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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