Single-mode squeezing in arbitrary spatial modes

1.4 MB·

by  Marion Semmler

Single-mode squeezing in arbitrary
spatial modes
MarionSemmler1,2,StefanBerg-Johansen1,2, ,VanessaChille1,2,
∗
ChristianGabriel1,2,PeterBanzer1,2,3,AndreaAiello1,2,
ChristophMarquardt1,2andGerdLeuchs1,2,3
1MaxPlanckInstitutefortheScienceofLight,Guenther-Scharowsky-Str.1/Bldg.24,
91058Erlangen,Germany
6 2InstituteforOptics,InformationandPhotonics,UniversityErlangen-Nuremberg,
1 Staudtstr.7/B2,91058Erlangen,Germany
0 3DepartmentofPhysics,UniversityofOttawa,25Templeton,Ottawa,Ontario,K1N6N5
Canada
2
∗[email protected]
n
a
J
Abstract: As the generation of squeezed states of light has become a
5
standardtechniqueinlaboratories,attentionisincreasinglydirectedtowards
1
adapting the optical parameters of squeezed beams to the specific require-

] ments of individual applications. It is known that imaging, metrology, and
h
quantuminformationmaybenefitfromusingsqueezedlightwithatailored
p
transverse spatial mode. However, experiments have so far been limited to
-
t generating only a few squeezed spatial modes within a given setup. Here,
n
a wepresentthegenerationofsingle-modesqueezinginLaguerre-Gaussand
u Bessel-Gauss modes, as well as an arbitrary intensity pattern, all from a
q
singlesetupusingaspatiallightmodulator(SLM).Thedegreeofsqueezing
[
obtained is limited mainly by the initial squeezing and diffractive losses

1 introducedbytheSLM,whilenoexcessnoisefromtheSLMisdetectable
v at the measured sideband. The experiment illustrates the single-mode
1
concept in quantum optics and demonstrates the viability of current SLMs
6
asflexibletoolsforthespatialreshapingofsqueezedlight.
9
3
© 2016 OpticalSocietyofAmerica
0
. OCIScodes:(270.6570)Squeezedstates;(230.6120)Spatiallightmodulators.
1
0
6
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1. Introduction
Squeezed states of the electromagnetic field [1, 2, 3, 4] have been the subject of intense the-
oretical and experimental study over the past four decades. As a generalisation of Glauber’s
coherentstates[5]tominimumuncertaintystates[6,7],theirexperimentalrealisation[8]has
providedastrikingconfirmationofthequantumtheoryoflight.Fromanappliedpointofview,
squeezedstatesallowthelimitationsimposedbyquantumuncertaintiesontheaccuracyofop-
ticalmeasurementstobeovercome.ItwaspointedoutearlyonbyCaves[9]thatthesensitivity
ofgravitationalwaveinterferometerscanbeenhancedbysqueezingthevacuumstateentering
theinterferometer’sunusedport.Inrecentyears,thisideabecamerealityintheGEO600inter-
ferometer,wheresqueezingiscurrentlyusedtoenhancelong-termsensitivityby2.0dB[10].In
quantuminformationscience,squeezedstatesarerelevantasaresourceforcontinuous-variable
(CV)entanglement[11],aswellasCVquantumkeydistributionprotocols[12,13,14].
Meanwhile,therecognitionoflight’stransversespatialdegreesoffreedomasaninformation
carrier[15]hasmadespatialmodesrelevantforopticalimplementationsofquantuminforma-
tion protocols [16, 17, 18]. The consideration of the effects of quantum noise on the spatial
statisticsofphotonsalsoledtothefieldofquantumimaging[19,20],anditwasfoundthatthe
displacementandtiltofalaserbeamcanbemeasuredmoreaccuratelybyinterferingsqueezed
lightwithhigher-orderspatialmodes[21].Finally,itwasshownthatthermalnoisefrommirror
coatingsingravitationalwaveinterferometerscanbereducedbyusinghigher-orderLaguerre-
Gauss (LG) modes instead of the fundamental mode, allowing for higher optical powers and
thusanimprovedsignal-to-noiseratio[22,23].Combiningthistechniquewithsqueezedlight
withinthestringentparameterregimerequiredbygravitationalwaveinterferometrypresentsa
formidablechallenge,butwouldallowthephasesensitivitytoincreaseevenfurther.
Here, we present a proof-of-principle experiment to generate amplitude squeezing in light
beamswitharbitraryspatialintensitypatterns.Asademonstrationofthesetup’sversatility,we
generatesqueezedLGandBessel-Gauss(BG)beamsofdifferentorders,aswellasacomplex
patterncontaininghighspatialfrequencies.Allmodesaregeneratedwithoutmodificationsto
the setup. The presented experiment showcases the possibility of generating practically arbi-
trarytwo-dimensionalspatialmodesthataresingle-modesqueezed.
Beforedetailingtheexperimentalsetup,wefirstreviewtherelationbetweenspatialmodes
and quantum states of light, and describe some of the existing approaches for spatial mode
squeezing.
1.1. Quantumstatesoflightandtransversespatialmodes
In order to make more precise the relationship between spatial modes of the electromagnetic
fieldandphotonstatistics,werecallthecanonicalquantisationofthetransverseelectromagnetic
field,whichconsidersthemodesofafinitevolumewitheitherperiodicorreflectingboundaries
(eventuallytobetakentoinfinity).Theelectricfieldoperatorcanbeexpandedas
(cid:114)
(cid:126)Eˆ((cid:126)r,t)=i∑ h¯ωk(cid:16)(cid:126)u ((cid:126)r,t)aˆ (cid:126)u ((cid:126)r,t)aˆ†(cid:17), (1)
2 k k− ∗k k
k
whereaˆandaˆ†arebosonicladderoperatorssatisfying[aˆ,aˆ†]=δ ,andthefunctions(cid:126)u denote
i j ij k
mutually orthogonal solutions of the Helmholtz equation that describe transverse oscillation
ofthetransverseelectricfieldsandcanbedirectlyderivedfromclassicalMaxwellequations.
However,thegeneraliseddefinitionofanopticalmodepermitssuperpositionsofsuchsolutions
to be treated as a single-mode excitation as long as the field possesses first-order coherence
[24,25,26,27].Thus,aslongastheconditionoffirst-ordercoherenceismet,abeamwitha
complicatedspatialstructuremaybetreatedasasingle-modeexcitationofthefield.
When a squeezed beam interacts with a diffractive optical element (such as an SLM in the
presentwork),theresultingmodepatterncanbedeterminedfromtheclassicaltheoryofdiffrac-
tionbyconsideringtheplanewavespectrum.Diffractiondoesnotaffectthequantumstatistics
ofthemodeperse.Rather,weobserveareductionofthedegreeofsqueezingsinceweareno
longerabletointegrateoverthefullplanewavespectrumwithourdetector(i.e.,highdiffrac-
tion orders result in losses) [28]. The effect of such losses on the single-mode squeezing can
then be found from the beam splitter relation to be Var =η Var +(1 η) Var . Here,
out in vac
· − ·
η is the efficiency, Var and Var represent the variances of the input and output beam of
in out
thesqueezedquadrature,andVar isthevacuumvariance(shot-noise).Apartfromlossesin-
vac
ducedbyimperfectreflectivityandabsorptiononealsohastotakeintoaccountpossiblesources
of additional classical noise. If the diffractive element were to impose an unwanted temporal
modulationatafrequency f (forexample,duetoelectronicflickernoiseinthecaseofaliquid
N
crystalSLM),thequantumnoiseofthedetectedlightmodewouldbemaskedbyexcessnoise
attheopticalsidebandsat f f ,rapidlydegradingtheobservablesqueezing.
0 N
±
Therearevariousstudiesofnonclassicalbeamsinamultimodesetting[29,30,31,32,33,
34].Here,weconcentrateonsingle-modesqueezing,whichisparticularlysuitedforapplica-
tionssuchasquantum-enhancedinterferometry.
1.2. Existingexperimentalapproaches
Twomainapproachestosqueezingasinglespatialmodehavebeendemonstratedsofar:
(1) Reshaping, where a squeezed fundamental TEM mode is generated first and subse-
00
quentlyconvertedintothedesiredspatialmode.Anyconversionlossnecessarilyreduces
thesqueezingfromtheinitialvalue.Thishasbeenachievedwithphaseplates[21,35,36],
wherethewavelengthanddesignatedmodearefixed,orwithspecial-purposeliquidcrys-
taldevices[37],whichallowmoreflexibilityinthechoiceofwavelengthandmodepa-
rameters.Anotherapproachusesprogrammableadaptiveoptics,forwhich,althoughca-
pableinprincipleofgeneratinganymode,squeezinghassofaronlybeendemonstrated
in1DwithHermite-GaussHG modes[38].
n0
(2) Direct squeezing, where the nonlinear medium is either resonant for the desired spa-
tial mode, or transmissive, as in the case of a traveling-wave, or single-pass scheme,
a squeezed spatial mode can be generated directly. Examples include misaligned OPO
cavities[39,40]andphotoniccrystalfibers[41].Themulti-modesqueezingmentioned
abovecanbeachievedwiththisapproachwhenthenonlinearmediumdoesnotenforce
aparticularspatialmode.
Inthisworkwetakethereshapingapproach,usinganasymmetricfiberSagnacinterferometer
asasqueezingsourceforTEM modes[42]andaspatiallightmodulatorforthesubsequent
00
modeconversion.
2. Experimentalsetup
2.1. Squeezing
Our light source is a shot-noise limited laser (Origami, Onefive GmbH) emitting linearly po-
larisedlightin220fspulses,centeredatawavelengthofλ =1558nm.Fig.1showstheasym-
0
metricSagnacinterferometerusedtogenerateamplitudesqueezedlightintheinitialGaussian
mode. The laser beam is split on an asymmetric beam splitter with a splitting ratio of 90:10.
This results in a strong and a weak pulse counter-propagating in the polarisation-maintaining
single-modefiber(FSPM7811by3M).Duetothefiber’snonlinearKerreffect,aquadrature
squeezingisachievedinthebrightpulsethat,bymeansofthecounter-propagatingweakpulse,
isadjustedtooccurintheamplitudequadrature[42].
2.2. ModeConversion
ThesqueezedGaussianbeam,havingawaistofw =1.32mm,isconvertedintoahigher-order
0
spatialmodebyareflectiveliquid-crystal-on-siliconspatiallightmodulator(LCoS-SLM,Pluto,
Holoeye Photonics AG, 1920x1080 pixels, display optimised for 1550nm, no anti-reflection
coating).ThisSLMisdesignedforphase-onlymodulationanddoesnotdirectlymodulatethe
amplitude. The local refractive index is modulated due to the preferential alignment of the
rod-shaped LC molecules with the electric field at each pixel. This affects only the polarisa-
tion component along the long axis of the LC molecules, leaving the orthogonal polarisation
componentunmodulated.
WeprogramourSLMwithphasepatternsconsistingoffourcontributions:First,thetrans-
versephasepatternofthetheoreticalmodefunctionofthedesiredmodeasdescribedlaterin
this section. Second, a blazed grating phase which diffracts the modulated beam away from
thezerothorderandtransferstheenergymostlyintothefirstdiffractionorder.Thisstepisre-
quiredtospatiallyseparatethemodulatedlightfromtheapproximately20%ofincominglight
whichtheSLMeffectivelydoesnotmodulateduetoitslimiteddiffractionefficiency.Thegrat-
ingperiodof35px 8µm/px 180λ ischosenempiricallytomaximisediffractionintothe
0
× ≈
first order while enabling sufficient transverse separation from other diffraction orders in the
detection plane at a distance of 45cm from the SLM (corresponding to 1/8th of the Rayleigh
lengthbeforeconversion).Additionally,alensphaseisaddedtothehologram.Andfinally,a
binarycircularaperturepatternismultipliedtotheentirehologram,restrictingmodulationto
Fig. 1: Experimental setup. A femtosecond laser emits pulses of 220fs duration centered at
λ =1558nm. For squeezed light generation, the pulses are split up on a 90:10 beam splitter
0
and launched into a Kerr fiber (χ(3) nonlinearity) of length 3.8m in a counter-propagating
configuration.Theexitingpulsestypicallyexhibit 3.0dBofamplitudesqueezingpriortothe
−
SLM. A pair of folding mirrors (FM , FM ) allow the squeezer to be bypassed to obtain a
a b
coherentshotnoisereferenceforsqueezingmeasurements.Thebeamimpingesonareflective
SLM. An iris aperture selects the 1. diffraction order (see text for details). Another folding
mirror (FM ) is used to direct the beam either at a InGaAs camera for mode inspection or at
c
adetector,whose9MHzsidebandfluctuationsandDCamplitudearerespectivelyrecordedby
anelectronicspectrumanalyserandavoltmeter.
2π
e
π as
h
p
0
(a) LGl=1 (b) BG (n=1)
p=1
Fig.2:Examplephasepatterns.Basicphasepatternsforgenerating(a)aLaguerre-Gaussbeam
and (b) a Bessel-Gauss beam. In addition, a blazed grating, kinoform lens and aperture are
addedtoeachpattern(notshown,seetextfordetails).
thecentralregiononly.
AnSLM’simportantadvantageisthatitallowsforthegenerationofarbitrarypatterns,i.e.su-
perpositionsofverymanybasismodeswithalmostanycombinationofcoefficients.Toshow
the versatility of the setup, we generate amplitude squeezed Laguerre-Gauss beams, Bessel-
Gauss beams as well as an arbitrary pattern. Laguerre-Gauss (LG) modes are of particular
interest as they represent a natural basis for optical orbital angular momentum [43]. Fig. 2(a)
showsthephasepatternrequiredtogenerateanLGbeamwithradialindex p=1andhelical
indexl=1,whereeachphotoncarriesanorbitalangularmomentumofh¯.Besselbeams,too,
exhibitremarkablefeatures:theyarenon-diffractiveandself-healing[44].Theseidealbeams
extendtransverselytoinfinityandcontainaninfiniteamountofenergy,similartoplanewaves.
InarealsettingitishenceonlypossibletogenerateBessel-Gauss(BG)beams,forwhichthe
idealmodefunctionismultipliedbyaGaussianenvelope,while,however,retainingsomeofits
favourableproperties.Fig.2(b)displaysthephasepatternusedtogenerateaBGbeamoforder
n=1.
The two-dimensional spatial Fourier transform of the desired beam is used as the phase
patternontheSLM.ThepatternsemployedtogeneratebothBGandLGbeamsaredominated
by their mode functions’ defining polynomials, i.e. the generalized Laguerre polynomial [15]
and the nth-order Bessel function of the first kind [44]. Every zero in the radial direction of
thepolynomialdefiningthemodesresultsinaphasediscontinuityofπ ofthephasemask(see
Fig.2).Theazimuthalphaseconsistsinarepeatedcontinuousgradientfrom0to2π.
2.3. Measurements
The generated modes are analysed with respect to the quality of the spatial modes and the
quantumnoisereduction.Thetransverseintensitydistributionsoftheexperimentallygenerated
modesarerecordedwithaXenicsXS-1.7-320InGaAscamera.Asameasureofmodequality,
theintensitydistributionasinferredfromthetheoreticalmodefunctionisfittedtoalinesection
ofthemeasuredmode.
The amplitude squeezing is measured by direct photodetection at a sideband frequency of
9MHzusinganelectronicspectrumanalyserwithresolutionbandwidth1MHzandvideoband-
width 3kHz. The shot-noise reference level is determined by measuring the fluctuations of a
coherentbeamofthesamecontinuous-waveequivalentopticalpowerinthesamespatialmode.
Thefinalsqueezingfigureisdeterminedbyformingthedifferencebetweentherespectivetime-
averagedvalues.
Foreachmodepattern,squeezingismeasuredinthiswaybothpriortothemodeconversion
andinthefinalconvertedmode.Themodeconversionefficiencyη isestimatedbycomparing
[mm]
l=1 l=2 l=3 -1.5 0 1.5
1
p=1 n=0
u.]
a.
r,
a
e
p=2 n=1 n
[li
2
|
E
|
∝
p=3 n=2
0
(a) Laguerre-Gauss (LGl) (b) Bessel-Gauss
p
( 1.31 0.31)dB squeezing ( 1.17 0.31)dB squeezing
− ± − ±
(c) arbitrary pattern
( 0.4 0.3)dB squeezing
− ±
Fig. 3: Squeezed spatial modes. The average measured amplitude squeezing for each family
ofmodesisshownbelowthepanels.Eachpanelrepresentsa3 3mm2 regioninthecamera
×
plane.Relativeintensitybetweenpanelsisarbitrary.
thecontinuous-waveequivalentpowerofeachdiffractedbeambehindtheSLMtothatofthe
squeezedGaussianinputbeambeforetheSLM.Theobservedreductioninsqueezingiscom-
pared to the expected reduction from the variance relation in Sec. 1.1. Agreement of the two
valueswithintheirexperimentalerrorindicatesthatanyexcessnoisepoweraddedbytheSLM
atthemeasuredsidebandliesbelowthedetectablethreshold.
3. ResultsandDiscussion
3.1. Qualityofthegeneratedspatialmodes
The observed intensity distributions for LG modes with p,l 1,2,3 and BG modes with
∈{ }
n=0,1,2 are shown in Fig. 3(a) and (b), respectively. The intensity distributions are in very
goodagreementwiththeexpectedbeamprofiles.Forthesamemodes,Fig.4(a)and(b)show
thecross-sectionalintensitydistributions(blackdots)andcorrespondingfittedcurves(redline).
The fitted curves, derived from the theoretical mode functions, generally agree well with the
observed line sections. However, for LG modes with larger values of p the overlap decreases
[mm]
l=1 l=2 l=3 -2.4 0 2.4
u.]
p=1 [a. n=0
2
E|
|
p=2 n=1
p=3 n=2
(a) Laguerre-Gauss (b) Bessel-Gauss
Fig.4:Modequality.Measuredcross-sectionalintensitydistributions(dots)withfittedtheoret-
icalcurves(lines)for(a)Laguerre-Gaussbeamsand(b)Bessel-Gaussbeams.
duetotheorderoftheradialLaguerrepolynomialgrowingwith pandbecomingincreasingly
hardtoapproximatebyphase-onlymodulation.Amethodforamplitudeandphasemodulation
via a single phase-only SLM has been demonstrated by Bolduc et al. [45], based on careful
spatial modulation of the blazed grating depth while compensating the resulting phase aber-
rations. However, the technique introduces some additional loss, so that the suitable trade-off
betweenmodequalityandsqueezingmustbefoundinaccordancewiththerequirementsofa
givenapplication.Moregenerally,itmaybepossibletoachieveanimprovementbymeasuring
andcompensatingforanymechanicaldistortionsinthesiliconbackplaneoftheSLM.
3.2. Opticalconversionlosses
Wefindη =0.90(3)forthediffractionefficiencyandη =0.61(2)forthereflectivityofthe
d r
SLM.Thegratingefficiencywasfoundtobeη =0.91(3)inthefirstorder,leadingtoatotal
g
efficiency of η =η η η =0.50(3) with no phase pattern applied. As shown in tables 1 and
d r g
2,thetotalefficiencyisreducedfurtherbyafewpercentformodeswithhighorders.Forthe
arbitrarypattern,theefficiencywas0.15.
3.3. Squeezinginthegeneratedhigher-ordermodes
BeforetheSLM,wetypicallyobserveVar =( 3.0 0.3)dBofamplitudesqueezinginthe
in
− ±
fundamental Gauss beam. The procedure described in 2.3 to quantify squeezing is demon-
strated by example of an LG1 mode in Fig. 5. Here, a noise reduction of ( 1.30 0.30)dB
1 − ±
belowtheshot-noiselevelcanbeseen.Tables1and2showacompletelistofsqueezingvalues
for the LG and BG beams, respectively. For the arbitrary mode pattern, a noise reduction of
( 0.4 0.3)dBbelowshotnoisewasobserved.Tables1and2showthatforlower-ordermodes
− ±
theefficienciesaremuchhigher( 50%),allowingatypicalsqueezingof( 1.3 0.3)dB.All
≈ − ±
quoted squeezing figures were verified by attenuation measurements to result from quantum
m] −52.5
B
d
[ 53.0
e −
c
n
a 53.5
ri −
a
v
54.0
se − ∆ = -1.30 dB
oi
n 54.5
de − u.]
mplitu −55.0 2E[a.|
a −55.50.0 0.2 0.4 0.6 0.8 1.0 |-1.5 -0.75 0 0.75 1.5
time [s] [mm]
Fig.5:Amplitudesqueezing.Noisetraceshowingtypicalamplitudesqueezingat9MHzinthe
LG1beam.Shadedareascorrespondtoonestandarddeviation.
1
noise reduction. For each mode, the measured squeezing matches the expected value corre-
spondingtothemeasuredtotalefficiencyη oftheSLMfortherespectivemodetowithinthe
experimental accuracy. In other words, the upper bound for excess noise at this frequency is
lowerthantheerrorbarsofthemeasurement.Weconcludethatnodetectableexcessnoisewas
addedbytheSLMinthemodeconversionprocessatthe9MHzsideband.
Radialindex Helicalindex Efficiency Squeezing
p l η ∆
1 1 52.0% ( 1.34 0.32)dB
− ±
1 2 51.3% ( 1.32 0.31)dB
− ±
1 3 50.1% ( 1.34 0.31)dB
− ±
2 1 51.2% ( 1.34 0.30)dB
− ±
2 2 51.2% ( 1.31 0.32)dB
− ±
2 3 51.0% ( 1.27 0.32)dB
− ±
3 1 50.3% ( 1.30 0.30)dB
− ±
3 2 50.0% ( 1.30 0.31)dB
− ±
3 3 50.0% ( 1.28 0.31)dB
− ±
Table1:ExperimentalconversionefficiencyandmeasuredsqueezinginLGl beams.
p
Order Efficiency Squeezing
n η ∆
0 47.0% ( 1.17 0.31)dB
− ±
1 47.3% ( 1.17 0.31)dB
− ±
2 47.5% ( 1.17 0.31)dB
− ±
Table2:ExperimentalconversionefficiencyandmeasuredsqueezinginBGbeams.
4. Summary
WehaveshownthatacommerciallyavailableSLMcanbeusedtotransfersqueezingfromthe
fundamental transverse mode of an optical field into arbitrary higher-order modes. With this
approach,differentspatialmodescanbegeneratedsimplybyapplyingadifferentphasepattern
to the SLM with no further modifications to the setup. In principle, the range of achievable
spatial modes is unlimited (up to the resolution of the SLM), but there is a trade-off between
mode quality and conversion efficiency, which ultimately affects the observable squeezing in
the output mode. In all cases, the observed reduction of squeezing was consistent with linear
losses, ruling out excess noise from the SLM at the 9MHz sideband investigated. Our work
provides a direct illustration of the generalised single-mode concept in quantum optics and
shows that applications requiring squeezed light in tailored spatial modes are within reach of
commerciallyavailabletechnology.