Table Of ContentComment on Afshar’s expriments
Daniel Reitzner
Research Center for Quantum Information, Institute of Physics,
Slovak Academy of Sciences, Du´bravsk´a cesta 9, 811 45 Bratislava, Slovakia
(Dated: February 1, 2008)
Results of theexperimentscarried out in [1] and [2] are reviewed and theirinterpretation by the
authors is questioned. Argumentsare supported by numerical simulations.
I. INTRODUCTION 1.4e-008
both slits open
upper slit open
lower slit open
1.2e-008
In[2]atwo-slitexperimentsupposedlyshowingthevi-
7 olationofthepricipleofcomplementarityisrealized. Au-
0 1e-008
thors suggest that a which-way information is obtained
0
from the position of incident photon/light while at the 8e-009
2
same time the existence of interference is testified. In I1
n severalarticlesthis experimentis being criticised. These 6e-009
a
critiquesarebasedusuallyonexplanationsdwellingdeep
J
4e-009
insidethebasicsofquantummechanics. Herewepresent
2
numerical results showing that there is no need to go so
2 2e-009
deep to show that the principle of complementarity hold
1 alsointhiscase. In[3]asimpleanalyticanalysisisshown 0
-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008
v to support our arguments and numerical simulation. y1
2 The claim of Afshar, that the law of conservation of
5 FIG.1: Resultsforlight passingthroughselected slits(both,
linear momentum compels us to accept that a photon in
1 upper,or lower one) — theintensity of light as a function of
aparticularspotonthephotosensitivesurfacemusthave
1 position on the first interference plate. With both slits open
originatedfromthe correspondingpinhole,isindeedcor-
0 theinterference pattern is clearly visible.
7 rect. While having only one slit open the momentum of
0 thephotonforcesittoendinastatefromwhichaninfor-
h/ mation about pinhole can be obtained. However having propagation of the wave we alsohave to know, how to
p both slits open transverse momentum of photon (as a implementthelens. Lightpassingthroughalenschanges
- quantum-mechanicalobject)is zero andatthe endmust its phase depending on the distance y from the center
t
n stay zero. The final state of the photon is thus sym- of the lens, since the thickness of the lens depends on
a metric and holds no evidence about the slits. In other this distance. This phase-shift (up to the unsignificant
u wordsthe two-peakeddistributionis aninterferencepat- constant) can be expressed as:
q
tern and the photon behaves as a wave and exhibits no
:
v particle properties until it hits the plate. As a result a δ(y)=−2k 4f2+y2, (2)
Xi which-wayinformationcanneverbeobtainedinthisway. p
where f is the focal length and k = 2π is the circular
r λ
a wave-number of the wave, with λ being its wave-length.
II. NUMERICAL SIMULATION
Including this in the simulation is done by changing the
exponent in the intetgral Eq.(1) to ikr+δ(y) and y is
Tu support previous arguments we performed numer- the position on the originating plane — lens. According
ical simmulations following set-up used in [1]. In the to previous scheme we obtained following results.
simulation process we utilized Huygens-Fresnelprinciple
in the following form. The wave function on a chosen
plane (surface) can be expressed in the form III. RESULTS
eikr
Ψ(r)∝ Ψ0(σ′) dσ′, (1) First we simulated the intensity of light on the first
Z r
photosensitive surface (which is in the experiment after-
σ
wardstakenaway). Ifthelightispassingonlythrougone
whereσ representsthesurfaceofthepreviousplanefrom slit (Fig. 1), we see that the resulting intensity of light
which the wave originates, e.g. for the wave function on — in quantum mechanical sense the probability density
thefirstinterferenceplaneσ representsthesurfaceofthe function of the incident photon — has only one peak.
two pinholes; r represents the distance from the point of Openingbothslitscreatesaninterferencepatternthatis
originandΨ0 is the initial wave-function(on the surface clearlyobservableandcorrespondsalsoquantitativelyto
of the slits being constant). To succesfully simulate the Afshar’s results.
2
3.5e-013 obtained without wires (see Fig. 3) for both slits open,
both slits open
upper slit open since the minima in the interference pattern mean that
3e-013 the probability of finding photon in thet place is zero
and thus no photon is intercepted. Situation of course
2.5e-013
changes when only one slit stays open. Since there is a
non zero probability to find a photon in the positiion of
2e-013
I2 thewires,thesinterceptsomeofthelightsotheobtained
1.5e-013 image has lower peak intensity than that obtained with-
out wires. This decrease in intensity is as large as 10%
1e-013 for collected data.
3.5e-013
5e-014 without wires, both slits open
with wires, both slits open
with wires, only upper slit open
3e-013
0
-0.0005-0.0004-0.0003-0.0002-0.0001 0 0.0001 0.0002 0.0003 0.0004 0.0005
y2 2.5e-013
FIG. 2: Results for light passing through selected slits (both
2e-013
or only upper one) and lens. As geometrical optics predicts
the image of opened slits is obtained in the predicted posi- I2
1.5e-013
tions. However which-way information cannot be obtained
sincetheresults(both numericaland experimental) emanate
1e-013
from wave properties of thelight.
5e-014
By replacing the photosensitive surface by a convex 0
-0.0005-0.0004-0.0003-0.0002-0.0001 0 0.0001 0.0002 0.0003 0.0004 0.0005
lens (in experiment the lens was positioned before this y2
point, however this has no great implication for the re-
FIG. 3: When the wires are placed on the interference pat-
sults) we can now obtain the distribution of the inten-
ternminimafrom thefirstsimulation andbothslitsareopen
sity/photons behind lens in the position where the geo-
the resulting interferrence pattern bihind the lens stays the
metrical optics predicts focused image of slits. The re- same. Howeverwhenonlyoneslitisopenthewirescausethe
sults are depicted on the Fig. 2. When only one slit is decrease in the light intensity since in this case there are no
open, it is clear that the incident photonoriginates from minima in intensity of light on the position of wires and this
the open slit (upper slit thus corresponds to the lower portion of light is intercepted.
peakandlowerslitcorrespondstotheupper slit). When
both slits are open, Afshar claims, that previous results
stillholdandhe canthus obtainthe which-wayinforma- IV. CONCLUSION
tion. Thisishoweverfalsebecausehavingbothslitsopen
is a differentexperiment fromthe one with a singleopen In conclusion, we have numerically simulated experi-
slit. Numerical results confirm this, since to obtain the ment from [1]. Since in the simulation only wave prop-
double-peaked interference pattern in Fig. 2 correspond- erties of light were used and the results correspond to
ing tothe focusedimageofbothslits, weusedonlywave those obtained by Afshar, we conclude that his claim,
propertiesofthelight. Theresultingimageisthusanin- that he can obtain the which-way information from the
terference pattern that does not contain any which-way positionofthephotonincidentonthephotosensitivesur-
information. faceplacedbehindthelens,isfalse. Thewhich-wayinfor-
To further support this claim we present results of mationinthecaseofbothslitsopenisthusunobtainable
simulation when the wires are put on the places where andthe sameprobablyholdsalsoforAfshar’ssecondex-
the interference pattern from the first simulation had its periment carried out in [2].
minima. Now it is clear that this cannot change results
[1] Shahriar S. Afshar: Violation of the principle of Comple- quant-ph/0701039
mentarity, and its implications, Proc. SPIE 5866 (2005) [3] Tabish Qureshi: Complementarity and the Afshar Exper-
229-244, quant-ph/0701027 iment, quant-ph/0701109
[2] ShahriarS.Afshar: ViolationofBohr’sComplementarity:
One Slit or Both?, AIP Cof. Proc. 810, (2006) 294-299,