The Wheeler's Delayed Choice Experiment in an ideal
Mach-Zehnder Interferometer
Juan María Fernández
Physics Teacher, IES Jorge Manrique de Palencia (España)
juanmariafernandez@gmail.com
December 30, 2014
Abstract
This document is a description of the EJS java simulator WheelersDelayedChoice (the jar le is
), an implementation of John Archibald Wheeler's
Delayed Choice Gedanken Experiment. The striking dierence between the Classical particles
and Photons, the subtle nature of the Particle-Wave duality of matter, the Quantum Superposition of States and the Probabilistic Interpretation of Quantum Mechanics, are all shown in
a model, the Mach-Zehnder interferometer, where the Quantum fundamental ideas emerge
without any fuss of mathematical elaboration: only a little matrix algebra, and the Quantum
Postulates; there are no dierential equations, no Fourier transforms, no existence and uniqueness
theorems, only Quantum Mechanics behaviour in the simplest powerful way. This document is also
related to a video tutorial on the same simulator. You can nd both in the Open Source Physics
repository.
ejs_model_WheelersDelayedChoice.jar
This document is intended to be read on screen. It can be printed, but color and hyperlinks will be
lost in that case.
If you nd any error in the simulator or in this document, wrong concepts or language, please,
feel free to use the above email and let my know.
This document, the video tutorial and the WheelersDelayedChoice EJS [10] java simulator are
licensed with Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BYNC-SA 4.0)
1
Contents
1
The Mach-Zehnder interferometer
3
2
Laser interference. Classical Wave theory explanation
5
3
Classical Particles
6
4
The Wheeler's Delayed Choice Experiment
7
5
Single Photon interference
10
5.1
11
6
7
Single Photon interference with a PathFinder detector before Beam Splitter 2 . . . . . . .
Quantum formalism for Beam Splitters, Mirrors and Phase Shifters
13
6.1
The Beam Splitter 1 representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
6.2
The Beam Splitter 2 representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
6.3
Mirror 2 and Mirror 1 representations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
6.4
The Phase Shifter representation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Calculating the Detectors countings with the Quantum Formalism
2
17
1
The Mach-Zehnder interferometer
Figure 1: The Mach-Zehnder interferometer with a red Laser.
This simulator ideal interferometer can be seen in Fig.1. In the basic version with a red laser, it has:
1. A source
2. Beam Splitter 1, that divides the incident beam in two equal Intensity beams
3. In the
Way UP route: a Mirror 2, that sends the beam to
4. Detector 2
5. In the
Way Down route: a Mirror 1, that sends the beam to
6. Detector 1.
More elements are added to this basic set, they will be discussed in their corrrespondent sections.
Activating the Laser option after booting the simulator, and then pressing the Play-Laser button (the
one with the red asterisk), and afterwards choosing the Laser tab, you can see the detectors ashing each
time a Photon is detected.
When two Photons are detected is a short time interval, lower than some
threshold, the Coincidence detector also ash a light. This simulator mode is intended to show that a
normal laser light can be conceived as composed by many Photons. In the previous Fig.1, the Up and
Down Beams are of equal Intensity (50 % of the BeamSplitter 1 incident beam), and each detector is
supossed to count the same number of particles in the long run.
Adding a second Beam Spliter 2, and a Phase Shifter, the beams coming to the detectors can be splitted.
The Phase Shifter can change the dierence of phase between the Up and Down beams, thus changing
the Intensities in both detectors.
3
Figure 2: Laser interference with phase dierence equal to 0. All the Intensity goes to Detector 2.
Figure 3: Laser interference with phase dierence equal to 180 degrees. All the Intensity goes to Detector
1.
In Figs.2 and 3 the extreme cases are shown, with all the Intensity going to Detector 1 or Detector 2. If
the waves are equal in intensity and equally linearly polarized, the interference results can be explained
in the Classical Wave Theory, for example in Chapter VII of [2].
4
2
Laser interference. Classical Wave theory explanation
Only the result in Fig.2 will be discussed. Other settings with PhaseShifter at any
out easily.
ϕ value can be worked
Please note that each Beam Splitter has clearly marked the reectant surface with a ligth
Gray band in the blue overall container of the Beam Splitter. There are two routes after Beam Splitter
1:
ˆ Way Up:
from Beam Splitter 1 to Mirror 2, to Beam Splitter 2, to Detectors 1 and 2.
ˆ Way Down:
from Beam Splitter 2 to Mirrror 1, to Beam Splitter 2, to Detectors 1 and 2.
Please remenber that a Phase change occurs when the electromagnetic eld propagates from a medium
to another. The frontier conditions impose, see [2],
ˆ
a Phase change of
nmedium =
ˆ
π,
c
or 180º, when the beam reects from a medium of higher refractive index ,
vlight
a Phase change of 0 when the beam trasmits.
In the
Way Up (phase changes in degrees):
Element
Change of phase
Beam Splitter 1
180
Mirror 2
180
Beam Splitter 2 to Detector 1
0
Beam Splitter 2 to Detector 2
0
ϕ
Explanation
reection on medium of higher
n
reection on medium of smaller or equal
trasmission
Table 1: Phase changes in Way Up.
The total change in the
In the
Way Up is 360º.
Way Down:
Element
Change of phase
Beam Splitter 1
0
Mirror 1
180
Beam Splitter 2 to Detector 1
0
Beam Splitter 2 to Detector 2
180
ϕ
Explanation
trasmission
reection on medium of higher
trasmission
reection on medium of higher
Table 2: Phase changes in Way Down.
The phase dierences between Up and Down depends on the Detector
5
n
n
n
1.
Detector 1
(a) Total phase change in
Way Up: 180+180=360
(b) Total phase change in
Way Down: 180
(c) Dierence 360-180=180
In Detector 1 the phase dierence is 180, this means desctructive interference of waves, and
no signal
goes to Detector 1.
1.
Detector 2
(a) Total phase change in
Way Up: 180+180=360
(b) Total phase change in
Way Down: 180+180
(c) Dierence 360-360=0
In Detector 2 the phase dierence is 0, this means constructive interference of waves, and
all the signal
goes to Detector 2.
3
Classical Particles
Figure 4: Classical Particles experiment. Each Detector counts 50% of total input.
In Fig.4 the source emits classical particles. Both Beam Splitters have 50% chance in trasmission and
reection. A quick calculation makes obvious that both detectors will have 50 % of the total input. This
result is independent of having the Beam Splitter 2 active or not.
6
4
The Wheeler's Delayed Choice Experiment
The controversy about the interpretation of Quantum Mechanics dates from the very beginnings of Quantum Theory, a quick look to [1, 9] or to any estandard reference with a good bibliography, as [6], will
suce to convince anyone. Among the older and powerful gedanken experiments, the Wheeler's Delayed Choice shows in the most direct terms the non-Classical and new characteristics of Quantum
Mechanics: the Probabilistic Interpretation of the Wave funcion, the Interference of states,
and the Particle-Wave duality. Wheeler himself gives a crisp concise description of the experiment
in the article Law without Law , included in [1]. His presentation also remarks the paradoxical conclusions reached if the Classical Mechanics language and assumptions are mantained: if the particle's
properties are conceived as being ontologicaly owned-the particle has such and such property- against the
Quantum Mechanical dictum of properties only existing when measured, intrincate paradoxes appear in
the description of the experiment.
set a Wheeler's Delayed Experiment, the simulator is prepared to go.
Play button for Particles, you can get a situation similar to the next Figure:
Activating the CheckBox
Pressing the
Figure 5: After passing Beam Splitter 1, the Photon, as a wave, is in a Superposition of two states.
After passing Beam Splitter 1, the Photon is in its way to Beam Splitter 2. The Photon is in a superposition of two states, superposition represented by a spring connecting both orthogonal states. This is
the Photon in its Wave disguise, and the result of the experiment, if this situation is mantained, will be
a Single Photon Interference, with result dependent on the dierence of phase
ϕ
between the two states.
A Classical version of this would be: the Photon, as a wave, has arrived to the detectors both by the
Way Up and the Way Down routes.
With no Beam Splitter 2, as in Fig.6, the Photon will be detected only by one of the Detectors. Classical
this equivalent to have been travelling in the
Way Up route, or, exclusive or, in the Way Down route.
7
Figure 6: After passing Beam Splitter 1, the Photon, as a particle, is in
Way Up route, or in Way
Down route, heading to the Detectors.
Way Down, if
Way Up. Wich one? There is no answer, until Detector 1, or
The Photon, now in its Particle disguise, if found by Detector 1, was comming in from
found by Detector 2, was coming in from
Detector 2 ashes a light.
Now enters Wheeler's idea: Let's change the experimental setting, just before the Photon is arriving to
the Detectors, making suddenly appear the Beam Splitter 2. See the next Figure.
Figure 7: The Photon, previously a Particle, goes to a Wavelike state after Beam Splitter 2 reappearance.
8
The Photon,
which was a Particle just a moment before, now is a Wave, and will show a wavelike
interference when detected. If a description using the Particle and Wave concepts has to be made, in a
Classical explanation, the experiment makes unnavoidable the conclusion that the Photon has changed
shall have come by Way
Up, Way Down, or both routes- paraphrasing Wheeler in [1]- after the travel is already done .
from Particle to Wave after the travel is already done. Thus, the Photon
Amazing, and paradoxical!
The experiment goes on, with Beam Splitter 2 being part of the experiment, or not, according to some
random variable. The Single Photons swap consequently their Wavelike and Particlelike behavior. This
is also an strong evidence of the non locality of Quantum Mechanics.
This Gedanken Experiment has been made in real world laboratories, each time more faithfully to
Wheeler's idea . Only in recent years the experimentalists could work with Single Photons, and swap the
Beam Splitter 2 in and out quickly enough: Jacques, V., et all, see reference [4], in the Laboratoire de
Photonique Quantique et Moleculaire, ENS de Cachan, did it. One of the master of masters in the modern Quantum Optique, Alain Aspect, also an author of [4], lectured on the experiment, with his unique
sympathy and good mood, when he was awarded the Bohr Medal in 2013, as can be seen and enjoyed in [3].
In the next section we will use always the notion of superposed states, after passing Beam Splitter 1, and
the interference in Beam Splitter 2, and the reduction of the Wave Packet, nomenclature of [6], where
this principle is presented as the Fifth Postulate of Quantum Mechanics.
9
5
Single Photon interference
Figure 8: Photon experiment.
After Beam Splitter 1, the unique Photon is in a Superposed state of
Reection and Trasmission, with 50% of probability.
Figure 9: Photon experiment. Beam Splitter 2 is active with
ϕ = 0.
The Superposition of states after
Beam Splitter 1 collapses in Beam Splitter 2, and all the input goes to Detector 2.
Fig.8 shows a Photon, just before being detected. After Beam Splitter 1, the Photon is in a superposition
of states, Reection and Trassmission, both with 50 % of probability ( the superposition is graphically
10
depicted by a spring linking both states ). Only one of the detectors will count the whole Photon, there
is no splitting in the total energy of the Photon, and each detector has 50% of chances of detecting
the whole Photon as an unique particle. In fact, this is the experimental character of a particle: being
detected as a single object.
Fig.9 shows a Photon experiment with Beam Splitter active with phase dierence (between
and
Way Down )ϕ
= 0.
Way Up
The Superposition of states made after Beam Splitter 1 collapses after Beam
Splitter 2, and all the counts are made by Detector 2.
Figure 10: Photon experiment. Beam Splitter 2 is active with
ϕ = 180.
The Superposition of states after
Beam Splitter 1 collapses in Beam Splitter 2 and goes to Detector 1.
In Fig.10 Beam Splitter 2 is active with Phase Shifter
ϕ = 180º
value for phase dierence. This makes
the Superposition collapsing in Beam Splitter 2, this time to Detector 1, wich gets now all the counts.
These results in the Single Photon Experiments show what Feyman called [the] hearth of quantum
mechanics, see [5], in an arguably simpler experimental device (this Interferometer and Detectors are
less complicated than Feynman's double slit, hope so!). The Quantum Particle-Wave duality is cristal
clear:
ˆ
Each Photon is a single identity, and is detected only in one of the Detectors. This is the Particle.
ˆ
Each Photon is transformed, by Beam Splitter 1, in a Superposition of Photon states, where additional phase dierence can be added by a Phase Shifter.
Then the Superposition of states is
recombined in Beam Splitter 2, with an Interference of the Superposed States, whose results depends on the phase dierence
5.1
ϕ.
This is the Wave.
Single Photon interference with a PathFinder detector before Beam
Splitter 2
Path Finder
Way is the Photon's, Up or Down, before entering
What if before the Photon, best said, the Superposed states, enter Beam Splitter 2, a
apparatus is placed? The Pathnder measures which
Beam Splitter 2. We use
Path Finder Check Box for this experiment.
11
Figure 11: Passing the PathFinder the Photon is obliged to choose wich Way to be in.
Figure 12: The Photon Superposed state collapsed to the Way Down state and now is heading to Beam
Splitter 2.
As shown in Fig.12, the Superposition of quantum States collapses, and goes to the
Way Down ( this is
known because the Photon is entering from below in Beam Splitter, as result of the Path Finder Detector
measurement).
In Beam Splitter 2 such a Photon has 50 % probabilities of going to Detector 1, and
50 % of going to Dectector 2: With those settings, is obvious that including a Path Finder after Beam
Splitter 2 will give each Detector probability 50% of making a count. This beavior is typical of Quantum
Mechanical objects. More on the measurement and the collapse of a superposition of states can be read
12
wave packet reduction )
Other proposed resolutions of the measurement paradox , in Wigner's
article Interpretation of Quantum Mechanics, included in [1]in Chapter III of [6] (there Cohen-Tannoudji et coauthors describe this as
and in the section named
6
Quantum formalism for Beam Splitters, Mirrors and Phase
Shifters
The formalism of Quantum Mechanics can be used to explain all the above results in the Single Photon
experiments. Only the simplest mathematical tools are used: Linear Algebra and Complex Arithmetic.
We will use two vectorial spaces of dimension 2: The input vectorial space, and the output vectorial
space, both with two basis ortoghonal states, one in the
Way Up and other in the Way Down. Each
element in the interferometer acts on an input state and transforms it in an output state.
6.1
The Beam Splitter 1 representation
Figure 13: Beam Splitter 1 action:
from
d reected is 180º phase shifted from a; c and d are not phase shifted
b.
In Beam Splitter 1 the reectant inner surface is in the left side. So, the Up incoming beam,
|d >with
to a superposition of two states, one a reected state
the other a trasmitted state
|c >,
0.5
|a >
goes
in radians, and
|a >is
|BS1 acting on |a
√
π
with no phase change. This way, in the estándard notation of [5, 6] ,
the action of Beam Splitter 1, BS1, on
The coecients
a phase change of 180º,
√
√
0.5 |eiπ |d > + 0.5 |c >
√
= 0.5 (−|d > +|c >)
> −→
(1)
(2)
denote the 50% probability of the Photon of being in each state if a measurement
is made. The conservation of probability requires, for the coecients
2
X
2
|ci | = 1
i=1
13
ci
in a superposition of two states
(3)
In a similar way, with no phase changes due to trasmission or reection in a medium of lower or equal
index of refraction, the action of Beam Splitter 1, BS1, on
|BS1 acting on |b
The basic main input states are
|b >
is
√
0.5 |d > + 0.5 |c >
√
= 0.5 (|d > +|c >)
> −→
√
(4)
(5)
|a >, input in the Way Up, and |b >, input from the Way Down.
They
are orthogonal, because the are exclusive and none has part in the other, so, it is possible to identify
|a >=
|b >=
In a simple election of states, being in
Way Up , the
1
0
0
1
(6)
(7)
|d >
state, and being in the
Way Down, the
|c >
state, after Beam Splitter 1, can be represented by
|d >=
|c >=
1
0
0
1
(8)
(9)
The matrix operator representing Beam Splitter 1 is
BS1 =
It is a trivial task to prove that
6.2
BS1 · BS1 = I
√
0.5
−1
1
, and that
1
1
BS1
(10)
is Unitary and Hermitian.
The Beam Splitter 2 representation
Figure 14: Beam Splitter 2 action:
b; c is 180º phase shifted from b.
d reected is 0º phase shifted from a and c; d is 0º phase shifted from
14
In Beam Splitter 2 the reectant inner surface is in the right side. So, the Up incoming beam,
reected to a reected state
|d >with
a phase change of 0º, and trasmited to
|c >,
change. This way, in the estándard notation of [5, 6] , the action of Beam Splitter 2, BS2, on
|BS2 acting on |a
>
|a >
|a >is
√
0.5 |d > + 0.5 |c >
√
= 0.5 (|d > +|c >)
−→
is
also with no phase
√
(11)
(12)
Again the Probability conservation requires
2
X
2
|ci | = 1
(13)
i=1
In a similar way, with no phase changes due to trasmission and phase change
ϕ = 180º, π
in radians, for
reection in a medium of higher index of refraction, the action of Beam Splitter 2, BS2, on
|BS2 acting on |b
√
0.5 |d > − 0.5 |c >
√
= 0.5 (|d > −|c >)
> −→
|b >is
√
(14)
(15)
As was made in BS1, it is possible to identify vectors and states
|a >=
|b >=
1
0
0
1
1
0
0
1
(16)
(17)
and
|d >=
|c >=
(18)
(19)
The matrix operator representing Beam Splitter 2 is
BS2 =
It is a trivial task to prove that
BS2 · BS2 = I
√
0.5
1
1
, and that
15
1
−1
BS2
(20)
is Unitary and Hermitian.
6.3
Mirror 2 and Mirror 1 representations
Figure 15: Mirror 2 action:
x' reected is 180º phase shifted from x.
Fig.15 shows the action of Mirror 2 on the below incident state.
|M irror2 acting on|x >−→ eiϕ |x0 >
(21)
Because there is a reection in a medium of higher refraction index, the phase change is 180º, or
radians. The matrix representation of Mirror 2,
M2,
M2 =
π
is
−1
0
0
1
(22)
The meaning of the +1 in the second row/second column is because the Mirror 2 does not act or change
the
Way Down beam or Superposed state. The doubt could be to put a 0 in this matrix element. But
Way Down beam or Superposed state, and that
this is nonsense, because a 0 means the elimination of
is not the action of the Mirror.
It is quite easy to obtain with a similar reasoning the matrix representation of Mirror 1, where an
is changed in a
|d >
with a phase change of
|a >state
π:
M1 =
1
0
0
−1
The meaning of the +1 in the rst row/rst column is because Mirror 1 does not act or change the
Up beam or Superposed State.
16
(23)
Way
6.4
The Phase Shifter representation
Figure 16: Phase Shifter action:
x' ouput
ϕ
phase shifted from
x imput.
The matrix representation of a Phase Shifter depends on where is placed. In our interferometer, it is in
the
Way Down, from Beam Splitter 1 to Mirror 1, see Fig.3, and its action is
|P hase Shif ter acting on| c >−→ eiϕ |c >
(24)
So, the matrix representation of this Beam Splitter, is
P hS =
1
0
0
(25)
eiϕ
The +1 in the rst row/rst column is because the Phase Shifteer does not act or change the
Way Up
beam or Superposed State.
7
Calculating the Detectors countings with the Quantum Formalism
The above matrix representation can be used to calculate the countings in the detectors. Being
|a >
the
input state in the Beam Splitter 1, the Detector's results are represented by a vector superposition of
states. This is a direct use ot the sum of amplitudes postulate of Quantum Mechanics, see [5, 6].
In the
Way Down, the detectors counting are the result of Beam Splitter 1, Phase Shifter, Mirror1 and
Beam Splitter 2 acting on
|a >:
BS2 · M 1 · P hS · BS1 |a >
(26)
In the Way Up, the dectectors counting are the result of Beam Splitter 1, Mirror 2 and Beam Splitter 2
acting on
|a >:
BS2 · M 2 · BS1 |a >
The global result in the detectors the superposition of both states
17
(27)
BS2 · (M 1 · P hS + M 2) · BS1 |a >
(28)
In matrix representation
√
0.5
1
1
1
−1
1
·
0
0
−1
1
·
0
0
eiϕ
0
1
+
−1
0
√
−1
1
·
√
0.5
−1
1
1
1
1
0
(29)
Calculating the parenthesis
√
0.5
1
1
1
−1
−1
·
0
0
·
−eiϕ
0.5
1
1
1
0
(30)
Multiplying the matrices
−1 −eiϕ
−1 1
1
0.5
·
−1
eiϕ
1 1
0
iϕ
iϕ
1−e
−1 − e
1
= 0.5
1 + eiϕ −1 + eiϕ
0
(31)
(32)
Multiplying the matrix and the column vector
1
2
ϕ
=
ei 2
2
And the nal result is
e
iϕ
2
1 − eiϕ
1 + eiϕ
ϕ
ϕ
e−i 2 − ei 2
ϕ
ϕ
e−i 2 + ei 2

−i sin( ϕ2 )


cos( ϕ2 )

(33)
(34)
(35)
The probabilities are
ˆ for Detector 1
P1 = sin2
ϕ
P2 = cos2
ϕ
2
(36)
ˆ for Detector 2
2
(37)
These values are the Quantum Mechanics predictions, in good agreement with recent experiments, see,
for example, [3, 4]. In these references the
Single Photon technology and experiments are discussed.
18
References
[1]
"Quantum Theory and Measurement", John Archibald Wheeler and Wojciech Hubert Zurek,
Editors, Princeton Series in Physics, (1984). The article is "Law without Law", and the relevant
point is stated in Figure 4, page 183.
[2]
"Principles of Optics", Chapter VII,"Elements of the Theory of Interference and Interferometers", Max Born & Emil Wolf, Sixth Edition, Pergamon Press, Oxford (1993).
[3]
The 2013 Bohr Medal Lesson, Particle-Wave Duality in One Photon experiments,
and Wheeler's Delayed-Choice experiment , given by Alain Aspect, 11-October-2013, see the
Youtube HD movie.
Sixty-Two Years of Uncertainty,
Historical, Philosophical and Physical Inquiries into the Foundations of Quantum Mechanics, Edited by A.I. Miller, Plenum Press (1990)
Some of the Aspect's remarks in this lecture are also in
The article is Wave-Particle Duality: a case study, authors A. Aspect and Ph. Grangier.
[4]
"Experimental Realization of Wheeler's Delayed-Choice Gedanken Experiment", V.
Jacques, E. Wu, F. Grosshans, F. Treussat, Ph. Grangier, A. Aspect, J.F. Roch, Science, Volume
315, Issue 5814, pp. 966- (2007). Can be read in arXiv.
[5]
Online edition of the Feynman Lectures on Physics, Vol II Quantum Mechanics, or: R Feynman, R.
Leighton, M. Sands,
"The Feynman Lectures on Physics, Vol III Quantum Mechanics",
Fondo Educativo Interarmericano, Bilingual Edition English-Spanish, Bogotá (1971). The english
version is the 1965 edition of Addison-Wesley. The relevant part to this document is ChapterI Quantum Behavior, An Experiment with bullets to An Experiment with electrons and The Interference
of electron waves. The Feynman himself can be seen in the Cornell University 1964 Master Lectures,
see him in Youtube.
[6]
Mécanique Quantique vols. I and II,Claude Cohen-Tannoudji, Bernard Diu and Frank Laloë,
Collection Enseignement des sciences, Paris (1973).
[7]
Interferomenter Experiments of the University of St. Andrews, Quantum Mechanics Visualization
Proyect (QuVis) (requires Flash pluggins installed in the Internet browser).
[8]
3D Mach-Zehnder interferometer, in the Instituto to Física, Universidade do Rio Grande do
Sul (Brasil): http://www.if.ufrgs.br/∼fernanda/IMZ/Mach-Zehnder.exe ( a Windows executable).
[9]
The Philosophy of Quantum Mechanics, Max Jammer, Wiley, NY (1974)
[10]
Creación de Simulaciones Interactivas en Java , Francisco Esquembre, Pearson-Prentice Hall
ISBN 84-205-4009-9.
EJS can be downloaded in www.um.es/fem/Ejs/
19