Self-calibrating Tomography for Angular Schmidt Modes in Spontaneous Parametric Down-Conversion

1.1 MB·English

by  Stanislav Straupe

Spatial Correlations and Angular Schmidt Modes in Spontaneous Parametric
Down-Conversion
S.S.Straupe∗ and D.P.Ivanov, A.A.Kalinkin, I.B.Bobrov, S.P.Kulik
Faculty of Physics, M.V.Lomonosov Moscow State University, 199001, Moscow, Russia
(Dated: December 19, 2011)
Wereportanexperimentalapproachtoentanglementcharacterization inhigh-dimensionalquan-
tumsystemsusingSchmidtdecompositiontechniques. Aparticularexperimentusesspatialdegrees
of freedom of biphotons. We present a technique to realize projective measurements in Schmidt
basis, allowing us to measure thecoefficients in Schmidt decomposition and to estimate the degree
of entanglement directly. Issues of modeling the spatial part of biphoton amplitude with simple
1 double-gaussian functions are discussed and shown to be in good agreement with experimental
1 results.
0
PACSnumbers: 03.67.Bg,03.67.Mn,42.65.Lm
2

c
e I. INTRODUCTION in SPDC, and the structure of Schmidt decomposition
D
revealed in our experiments.
The paper is organized as follows: in Section II we
6 Spatialentanglementinspontaneousparametricdown-
1 conversion(SPDC)wasasubjectofintenseresearchdur- briefly review the main features of SPDC angular spec-
trumwith emphasis onspatialentanglement,Section III
ing the last decade. Besides fundamental issues, spa-
] reminds the main ideas behind Schmidt decomposition
h tial states of biphoton pairs offer a platform for high-
analysis of spatial entanglement, explicit expressions for
p dimensio-nalquantumstatesengineeringmotivatingthis
- interest. One can distinguish two complementary ap- SchmidtmodesinbothHermite-GaussianandLaguerre-
t GaussianbasisarediscussedinSectionIVwithemphasis
n proaches to spatial qudit engineering with biphotons:
a one using ”pixel entanglement” and similar schemes [1– onHermite-Gaussiancase,SectionVgivesabriefreview
u ofexperimentalapproachestomodetransformationsand
4], and another one based on using high-order coher-
q some recent results on estimating the degree of spatial
ent (usually Laguerre-Gaus-sian) modes [5–16]. In both
[
entanglementand,finally, the detaileddescriptionofour
approaches achievable dimensionality and collection ef-

1 ficiency are figures of merit. Dimensionality of effective experiments with spatial Schmidt modes is given in Sec-
v tion VI.
Hilbert space is limited by degree of spatial entangle-
6
ment. In pixel entanglement schemes, for example, the
0
pixel size should be made smaller than the coherence ra-
8 II. SPDC ANGULAR SPECTRUM AND
3 dius of the pump in the far zone, and since the pump is
SPATIAL CORRELATIONS OF PHOTONS
. alwaysdivergent,evenaplainwave,selectedbypoint-like
2
aperturewouldbecorrelatedtoawholesetofplain-wave
1
BiphotonsgeneratedintheSPDCprocesshavecontin-
1 modesintheconjugatebeam. Thesameholdsingeneral
uousfrequencyandangularspectrum. Letusconsiderits
1 for other possible choices of modes. A natural question
structure in some details. SPDC can be phenomenolog-
: to ask is whether there exist a ”preferred” basis among
v ically described using the following effective interaction
the multitude of possible coherent spatial modes, which
i
X consists only of pairwise correlated modes and allows us Hamiltonian [19]:
r to fully exploit the spatial correlations in SPDC?
a Theansweris,ofcourse,positiveandwellknown–the H = d3~rχ(2)(~r)Ep(−)(~r)E(+)(~r))E(+)(~r)+H.c. (1)
decomposition into a set of Schmidt modes has all the Z
V
required properties. Since it was used for the first time
Here E is the classicalamplitude of the pump field and
by Law and Eberly [17], it has become a common tool p
E is the scattered field operator. Considering pump to
forentanglementanalysisofinfinite dimensionalsystems
bemonochromatic,thefirstorderofperturbationtheory
in general, and of spatial states of photons in particu-
gives the following expression for the state of the scat-
lar. Nevertheless,directexperimentalattempttoaddress
tered field:
spatial entanglement of SPDC biphotons in Schmidt ba-
sis was not reported until the recent work of authors
Ψ = vac + dk~ dk~ Ψ(k~ ,k~ ) 1 1 ,
[18]. Here we report a detailed description of experi- | i | i 1 2 1 2 | ik1| ik2
Z
mental techniques used for Schmidt modes filtering and (2)
detection, models used to describe spatial entanglement Ψ(k~1,k~2)= d3~rχ(2)(~r)Ep(−)(~r)exp i∆~~r ,
VZ h i
where ∆~ = k~ + k~ k~ , ω + ω = ω . In the case
1 2 p 1 2 p
−
∗Electronicaddress: [email protected] of collinear phase-matching and under wide crystal ap-
2
of scanned detector’s motion. One can also study two-
dimensional distributions by changing the scanning di-
rection. These distributions are related to the biphoton
amplitude in the following way:
dwuncond
= dk Ψ(k ,k )2
2⊥x 1⊥x 2⊥x
dk | |
1⊥x Z
(6)
dwcond Ψ(k ,k )2
1⊥x 2⊥
= | | .
dx1 dk1⊥x Ψ(k1⊥x,k2⊥x)2(cid:12)
Figure 1: A simple experiment to observe correlations in | | (cid:12)(cid:12)k2⊥x=const
SPDC angular spectrum. If the biphoRton amplitude may be (cid:12)factorized in two
(cid:12)
terms depending on the wavevectorsof each photon sep-
arately: Ψ(k~ ,k~ ) = Ψ (k~ ) Ψ(k~ ), i.e. if the
1⊥ 2⊥ 1 1⊥ 2⊥
×
proximation one can obtain the following biphoton field biphoton state is separable, both distributions coincide.
amplitude [20–22]: If this is not the case, and the state is entangled, the
width of conditional distribution ∆k(c) is less then the
1
Ψ(k~1,k~2)= p(k~1⊥+k~2⊥) (k~1⊥ k~2⊥), (3) width of unconditional one ∆k(s). It was proposed to
E F − 1
use the ratio of these quantities R = ∆k(s)/∆k(c) as a
where (k~ +k~ ) stands for angular spectrum of the k 1 1
p 1⊥ 2⊥ quantitativemeasureofspatialentanglement(whichwas
E
pump, and (k~1⊥ k~2⊥) is a geometrical factor deter- laterreferredtoasFedorov ratio)[24]. Experimentshave
F −
mined by phase-matching conditions. shown, that the value of R for SPDC biphoton states
k
Authorsof[21,22]givethefollowingexpressionfor : may reachvalues as large as R 100, correspondingto
k
F ∼
high degree of entanglement [1, 4].
L(k~ k~ )2
Ψ(k~ ,k~ )= (k~ +k~ )sinc 1⊥− 2⊥ , (4)
1 2 p 1⊥ 2⊥
NE " 4kp # III. ANALYZING SPATIAL ENTANGLEMENT
WITH SCHMIDT DECOMPOSITION
with L being the crystal length and – a normaliza-
N
tionconstant. Thisexpressionisvalidonlyinthecaseof
The most developedapproachto quantitative analysis
small pump divergence. A detailed analysis shows that
ofspatial (andfrequency) entanglementof SPDC bipho-
inthecaseofhighlydivergentpumponeshouldtakeinto
ton states is based on using coherent modes decomposi-
account the dependence of refractive index for extraor-
tion. Biphoton spatial state space is ”discretized” by
dinary waveon the propagationdirection. This question
switching from continuous distributions in plane-wave
was addressed in [4, 23], where it was shown that for
basis of the previous section to discrete distributions in
highly divergent pump experimental results are better
a chosen basis of spatial mode functions ξ (~k ). For
described by the following biphoton amplitude: i 1⊥,2⊥
an arbitrary choice of mode functions in the decomposi-
tionofspatialstateforeachofthephotons,thebiphoton
Ψ(~k ,~k ) E∗(~k +~k )
1⊥ 2⊥ ∝ p 1⊥ 2⊥ × amplitude takes the following form:
sinc L ω(~k +~k ) ∂np + (~k1⊥−~k2⊥)2 , ∞
(2 "c 1⊥ 2⊥ · ∂~kp⊥ 2kp #) Ψ(~k1⊥,~k2⊥)= Cijξi(~k1⊥)ξj(~k2⊥). (7)
(5) iX,j=0
It turns out, that by appropriate choice of the basis
where k and ∂np are taken at~k =0. mode functions one can transform the expression (7) to
p ∂~kp⊥ p⊥ a single-sum form
Spatial correlations in SPDC angular spectrum mani-
fest themselves in significant difference of single-particle ∞
and two-particle (conditional) distributions obtained in Ψ(~k1⊥,~k2⊥)= λiψi(~k1⊥)ψi(~k2⊥) (8)
experimental scheme of Fig. 1. i=0
Xp
The setup consists of two point-like detectors posi- which is called Schmidt decomposition. In this case
tioned in the far field and a coincidence circuit. One the basis functions ψ (~k ) should be eigenfunctions of
i 1,⊥
of the detectors is fixed, the other one may be scanned.
single-photon density matrix ρ (~k ,~k ), and λ are
The distribution of photocounts of the scanned detector 1,2 1⊥ 1⊥ i
the corresponding eigenvalues. It means that the ap-
corresponds to unconditional single-particle distribution
propriate mode functions may be found by solving the
dwuncond/dk , while the coincidence distribution in
1⊥x following integral equation:
such a scheme corresponds to a conditional distribution
dwcond/dk , where k is a transverse wavevector
1⊥x 1⊥x ρ (~k ,~k′ )ψ (~k′ )d~k′ =λ ψ (~k ). (9)
component of one of the photons along the direction 1,2 ⊥ ⊥ i ⊥ ⊥ i i ⊥
Z
3
It is straightforwardto note that for a factorizedstate A. Laguerre-Gaussian modes and orbital angular
onlyasinglecoefficientλ inthe Schmidtdecomposition momentum entanglement
0
is nonzero. Fora highly entangledstate, to the contrary,
λi only slowly decrease with i. Since the biphoton state Letusintroducepolarcoordinates k⊥,φ ontheplane
is pure, it is natural to use single-photon fon Neuman of transverse wave-vector compone{nts k~}. Laguerre-
⊥
entropy as a measure of entanglement [25]:
Gaussian modes are described by functions of the fol-
∞ lowing form:
E(Ψ )= λ log λ . (10)
| i − i 2 i k2 k2
i=1 LG (k~ ) L|l| ⊥ exp ⊥
X pl ⊥ ∝ p (∆k )2 −2(∆k )2 ×
Calculating entropy is a rather demanding task, requir- (cid:18) ⊥ (cid:19) (cid:18) ⊥ (cid:19) (14)
ing the knowledge of all eigenvalues in the Schmidt de- l
exp ilφ+i p | | π .
composition. Itismuchmoreconvenientto use”average − 2
(cid:18) (cid:18) (cid:19) (cid:19)
numberofSchmidtmodes”determinedbySchmidt num-
ber as an operational measure of entanglement[? ]: Here L|pl|(x) are associated Laguerre polynomials, ∆k⊥
is angular width of the mode. Radial index p corre-
1
K = . (11) sponds to number of zeros in radial direction and az-
∞ λ2
i=0 i imuthal index l – to the phase shift on the closed loop
DegreeofspatialentanglementofSPDCbiphotonsde- around k = 0 point (topological charge of the beam).
P ⊥
scribed by wavefunction (4) was analyzed by Law and One can show, that Laguerre-Gaussianbeams with non-
Eberly in [17]. The pump was assumed to be gaus- zerolpossessorbitalangularmomentum(OAM)[27,28].
sian = exp |k~1⊥+k~2⊥| . Authors derived an ana- This is also valid for single-photon beams: a photon in
Ep − σ2 Laguerre-Gaussianmode has OAM of l~.
lytical expressiohnfor Schmiidt number by approximating In Laguerre-Gaussian basis decomposition (7) takes
(k~1⊥ k~2⊥) function of (4) by a gaussian function: the following form:
F −
2
Kg = 41 bσ+ b1σ , (12) Ψ(k~1⊥,k~2⊥)= Cpl11lp22LGp1l1(k~1⊥)LGp2l2(k~2⊥).
(cid:18) (cid:19) lX1,p1lX2,p2
(15)
where b = L is the waist of the gaussian function
4kp It was shown in [29, 30] that OAM is conserved in the
modeling (qk~1⊥ k~2⊥). Inthe followingwewillcallthis SPDC process, i.e. in the subspace of fixed p1 and p2
F −
procedure”adoublegaussianapproximation”. Thevalue the identity l0 = l1+l2 holds, where l0 is azimuthal in-
of K is determined by a single parameter bσ and may dex of the pump beam.[48] This conservation law gives
g
reachveryhighvaluesforbσ 1andbσ 1. Numerical rise to OAM entanglement of photons in the same sense
≫ ≪
calculations performed for as transverse momentum conservation leads to spatial
correlations considered in Sect. II. In the subspace of
(k~1⊥ k~2⊥)=sinc b(k~1⊥ k~2⊥)2 (13) fixed radial indexes decomposition in terms of Laguerre-
F − −
h i Gaussian modes takes the form of Schmidt decomposi-
showedthat realvalue of K is largerthan K for all val-
g tion:
ues of bσ. Authors also proposed a method to increase
the degreeofentanglementbyspatialfiltering. Itturned Ψ(~k ,~k ) =
1⊥ 2⊥ p1p2
outthatthenumberofsignificantlynonzerotermsinthe Cl(l0−l)LG (k~ )LG (k~ ). (16)
Schmidt decomposition increases if one cuts off part of p1p2 p1l 1⊥ p2(l0−l) 2⊥
angular spectrum with small values of k⊥. Influence of Xl
spatial filtering on Schmidt number was further investi- Properties of such decomposition were studied in [33]
gatedin[26], whereauthorsstudy the effectofirisesand whereauthorsproposetouseitswidth(whichtheycalled
finite detection apertures. ”quantum spiral bandwidth”) to quantify the degree of
OAM entanglement. Quantum spiral bandwidth esti-
matestheeffectivedimensionalityofOAMHilbertspace,
IV. SPATIAL SCHMIDT MODES STRUCTURE butitisdefinedonlyforsubspacesoffixedradialindexes.
Togetridofdoublesumandtransform(16)toaSchmidt
The choice of mode functions set in (7) is quite ar- decomposition form one should use double-gaussian ap-
bitrary and may be determined by convenience in an- proximation.
alyzing a particular physical situation. If the situation
corresponds to a beam propagating in free space, it is
naturalto chose the solutions of paraxialwaveequation, B. Hermite-Gaussian modes as approximate
i.e. Hermite-Gaussianor Laguerre-Gaussianmodes, as a Schmidt modes for SPDC biphoton field
set of mode functions. Moreover it turns out, that these
functions form the Schmidt decomposition for SPDC Another convenient set of modes which is somewhat
with the gaussian pump of moderate divergence. less used in literature is the set of Hermite-Gaussian
4
modes:
HG (k ,k )
nm x y
∝
k2 k2 k2+k2 (17)
H x H y exp x y ,
n(cid:18)(∆kx)2(cid:19) m (∆ky)2! −2(∆k⊥)2!
where H (x) are Hermite polynomials, and k ,k are
n x y
{ }
transverse wave-vectorcomponents. This decomposition
was studied in [34]. Authors investigate dependence of
coefficients of the decomposition on modal structure of
the pump (which is also expressed in terms of Hermite-
Gaussian modes) and on the relative width of the gaus-
sian function in (17) and the pump mode. In this case Figure 2: Phase holograms transforming a Gaussian beam
the spatial parity of the beam is the conservedquantity: into Hermite-Gaussian HGnm (a) and Laguerre-Gaussian
parity of product HG (k ,k ) HG (k ,k ) LGpl modes (b). Color range corresponds to phase change
n1m1 1x 1y × n2m2 2x 2y from zero toπ.
equals to the parity of pump mode.
As well as for Laguerre-Gaussian modes, one can get
rid of two indexes in decomposition (17) and transform Degree of entanglement for this two-dimensional wave-
it to a form of Schmidt decomposition using double- packet is given by Schmidt number:
gaussianapproximation[17,35]. Letusconsiderthecase
of moderately divergent pump, when one may not con- K =K K =(a2+b2)2/4a2b2. (23)
x y
×
sider the linear term in (5).[49]For small σ we canmake
a substitution sinc(x2) exp( γx2), where γ is a co- Let us note once again, that (22) is not the only pos-
σ2 → − σ2 sible form of Schmidt decomposition. It may as well
efficient chosen to make both functions ”close” to each
be described in terms of Laguerre-Gaussianmodes as in
other. A good approximation is provided by choosing a
[17,36]. ThevalueofSchmidtnumberis,ofcourse,basis
value of γ = 0.86.[50] The biphoton wavefunction now
independent,aswasexplicitlyshowninarecentpreprint
takes the following form:
by Miatto et al. [37]. In fact, the difference between
(k~ +k~ )2 (k~ k~ )2 these two representations corresponds to the choice of
Ψ(k~1,k~2)∝exp(− 1⊥2a22⊥ )exp − 1⊥2−b22⊥ !, polar or cartesiancoordinates on the plane of transverse
momentum components.
(18)
where a determines the angularbandwidth of the pump,
and b = 4k /γL – the phase-matching bandwidth.
p V. EXPERIMENTAL METHODS OF SPATIAL
Since the wavefunctionisa productoffunctions depend-
MODES TRANSFORMATION
p
ing only on k and only on k , it is sufficient to
1,2x 1,2y
consider the problem in one dimension:
Thereareseveralwelldevelopedmethodstotransform
Ψ(k ,k )= a gaussian laser beam to higher Laguerre-Gaussian and
1x 2x
Hermite-Gaussian modes. The general idea is based on
2 (k +k )2 (k k )2 (19)
exp( 1x 2x )exp 1x− 2x . introducingaspatiallyvaryingphaseshiftcorresponding
πab − 2a2 − 2b2
r (cid:18) (cid:19) to the desired mode to a beam [38–40]. This is achieved
One can show, that solutions of (9) for such a wave- when a beam is diffracted on a phase mask or a holo-
function have the form [35]: gram, acquiring a desired phase structure. Examples of
phasehologramscorrespondingtoHermite-Gaussianand
1/4
2 2 Laguerre-Gaussian modes of lower orders are shown in
ψ (k )= φ k , (20)
n 1x,2x ab n ab 1x,2x Fig. 2. Usually these masks are combined with blazed
(cid:18) (cid:19) r !
diffractiongrating,”highlighting”thefirstdiffractionor-
where φ (x) = (2nn!√π)−1/2e−x2/2H (x). For corre- der, which allows to get rid of nondiffracted components
n n
sponding eigenvalues and Schmidt number we obtain: increasing the quality of mode transformation. Using
this type of hologramsallows effective transformationto
(a b)2n a2+b2
λ =4ab − , K = . (21) relatively high order modes [41].
n (a+b)2(n+1) x 2ab
Inmodernexperimentsphasehologramsareusuallyre-
So we havethe following formof Schmidt decomposition alizedusingactiveliquidcrystalspatiallightmodulators
for SPDC biphoton state under the double-gaussian ap- (SLM). Devices based on parallel-aligned nematic liquid
proximation: crystals (VAN) matrices on silicon substrate (LCoS) are
mostcommonlyusedformodetransformationtasks[42].
Ψ(k~ ,k~ )=
1 2 Every pixel of such matrix works as a birefringent phase
λ λ ψ (k )ψ (k ) ψ (k )ψ (k ). (22) plateintroducingaphasedelayforapolarizationcompo-
n m n 1x m 1y n 2x m 2y
× nent parallel to the crystals alignment plane. The delay
mn
Xp
5
SPDC PM L D basis were reported. The question of properly adjusting
SMF the experimental conditions to make the detected set of
modesasclosetoSchmidtdecompositionaspossiblewas
not addressed as well. These were the main concerns of
our work.
Figure 3: Experimental scheme realizing projective measure-
ments in spatial modes basis. PM - phase hologram, SMF -
VI. EXPERIMENTAL REALIZATION OF
single mode optical fiber, L - lens coupling gaussian mode of
MEASUREMENTS IN SCHMIDT BASIS
thefiber with gaussian mode of theentrance beam.
Our main goal was to demonstrate the possibility of
is determined by the orientation of the crystals and is performingdirectmeasurementsinspatialSchmidtbasis
controlledby voltageappliedto the pixel. Commercially and to experimentally demonstrate all features specific
available devices have pixels of 10µm size. to Schmidt decomposition. We have chosen to work in
Adapting the technique of mode transformations to Hermite-Gaussian basis. In this basis Schmidt decom-
the single photon level allows to realize both transfor- position is symmetric in indexes m and n, making it
mations and measurements in spatial modes basis. A convenient for characterizing complete two-dimensional
principalsketchofexperimentalsetupforprojectivemea- entanglement structure.
surementsisshowninFig.3. Thecrucialelementhereis First of all let us address the issue of applicability of
a single mode fiber guiding only the fundamental gaus- double-gaussianapproximationin our experimental con-
sian HG LG mode. This mode is coupled with a ditions. We used a 2 mm BBO crystal pumped by a
00 00
≡
lens to a gaussian part of incident light, thus realizing CW He-Cd laser with λ = 325 nm wavelength. The
p
a projection on gaussian mode. To realize higher order crystalwascutforcollinearfrequency-degenerateType-I
projections the hologram transforming a desired mode phase-matching. Angular bandwidth of phase-matching
to a gaussian one should be used. In such a setup only in such crystal (neglecting the pump divergence) – b pa-
the mode corresponding to a particular hologram passes rameter in (18) is b = 0.037. It was convenient for our
through the fiber and gives a click in the single photon purposes to select the value of pump divergence corre-
counter, thus realizing a projective measurement. sponding to a moderate Schmidt number. We focused
Asfarasweknow,thisschemewasrealizedforthefirst the pump inside the crystal with a 150 mm quartz lens
time at the single photon level in the work of Zeilinger’s and measured the divergence – a parameter in (18) to
group [5]. Authors have experimentally demonstrated be a=(5.8 0.1) 10−3.[51] We calculated eignevalues
± ×
OAM entanglement of SPDC photons. Following works andeigenfunctions for the reducedsingle-photondensity
of the same group showed the possibility to engineer matrix, corresponding to the precise SPDC wavefunc-
qutrit[8,9]andhigher dimensionalqudit[43] states and tion (5) numerically. The calculation was performed as
torealizequantumkeydistributionprotocolswithspatial follows: Hermite-Gaussian modes corresponding to the
encoding. approximate function (19) were chosen as a basis, we
Experimental preparation of photons with OAM and have restricted ourselves to 10 lower order modes (giv-
study of OAM-entanglement of photon pairs is a rather ing the Schmidt number with 3 decimal digits precision)
developed field with numerous experimental contribu- and calculated the matrix elements of the precise den-
tions [6, 7, 10–16] (see review [44] for a much more com- sity matrix in this basis. Diagonalizing the calculated
prehensive list of references). In context of our work it matrix,weobtainedeigenvaluesandeigenfunctions. The
is important to mention recent works of Boyd’s group results for lower order modes are shown in Fig. 4. One
[16, 45], where the quantum spiral bandwidth was di- can see reasonable correspondence, at least, we should
rectly measured. The same quantity was estimated with expect that phase holograms for Schmidt modes should
excellent precision using indirect measurements in [15]. be close to those of Hermite-Gaussian modes of appro-
Authors obtained the coefficients of decomposition in priate divergence. We should note that measured waist
Laguerre-Gaussianbasisby Fouriertransformingthe de- size[52] of the pump beam in the focal plane of the lens
pendenceofHong-Ou-Mandel-typeinterferencevisibility was w = (25 1)µm, corresponding to M2 = 1.4.
p
±
on azimuthal rotation of one of the interfering beams. That means, the pump beam is aberrated and is not re-
Quantum tomography of states with OAM was carried ally gaussian, and that may cause some deviations from
out in [46], states up to l . 30 were realized, but Hermite-Gaussian shape of Schmidt modes as well.
only a subspace spanned by l, l , l,l , l, l , l,l We have used an LCoS SLM with VAN matrix pro-
|− − i |− i | − i | i
was used for tomographical reconstruction. Same au- ducedbyCambridgeCorrelators. Thematrixhas1027
×
thors succeeded in preparing arbitrary states in l = 2 768 pixels of 10µm size each. It is an 8 bit device, ca-
±
subspace using superposition of phase holograms for pable of introducing a phase shift of up to 0.8π. Since
Laguerre-Gaussianand Hermite-Gaussian modes [13]. largerphaseshiftsarerequiredforourhologramsweused
Uptoourknowledge,noexperimentalattemptstoan- a double reflection scheme. We used two polymer film
alyzespatialentanglementinSPDCinHermite-Gaussian polarizers in front and after the SLM to reduce the un-
6
Figure 4: Numerically calculated Schmidt modes (blue) and
approximateHermite-Gaussianmodesofdouble-gaussianap-
proximation (magenta) and corresponding eigenvalues.
Figure6: Detectorcountingrateforanattenuatedlaserbeam
transformedtoHGnm modes. Thefibertipissettoposition,
corresponding to x=0 in Fig. 7.
transformations as the ratio of counting rates with holo-
gramsfor HG modes to that for untransformedgaus-
nm
Figure5: Experimentalsetupfortransforminganattenuated
sian mode: V = (R R )/(R + R ), than for
00 mn 00 mn
laser beam to higher Hermite-Gaussian modes. SMF - single −
almost all of the modes with 0 m,n 4 it exceeds
mode fiber; O1,2,3 - microscope objectives; L - 145 mm lens; ≤ ≤
97%, corresponding to high quality of mode transforma-
P - polarizers; D - single photon counter. A phase modula-
tions. ThevisibilityisslightlylowerforHG andHG
tor is shown as a transmitting mask, while in reality it is a 01 11
combination of SLM and a dielectric mirror M (see text for modes (still exceeding90%), whichwe explainby poorer
details). adjustment in vertical direction due to the design of op-
tomechanical components used. Histogram of counting
rates for various modes is shown in Fig. 6.
To check whether the spatial structure of transformed
wanted polarization rotations by an additional dielectric
modes is really close to Hermite-Gaussian, we scanned
mirror necessary in such scheme (see insets in Fig. 5,9).
the fiber tip in the focal plane of O3 objective. The
To check the quality ofmode transformationwith this
countingratedependence onfiberpositionisdetermined
device we used the setup sketched in Fig. 5. We used an
by the convolutionof a correspondingHermite-Gaussian
attenuated650nmdiodelaserasasource. Thebeamwas
function and a fundamental gaussian mode of the fiber:
mode filtered with single mode fiber and focused with a
20 microscope objective to obtain the divergence simi-
R(x)
×
lar to that of an HG Schmidt mode in Fig. 4, and the ∝
waistatthepositiono00fthecrystal. Soweobtainedasin- ∞ x˜2 (x x˜)2 2
H (√2x˜/w)exp exp − dx˜ ,
glemodegaussianbeammodelingthezeroorderSchmidt nm −w2 − w2
(cid:12)Z−∞ (cid:18) (cid:19) (cid:18) (cid:19) (cid:12)
mode of SPDC beam. The beam was collimated with (cid:12) (2(cid:12)4)
(cid:12) (cid:12)
145 mm lens and after reflection from SLM was focused (cid:12) (cid:12)
with a 8 microscope objective to a single mode fiber where w is the gaussian mode waist. Experimental de-
×
followedby asinglephotoncounter(PerkinElmer). The pendencies are shown in Fig. 7 and have a character-
beam waist exactly coincided with the mode size of the istic shape of double-peak curves. Distance between
fiber ( 4µm). We should stress, that we paid special at- maxima depends on the mode number, and is shown
tentiontomodematchingandopticswerechoseninsuch in Fig. 8 for ”horizontal” HG and ”vertical” HG
n0 0m
a way that detection mode exactly coincides with calcu- modes,togetherwiththeoreticalpredictionsforHermite-
lated HG Schmidt mode. Parameters of phase holo- Gaussianmodes. To plotthe theoreticalpredictions cor-
00
grams were adjusted to minimize the detector counting rectly we estimated the waist size by fitting the convo-
rate, i.e. to ensure orthogonality of transformed modes lution for HG mode with a gaussian curve, obtaining
00
to a fundamental gaussian one. We have actually ad- w =(3.87 0.07)µm.
±
justed three parameters: the position of phase step for When the attenuated laser beam is substituted with
HG modes, which is determined by the beam po- SPDCradiation,thedescribedschemerealizesprojective
10(01)
sition at the SLM (in horizontal and vertical directions, measurements in Hermite-Gaussian basis. Full scheme
respectively), and the distance between phase steps for of experimental setup is shown in Fig. 9. Pump was fo-
HG modesdeterminedbybeamsizeattheSLMplane. cused to a 2 mm BBO crystal with a 150 mm quartz
20
Theseparametersdefinetheshapeofhologramsforother lens L1, a second lens L2 with F = 145 mm was set
modesinaunique way. Ifwedefine”visibility”formode confocal with L1 to collimate the beam. Pump radi-
7
Figure 7: Counting rate dependence on the position of fiber Figure 9: Experimental setup. L1 – 150 mm quartz lens; L2
tip for an attenuated laser beam transformed to various –145 mmlens; BBO -2mm BBO crystalplaced inthejoint
modes. Fiber is scanned in horizontal direction (1) and in focus ofL1and L2;UVM–UVmirror cuttingoffthepump;
vertical one (2). IF–interferencefilterfor650nmwith40nmbandwidth;BS
– non-polarizing 50/50 beam-splitter; 01,2 – 8× microscope
objectives; PM – spatial light modulator (is shown as trans-
mitting mask for simplicity, real alignment is shown on the
inset); PM2 – phase mask made of thin glass plates; SMF -
single mode fiber; SMF/MMF – single or multi-mode fiber
depending on the experiment (see text for details); D1,2 –
singlephotoncounters(PerkinElmer). A200µmverticalslit
S was used in ”ghost” imaging experiments.
Schmidt modes are really close to Hermite-Gaussian
ones, we expect coincidences to appear only when simi-
lar modes are selected in both channels of the setup. If
different modes are selected in the transmitted and the
reflected channels, no coincidences should appear, since
Figure 8: Dependence of maxima positions for fiber tip scan
only terms with equal indexes are present in (8). This
on mode number for laser beam transformations. ”Horizon-
fact may be used as an experimental criteria of how well
tal”HGn0(bluebars),”vertical”HG0m(redbars)modesand Hermite-Gaussian modes detected in our setup approxi-
theoretical predictions (solid black line and dots). Theoreti-
mate precise spatialSchmidt modes. Fig. 10 (1) shows a
cal predictions are calculated for gaussian waist a = 3.9µm
(see text for details). histogramofcoincidencecountingrateforthe casewhen
HG with 0 m,n 4 modes were consequently se-
mn
≤ ≤
lected in the transmitted channel and HG mode was
00
selected in the reflected channel. We obtained values of
ation was cut off with a UV-mirror UVM, and SPDC
visibility V =(R R )/(R +R ), where R is
radiation was frequency filtered with an interference fil- 00 mn 00 mn mn
−
a counting rate for HG mode, over 94% for all modes
ter IF. We used filters with central wavelength of 650 mn
except HG and HG modes. Comparing the results
nm and bandwidth of 40 nm and 10 nm and did not 01 11
to those of Fig. 6, we conclude that non-100% visibility
observeany significantimprovementof visibility for nar-
for these modes is rather a result of technical imperfec-
rower filter. All the following results were obtained with
tions, common for schemes with attenuated laser beam
awide40nmfilter. Photonpairsweresplitwitha50/50
and SPDC, than some physical discrepancy between de-
non-polarizingbeam-splitter. An SLMwas placedinthe
tected modes and Schmidt modes. So we can state, that
transmittedchannelandafterreflectiontheradiationwas
our scheme indeed realizes projective measurements in
focused into single mode fiber placed in the focal plane
Schmidt basis.
of 8 microscope objective. In the reflected channel the
×
beam was focused into the same single mode fiber or, As is expected, when a phase mask corresponding to
alternatively, in a multi-mode fiber with core diameter HG10 mode is inserted in the second channel, maximal
of 50µm. Multi-mode fiber allowed all spatial modes to coincidencerateappearsforthesamemode inthe trans-
be detected simultaneously. To transform modes in the mitted channel, which is clearly seen as a peak shift on
reflected channel we used phase masks made of thin mi- the histogramof Fig. 10 (2). We believe lower quality of
croscope cover glass plates. This construction was used phasemaskusedinthe reflectedchanneltobe theorigin
instead of the second SLM, although it has limited per- of somewhat lower visibility in this case.
formanceforbroadbandSPDCradiationandisonlysuit- Further evidence of similarity of Schmidt modes to
able for modes with n,m 1. Signals in both channels Hermite-Gaussian ones may be obtained by analyzing
≤
were detected by single photon counters connected to a thedependenciesofsinglecountsandcoincidencesonthe
coincidence circuit. fiber tip position in the focal plane of the focusing mi-
If the double-gaussian approximation is valid and croscopeobjective. Weexpectthedependenceforcoinci-
8
Figure 10: Coincidence countingrate for SPDCradiation for
various masks in the channel with SLM. (1) – HG00 mode is Figure12: Coincidencecountingratedependenceonthefiber
selectedinthereflectedchannel,(2)–HG01 isselectedinthe tippositioninthereflectedchannel(withoutSLM)fordiffer-
reflected channel. ent modes. Fiber tip is scanned in horizontal direction.
m)12
a (
m10
axi
m 8
n
e
we 6
et
e b 4
nc
dista 2
0
1 2 3 4
Figure 11: Coincidence (1) and single counts (2) rate depen- mode number, m
denceonthefibertippositioninthefocalplaneofthemicro-
scopeobjective in thechannelwith SLMfordifferentmodes. Figure 13: Dependence of maxima positions for coincidence
Fiber is scanned in horizontal direction. distributionsofFig. 11 onthemodenumber(redbars). Cal-
culated dependence for Hermite-Gaussian modes with w =
(3.0±0.1)µm corresponding to HG00 waist size (grey bars)
dences to be described by (24). Experimental curves for is provided for comparison.
thecasewhenfiberinthetransmittedchannelisscanned
are shown in Fig. 11. Note that the double-peak struc-
ture characteristic for Hermite-Gaussian modes appears reflected channel with a multi-mode one, which serves
only in coincidences dependence, while single counts be- as a ”bucket” detector, collecting all spatial modes. A
have monoto-nously, as is expected for spatially multi- 200µm vertical slit was inserted in front of the objec-
mode radiation. The maximal value of single counting tive in the reflected channel. Dependence of coincidence
rate, however, decreases with increasing value of mode counting rate on the slit position forms a ”ghost im-
indexes. Wewillpayspecialattentiontothisdependence age”ofthe mode selectedinthe conjugate (transmitted)
later. Distance between maxima behaves analogously to channel. Experimental results for the first three modes
the case of attenuated laser beam, as shown in Fig. 13. HG ,HG ,HG areshowninFig.14. Solidcurvesare
00 10 20
We obtained same dependencies of coincidence count- fit with Hermite-Gaussian functions[53].
ing rate when the fiber tip was scanned in the reflected
channel (see Fig. 12). In this case single counts, obvi-
ously, do not depend on the mode selected in the conju- (a) (b) (c)
. . .
gate channel at all. This effect is a straightforward con-
. . .
sequence of intermodal correlations in SPDC and may
. .
.
be thought of as a sort of ”ghost interference” [47]. We . .
shouldnotethatalmostzerocoincidencecountingratein . . .
. . .
the central position of the fiber is an interference effect
demonstrating spatial coherence of detected modes. So
thisresultmaybe consideredasanexperimentaldemon- Figure 14: Coincidence counting rate dependence on the
stration of one of the main features of Schmidt modes – slit position with multi-mode fiber in the reflected channel.
their spatial coherence. ”Ghost images” of the modes selected in the transmitted
One can also observe the shape of Schmidt modes di- channel are observed. Results for first three modes: HG00
rectly. For this purpose we substituted the fiber in the – (a), HG10 – (b), and HG20 – (c) are shown.
9
Figure 15: Distribution of eigenvalues in Schmidt basis. (1)
SinglecountsrateofdetectorD1inthechannelwithSLMde-
pendenceontheselectedmode. (2)Calculatedeigenvaluesof
single-photon density matrix in Hermite-Gaussian basis (i.e. Figure16: Normalized distributionof eigenvaluesinSchmidt
in a doublegaussian approximation). basis. Experimental results (red) and exponentially decreas-
ing calculated values under double-gaussian approximation
(grey). Both distributions are normalized to unit Schmidt
Aswenotedabove,singlecountingrateinthe channel number(i.e. divided byP λ2).
i i
withSLMforthecentralpositionofthesinglemodefiber
decreasesforhigherorderdetectedmodes. Indeedsstate
of the single photon of a pair is described by a reduced
densitymatrix,whichhasthe followingforminthe basis
of Schmidt modes ψ (k~): Using experimental results for HG modes we estimate
nm i 0m
the ”horizontal” Schmidt number to be K = 3.1 0.9.
ρ(k~,k~′ )= λ ψ (k~)ψ (k~′ ), (25) x ±
i i nm nm i nm i Analogous procedure for ”vertical” modes HG0n gives
n,m K =2.7 0.5. Both values are in agreement with K =
X y
±
where index i =1,2 numbers photons. Counting rate of 2.97 calculated from (21) for experimental values of a =
detector D1: R ψ ρ ψ λ is propor- 3.5 10−3 and b = 0.020 (values are taken inside the
tional to the eig1emnvnal∼ueh–m”wn|eig|htm”noif∼the cmonrresponding crys×tal).
mode in the Schmidt decomposition. Analyzing single
counts rate in the channel with SLM we can determine Experimental dependencies of eigenvalues are in good
the eigenvalues in Schmidt decomposition. Histogramof qualitativecorrespondencebothwithexponentialbehav-
single counting rate for detector D1 is shown in Fig. 15 ior for double-gaussian approximation and with the re-
(1). It should be compared to the calculated eigenval- sultsofnumericalcalculations. Smallqualitativediscrep-
ues ofsingle-photondensity matrix in Hermite-Gaussian ancies are still above the level of statistical fluctuations.
basis shown on Fig. 15 (2). We can use fidelity: They can be partially explained by systematic fluctua-
tions of signal level on long time scales caused primarily
F =Tr √ρρ(exp)√ρ= λ λ(exp), (26) by temperature fluctuations. Experiments have shown,
mn mn
that both coincidence and single counts rates are ex-
q Xm,nq
tremely sensitive to small beam displacements. So work
asaquantitativemeasureofcorrespondencebetweenthe
of an air-conditioning system stabilizing the room tem-
measuredandcalculateddistributions. Experimentales-
perature cau-sed smallbut noticeable periodic change of
timate foreigenvaluesλ(mexnp) =(R1mn−R0)/ mnR1mn signal with a period of 10 min. Error bars shown on
takes into account normalization, implying unit trace Fig. 16 correspond to the amplitude of this fluctuations
P
for the density matrix, and substitution of a constant since the exposition time was comparable with their pe-
noise level R0, originatedboth from stray light and non- riod. Complete thermal isolation of the setup should re-
diffractedSPDCradiation. Experimentaldatagivevalue move this source of errors. Another significant source
of fidelity F =0.92 0.03. of errors may be the difference between detected angu-
±
Using directly measured eigenvalues we may estimate larapertureandfull angularbandwidthofSPDC.Sharp
theSchmidtnumber. Letusanalyzetheone-dimensional dependence of counting rates on the position of fiber tip
section of eigenvalues distribution shown in Fig. 16 for relatively to the microscope objective shows maximum
HGm0 modes. This distribution is excellently approxi- in the position slightly different from the focal plane.
mated with a gaussian one: R1m0 = R0 +Cexp( αn) Moreover, the pump, assumed to be gaussian, was ac-
−
(correlation coefficient R2 = 0.993). Using expression tually aberrated to M2 = 1.4, which may also slightly
(21) for the Schmidt number it follows that λn+1/λn = changetherealdistribution. Nevertheless,webelieveour
(Kx 1)/(Kx +1), and it is straightforward to obtain results to be clear evidence of possibility to realize pro-
−
the following estimate for the Schmidt number: jective measurements in spatial Schmidt modes basis for
eα+1 SPDC biphotons, and the implemented scheme is useful
K = . (27)
x eα 1 for quantitative measurements and state reconstruction.
−
10
VII. CONCLUSION properly. One can both obtain single-mode biphotons
and highly entangled EPR-like states as extreme cases.
We have analyzed spatial entanglement in SPDC in However, intermediate case, considered in our work also
terms of spatial Schmidt decomposition and shown that offers interesting possibilities. We foresee at least two
under reasonable assumptions, applicable to the par- lines of researchin this area: quantum tomography with
ticular experiment, these modes are close to Hermite- high-dimensional spatial states and quantum state engi-
Gaussianmodes. Using this fact we have experimentally neering of non-trivial states. Schmidt modes measure-
realizedaschemeofprojectivemeasurementsinSchmidt ment techniques developed here should be a useful tool
basisusinganactivespatiallightmodulator. Experimen- for both tasks.
tal results prove high quality of gaussian beam transfor-
Another interesting question is the behavior of
mation to higher order Hermite-Gaussian modes. For a
Schmidt-like intermodal correlations in the classical
spatialmulti-mode SPDC radiationsuchspatialfiltering
limit. What would stand for the Schmidt number, and
allowedustorealizeprojectivemeasurementsinSchmidt
whetherthemodalstructuremaybedescribedintermsof
basis. We have experimentally verified similarity of spa-
somesimpledecompositionslikeinthecaseofSPDC,are
tialSchmidtmodesforSPDCtoHermite-Gaussianones.
interestingissuestoaddressboththeoreticallyandexper-
We have performed direct measurement of eigenvalues
imentally. Ourresultsforquasi-thermallightsourceswill
in Schmidt decomposition and experimentally estimated
be reported elsewhere.
theSchmidtnumber,obtaininggoodagreementwithpre-
dictionsofasimpledouble-gaussianmodelofSPDCspec- We are grateful to M. V. Fedorov for stimulating dis-
trum. cussions. This work has been founded by RFBR grants
Spatialmodesareinterestingobjectsforquantumstate 10-02-00204and12-02-00288. S.S.StraupeandI.B.Bo-
engineering. The Hilbert space is vast and its dimen- brov are grateful to the ”Dynasty” foundation for finan-
sion may be chosen at will by simply focusing the pump cial support.
[1] J. C. Howell, et. al., Phys.Rev.Lett.92, (2004) 210403. [22] S. P. Walborn, A. N. de Oliveira, and C. H. Monken,
[2] M. N. OSullivan-Hale et al., Phys.Rev.Lett. 94, (2005) Phys. Rev.Lett. 90, (2003) 143601.
220501. [23] M. V.Fedorov, et al., Phys.Rev.A., 77, (2008) 032336.
[3] L. Neveset al., Phys.Rev.Lett.94, (2005) 100501. [24] M. V.Fedorov, et al., Phys.Rev.A., 69, (2004) 052117.
[4] M.V.Fedorovet al., Phys.Rev.Lett.99,(2007) 063901. [25] S. Parker, S. Bose and M. B. Plenio, Phys. Rev. A., 61,
[5] A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, Nature, (2000) 032305.
412, (2001) 313. [26] M.P.vanExter,etal.,Phys.Rev.A.,74,(2006)012309.
[6] S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi and [27] L. Allen, M. W. Beijersbergen, R. J. Spreeuw and
C. H. Monken,Phys. Rev.A., 69, (2004) 023811. J. P. Woerdman, Phys.Rev.A, 45, (1992) 8185.
[7] J. Leach et al.,Phys. Rev.Lett., 88, (2002) 257901. [28] L.Allen,M.J. Padgettand M.Babiker,Prog. Opt.,39,
[8] A.Vaziri, G.WeihsandA.Zeilinger, Journal ofModern (1999), 291.
Optics, 89, (2002) 240401. [29] S. Franke-Arnold, S. M. Barnett, M. J. Padgett and
[9] A.Vaziri et al., Phys.Rev.Lett.,91, (2003) 227902. L. Allen, Phys. Rev.A, 65, (2002) 033823.
[10] G. Molina-Terriza, J. P. Torres and L. Torner, [30] H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett., 85,
Phys.Rev.Lett.,88, (2002) 013601. (2000) 286.
[11] N. K. Langford et al., Phys. Rev. Lett., 93, (2004) [31] G. A.Barbosa, Phys.Rev.A., 80, (2009) 063833.
053601. [32] Sheng Feng and Prem Kumar, Phys. Rev. Lett., 101,
[12] S.S.R.Oemrawsinghetal.,Phys.Rev.Lett.,95,(2005) (2008) 163602.
240501. [33] J. P.Torres, A.Alexandrescuand L.Torner, Phys.Rev.
[13] B. Jack et al.,Phys. Rev.A., 81, (2010) 043844. A., 68, (2003) 050301(R).
[14] W. H. Peeters, E. J. K. Verstegen and M. P. van Exter, [34] S. P. Walborn, S. Padua and C. H. Monken, Phys. Rev.
Phys.Rev.A., 76, (2007) 042302. A., 71, (2005) 053812.
[15] H.DiLorenzoPires,H.C.B.FlorijnandM.P.vanExter, [35] M. V. Fedorov, Yu. M. Mikhailova and P. A. Volkov,
Phys.Rev.Lett., 104, (2010) 020505. J.Phys.B:At.Mol.Opt.Phys., 42, (2009) 175503.
[16] Anand Kumar Jha et al., Phys. Rev. Lett., 104, (2010) [36] G.F.Calvo,A.PiconandA.Bramon,Phys.Rev.A,75,
010501. (2007) 012319.
[17] C.K.Law,andJ.H.Eberly, Phys.Rev.Lett.,92,(2004) [37] F. M. Miatto, T. Brougham and A. M. Yao,
127903. arXiv:1111.6449 [quant-ph],(2011).
[18] S.S.Straupeet al.,Phys.Rev.A,83,(2011)060302(R). [38] M.W.Beijersbergen,R.P.C.Coerwinkel,M.Kristensen
[19] D.N.Klyshko. Photons and Nonlinear Optics, (Gordon and J. P. Woerdman, Opt.Commun., 112, (1994) 321.
and Breach, London,1988). [39] G. A.Turnbull,et al.,Opt.Commun., 127, (1996) 183.
[20] A.V. Burlakov et al., Phys. Rev.A 56, (1997) 3214. [40] N. R. Heckenberg, et al., Opt. Quant. Electron., 24,
[21] C. H. Monken, P. H. S. Ribeiro, and S. Padua, Phys. (1992) S951.
Rev.A.57, (1998) 3123. [41] J.Arlt,K.Dholakia,L.AllenandM.J.Padgett,J.Mod.