classical and quantum interference

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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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

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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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■

■

■

■

■

■

■

■

■

■

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