double slit

The double-slit experiment is used as the
archetypal interference experiment to dramat-
ically demonstrate the quantum weirdness of
wave-particle duality. It is a long-held assump-
tion that whether which-way detectors are placed
at one slit or both will yield the same results, ren-
dering the second detector redundant. Here we
show that the presence of the second detector can
have measurable effects on the interference pat-
tern, violating this assumption. We show that
this effect forms the basis for a new measure of
Born rule violation which is correct to all orders.
It opens new experimental opportunities for the
accurate testing of the Born rule.
Particle nature is demonstrated in double-slit experi-
ments by placing which-way detectors at the slits. The
reason that a second which-way detector is believed to be
redundant, is that a detection at one slit means that the
particle is assumed to not have gone through the other
slit. Now on the other hand, in the Feynman path inte-
gral formulation of quantum mechanics all possible paths
between points contribute to the wave function; this even
includes paths that go through one slit then the other as
depicted in Fig. 1. The inclusion of these non-classical
provides higher-order corrections to the interference pat-
terns.
The Born rule states that if a quantum object is rep-
resented by a wave function ψ(r, t), then the probability
density of detecting it at position r and time t is given
by the absolute square of the wave function [1],
P (r, t) = ψ ∗ (r, t)ψ(r, t) = |ψ(r, t)| 2 .
reason for this is that underlying the Sorkin parameter
is the assumption that the multi-slit wave function is the
superposition of single-slit wave functions, i.e. composed
of paths that only go through either one slit or the other.
Correcting this approximation by including non-classical
paths renders the Sorkin parameter finite, without vio-
lating the Born rule; in fact in some regimes these cor-
rections can be significant [7–9]. Given the success of
quantum mechanics any violation of the Born rule is ex-
pected to occur at the higher-orders. Therefore for a
parameter to be a useful measure of Born rule violation,
it is important that it is accurate to higher-orders, where
the Sorkin parameter is not.
In this letter we will show that by accounting for non-
classical paths, the which-way double-slit experiment
with one and two which-way detectors will produce dif-
ferent inference patterns, contrary to commonly held as-
sumptions. Making use of this difference, we give a new
parameter that completely accounts for higher-order cor-
rections, to exactly test the Born rule: the parameter will
be exactly zero if the Born rule is not violated, finite oth-
erwise.
Let us consider a double-slit experiment with two types
of which-way detectors: one detects whether a particle
Slits
(D, y D )
(0, d/2)
(-S, 0)
Screen
Source
(0, -d/2)
y
(1)
x
In attempts to unify quantum mechanics with general
relativity, it has been proposed that the Born rule be gen-
eralised to include beyond pair-wise interference. Up un-
til now, direct tests of Born rule violation have been mea-
surements of the Sorkin parameter [2], which quantifies
non-pairwise interference. As the Born rule only allows
for pairwise interference, a finite Sorkin parameter would
suggest a violation. The first experiment to measure the
Sorkin parameter was conducted by Sinha et al. [3] in a
triple-slit experiment. The experiment found the Sorkin
parameter to be zero, within experimental error bounds,
and concluded no Born rule violation. Subsequent more
precise measurements of the Sorkin parameter also came
to the same conclusion [4–6].
Shortly after the Sinha et al. experiment however, it
was pointed out that the Sorkin parameter was not an
exact measure and there are regimes where it could dra-
matically fail as a test of Born rule violation [7, 8]. The
(a)
(b)
(c)
(d)
(e)
FIG. 1. Non-classical paths and type I/II detector
gedanken experiments. Top: Side view of one of the myr-
iad of non-classical paths that enters both slits: the path en-
ters slit A, makes an abrupt turn, briefly enters slit B, before
hitting the detecting screen. Bottom: gedanken experiments
with balls serving as detectors to illustrate different which-
way detector setups. (a) setup 1: no which-way detectors,
(b) setup 2: a type I which-way detector at slit A, (c) setup
3: a type I which-way detector at slit B, (d) setup 4: type I
which-way detectors at slit A and B, (e) setup 5: a type II
which-way detector.

has gone through slit A or B, the other detects that a
particle has gone through one or both slits, but does not
know which one. Such detectors can be illustrated in
a gedanken experiment, with light balls placed precari-
ously in the slits to serve as the detectors (Fig. 1). In
one type of detector a ball is placed in slit A, such that
a particle entering the slit will cause it to fall into a tray.
We look into the tray to reveal which slit the particle
went through: if we see a ball in the tray then the par-
ticle went through slit A, if we do not see a ball, then
the particle went through slit B. Implicit here is the as-
sumption that we can not directly view the ball until it
hits the tray; in this sense the tray can be thought of as
representing a signal amplifier. In another detector, in-
distinguishable balls are placed in each slit. In this case,
peering into the tray we will see either one or two balls.
One ball indicates that the particle had gone through
one slit but it does not reveal which one, whereas two
balls indicate that the particle had gone through both
slits. We will call the former type I and latter type II
which-way detectors. Such types of detectors have been
realised in neutron [10] and molecular [11] interference se-
tups, electrons in semiconductors [12], inelastic electron
holography [13], electron interferometers [14], and with
ion and electron beam nanofabrication [15]. Of note is
Bertet et al. atomic double-pulse Ramsey interferometer
experiments which had both type I and type II detec-
tors [16].
If ψ A represents the wave function from a single slit A
and ψ B from a single slit B, it is widely taught that
the intensity or probability distribution in the double
slit experiment is P AB = |ψ A + ψ B | 2 . This approxi-
mation is corrected by including paths that go through
both slits. Labelling the contribution from paths that
go through both slits as ψ AB , the probability distribu-
tion for the double-slit experiment with no which-path
detectors [Fig. 1(a): setup 1] is
P AB = |ψ A + ψ B + ψ AB | 2 .
P D B = |ψ B + ψ AB | 2 + |ψ A | 2 .
(4)
Now if we place a second type I detector at slit B, rep-
resented in our gedanken experiment by a second distin-
guishable ball at slit B [Fig. 1(d): setup 4], three types of
paths are distinctly detected: paths that go through slit
A (ψ A ) or B (ψ B ) only, and non-classical paths that go
through both before hitting the detection screen (ψ AB ).
The probability density when there are type I detectors
in both slits is therefore,
P D A D B = |ψ A | 2 + |ψ B | 2 + |ψ AB | 2 .
(5)
When there is a type II which-path detector [Fig. 1(e):
setup 5] one is not able to distinguish paths that con-
tribute to ψ A from ψ B . However we can distinguish paths
that went through one slit from paths that went through
both slits before hitting the detecting screen. The prob-
ability density with a type II detector is
P D AB = |ψ A + ψ B | 2 + |ψ AB | 2 .
(6)
We would like now to quantify the difference in the
probability distribution of the single (setup 2) and dou-
ble (setup 4) which-way detector double-slit experiments.
Let the source be at position r S = (−S, 0), the screen de-
tector at r D = (D, y D ), and slit centers are at (0, ±d/2)
(Fig. 1). The slits have w width. The setup is symmet-
ric in the z-direction, which means we can ignore this
component as it only introduces an irrelevant constant,
effectively becoming a 2-dimensional problem [8]. We as-
sume a monochromatic source and consider a duration
much larger than the time of flight, so that the probabil-
ity distribution at the screen detector can be quantified
with the time-independent 2-point propagator, which is
attained by summing over all possible paths between r 1
and r 2 ,
(2)
Now let us place a type I which-path detector at slit
A [Fig. 1(b): setup 2]. Conventionally, a detection at
slit A means that the particle did not go through slit B,
otherwise the particle went through slit B, so that the
probability density is P D A = |ψ A | 2 + |ψ B | 2 . However if
one accounts for non-classical paths, then a detection at
slit A includes paths that only go through slit A as well as
paths that go through slit A and B; as they are indistin-
guishable, we must sum both types of paths (ψ A + ψ AB ).
A non-detection at slit A means that the path must have
gone through slit B only (ψ B ). Thus the probability den-
sity when there is a type I which-path detector at slit A
is
P D A = |ψ A + ψ AB | 2 + |ψ B | 2 ;
B [Fig. 1(c): setup 3] the probability density is,
(3)
and similarly when there is a which-way detector at slit
K(r 1 , r 2 ) =
D[x(s)] exp(ik
ds) ,
(7)
where D[x(s)] is the usual path integral measure of paths
x(s) with contour length s. For a point source at r S ,
ψ(r D ) = K(r D , r S ), where K(r D , r S ) is the correspond-
ing propagator from source to screen.
The contributions from non-classical paths become
more pronounced as the operating wavelength increases
relative to slit spacing. The reason for this is that longer
wavelengths mean more overlap between the single-slit
wave functions, so that non-classical paths which enter
both slits are more likely. Sawant et al. propose work-
ing in the microwave regime (which has a longer wave-
length than the optical regime) to detect non-classical
paths; however single photon detectors in the microwave
regime are relatively inefficient due to the low energy at
these frequencies (although there has been progress in

this area [17]). As a case-study we consider the optical
regime with the following parameters: photon source of
λ = 810 nm wavelength, slit width w = 500 nm, inter-
slit spacing of d = 2000 nm, and source and detector
distances S = D = 1 mm. We work in the thin slit limit.
Using equations (13) and (15) (Methods section),
Fig. 2(a) plots the intensity of the single which-way de-
tector experiment (setup 2), P D A . Fig. 2(b) plots the
intensity of the double which-way detector experiment
(setup 4), P D A D B . Fig. 2(c) plots the difference in
the interference pattern between the two experiments,
∆ 1 ≡ P D A − P D A D B . [For completeness we have also
plotted the intensities of P AB and P D AB , and their dif-
ference, ∆ 2 , in Fig. 2(d)-(f)]. ∆ 1 /P AB (0) is on the order
of 10 −2 . The Sinha et al. experiment achieved inten-
sity accuracies of order 10 −2 normalised to the expected
two-path interference, and subsequent experiments with
multi-path interferometers [4, 6] and nuclear magnetic
resonance [5] have claimed accuracies of at least an order
of magnitude better. If the same sort of accuracy can
be achieved in double slit experiments with which-way
detectors, the effects of the second which-way detector
on the interference pattern due to non-classical paths,
should be detectable.
It is important to point out that the higher-order inter-
ference in classical slit experiments (i.e. without which-
way detectors) of previous works [6, 7, 9, 18], is a classi-
cal wave effect as pointed out by Ref [7] using Maxwell’s
equation. The experiment proposed here caot be ex-
plained with classical equations, explicitly requiring a
quantum mechanical description.
The Sorkin parameter for the triple-slit experiment is
defined as
I ABC ≡ P ABC −P AB −P AC −P BC +P A +P B +P C , (8)
where P ABC is the probability of detection when all 3
slits (A, B, C) are open, P AB is the probability of de-
tection when 2 slits (A, B) are open, and so on. If one
assumes that the probabilities are simply given by the lin-
ear superposition of the individual wave functions of the
constituent single-slit setups (P ABC = |ψ A + ψ B + ψ C | 2 ,
P AB = |ψ A + ψ B | 2 , and so on) then by rewriting proba-
bilities in equation (8) in terms of wave functions, it can
be shown that I ABC = 0 if the Born rule is correct. The
proposed advantage of using the Sorkin parameter to ex-
perimentally test the Born rule is that one does not need
to know the theoretical values of these probabilities, one
need only measure them. If one accounts for non-classical
paths however (i.e. P ABC = |ψ A + ψ B + ψ C + ψ ABC | 2 ,
P AB = |ψ A + ψ B + ψ AB | 2 , and so on), I ABC = 0, and
therefore a finite Sorkin parameter does not immediately
signal Born rule violation. As a solution to this problem
we introduce the following parameter which is exactly
zero in all regimes when the Born rule is not violated,
I AB ≡ P AB − P D A − P D B − P D AB + 2P D A D B .
(9)
Substitution of equations (2)-(6) into equation (9)
shows that I AB = 0. Like the Sorkin parameter, I AB sub-
tracts from P AB all possible combinations of pair-wise in-
teractions terms yielding I AB = 0 if the Born rule holds;
if the probability of detection is anything other than the
absolute square of the wave function, then I AB = 0 in
general. Different from the Sorkin parameter however,
I AB exactly accounts for the higher-order corrections to
all orders. This allows an exact direct test for Born rule
violation, limited only to experimental uncertainty.
The experiment to test the Born rule will involve re-
peating the double-slit experiments 5 times, each time
the only thing to be changed is the which-path detector
configuration, to get values for P AB , P D A , P D B , P D AB ,
and P D A D B . One may then measure I AB using equa-
tion (9): any value other than I AB = 0 signals Born rule
violation.
We have shown that the inclusion of higher-order or
non-classical paths will lead to different interference pat-
terns for which-way double-slit experiments with one and
two which-way detectors. These differences should be
measurable in regimes where the operating wavelength is
commensurate to or larger than slit spacing. Previously,
tests for Born rule violation have been triple-slit experi-
ments measuring the Sorkin parameter. The Sorkin pa-
rameter however is only an approximate measure of Bo
rule violation, and can only be applied in regimes where
the operating wavelength is much smaller than slit spac-
ing. By explicitly accounting for higher-order correction,
we have given an alternative parameter which is an exact
test of the Born rule for all wavelengths and slit spacing.
This should open up a new suite of experiments based on
which-path double-slit experiments, to test the Born rule
to accuracies limited only by experimental uncertainties
and not theoretical ones.
METHODS
The problem of summing over all possible paths with
the boundary conditions imposed by the slit plane is un-
wieldy and has yet to be exactly solved. Nevertheless,
Sawant et al. [8] argues that a good approximation can
be achieved by considering two types of paths: paths with
straight trajectory segments from source to slit, then to
detecting screen; and paths composing of straight trajec-
tory segment from source to slit, then to another slit, be-
fore hitting the detection screen (Fig. 1) - this conjecture
has subsequently been supported with finite-difference
time-domain simulations (FDTD) [9]. Following the con-
vention of Sawant et al. we will call the former classical
paths, and the latter non-classical paths (as we have been

FIG. 2. The effects of a second which-way detector is measurable. (a) plots the normalised intensity of the single
type I which-way detector experiment (setup 2). (b) plots the normalised intensity of the double type I which-way detector
experiment (setup 4). (c) plots the difference in the interference pattern produced by the single and double type I which-way
detector experiments, ∆ 1 ≡ P D A − P D A D B . (d) plots the normalised intensity of the double-slit experiment without which-way
detectors (setup 1). (e) plots the normalised intensity of the type II which-way detector experiment (setup 5). (e) plots the
difference in the interference pattern between setup 1 and setup 5, ∆ 2 ≡ P AB −P D AB . Parameters: photon source of λ = 810 nm
wavelength, slit width w = 500 nm, inter-slit spacing of d = 2000 nm, and source and detector distances S = D = 1 mm. All
plots are normalised to the maximum central intensity of the double-slit experiment, P AB (0).
doing). One notes that consideration of just the classical
paths directly leads to Fresnel’s theory of diffraction by a
slit [19]. Using the following free propagator for straight
paths [20]