, 3600 (1995)GHOST

DIFFRACTION

112211221122 xIxIxIxIxIxI

Imaging information

no information, background

THE IMAGING INFORMATION IS CONTAINED IN THE CORRELATION FUNCTION

OF INTENSITY FLUCTUATIONS .iii III

Correlation function of intensities

1122 xIxI

POINT-LIKE DETECTOR, FIXED POSITION

2

1

(2)

1x

Pump

ARRAYOF

DETECTORS2x

h 1(x 1 , x

)

h2(x2 , x)

OBJECT

Generalization to the regime of many photon pairs: signal-idler intensity correlation function [Gatti, Brambilla, Lugiato, PRL 90, 133603 (2003)]

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”

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...

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.”

No,but..

2f-2f scheme:ghost image

10000 SHOTS

numerics 1122 )x(I)x(I

f-f scheme:ghost diffraction

-15 -10 -5 0 5 10 150

100

200

300

400

500

600

corr

elat

ion

func

tion

(a.u

.)

x2/lcoh

-8 -6 -4 -2 0 2 4 6 80

50

100

150

200

250

corr

elat

ion

func

tion

(a.u

.)

x2/x

0

10000 SHOTS

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 far-field

of the PDC beams. Feature that distinguishes the entangled from the classical source ?

numerics 1122 )x(I)x(I

reference beam 2

test beam 1

(2)

ff

ff x

reference beam 2

test beam 1

(2)

ff

2f2fx

DOUBLE SLIT DOUBLE SLIT

Intensity operators in the far field and in the near field of each beam

ARE NON COMMUTING OBSERVABLES

,qbqbqI iii xbxbxI iii

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

BUT no information about the image in the 2f-2f scheme

Replace the pure EPR state with a statistical mixture that exactly preserves the near-field spatial correlation the far-field spatial correlation is completely lost

2f-2f scheme: image of the object

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” 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)}

wP=160m FAR FIELD INTENSITY CORRELATION

Directions of propagation of twin photons are correlated because of phase matching

Momentum q of signal photon determined from a measurement of the momentum -q of the idler photon

(2)

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

q’

IDLER

SIGNAL

IDLER SIGNAL

-q’

-q

q

EPR-like inequality for the conditional variancies of position and momentum EPR-like inequality for the conditional variancies of position and momentum of two photonsof two photons

satisfied only by entangled (nonseparable) statessatisfied only by entangled (nonseparable) states

Bennink, Bentley, Boyd, PRL 92 033601 (2004) ; see also D’Angelo Kim Kulik Shih PRL 92, 233601 (2004)

1 qx

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 (i.e. classically, or at shot noise)

coherent state (no correlation)

21

1

22

21

21 1N

N

NN

NNC

10 CCauchy-Schwartz

(perfect correlation)

b1

b2

vacuum

50:50 BS

Beam in a thermal-like state

N1

N2

LEVEL NOISE-SHOT

212 NNN

21 NNN

1 beam thermalIntense

01 beam Thermal

0 beamCoherent

121

1121

121

CNN

CNNN

CNN

HIGH LEVEL OF CORRELATION BUT STILL CLASSICAL!

LASER

BSROTATING GROUND GLASS

TO CCD

“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

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

h 1(x 1 , x

)b1

x 1

point-like detector

h2 (x

2 , x)b2

x2

array of pixel detectors

x 2211 xIxI

vacuum

50:50 BS

beam in a thermal (or pseudo-thermal) state

OBJECT

Correlated imaging : parallel between the use of

(a) ENTANGLED PDC BEAMS and (b) CLASSICALLY CORRELATED BEAMS BY SPLITTING THERMAL RADIATION

2

221122211121

2211

'xb'xb'x,xh'x,xh'xd'xd

xIxI

Correlation function of intensity fluctuations at the detection planes

2

2122211121

2211 41

'xa'xa'x,xh'x,x*h'xd'xd

xIxI

Correlation function of intensity fluctuations at the detection planes

'xb'xb

21 'xaxa

Signal-idler field cross-correlation (two-photon propagator)

Second order correlation of the thermal radiation

correlation length = coherence length of PDC beams1/q

correlation length = coherence length of thermal radiation1/q

Correlation length in the far field: inversely proportional to the

pump beam-waist

Correlation length in the far field: inversely proportional to the cross-

section of the thermal source

Gatti et al. quant-phys/0307187 (2003), PRL 93, 093602 (2004), Phys. Rev. A 70, 013802 (2004)

(b) CORRELATEDTHERMAL BEAMSa) ENTANGLED PDC BEAMS

RELEVANT DIFFERENCE: VISIBILITY OF THE INFORMATION RETRIEVED VIA CORRELATION MEASUREMENTS

n n x I x I 1 as scales 1 1 2 2

21 1 2 2 as scalesn x I x I

2

1 1 2 2 as scalesn x I x I 2

1 1 2 2 as scalesn x I x I

Background term is negligible in the coincidence count regime

Background term is never negligible 1n

The entangled configuration, in the regime of coincidence counts, offers a better visibility of the information

112211221122 xIxIxIxIxIxI Imaging

informationno information,

background

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

CCD

He-Ne LASER

BSGROUND GLASS

F'

OBJECT

2p2q

FF

D=3mm

near-field plane

TURBID MEDIUM

mm 400

system lens two theof focal

111

22

eff

eff

F

Fpq

IMAGE OBTAINED BY SHINING LASER LIGHT

IMAGES OF A DOUBLE SLIT (190 m needle inside a 690 m aperture) OBTAINED BY CROSS-CORRELATING THE REFERENCE ARM INTENSITY

DISTRIBUTION WITH THE TOTAL LIGHT IN THE OBJECT ARM

5000 FRAMES -750 -500 -250 0 250 500 750

0,0

0,2

0,4

0,6

0,8

1,0x10-3

correlation laser illumination (a.u.)

x m30000 FRAMES SECTION

FRINGES OBTAINED BY SHINING LASER LIGHT

BY SIMPLY REMOVING THE LENS F’ IN THE REFERENCE ARM: DIFFRACTION PATTERN OF THE DOUBLE SLIT

SECTION FRINGES OBTAINED BY

CROSS CORRELATION (500 FRAMES)

-200 -150 -100 -50 0 50 100 150 200

0

10

20

30

40

50a.u.

correlation function G(x

2-x

1)

laser illumination

x m

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.

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.

xn q = 0.066 < 1

0 10 20 30 40 50 60 70 80 90 1000,8

1,0

1,2

1,4

1,6

1,8

2,0 far-field correlation Gaussian fit =7.8 m near-field correlation Gaussian fit =14.3 m

|x -x'| m

We find xn = 34.3 m

xf =15.6 m

13 m109312 .xf

q f

SUMMARY AND CONCLUSIONS

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

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

(b) CORRELATED THERMAL BEAMSa) ENTANGLED PDC BEAMS

1

2

22 1 1 2 21

xf

q

objq n q n ) x ( T x I x I

1

22 2

2 1 1 2 2x

fq

objq n ) x ( T x I x I

2f-2f scheme: image 2f-2f scheme: image

spatial resolution determined by the near-field PDC

coherence length

spatial resolution determined by the near-field thermal coherence length

2

2

2

2 1

1 1 2 2

12

xf

q

objq n q n x xf

q P

) x( I ) x( I

2

22

2

1 2

1 1 2 2

2

xf

qobjq n x x

fq P

) x( I ) x( I

f-f scheme: diffraction pattern f-f scheme: diffraction pattern

mean photon number per mode

mean photon number per mode

spatial resolution determined by the far-field coherence length , inverse of the pump waist

spatial resolution determined by the far field coherence length , inverse of the source cross-section

Gatti et al. quant-phys/0307187 (2003), PRL 93, 093602 (2004), Phys. Rev. A 70, 013802 (2004)

-2 -1 0 1 2

0

100

200

300

400 entangled thermal analytic (a.u)

1000 shots

<I 2(

x 2)I 1(

x 1=0)

>

x2/x

0

1D NUMERICAL SIMULATION FOR THE RECOSTRUCTION OF THE INTERFERENCE

FRINGES VIA IN THE f-f SCHEME)x(I)x(I 1122

10000 shots

1000 shots

-6 -4 -2 0 2 4 60

1000

2000

3000

4000

5000

<I2(x

2)I

1(x

1=0)>

entangled thermal

<I 2(

x 2)I 1(

x 1=0)

>

x2/x

0

FRINGE VISIBILITY 5

IN BOTH CASES

HOWEVER, EFFICIENT RECONSTRUCTION AFTER A REASONABLE NUMBER

OF PUMP SHOTS

00

0

11221122

1122

xIxIxIxI

xIxI 01122 xIxI

-2 -1 0 1 20

50

100

150

200

250

300

350 entangled thermal analytic (a.u.)

<I 2(

x 2)I 1(

x 1=0)

>

x2/x

0

ONLY RELEVANT DIFFERENCE: VISIBILITY OF THE INFORMATION

(b) CORRELATED THERMAL BEAMSa) ENTANGLED PDC BEAMS

n n x I x I 1 as scales 1 1 2 2

21 1 2 2 as scalesn x I x I

2

1 1 2 2 as scalesn x I x I 2

1 1 2 2 as scalesn x I x I

Background term is negligible in the coincidence count regime

Background term is never negligible 1n

The entangled configuration, in the regime of coincidence counts, offers a better visibility of the information

112211221122 xIxIxIxIxIxI Imaging

informationno information,

background

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)