FEATURE ARTICLE
A TRIPLE
BY
SLIT TEST FOR
QUANTUM MECHANICS
URBASI SINHA, CHRISTOPHE COUTEAU, FAY DOWKER, THOMAS JENNEWEIN, GREGOR WEIHS,
AND RAYMOND LAFLAMME
“If Born’s rule fails, everything goes to hell”. [1]
Q
uantum mechanics, one of the pillars of theoretical physics in the 20th century, has been
tremendously successful at describing the
world around us. The theory has been able to
describe the world of atoms and molecules,
solid state physics, particle physics, allowing us to understand the photoelectric effect, superconductivity and much
more. It has led to new technologies which have transformed our lives, from Magnetic Resonance Imaging
(MRI) to the laser and the transistor and might lead to new
ones such as quantum computers and quantum cryptography. Despite having these resounding successes, the theory still predicts phenomena that are very much counterintuitive. Quantum mechanics seems to fundamentally
change the way we understand the world, opening the
door to many potential interpretations.
All other theories of physics that we have encountered
have ultimately disagreed with observations or predicted
their own demise; it would thus be surprising if quantum
mechanics were to be a final theory of nature. Many
attempts have been made to complement it or generalize
it, by modifying some of its axioms, such as adding hidden variables, non-linear evolution etc. One axiom of
quantum mechanics is that the probability is proportional
to the modulus of the wave function squared [2]. This
paper describes a program to test this axiom using a generalization of the famous double slit experiment.
THE TRIPLE SLIT EXPERIMENT
Many people encounter quantum mechanics for the first
time when reading the third volume of the Feynman
SUMMARY
As one of the postulates of quantum
mechanics, Born’s rule tells us how to get
probabilities for experimental outcomes
from the complex wave function of the system. Its quadratic nature entails that interference occurs in pairs of paths. We present an
experiment that sets out to test the correctness of Born’s rule by testing for the presence or absence of genuine three-path interference. This is done using single photons
and a three slit aperture.
Lectures in Physics [3] in which Feynman describes the
double slit experiment and comments that it “has in it the
heart of quantum mechanics”. This view is widely shared
in the physics community so it will come as a surprise to
many that, though indeed the double slit experiment
exhibits, beautifully, the phenomenon of interference
between two histories of a single system B the photon trajectories in this case B this phenomenon does not fully
characterize “quantumness”. Quantumness, it has been
discovered [4], consists not only of the existence of interference between pairs of histories but precludes interference between triples of histories.
The double slit experiment demonstrates two-way interference. The probability P for the photon to be detected in
an experiment at a particular position on the screen when
both slits, call them A and B, are open is not equal to the
probability of the photon being detected at that particular
band when only A is open plus the the probability for the
photon to be detected there when only B is open:
P (A c B) − P (A) − P (B) /= 0
(1)
where A c B is shorthand for “the photon is detected at the
band when both A and B are open” etc. We call this nonzero quantity, the interference I2(A, B) and it is responsible
for the famous pattern of light and dark bands. It describes
the interference between two histories composed of the
photon passing through slit A or slit B.
If we now consider an experiment with three slits, A, B
and C, we can generalize I2 to I3(A,B,C). Now, there are
seven experimental situations to consider: all three slits
open, any pair of slits open or any single slit open. If we
write the probability of a photon landing at a specified
position on the screen for these seven different experiments as P (A c B c C), P (A c B) etc. and P (A) etc.
(respectively) then Quantum Mechanics tells us that the
following combination is zero:
P (A c B c C) − P (A c B) − P (B c C) − P (C c A)
+ P (A) + P (B) + P (C) = 0.
(2)
This is a consequence of Born’s rule. Remarkably, until
the present work, this prediction of quantum theory had
not yet been put to direct experimental test.
PHYSICS
IN
U. Sinha <usinha@
uwaterloo.ca>,
Institute for Quantum
Computing, University
of Waterloo, Waterloo,
ON N2L 3G1
C. Couteau,
Université de
Technologie de
Troyes, France
F. Dowker, Imperial
College, UK
T. Jennewein,
Institute for Quantum
Computing, University
of Waterloo
G. Weihs, Universität
Innsbruck, Austria
R. Laflamme,
Perimeter Institute
and Director,
Institute for Quantum
Computing, University
of Waterloo
CANADA / VOL. 66, NO. 2 ( Apr.-June 2010 ) C 83
A TRIPLE SLIT TEST ... (SINHA ET AL.)
Although Born’s rule has been indirectly verified to high accuracy in other experiments, the consequences of a detection of
even a small three-way interference in the quantum mechanical
null prediction would be tremendous. If a non-zero result were
to be obtained, it would mean that quantum mechanics is only
approximate, in the same way that the double slit experiment
proves that classical physics is only an approximation to the
true laws of nature.
This would give an important hint on how to generalize quantum mechanics and open a new window to the world. Currently
we have no idea what such a theory could look like but research
is already being done to explore the characteristics of and alternative ways to understand such a theory [7]. It might even give
a hint towards unifying quantum mechanics and gravity, a
major goal of fundamental physics today.
Obviously the discovery of a three path interference would lead
to the question: Is there four-way interference? There is indeed
a whole hierarchy of theory types: a level k theory being one in
which there is k-way interference but no k+1-way interference [4].
An interesting consequence of the violation of Born’s rule
would be for computer science. In the last 40 years, computer
scientists have classified sets of problems according to the difficulty with which they can be solved. They look at how these
sets relate to each other and have conjectured many relationships. A well-known example is the famous question of
whether or not the classes P and NP are the same [5] C finding
a proof to resolve this longstanding question would earn a million dollar prize from the Clay Foundation [6]. Aaronson has
shown that violating Born’s rule would have a dramatic effect
on computational complexity because it would allow one to
efficiently distinguish two states that are exponentially close.
This would relate two complexity classes implying that NPcomplete problems could be solved in polynomial space [8]
something which is not believed to be true with either classical
or quantum computers and would surprise many computer scientists.
A similar conclusion was reached by Meyer in [9]. He has suggested that a task that takes two steps with quantum C level
k = 2 C resources could be achieved in one step with level 3
resources and so on. To realise this intriguing idea would
require models for level 3 and higher k physical systems to be
discovered but it shows that the implications of a detection of
super-quantum theories would be very far reaching indeed,
even beyond the boundaries of physics itself.
BRINGING THEORY TO THE LAB ...
The triple slit experiment is being performed at the Institute for
Quantum Computing in the University of Waterloo, Canada. In
this experiment, we evaluate the triple slit interference term
given by equation (2). If Quantum Mechanics is correct, this
term will be zero, if there is a further generalization to the theory, then we would get a non zero result which cannot be
explained by experimental errors.
Fig. 1
Pictorial representation of how the different probability
terms are measured. The leftmost configuration has all slits
open, whereas the rightmost has all three slits blocked. The
black bars represent the slits, which are never changed or
moved throughout the experiment. The thick grey bars represent the opening mask, which is moved in order to make
different combinations of openings overlap with the slits,
thus switching between the different combinations of open
and closed slits.
The experiment consists of measuring the seven probability
terms in equation (2) along with an eighth term P(0) which
gives the background probability (this takes care of any experimental background such as detector dark counts i.e. spurious
counts measured by the detector even in the absence of a
source of photons). We define a quantity ε as
ε = P (A c B c C) − P (A c B) − P (B c C) − P (C c A)
+ P (A) + P (B) + P (C) − P (0)
(3)
Figure 1 shows how the various probabilities are measured in a
triple slit configuration. For better comparison between possible realizations of such an experiment, we further define a normalized variant of ε called ρ,
ε
,where
δ
δ= | IAB | + | IBC | + | ICA |
= | P (A c B) − P (A) − P (B) + P (0)|
+ | P (B c C) − P (B) − P (C) + P (0)|
+ | P (A c C) − P (A) − P (C) + P (0)| .
ρ
=
(4)
(5)
Since δis a measure of the regular interference contrast, ρ can
be seen as the ratio of three-path interference over the regular
two-path interference. (If δ= 0 then ε = 0 trivially, and we really are not dealing with quantum behavior at all, but only classical probabilities.)
EXPERIMENTAL SET UP
Figure 2 shows a schematic of the complete experimental setup. The laser beam passes through an arrangement of mirrors
and collimators before being incident on a 50/50 beam splitter.
The beam then splits into two, one of the beams is used as a reference arm for measuring fluctuations in laser power whereas
the other beam is incident on the thin metal membrane , which
has the slit pattern cut into it using commercial laser cutting.
The beam height and wast is adjusted so that it is incident on a
set of three slits, the slits being centered on the beam. There is
84 C LA PHYSIQUE AU CANADA / Vol. 66, No. 2 ( avr. à juin 2010 )
A TRIPLE SLIT TEST ... (SINHA ET AL.)AA
Fig. 2
Schematic of experimental set-up.
another membrane in front which has the corresponding blocking designs on it such that one can measure the seven probabilities in equation (2). The slit plate remains stationary whereas
the blocking plate is moved up and down in front of the slits to
yield the various combinations of opened slits required to
measure the seven probabilities. As mentioned above, in our
experimental set-up, we also measure an eighth probability
which corresponds to all three slits being closed in order to
account for dark counts and any background light. Figure 2
shows this pictorially. A multi-mode optical fiber is placed at a
point in the diffraction pattern and connected to an avalanche
photodiode (APD) single photon detector which measures the
photon counts corresponding to the various probabilities. The
optical fiber can be moved to different positions in the diffraction pattern in order to obtain the value of ρ at different positions in the pattern. Some of our preliminary results as well as
experimental details have been published in [10]. At present we
are working on using single photons as our incident photons.
We have a heralded single photon source (HSPS) [11] based on
parametric down conversion (a method by which a blue photon
splits into two red photons when it is incident on a non linear
crystal to maintain energy conservation) and the use of single
photons enables us to know the exact number of events and
also gives us a means of performing the same experiment using
two independent sources of incidence with different statistics.
Figure 3 shows a comparison between interference with three
slits open obtained by using a Titanium Sapphire laser at
810 nm on one hand and a heralded single photon source on the
other.
CONCLUSION
Quantum mechanics has been one of the most elegant and
important theories in 20th century physics and has been suc-
Fig. 3
Comparison between experimentally obtained triple slit
interference patterns. The blue dots indicate the laser pattern
and the red dots indicate the single photon pattern. The
black line has been drawn to aid the eye.
cessful in explaining and motivating numerous applications.
However, in spite of its successes, there are still mysteries associated with the theory which hint at the possibility of the existence of more generalized versions. This makes it important to
test the fundamental postulates of quantum mechanics through
dedicated experiments. In this paper we have described an
experiment to test Born’s rule. Some of our preliminary observations have been reported in [10] giving non-zero result for ρ,
as defined in equation (5). However, this could be caused by
some systematic errors that have not yet been controlled.
Improvements to the experiment set-up have since been made,
including performing the experiment using single photons [12].
The future will tell us whether we understand these errors or
perhaps that there could be a discrepancy with the predictions
of quantum mechanics. We just have to wait and watch!
ACKNOWLEDGEMENTS
Research at IQC and Perimeter Institute was funded in part by
the Government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Research and
Innovation. Research at IQC was also funded in part by CIFAR
and NSERC. This research was partly supported by NSF grant
PHY-0404646. U.S. Thanks to Aninda Sinha for useful discussions.
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IN
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