Table Of ContentContinuous-Variable Spatial Entanglement for Bright Optical Beams
Magnus T.L. Hsu,1 Warwick P. Bowen,1,∗ Nicolas Treps,2 and Ping Koy Lam1,†
1Australian Centre for Quantum-Atom Optics, Department of Physics,
Australian National University, ACT 0200, Australia
2Laboratoire Kastler Brossel, Universit´e Pierre et Marie Curie, case 74, 75252 Paris cedex 05, France
(Dated: February 1, 2008)
A light beam is said to be position squeezed if its position can be determined to an accuracy
beyondthestandardquantumlimit. Weidentifythepositionandmomentumobservablesforbright
opticalbeamsandshowthatpositionandmomentumentanglementcanbegeneratedbyinterfering
two position, or momentum, squeezed beams on a beam splitter. The position and momentum
measurements of these beams can be performed using a homodyne detector with local oscillator of
an appropriate transverse beam profile. We compare this form of spatial entanglement with split
5 detection-based spatial entanglement.
0
0
2 I. INTRODUCTION Our proposal considers the possibility of entangling
n the position and momentum of a free propagating beam
a of light, as opposed to the entanglement of local ar-
The concept of entanglement was first proposed by
J eas of images, considered in previous proposals. Our
Einstein,PodolskyandRoseninaseminalpaperin1935
5 scheme is based on the concept of position squeezed
[1]. The original Einstein-Podolsky-Rosen(EPR) entan-
2 beamswhere wehaveshownthatwehaveto squeezethe
glement, as discussed in the paper, involvedthe position
transverse mode corresponding to the first order deriva-
1 and momentum of a pair of particles. In this article, we
tive of the mean field in order to generate the posi-
v drawananalogybetweentheoriginalEPRentanglement
4 and the position and momentum (x-p) entanglement of tion squeezed beam [17]. Similarly to the generation
4 bright optical beams. of quadrature entangled beams, the position squeezed
1 beams are combined on a beam-splitter to generate x-p
Entanglementhas been reported in various manifesta-
1 entangledbeams. Wealsoproposetogeneratespatialen-
tions. For continuous wave (CW) optical beams, these
0 tanglement for split detection, utilising spatial squeezed
5 include, quadrature [2, 3] and polarisation [4] entangle-
beams reportedby Trepset al. [19, 20,21]. This formof
0 ment. Spatialformsofentanglement,althoughwellstud-
spatial entanglement has applications in quantum imag-
/ iedinthesinglephotonregime,havenotbeenstudiedsig-
h ing systems.
p nificantly in the continuous wave regime. Such forms of
We first define the position and momentum of an op-
- entanglement are interesting as they span a potentially
t tical beam by performing a multi-modal decomposition
n infinite Hilbert space. Spatial EPR entanglement [11]
onadisplacedandtiltedbeam,respectively. Weconsider
a haswide-rangingapplicationsfromtwo-photonquantum
u imaging[12,13]toholographicteleportation[14,15]and thecaseofaTEM00beamandshowthatthecorrespond-
q ing position and momentum observables are conjugate
interferometricfaint phaseobjectquantum imaging[16].
: observables which obey the Heisenberg commutation re-
v Current studies are focused on x-p entanglement for
lation. We then propose a scheme to produce x-p entan-
i the few photons regime. Howell et al. [5] observed
X glement for TEM optical beams. Finally, we consider
near and far-fieldquantum correlation,corresponding to 00
r spatialsqueezedbeams for split detectors and show that
a thepositionandmomentumobservablesofphotonpairs.
it is also possible to generate spatial entanglement with
Gatti et al. [6] have also discussed the spatial EPR as-
such beams.
pects in the photons pairs emitted from an optical para-
metric oscillator below threshold. Other forms of spatial
entanglementwhicharerelatedtoimagecorrelationhave
alsobeeninvestigated. Aschemetoproducespatiallyen- II. POSITION-MOMENTUM ENTANGLEMENT
tangledimagesbetweenthesignalandidlerfieldsfroman
optical parametric amplifier has been proposed by Gatti A. Definitions - Classical Treatment
et al. [7,8]. Theirworkwasextendedtothemacroscopic
domain by observing the spatial correlationbetween the
Let us consider an optical beam with x- and y- sym-
detected signal and idler intensities, generated via the
metric transverse intensity profile propagating along the
parametric down conversion process [10].
z-axis. Since the axes of symmetries remain well de-
fined during propagation, we can relate the beam po-
sition relative to these axes. To simplify our analysis
we henceforth assume without loss of generality, a one-
∗now at the Quantum Optics Group, MC 12-33, Norman Bridge
dimensional beam displacement, d, from the reference
LaboratoryofPhysics,CaliforniaInstituteofTechnology,Pasadena
CA91125. x-axis(see Fig.1(a)). We denote the electric fieldprofile
†Email: [email protected] of the beam by E(x). For a displaced beam, the electric
2
(a) Displaced beam probabilitydensityofthephotoninthefar-field,orequiv-
k
alently by taking the Fourier transform of x . These
d h i
Reference definitions are consistent with our definitions of position
Axis and momentum for bright optical modes.
(b) Tilted beam
q p
Reference B. TEMpq Basis
Axis
k In theory, spatial entanglement can be generated for
fields with any arbitrary transverse mode-shape. How-
FIG.1: (a) Beam displacementd,and(b)beamtiltbyangle ever,aswithotherformsofcontinuous-variableentangle-
θ, from a reference axis. ment, the efficacy of protocols to generate entanglement
is highest if the initial states are minimum uncertainty.
Forpositionandmomentumvariables,the minimumun-
certainty states are those which satisfy the Heisenberg
field profile is given by
uncertainty relation ∆2xˆ∆2pˆ ¯h, in the equality. This
≥
∂E(x) d2∂2E(x) equality is only satisfied by states with Gaussian trans-
Ed(x)=E(x)+d ∂x + 2 ∂x2 +··· (1) versedistributions[18],thereforewelimitouranalysisto
that of TEM modes.
00
In the regime where displacement is much smaller than A field of frequency ω can be represented by the posi-
the beam size, we can utilise the linearised approxima- tive frequency partof the meanelectric field +eiωt. We
E
tion where only the zeroth and first order terms are sig- are interested in the transverse information of the beam
nificant. We see from this expression that the zeroth described fully by the slowly varying field envelope +.
E
order term is not dependent on d, and that the displace- We express this field in terms of the TEMpq modes. For
mentisdirectlyproportionaltothederivativeofthefield a measurement performed in an exposure time T, the
amplitude ∂E(x)/∂x [17]. mean field for a displaced TEM beam can be written
00
The transversebeam momentum p onthe other hand, as
can be obtained from the transverse component of the
¯hω d
wave-number of the beam, p = ksinθ, where k = 2π/λ +(x)=i √N u (x)+ u (x) (5)
and the beam tilt is θ. This beam tilt is defined with Ed r2ǫ0cT (cid:16) 0 w0 1 (cid:17)
respect to a pivot point at the beam waist, as shown in where the first term indicates that the power of the dis-
Fig. 1(b). placed beam is in the TEM00 mode while the second
Theelectricfieldprofileforatiltedbeamwithuntilted term gives the displacement signal contained in the am-
electric field profile E(x) and wavelength λ is given by plitudeoftheTEM10modecomponent. Thecorrespond-
ing mean field for a tilted TEM beam can be written
00
i2πxsinθ as
E (x)=exp E(xcosθ) (2)
θ
(cid:20) λ (cid:21)
¯hω iw p
+(x)=i √N u (x)+ 0 u (x) (6)
We can again simplify Eq. (2) by taking the zeroth and Ep r2ǫ0cT (cid:16) 0 2 1 (cid:17)
first order Taylor expansion terms to get a transverse where the second term describes the beam momentum
beam momentum of p kθ. In the case of small dis- signal, contained in the π/2 phase-shifted TEM mode
≃ 10
placementortilt, we thereforeobtaina pairofequations component.
∂E(x)
E (x) E(x)+ d (3)
d
≈ ∂x C. Definitions - Quantum Treatment
E (x) E(x)+ p ixE(x) (4)
p
≈
We now introduce a quantum mechanical representa-
Eqs. (3) and (4) give the field parameters that relate to
tion of the beam by taking into account the quantum
thedisplacementandtiltofabeam. Forfreelypropagat-
noise of optical modes. We can write the positive fre-
ingopticalmodes,theFouriertransformofthederivative
quencypartoftheelectricfieldoperatorintermsofpho-
ofthe electricfield, (∂E(x)/∂x), is ofthe formixE(x).
F ton annihilation operators aˆ. The field operator is given
Hence, the Fourier transform of displacement is tilt.
by
In the case of a single photon, the position and mo-
∞
mentum are defined by considering the spatial proba- ¯hω
bility density of the photon, given by E(x)2/I, where Eˆi+n =ir2ǫ cT aˆnun(x) (7)
I = E(x)2dx is the normalisation fa|ctor.| The mean 0 nX=0
| |
positRion obtained from an ensemble of measurements on whereun(x)arethetransversebeamamplitudefunctions
single photons is then given by x = 1 xE(x)2dx. for the TEM modes and aˆ are the corresponding an-
h i I | | pq n
The momentum of the photon is defined bRy the spatial nihilation operators. aˆn is normally written in the form
3
of aˆ = aˆ +δaˆ , where aˆ describes the coherent anequivalentdependence of√N. This validatesthe fac-
n n n n
h i h i
amplitude part and δaˆ is the quantum noise operator. tor of N in the Heisenberg inequality and the commuta-
n
In the small displacement and tilt regime, the TEM tion relation for a CW optical beam.
00
andTEM modesarethe mostsignificant[17],withthe
10
TEM mode contributing to the displacement and tilt
10
signals. We can rewrite the electric field operator for E. Entanglement Scheme
mean number of photons N as
We have shown that the position and momentum ob-
ˆ+ = i ¯hω √Nu (x)+ δXˆa+0 +iδXˆa−0 servablesofCWTEM00opticalbeamssatisfytheHeisen-
Ein r2ǫ0cT(cid:16) 0 2 berg commutation relation. Consequently, EPR entan-
∞ Xˆ+ +iXˆ− glementforthepositionandmomentumofTEM00beams
+ ai ai u (x) (8) is possible. Experimentally, the usual quadrature entan-
i
2
nX=1(cid:16) (cid:17) (cid:17) glement is generated by mixing two amplitude squeezed
beams on a 50:50 beamsplitter. The same idea can be
wheretheannihilationoperatorisnowwrittenintermsof
applied to generate EPRx-p entanglement,by using po-
the amplitude Xˆ+ and phase Xˆ− quadrature operators.
a a sition squeezed beams [17]. Our scheme to produce x-p
The displacement and tilt of a TEM beam is given
00 entanglement between two CW TEM optical beams is
00
bytheamplitudeandphaseoftheu (x)modeinEqs.(5)
1 shown in Figure 2. The position squeezed beams in (a)
and (6), respectively. We can, therefore, write the beam
position and momentum operators as
(c) (d)
TEM
w 10
xˆ = 0 Xˆ+ (9) OPA TEM10LO
2√N a1 f
1
pˆ = w0√NXˆa−1 (10) (a) TEM00 + 1 3 A B
HD
BS
D. Commutation Relation HD
(b) TEM00 + A B
Twoobservablescorrespondingtothepositionandmo- 2 4
mentum of a TEM beam have been defined. We have f
00
shownthatthepositionandmomentumobservablescor-
respondtonear-andfar-fieldmeasurementsofthebeam, OPA TEM TEM10LO
10 (e)
respectively. Hence, we expect from Eqs. (9) and (10),
thatthepositionandmomentumobservablesdonotcom-
mute. Indeed,thecommutationrelationbetweenthetwo FIG.2: Schemefor generatingposition-momentum entangle-
quadraturesoftheTEM10 modeis[Xˆa+1,Xˆa−1]=2i. This ment for continuous wave TEM00 optical beams. OPA:opti-
leads to the commutation relation between the position calparametricamplifiersforthegeneration ofsqueezedlight,
and momentum observables of an optical beam with N BS: 50:50 beam-splitter, HD: homodyne detectors, LO: local
oscillator beams, φ: phaseof local oscillator beam.
photons
i
[xˆ,pˆ]= (11) and (b) are generated via the in-phase combination of a
N
vacuum squeezed TEM beam with a coherent TEM
10 00
This commutation relation is similar to the position- beam. Such beam combination can be achieved experi-
momentum commutation relation for a single photon, mentally, for example using an optical cavity or a beam-
aside from the 1/N factor. The 1/N factor is related to splitter [21]. The result of the combination is a position
the precisionwith which one canmeasure beam position squeezed beam. To generate entanglement, we consider
andmomentum. RewritingtheHeisenberginequalityus- beams with zero mean position and momentum, but we
ing the commutation relation, gives are interested in the quantum noise of the position and
momentumofthebeam. Withthisassumption,theelec-
1 tricfieldoperatorsforthe twoinput beamsatthe beam-
∆2xˆ∆2pˆ . (12)
≥ 4N splitter are given by
∞
The position measurement of a coherent optical beam ¯hω
ˆ+ =i √Nu (x)+ δaˆ u (x) (13)
tguivmesnaosisigenoafltwhheicphossictaiolensmweitahsuNre.mTehnetascssaolecsiawteitdhq√uaNn-. E1 r2ǫ0cT(cid:16) 0 nX=0 n n (cid:17)
Hence the positioning sensitivity of a coherent beam
∞
scalesas√N [17,19]. Thesameconsiderationappliedto ˆ+ =i ¯hω √Nu (x)+ δˆb u (x) (14)
the sensitivity of beam momentum measurement shows E2 r2ǫ0cT(cid:16) 0 nX=0 n n (cid:17)
4
where in both equations, the first bracketed term de- Inthecasewheretwobeamsareperfectlyinterchange-
scribesthecoherentamplitude oftheTEM beam. The able and have symmetricalfluctuations in the amplitude
00
second bracketed terms describe the quantum fluctua- and phase quadratures, the inseparability criterion has
tions present in all modes. For position squeezed states, beengeneralisedandnormalisedtoaproductformgiven
onlytheTEM modeisoccupiedbyavacuumsqueezed by [4, 27, 28, 29, 30, 31]
10
mode. All other modes are occupied by vacuum fluctua-
tions. It is also assumed that the number of photons in (xˆ3+xˆ4)2 (pˆ3 pˆ4)2
(xˆ,pˆ)= h ih − i (19)
eachofthe twobeams,duringthe measurementwindow, I [xˆ,pˆ]2
| |
isequaltoN. Thetwopositionsqueezedbeams(1,2)are
combined in-phase on a 50:50 beam-splitter (BS) in (c). for anypair of conjugateobservablesxˆ and pˆ, anda pair
The usual input-output relations of a beam-splitter of beams denoted by the subscripts 3 and 4. For states
apply. The electric field operators describing the two which are inseparable, (δxˆ,δpˆ) < 1. By using observ-
I
output fields from the beam-splitter are given by ˆ = ables δxˆ and δpˆ from Eqs. (15), (16), (17) and (18) as
3
(ˆ + ˆ )/√2 and ˆ = (ˆ ˆ )/√2. To demonsEtrate well as the commutation relation of Eq. (11) the insepa-
1 2 4 1 2
tEheexiEstenceofentaEnglemEent−,wEeseekforquantumcorre- rability criterion for beams 3 and 4 is given by
lationandanti-correlationbetweenthe positionandmo-
16N2
mentumquantumnoiseoperators. Thepositionoperator (δxˆ,δpˆ) = (δxˆ+)2 (δxˆ+)2
correspondingtobeams3and4aregivenrespectivelyby I ω04 h a1 ih b1 i
= (δXˆ+)2 (δXˆ+)2
δxˆ = w0 δXˆ+ +δXˆ− h a1 ih b1 i
3 2√2√N(cid:16) a1 b1(cid:17) < 1 (20)
1 w2
= δxˆa+ 0δpˆb (15) wherewehaveassumedthattheTEM10 modesofbeams
√2(cid:16) 2 (cid:17) 1 and 2 are amplitude squeezed (i.e. (δXˆ+)2 < 1 and
w h a1 i
δxˆ = 0 δXˆ+ δXˆ− (δXˆ+)2 <1).
4 2√2√N(cid:16) a1 − b1(cid:17) h Thbu1s wie have demonstrated that CV EPR entangle-
1 w2 ment between the position and momentum observables
= δxˆ 0δpˆ (16)
√2(cid:16) a− 2 b(cid:17) of two CW beams can be achieved.
The momentum operator corresponding to the photo-
current difference for beams 3 and 4 are given by G. Detection Scheme
1
δpˆ = δXˆ− +δXˆ+ Ref. [17] has shown that the optimum small displace-
3 w0 2√N(cid:16) a1 b1(cid:17) mentmeasurementishomodynedetectionwithaTEM10
1p 2 local oscillator beam (see Fig. 2 (d)). When the input
= δpˆ + δxˆ (17)
√2(cid:16) a w02 b(cid:17) beam is centred with respect to the TEM10 local oscil-
1 lator beam, no power is contained in the TEM10 mode.
δpˆ4 = w 2√N(cid:16)δXˆa−1 −δXˆb+1(cid:17) Due to the orthogonality of Hermite-Gauss modes, the
0 TEM local oscillator only detects the TEM vacuum
10 10
1p 2 noise component. As the input beamis displaced,power
= δpˆ δxˆ (18)
√2(cid:16) a− w02 b(cid:17) is coupled into the TEM10 mode. This coupled power
interferes with the TEM local oscillatorbeam, causing
10
In our case where the two input beams are position
a change in photo-current observed at the output of the
squeezed, the sign difference between the position noise
homodynedetector. Thusthedifferencephoto-currentof
operators in Eqs. (15) and (16) as well as that between
the TEM homodyne detector is given by [17]
10
the momentum noise operatorsin Eqs.(17) and (18) are
signatures of correlation and anti-correlation for δxˆ and 2√N√N
δpˆ. nˆd− = w LOxˆ (21)
0
whereN andN arethetotalnumberofphotonsinthe
LO
F. Inseparability Criterion local oscillator and displaced beams, respectively, with
N N. The linearised approximation is utilised,
LO
≫
Manycriterionsexisttocharacteriseentanglement,for where secondorder terms in δaˆ are neglected since N
≫
example the inseparability criterion [22] and the EPR δaˆ2 for all n.
|h ni|
criterion [23]. We have adopted the inseparability cri- Inordertomeasuremomentum,onecouldusealensto
teriontocharacteriseposition-momentumentanglement. Fourier transform to the far-field plane, where the beam
ForstateswithGaussiannoisestatistics,Duanet al. [22] is then measured using the TEM homodyning scheme.
10
haveshownthattheinseparabilitycriterionisanecessary However, we have shown that the the position and mo-
and sufficient criterion for EPR entanglement. mentum of a TEM beam differs by the phase of the
00
5
TEM mode component. Indeed for a tilted TEM of photons in the beam. It has been shownthat the cor-
10 00
beam, the TEM mode component is π/2 phase shifted responding photon number difference operator for split
10
relative to the TEM mode component. Consequently detection is given by [17]
00
the phase quadrature of the TEM mode has to be in-
10
terrogated. This can be achieved by utilising a TEM10 nˆ(−+) =√NδXˆc+0 (24)
localoscillatorbeamwitha π/2phasedifference relative
totheTEM10modecomponentoftheTEM00beam. The The beam is spatially squeezed if the state of the
resultingphoto-currentdifferencebetweenthetwohomo- flipped mode is vacuum squeezed and in phase with the
dyning detectors, for NLO N, is given by mean field mode (see Fig. 3 (a) and (b)).
≫
nˆp =w √N N pˆ (22)
− 0 LO
p
B. Spatial Homodyne
III. SPATIAL ENTANGLEMENT FOR SPLIT
DETECTION Since split detection is commonly used as a detection
device for beam position, one would naturally consider
taking the Fourier transform of a spatial squeezed beam
The entanglement presented in the previous section
to obtain the conjugate observable for the beam. How-
is analogous to x-p entanglement in the single photon
ever, we have shown that split detection does not cor-
regime. However, the choice of the mean field mode is
respond exactly to beam position measurement. Thus
restrictedtotheTEM mode. Thislimitstherichnessof
00
the Fourier plane of the spatial squeezed beam does not
aspatialvariableandthusexcludesthepossibilityofgen-
provide the conjugate observable. More practically, the
erating an infinite Hilbert space. To exploit the proper-
flipped mode is not mode-shape invariant under Fourier
tiesofspatialvariables,wenowconsidermoretraditional
transformation. In the far-field, each odd-ordered mode
forms of spatial squeezing. Consequently, we study the
componentoftheflippedmodeobtainsa(2n+1)π Gouy
possibility of generating spatial entanglement for array
phase difference, compared to the near-field. Thus the
detection devices, based on spatial squeezed beams.
mode-shape in the far-field is no longer a flipped mode.
Consequently, far and near-field measurements of a spa-
tial squeezed beam will not give the conjugate observ-
A. Spatial Squeezing
ables.
However, we can find the conjugate observables of a
SpatialsqueezingwasfirstintroducedbyKolobov[25].
spatial squeezed beam by drawing an analogy to stan-
Thegenerationofspatialsqueezedbeamsforsplitandar-
dard homodyne detection. In split detection, the equiv-
raydetectorswasexperimentallydemonstratedbyTreps
alent local oscillator mode is the mean field v (x) mode.
et al. [19, 20, 21]. A one-dimensional spatial squeezed 0
Themodeunderinterrogationbythesplitdetectoristhe
beam has a spatially ordered distribution, where there
flipped mode v (x). In the case of homodyne detection,
exists correlation between the photon numbers in both 1
the phase of the local oscillator beam is varied to mea-
transversehalvesofthebeam. Adisplacementsignalap-
surethe conjugateobservables(i.e. amplitude andphase
plied to this beam can thus be measured to beyond the
quadratures) of the input beam. Adapting this concept
QNL.
to the split detector, the conjugate observables for the
We consider a beam of normalised transverse ampli-
spatial squeezed beam is thus the amplitude and phase
tude function v (x) incident on a split detector. The
0 quadratures of the flipped mode, while the mode-shape
noise of split detection has been shown to be due to the
of the flipped mode remains unaltered. This is further
flipped mode [24], given by
verified upon inspection of Eq. (24).
v (x) for x>0 Our scheme to perform a phase measurement of the
v1(x)=(cid:26) 0v0(x) for x<0 flippedmodeisshowninFig.3(d). Inourschemeweas-
− sumethatthemeanfieldisaTEM mode. Notethatin
00
When the field is centred at the split-detector, such principle,thisanalysiscouldbeperformedforanymode-
that the mean value of the measurement is zero, the shape. ThecoherentTEM modecomponentprovidesa
00
flipped mode is thus orthogonalto the mean field mode. phasereferencefortheflippedmode,analogoustothatof
In this instance, modes v (x) (for i > 1) can be derived alocaloscillatorbeam inhomodyne detection. Thus the
i
to complete the modal basis. The electric field operator phasequadratureoftheflippedmodecanbe accessedby
written in this new modal basis is given by applyingaπ/2phaseshiftbetweenthetheTEM mode
00
and the flipped mode noise component. Experimentally,
∞
ˆ+ =i ¯hω √Nv (x)+ δcˆ v (x) (23) this is achievable using an optical cavity. When the cav-
E r2ǫ0cT(cid:16) 0 nX=0 n n (cid:17) ity is non-resonant for the v0(x) and v1(x) modes it will
reflect off the two modes, in phase, onto the split de-
where the first term describes the coherent excitation of tector. This will give a measurement of the amplitude
the beam in the v (x) mode and N is the total number quadratureofthe flippedmode. However,the cavitycan
0
6
betunedtobepartiallyresonantonthev (x)modewhile and 2 are labelled by cˆ and dˆ , respectively. By fol-
0 n n
reflecting the flipped mode. A π/2 phase difference can lowinga similarprocedureas before,the photonnumber
thenbeintroducedbetweenthereflectedv (x)andv (x) difference operator for output beams 3 and 4 from the
0 1
modes,givinga measurementofthe phasequadratureof beam-splitter are calculated.
theflippedmode. Thecorrespondingphotonnumberdif- For the amplitude quadraturemeasurement, the addi-
ference operator is tionofthedifferencephoto-currentbetweenbeams3and
4 yields
nˆ(−) =√NδXˆ− (25)
− c0
nˆ(+)+nˆ(+) =√N(δXˆ+ +δXˆ+) (26)
which is the orthogonal quadrature of the spatial 3− 4− c0 d0
squeezed beam. The photon number operators corre- For the phase quadrature measurement, the subtraction
sponding to the two measurementsin Eqs. (24) and (25) of the difference photo-current between beams 3 and 4
are conjugate observables and satisfy the commutation gives
relation [nˆ(+),nˆ(−)]=2iN.
− −
It is important to realise that the number of photons nˆ(−) nˆ(−) =√N(δXˆ+ δXˆ+) (27)
3− − 4− d0 − c0
N in Eqs. (24) and (25) are only approximately equal.
This is due to the fact that partial power in the TEM To verify spatial entanglement, the inseparability cri-
00
mode is transmitted by the cavity, when the cavity is terion is utilised. The substitution of Eqs. (26), (27)
partially resonant on the TEM mode. Although it is and the commutation relation between the photon num-
00
possible to implement a scheme that conserves the total ber difference operators into the generalised form of the
numberofphotonsatdetection(e.g. losslesslyseparating inseparability criterion gives
the mean field and flipped modes and recombining them
N V2 +2V V +V2
withaphasedifference),wewouldliketoemphasisethat (δnˆ(+),δnˆ(−)) = c0 c0 d0 d0
our schemeis moresimple andintuitive, as wellasbeing I − − (cid:0) 4N (cid:1)
valid when N is large. < 1 (28)
where V = (δXˆ+)2 and V = (δXˆ+)2 are the
c0 h c0 i d0 h d0 i
C. Entanglement Scheme variances for the flipped mode component of the spatial
squeezed beams 1 and 2. The inseparability criterion is
satisfied for amplitude squeezed flipped modes V < 1
In order to generate spatial entanglement for split de- c0
and V <1.
tection, two spatial squeezed beams labelled 1 and 2 d0
We have proposeda scheme to generatespatial entan-
are combined on a 50:50 beam-splitter, as shown in Fig-
glementforsplitdetection,usingspatialsqueezedbeams.
ure 3 (c).
Spatial squeezing has been defined for any linear mea-
surement performed with an array detector [26]. Sim-
(c) Optical (d) ilarly, spatial entanglement corresponding to any linear
OPA cavity measurement, can be obtained. For an infinite span ar-
PP p 0 ray detector with infinitessimally small pixels, it is thus
possibletogeneratemulti-modespatialentanglement,in-
+
1 3 creasing the Hilbert space to being infinite-dimensional.
(a) SD
BS
(b) SD IV. CONCLUSION
+
2 4
We have identified the position and momentum of a
PP p 0 TEM optical beam. By showing that xˆ and pˆare con-
00
OPA Optical jugate observables that satisfy the Heisenberg commu-
cavity (e) tation relation, a continuous variable x-p entanglement
schemeisproposed. Thisproposedentanglement,ascon-
sideredbyEPR[1],wascharacterisedusingageneralised
FIG. 3: Scheme for generating spatial entanglement for
form of the inseparability criterion.
TEM00 continuouswavelight beams. OPA:opticalparamet-
We further explored a form of spatial entanglement
ricamplifiersforgeneratingsqueezedbeams,PP:phaseplates
which has applications in quantum imaging. The detec-
for producing flipped modes, BS: 50:50 beam-splitter.
tionschemesforquantumimagingaretypicallyarrayde-
tectors. In this article, we considered the split detector.
The electric field operators for the two input spatial We utilised the one-dimensional spatial squeezing work
squeezed beams at the beam-splitter are described in a of Treps et al. [19] and proposed a spatial homodyning
form identical to that of Eq. (23). The annihilation op- scheme for the spatial squeezed beam. By identifying
erators of the electric field operators for input beams 1 the conjugate observables for the spatial squeezed beam
7
as the amplitude and phase quadratures of the flipped portedbytheAustralianResearchCouncilCentreofEx-
mode, we showed that split detection-based spatial en- cellence Programme. The LaboratoireKastler-Brosselof
tanglement can be obtained. the Ecole Normale Superieure and University Pierre et
Marie Curie is associated via CNRS.
Acknowledgments
We would like to thank Vincent Delaubert and Hans-
A. Bachor for fruitful discussions. This work was sup-
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