On the superposition principle in interference experiments

7/25/2019 On the superposition principle in interference experiments

1/91SCIENTIFICREPORTS| 5:10304 | DOI: 10.1038/srep10304

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On the superposition principle in

interference experimentsAninda Sinha1, Aravind H. Vijay1,2& Urbasi Sinha2,3

The superposition principle is usually incorrectly applied in interference experiments. This has

recently been investigated through numerics based on Finite Diference Time Domain (FDTD)

methods as well as the Feynman path integral formalism. In the current work, we have derived

an analytic formula for the Sorkin parameter which can be used to determine the deviation

from the application of the principle. We have found excellent agreement between the analytic

distribution and those that have been earlier estimated by numerical integration as well as resource

intensive FDTD simulations. The analytic handle would be useful for comparing theory with future

experiments. It is applicable both to physics based on classical wave equations as well as the non-

relativistic Schrdinger equation.

It is not widely appreciated that the superposition principle is incorrectly applied in most textbook expo-sitions o intererence experiments both in optics and quantum mechanics14. For example, in a doubleslit experiment, the amplitude at the screen is usually obtained by adding the amplitudes correspondingto the slits open one at a time. However, the conditions described here correspond to different boundaryconditions (or different Hamiltonians) and as such the superposition principle should not be directlyapplicable in this case. Tis incorrect application was pointed out in a physically inaccessible domain

by5

and in a classical simulation o Maxwell equations by6

. More recently7

, dealt with the quantificationo this correction in the quantum mechanical domain where the Feynman path integral ormalism 8wasused to solve the problem o scattering due to the presence o slits . According to the path integral or-malism, the probability amplitude to travel rom point A to B should take into account all possible pathswith proper weightage given to the different paths. In the nomenclature used, paths which extremize theclassical action are called classical paths whereas paths which do not extremize the action are callednon-classical paths. In the usual Fresnel theory o diffraction, the assumption is that the wave ampli-tude at a particular slit would be the same as it would be away rom the slits. Adding to Fresnel theory,we take into account a higher order effect and also account or influence by waves arriving throughneighboring slits. Tis way the naive application o the superposition principle is violated. Tis discus-sion makes it clear that our approach is equally applicable to physics described by Maxwell theory andSchrdinger equationsee supplementary material o Re. 7or urther details.

In Res. 7, 9, 10, the normalized version o the Sorkin parameter (defined later) was estimated. Tiswould be zero i only the classical paths contribute and would be non-zero when the non-classical paths

are taken into account. Te proposed experiment in Re. 7to detect the presence o the non-classicalpaths uses a triple slit configuration as shown in fig. (1). However7, was restricted to semi-analytic andnumerical methods. Te analysis using path integrals was restricted to the ar field regime i.e., theFraunhoer regime in optics and considered cases in which the thickness o the slits is negligible. Onlythe first order correction term was considered in which paths o the kind shown in the inset o fig. (1)contribute. In the current work, we have derived an analytic ormula or as a unction o detectorposition in the Fraunhoer regime. We find that the quantity is very sensitive to certain length param-eters. Tus having an analytic handle is very important as this makes it a much more accessible quantityto experimentalists. Tis would enable experimentalists to have a eel or how errors in the precise

1Centre for High Energy Physics, Indian Institute of Science, Bangalore, India. 2Raman Research Institute,

Sadashivanagar, Bangalore, India. 3Institute for Quantum Computing, 200 University Avenue West, Waterloo,

Ontario, Canada. Correspondence and requests for materials should be addressed to U.S. (e-mail: [email protected].

in) or A.S. (e-mail: [email protected])

Received: 29 December 2014

Accepted: 08 April 2015

Published: 14 May 2015

OPEN
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2SCIENTIFICREPORTS| 5:10304 | DOI: 10.1038/srep10304

knowledge o various parameters can affect the distribution on the detector plane thus making it easierto compare theory with experiments. Te analytic ormula now makes the understanding o the devia-tion rom the naive application o the superposition principle more tractable, shedding more light on theblack-box like understanding that numerical simulations could afford. We have showed that the ana-lytic ormula gives us an excellent match with both photon and electron parameters used in Re. 7.Moreover, it compares very well with Re. 6, where a classical simulation o Maxwell equations usingFinite Difference ime Domain (FDD) methods was done. An important point to note here is that anFDD simulation o in Re. 6needed several days o computation time o a supercomputer and severalterabytes o memory, while our analytic ormula gives us a distribution almost immediately on a stand-ard laptop using Mathematica. O course, an FDD simulation will be able to capture effects due tomaterial properties o the absorber as well as be applicable to near field regimes. However, our analytic

approximation now makes it easy to describe the effect o non-classical paths in different experimentalscenarios without having to go through resource intensive detailed numerics.

In addition to the analytic handle on , we have done a path integral based simulation using a differ-ent numerical approach rom Re. 7 based on Riemannian integration. Tis has enabled us to includeboth ar field and near field regimes in our analysis. We have also verified the effect o increasing thenumber o kinks in the non-classical paths. Our current results now make the experimental conditionsrequired less restrictive in terms o length parameters other than o course providing urther verificationor the results obtained in Res. 6and 7.

The Sorkin parameterConsider the triple slit configuration shown in fig. (1). Let the three slits be labelled A, B and C respec-tively. Te wave unction corresponding to slit A being open is A , that corresponding to slit B beingopen is B and that corresponding to slit C being open is C . Similarly, or both A and B open, it will

be AB , or A, B and C open, it will be ABC and so on. Now, a naive application o the superpositionprinciple will dictate that AB A B = + . However as pointed out in Res. 5, 6, 7, this approximationis strictly not true as the situations described correspond to three different boundary conditions and thesuperposition principle cannot be applied to add solutions to different boundary conditions to arrive ata solution or yet another one. Tis leads to a modification o the wave unction at the screen which nowbecomes:

1AB A B nc = + + ( )

where nc is the contribution due to the kinked i.e., non-classical paths7. uses the Feynman path integral

ormalism to quantiy the effect due to non classical paths in intererence experiments which helps ingetting an idea about the correction nc . Te normalized version o the Sorkin parameter called wasused to propose experiments which can be done to measure such deviations. Te quantity has a specialsymmetry in its ormulation which ensures that it evaluates to zero in the absence o any contribution

Figure 1. Te triple slit set-up with a representative non-classical path. A grazing path has been illustratedin the inset where a path enters slit A, goes to slit C, just enters it and then goes to the detector. Weintegrate over the widths o slits A and C or this path.
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rom the correction term but assumes a finite non-zero value when the correction term is present. Tenumerator o which we call is defined as ollows:

I I I I I I I 2ABC AB BC CA A B C= ( + + ) + ( + + ) ( )

where IABCis the probability or the intensity at the screen when all three slits are open, IABis the inten-sity at the screen when slits A and B are open and so on. aking into account eqn. (2), is defined asollows:

3

= ( )

where is defined as the value o the intensity at the central maximum o the triple slit intererencepattern. I the correction term in eqn. (1) is not taken into account, then will evaluate to zero romalgebra (Tis is under the assumption that Borns rule or probability i.e. Probability 2 is true). Tepresence o the correction term nc makes maniestly non-zero (as explicitly shown in the next section)thus making it a perect tool to investigate such correction effects to the application o the superpositionprinciple in intererence experiments. One has to note here that such correction effects are not a purviewo quantum mechanics alone. Even i one considers classical Maxwell equations and then applies thedifferent boundary conditions corresponding to slits being open one at a time and then all together, oneis able to get a difference in the two situations as per eqn. (1). Tis was shown through FDD solutionsin Re. 6. In Re. 7, we used Feynman Path integral ormalism to analyze the problem and this made theanalysis applicable to the domain o single particles like electrons, photons and neutrons. One has to notethat the path integral analysis is also applicable to the classical domain as we had used the time inde-pendent Helmholtz equation propagator which is applicable to both Maxwell equations and or instancethe time independent Schrodinger equation.

Analytic approximation for in the thin slit case. In this section we will discuss how to get ananalytic expression or the normalized Sorkin parameter in the thin slit approximation (t in fig.1 ismuch smaller than other length parameters in the problem) and in the Fraunhoer limit. We begin byreviewing the logic in Re. 7. Te wave unction at the screen gets contributions rom several paths. Wecan subdivide the paths into ones which involve the classical straight paths going rom source to slit andthen slit to screen, ones which go rom source to slit P then rom slit P to slit Q and then to the screenand so on. Te second category o paths can urther have kinks in them. We will assume that the dom-inant path rom source to slit P is the straight line one and then rom slit P to slit Q is also a straightline one (since the classical path rom P to Q is a straight line one) and rom slit Q to the screen is againthe straight path. Tis approximation seems reasonable to us and we have confirmed this by explicitly

adding kinks to these paths and numerically checking that their contribution is negligible. Keeping thisdiscussion in mind we can or example write the wave unction at the screen as:

A B Copen 4A B C A B A C B A B C C A C B = + + + + + + + + , , , ( ), , , , , ,

A B open 5A A B B A = + + , , ( ), ,

A open 6A = , ( )

where A denotes the classical contribution, A B , denotes the non-classical contribution correspondingto a path going rom source to A, A to B and then B to detector, A C , denotes the non-classical contri-bution corresponding to a path going rom source to A, A to C and then C to detector. Similarly or theother cases. Using this it is easy to check that the numerator in becomes

e2R [ ] 7A B C C B B A C C A C A B B A ( + ) + ( + ) + ( + ) , ( ), , , , , ,

where we have ignored the second order terms like B C C A , , since these turn out to be much smaller

compared to the terms displayed above. An important point to note here is that we could replace P by

P Pk + where P

k denote contributions due to kinks in the paths going rom the source to the slit Pand then to the detector. However these contributions cancel out in as can be easily checked. Tis isone o the main reasons why the triple-slit set up is preerred over the double-slit intererenceI I IAB A B in our discussion o non-classical paths, since in the latter, contributions rom P

k cannotbe ignored and these are difficult to estimate. As in Re. 7we will use the ree particle propagator or aparticle with wave number kgoing rom rto r
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K r rk

i

e

r r2 8

ik r r

(,)=

.( )

Here the normalization actor has been fixed by demanding the composition rule ollowing11

K r r d r K r r K r r 1 3 2 1 2 2 3( , )= ( , ) ( , )

where the integration is over a plane perpendicular to

r r1 3 . We will consider the evolution o the waveunction rom the source to the detector which isgiven in the Feynman path integral ormulation o quantum mechanics by summing over all paths that

go rom the source to the detector. Any path can be thought to be made by integrating small straight linepropagators. Te y-coordinate extents o the slits are A d w d w: 2 2 / , + / , B w w: 2 2 / , / andC d w d w: 2 2 / , + / . Namely the centre to centre distance is d and the width is w. We willassume that the slit has negligible thickness and hence there is no need or an x-integration. Further asargued in Re. 7, afer the stationary phase approximation, the z-integration along the height o the slitin the numerator and denominator are the same and hence we will not have to worry about this eitherand we will drop the zintegrals rom the beginning. We will urther assume a Fraunhoer regime so thatthe source to slit distance Land slit to detector/screen distance are both large compared to any otherscale in the problem. Using the results o Re. 7, we have

kdye

kdye

4 4 9B w

w iky

L

y y

D

w

wik

y y

D2

2

2 2 2

22

2D

D

2 2

= ,( )

+( )

( )

e 10Aikd

y

D B

D

= , ( )

e 11Cikd

y

D B

D

= , ( )

where e LDik L D= /( + ) . In the first line, we have dropped the quadratic terms since y L y D2 2/ , / arevery small in the domain o integration while we have retained the linear term y y D

D/ in order to be

able to compute what happens at the detector screen ar away rom the centre. Afer doing the stationaryphase approximation as explained in Re. 7, we find that

ik

d y d y y y e2 12

P Qik y y ik y

y

D3 2

5 2

1 2 2 1

1 2 D2 1 2

, ( ),

//

/

where the y1 integral runs over slit P and the y

2 integral runs over slit Q. At this stage we observe that

i y y2 1

is large in the domain o integration, then we can approximate the y y1 2, integrals by retain-

ing the leading order term obtained by integrating by parts--this is the standard technique used toapproximate such integrals leading to an asymptotic series. Explicitly, we use denoting y D

D / = and

rescalingy y1 2, by kwe have

d y d y y y e

b ye

b yy b

e

y b

a ye

a yy a

e

y a

Ob y a y

1 1

13

a

b

p

qi y y i y

i b y iy

p

q

i b y i y

p

q

i a y i y

p

qi a y iy

p

q

1 2 2 1

1 2

2

2

1 2 2

2

1 2

2

21 2 2

21 2

2

3 2

2

3 2

2 1 2

2 2 2 2

2 2 2 2

( )( )

( )( )

=

+

( )( )

( )( )

+

,

,

( )

( ) ( )

/

/

/

( )

/

( )

/

/ /

where x( )is the Heaviside step unction and takes into account the modulus sign in the integrand (wehave assumed that 1 ). Using this we finally find (here all length variables have been rescaled by k)
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d

w

wf d w

1

9 2

sin

142

( ) = ( , , ) ,

( )

where

f d w d d d d d d[2 cos 2 cos 4 2 cos cos 2 4 2 cos cos 4 ] ( , , ) = ( ) ( / ) + ( ) ( / ) + ( ) ( / )

dd w

d ww

d d w2 cos 2cos 2

cos 3 2 cos 4

(( ) / )( / )

( / ) ( / )

d

d w

d w

wd d w2

cos 2

cos 2cos 3 2 cos 4

+

(( + ) / )

( / ) ( / ) ( + / )

d

d w

d w

wd w

2

cos 2 2

cos 2cos 2 4

(( ) / )

( / ) ( / )

d

d w

d w

wd w

2

cos 2 2

cos 2cos 2 4

+

(( + ) / )

( / ) ( + / )

In deriving the above expression, we have put in an extra actor o 1 4/ that arises rom inclinationactors as argued in Re. 7. One important cross check that this ormula satisfies is that ( ) becomeszero when wgoes to zero. We point out that the above result is a very good approximation to in the

limit when d w 1 since the term that is neglected in eqn. (13) is O d w23 2 3 2 3 2 ( /( ) ( ) )/ / / afer

reinstating actors o k.Using this approximate expression we can compare with the results o numerical integration in Re.

7. Tis is shown in fig.2(a) and fig.2(b). We find that the agreement with the numerics is excellent inthe ar field regime.

Now using the analytic expression, we can derive a bound on in the regime kw d w1, . Bysetting the trigonometric unctions to their maximum value and adjusting all the relative signs to be thesame, we find afer reinstating actors o k

Figure 2. Comparison between numerics in Re. 7and the analytic approximation. Figure on the lef showsas a unction o angle (where

y

D

180D=

in degrees) or the photon parameters i.e.,slit width =30 m ,inter- slit distance =100 m and wavelength o incident photon=810nm. Te red dotted line indicates theresult o numerical integration where source-slit distance and slit-detector distance =18.1cm as in7. Tiscorresponds to a Fresnel number o 0.006. Te blue line indicates the result o application o the analytic

ormula in eqn. (14) and the blue dots show the result o numerical integration when the Fresnel numberhas been adjusted to 0.0002. Tis implies that in the ar field regime, the analytic ormula and numericalintegration results show perect overlap. We find that or Fresnel number 0 001. leads to a discrepancy o

10% at the centre. Figure on the right shows as a unction o detector position or the electronparameters13i.e.,slit width =62nm, inter-slit distance =272 nm, distance between source and slits =30.5cmand distance between slits and detector =24 cm and de Broglie wavelength o incident electrons=50 pm.Tered dots indicate the result o numerical integration as per7. Te blue line indicates the result o applicationo the analytic ormula in eqn. (14). Te Fresnel number is 0.0002.
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d w0 03

15max

3 2

1 2

. .

( )

/

/

In all examples we have numerically verified that this is a strict upper bound. It will be a useul simpleormula to remember.

Comparison with FDTD. We can use the analytic expression to compare with the FDD results inRe. 6. Although the FDD simulations were done or non-zero thickness or the slits and or non-idealmaterials, we will find that the analytic ormula agrees remarkably well with the FDD results. Te rea-son or this agreement is the ollowing. One can repeat the steps outlined above but now with thickness.o handle the thick slit case, we can consider two slit planes instead o one where the separation betweenthe two planes is given by the thickness o the slit. Ten there are paths that reach rom the source tothe first slit plane, rom the first plane to the second plane, and finally rom the second plane to thedetector. We can as beore use the stationary phase approximation. In the end we find that again the z-integrals cancel out and we are lef with expressions o the orm

Kk

e

d y d y d y

ik y y t y y

y y y y t

2

exp

16

ncA i

y

L

y y

D

32

1 2 3

2 2 12 2

2 3 2

3 2

1 2

2 1

2 2 1 4

D12

32

=

+ ( ) + + +

( ) +

,

( )

/

( )

/ /

where there is the thickness. y y1 2, are the y-coordinates on the first and second slit plane respectively

involving slit A and y3 is on the second slit plane involving slits B or C--this denotes a path where the

kink in the path occurs at the second slit plane. Now there are two observations to make. First, thereshould also be a contribution rom a path that has a kink in the first slit plane. When the thickness issmall (t ) then these two paths will approximately be in phase and hence there will be an overallactor o 2 42 = in compared to the thin slit approximation. Second, when t is small, the actor

y y t12 1

2 2 1 4/ ( ) +

/

will be sharply peaked aroundy y2 1

= and hence the result o they2integral will

lead to an expression which is the same as in the thin slit case. Now i we wanted to compare with theFDD simulations in6, we note that the material making the slits in the simulations was considered tobe steel with a complex reractive index. Te effect o the imaginary part o the reractive index is tomake the effective slit width bigger compared to the idealized scenario we are considering. By consider-ing a slightly bigger w, we find that the agreement o the analytic expression or with the FDD sim-ulation or d w t3 4 = , = , = as considered in6is remarkably good as shown in fig.(3). Te complexreractive index or steel as used in 6 or FDD simulation is n n n i i2 29 2 61R I= + = . + . . Using the

Figure 3. Comparison with the FDD simulations in6or the d w t3 4 = , = , = case. In the figure onthe lef, the black dots indicate the FDD values which have been read off rom fig.(2b)in Re. 6. Teorange line indicates the analytic expression while the blue line which leads to an agreement with the FDDresult is the analytic expression with d w3 1 15 = , = . . Tis choice o win the analytic expression hasbeen justified in the text. In the figure on the right, we compare the analytic ormula with the numerical

integration as in Re. 7or the d w t3 1 15 4 = , = . , = case. Te red dots indicate the result romnumerical integration while the blue line indicates the analytic expression. Te good agreement, especiallyclose to the central region justifies our usage o the analytic approximation or this choice o parameters.
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act that the wave gets attenuated by n xexp 2 I ( / )12at a distance o xinside the material, we find that

or x 0 075= . the attenuation actor is 30%. Tis gives an effective increase in the slit width which wewill take to be 0 075 0 15 . = . .

The Sorkin parameter in the Fresnel regime. In Re. 7as well as in deriving the analytic approx-imation in our current work, we needed to be in the Fraunhoer regime. Tis enabled us to expand thepropagator distance or example in eqn. (9) which was crucial in the simplifications arising rom thestationary phase approximation namely the integral over the height o the slits cancelled between thenumerator and denominator in . However, in order to consider the Fresnel regime, we can no longerappeal to this simplification and will need to consider a different numerical approach. While FDD canenable us to address the same question, as pointed out in the Introduction, it is computationally resourceintensive. Te approach we will outline below is more efficient in addressing this issue. We will use thecommon technique or numerical integration which is Riemannian integration14. Te technique involvesdividing a certain domain into many smaller sub-domains and assuming that the integrand unction is

constant across the domain. One then sums up the constants multiplied by the area o the sub-domainsto get the integral o the unction over the whole domain. Our code to evaluate was written in theC++ programing language. We retained the exact propagator distances and integrated over the lengtho the slit along the z-axis in fig. 1. We used the same parameters that were used to generate fig.(3a)oreerence7and in addition chose the height to be 300m.

Figure 4shows as a unction o distance between slit plane and detector plane . We find that usingthis approach, the value o at cm20= is 6 10 7 while the analytic ormula gives 5 6 10 7 . which means a deviation o around 7%. Tus, already or a Fresnel number o 0.005 (which correspondsto D=20cm), the agreement between our numerical integration approach and analytic approximation is

very good. As the distance between the slit plane and the detector plane decreases, the value o startsincreasing which can be explained by the decrease o the value o the denominator o . However, thesudden dip at very close distances may be an arteact o our approximation and the act that the parax-ial approximation breaks down in the extreme near field regime.

Can we use the new numerical approach to compare with FDD results? In our current numerical

approach, we have made certain assumptions which are summarized next. Te first assumption is o asteady source which ollows scalar electrodynamics. Tis approximation will break down in case o polar-ized radiation but in construction o the quantity or unpolarized light, the polarization sums cancelin the numerator and denominator. Moreover, induction o currents in materials used or building theapparatus and scattering o radiation are not accounted or in the above derivations i.e.,we have assumedthat our material is a perect absorber. Tis approximation will break down when the material scattersradiation to a significant effect and behaves as a secondary source due to induction. We have also usedan approximate orm or Kirchoffs integral theorem1 whereby it is important that length scales in theproblem are much larger than the wavelength o incident radiation. One could use the complete Kirchoffsboundary integral in order to investigate problems where the length scales are comparable to wavelength.A final approximation used in this section is that o paraxial rays. Tis one breaks down when the dis-tance between the slit plane and source or screen is not large compared to the vertical position on thescreen or the slit width i.e.,when we want to plot as a unction o detector positions which are veryar rom the central region. We leave more careul investigation o the Fresnel regime or the comparison

Figure 4. (at the central maximum o the triple slit intererence pattern) as a unction o distance

between slit plane and detector screen plane D. Te parameters used are slit width w 30= m, inter slitdistance d 100= m, height o the slit =300m, incident wavelength =810 nm, source to slit planedistance L cm20= .
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between the Riemannian integration based technique outlined above and the resource intensive FDDapproach or uture work.

DiscussionIn this paper, we have derived an analytic expression or the Sorkin parameter which has been used toquantiy deviations rom the naive application o the superposition principle in slit-based intererenceset-ups. Our main ormula in eqn. (14) can be trusted in the Fraunhoer regime and in a thin-slit approx-imation. When the thickness o the slit plane is not too big, we have given an argument on how to useour analytic ormula which led to impressive agreements with the FDD simulations o Re. 6as well as

numerical integrations o Re. 7. In the uture, it will be interesting to develop systematics o the thick-slitscenario ollowing some o the techniques used in this paper. One important point to note is that in thefinal expression or . i.e. eqn. (14), appears only indirectly through the de Broglie wavelength. Tisis in keeping with our claim that our analytic ormula should be applicable or both Maxwells equationsas well as the Schrdinger equation. Te non-zeroness o is essentially due to boundary conditionconsiderations and should affect both classical as well as quantum physics. In existing experimentalresults in literature which measure or example Res. 10, 15, 16, 17,the experimental inaccuracies haveprevented us rom concluding that is non-zero. As is easy to see in our analytic ormula, is verysensitive to experimental parameters. Future experimental attempts will benefit rom our analytic handleas it would be much easier to compare experimental data with theoretical expectations. One has to notehere that the quantity has been measured previously to test or a possible deviation rom Born rule.So, what do our findings imply or using as a test or Born rule? should be used or a Born rule testin experimental situations where the non-zeroness due to the correction to the superposition principle

is very small. For instance, a set-up like in

15

has a very small correction rom non classical paths andcould be a good experiment still to test Born rule. Tus, any potentially detectable violation o the Bornrule should be bigger than that due to non-classical paths and any uture test should take this intoaccount.

References1. Born, M. & Wol, E. Principles of Optics(Cambridge University Press, Seventh expanded edition, 1999).2. Feynman, . P., Leighton, . & Sands, M. Te Feynman Lectures on PhysicsVol. 3 (Addison - Wesley, eading Mass, 1963).3. Cohen-annoudji, C., Diu, B. & Laloe, F. Quantum Mechanics I(Wiley - VCH, 2nd edition, 2005)4. Shanar, . Principles of Quantum Mechanics(Springer, 2nd edition, 1994).5. Yabui, H. Feynman path integrals in the young double-slit experiment. Int. J. Teor. Ph. 25,No.2, 159174 (1986).6. aedt, H. D., Michielsen, . & Hess, . Analysis o multipath intererence in three-slit experiments. Phys. ev. A.85, 012101

(2012).7. Sawant, ., Samuel, J., Sinha, A., Sinha, S. & Sinha, U. Non classical paths in quantum intererence experiments. Phys.ev.Lett.

113,120406 (2014).8. Feynman, . P. & Hibbs, A. . Quantum Mechanics and Path Integrals (McGraw-Hill, New Yor, 3rd. ed. 1965).

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AcknowledgmentsWe thank Joseph Samuel, Barry Sanders, Diptiman Sen and Supurna Sinha or useul discussions andAnimesh Aaryan or assistance with fig. 1. We thank Anthony Leggett, Barry Sanders and Diptiman

Sen or comments on the draf. US wishes to thank Nicolas Copernicus University, Poland or kindhospitality while a part o the work was being carried out. AS acknowledges partial support rom aRamanujan ellowship, Govt. o India.

Author ContributionsA.S. and U.S. ormulated the questions, A.S. perormed the analytic calculations with inputs rom A.H.V.and U.S., A.H.V. perormed the numerical simulations in the Fresnel regime. A.S. and U.S. wrote themanuscript and all authors reviewed it.

Additional InformationSupplementary informationaccompanies this paper at http://www.nature.com/srep

Competing financial interests: Te authors declare no competing financial interests.
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9SCIENTIFIC REPORTS | 5:10304 | DOI: 10 1038/srep10304

How to cite this article: Sinha, A. et al. On the superposition principle in intererence experiments.Sci. Rep.5, 10304; doi: 10.1038/srep10304 (2015).

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On the superposition principle in interference experiments

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