X-entanglement of PDC photon pairs

Italian Quantum Information Science Conference 2008Camerino, Ducal Palace 24th-29th October 2008

Alessandra Gatti

Lucia Caspani

Enrico Brambilla

Luigi Lugiato

Ottavia Jedrkiewicz

INFM-CNR-CNISM, Università dell’Insubria, Como, Italy

Introduction

Usually, bi-photon entanglement in PDC explored only from points of view either purely

temporal or purely spatial.

Non factorability in space and time of the spatio-temporal structure of PDC bi-

photon entanglementInvestigation aimed at disclosing its X-shaped geometry,

non-separable in space and time.

Key elements of novelty: possibility of tailoring the spectral bandwidth of bi-

photons by acting on spatial degrees of freedom, access to an ultra-broad

bandwidth source of entangled photons

Nonlinear X-waves

Nonlinear X-wave schematic view

A Nonlinear X-wave (NLX) is a multi-dimensional wave-packet which can travel without distortionin dispersive media.The distinctive feature of an NLX is its "biconical" shape, (see figure) which appears as an "X" in any section plain containing the wave peak and the direction of propagation.

A Nonlinear X-wave does exist in the presence of nonlinearity, and in many cases it self-generates from a gaussian (in any direction) wave-packetSo far, Nonlinear X-waves have been predicted in a variety of nonlinear media including Bose-Einstein condensates.

x

time

y

C. Conti et al. Nonlinear electromagnetic X-waves,

Phys. Rev. Lett. 90 , 170406 (2003)

C. Conti and S. Trillo, Nonspreading Wave Packets in Three

Dimensions Formed by an Ultracold Bose Gas in an Optical

Lattice, Phys. Rev. Lett. 92 , 120404 (2004)

diameter~ 10 µm

FWHM ~ 20 fs

diameter~45 µm

FWHM 100-200 fs

X-wave description adopted to describe the spatio-temporal structure of bi-photon

entanglement

Bi-photon amplitude: ),(),(),;,( 2221112211 txatxatxtxrrrr

=ψ

PDC crystal

BS (type I) or

PBS (type II)Images of crystal

output face

),( signal 111 txav

),(idler 222 txar

222112221112221112211

)2( ),;,(),(),(),(),(),;,( txtxtxItxItxItxItxtxGrrrrrrrr

ψδ ≈−=

Low-gain regime: coincidence rate

∝ Probability density of finding an idler photon at position provided a signal

photon was detected at position

timeand 22 txr

11 timeand txr

Near-field

position

Outline

•Example 1- Type I PDC

-Model for PDC and approximations

-The bi-photon amplitude: numerical and analytical results

-Temporal and spatial localization, tailoring the spectral

bandwidth of bi-photons

•Conclusions

Modelling type I parametric down-conversion (any gain regime)

Starting point: propagation equation along the crystal for the signal envelope operator :

PUMP

χ(2) emission cones

0 lc

z

z

kv

qr

zkv

( ) ( ) ( ) ( )zqqieqaqqAdqdqa

dz

dp

ωωωωωωχω

′′∆−′′+′+′′= +

∫ ∫,;,

,ˆ,~

,ˆ '

rrrrrrr

Phase mismatch function: includes

the effects of linear propagation:

temporal and spatial walk-off,

diffraction, dispersion…

q+q’, ω +ω’

q’, ω’

q, ω

pump

( ) ( ) ( ) ( )ωωωωωω ′+′+−′′+=′′∆ ,,,,;, qqkqkqkqq pzzz

rrrrrr

spatio-temporal Fourier transform of a classical, undepleted, pump field

(parametric regime)

a

),(~

ωqAp

r

ωp

ωω

−2

p

ωω

+2

p

1 :regimegain -Low <<= cplAg χ

Perturbative expansion of the solution. At first order in g, the biphoton amplitude in the Fourier

domain (transverse wave-vector, frequency) reads :

Limit of a nearly plane-wave pump:

For a broad enough pump (in practice pump diameter > pump-signal spatial walk-off along the crystal

~300 µm, pump duration > pump-signal temporal delay ~ picosecond),

( ) 2),;,(

e 2

,;,sinc),,(~

),(),(),;,(

clqqic

cp

lqqlqqAg

qaqaqq

ωωωωωω

ωωωωψ

′′∆+

′′∆′+′+=

′′=′′rr

rrrr

rrrr

( ) ( ) ( ) mismatch phase pump-wave-plane 0,0,,

22),;,(),;,(

pzzz kQkQk

qqQQQqq

−Ω−−+Ω=

′−=Ω

′−=Ω−Ω−∆≈′′∆

vv

rrrrrrr ωωωω

1 :regimegain -Low <<= cplAg χ

The biphoton amplitude in the spatio-temporal domain reads:

),(2

,2

),(),(),;,( 12122121

22112211 ttxxttxx

gAtxatxatxtx PWp −−×

++==

rrrr

rrrrψψ

→As a function of the mean spatio-temporal

point: pump profile

→As a function of position shift, time delay it is the plane-wave

pump bi-photon amplitude

Spatio-temporal Fourier transform of gain function V, strongly

peaked around the phase matching curves where .

e 2

),;,(sinc),(

),(ee22

),(

2),;,(

)(i)(i-2

12121212

Ω−Ω−∆=Ω

ΩΩ

=−−

Ω−Ω−∆+

−⋅−Ω∫∫

clQQic

xxQttPW

lQQQV

QVdQd

ttxx

rr

rrr

rrr

rrr

ππψ

0),( =Ω∆ Qr

Frequency filter

centered around

degeneracy

4 mm type I BBO crystal, cut for collinear phase matching at λ=704 nm.

Phase matching curves (angular dispersion relations)

“Exact” calculation, includes any order of Q,Ω: ( ) ( ) z alongvector of projection2,, k-QQkQk iiiz −Ω+=Ω ω

vv

( )c

ΩωΩ,ωQn iii )( ++r

Material Sellmaier

relations

Phase matching curves in

the (λ, angle) domain

Frequency filter

λ=550 →950nm

Biphoton amplitude

4 mm type I BBO crystal, cut for collinear phase matching at λ=704 nm. Low gain (g=10-3)

Biphoton amplitude as a function of time delay t=t2-t1, position shift x=x2 –x1),;,( 2211 txtxψ

Non-trivial geometrical structure of the spatio-temporal correlation:

microscopic counter-part of the macroscopic X-waves

)0,0(/),( ΨΨ tx

Type I BBO: X-entanglement at various crystal angles

θp=33.30

θp=33.40

θp=33.50

cc l

kq

lk=

′′=Ω 0

''0 ,

1

Toy model

Expand quadratically the phase mismatch function around Q=0, Ω=0

and extend the approximation to the entire (Q, Ω) domain → the biphoton amplitude can be

analytically calculated as:

→constant on the hyperboloids

Evidences the X-geometry, but diverges as

k

QkkQkQkQQ pzz

2),(),(),;,(

22

0 −Ω′′+∆≈−Ω−−+Ω=Ω−−Ω∆rrrr

( )si

qxts

i

ees

dstx 0

20

220

2

41

0 2/3),(

∆−Ω−

∫∝Ψr

.20

220

2 const=Ω− tqx

cc l

kq

lk=

′′=Ω 00 ,

1

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

0

2

4

6

8

10

q0x

Ω0t

diffraction

temporal dispersion

kktx

′′=

1

)asymptotes (on the0for1 2

022

02

20

220

2→Ω−

Ω−∝ tqx

tqx

Biphoton localization

-40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

t (fs)

τFWHM

=4.4fs

|ψ(x

=0,t

)/ψ

(0,0

)|2

-40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

|ψ(x

,t=

0)/

ψ(0

,0)|

2

x (µm)

xFWHM

=2.9µm

Unusually small width of the correlation peak both in space (few microns)

and time (few fs): extreme relative localization of photon pairs

Ultra-short localization of photon pairs that

emerges spontaneously from a nearly plane-wave

pump due to the ultra-broad natural band of PDC

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Ω (1015 Hz)

~1.5 1014 Hz ~1.5 1013 Hz

Typical far-field coincidence detection: correlation time of photon pairs ~ 100 fs

determined by the inverse of bandwidth of phase matching at a fixed angle, as in the

original HOM experiment

Near-field coincidence detection: correlation time of photon pairs ~ few fs

determined by the inverse of full bandwidth of phase matching (the optical pump

frequency) or more practically by the bandwidth intercepted in the measurement

1.4 1015 Hz

ωp ~5 1015 Hz

The few fs temporal localization of twin photons definitely relies on the ability

to resolve their near-field positions

-40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

t (fs)

|ψ(x

=0,t

)/ψ

(0,0

)|2

τFWHM=96 fs

τFWHM= 4.4 fs

---- collecting the photons from the whole cross-section

without discriminating their positions

→ integrated coincidence rate

Temporal localization ~100 fs as in far-field detection

2222 ),(),( tqqdtxxdrr

Ψ=Ψ ∫∫

__ resolving the near-field positions of photons with

small detectors located at the same position

→

Temporal localization ~ few femtoseconds

22),(),( xqietqqdtx

rrrrr ⋅Ψ=Ψ ∫

→X-structure of entanglement: key to access the ultra-broad bandwidth of PDC

Non-factorability in space time: possibility of tailoring the temporal bandwidth of

bi-photons by acting on the spatial degrees of freedom.

Diaphragm:

cut the angular

spectrum at

angle αmax

Spatial filtering

PDC

f fff Detection

plane

αmax=40 αmax=80

αmax

Effect of spatial filtering on the temporal peak of photon pair correlation

-40 -30 -20 -10 0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

no spatial filter, τFWHM

=4.4fs

amax

=40, τ

FWHM=10.5fs

amax

=20, τ

FWHM=20.6fs

|ψ(x

=0,t

)/ψ

(0,0

)|2

t (fs)

X-entanglement: microscopic counterpart of macroscopic nonlinear X-

wavesInvestigations aimed at disclosing the genuine X-shaped geometry, non-separable in

space and time, of the bi-photon entangled state.

Key elements of novelty

a) Extreme relative localization of photon pairs in space and time

b) Non-factorability of the state in space and time

Strong localisation in space (few microns) and time (few fs), which is not present in typical

far-fields measurements that select a small angular bandwidth, nor in a “bucket” detection

New perspectives regarding the bi-photon localization in space and time, novel

applications to high precision measurements in the temporal domain (QOCT, clock

synchronisation), or in the spatial domain (precise position measurements)

Opens the possibility of tailoring the temporal bandwidth of single-photons/photon-pairs

by acting on the spatial degrees of freedom. Gives access to an ultra-broad bandwidth

photonic source

Conclusions

“This project aims at a breakthrough in the information capacity of quantum communication,

by exploiting the intrinsic multivariate and multi-modal character of the radiation field,

which involves spatial, temporal and polarization degrees of freedom. The long term vision

underlying our project is that of a broadband quantum communication, where all the

physical properties of the photons are utilized to store information”

FP7 Fet Open project of the European Community

HIDEAS

High Dimensional Entangled Systems

Start date: October 1, 2008

Coordinated by University of Insubria:

-L.Lugiato (admistrative coordinator)

-A.Gatti (scientific coordinator)

UPMC (C.Fabre, E. Giacobino)

Strathclyde University (S. Barnett, M. Padgett)

Lille University (M. Kolobov)

Leiden University (H. Woerdman)

Copenhagen University (E. Polzik)

St. Petersbug University (I. Sokolov)

Camberra University (H. Bachor)

Multimode quantum memories

for light

Sources of broadband photon

pairs: X-entanglement

Orbital angular momentum

entanglement of photon pairs

for q.cryptografy

Quantum broadband

light: quantum frequency

combs

Quantum spatial correlation of

light beams for parallel

quantum communication

The nodes of HIDEAS network

Hope to be able to show some interesting

experimental results for the next IQIS..

THANK YOU

lc=0.5mmlc=0.1mm lc=1mm lc=4mm

Development of the X-structure along the crystal length