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TheinterferometricsignaturesofquantumandclassicalstatesoflightSarooshShabbirQuantumElectronics&QuantumOp�csKTHRoyalIns�tuteofTechnology,Stockholm
Interferometric signatures of quantum and classical states
Dointerferometricsignalsofquantumstatesdifferfundam-entallyfromclassicalstates,intermsofshapeandvisibility?
Howdotheinterferometricsignalsvaryasstatesaretrans-formedfromquantumtoclassical?
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 1 / 38
Outline
Interference-FromHuygens&YoungtoHanburyBrown&Twiss
Higherorderinterference
Two-modeprojec�onmeasurements
Quantuminterferencefromsemi-classicalstates
Engineeredinterference
Projec�onmeasurementsofincreasinglydis�nguishablestates
SummaryandConclusions
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 2 / 38
Classical Interference - First order
Unfiltered 605nm,FWHM5nm HeNelaser
Chris�anHuygens(1629-1659)
ThomasYoung(1773-1829)
Allsinglemodestatesdisplayfirstorderinterference.Firstorderinterferencedoesnotdiscriminatebetweenstates.Itthusdoesnotseparateclassicalfromquantum.
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 3 / 38
Second order interference
RobertHanburyBrown(1916-2002)
RichardTwiss(1920-2005)
Intensi�escanalsobecorrelatedandhaveacoherencelengthassociatedtotheemi�nglightsource.
R.HanburyBrown&R.Twiss,Nature178,1046-1048(1956)
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 4 / 38
Quantum Interference - Hong-Ou-Mandel effect
2,0 1,1 0,2
25% 50% 25%
Classicallywewouldget:
2,0 1,1 0,2
50% 0% 50%
Whenwavepacketsoverlap:
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 5 / 38
Quantum Interference - Hong-Ou-Mandel effect
2,0 1,1 0,2
25% 50% 25%
Classicallywewouldget:
2,0 1,1 0,2
50% 0% 50%
Whenwavepacketsoverlap:
Hongetal.,PRL(1987)
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 6 / 38
Higher order (multiphoton) quantum interference
Y.-SKimetal.,Opt.Express19,24956(2011)
Boththegenera�onandthedet-ec�onofmul�-photonstatesiscomplicated.
Non-linearop�csisrequiredtogeneratestates.
Polarisa�onop�csandcoincide-ncedetec�onisrequiredtodetectstates.
Themeasurementisprobabilis�c.OnlywhenNphotodetectorsclickincoincidencetheresultisrec-orded.
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 7 / 38
N00N states, de Broglie waves and quantum phasesuper-resolution
1 2 3 4 5 6
0.2
0.4
0.6
0.8
1.0
phaseshi�
No.ofcou
nts
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 8 / 38
N00N states, de Broglie waves and quantum phasesuper-resolution
phaseshi�
No.ofcou
nts
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
Phase difference
Countrate
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 9 / 38
N00N states, de Broglie waves and quantum phasesuper-resolution
2 oscillations where we
would classically expect 1!
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 10 / 38
N00N states, de Broglie waves and quantum phasesuper-resolution
N oscillations where we
would classically expect 1!
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 11 / 38
N00N states, de Broglie waves and quantum phasesuper-resolution
N oscillations where we
would classically expect 1!
Phasesuper-resolu�on:Resolvefeatures�messmallerthanwithordinarylight Beyond Rayleigh diffraction limit
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 12 / 38
N00N states, de Broglie waves and quantum phasesuper-resolution
N oscillations where we
would classically expect 1!
Phasesuper-sensi�vity:Uncertaintyinphasemeasurement
Phasesuper-resolu�on:Resolvefeatures�messmallerthanwithordinarylight Beyond Rayleigh diffraction limit
Heisenberg limit
J.Jacobson,G.Björk,I.Chuang,andY.Yamamoto,PRL74,4835-4838(1995)Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 13 / 38
Measurement post-selection of N00N states
giventhatwehaveonly available.
Supposewewanttoprojectoutthestate
Writethewantedstateas
Formthepolynomialandfactoriseovercomplexnumbers
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 14 / 38
Measurement post-selection of N00N states
giventhatwehaveonly available.
Supposewewanttoprojectoutthestate
Writethewantedstateas
Formthepolynomialandfactoriseovercomplexnumbers
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 15 / 38
Measurement post-selection of N00N states
giventhatwehaveonly available.
Supposewewanttoprojectoutthestate
Writethewantedstateas
Formthepolynomialandfactoriseovercomplexnumbers
D A R L
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 16 / 38
Measurement post-selection of N00N states
giventhatwehaveonly available.
Supposewewanttoprojectoutthestate
Writethewantedstateas
Formthepolynomialandfactoriseovercomplexnumbers
D A R L
Abeamspli�erhasthetransforma�onlaw
andaddi�onalphase-shi�givesthetransforma�on
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 17 / 38
Measurement post-selection of N00N states
D A R L
Abeamspli�erhasthetransforma�onlaw
andaddi�onalphase-shi�givesthetransforma�on
Coincident detection in all 4 SPDs
projects outs the NOON4 state
from the input!
R
L
A
D
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 18 / 38
Post-selection using coherent state input
A linearly polarised coherent state
also has a non-zero overlap with
NOON4 state!
R
L
A
D
Withveryweakexcita�on,probabilityofhaving5ormorephotons<<probabilityofhavingexactly4photons
If4detectorsclickincoincidence,wearepre�ysurewe'vedetectedNOON4state!
K.J.Reschetal.,Phys.Rev.Le�.98,223601(2007)
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 19 / 38
Quantum optics from semi-classical states
A linearly polarised coherent state
also has a non-zero overlap with
NOON4 state!
R
L
A
D
Withveryweakexcita�on,probabilityofhaving5ormorephotons<<probabilityofhavingexactly4photons
If4detectorsclickincoincidence,wearepre�ysurewe'vedetectedNOON4state!
K.J.Reschetal.,Phys.Rev.Le�.98,223601(2007)
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 20 / 38
Generalising the projection measurement method
Anycomplexpolynomialcanbefactoredoverthefieldofcomplexnumbers.
Mathematical Theorem:
Implication:
ThecorrespondingprojectortoanyN-photon,two-modestatecanbeimplementedthroughaseriesofbeam-spli�ers,polarisingop�cs,andsignlephotoncoincidencemeasurements!
Example:
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 21 / 38
Coherent state - temporal instead of spatial splitting
Uncorrelated (product state)!
R.J.Glauber,Phys.Rev.131,2766(1963)
where
Laser
LaserLaser
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 22 / 38
Coherent state - temporal instead of spatial splitting
Uncorrelated (product state)!
R.J.Glauber,Phys.Rev.131,2766(1963)
where
Laser
LaserLaser
Switchspa�alspli�ngfortemporalspli�ng!
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 23 / 38
Coherent state - temporal instead of spatial splitting
R
L
A
D
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 24 / 38
N00N states projected from a coherent state
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0 N=30
N=60
Cou
nt r
ate
(arb
. un
its) 0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0 N=15
Phase difference (π radians)
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0.6 0.7 0.8 0.9 1.0
1.6
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1.7 1.8 1.9 2.0
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0.6 0.7 0.8 0.9 1.0
Visibility Max 88 %Min 57.5 %
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0 N=30
N=60
Cou
nt r
ate
(arb
. un
its) 0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0 N=15
Phase difference (π radians)
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0.6 0.7 0.8 0.9 1.0
1.6
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0.6 0.7 0.8 0.9 1.0
Visibility Max 88 %Min 57.5 %
phaseshi� radians)(
S.Shabbir,M.Swillo,G.Björk,Phys.Rev.A87,053821Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 25 / 38
Arbitrary interference a using coherent state
S.Shabbir,M.Swillo,G.Björk,Phys.Rev.A87,053821
Birefringence
N-photoncoincidentdetec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 26 / 38
Arbitrary interference a using coherent state
S.Shabbir,M.Swillo,G.Björk,Phys.Rev.A87,053821
Fourier series
Birefringence
N-photoncoincidentdetec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 27 / 38
Arbitrary interference a using coherent state
S.Shabbir,M.Swillo,G.Björk,Phys.Rev.A87,053821
Fourier series
Birefringence
N-photoncoincidentdetec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Engineeranyinterferencepa�ern!
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 28 / 38
Engineered interference
S.Shabbir,M.Swillo,G.Björk,Phys.Rev.A87,053821
Fourier series
Birefringence
N-photoncoincidentdetec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Engineeranyinterferencepa�ern!
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■
■
■
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
Phase difference (π radians)
Cou
ntrate
(arb.un
its) 31 term Fourier expansion
of Saw function
Raw data
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 29 / 38
Engineered interference
S.Shabbir,M.Swillo,G.Björk,Phys.Rev.A87,053821
Fourier series
Birefringence
N-photoncoincidentdetec�on
General two-mode state:
Overlap with phase-shifted coherent state:
Engineeranyinterferencepa�ern!
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■
■
■
■
■
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■
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
Phase difference (π radians)
Cou
ntrate
(arb.un
its)
■
■ ■
■
■
■
■
■
■
■
■■
■
■
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
Phase difference (π radians)
Cou
ntrate
(arb.un
its)31 term Fourier expansion
of Saw function
Raw data 31 term Fourier expansion
of Rectangular function
Raw data
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 30 / 38
Distinguishability transitions
Normalize
dcoun
ts
Pathdelay
Completely
indistinguishable
Completely
distinguishable
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 31 / 38
Distinguishability transitions
Normalize
dcoun
ts
Completely
indistinguishable
Completely
distinguishable
Normalize
dcoun
ts
Pathdelay()
Pathdelay()
Y-S.Raetal.,PNAS110,1227(2013)Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 32 / 38
Distinguishability transitions
Normalize
dcoun
ts
Completely
indistinguishable
Completely
distinguishable
Normalize
dcoun
ts
Pathdelay()
Pathdelay()
Y-S.Raetal.,PNAS110,1227(2013)Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 33 / 38
Distinguishability transitions
Normalize
dcoun
ts
Completely
indistinguishable
Completely
distinguishable
Normalize
dcoun
ts
Pathdelay()
Pathdelay()
Y-S.Raetal.,PNAS110,1227(2013)
Non-monotonicquantumtoclassicaltransi�on?
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 34 / 38
Distinguishability transitions
Normalize
dcounts
Completely
indistinguishable
Completely
distinguishable
Pathdelay()
G.Björk,S.Shabbir,NewJ.Phys.16,013006(2014)
Coincidencedetec�onwindowprojectstheoutputonto
Infact,onecouldwriteprojectorsforsinglephotonandclassicalstatesthatalsoshownon-monotonicbehaviour.
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 35 / 38
Distinguishability transitions
Normalize
dcounts
Completely
indistinguishable
Completely
distinguishable
Pathdelay()
G.Björk,S.Shabbir,NewJ.Phys.16,013006(2014)
Coincidencedetec�onwindowprojectstheoutputonto
Infact,onecouldwriteprojectorsforsinglephotonandclassicalstatesthatalsoshownon-monotonicbehaviour.
Non-monotonicprojec�onprobabili�esasafunc�onofdis�nguishabilitydonotsignalquantumtoclassicaltransi�on.
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 36 / 38
Summary & Conclusions
Itispossibletodemonstratehighly"non-classical"interferenceeffectsusingcoherentstateinput.Thespecialcharacterofthecoherentstateallowsthemeasurementtobedone"inseries"ratherthan"inparallel",saving�meandmaterialresources.
Themeasurementnon-linearitycreatesthedesired"non-classical"interference.
Mul�-photoninterferencecangivehighlyunusualinterferenceeffects/pa�erns.
Usinglinearop�csandsinglephotoncountersonecansynthesizeanytwo-modeprojec�onmeasurement.
Itisalsopossibletoimplementengineeredinterference.Any"Fourierspectrum"canbeobtained.
However,themeasurementisprobabilis�c,whichmeansthatit'snotanefficientmethodintermsofinputphotons.Photonnumberresolvingdetectorswouldimprovethedetec�onefficiency.
Ingeneral,neithertheshapeoftheinterferencepa�ernnorthevisibilityaresignaturesofquantumstates.
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 37 / 38
Acknowledgements
GunnarBjörk MarcinSwillo
Thankyou!
Saroosh Shabbir, Gunnar Bjork Interferometric signatures June 3, 2014 38 / 38