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


Classical Interference - First order

Unfiltered 605nm,FWHM5nm HeNelaser

Chris�anHuygens(1629-1659)

ThomasYoung(1773-1829)

Allsinglemodestatesdisplayfirstorderinterference.Firstorderinterferencedoesnotdiscriminatebetweenstates.Itthusdoesnotseparateclassicalfromquantum.


Second order interference

RobertHanburyBrown(1916-2002)

RichardTwiss(1920-2005)

Intensi�escanalsobecorrelatedandhaveacoherencelengthassociatedtotheemi�nglightsource.

R.HanburyBrown&R.Twiss,Nature178,1046-1048(1956)


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:


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)


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.


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



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



2 oscillations where we

would classically expect 1!



N oscillations where we






Phasesuper-resolu�on:Resolvefeatures�messmallerthanwithordinarylight Beyond Rayleigh diffraction limit





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







D A R L







D A R L

Abeamspli�erhasthetransforma�onlaw

andaddi�onalphase-shi�givesthetransforma�on



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


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)


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)


Generalising the projection measurement method

Anycomplexpolynomialcanbefactoredoverthefieldofcomplexnumbers.

Mathematical Theorem:

Implication:

ThecorrespondingprojectortoanyN-photon,two-modestatecanbeimplementedthroughaseriesofbeam-spli�ers,polarisingop�cs,andsignlephotoncoincidencemeasurements!

Example:


Coherent state - temporal instead of spatial splitting

Uncorrelated (product state)!

R.J.Glauber,Phys.Rev.131,2766(1963)

where

Laser

LaserLaser



Uncorrelated (product state)!

R.J.Glauber,Phys.Rev.131,2766(1963)

where

Laser

LaserLaser

Switchspa�alspli�ngfortemporalspli�ng!



R

L

A

D


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

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


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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:




Fourier series

Birefringence







Fourier series

Birefringence




Engineeranyinterferencepa�ern!


Engineered interference


Fourier series

Birefringence





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0.0 0.5 1.0 1.5 2.00.0

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Cou

ntrate

(arb.un

its) 31 term Fourier expansion

of Saw function

Raw data


Engineered interference


Fourier series

Birefringence





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(arb.un

its)

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Cou

ntrate

(arb.un

its)31 term Fourier expansion

of Saw function

Raw data 31 term Fourier expansion

of Rectangular function

Raw data


Distinguishability transitions

Normalize

dcoun

ts

Pathdelay

Completely

indistinguishable

Completely

distinguishable



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


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


Normalize

dcoun

ts

Completely

indistinguishable

Completely

distinguishable

Normalize

dcoun

ts

Pathdelay()

Pathdelay()

Y-S.Raetal.,PNAS110,1227(2013)

Non-monotonicquantumtoclassicaltransi�on?



Normalize

dcounts

Completely

indistinguishable

Completely

distinguishable

Pathdelay()

G.Björk,S.Shabbir,NewJ.Phys.16,013006(2014)

Coincidencedetec�onwindowprojectstheoutputonto

Infact,onecouldwriteprojectorsforsinglephotonandclassicalstatesthatalsoshownon-monotonicbehaviour.



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.


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.


Acknowledgements

GunnarBjörk MarcinSwillo

Thankyou!


Classical and Quantum Interference

Science

Transcript of Classical and Quantum Interference