Increasing entanglement between Gaussian states by coherent photon subtraction

0.35 MB·English

by  Alexei Ourjoumtsev

Increasing entanglement between Gaussian states by coherent photon subtraction
Alexei Ourjoumtsev, Aur´elien Dantan, Rosa Tualle-Brouri, and Philippe Grangier
Laboratoire Charles Fabry de l’Institut d’Optique, CNRS UMR 8501, 91403 Orsay, France∗
(Dated: February 1, 2008)
We experimentally demonstrate that the entanglement between Gaussian entangled states can
be increased by non-Gaussian operations. Coherent subtraction of single photons from Gaussian
quadrature-entangled light pulses, created by a nondegenerate parametric amplifier, produces de-
localized states with negative Wigner functions and complex structures more entangled than the
initial states in terms of negativity. The experimental results are in very good agreement with the
theoretical predictions.
PACSnumbers: : 03.67.-a,03.65.Wj,42.50.Dv
7
0
0 Entanglementplaysakeyroleinquantuminformation wesubtractasingle photondelocalizedinthetwobeams
2 processing (QIP). Entanglement distillation [1], demon- and prepare a complex quantum state with a negative
n strated for discrete-variable systems (ebits) in recent ex- two-mode Wigner function. We determine a range of
a periments[2,3,4],allowsonetoproducestrongentangle- experimental parameters where the entanglement of the
J ment between distant sites, initially sharing a larger set prepared state, quantified by the negativity [22], is sig-

0 of weakly entangled states, and constitutes the basis of nificantlyhighercomparedtothatoftheinitialGaussian
3
quantum repeaters, essential for long-distance quantum state.

communications. An interesting alternative to discrete- Operating with single photon counts rather than with
4
v levelsystems arequantum continuous variables(QCVs). coincidences as proposed e. g. in [10, 11, 12, 13, 23, 24],
0 In this case the information is encoded in the quadra- thisprotocolallowsformuchhighergenerationratesand
3 tures xˆ and pˆof traveling light fields, which can be effi- produces states more robust to experimental imperfec-
2
ciently measured by homodyne detection. Optical para- tions(seebelow). Besides,itismoreefficientatmoderate
8
metric amplification allows one to produce quadrature- OPAgain: in the zero-gainlimit, the detection ofa pho-
0
6 entangled beams, used in many QIP protocols. To- tontransformsa state with almostno entanglementinto
0 gether with linear optics, these tools preserve the Gaus- a maximally entangledebit state (10 + 01 )/√2. With
/ sian character of the states involved in most of QCV ex- the higher gain (up to 3 dB) used|inithe|priesent experi-
h
p periments : the quasi-distributions (Wigner functions) ment, the generated states have a much richer structure
- of their quadratures remain Gaussian. However, it has as is shown below.
t
n been shown that Gaussian entanglement distillation re- Our experimental setup is presented in Fig. 2. Nearly
a quires non-Gaussian operations [5, 6, 7]. Among sev- Fourier-limited femtosecond pulses (180 fs, 40 nJ), pro-
u
eral proposals [8, 9], one of the simplest is the condi- duced by a Ti:sapphire laser with a 800 kHz repeti-
q
: tional subtraction of photons from Gaussian entangled tion rate, are frequency doubled by a single pass in a
v beams [10, 11, 12, 13], by reflecting a small partof these 100 µm-thick type I noncritically phase-matched potas-
i
X beams towards two photon-counting avalanche photodi- sium niobate (KNbO ) crystal. The frequency-doubled
3
r odes (APDs). If the reflectivity is low, a simultaneous beampumpsanidenticalcrystalusedasanopticalpara-
a detection of photons by the APDs heralds the subtrac- metricamplifier(OPA),generatingGaussianquadrature-
tion of exactly one photon from each beam. Recently, entangled pulses spatially separated by an angle of 10 .
◦
such methods allowed the preparation and analysis of Adjusting the pump power allows us to vary the two-
several states with negative Wigner functions, including modesqueezingbetween0and3.5dB.Thephotonpickoff
one- and two-photonFock states [14, 15, 16], delocalized beam splitters are realizedwith a single polarizingbeam
single photons [17, 18] and photon-subtracted squeezed splitter (PBS) cube, where the signal and idler beams
states, very similar to quantum superpositions of coher- are recombined spatially but remain separated in polar-
ent states with small amplitudes [19, 20]. ization. A small adjustable fraction R of both beams
is sent into the APD channel, where they interfere on a
In this Letter, we experimentally demonstrate that
non-Gaussian operations allow us to increase the entan-
glement between Gaussian states, with a protocol pre-
sentedonFig. 1. An opticalparametricamplifier (OPA)
produces Gaussian quadrature-entangled light pulses,
known as two-mode squeezed states [21]. We pick off
small fractions of these beams, which interfere with a
well-defined phase on a 50/50 beam splitter (BS), and FIG. 1: Coherent photon subtraction from Gaussian entan-
we detect photons in one of the BS outputs. This way, gled beams.
2
deterioratethemeasureddata. However,theyarenotin-
volvedinthegenerationprocessbutonlyintheanalysis,
andwe cancorrectfor their effects inorderto determine
the actual Wigner function of the generated state. Even
withnoneoftheseimperfections,thisprotocolwouldstill
belimitedbythefinitepick-offBSreflectivityR,required
forasufficientAPDcountratebutinducinglossesonthe
transmittedbeam. The limited overallefficiency µ=5%
of the APD channel has little effect in this experiment.
A detailed analytic model [16, 19] including all these
imperfections yields an expression for the Wigner func-
tion W of the state studied in our experiment :
FIG. 2: Experimental setup.
W(x ,p ,x ,p )=W (x ,p )W (x ,p ) (1)
1 1 2 2 s + + c
− −
50/50BS.Atiltedhalf-waveplatecompensatesforresid- wtiohneroefxa±si=nglxe1-√±m2xo2d,eps±qu=eepz1√e±d2p2st,aWtes,aisndthWe Wcoigrnreesrpfounndcs-
c
ual birefringence. An APD detects one of the 50/50 BS to a photon-subtracted squeezed state analyzed in [19].
outputs after spatial and spectral filtering. The signal More explicitly :
and idler beams transmitted through the pickoff beam
splitter are projected into a non-Gaussian state by an Ws(x,p) = exp x2/a p2/b /(π√ab)
− −
APD detection. They are spatiallyseparatedonanother (cid:0) 2A (cid:1)2B A B
W (x,p) = W (x,p) x2+ p2+1
PBS, where they are combined with bright local oscilla- c s (cid:20)a2 b2 − a − b (cid:21)
tor beams. A quarter-wave and a half-wave plate allow
a(s) = b(1/s) = 1+e+η(1 R)(hs+h 2)
us to prepare two local oscillators with equal intensities − −
η ξ (1 R)(hs+h 2)2
and a well-defined relative phase. The signal and idler A(s) = B(1/s) = − −
h(s+1/s)+2h 4
beams are analyzed by two time-resolved homodyne de-
−
tections, which sample each individual pulse, measuring In this experiment the photon-subtracted state is “delo-
one quadrature x1,2(θ1,2) in phase with the local oscilla- calized” into two spatially separated modes 1 and 2 and
tor. revealedby measuring the correlationsbetweenidentical
In this setup, all the relative phases except φ1 and quadratures,the anticorrelationsremainingin the initial
φ (see Fig. 2) are precisely adjusted with wave plates. squeezed state.
2
Phase fluctuations concern only the initial two-mode Without assuming any particular shape for W and
s
squeezed state, where the phase difference is not defined W , we can experimentally show that the state becomes
c
and plays no role, and the slow (thermal and acous- separable if we make a joint measurement, transform-
tic) phase sum fluctuations simply rotate the two-mode ing x into x by rotating the polarizations by 45
1,2 ◦
squeezingellipseandcanbecompensatedbyshiftingthe with the option±al half-wave plate shown on Fig. 2. We
commonphaseφ2 ofthelocaloscillators. Thisphasecan then observe that the quadratures measured by one de-
bescannedwithapiezotranslator,andrapidlymeasured tection do not depend on the other (see Fig. 3). For
using the unconditioned two-mode squeezing variance. every θ the joint distribution, and hence the Wigner
±
Quantum states with negative Wigner functions are function, becomes factorable : P(x (θ ),x (θ )) =
+ +
− −
very sensitive to experimental imperfections. In our P (x (θ ))P (x (θ )). It means that one can fix θ =
s + + c
− − −
case, the most important issue are spurious APD trig- θ =θ and scanθ to perform a complete tomographyof
+
ger events, due to imperfect filtering, limited qualities of this state.
theopticalbeams,imperfectmode-matchingbetweenthe Thispropertyconsiderablysimplifiestheexperimental
subtractedbeams,andAPDdarkcounts. AnAPDcount analysis. We perform direct homodyne measurements of
corresponds to the desired subtraction event with a suc- theentangledquadraturesx (θ), keepingtheentangled
1,2
cess probability ξ < 1. This explains why single-photon modes 1 and 2 separated without mixing them. We use
protocols are more robust than two-photon ones, where thefactthatthestatefactorizesinthex ,x basistore-
+
the total success probability is only ξ2. Another issue constructitfromalimitedsetofdata: inste−adofatime-
is the OPA excess noise. To describe it, we can consider consuminggeneraltwo-modetomography,whichrequires
thatafirstamplificationprocesscreatesapureentangled to measure x (θ ),x (θ ) with all possible combinations
1 1 2 2
state with a two-mode squeezing variance s=e 2r, and of phases, we can restrict ourselves to θ = θ = θ. In
− 1 2
that each of the resulting modes is independently ampli- practice, we set the relative phase between the local os-
fied with a gain h = cosh2(γr) by a phase-independent cillatorstozero,scanthecommonphaseφ =θ,measure
2
amplifier with a relative efficiency γ. The finite homo- x (θ) andcalculate x (θ). We reconstructthe distri-
1,2 +,
−
dyne efficiency η and the homodyne excess noise e also butions P (x (θ)) and P (x (θ)) for severalphases. We
c s +
−
3
(b) P(x ) (a)
+
0.15 −2<x <2
− θ=90°
0.1 θ=72°
0.05 −P2<(xx−)< 2 P(x)c θ=54°
+ θ=36°
−06 −4 −2 0 2 4 6 θ=18°
θ=0°
−4 −2 0 2 4
FIG.3: Stateseparability testafterinterferencebetween sig- x
nal and idler beams : (a) Joint distribution P(x+,x−), (b)
Distributions Ps(x+) and Pc(x−), for 11 values of x− (resp.
x+), chosen between -2 and +2. This separability was ver-
θifi+ed=f2o0r◦seavnedraθl−ra=nd50o◦m)l.y chosen values of θ+ and θ− (here P(x)s θθ==9702°°
θ=54°
θ=36°
θ=18°
observethatthemeasureddistributionsareinvariantun- θ=0°
derθ π θ,sowerestricttheanalysisto0 θ π/2. −4 −2 x0 2 4
→ ± ≤ ≤
Typically, we measure 6 to 12 different quadrature dis-
tributions, with 10000 to 20000 data points each. A nu- FIG. 4: (a) Set of experimentally measured quadrature dis-
tributions (dots), compared to those reconstructed from our
meric Radon transform allows us to reconstruct the un-
model (solid line). (b) Wigner function corrected for homo-
corrected Wigner functions W and W . We can correct
c s dyne detection losses, obtained with a standard maximal-
for the homodyne detection losses (η = 70%, e = 1%
mikelihood algorithm (MaxLike), compared to the result of
of the shot noise) using a maximal-likelihood algorithm our model. This state is produced with 1.8 dB of squeezing
[25,26]toobtaintheWignerfunctionW ofthegenerated and R=5%.
state. We use W to calculatethe density matrix ρ ofthe
state and obtain its entanglement, given by the negativ-
ity N = kρT12k1−1, where T1 is the partial transposition wstoartkesisfoarlreaandayrsbtirtornargylyseqnuteaenzginlegd,obryifp,ewrfhoernmitnhgeainniitmia-l
operation [22].
perfect photon subtraction we actually lose more entan-
Figure 4 presents the tomography of a state produced
glement than we gain. It has already been shown that,
with 1.8 dB of squeezing and a BS reflectivity R = 5%.
whenthe pick-offbeamsplittershaveafinite reflectivity,
The Wigner function, corrected for detection losses, is
subtracting one photon from each of the Gaussian en-
clearly negative : W (0) = 0.13 0.01 (0.01 0.01
c − ± ± tangled beams may actually decrease the entanglement
beforecorrection). Theentanglementofthisstateis =
N [10]. Using our model, we can take into account all the
0.34 0.02,whereasfortheinitialstate(beforethepickoff
± other experimental parameters to derive an analytic ex-
BS) =0.24 0.01.
N0 ± pression for ρT1, which can be diagonalized numerically
In Refs. [16, 19] we demonstrated another analysis
in a few seconds to obtain the expected negativity of a
method,moreconstrainedbutalsomuchfasterandcloser
given state. We found that the experimental imperfec-
to the physics of the experiment. If we assume that the
tionshaveaverystrongeffect. Forexample,foraninitial
Wigner function has the form defined in Eq. 1, we can
squeezing of 3 dB, the negativity increases ideally from
easily extractthe parameters a, A, b, B from the second
= 0.50 to = 0.90. If we assume R = 3% for the
0
and fourth moments of the measured distributions, and N N
determine the Wigner function, the density matrix, and
the quadrature distributions of the measured state. For
a given squeezing, one can also obtain from a, A, b and
B the values of all the experimental parameters intro-
duced above. To correct for homodyne losses, we sim-
ply calculate the Wigner function that we would mea-
sure with an ideal detection (η = 1 and e = 0), using
the values extracted from the experimental data for all
the other parameters. As shown in Fig. 4, the distri-
butions reconstructed with this method are in excellent
agreement with those directly extracted from the data,
andtheWignerfunctionisalmostindistinguishablefrom
the oneobtainedwiththe maximal-likelihoodalgorithm. FIG.5: Tomographyoftwostatesproducedwith1.3dB(first
Both methods give the same values for the negativity. row) and 3.2 dB (second row) of squeezing and R = 10%:
A natural question to ask is whether this protocol threedifferent cuts of the two-mode Wigner function.
4
tweensubtractedphotonswasnotanissue,ξ reached0.9
[16]).
0.5 Initial Gaussian State Inconclusion,the presentphotonsubtractionprotocol
ativity)0.4 Final Photon−Subtracted State asilalonwsstaotnees wtoitihncurpeatsoe3thdeBeonftasnqguleeemzienngt, banetdweeveennGsmauasl-l
eg R=3%
N experimental improvements should significantly increase
ent (0.3 R=5% thislimit. ForQIPprotocolsspecificallyrequiringGaus-
m
gle R=10% sian entanglement, these non-Gaussian states could in
ntan0.2 R=20% principle be used as a starting point for a “Gaussifica-
E
tion” procedure [9]. This demonstrates one of the key
0.1
steps required for long distance quantum communica-
tions with continuous variables.
0
0 0.5 1 1.5 2 2.5 3 3.5
Squeezing (dB) This work is supported by the EU IST/FET project
COVAQIAL, and by the French ANR/PNANO project
FIG.6: Entanglementnegativityoftheinitialandfinalstates IRCOQ.
asafunction of squeezingfor severalpickoffBS reflectivities,
correctedforhomodynedetectionlosses. Solidlinesaretheo-
reticalcalculationsusingtheaveragevaluesoftheexperimen-
tal parameters.
∗
Electronicaddress: [email protected]
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