Illustration of quantum complementarity using single photons interfering on a grating

Abstract

A recent experiment performed by Afshar et al (2007 Found. Phys. 37 295–305) has been interpreted as a violation of Bohr's complementarity principle between interference visibility and which-path information (WPI) in a two-path interferometer. We have reproduced this experiment, using true singlephoton pulses propagating in a two-path wavefront-splitting interferometer realized with a Fresnel's biprism, and followed by a grating with adjustable transmitting slits. The measured values of interference visibility V and WPI, characterized by the distinguishability parameter D, are found to obey the complementarity relation V2 + D2 6 1. This result demonstrates that the experiment can be perfectly explained by the standard interpretation of quantum mechanics.

Full text

Illustration of quantum complementarity using
single photons interfering on a grating
V Jacques1, N D Lai1, A Dr´eau1, D Zheng1, D Chauvat1,
F Treussart1, P Grangier2, and J-F Roch1
1Laboratoire de Photonique Quantique et Mol´eculaire, Ecole Normale Sup´erieure de
Cachan, UMR CNRS 8537, Cachan, France
2Laboratoire Charles Fabry de l’Institut d’Optique, UMR CNRS 8501, Palaiseau,
France
E-mail: roch@physique.ens-cachan.fr
Abstract. A recent experiment performed by S. S. Afshar et al. has been interpreted
as a violation of Bohr’s complementarity principle between interference visibility
and which-path information in a two-path interferometer. We have reproduced this
experiment, using true single-photon pulses propagating in a two-path wavefront-
splitting interferometer realized with a Fresnel’s biprism, and followed by a grating
with adjustable transmitting slits. The measured values of interference visibility V
and which-path information, characterized by the distinguishability parameter D, are
found to obey the complementarity relation V2+D2≤1. This result demonstrates
that the experiment can be perfectly explained by the Copenhagen interpretation of
quantum mechanics.
PACS numbers: 03.65.Ta, 42.50.Ar, 42.50.Xa
Submitted to: New J. Phys.
arXiv:0807.5079v2 [quant-ph] 10 Feb 2009

Illustration of quantum complementarity using single photons interfering on a grating2
1. Introduction
Bohr’s principle of complementarity states that every quantum system has mutually
incompatible properties which cannot be simultaneously measured [1]. This principle is
commonly illustrated by considering single particles in a two-way interferometer where
one chooses either to observe interference fringes, associated to a wave-like behavior, or
to know which path of the interferometer has been followed, according to a particle-like
behavior [2]. In such an experiment, any attempt to obtain some which-path information
(WPI) unavoidably reduces interference, and reciprocally. The incompatibility between
these two measurements is then ensured by the complementarity inequality [3,4]:
V2+D2≤1 (1)
which puts an upper bound to the maximum values of independently determined
interference visibility Vand path distinguishability D, the parameter that quantifies
the available WPI on the quantum system [4].
The two all-or-nothing cases (V= 1, D = 0) and (V= 0, D = 1) have been clearly
confirmed by experiments performed with a wide range of quantum objects [5–15], as well
in the quantum eraser configuration [16–18] or in Wheeler’s delayed-choice regime [19].
The complementarity inequality (1) has also been successfully verified in intermediate
regime, corresponding to partial WPI and reduced visibility, with atoms [18,20], nuclear
spins [21], and single photons in the delayed-choice regime [22]. Although recent
discussions focused on the mechanism which enforces complementarity, by discussing
its relation with Heisenberg’s uncertainty relations [23–26], it is well established that
Bohr’s complementarity principle is a cornerstone of quantum mechanics [27].
Recently, Afshar et al. have claimed to be able to violate this principle [28, 29].
Their experimental scheme, depicted on figure 1, can be summarized as follows:
Attenuated laser light illuminates a Young’s double-pinhole screen which produces an
interference pattern at a distance behind the two pinholes S1and S2where the two
diffracted beams overlap. Using a lens, each pinhole is imaged on an associated detector,
i.e. S1on P1and S2on P2. Each detector is then univocally associated to a given path
of the interferometer, leading to the full knowledge of the WPI and corresponding to
D= 1.
In order to simultaneously recover the complementary wave-like information, a grid
of thin wires is inserted close to the imaging lens. The wires are exactly superposed
on the dark fringes of the interference pattern (see figure 1). Using a particle-like
description, Afshar et al. claim that no photon is blocked by the grid and the
signals associated to the output detectors P1and P2remain almost unchanged, as
experimentally verified. Their conclusion is that the grid of wires perfectly reveals
the interference pattern while keeping a perfect WPI, corresponding to combined
measurements of V= 1 and D= 1. This result, in clear contradiction with
inequality (1), is interpreted as a violation of the complementarity principle.
Different papers have pointed out the flaws in the interpretation of the experiment
and explained why there is no contradiction with Bohr’s complementarity [30]. In this

Illustration of quantum complementarity using single photons interfering on a grating3
laser
attenuated
G
L
S1
S2
P2
P1
Figure 1. Simplified representation of Afshar’s experiment [28, 29]. An attenuated
laser illuminates a Young’s double-pinhole interferometer. A lens (L) images each
pinhole S1and S2on two detectors P1and P2. A grid of thin wires (G) with a period
matching the interfringe is positioned after the lens so that the wires of the grid are
exactly superimposed on the dark fringes of the interference pattern.
paper, we report an experiment designed to check the complementarity inequality using
a setup similar to the one of figure 1, the Young’s double-pinhole being replaced by a
Fresnel’s biprism. To be meaningful, the experiment is realized with true single-photon
pulses for which full and unambiguous WPI can be obtained, complementary to the
observation of interference [31].
The paper is organized as follows: We start with a wave-like analysis of the
experiment, allowing us to determine the interference visibility Vand the path
distinguishability parameter D. We demonstrate that the set of these two parameters
obeys inequality (1). This analysis is then compared to the experiment. The results
correspond to the almost ideal case, close to the upper bound of inequality (1).
2. Afshar’s setup with a Fresnel’s biprism: A wave-optical analysis
Figure 2-(a) shows the setup corresponding to two separated incident beams at normal
incidence on a Fresnel’s biprism, with two output detectors P1and P2positioned far
away from the overlapping region of the two deviated beams [15]. Each detector is then
unambiguously associated to a given path of the interferometer, i.e. detector P1to path
1 and detector P2to path 2. The experiment depicted on figure 1 can then be reproduced
by introducing a transmission grating inside the interference zone corresponding to the
overlap of the two beams refracted by the biprism.
A strong assumption in Afshar’s interpretation is that positioning the wires of the
grid at the dark-fringe locations is enough to reveal the existence of the interference
pattern, without inducing any further perturbation on the transmitted light field.
However, the grid has an unavoidable effect due to diffraction, which redirects some
light from path 1 to detector P2and, reciprocally, from path 2 to detector P1. The
introduction of the grid has then partially erased the WPI since it becomes impossible
to univocally associate each output detector to a given path of the interferometer.
We first need to evaluate the influence of diffraction due to the grating G. As shown

Illustration of quantum complementarity using single photons interfering on a grating4
(c)
a= 80 µm
a= 20 µm
(d)
u/u0
=
α/α0
x/Λ
P1
u/u0
=
α/α0
x/Λ
P2
P2
P1
α0
α
t(x)
a
1
0
x
Λ
P2
P1
FB G
x
(a) (b)
path 1
path 2
β
Figure 2. Modified Afshar’s experiment with a Fresnel’s biprism (FB) of summit angle
βand two interfering paths 1 and 2. (a)-Two detectors P1and P2are positioned far
away from the interference area and are therefore each univocally associated to a given
path of the interferometer. A grating (G) is then introduced in the interference area
and can be moved along the x-axis of the interference pattern. (b)-G is modelized as an
amplitude transmission function t(x) with periodicity Λ and transmitting slits of width
a. (c)-(d) Light intensity distribution after diffraction by the grating G as a function
of angle αand grating position x, for transmitting slit width values a= 80(c)µm
and 20 µm (d). Light intensities of all diffraction orders undergo maxima and minima
when G is translated from a bright interference fringe (x=pΛ, p = 0,1,2. . .) to a
dark interference fringe (x=pΛ + Λ/2, p = 1,2. . .). The detectors P1and P2are
respectively associated with propagation at oblique angle α=−α0(u=−u0) and
α=α0(u=u0)( black arrows). The calculation is done with β= 7.5×10−3rad,
Λ = 87 µm, and N= 20, corresponding to the values of the experiment described in
the latter.
in figure 2-(b), G corresponds to transmitting slits of width awith a periodicity equal
to the interfringe Λ of the interference pattern obtained with monochromatic light of
wavelength λ. The interfringe depends on the deviation angle α0= (n−1)βcaused by
the Fresnel’s biprism of refraction index nand summit angle β:
Λ = λ
2α0
=1
2u0
,(2)
when expressed as a function of the associated spatial frequency u0=α0/λ.
As well known from quantum optics [32], all optical phenomena like interference,
diffraction, and propagation, can be calculated using the classical theory of light even
in the single-photon regime. Then, using classical-wave Fraunhofer diffraction, the

Illustration of quantum complementarity using single photons interfering on a grating5
diffracted wave amplitudes S1(u) and S2(u) associated to path 1 and path 2 of the
interferometer are:
S1(u) = S0sinc[π(u+u0)a]sin[Nπ(u+u0)Λ]
sin[π(u+u0)Λ] eiπ(N−1)hu−u0
2u0ie−2iπ(u−u0)x(3)
S2(u) = S0sinc[π(u−u0)a]sin[Nπ(u−u0)Λ]
sin[π(u−u0)Λ] eiπ(N−1)hu+u0
2u0ie−2iπ(u+u0)x(4)
where u=α/λ is the spatial frequency associated to propagation with oblique angle α,
xis the position of the grating along the x-axis and Nis the number of transmitting
apertures illuminated by the incident beams of equal amplitude S0.
Consequently, detector P1(resp. P2) positioned in direction u=−u0(resp. at
u=u0) is associated to the zero-order diffraction (resp. first-order) from path 1 and also
to the first-order diffraction (resp. zero-order) from path 2. The WPI on the behavior
of a single-photon in the interferometer is then partially erased as each detector cannot
be associated to a given path.
To test inequality (1), a value of the distinguishability parameter Dis required,
to quantify the amount of WPI that can be extracted in the experiment. Following
the discussion of reference [22], we introduce the parameters D1and D2, respectively
associated to the WPI on path 1 and on path 2:
D1=|p(P1,path 1) −p(P2,path 1)|,(5)
D2=|p(P1,path 2) −p(P2,path 2)|,(6)
where p(Pi,path j) is the probability that the particle follows path j and is detected on
detector Pi. The distinguishability parameter Dis then finally defined as [4]:
D=D1+D2.(7)
Using true single-photon pulses and photodetectors operating in the photon
counting regime, the values of D1and D2can be estimated by blocking one path of
the interferometer and measuring the corresponding number of detections N1and N2
on detectors P1and P2. This quantities are statistically related to D1and D2according
to [20, 22] :
D1=1
2



N1−N2
N1+N2


path 2 blocked
,(8)
D2=1
2



N1−N2
N1+N2


path 1 blocked
.(9)
Using equations (3) and (4), the distinguishability parameter Dis then equal to:
D=1−sinc2(2πu0a)
1 + sinc2(2πu0a).(10)
In the extreme case of a grating consisting of Dirac transmission peaks (equivalent
to the limit case a= 0), Dis equal to zero and no WPI can be obtained. Conversely,

Illustration of quantum complementarity using single photons interfering on a grating6
when the grating is absent, equivalent to the case where a= Λ, we obtain D= 1.
In order to retrieve the complementary wave-like information associated to the
single-photon detections on detectors P1and P2, a quantitative measurement of the
interference visibility is required. Note that such measurement cannot be realized
by positionning the grating as described in references [28, 29]. Indeed, the visibility
inferred using such method is related to photons intercepted by the grid for which no
WPI is available. We stress that the complementarity inequality is only meaningful if
both complementary measurements are performed for the same photons, i.e. either for
photons transmitted behind the grating or for photons intercepted by the grating.
Using detectors P1and P2, the wave-like information complementary to the WPI
defined by equation (10) can be measured by translating the grating along the x-direction
(see figure 2-(a)). The intensity I(u) of diffracted light behind the grating is given by:
I(u) = |S1(u) + S2(u)|2(11)
=|S1(u)|2+|S2(u)|2+ 2S1(u)S∗
2(u) cos(4πu0x) (12)
The counting rates on detectors P1and P2are then modulated as a function of the
grating position x(see figure 2-(c)), corresponding to an interference visibility:
V=2sinc(2πu0a)
1 + sinc2(2πu0a)(13)
In the limit a= 0, the visibility is equal to unity whereas it becomes null in the
absence of grating (a= Λ).
Combining equations (10) and (13) leads to V2+D2= 1, in agreement with
inequality (1). Opposite to the conclusion that it leads to a violation of Bohr’s
complementarity, the experimental setup proposed in references [28,29] provides a nice
and clever illustration of the balance between which-path information and interference
visibility in a two-path interferometer. Note that this experiment differs from usually
considered which-way schemes consisting of an interferometer where one tries to get
(either a priori or a posteriori) information about the path followed by the particle,
leading to a degradation of the interference visibility according to inequality (1). In our
experiment, we start from a perfect which-way knowledge and we try to conversely
retrieve the wave-like information. As it can be expected, this slight change in
perspective does not affect the complementarity argument, and the usual reasoning
made above with the quantities Vand Dremains therefore valid.
3. Experimental results
The above predictions are now compared with the experiment which setup is shown in
figure 3. The experiment starts from a clock-triggered single-photon source based on
the photoluminescence of a single NV colour centre in a diamond nanocrystal [19, 33].
Since the photoluminescence spectrum of NV colour centre is very broad (about 100-
nm FWHM at room temperature), we use a 10-nm-bandwidth bandpass filter centered
at λ= 670 nm, corresponding to the emission peak of the NV centre. This spectral

Illustration of quantum complementarity using single photons interfering on a grating7
GP
single
FB
x
G
BD
BD
Single-photon beam preparation
Interferometer
photon
P2
P1
NV colour
centre
Pulsed
4 MHz
F
λ/2 λ/2
λ/2
laser
Figure 3. Experimental realization of the modified Afshar’s experiment based on a
Fresnel’s biprism and single-photon pulses emitted by an individual NV colour centre in
a diamond nanocrystal, excited in pulsed regime at a 4-MHz repetition rate. λ/2: half-
wave plate. BD: YVO4polarized beam displacer. GP: glass plate. F: 10-nm-bandwidth
bandpass filter centered at λ= 670 nm. FB: Fresnel’s biprism. G: transmission grating
inserted in the interference area and translated along the interference x-axis. P1and
P2: avalanche photodiodes positionned in the zero-order diffraction direction of each
beam and operated in the photon-counting regime (Perkin Elmer, AQR14).
filtering allows us to extend the coherence length of the single-photon pulses. The
linearly-polarized single-photon pulses are divided into two spatially separated paths of
equal amplitudes, using polarization beam displacers (BD) and half-wave plates. The
experimental configuration leads to a 5.6-mm beam separation while keeping zero optical
path difference between the two interfering channels. A third half wave plate is then
selectively placed in one beam, in order to obtain two beams with identical polarizations
(see figure 3). The optical path difference induced by this half-wave plate is compensated
by a piece of glass (GP) introduced on the other beam. After preparation, the two
single-photon beams have then identical linear polarization state, equal amplitude, no
optical path difference between them and a spatial separation high enough to avoid
any diffraction effect at the apex of the Fresnel’s biprism. The prepared single-photon
beams are finally sent at normal incidence through the Fresnel’s biprism followed by the
transmission grating. The diameter of each beam is around 2 mm, corresponding to the
illumination of approximately 20 slits of the grating.
As meaningful illustration of complementarity requires the use of single
particles [14], the quantum behavior of the light field is first tested using the two
output detectors feeding single and coincidence counters without the grating. In this
situation, we measure the correlation parameter α[14, 15] which is equivalent to the
second-order correlation function at zero delay g(2)(0). For an ideal single-photon source,
quantum optics predicts a perfect anticorrelation α= 0, in agreement with the particle-
like image that the photon cannot be detected simultaneously in the two paths of the
interferometer. With our source, we find α= 0.14 ±0.02. This value, much smaller
than unity, shows that we are close to the pure single-photon regime [34].
With the parameters of the Fresnel’s biprism (β= 7.5×10−3rad and n= 1.51),
the interfringe is Λ = 87 µm at λ= 670 nm (see equation (2)). Following the discussion

Illustration of quantum complementarity using single photons interfering on a grating8
of section 2, we then use a set of gratings with the same period Λ but with different
widths aof the transmitting slits. The experiment then consists in measuring Dand V
for each value of the parameter a.
Each grating is introduced into the interference area and translated along the
x-axis of the interference pattern, using a computer-driven translation stage with
sub-micrometer accuracy positioning. As shown in figure 4, a modulation of the
counting rates is observed for detectors P1and P2, allowing us to estimate the wave-
like information by measuring the visibility of the modulation for each grating. As
expected, the visibility Vdecreases when the width aof the transmitting slits increases,
the dependance being in good agreement with equation (10) (see figure 5-(a)).
For each grating, we independently measure the distinguishability parameter Dto
quantify the available WPI. This is experimentally realized by consecutively blocking
one arm of the interferometer and the other, and by measuring the quantity D1and
D2defined by equations (8) and (9). The final results, shown on figure 5-(a), lead
to V2+D2= 0.96 ±0.03 (see figure 5-(b)), close to the maximal value permitted by
inequality (1) even though each quantity varies from zero to unity.
Note that when the width aof the transmitting slits is wide (a= 80 µm),
Counts (s
−1)
Counts (s
−1)
Counts (s
−1)
Counts (s
−1)
(a)
(c) (d)
(b)
a= 20 µm, V = 97 ±2%
a= 50 µm, V = 81 ±2%
a= 70 µm, V = 41 ±2%
a= 80 µm, V = 23 ±2%
150
100
50
0
300200100
400
200
0
200100
600
400
200
0
200100
600
400
200
0
200100
x(µm)
x(µm)
x(µm)
x(µm)
Figure 4. Photocounts recorded on detector P1while translating the grating G along
the x-axis, for different widths aof the transmitting slits: (a) a= 20 µm, (b) 50 µm,
(c) 70 µm and (d) 80 µm. Identical result are recorded on detector P2. The grating
is translated by 4-µm steps and each point is recorded with 3-s acquisition time. A
constant averaged background due to detector dark count rate (about 180 counts.s−1)
has been subtracted from the data. The visibility is evaluated using a fit by a cosine
function shown in solid line.

Illustration of quantum complementarity using single photons interfering on a grating9
(a)
a (µm)
(b)
1.0
0.5
0.0
80604020
1.0
0.5
0.0
V2
D2
2.0
1.5
1.0
0.5
0.0
80604020
V2+D2
a (µm)
a(µm)
a(µm)
Figure 5. (a) Wave-like information V2and particle-like information D2as a function
of the width aof the transmitting slits. The solid lines are the theoretical expectations
given by equations (10) and (13) without any fitting parameter. (b) D2+V2as a
function of a.
which corresponds to the setup of references [28, 29] with very thin wires, the wave-
like information associated to single-photon detection on detectors P1and P2is equal
to V2= 0.05 ±0.01, very far from unity. This result illustrates that a quantitative
measurement of the visibility of the transmitted light is necessary to qualify the wave-
like information in a two-path interference experiment, which the simple positioning of
the grid of wires at the dark fringes of the interference pattern does not realize.
4. Conclusion
We have reported the two complementary measurements of “interference vs. which-path
information” using single-photon pulses and a setup close to the proposal by Afshar
et al [28, 29]. By investigating intermediate situations corresponding to partial path
distinguishability and reduced interference visibility, we have shown that the results are
in perfect agreement with the complementarity inequality. While the results may not
appear as a big surprise, we hope that our experiment can contribute to clarify confuse
debates around the dual behavior of the lightfield, which in Feynman’s words contains
“the only mystery of quantum mechanics” [2]. So far, Bohr’s complementarity principle
has thus never been violated.
Acknowledgments
The authors are grateful to A Aspect and F Grosshans for many fruitful discussions and
to S R´egni´e and F Record for their contributions to early stages of the experiment. We
thank G Colla and R Mercier (Institut d’Optique Graduate School) for the realization of
the Fresnel’s biprism, and J-P Madrange (ENS Cachan) for all mechanical realizations.
This work is supported by Institut Universitaire de France and European projects
EQUIND (FP6 project number IST-034368) and NEDQIT (ERANET Nano-Sci).

Illustration of quantum complementarity using single photons interfering on a grating10
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