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Plane wave representing a particle passing through two slits, resulting in an interference pattern on a screen some distance away from the slits. [1] .

The double-slit experiment is an experiment in quantum mechanics and optics demonstrating the wave-particle duality of electrons , photons , and other fundamental objects in physics. When streams of particles such as electrons or photons pass through two narrow adjacent slits to hit a detector screen on the other side, they don't form clusters based on whether they passed through one slit or the other. Instead, they interfere: simultaneously passing through both slits, and producing a pattern of interference bands on the screen. This phenomenon occurs even if the particles are fired one at a time, showing that the particles demonstrate some wave behavior by interfering with themselves as if they were a wave passing through both slits.

Niels Bohr proposed the idea of wave-particle duality to explain the results of the double-slit experiment. The idea is that all fundamental particles behave in some ways like waves and in other ways like particles, depending on what properties are being observed. These insights led to the development of quantum mechanics and quantum field theory , the current basis behind the Standard Model of particle physics , which is our most accurate understanding of how particles work.

The original double-slit experiment was performed using light/photons around the turn of the nineteenth century by Thomas Young, so the original experiment is often called Young's double-slit experiment. The idea of using particles other than photons in the experiment did not come until after the ideas of de Broglie and the advent of quantum mechanics, when it was proposed that fundamental particles might also behave as waves with characteristic wavelengths depending on their momenta. The single-electron version of the experiment was in fact not performed until 1974. A more recent version of the experiment successfully demonstrating wave-particle duality used buckminsterfullerene or buckyballs , the \(C_{60}\) allotrope of carbon.

Waves vs. Particles

Double-slit experiment with electrons, modeling the double-slit experiment.

To understand why the double-slit experiment is important, it is useful to understand the strong distinctions between wave and particles that make wave-particle duality so intriguing.

Waves describe oscillating values of a physical quantity that obey the wave equation . They are usually described by sums of sine and cosine functions, since any periodic (oscillating) function may be decomposed into a Fourier series . When two waves pass through each other, the resulting wave is the sum of the two original waves. This is called a superposition since the waves are placed ("-position") on top of each other ("super-"). Superposition is one of the most fundamental principles of quantum mechanics. A general quantum system need not be in one state or another but can reside in a superposition of two where there is some probability of measuring the quantum wavefunction in one state or another.

Left: example of superposed waves constructively interfering. Right: superposed waves destructively interfering. [2]

If one wave is \(A(x) = \sin (2x)\) and the other is \(B(x) = \sin (2x)\), then they add together to make \(A + B = 2 \sin (2x)\). The addition of two waves to form a wave of larger amplitude is in general known as constructive interference since the interference results in a larger wave.

If one wave is \(A(x) = \sin (2x)\) and the other is \(B(x) = \sin (2x + \pi)\), then they add together to make \(A + B = 0\) \(\big(\)since \(\sin (2x + \pi) = - \sin (2x)\big).\) This is known as destructive interference in general, when adding two waves results in a wave of smaller amplitude. See the figure above for examples of both constructive and destructive interference.

Two speakers are generating sounds with the same phase, amplitude, and wavelength. The two sound waves can make constructive interference, as above left. Or they can make destructive interference, as above right. If we want to find out the exact position where the two sounds make destructive interference, which of the following do we need to know?

a) the wavelength of the sound waves b) the distances from the two speakers c) the speed of sound generated by the two speakers

This wave behavior is quite unlike the behavior of particles. Classically, particles are objects with a single definite position and a single definite momentum. Particles do not make interference patterns with other particles in detectors whether or not they pass through slits. They only interact by colliding elastically , i.e., via electromagnetic forces at short distances. Before the discovery of quantum mechanics, it was assumed that waves and particles were two distinct models for objects, and that any real physical thing could only be described as a particle or as a wave, but not both.

In the more modern version of the double slit experiment using electrons, electrons with the same momentum are shot from an "electron gun" like the ones inside CRT televisions towards a screen with two slits in it. After each electron goes through one of the slits, it is observed hitting a single point on a detecting screen at an apparently random location. As more and more electrons pass through, one at a time, they form an overall pattern of light and dark interference bands. If each electron was truly just a point particle, then there would only be two clusters of observations: one for the electrons passing through the left slit, and one for the right. However, if electrons are made of waves, they interfere with themselves and pass through both slits simultaneously. Indeed, this is what is observed when the double-slit experiment is performed using electrons. It must therefore be true that the electron is interfering with itself since each electron was only sent through one at a time—there were no other electrons to interfere with it!

When the double-slit experiment is performed using electrons instead of photons, the relevant wavelength is the de Broglie wavelength \(\lambda:\)

\[\lambda = \frac{h}{p},\]

where \(h\) is Planck's constant and \(p\) is the electron's momentum.

Calculate the de Broglie wavelength of an electron moving with velocity \(1.0 \times 10^{7} \text{ m/s}.\)

Usain Bolt, the world champion sprinter, hit a top speed of 27.79 miles per hour at the Olympics. If he has a mass of 94 kg, what was his de Broglie wavelength?

Express your answer as an order of magnitude in units of the Bohr radius \(r_{B} = 5.29 \times 10^{-11} \text{m}\). For instance, if your answer was \(4 \times 10^{-5} r_{B}\), your should give \(-5.\)

Image Credit: Flickr drcliffordchoi.

While the de Broglie relation was postulated for massive matter, the equation applies equally well to light. Given light of a certain wavelength, the momentum and energy of that light can be found using de Broglie's formula. This generalizes the naive formula \(p = m v\), which can't be applied to light since light has no mass and always moves at a constant velocity of \(c\) regardless of wavelength.

The below is reproduced from the Amplitude, Frequency, Wave Number, Phase Shift wiki.

In Young's double-slit experiment, photons corresponding to light of wavelength \(\lambda\) are fired at a barrier with two thin slits separated by a distance \(d,\) as shown in the diagram below. After passing through the slits, they hit a screen at a distance of \(D\) away with \(D \gg d,\) and the point of impact is measured. Remarkably, both the experiment and theory of quantum mechanics predict that the number of photons measured at each point along the screen follows a complicated series of peaks and troughs called an interference pattern as below. The photons must exhibit the wave behavior of a relative phase shift somehow to be responsible for this phenomenon. Below, the condition for which maxima of the interference pattern occur on the screen is derived.

Left: actual experimental two-slit interference pattern of photons, exhibiting many small peaks and troughs. Right: schematic diagram of the experiment as described above. [3]

Since \(D \gg d\), the angle from each of the slits is approximately the same and equal to \(\theta\). If \(y\) is the vertical displacement to an interference peak from the midpoint between the slits, it is therefore true that

\[D\tan \theta \approx D\sin \theta \approx D\theta = y.\]

Furthermore, there is a path difference \(\Delta L\) between the two slits and the interference peak. Light from the lower slit must travel \(\Delta L\) further to reach any particular spot on the screen, as in the diagram below:

Light from the lower slit must travel further to reach the screen at any given point above the midpoint, causing the interference pattern.

The condition for constructive interference is that the path difference \(\Delta L\) is exactly equal to an integer number of wavelengths. The phase shift of light traveling over an integer \(n\) number of wavelengths is exactly \(2\pi n\), which is the same as no phase shift and therefore constructive interference. From the above diagram and basic trigonometry, one can write

\[\Delta L = d\sin \theta \approx d\theta = n\lambda.\]

The first equality is always true; the second is the condition for constructive interference.

Now using \(\theta = \frac{y}{D}\), one can see that the condition for maxima of the interference pattern, corresponding to constructive interference, is

\[n\lambda = \frac{dy}{D},\]

i.e. the maxima occur at the vertical displacements of

\[y = \frac{n\lambda D}{d}.\]

The analogous experimental setup and mathematical modeling using electrons instead of photons is identical except that the de Broglie wavelength of the electrons \(\lambda = \frac{h}{p}\) is used instead of the literal wavelength of light.

• Lookang, . CC-3.0 Licensing . Retrieved from https://commons.wikimedia.org/w/index.php?curid=17014507
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• Quantum mechanics
• Opinion and reviews

The double-slit experiment

This article is an extended version of the article “The double-slit experiment” that appeared in the September 2002 issue of Physics World (p15). It has been further extended to include three letters about the history of the double-slit experiment with single electrons that were published in the May 2003 issue of the magazine.

What is the most beautiful experiment in physics? This is the question that Robert Crease asked Physics World readers in May – and more than 200 replied with suggestions as diverse as Schrödinger’s cat and the Trinity nuclear test in 1945. The top five included classic experiments by Galileo, Millikan, Newton and Thomas Young. But uniquely among the top 10, the most beautiful experiment in physics – Young’s double-slit experiment applied to the interference of single electrons – does not have a name associated with it.

Most discussions of double-slit experiments with particles refer to Feynman’s quote in his lectures: “We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.” Feynman went on to add: “We should say right away that you should not try to set up this experiment. This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in. We are doing a “thought experiment”, which we have chosen because it is easy to think about. We know the results that would be obtained because there are many experiments that have been done, in which the scale and the proportions have been chosen to show the effects we shall describe”.

It is not clear that Feynman was aware that the first double-slit experiment with electrons had been carried out in 1961, the year he started his lectures (which were published in 1963). More surprisingly, perhaps, Feynman did not stress that an interference pattern would build up even if there was just one electron in the apparatus at a time. (This lack of emphasis was unusual because in the same lecture Feynman describes the electron experiment – and other double-slit experiments with water waves and bullets – in considerable detail).

So who actually carried out the first double-slit experiment with single electrons? Not surprisingly many thought or gedanken experiments are named after theorists – such as the Aharonov-Bohm effect, Bell’s inequality, the Casimir force, the Einstein-Podolsky-Rosen paradox, Schrödinger’s cat and so on – and these names rightly remain even when the experiment has been performed by others in the laboratory. However, it seems remarkable that no name whatsoever is attached to the double-slit experiment with electrons. Standard reference books are silent on this question but a study of the literature reveals several unsung experimental heroes.

Back to Young

Young carried out his original double-slit experiment with light some time in the first decade of the 1800s, showing that the waves of light from the two slits interfered to produce a characteristic fringe pattern on a screen. In 1909 Geoffrey Ingram (G I) Taylor conducted an experiment in which he showed that even the feeblest light source – equivalent to “a candle burning at a distance slightly exceeding a mile” – could lead to interference fringes. This led to Dirac’s famous statement that “each photon then interferes only with itself”.

In 1927 Clinton Davisson and Lester Germer observed the diffraction of electron beams from a nickel crystal – demonstrating the wave-like properties of particles for the first time – and George (G P) Thompson did the same with thin films of celluloid and other materials shortly afterwards. Davisson and Thomson shared the 1937 Nobel prize for “discovery of the interference phenomena arising when crystals are exposed to electronic beams”, but neither performed a double-slit experiment with electrons.

In the early 1950s Ladislaus Laszlo Marton of the US National Bureau of Standards (now NIST) in Washington, DC demonstrated electron interference but this was in a Mach-Zehnder rather than a double-slit geometry. These were the early days of the electron microscope and physicists were keen to exploit the very short de Broglie wavelength of electrons to study objects that were too small to be studied with visible light. Doing gedanken or thought experiments in the laboratory was further down their list of priorities.

A few years later Gottfried Möllenstedt and Heinrich Düker of the University of Tübingen in Germany used an electron biprism – essentially a very thin conducting wire at right angles to the beam – to split an electron beam into two components and observe interference between them. (Möllenstedt made the wires by coating fibres from spiders’ webs with gold – indeed, it is said that he kept spiders in the laboratory for this purpose). The electron biprism was to become widely used in the development of electron holography and also in other experiments, including the first measurement of the Aharonov-Bohm effect by Bob Chambers at Bristol University in the UK in 1960.

But in 1961 Claus Jönsson of Tübingen, who had been one of Möllenstedt’s students, finally performed an actual double-slit experiment with electrons for the first time ( Zeitschrift für Physik 161 454). Indeed, he demonstrated interference with up to five slits. The next milestone – an experiment in which there was just one electron in the apparatus at any one time – was reached by Akira Tonomura and co-workers at Hitachi in 1989 when they observed the build up of the fringe pattern with a very weak electron source and an electron biprism ( American Journal of Physics 57 117-120). Whereas Jönsson’s experiment was analogous to Young’s original experiment, Tonomura’s was similar to G I Taylor’s. (Note added on 7 May: Pier Giorgio Merli, Giulio Pozzi and GianFranco Missiroli carried out double-slit interference experiments with single electrons in Bologna in the 1970s; see Merli et al. in Further reading and the letters from Steeds, Merli et al. , and Tonomura at the end of this article.)

Since then particle interference has been demonstrated with neutrons, atoms and molecules as large as carbon-60 and carbon-70. And earlier this year another famous experiment in optics – the Hanbury Brown and Twiss experiment – was performed with electrons for the first time (again at Tübingen!). However, the results are profoundly different this time because electrons are fermions – and therefore obey the Pauli exclusion principle – whereas photons are bosons and do not.

Credit where it’s due

So why are Jönsson, Tonomura and the other pioneers of the double-slit experiment not well known? One obvious reason is that Jönsson’s results were first published in German in a German journal. Another reason might be that there was little incentive to perform the ultimate thought experiment in the lab, and little recognition for doing so. When Jönsson’s paper was translated into English 13 years later and published in the American Journal of Physics in 1974 (volume 42, pp4-11), the journal’s editors, Anthony (A P) French and Edwin Taylor, described it as a “great experiment”, but added that there are “few professional rewards” for performing what they describe as “real, pedagogically clean fundamental experiments.”

It is worth noting that the first double-slit experiment with single electrons by Tonomura and co-workers was also published in the American Journal of Physics , which publishes articles on the educational and cultural aspects of physics, rather than being a research journal. Indeed, the journal’s information for contributors states: “We particularly encourage manuscripts on already published contemporary research that can be used directly or indirectly in the classroom. We specifically do not publish articles announcing new theories or experimental results.”

French and Taylor’s editorial also confirms how little known Jönsson’s experiment was at the time: “For decades two-slit electron interference has been presented as a thought experiment whose predicted results are justified by their remote and somewhat obscure relation to real experiments in which electrons are diffracted by crystals. Few such recent presentations acknowledge that the two-slit electron interference experiment has now been done and that the results agree with the expectation of quantum physics in all detail.”

However, it should be noted that the history of physics is complicated and that events are rarely as clear-cut as we might like. For instance, it is widely claimed that Young performed his double-slit experiment in 1801 but he did not publish any account of it until his Lectures on Natural Philosophy in 1807. It also appears as if Davisson and a young collaborator called Charles Kunsman observed electron diffraction in 1923 – four years before Davisson and Germer – without realising it.

Final thoughts

Gedanken or thought experiments have played an important role in the history of quantum physics. It is unlikely that the whole area of quantum information would be as lively as it is today – both theoretically and experimentally – if a small band of physicists had not persevered and actually demonstrated quantum phenomena with individual particles.

At one time the Casimir force, which has yet to be measured with an accuracy of better than 15% in the geometry first proposed by Hendrik Casimir in 1948, might also have been viewed as purely a pedagogical experiment – a gedanken experiment with little relevance to real experimental physics. However, it is now clear that applications as varied as nanotechnology and experimental tests of theories of “large” extra dimensions require a detailed knowledge of the Casimir force .

The need for “real, pedagogically clean fundamental experiments” is clearly as great as ever.

This is a longer version of the article “The double-slit experiment” that appeared in the print version of the September issue of Physics World, on page 15. Three letters that appeared in the May 2003 issue of the magazine have been added to the end of this version of the article.

T Young 1802 On the theory of light and colours (The 1801 Bakerian Lecture) Philosophical Transactions of the Royal Society of London 92 12-48

T Young 1804 Experiments and calculations relative to physical optics (The 1803 Bakerian Lecture) Philosophical Transactions of the Royal Society of London 94 1-16

T Young 1807 A Course of Lectures on Natural Philosophy and the Mechanical Arts (J Johnson, London)

G I Taylor 1909 Interference fringes with feeble light Proceedings of the Cambridge Philosophical Society 15 114-115

P A M Dirac 1958 The Principles of Quantum Mechanics (Oxford University Press) 4th edn p9

R P Feynman, R B Leighton and M Sands 1963 The Feynman Lecture on Physics (Addison-Wesley) vol 3 ch 37 (Quantum behaviour)

A Howie and J E Fowcs Williams (eds) 2002 Interference: 200 years after Thomas Young’s discoveries Philosophical Transactions of the Royal Society of London 360 803-1069

R P Crease 2002 The most beautiful experiment Physics World September pp19-20. This article contains the results of Crease’s survey for Physics World ; the first article about the survey appeared on page 17 of the May 2002 issue.

Electron interference experiments

Visit www.nobel.se/physics/laureates/1937/index.html for details of the Nobel prize awarded to Clinton Davisson and George Thomson

L Marton 1952 Electron interferometer Physical Review 85 1057-1058

L Marton, J Arol Simpson and J A Suddeth 1953 Electron beam interferometer Physical Review 90 490-491

L Marton, J Arol Simpson and J A Suddeth 1954 An electron interferometer Reviews of Scientific Instruments 25 1099-1104

G Möllenstedt and H Düker 1955 Naturwissenschaften 42 41

G Möllenstedt and H Düker 1956 Zeitschrift für Physik 145 377-397

G Möllenstedt and C Jönsson 1959 Zeitschrift für Physik 155 472-474

R G Chambers 1960 Shift of an electron interference pattern by enclosed magnetic flux Physical Review Letters 5 3-5

C Jönsson 1961 Zeitschrift für Physik 161 454-474

C Jönsson 1974 Electron diffraction at multiple slits American Journal of Physics 42 4-11

A P French and E F Taylor 1974 The pedagogically clean, fundamental experiment American Journal of Physics 42 3

P G Merli, G F Missiroli and G Pozzi 1976 On the statistical aspect of electron interference phenomena American Journal of Physics 44 306-7

A Tonomura, J Endo, T Matsuda, T Kawasaki and H Ezawa 1989 Demonstration of single-electron build-up of an interference pattern American Journal of Physics 57 117-120

H Kiesel, A Renz and F Hasselbach 2002 Observation of Hanbury Brown-Twiss anticorrelations for free electrons Nature 418 392-394

Atoms and molecules

O Carnal and J Mlynek 1991 Young’s double-slit experiment with atoms: a simple atom interferometer Physical Review Letters 66 2689-2692

D W Keith, C R Ekstrom, Q A Turchette and D E Pritchard 1991 An interferometer for atoms Physical Review Letters 66 2693-2696

M W Noel and C R Stroud Jr 1995 Young’s double-slit interferometry within an atom Physical Review Letters 75 1252-1255

M Arndt, O Nairz, J Vos-Andreae, C Keller, G van der Zouw and A Zeilinger 1999 Wave-particle duality of C 60 molecules Nature 401 680-682

B Brezger, L Hackermüller, S Uttenthaler, J Petschinka, M Arndt and A Zeilinger 2002 Matter-wave interferometer for large molecules Physical Review Letters 88 100404

Review articles and books

G F Missiroli, G Pozzi and U Valdrè 1981 Electron interferometry and interference electron microscopy Journal of Physics E 14 649-671. This review covers early work on electron interferometry by groups in Bologna, Toulouse, Tübingen and elsewhere.

A Zeilinger, R Gähler, C G Shull, W Treimer and W Mampe 1988 Single- and double-slit diffraction of neutrons Reviews of Modern Physics 60 1067-1073

A Tonomura 1993 Electron Holography (Springer-Verlag, Berlin/New York)

H Rauch and S A Werner 2000 Neutron Interferometry: Lessons in Experimental Quantum Mechanics (Oxford Science Publications)

The double-slit experiment with single electrons

The article “A brief history of the double-slit experiment” (September 2002 p15; correction October p17) describes how Claus Jönsson of the University of Tübingen performed the first double-slit interference experiment with electrons in 1961. It then goes on to say: “The next milestone – an experiment in which there was just one electron in the apparatus at any one time – was reached by Akira Tonomura and co-workers at Hitachi in 1989 when they observed the build up of the fringe pattern with a very weak electron source and an electron biprism ( Am. J. Phys. 57 117-120)”.

In fact, I believe that “the first double-slit experiment with single electrons” was performed by Pier Giorgio Merli, GianFranco Missiroli and Giulio Pozzi in Bologna in 1974 – some 15 years before the Hitachi experiment. Moreover, the Bologna experiment was performed under very difficult experimental conditions: the intrinsic coherence of the thermionic electron source used by the Bologna group was considerably lower than that of the field-emission source used in the Hitachi experiment.

The Bologna experiment is reported in a film called “Electron Interference” that received the award in the physics category at the International Festival on Scientific Cinematography in Brussels in 1976. A selection of six frames from the film ( see figure ) was also used for a short paper, “On the statistical aspect of electron interference phenomena”, that was submitted for publication in May 1974 and published two years later (P G Merli, G F Missiroli and G Pozzi 1976 Am. J. Phys. 44 306-7).

John Steeds Department of Physics, University of Bristol [email protected]

The history of science is not restricted to the achievements of big scientists or big scientific institutions. Contributions can also be made by researchers with the necessary background, curiosity and enthusiasm. In the period 1973-1974 we were investigating practical applications of electron interferometry with a Siemens Elmiskop 101 electron microscope that had been carefully calibrated at the CNR-LAMEL laboratory in Bologna, where one of us (PGM) was based ( J. Phys. E7 729-32).

These experiments followed earlier work at the Istituto di Fisica in 1972-73 in which the electron biprism was inserted in a Siemens Elmiskop IA and then used both for didactic ( Am. J. Phys. 41 639-644) and research experiments ( J. Microscopie 18 103-108). We used the Elmiskop 101 for many experiments including, for instance, the observation of the electrostatic field associated with p-n junctions ( J. Microscopie 21 11-20).

During this period we learnt that Professors Angelo and Aurelio Bairati in the Institute of Anatomy at the University of Milan had bought an image intensifier that could be used with the Elmiskop 101. Out of curiosity, and also realizing the conceptual importance of interference experiments with single photons or electrons, we asked if we could attempt to perform an interference experiment with single electrons in the Milan laboratory. Our results formed the basis of the film “Electron interference” and were also published in 1976 ( Am. J. Phys. 44 306-7).

Following the publication of the paper by Tonomura and co-workers in 1989, which did not refer to our 1976 paper (although it did contain an incorrect reference to our film), the American Journal of Physics published a letter from Greyson Gilson of Submicron Structures Inc. The letter stated: “Tonomura et al. seem to believe that they were the first to perform a successful two-slit interference experiment using electrons and also that they were the first to observe the cumulative build-up of the resulting electron interference pattern. Although their demonstration is very admirable, reports of similar work have appeared in this Journal for about 30 years (see, for examples, Refs. 2-7.) It seems inappropriate to permit the widespread misconception that such experiments have not been performed and perhaps cannot be performed to continue.” (G Gilson 1989 Am. J. Phys. 57 680). Three of the seven papers that Gilson refers to were from our group in Bologna.

The main subject of our 1976 paper and the 1989 paper from the Hitachi group are the same: the single-electron build-up of the interference pattern and the statistical aspect of the phenomena. Obviously the electron-detection system used by the Hitachi group in 1989 was more sophisticated than the one we used in 1974. However, the sentence on page 118 of the paper by Tonomura et al. , which states that in our film we “showed the electron arrival in each frame without recording the cumulative arrivals”, is not correct: this can be seen by watching the film and looking at figure 1 of our 1976 paper (a version of which is shown here ).

Finally, it is also worth noting that the first double-slit experiment with single electrons was actually a by-product of research into the practical applications of electron interferometry.

Pier Giorgio Merli LAMEL, CNR Bologna, Italy [email protected] Giulio Pozzi Department of Physics, University of Bologna [email protected] GianFranco Missiroli Department of Physics, University of Bologna [email protected]

The Bologna group photographed the monitor of a sensitive TV camera as they changed the intensity of an electron beam. They observed that a few light flashes of electrons appeared at low intensities, and that interference fringes were formed at high intensities. They also mentioned that they were able to increase the storage time up to “values of minutes”. Historically, they are the first to report such experiments concerning the formation of interference patterns as far as I know.

Later, similar experiments were conducted by Hannes Lichte, then at Tübingen and now at Dresden. Important experiments on electron interference were also carried out by Valentin Fabrikant and co-workers at the Moscow Institute for Energetics in 1949 and later by Takeo Ichinokawa of Waseda University in Tokyo.

Our experiments at Hitachi (A Tonomura, J Endo, T Matsuda, T Kawasaki and H Ezawa 1989 Demonstration of single-electron buildup of an interference pattern Am. J. Phys. 57 117–120) differed from these experiments in the following respects:

(a) Our experiments were carried out from beginning to end with constant and extremely low electron intensities – fewer than 1000 electrons per second – so there was no chance of finding two or more electrons in the apparatus at the same time. This removed any possibility that the fringes might be due to interactions between the electrons, as had been suspected by some physicists, such as Sin-Itiro Tomonaga.

(b) We developed a position-sensitive electron-counting system that was modified from the photon-counting image acquisition system produced by Hamamatsu Photonics. In this system, the formation of fringes could be observed as a time series; the electrons were accumulated over time to gradually form an interference pattern on the monitor (similar to a long exposure with a photographic film). The electrons arrived at random positions on the detector only once in a while and it took more than 20 minutes for the interference pattern to form (see figure). To film the build-up process, the electron source, the electron biprism and the rest of the experiment therefore had to be extremely stable: if the interference pattern had drifted by a fraction of fringe spacing over the exposure time, the whole fringe pattern would have disappeared.

(c) The electrons arriving at the detector were detected with almost 100% efficiency. Counting losses and noise in conventional TV cameras mean that it is difficult to know if each flash of the screen really corresponds to an individual electron. Therefore, the detection error in our experiment was limited to less than 1%.

We believe that we carried out the first experiment in which the build-up process of an interference pattern from single-electron events could be seen in real time as in Feynman’s famous double-slit Gedanken experiment under the condition, we emphasize, that there was no chance of finding two or more electrons in the apparatus.

Akira Tonomura Hitachi Advanced Research Laboratory, Saitama, Japan [email protected]

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1 Quantum Behavior

1–1 Atomic mechanics

“Quantum mechanics” is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.

Newton thought that light was made up of particles, but then it was discovered that it behaves like a wave. Later, however (in the beginning of the twentieth century), it was found that light did indeed sometimes behave like a particle. Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither. Now we have given up. We say: “It is like neither .”

There is one lucky break, however—electrons behave just like light. The quantum behavior of atomic objects (electrons, protons, neutrons, photons, and so on) is the same for all, they are all “particle waves,” or whatever you want to call them. So what we learn about the properties of electrons (which we shall use for our examples) will apply also to all “particles,” including photons of light.

The gradual accumulation of information about atomic and small-scale behavior during the first quarter of the 20th century, which gave some indications about how small things do behave, produced an increasing confusion which was finally resolved in 1926 and 1927 by Schrödinger, Heisenberg, and Born. They finally obtained a consistent description of the behavior of matter on a small scale. We take up the main features of that description in this chapter.

Because atomic behavior is so unlike ordinary experience, it is very difficult to get used to, and it appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not understand it the way they would like to, and it is perfectly reasonable that they should not, because all of direct, human experience and of human intuition applies to large objects. We know how large objects will act, but things on a small scale just do not act that way. So we have to learn about them in a sort of abstract or imaginative fashion and not by connection with our direct experience.

In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.

1–2 An experiment with bullets

To try to understand the quantum behavior of electrons, we shall compare and contrast their behavior, in a particular experimental setup, with the more familiar behavior of particles like bullets, and with the behavior of waves like water waves. We consider first the behavior of bullets in the experimental setup shown diagrammatically in Fig.  1–1 . We have a machine gun that shoots a stream of bullets. It is not a very good gun, in that it sprays the bullets (randomly) over a fairly large angular spread, as indicated in the figure. In front of the gun we have a wall (made of armor plate) that has in it two holes just about big enough to let a bullet through. Beyond the wall is a backstop (say a thick wall of wood) which will “absorb” the bullets when they hit it. In front of the backstop we have an object which we shall call a “detector” of bullets. It might be a box containing sand. Any bullet that enters the detector will be stopped and accumulated. When we wish, we can empty the box and count the number of bullets that have been caught. The detector can be moved back and forth (in what we will call the $x$-direction). With this apparatus, we can find out experimentally the answer to the question: “What is the probability that a bullet which passes through the holes in the wall will arrive at the backstop at the distance $x$ from the center?” First, you should realize that we should talk about probability, because we cannot say definitely where any particular bullet will go. A bullet which happens to hit one of the holes may bounce off the edges of the hole, and may end up anywhere at all. By “probability” we mean the chance that the bullet will arrive at the detector, which we can measure by counting the number which arrive at the detector in a certain time and then taking the ratio of this number to the total number that hit the backstop during that time. Or, if we assume that the gun always shoots at the same rate during the measurements, the probability we want is just proportional to the number that reach the detector in some standard time interval.

For our present purposes we would like to imagine a somewhat idealized experiment in which the bullets are not real bullets, but are indestructible bullets—they cannot break in half. In our experiment we find that bullets always arrive in lumps, and when we find something in the detector, it is always one whole bullet. If the rate at which the machine gun fires is made very low, we find that at any given moment either nothing arrives, or one and only one—exactly one—bullet arrives at the backstop. Also, the size of the lump certainly does not depend on the rate of firing of the gun. We shall say: “Bullets always arrive in identical lumps.” What we measure with our detector is the probability of arrival of a lump. And we measure the probability as a function of $x$. The result of such measurements with this apparatus (we have not yet done the experiment, so we are really imagining the result) are plotted in the graph drawn in part (c) of Fig.  1–1 . In the graph we plot the probability to the right and $x$ vertically, so that the $x$-scale fits the diagram of the apparatus. We call the probability $P_{12}$ because the bullets may have come either through hole $1$ or through hole $2$. You will not be surprised that $P_{12}$ is large near the middle of the graph but gets small if $x$ is very large. You may wonder, however, why $P_{12}$ has its maximum value at $x=0$. We can understand this fact if we do our experiment again after covering up hole $2$, and once more while covering up hole $1$. When hole $2$ is covered, bullets can pass only through hole $1$, and we get the curve marked $P_1$ in part (b) of the figure. As you would expect, the maximum of $P_1$ occurs at the value of $x$ which is on a straight line with the gun and hole $1$. When hole $1$ is closed, we get the symmetric curve $P_2$ drawn in the figure. $P_2$ is the probability distribution for bullets that pass through hole $2$. Comparing parts (b) and (c) of Fig.  1–1 , we find the important result that \begin{equation} \label{Eq:III:1:1} P_{12}=P_1+P_2. \end{equation} The probabilities just add together. The effect with both holes open is the sum of the effects with each hole open alone. We shall call this result an observation of “ no interference ,” for a reason that you will see later. So much for bullets. They come in lumps, and their probability of arrival shows no interference.

1–3 An experiment with waves

Now we wish to consider an experiment with water waves. The apparatus is shown diagrammatically in Fig.  1–2 . We have a shallow trough of water. A small object labeled the “wave source” is jiggled up and down by a motor and makes circular waves. To the right of the source we have again a wall with two holes, and beyond that is a second wall, which, to keep things simple, is an “absorber,” so that there is no reflection of the waves that arrive there. This can be done by building a gradual sand “beach.” In front of the beach we place a detector which can be moved back and forth in the $x$-direction, as before. The detector is now a device which measures the “intensity” of the wave motion. You can imagine a gadget which measures the height of the wave motion, but whose scale is calibrated in proportion to the square of the actual height, so that the reading is proportional to the intensity of the wave. Our detector reads, then, in proportion to the energy being carried by the wave—or rather, the rate at which energy is carried to the detector.

With our wave apparatus, the first thing to notice is that the intensity can have any size. If the source just moves a very small amount, then there is just a little bit of wave motion at the detector. When there is more motion at the source, there is more intensity at the detector. The intensity of the wave can have any value at all. We would not say that there was any “lumpiness” in the wave intensity.

Now let us measure the wave intensity for various values of $x$ (keeping the wave source operating always in the same way). We get the interesting-looking curve marked $I_{12}$ in part (c) of the figure.

We have already worked out how such patterns can come about when we studied the interference of electric waves in Volume I. In this case we would observe that the original wave is diffracted at the holes, and new circular waves spread out from each hole. If we cover one hole at a time and measure the intensity distribution at the absorber we find the rather simple intensity curves shown in part (b) of the figure. $I_1$ is the intensity of the wave from hole $1$ (which we find by measuring when hole $2$ is blocked off) and $I_2$ is the intensity of the wave from hole $2$ (seen when hole $1$ is blocked).

The intensity $I_{12}$ observed when both holes are open is certainly not the sum of $I_1$ and $I_2$. We say that there is “interference” of the two waves. At some places (where the curve $I_{12}$ has its maxima) the waves are “in phase” and the wave peaks add together to give a large amplitude and, therefore, a large intensity. We say that the two waves are “interfering constructively” at such places. There will be such constructive interference wherever the distance from the detector to one hole is a whole number of wavelengths larger (or shorter) than the distance from the detector to the other hole.

At those places where the two waves arrive at the detector with a phase difference of $\pi$ (where they are “out of phase”) the resulting wave motion at the detector will be the difference of the two amplitudes. The waves “interfere destructively,” and we get a low value for the wave intensity. We expect such low values wherever the distance between hole $1$ and the detector is different from the distance between hole $2$ and the detector by an odd number of half-wavelengths. The low values of $I_{12}$ in Fig.  1–2 correspond to the places where the two waves interfere destructively.

You will remember that the quantitative relationship between $I_1$, $I_2$, and $I_{12}$ can be expressed in the following way: The instantaneous height of the water wave at the detector for the wave from hole $1$ can be written as (the real part of) $h_1e^{i\omega t}$, where the “amplitude” $h_1$ is, in general, a complex number. The intensity is proportional to the mean squared height or, when we use the complex numbers, to the absolute value squared $\abs{h_1}^2$. Similarly, for hole $2$ the height is $h_2e^{i\omega t}$ and the intensity is proportional to $\abs{h_2}^2$. When both holes are open, the wave heights add to give the height $(h_1+h_2)e^{i\omega t}$ and the intensity $\abs{h_1+h_2}^2$. Omitting the constant of proportionality for our present purposes, the proper relations for interfering waves are \begin{equation} \label{Eq:III:1:2} I_1=\abs{h_1}^2,\quad I_2=\abs{h_2}^2,\quad I_{12}=\abs{h_1+h_2}^2. \end{equation}

You will notice that the result is quite different from that obtained with bullets (Eq.  1.1 ). If we expand $\abs{h_1+h_2}^2$ we see that \begin{equation} \label{Eq:III:1:3} \abs{h_1+h_2}^2=\abs{h_1}^2+\abs{h_2}^2+2\abs{h_1}\abs{h_2}\cos\delta, \end{equation} where $\delta$ is the phase difference between $h_1$ and $h_2$. In terms of the intensities, we could write \begin{equation} \label{Eq:III:1:4} I_{12}=I_1+I_2+2\sqrt{I_1I_2}\cos\delta. \end{equation} The last term in ( 1.4 ) is the “interference term.” So much for water waves. The intensity can have any value, and it shows interference.

1–4 An experiment with electrons

Now we imagine a similar experiment with electrons. It is shown diagrammatically in Fig.  1–3 . We make an electron gun which consists of a tungsten wire heated by an electric current and surrounded by a metal box with a hole in it. If the wire is at a negative voltage with respect to the box, electrons emitted by the wire will be accelerated toward the walls and some will pass through the hole. All the electrons which come out of the gun will have (nearly) the same energy. In front of the gun is again a wall (just a thin metal plate) with two holes in it. Beyond the wall is another plate which will serve as a “backstop.” In front of the backstop we place a movable detector. The detector might be a geiger counter or, perhaps better, an electron multiplier, which is connected to a loudspeaker.

We should say right away that you should not try to set up this experiment (as you could have done with the two we have already described). This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in. We are doing a “thought experiment,” which we have chosen because it is easy to think about. We know the results that would be obtained because there are many experiments that have been done, in which the scale and the proportions have been chosen to show the effects we shall describe.

The first thing we notice with our electron experiment is that we hear sharp “clicks” from the detector (that is, from the loudspeaker). And all “clicks” are the same. There are no “half-clicks.”

We would also notice that the “clicks” come very erratically. Something like: click ….. click-click … click …….. click …. click-click …… click …, etc., just as you have, no doubt, heard a geiger counter operating. If we count the clicks which arrive in a sufficiently long time—say for many minutes—and then count again for another equal period, we find that the two numbers are very nearly the same. So we can speak of the average rate at which the clicks are heard (so-and-so-many clicks per minute on the average).

As we move the detector around, the rate at which the clicks appear is faster or slower, but the size (loudness) of each click is always the same. If we lower the temperature of the wire in the gun, the rate of clicking slows down, but still each click sounds the same. We would notice also that if we put two separate detectors at the backstop, one or the other would click, but never both at once. (Except that once in a while, if there were two clicks very close together in time, our ear might not sense the separation.) We conclude, therefore, that whatever arrives at the backstop arrives in “lumps.” All the “lumps” are the same size: only whole “lumps” arrive, and they arrive one at a time at the backstop. We shall say: “Electrons always arrive in identical lumps.”

Just as for our experiment with bullets, we can now proceed to find experimentally the answer to the question: “What is the relative probability that an electron ‘lump’ will arrive at the backstop at various distances $x$ from the center?” As before, we obtain the relative probability by observing the rate of clicks, holding the operation of the gun constant. The probability that lumps will arrive at a particular $x$ is proportional to the average rate of clicks at that $x$.

The result of our experiment is the interesting curve marked $P_{12}$ in part (c) of Fig.  1–3 . Yes! That is the way electrons go.

1–5 The interference of electron waves

Now let us try to analyze the curve of Fig.  1–3 to see whether we can understand the behavior of the electrons. The first thing we would say is that since they come in lumps, each lump, which we may as well call an electron, has come either through hole $1$ or through hole $2$. Let us write this in the form of a “Proposition”:

Proposition A: Each electron either goes through hole $1$ or it goes through hole $2$.

Assuming Proposition A, all electrons that arrive at the backstop can be divided into two classes: (1) those that come through hole $1$, and (2) those that come through hole $2$. So our observed curve must be the sum of the effects of the electrons which come through hole $1$ and the electrons which come through hole $2$. Let us check this idea by experiment. First, we will make a measurement for those electrons that come through hole $1$. We block off hole $2$ and make our counts of the clicks from the detector. From the clicking rate, we get $P_1$. The result of the measurement is shown by the curve marked $P_1$ in part (b) of Fig.  1–3 . The result seems quite reasonable. In a similar way, we measure $P_2$, the probability distribution for the electrons that come through hole $2$. The result of this measurement is also drawn in the figure.

The result $P_{12}$ obtained with both holes open is clearly not the sum of $P_1$ and $P_2$, the probabilities for each hole alone. In analogy with our water-wave experiment, we say: “There is interference.” \begin{equation} \label{Eq:III:1:5} \text{For electrons:}\quad P_{12}\neq P_1+P_2. \end{equation}

How can such an interference come about? Perhaps we should say: “Well, that means, presumably, that it is not true that the lumps go either through hole $1$ or hole $2$, because if they did, the probabilities should add. Perhaps they go in a more complicated way. They split in half and …” But no! They cannot, they always arrive in lumps … “Well, perhaps some of them go through $1$, and then they go around through $2$, and then around a few more times, or by some other complicated path … then by closing hole $2$, we changed the chance that an electron that started out through hole $1$ would finally get to the backstop …” But notice! There are some points at which very few electrons arrive when both holes are open, but which receive many electrons if we close one hole, so closing one hole increased the number from the other. Notice, however, that at the center of the pattern, $P_{12}$ is more than twice as large as $P_1+P_2$. It is as though closing one hole decreased the number of electrons which come through the other hole. It seems hard to explain both effects by proposing that the electrons travel in complicated paths.

It is all quite mysterious. And the more you look at it the more mysterious it seems. Many ideas have been concocted to try to explain the curve for $P_{12}$ in terms of individual electrons going around in complicated ways through the holes. None of them has succeeded. None of them can get the right curve for $P_{12}$ in terms of $P_1$ and $P_2$.

Yet, surprisingly enough, the mathematics for relating $P_1$ and $P_2$ to $P_{12}$ is extremely simple. For $P_{12}$ is just like the curve $I_{12}$ of Fig.  1–2 , and that was simple. What is going on at the backstop can be described by two complex numbers that we can call $\phi_1$ and $\phi_2$ (they are functions of $x$, of course). The absolute square of $\phi_1$ gives the effect with only hole $1$ open. That is, $P_1=\abs{\phi_1}^2$. The effect with only hole $2$ open is given by $\phi_2$ in the same way. That is, $P_2=\abs{\phi_2}^2$. And the combined effect of the two holes is just $P_{12}=\abs{\phi_1+\phi_2}^2$. The mathematics is the same as that we had for the water waves! (It is hard to see how one could get such a simple result from a complicated game of electrons going back and forth through the plate on some strange trajectory.)

We conclude the following: The electrons arrive in lumps, like particles, and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves “sometimes like a particle and sometimes like a wave.”

Incidentally, when we were dealing with classical waves we defined the intensity as the mean over time of the square of the wave amplitude, and we used complex numbers as a mathematical trick to simplify the analysis. But in quantum mechanics it turns out that the amplitudes must be represented by complex numbers. The real parts alone will not do. That is a technical point, for the moment, because the formulas look just the same.

Since the probability of arrival through both holes is given so simply, although it is not equal to $(P_1+P_2)$, that is really all there is to say. But there are a large number of subtleties involved in the fact that nature does work this way. We would like to illustrate some of these subtleties for you now. First, since the number that arrives at a particular point is not equal to the number that arrives through $1$ plus the number that arrives through $2$, as we would have concluded from Proposition A, undoubtedly we should conclude that Proposition A is false . It is not true that the electrons go either through hole $1$ or hole $2$. But that conclusion can be tested by another experiment.

1–6 Watching the electrons

We shall now try the following experiment. To our electron apparatus we add a very strong light source, placed behind the wall and between the two holes, as shown in Fig.  1–4 . We know that electric charges scatter light. So when an electron passes, however it does pass, on its way to the detector, it will scatter some light to our eye, and we can see where the electron goes. If, for instance, an electron were to take the path via hole $2$ that is sketched in Fig.  1–4 , we should see a flash of light coming from the vicinity of the place marked $A$ in the figure. If an electron passes through hole $1$, we would expect to see a flash from the vicinity of the upper hole. If it should happen that we get light from both places at the same time, because the electron divides in half … Let us just do the experiment!

Here is what we see: every time that we hear a “click” from our electron detector (at the backstop), we also see a flash of light either near hole $1$ or near hole $2$, but never both at once! And we observe the same result no matter where we put the detector. From this observation we conclude that when we look at the electrons we find that the electrons go either through one hole or the other. Experimentally, Proposition A is necessarily true.

What, then, is wrong with our argument against Proposition A? Why isn’t  $P_{12}$ just equal to $P_1+P_2$? Back to experiment! Let us keep track of the electrons and find out what they are doing. For each position ($x$-location) of the detector we will count the electrons that arrive and also keep track of which hole they went through, by watching for the flashes. We can keep track of things this way: whenever we hear a “click” we will put a count in Column $1$ if we see the flash near hole $1$, and if we see the flash near hole $2$, we will record a count in Column $2$. Every electron which arrives is recorded in one of two classes: those which come through $1$ and those which come through $2$. From the number recorded in Column $1$ we get the probability $P_1'$ that an electron will arrive at the detector via hole $1$; and from the number recorded in Column $2$ we get $P_2'$, the probability that an electron will arrive at the detector via hole $2$. If we now repeat such a measurement for many values of $x$, we get the curves for $P_1'$ and $P_2'$ shown in part (b) of Fig.  1–4 .

Well, that is not too surprising! We get for $P_1'$ something quite similar to what we got before for $P_1$ by blocking off hole $2$; and $P_2'$ is similar to what we got by blocking hole $1$. So there is not any complicated business like going through both holes. When we watch them, the electrons come through just as we would expect them to come through. Whether the holes are closed or open, those which we see come through hole $1$ are distributed in the same way whether hole $2$ is open or closed.

But wait! What do we have now for the total probability, the probability that an electron will arrive at the detector by any route? We already have that information. We just pretend that we never looked at the light flashes, and we lump together the detector clicks which we have separated into the two columns. We must just add the numbers. For the probability that an electron will arrive at the backstop by passing through either hole, we do find $P_{12}'=P_1'+P_2'$. That is, although we succeeded in watching which hole our electrons come through, we no longer get the old interference curve $P_{12}$, but a new one, $P_{12}'$, showing no interference! If we turn out the light $P_{12}$ is restored.

We must conclude that when we look at the electrons the distribution of them on the screen is different than when we do not look. Perhaps it is turning on our light source that disturbs things? It must be that the electrons are very delicate, and the light, when it scatters off the electrons, gives them a jolt that changes their motion. We know that the electric field of the light acting on a charge will exert a force on it. So perhaps we should expect the motion to be changed. Anyway, the light exerts a big influence on the electrons. By trying to “watch” the electrons we have changed their motions. That is, the jolt given to the electron when the photon is scattered by it is such as to change the electron’s motion enough so that if it might have gone to where $P_{12}$ was at a maximum it will instead land where $P_{12}$ was a minimum; that is why we no longer see the wavy interference effects.

You may be thinking: “Don’t use such a bright source! Turn the brightness down! The light waves will then be weaker and will not disturb the electrons so much. Surely, by making the light dimmer and dimmer, eventually the wave will be weak enough that it will have a negligible effect.” O.K. Let’s try it. The first thing we observe is that the flashes of light scattered from the electrons as they pass by does not get weaker. It is always the same-sized flash . The only thing that happens as the light is made dimmer is that sometimes we hear a “click” from the detector but see no flash at all . The electron has gone by without being “seen.” What we are observing is that light also acts like electrons, we knew that it was “wavy,” but now we find that it is also “lumpy.” It always arrives—or is scattered—in lumps that we call “photons.” As we turn down the intensity of the light source we do not change the size of the photons, only the rate at which they are emitted. That explains why, when our source is dim, some electrons get by without being seen. There did not happen to be a photon around at the time the electron went through.

This is all a little discouraging. If it is true that whenever we “see” the electron we see the same-sized flash, then those electrons we see are always the disturbed ones. Let us try the experiment with a dim light anyway. Now whenever we hear a click in the detector we will keep a count in three columns: in Column (1) those electrons seen by hole $1$, in Column (2) those electrons seen by hole $2$, and in Column (3) those electrons not seen at all. When we work up our data (computing the probabilities) we find these results: Those “seen by hole $1$” have a distribution like $P_1'$; those “seen by hole $2$” have a distribution like $P_2'$ (so that those “seen by either hole $1$ or $2$” have a distribution like $P_{12}'$); and those “not seen at all” have a “wavy” distribution just like $P_{12}$ of Fig.  1–3 ! If the electrons are not seen, we have interference!

That is understandable. When we do not see the electron, no photon disturbs it, and when we do see it, a photon has disturbed it. There is always the same amount of disturbance because the light photons all produce the same-sized effects and the effect of the photons being scattered is enough to smear out any interference effect.

Is there not some way we can see the electrons without disturbing them? We learned in an earlier chapter that the momentum carried by a “photon” is inversely proportional to its wavelength ($p=h/\lambda$). Certainly the jolt given to the electron when the photon is scattered toward our eye depends on the momentum that photon carries. Aha! If we want to disturb the electrons only slightly we should not have lowered the intensity of the light, we should have lowered its frequency (the same as increasing its wavelength). Let us use light of a redder color. We could even use infrared light, or radiowaves (like radar), and “see” where the electron went with the help of some equipment that can “see” light of these longer wavelengths. If we use “gentler” light perhaps we can avoid disturbing the electrons so much.

Let us try the experiment with longer waves. We shall keep repeating our experiment, each time with light of a longer wavelength. At first, nothing seems to change. The results are the same. Then a terrible thing happens. You remember that when we discussed the microscope we pointed out that, due to the wave nature of the light, there is a limitation on how close two spots can be and still be seen as two separate spots. This distance is of the order of the wavelength of light. So now, when we make the wavelength longer than the distance between our holes, we see a big fuzzy flash when the light is scattered by the electrons. We can no longer tell which hole the electron went through! We just know it went somewhere! And it is just with light of this color that we find that the jolts given to the electron are small enough so that $P_{12}'$ begins to look like $P_{12}$—that we begin to get some interference effect. And it is only for wavelengths much longer than the separation of the two holes (when we have no chance at all of telling where the electron went) that the disturbance due to the light gets sufficiently small that we again get the curve $P_{12}$ shown in Fig.  1–3 .

In our experiment we find that it is impossible to arrange the light in such a way that one can tell which hole the electron went through, and at the same time not disturb the pattern. It was suggested by Heisenberg that the then new laws of nature could only be consistent if there were some basic limitation on our experimental capabilities not previously recognized. He proposed, as a general principle, his uncertainty principle , which we can state in terms of our experiment as follows: “It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern.” If an apparatus is capable of determining which hole the electron goes through, it cannot be so delicate that it does not disturb the pattern in an essential way. No one has ever found (or even thought of) a way around the uncertainty principle. So we must assume that it describes a basic characteristic of nature.

The complete theory of quantum mechanics which we now use to describe atoms and, in fact, all matter, depends on the correctness of the uncertainty principle. Since quantum mechanics is such a successful theory, our belief in the uncertainty principle is reinforced. But if a way to “beat” the uncertainty principle were ever discovered, quantum mechanics would give inconsistent results and would have to be discarded as a valid theory of nature.

“Well,” you say, “what about Proposition A? Is it true, or is it not true, that the electron either goes through hole $1$ or it goes through hole $2$?” The only answer that can be given is that we have found from experiment that there is a certain special way that we have to think in order that we do not get into inconsistencies. What we must say (to avoid making wrong predictions) is the following. If one looks at the holes or, more accurately, if one has a piece of apparatus which is capable of determining whether the electrons go through hole $1$ or hole $2$, then one can say that it goes either through hole $1$ or hole $2$. But , when one does not try to tell which way the electron goes, when there is nothing in the experiment to disturb the electrons, then one may not say that an electron goes either through hole $1$ or hole $2$. If one does say that, and starts to make any deductions from the statement, he will make errors in the analysis. This is the logical tightrope on which we must walk if we wish to describe nature successfully.

If the motion of all matter—as well as electrons—must be described in terms of waves, what about the bullets in our first experiment? Why didn’t we see an interference pattern there? It turns out that for the bullets the wavelengths were so tiny that the interference patterns became very fine. So fine, in fact, that with any detector of finite size one could not distinguish the separate maxima and minima. What we saw was only a kind of average, which is the classical curve. In Fig.  1–5 we have tried to indicate schematically what happens with large-scale objects. Part (a) of the figure shows the probability distribution one might predict for bullets, using quantum mechanics. The rapid wiggles are supposed to represent the interference pattern one gets for waves of very short wavelength. Any physical detector, however, straddles several wiggles of the probability curve, so that the measurements show the smooth curve drawn in part (b) of the figure.

1–7 First principles of quantum mechanics

We will now write a summary of the main conclusions of our experiments. We will, however, put the results in a form which makes them true for a general class of such experiments. We can write our summary more simply if we first define an “ideal experiment” as one in which there are no uncertain external influences, i.e., no jiggling or other things going on that we cannot take into account. We would be quite precise if we said: “An ideal experiment is one in which all of the initial and final conditions of the experiment are completely specified.” What we will call “an event” is, in general, just a specific set of initial and final conditions. (For example: “an electron leaves the gun, arrives at the detector, and nothing else happens.”) Now for our summary.

• The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number $\phi$ which is called the probability amplitude: \begin{equation} \begin{aligned} P&=\text{probability},\\ \phi&=\text{probability amplitude},\\ P&=\abs{\phi}^2. \end{aligned} \label{Eq:III:1:6} \end{equation}
• When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference: \begin{equation} \begin{aligned} \phi&=\phi_1+\phi_2,\\ P&=\abs{\phi_1+\phi_2}^2. \end{aligned} \label{Eq:III:1:7} \end{equation}
• If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost: \begin{equation} \label{Eq:III:1:8} P=P_1+P_2. \end{equation}

One might still like to ask: “How does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced.

We would like to emphasize a very important difference between classical and quantum mechanics . We have been talking about the probability that an electron will arrive in a given circumstance. We have implied that in our experimental arrangement (or even in the best possible one) it would be impossible to predict exactly what would happen. We can only predict the odds! This would mean, if it were true, that physics has given up on the problem of trying to predict exactly what will happen in a definite circumstance. Yes! physics has given up. We do not know how to predict what would happen in a given circumstance , and we believe now that it is impossible—that the only thing that can be predicted is the probability of different events. It must be recognized that this is a retrenchment in our earlier ideal of understanding nature. It may be a backward step, but no one has seen a way to avoid it.

We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen. If we could look more closely at the electron, we could be able to tell where it would end up.” So far as we know, that is impossible. We would still be in difficulty. Suppose we were to assume that inside the electron there is some kind of machinery that determines where it is going to end up. That machine must also determine which hole it is going to go through on its way. But we must not forget that what is inside the electron should not be dependent on what we do, and in particular upon whether we open or close one of the holes. So if an electron, before it starts, has already made up its mind (a) which hole it is going to use, and (b) where it is going to land, we should find $P_1$ for those electrons that have chosen hole $1$, $P_2$ for those that have chosen hole $2$, and necessarily the sum $P_1+P_2$ for those that arrive through the two holes. There seems to be no way around this. But we have verified experimentally that that is not the case. And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say “at the present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature really is .

1–8 The uncertainty principle

This is the way Heisenberg stated the uncertainty principle originally: If you make the measurement on any object, and you can determine the $x$-component of its momentum with an uncertainty $\Delta p$, you cannot, at the same time, know its $x$-position more accurately than $\Delta x\geq\hbar/2\Delta p$, where $\hbar$ is a definite fixed number given by nature. It is called the “reduced Planck constant,” and is approximately $1.05\times10^{-34}$ joule-seconds. The uncertainties in the position and momentum of a particle at any instant must have their product greater than or equal to half the reduced Planck constant. This is a special case of the uncertainty principle that was stated above more generally. The more general statement was that one cannot design equipment in any way to determine which of two alternatives is taken, without, at the same time, destroying the pattern of interference.

Let us show for one particular case that the kind of relation given by Heisenberg must be true in order to keep from getting into trouble. We imagine a modification of the experiment of Fig.  1–3 , in which the wall with the holes consists of a plate mounted on rollers so that it can move freely up and down (in the $x$-direction), as shown in Fig.  1–6 . By watching the motion of the plate carefully we can try to tell which hole an electron goes through. Imagine what happens when the detector is placed at $x=0$. We would expect that an electron which passes through hole $1$ must be deflected downward by the plate to reach the detector. Since the vertical component of the electron momentum is changed, the plate must recoil with an equal momentum in the opposite direction. The plate will get an upward kick. If the electron goes through the lower hole, the plate should feel a downward kick. It is clear that for every position of the detector, the momentum received by the plate will have a different value for a traversal via hole $1$ than for a traversal via hole $2$. So! Without disturbing the electrons at all , but just by watching the plate , we can tell which path the electron used.

Now in order to do this it is necessary to know what the momentum of the screen is, before the electron goes through. So when we measure the momentum after the electron goes by, we can figure out how much the plate’s momentum has changed. But remember, according to the uncertainty principle we cannot at the same time know the position of the plate with an arbitrary accuracy. But if we do not know exactly where the plate is, we cannot say precisely where the two holes are. They will be in a different place for every electron that goes through. This means that the center of our interference pattern will have a different location for each electron. The wiggles of the interference pattern will be smeared out. We shall show quantitatively in the next chapter that if we determine the momentum of the plate sufficiently accurately to determine from the recoil measurement which hole was used, then the uncertainty in the $x$-position of the plate will, according to the uncertainty principle, be enough to shift the pattern observed at the detector up and down in the $x$-direction about the distance from a maximum to its nearest minimum. Such a random shift is just enough to smear out the pattern so that no interference is observed.

The uncertainty principle “protects” quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, the quantum mechanics would collapse. So he proposed that it must be impossible. Then people sat down and tried to figure out ways of doing it, and nobody could figure out a way to measure the position and the momentum of anything—a screen, an electron, a billiard ball, anything—with any greater accuracy. Quantum mechanics maintains its perilous but still correct existence.

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1.1: The Double Slit Experiment

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What is the nature of light? You may have learned light is electromagnetic waves propagating through space. Also, you may have learned that light is made of a rain of individual particles called photons. But these two notions seem contradictory, how can it be both?

The debate over the nature of light goes deep into the history of science. The eminent physicist Isaac Newton believed that light was a rain of particles, called corpuscles. At the beginning of the nineteenth century, Thomas Young demonstrated with his famous double-slit interference experiment that light propagates as waves. With Maxwell's formulation of electromagnetism at the end of the nineteenth century, it was generally accepted that light is propagated as electromagnetic waves, and the debate seemed to be over. However, in 1905, Einstein was able to explain the photoelectric effect, by using the idea of light quanta, or particles which we now call photons. The consequences of this are explained in the video below

Similar confusion reigned over the nature of electrons, which behaved like particles, but then it was discovered in electron diffraction experiments, performed in 1927, that they exhibit wave behavior. So do electrons behave like particles or waves? And what about photons? This great challenge was resolved with the discovery of the equations of quantum mechanics. But the theory is not intuitive, and its description of matter is very different from our common experience.

To understand what seems to be a paradox, we look to Young's double-slit experiment. Here's the set up: a source of light is shone at a screen with two very thin, identical slits cut into it. Some light passes through the two slits and lands upon a subsequent screen. Take a look at Figure \(\PageIndex{1}\) for a diagram of the experiment setup.

First, think about what would happen to a stream of bullets going through this double slit experiment. The source, which we think of as a machine gun, is unsteady and sprays the bullets in the general direction of the two slits. Some bullets pass through one slit, some pass through the other slit, and others don't make it through the slits. The bullets that do go through the slits then land on the observing screen behind them. Now suppose we closed slit 2 . Then the bullets can only go through slit 1 and land in a small spread behind slit 1 .

If we graphed the number of times a bullet that went through slit 1 landed at the position \(y\) on the observation screen, we would see a normal distribution centered directly behind slit 1 . That is, most land directly behind the slit, but some stray off a little due to the small amount randomness inherent in the gun, and because of they ricochet off the edges of the slit. If we now close slit 1 and open slit 2, we would see a normal distribution centered directly behind slit 2.

Now let's repeat the experiment with both slits open. If we graph the number of times a bullet that went through either slit landed at the position \(y\), we should see the sum of the graph we made for slit 1 and a the graph for slit 2.

Another way we can think of the graphs we made is as graphs of the probability that a bullet will land at a particular spot \(y\) on the screen. Let \(P_{1}(y)\) denote the probability that the bullet lands at point \(y\) when only slit 1 is open, and similarly for \(P_{2}(y)\). And let \(P_{12}(y)\) denote the probability that the bullet lands at point \(y\) when both slits are open. Then \(P_{12}(y)=P_{1}(y)+P_{2}(y)\).

Next, we consider the situation for waves, for example water waves. A water wave doesn't go through either slit 1 or slit 2 , it goes through both. You should imagine the crest of 1 water wave as it approaches the slits. As it hits the slits, the wave is blocked at all places but the two slits, and waves on the other side are generated at each slit as depicted in Figure \(\PageIndex{1}\). The pFET simulation below shows wave interference. Click on each of the rectangles (Waves, Interference, silts, diffraction) to see difference demonstrations

When the new waves generated at each slit run into each other, interference occurs. We can see this by plotting the intensity (that is, the amount of energy carried by the waves) at each point \(y\) along the viewing screen. What we see is the familiar interference pattern seen in Figure \(\PageIndex{1}\). The dark patches of the interference pattern occur where the wave from the first slit arrives perfectly out of sync with wave from the second slit, while the bright points are where the two arrive in sync. For example, the bright spot right in the middle is bright because each wave travels the exact same distance from their respective slit to the screen, so they arrive in sync. The first dark spots are where the wave from one slit traveled exactly half of a wavelength longer than the other wave, thus they arrive at opposite points in their cycle and cancel. Here, it is not the intensities coming from each slit that add, but height of the wave. This differs from the case of bullets: \(I_{12}(y) \neq I_{1}(y)+I_{2}(y)\), but \(h_{12}(y)=h_{1}(y)+h_{2}(y)\), and \(I_{12}(y)=h(y)^{2}\), where \(h(y)\) is the height of the wave and \(I(y)\) is the intensity, or energy, of the wave.

Before we can say what light does, we need one more crucial piece of information. What happens when we turn down the intensity in both of these examples?

In the case of bullets, turning down the intensity means turning down the rate at which the bullets are fired. When we turn down the intensity, each time a bullet hits the screen it transfers the same amount of energy, but the frequency at which bullets hit the screen becomes less.

With water waves, turning down the intensity means making the wave amplitudes smaller. Each time a wave hits the screen it transfers less energy, but the frequency of the waves hitting the screen is unchanged.

Now, what happens when we do this experiment with light. As Young observed in 1802, light makes an interference pattern on the screen. From this observation he concluded that the nature of light is wavelike, and reasonably so! However, Young was unable at the time to turn down the intensity of light enough to see the problem with the wave explanation.

Picture now that the observation screen is made of thousands of tiny little photo-detectors that can detect the energy they absorb. For high intensities the photo-detectors individually are picking up a lot of energy, and when we plot the intensity against the position \(y\) along the screen we see the same interference pattern described earlier. Now, turn the intensity of the light very very very low. At first, the intensity scales down lower and lower everywhere, just like with a wave. But as soon as we get low enough, the energy that the photo-detectors report reaches a minimum energy, and all of the detectors are reporting the same energy, call it \(E_{0}\), just at different rates. This energy corresponds to the energy carried by an individual photon, and at this stage we see what is called the quantization of light.

ADAPT \(\PageIndex{1}\)

Turn down the intensity so low that only one photo-detector reports something each second. In other words, the source only sends one photon at a time. Each time a detector receives a photon, we record where on the array it landed and plot it on a graph. The distribution we draw will reflect the probability that a single photon will land at a particular point.

Query \(\PageIndex{1}\)

Logically we think that the photon will either go through one slit or the other. Then, like the bullets, the probability that the photon lands at a point should be \(y\) is \(P_{12}(y)=P_{1}(y)+P_{2}(y)\) and the distribution we expect to see is the two peaked distribution of the bullets. But this not what we see at all.

What we actually see is the same interference pattern from before! But how can this be? For there to be an interference pattern, light coming from one slit must interfere with light from the other slit; but there is only one photon going through at a time! The modern explanation is that the photon actually goes through both slits at the same time, and interferes with itself. The mathematics is analogous to that in the case of water waves. We say that the probability \(P(y)\) that a photon is detected at \(y\) is proportional to the square of some quantity \(a(y)\), which we call a probability amplitude. Now probability amplitudes for different alternatives add up. So \(a_{12}(y)=a_{1}(y)+a_{2}(y)\). But \(P_{12}(y)=\left|a_{12}(y)\right|^{2} \neq\left|a_{1}(y)\right|^{2}+\left|a_{2}(y)\right|^{2}=P_{1}(y)+P_{2}(y)\).

Logically, we can ask which slit the photon went through, and try to measure it. Thus, we might construct a double slit experiment where we put a photodetector at each slit, so that each time a photon comes through the experiment we see which slit it went through and where it hits on the screen. But when such an experiment is preformed, the interference pattern gets completely washed out! The very fact that we know which slit the photon goes through makes the interference pattern go away. This is the first example we see of how measuring a quantum system alters the system.

Here the photon looks both like a particle, a discreet package, and a wave that can have interference. It seems that the photon acts like both a wave and a particle, but at the same time it doesn't exactly behave like either. This is what is commonly known as the wave-particle duality, usually thought of as a paradox. The resolution is that the quantum mechanical behavior of matter is unique, something entirely new.

What may be more mind blowing still is that if we conduct the exact same experiment with electrons instead of light, we get the exact same results! Although it is common to imagine electrons as tiny little charged spheres, they are actually quantum entities, neither wave nor particle but understood by their wavefunction.The truth is that there is no paradox, just an absence of intuition for quantum entities. Why should they be intuitive? Things on our scale do not behave like wavefunctions, and unless we conduct wild experiments like this we do not see the effects of quantum mechanics. The following sections describe in more detail some of the basic truths of quantum mechanics, so that we can build an intuition for a new behavior of matter.

If you are reading this document on-line, there are a couple of links to Flash animations. To see them requires the Flash player, which is free and available from http://www.macromedia.com/ .

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Quantum Mechanics and the Famous Double-slit Experiment

Table of Contents

• Quantum mechanics is known for its strangeness, including phenomena like wave-particle duality, which allows particles to behave like waves.
• The double-slit experiment is a key demonstration of this duality, showing that even single particles, like photons, exhibit wave-like behavior.
• When the experiment measures which slit a particle goes through, it behaves like a particle. When this measurement is not made, it exhibits interference patterns typical of waves.
• Heisenberg’s uncertainty principle plays a crucial role, connecting the accuracy of position measurement to the momentum of particles.
• Matter waves, proposed by de Broglie, suggest that all matter, not just light, exhibits wave-like properties.
• Wheeler’s delayed-choice experiment demonstrates that making a decision about what to measure after a particle has passed through the slits can affect its behavior, effectively rewriting the past.
• This temporal editing could theoretically reach back to the beginning of the universe, the big bang.
• The interpretation of quantum phenomena like this varies, and different physicists may have differing views on the implications of these experiments.

Introduction

Quantum mechanics is famously strange. But Schroedinger’s dead and alive cat, Einstein’s spooky action at a distance, and de Broglie’s matter waves are mere trifles compared to the greatest strangeness: quantum mechanics can be used to edit events that should have already happened; in other words, to re-write history. However, before we can understand how this is possible, we first need to know how Heisenberg’s uncertainty principle accounts for wave-particle duality in the double-slit experiment.

The Double-Slit Experiment

The double-slit experiment is famous because it provides an unequivocal demonstration that light behaves like a wave. But the importance of the double-slit experiment extends far beyond that demonstration because, as Richard Feynman said in 1966:

In reality, it contains the only mystery…In telling you how it works, we will have to tell you about the basic peculiarities of all quantum mechanics .

Figure 1: The double-slit experiment. Waves travel from the source (top) until they reach the first barrier, which contains a slit. A semi-circular wave emanates from the slit until it reaches the second barrier, which contains two slits. The two semi-circular waves emanating from these slits interfere with each other, producing peaks and troughs along radial lines that form an interference pattern on a screen (bottom).

Based on wor k first described by Young in 1802, the experimental apparatus consists of two vertical slits and a screen, as shown in Figure 1. The light emanating from each slit interferes with light from the other slit to produce an  interference pattern  on the screen. This seems to prove conclusively that light consists of waves, even though the only entities detected at the screen are individual particles, which we now call  photons . The bright regions in the interference pattern correspond to areas where photons land with high probability and the dark regions to areas where photons land with low probability.

Figure 2: Emergence (a→d) of an interference pattern in a double-slit experiment8, where each dot represents a photon. Reproduced with permission from Tanamura (Wikimedia).

Remarkably, the same type of interference pattern is obtained even when the light is made so dim that only one photon at a time reaches the screen, as shown in Figure 2. With such a low photon rate, it can take several weeks for the interference pattern to emerge. However, the very fact that an interference pattern emerges at all implies that even a single photon behaves like a wave. This, in turn, seems to imply that each photon passes through both slits at the same time, which is clearly nonsense. Only a wave can pass through both slits, but only a photon can hit a single point on the screen. However, this single-photon interference behavior is not the most remarkable feature of the double-slit experiment.

If a light detector is used to measure which slit each photon passes through then the interference pattern (shown in Figures 2d and 3b) is replaced by a  diffraction envelope  (shown in Figure 3a), as if the light consists of two streams of particles which spread out and merge over time.

Figure 3: The double-slit experiment can produce either (a) a broad diffraction pattern, or, (b) an interference pattern. Here, the height of the curve indicates light intensity. This also shows how probability density evolves over distance after photons pass through two slits19. (a) If slit identity (photon position) is measured then the waves from the two slits do not interfere with each other. (b) If slit identity is not measured then the waves from the two slits produce an interference pattern as they travel further from the slits. For display purposes, the distribution at each distance has unit height.

Crucially, a diffraction envelope is exactly what is observed if the waves from different slits are prevented from interfering with each other. This can be achieved by using a photographic plate to record photons as the first slit is opened on its own, and then the other slit is opened on its own. In this case, the image captured by the photographic plate is the simple sum of photons from each slit (i.e. a diffraction envelope), with the guarantee that photons from the two slits could not possibly have interfered with each other. Thus, we can choose to use a light detector to find out which slit each photon passed through, but as soon as we do this, we see a diffraction envelope instead of an interference pattern.

Despite many ingenious experiments, any attempt to ascertain which slit each photon passed through forces the light to stop behaving like a wave (which yields an interference pattern) and to start behaving like a stream of billiard balls (which yields a diffraction envelope). The transformation from wave-like behavior to particle-like behavior is intimately related to Heisenberg’s uncertainty principle , as explained below. However, even this wave-particle duality is not the most remarkable feature of the double-slit experiment.

Matter Waves

If the light is replaced by a beam of electrons then an interference pattern is obtained, as if the electrons were waves (which looks like Figure 2d). This is especially surprising because only a few years before this result was obtained in 1927, electrons were thought to behave exactly like miniature billiard balls. So, just as light can be forced to behave like billiard balls, electrons can be forced to behave like waves.

The radical idea that electrons could behave like waves was proposed by de Broglie (pronounced de Broil) in 1923. In fact, de Broglie made the far more radical proposal that all matter has wave-like properties, now referred to as  matter waves . Since that time, the double-slit experiment has been used to demonstrate the existence of matter waves using whole atoms and even whole molecules of  buckminsterfullerene  (each buckminsterfullerene molecule is about 1 nanometre in diameter and contains 60 carbon atoms).

But I digress, so let’s return to the subject at hand.

Did the Photon Pass-Through One Slit or Two?

The problem is that a photon (or a molecule of buckminsterfullerene), unlike a wave, cannot pass through both slits, which raises the question: did the photon really pass through only one slit, and if so, which one?

To attempt to answer this question, consider what happens if we replace the screen with an array of long tubes, each of which points at just one slit, as shown in Figure 4a. At the end of each tube is a photodetector, such that any photon it detects could have come from only one slit. Note that there should be a pair of detectors at every screen position, with each member of the pair pointing at a different slit. Thus, irrespective of where a photon lands on the screen, the slit from which it originated is measured.

If we were to use this imaginary apparatus then we would find that the distribution of photons is a diffraction envelope, as in Figures 3a and 4a. Crucially, this envelope is identical to the pattern that would be obtained if the slits were opened one at a time, as described above.

Figure 4: An imaginary experiment for demonstrating wave-particle duality. a) If slit identity (photon position at the barrier) is measured using an array of oriented detectors at the screen then momentum (direction) precision is reduced, and a diffraction envelope is observed on the screen, as here and in Figure 5.9a. b) If slit identity is not measured then the screen is allowed to measure direction (photon momentum) with high precision, so an interference pattern is observed, as here. In Wheeler’s delayed-choice experiment, we decide to measure either a) slit identity, or b) photon momentum, but the decision is made after the photon has passed through the slit(s).

Thus, using detectors to measure each photon’s slit identity (i.e. which slit the photon passed through) prevents any wave-like behavior, just as if each photon had traveled in complete isolation as a single particle. If both slots are left open (and no photodetectors are used) then the original interference pattern is restored, as if the individual photons behave like waves (as in Figures 3b and 4b). This is the famous wave-particle duality .

Heisenberg’s Uncertainty Principle

As counter-intuitive as the result above seems, it is consistent with  Heisenberg’s uncertainty principle .  This ensures that when the slit identity is  not  measured, uncertainty in the particle’s screen position is roughly equal to the distance between successive maxima in the interference pattern (i.e. it is of the same order of magnitude as the width of each bright region in Figure 1).

More generally, Heisenberg’s uncertainty principle guarantees   that  any  reduction in the uncertainty in slit identity (position at the barrier) must increase uncertainty in the momentum of photons as they exit the slits; because momentum includes direction, uncertainty in momentum translates to uncertainty in screen position. This is the famous  position-momentum trade-off  traditionally associated with Heisenberg’s uncertainty principle, where position indicates slit identity and momentum indicates screen position here (i.e. position-momentum uncertainty translates to slit identity-screen position uncertainty in the double-slit experiment).

Heisenberg’s uncertainty principle means that, in practice, it is possible to adjust the amount of information gained regarding position by varying the accuracy of the measurement devices. As more information on position (slit identity) is gained, less information on the fine structure of the interference pattern is available. Consequently, as more information about the position is gained, the interference pattern is gradually replaced by a broad diffraction envelope.

As an aside, Heisenberg explained his uncertainty principle by showing that shining a light on an electron must alter the position and momentum of that electron, which therefore introduces uncertainty in the electron’s position and momentum. But Heisenberg’s uncertainty principle does not depend on the uncertainty of measurements (in fact, it is the other way round). Because both light and matter behave as if they are waves, they can be analyzed using Fourier analysis . And if position and momentum are waves then they must obey a key result from Fourier analysis:  Heisenberg’s inequality.  Speaking very loosely, this states that if position wavelengths are long then momentum wavelengths are short, and  vice versa . More precisely, Heisenberg’s inequality implies that reducing position uncertainty increases momentum uncertainty (and  vice versa ) so that the values of both cannot be known exactly, independently of whether or not they are measured.

Wheeler’s Delayed-Choice Experiment

So far, we have chosen to measure two different aspects of each photon: first, by measuring each photon’s position on the screen (which translates into momentum at a slit; see the previous section); and second, by using tube detectors to measure which slit each photon came from (which translates into the photon’s position at the barrier).

Clearly, when measuring photon position, the experimental setup does not change over time, so it seems plausible (or at least acceptable) that each photon could have passed through both slits. Similarly, when measuring slit identity, it seems plausible that each photon passes through only one slit.

But there is an alternative experiment, which involves changing the experimental setup while each photon is in transit between the slits and the screen.  If what we choose to measure alters how each photon behaves then it seems reasonable that we must make this decision before each photon reaches the slits.  However, what if the decision on whether to measure screen position or slit identity is made after each photon has passed through the slit(s) but before it has reached the screen or tube detectors? This is  Wheeler’s delayed-choice experiment , depicted in Figure 4.

Such an experiment is conceptually easy to set up. Once a photon is in transit between the slits and the screen, we can decide whether to measure the photon’s screen position (by leaving the screen in place, Figure 4b) or slit identity (by removing the screen so that the tube detectors can function, Figure 4a).

The results of an experiment that is conceptually no different from this were published by Jacques et al, 2007 (who used interferometers). Translating from the interferometer experiment performed by Jacques, when the screen is left in place, the photons’ screen positions are effectively measured, which yields an interference pattern on the screen. In contrast, when the screen is removed, an array of detectors is revealed that detects photons, and the pattern of detected photons is consistent with a broad diffraction envelope (which is the sum of two such envelopes; one per slit), as if no interference had occurred.

Crucially, the decision about whether to measure screen position (by leaving the screen in place) or slit identity (by removing the screen) was made (at random) after each photon had passed through the slit(s); so the behavior of each photon as it passed through the slit(s) depended on a decision made after that photon had passed through the slit(s). In essence, it is as if a decision made now about whether to measure slit identity or screen position of a photon (that is already in transit from the slits to the screen) retrospectively affects whether that photon passed through just one slit or both slits.

Incidentally, we can be certain that the decision was made after a photon passed through the slit(s), as follows. If the distance between the slits and the screen is S then the time taken for a photon to travel from the slit(s) to the screen is T = cS seconds, where c is the speed of light . Suppose it was decided at time t to leave the screen in place, and a photon arrived at the screen dt seconds later. Clearly, if dt < T then the photon must have been in transit between the slit(s) and the screen when the decision was made. Armed with this knowledge, we can select the subset P of photons for which dt<T , and record where they landed to find the pattern they collectively made on the screen. Using the same logic, if the decision was made to remove the screen then the tube detectors measure which slit each photon (in the subset P ) came from, with the guarantee that all these photons were in transit when the decision was made. Given a sufficiently large number of tube detectors, we could find the pattern collectively made by photons at the array of tube detectors.

In principle, the slit–screen distance can be made so large that it takes billions of years for each photon to travel from the slit(s) to the screen. In this case, a decision made now about whether to measure the slit identity or screen position of a photon seems to retrospectively affect whether that photon passed through just one slit or both slits billions of years ago.

As we should expect, these results are consistent with Heisenberg’s uncertainty principle. Regardless of when the decision is made, if the detectors measure slit identity (position) then this must increase uncertainty regarding the particle’s screen position (momentum), which leads to the disappearance of the interference pattern. Even though it is far from obvious how  any physical mechanism could produce this result, the fact remains that if such a mechanism did not exist then Heisenberg’s uncertainty principle would be violated.

Re-Writing Quantum History

So, suppose we wanted to re-write a little history. First, how could we do so, and second, how far back in time could that edit be?

Well, as we have seen, a decision made now about whether or not to leave the screen in place effectively determines how photons behaved in the past. Specifically, if we wish to ensure that each photon passed through just one slit then we should remove the screen (allowing the detectors to measure which slit each photon exited). Conversely, if we wish to ensure that photons passed through both slits then we should leave the screen in place. In both cases, this decision can be made after the photons have passed through the slit(s).

Second, the temporal range of our edit depends on how long the photons have been in transit from the double slits to the screen/detectors. This looks as if we need to have a double-slit experiment that has been set up in the distant past. However, there are natural phenomena, like gravitational lensing , which can be used to effectively mimic the double-slit experiment; as if slits are so far away that the photons we measure have been in transit for many years. Thus, in principle, temporal editing can reach back to the big bang , some 14 billion years ago.

Finally, we should acknowledge that not everyone (including Wheeler) agrees that Wheeler’s delayed-choice experiment edits the past. Like most quantum mechanical equations, the equations that define the results of the delayed choice experiment do not have a single unambiguous physical interpretation. However, in order to appreciate the nature of this ambiguity, it is necessary to understand the equations that govern quantum mechanics . For this, there is no shortcut, but there are many detailed maps and guide books.

James V Stone is an Honorary Associate Professor at the University of Sheffield, UK.

V Jacques, et al. Experimental realization of Wheeler’s delayed-choice gedanken experiment. Science, 315(5814):966–968, 2007.

Note :  This is an edited extract from  The Quantum Menagerie by James V Stone (published December 2020). Chapter 1, the table of contents, and book reviews can be seen here  The Quantum Menagerie .

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24.1: The Double-slit Experiment

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• Roberto Peverati
• Florida Institute of Technology

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The double-slit experiment is considered by many the seminal experiment in quantum mechanics. The reason why we see it only at this advanced point is that its interpretation is not as straightforward as it might seem from a superficial analysis. The famous physicist Richard Feynman was so fond of this experiment that he used to say that all of quantum mechanics can be understood from carefully thinking through its implications.

The premises of the experiment are very simple: cut two slits in a solid material (such as a sheet of metal), send light or electrons through them, and observe what happens on a screen position at some distance on the other side. The result of this experiment though are far from straightforward.

Let’s first consider the single-slit case. If light consisted of classical particles, and these particles were sent in a straight line through a single-slit and allowed to strike a screen on the other side, we would expect to see a pattern corresponding to the size and shape of the slit. However, when this “single-slit experiment” is actually performed, the pattern on the screen is a diffraction pattern in which the light is spread out. The smaller the slit, the greater the angle of spread. This behavior is typical of waves, where diffraction explains the pattern as being the result of the interference of the waves with the slit.

If one illuminates two parallel slits, the light from the two slits again interferes. Here the interference is a more pronounced pattern with a series of alternating light and dark bands. The width of the bands is a property of the frequency of the illuminating light. The pattern observed on the screen is the result of this interference, as shown in figure \(\PageIndex{1}\). 1

The interference pattern resulting from the double-slit experiment are observed not only with light, but also with a beam of electrons, and other small particles.

The individual particles experiment

The first twist in the plot is if we perform the experiment by sending individual particles (e.g, either individual photons, or individual electrons). Sending particles through a double-slit apparatus one at a time results in single particles appearing on the screen, as expected. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one by one (figure \(\PageIndex{2}\)) 2 . The resulting pattern on the screen is the same as if each individual particle had passed through both slits.

This variation of the double-slit experiment demonstrates the wave–particle duality: the particle is measured as a single pulse at a single position, while the wave describes the probability of absorbing the particle at a specific place on the screen.

“Which way” experiment

A second twist happens if we place particle detectors at the slits with the intent of showing through which slit a particle goes. The interference pattern in this case will disappear.

This experiment illustrates that photons (and electrons) can behave as either particles or waves, but cannot be observed as both at the same time. The simplest interpretation of this experiment is that the wave function of the photon collapses into a deterministic position due to the interaction with the detector on the slit, and the interference pattern is therefore lost. This result also proves that in order to measure (detect) a photon, we must interact with it, an act that changes its wave function.

The interpretation of the results of this experiment is not simple. As for other situations in quantum mechanics, the problem arise not because we cannot describe the experiment in mathematical terms, but because the math that we need to describe it cannot be related to the macroscopic classical world we live in. According to the math, in fact, particles in the experiment are described exclusively in probabilistic terms (given by the square of the wave function). The macroscopic world, however, is not probabilistic, and outcomes of experiments can be univocally measured. Several different ways of reediming this controversy have been proposed, including for example the possibility that quantum mechanics is incomplete (the emergence of probability is due to the ignorance of some more fundamental deterministic feature of nature), or assuming that every time a measurement is done on a quantum system, the universe splits, and every possible measurable outcome is observed in different branches of our universe (we only happen to live in one of such branches, so we observe only one non-probabilistic result). 3 The interpretation of quantum mechanics is still an unsolved problem in modern physics (luckily, it does not prevent us from using quantum mechanics in chemistry).

• This diagram is taken from Wikipedia by user Jordgette, and distributed under CC BY-SA 3.0 license.︎
• This diagram is taken from Wikipedia by user Alexandre Gondran, and distributed under CC BY-SA 4.0 license︎
• The interested student can read more about different interpretations HERE.

How Two Rebel Physicists Changed Quantum Theory

David Bohm and Hugh Everett were once ostracized for challenging the dominant thinking in physics. Now, science accepts their ideas, which are said to enrich our understanding of the universe.

The field of quantum mechanics dates to 1900, the year German scientist Max Planck (1858–1947) discovered that energy could come in discrete packages called quanta. It advanced in 1913, when Danish physicist Niels Bohr (1885–1962) used quantum principles to explain what had until then been inexplicable, the exact wavelengths of light emitted or absorbed by a gas of hydrogen atoms. And since the 1920s, when Werner Heisenberg (1901–1976) and Erwin Schrödinger (1887–1961) built new quantum theories, quantum mechanics has consistently proven its value as the fundamental theory of the nanoscale and as a source of technology, from computer chips and lasers to LED bulbs and solar panels.

One question, however, still puzzles: how does the quantum world relate to the more familiar human-scale one? For a century, the Copenhagen interpretation , chiefly developed by Bohr and Heisenberg in that city, has been the standard answer taught in physics courses. It posits that the quantum scale is indeterminate; that is, operates according to the laws of probability. This world is utterly different from the deterministic and predictable “classical” human scale, yet the Copenhagen interpretation doesn’t clearly explain how reality changes between the two worlds.

Heisenberg and Bohr developed the Copenhagen interpretation amidst the blossoming of new quantum theories in the first half of the twentieth century. In 1927, Heisenberg announced his important uncertainty principle : at the quantum level, certain pairs of quantities, such as momentum and position, cannot be simultaneously measured to any desired degree of precision. The more exactly you measure one, the less well you know the other. Thus, we can never fully know the quantum world, a key feature of the Copenhagen interpretation.

Indeterminism also appears in the Schrödinger wave equation at the heart of the Copenhagen view. Einstein had shown that light waves can act like swarms of particles, later called photons; in 1924, Louis de Broglie assumed the inverse, that tiny particles are also wave-like. In 1926, Schrödinger published his equation for these “matter waves.” Its solution, the “wave function” denoted by the Greek letter Ψ (psi), contains all possible information about a quantum entity such as an electron in an atom. But the information is indeterminate: Ψ is only a list of probable values for all the different physical properties, such as position or momentum, that the electron could have in its particular surroundings. The electron is said be in a superposition , simultaneously present in all its potential states of actual being.

This superposition exists until an observer measures the properties of the electron, which makes its wave function “collapse”; the cloud of possible outcomes yields just one result, a definite value emerging into the classical world. It is as if, asked to pick a card out of a deck, the instant you select the three of hearts, the other fifty-one cards fade away. In this case, we know that the rejected cards still physically exist with definite properties, but in the Copenhagen view, subatomic particles aren’t real until they’re observed. Another problem is that the notion of a sudden wave function collapse seems an arbitrary addition to the Copenhagen interpretation; it contradicts the smooth evolution in time built into the Schrödinger equation.

These troubling features, called “the measurement problem,” were hotly debated in the 1920s. But overwhelming any objections was the fact that the Copenhagen interpretation works! Its results agree precisely with experiments, the final test of any theory, and inspire real devices. Even so, David Joseph Bohm (1917–1992) and Hugh Everett III (1930–1982) sought equally valid theories without any incongruities. In the 1950s, these two American physicists dared to challenge the conventional Copenhagen interpretation with their “pilot wave” and “many-worlds” theories, respectively. Though from different backgrounds, Bohm and Everett shared characteristics that helped them seek answers: mathematical aptitude, necessary to manipulate quantum theory; and unconventional career paths, which separated them from the orthodoxy of academic physics.

Bohm was a second-generation American, born into an immigrant family from Europe that operated a furniture store in Wilkes-Barre, Pennsylvania. In high school, where his physics instructor described him as “outstanding” and “brilliant,” Bohm developed his own alternative ideas about Bohr’s hydrogen atom. After undergraduate work at Penn State, he began earning a PhD in nuclear physics in 1941 under J. Robert Oppenheimer (1904–1967) at the University of California, Berkeley. The United States was engaged in World War II at the time and was about to build an atomic bomb. Bohm’s doctoral research was classified, and he was awarded his degree in 1943 without writing a dissertation. Though Oppenheimer wanted Bohm to work with him at Los Alamos, Bohm couldn’t get security clearance as he had briefly been, in the early 1940s, a member of the Communist Party.

In 1947, supported by theorist John Wheeler, Bohm became an assistant professor at Princeton. There he taught quantum mechanics and wrote Quantum Theory (1951), in which he presented the Copenhagen interpretation, only to disavow it the next year, when he published his alternative theory in a pair of papers in the Physical Review (in 1957, he expounded his ideas further in his book Causality and Chance in Modern Physics ).

But in 1951, his life had taken a serious turn. In that Cold War era of McCarthyism, Bohm was brought before the House Committee on Un-American Activities (HUAC). He pleaded the Fifth Amendment against self-incrimination, and he was first indicted and jailed for contempt of Congress and then acquitted when the Supreme Court decriminalized this action. Still, the damage was done. Princeton didn’t renew Bohm’s contract and banned him from campus in June 1951. Unable to obtain a new academic position in the US, he began a life-long exile, taking temporary teaching positions in Brazil and elsewhere. Finally, in 1961, he accepted the offer of a chaired professorship in physics at Birkbeck College, London. He remained in that position until he retired in 1983, continuing to develop his new approach, the “pilot wave” theory.

When de Broglie postulated that tiny particles are also wave-like, he proposed the role of the waves as guiding or piloting the motions of real physical particles. Bohm fleshed out this insight by relating the pilot wave to Schrödinger’s wave function Ψ. In Bohm’s view, Ψ doesn’t collapse, but shepherds real subatomic particles into specific trajectories. This scenario yields the same results as the Schrödinger equation and resolves a great wave-particle quantum paradox. In the famous double-slit experiment , a stream of electrons or photons sent through two slits produces a pattern that could arise only from interfering waves, not particles. Bohm’s solution is that each particle traversing one of the slits rides a wave that pilots it into a complex path and generates an interference pattern from the swarm of particles.

For his part, Everett solved the measurement problem differently, as described by biographer Peter Byrne in an article , and later, a book . Born in Washington, DC, Everett showed an early interest in logical contradictions. At age twelve, he wrote to Einstein about the paradox “irresistible force meets immovable body,” and, as Everett reports, Einstein replied that there is no such paradox, but he noted Everett’s drive in attacking the problem. Everett graduated with honors from Catholic University as an engineer with strong backgrounds in math, operations research, and physics.

In 1953, Everett went to Princeton for graduate work. There he met Bohr, whose visit at the nearby Institute for Advanced Study sparked discussions about quantum mechanics. Later Everett said that the idea for his new theory came during a sherry-fueled session with one of Bohr’s assistants, among others. Everett was soon working out the consequences of his idea in a dissertation under John Wheeler, who had mentored Bohm and also Nobel Laureate Richard Feynman (1918–1988), and who called Everett “highly original.”

In the Copenhagen view, quantum reality as determined by the Schrödinger equation is separate from classical reality. Everett boldly asserted instead that the Schrödinger equation applies to everything, small or big, object or observer. The resulting universal wave function describes a reality without a boundary between microscopic and macroscopic or any need for the wave function to collapse. In his scheme, the measurement problem doesn’t exist.

This, however, comes at the cost of accepting a highly complex universe. If large objects and their observers obey the Schrödinger equation, then the universal wave function includes all observers and objects and their links in superposition. As Byrne explains: if the object could exist at either point A or B, in one branch of the universal wave function the observer sees the measurement result as “A,” and in another branch, a nearly identical person sees the result as “B.” (Everett called different elements of the superposition “branches.”) Further, without the jarring disruption of wave function collapse, the Schrödinger equation tells us that the branches go smoothly forward in time and do not interact, so each observer separately sees a normally unfolding macroscopic world.

In layman’s terms, this means that the universe, instead of being a unity that encompasses all reality, is filled with separate multiverses or bubbles of reality, each believed to be the entire universe by its inhabitants. The observer who saw result “A” now lives in that reality, and the person who saw “B” occupies a separately evolving reality according to their different outcome. Each of these unimaginable numbers of bubbles moves ahead into its own future, forming a totality filled with what have come to be called “parallel worlds.”

Bohr and his group scorned this grandiose idea as an answer to the measurement problem, one of his circle calling it “theology,” and another deriding Everett as “ stupid .” Wheeler had Everett rewrite his dissertation so it didn’t directly criticize the Copenhagen interpretation or its proponents. His thesis was published in 1957 and, according to Byrne, “slipped into instant obscurity.” All this should have been no surprise. As Olival Freire Jr. points out , Bohm’s earlier work—which Everett cited—had also been badly received, even with hostility, in a community dominated by Bohr and champions of the Copenhagen interpretation.

That was to change for both theories. In 1964, a bombshell result from theorist John Bell showed how to experimentally confirm the exceedingly strange quantum effect of entanglement, which Einstein called “ spooky action at a distance ”: the fact that two quantum entities can affect each other over arbitrary distances. Bell, it turns out, was strongly influenced by Bohm’s work, notes Freire. This shows that rethinking the foundations of quantum mechanics, downplayed by some physicists as only a philosophical exercise, can pay off in deep theoretical insights as well as in technology; entanglement today is used in quantum computation, communication, and cryptography.

Everett’s ideas too came to be more appreciated after The Many-Worlds Interpretation of Quantum Mechanics (Bryce DeWitt and Neill Graham, editors) was published in 1973. It included Everett’s original dissertation and related papers. This and DeWitt’s evocative phrase “many-worlds interpretation” brought new interest in Everett’s work and linked it to multiverse theory , which has been developed to solve certain problems in cosmology and as an outcome of string theory. Everett won further recognition—this time in popular culture—in 1976, when his work appeared in Analog , a leading science fiction magazine. (In fact, multiverses and parallel worlds have become staples of popular culture, as the film Everything Everywhere All at Once (2022) and the streaming series Dark Matter (2024), based on the novel by Blake Crouch, make clear.)

By 2023, Bohm’s and Everett’s seminal papers had each amassed tens of thousands of citations in the scientific literature. Surveys have also asked hundreds of physicists which interpretation of quantum mechanics they consider best. Many chose the Copenhagen view, but an equal number favor either the pilot wave or many-worlds interpretation. It’s striking that what in the 1950s were outlaw ideas, met with disbelief and antagonism, today have a significant degree of acceptance and have greatly expanded our view of the quantum world and the universe.

That Bohm and Everett could produce novel theories reflects their special circumstances and their times as well as their abilities. McCarthyism interrupted Bohm’s career but also freed him from conventional views of quantum mechanics. Historian of science Christian Forstner cites a 1981 interview in which Bohm acknowledges the upside to his departure from Princeton. It “liberated me,” Bohm admitted. “I was able to think more easily and more freely…without having to talk the language of other people.” Forstner notes that in exile, the physicist had the freedom to choose like-minded colleagues so that “the US-community and its thought-style lost importance for Bohm.” Indeed, Bohm’s exile was highly productive.

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Everett’s expertise in operations research brought him the offer of a position with the Pentagon’s Weapons Systems Evaluation Group (WSEG) to analyze nuclear warfare after finishing his PhD. Wheeler wanted him to continue at Princeton but also knew, Byrne writes, that the lack of recognition for Everett’s ideas had left him “disappointed, perhaps bitter.” Nor did Everett enjoy truncating his thesis to mollify Bohr, and he must have realized that advocating an unpopular theory would cloud his academic career. In the end, he chose WSEG and never again worked in theoretical physics, but perhaps having this alternate possibility stiffened his resolve in presenting and defending his audacious idea. His talents shone at WSEG, but, according to Byrne, he was an alcoholic and died of a heart attack at age fifty-one.

The co-existing Copenhagen, Bohm, and Everett interpretations give the same results for many different tests of quantum behavior; and so we await the subtle experiment that distinguishes among them, showing which one is physically true and might give philosophers new insight into the nature of reality. Bohm’s and Everett’s sagas provide another valuable lesson. Science prides itself on being self-correcting; wrong theories are eventually made right, as in the old notion of a geocentric universe giving way to the modern view. The Copenhagen interpretation became unquestioned orthodoxy, but Bohm and Everett challenged it even at personal cost. That reflects the highest aspirations of science and deserves to be recognized in 2025, the upcoming International Year of Quantum Science and Technology.

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You are here, phys 201: fundamentals of physics ii,  - quantum mechanics i: key experiments and wave-particle duality.

The double slit experiment, which implies the end of Newtonian Mechanics, is described. The de Broglie relation between wavelength and momentum is deduced from experiment for photons and electrons. The photoelectric effect and Compton scattering, which provided experimental support for Einstein’s photon theory of light, are reviewed. The wave function is introduced along with the probability interpretation. The uncertainty principle is shown to arise from the fact that the particle’s location is determined by a wave and that waves diffract when passing a narrow opening.

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Quantum Measurement

• Double-slit experiment：Quantum Measurement

Double-slit experiment

You may be familiar with an experiment known as the " double-slit experiment," as it is often introduced at the beginning of quantum-mechanics textbooks. The experimental arrangement can be seen in Fig. 1. Electrons are emitted one by one from the source in the electron microscope. They pass through a device called the "electron biprism", which consists of two parallel plates and a fine filament at the center. The filament is thinner than 1 micron (1/1000 mm) in diameter. Electrons having passed through on both sides of the filament are detected one by one as particles at the detector. This detector was specially modified for electrons from the photon detector produced by Hamamatsu Photonics (PIAS). To our surprise, it could detect even a single electron with almost 100 % detection efficiency.

Let's start the experiment and look at the monitor.

• Video clip 1

At the beginning of the experiment, we can see that bright spots begin to appear here and there at random positions (Fig. 2 (a) and (b)). These are electrons. Electrons are detected one by one as particles. As far as these micrographs show, you can be confident that electrons are particles. These electrons were accelerated to 50,000 V, and therefore the speed is about 40 % of the speed of the light, i. e., it is 120,000 km/second. These electrons can go around the earth three times in a second. So, they pass through a one-meter-long electron microscope in 1/100,000,000 of a second. It is all right to think that each electron is detected in an instant after it is emitted.

Interference fringes are produced only when two electrons pass through both sides of the electron biprism simultaneously. If there were two electrons in the microscope at the same time, such interference might happen. But this cannot occur, because there is no more than one electron in the microscope at one time, since only 10 electrons are emitted per second.

Please keep watching the experiment a little longer. When a large number of electrons is accumulated, something like regular fringes begin to appear in the perpendicular direction as Fig. 2(c) shows. Clear interference fringes can be seen in the last scene of the experiment after 20 minutes (Fig. 2(d)). It should also be noted that the fringes are made up of bright spots, each of which records the detection of an electron.

We have reached a mysterious conclusion. Although electrons were sent one by one, interference fringes could be observed. These interference fringes are formed only when electron waves pass through on both sides of the electron biprism at the same time but nothing other than this. Whenever electrons are observed, they are always detected as individual particles. When accumulated, however, interference fringes are formed. Please recall that at any one instant there was at most one electron in the microscope. We have reached a conclusion which is far from what our common sense tells us.

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What is the double-slit experiment, and why is it so important?

Reality will surprise you..

John Loeffler

Timm Weitkamp/Wikimedia Commons

Few science experiments are as strange and compelling as the double-slit experiment. 

Few experiments, if any, in modern physics are capable of conveying such a simple idea—that light and matter can act as both waves and discrete particles depending on whether they are being observed—but which is nonetheless one of the great mysteries of quantum mechanics.

It’s the kind of experiment that despite its simplicity is difficult to wrap your mind around because what it shows is incredibly counter-intuitive.

But not only has the double-slit experiment been repeated countless times in physics labs around the world, but it has also even spawned many derivative experiments that further reinforce its ultimate result, that particles can be waves or discrete objects and that it is as if they “know” when you are watching them.

What does the double-slit experiment demonstrate?

To understand what the double-slit experiment demonstrates, we need to lay out some key ideas from quantum mechanics.

In 1925, Werner Heisenberg presented his mentor, the eminent German physicist Max Born, with a paper to review that showed how the properties of subatomic particles, like position, momentum, and energy, could be measured. 

Born saw that these properties could be represented through mathematical matrices, with definite figures and descriptions of individual particles, and this laid the foundation for the matrix description of quantum mechanics .

Meanwhile, in 1926,  Edwin Schrödinger  published his wave theory of quantum mechanics which showed that particles could be described by an equation that defined their waveform; that is, it determined that particles were actually waves.

This gave rise to the concept of wave-particle duality, which is one of the defining features of quantum mechanics. According to this concept, subatomic entities can be described as both waves and particles, and it is up to the observer to decide how to measure them.

That last part is important since it will determine how quantum entities will manifest. If you try to measure a particle’s position, you will measure a particle’s position, and it will cease to be a wave at all.

If you try to define its momentum, you will find that behaves like a wave and you can’t know anything definitive about its position beyond the probability that it exists at any given point within that wave.

Essentially, you will measure it as a particle or a wave, and doing so decides what form it will take.

The double-slit experiment is one of the simplest demonstrations of this wave-particle duality as well as a central defining weirdness of quantum mechanics, one that makes the observer an active participant in the fundamental behavior of particles.

How does the double-slit experiment work?

The easiest way to describe the double-slit experiment is by using light. First, take a source of coherent light, such as a laser beam, that shines in a single wavelength, like purely blue visible light at 460nm, and aim it at a wall with two slits in it.  The distance between the slits should be roughly the same as the light’s wavelength  so that they will both sit inside that beam of light.

Behind that wall, place a screen that can detect and record the light that impacts it. If you fire the laser beam at the two slits, on the recording screen behind the wall you will see a stripey pattern like this:

This is probably not what you might have been expecting, and that’s perfectly rational if you treat light as if it were a wave. If the light was a wave, then when the single wave of light from the laser hit both slits, each slit would become a new “source” of light on the other side of the wall, and so you would have a new wave originating from each slit producing two waves.

Where those two waves intersect causes something known as interference, and it can be either constructive or destructive. When the amplitude of the waves overlaps at either a peak or a trough, it acts to boost the wavelength in either direction by adding its energy together. This is constructive interference, and it produces these brighter bars in this pattern.

When the waves cancel each other out, as when a peak hits a trough, the effect neutralizes the wavelength and diminishes or even eliminates the light, producing the blacked-out spaces in between the blue bars.

But in the case of quantum entities like photons of light or electrons, they are also individual particles. So what happens when you shoot a single photon through the double slits?

One photon alone reacting to the screen might leave a tiny dot behind, which might not mean much in isolation, but if you shoot many single photons at the double slits, those tiny dots that the photon leaves behind on our screen actually show up in that same stripey interference pattern produced by the laser beam hitting the double slits.

In other words, the individual photon behaves as if it passed through both slits like it was a wave.

Now, here’s where things get really weird.

We can set up a detector in front of one of the slits that can watch for photons and light up whenever it detects one passing through. When we do this, the detector will light up 50% of the time, and the pattern left behind on the screen changes, giving us something that looks like this:

And to make things even wilder, we can set up a detector behind the wall that only detects a photon after it has passed through the slit and we get the same result. That means that even if the photon passes through both slits as a wave, the moment it is detected, it is no longer a wave but a particle. And not just that, that second wave emerging from the other slit also collapses back into the particle that was detected passing through the other slit.

In practice, this means that somehow the universe “knows” that someone is watching and flips the metaphorical quantum coin to see which slit the particle passed through. The more individual photons you shoot through the double slit, the closer that photon detector comes to detecting photons 50% of the time, just as flipping a coin 10 times might give you heads 70% of the time while flipping it 100 times might give you tails 55% of the time, and flipping it 1 billion times gives you heads 50.0003% of the time.

This seems to show that not only is the universe watching the observer as well, but that the quantum states of entities passing through the double slits are governed by the laws of probability, making it impossible to ever predict with certainty what the quantum state of an entity will be.

Who invented the double-slit experiment?

The double-slit experiment actually predates quantum mechanics by a little more than a century.

During the Scientific Revolution, the nature of light was a particularly contentious topic, with many—like Isaac Newton himself—arguing in favor of a corpuscular theory of light that held that light was transmitted through particles.

Others believed that light was a wave that was transmitted through “aether” or some other medium, the way sound travels through air and water, but Newton’s reputation and a lack of an effective means to demonstrate the wave theory of light solidified the corpuscular view for just shy of a century after Newton published his  Opticks in 1704 .

The definitive demonstration came from the British polymath Thomas Young, who presented a paper to the Royal Society of London in 1803 that described a pair of simple experiments that anyone could perform to see for themselves that light was in fact a wave.

First, Young established that a pair of waves were subject to interference when they overlapped, producing a distinctive interference pattern.

He initially demonstrated this interference pattern using a ripple tank of water, showing that such a pattern is characteristic of wave propagation.

Young then introduced the precursor to the modern double-slit experiment, though instead of using a laser beam to produce the required light source, Young used reflected sunlight striking two slits in a card as its target.

The resulting light diffraction showed the expected interference pattern, and the wave theory of light gained considerable support. It would take another decade and a half before further experimentation conclusively refuted corpuscles in favor of waves, but the double-slit experiment that Young developed proved to be a fatal blow to Newton’s theory.

How to do the double-slit experiment

Young wasn’t lying when he said , “The experiments I am about to relate…may be repeated with great ease, whenever the sun shines, and without any other apparatus than is at hand to everyone.”

While it might be a stretch to say that you can use the double-slit experiment to demonstrate some of the more counterintuitive features of quantum mechanics (unless you have a photon detector handy and a laser that shoots individual photons), you can still use it to demonstrate the wave nature of light.

If you want to replicate Young’s experiment, you only need as large a box as is practical with a hole cut in it a little smaller than an index card. Then, take an Exacto knife or similar blade for fine cutting work and cut two slits into a piece of cardboard larger than the hole in your box. The slits should be between 0.1mm and 0.4mm apart, as the closer together they are, the more distinct the interference pattern will be. It’s better to create cards for this rather than cut directly into the box since you might need to make adjustments to the spacing of the slits.

Once you’re satisfied with the spacing, affix the card with the double-slit in it over the hole and secure it in place with tape. Just make sure sunlight isn’t leaking around the card.

You’ll also need to create some eye-holes in the box so you can look inside without getting in the way of the light hitting the double-slit card, but once you figure that out, you’re all set.

To accurately diffract sunlight using this box, you will need to have the sunlight more or less hitting the double-slit card dead on, so it might take some maneuvering to get it properly positioned.

Once it is, look through the eye holes and you can see the interference pattern forming on the inside wall, as well as different colors emerging as the different wavelengths interfering with each other change the color of the light being created.

If you wanted to try it out with something fancier, get yourself a laser pointer from an office supply store. Just like you’d do with a viewing box, create cards with slits in them, and when properly spaced, set up a shielded area for the card to rest on.

You’ll want to make sure that only the light from the laser pointer is hitting the double-slit, so shield the card however you need to. Then, set the laser pointer on a surface level with the slits and shine the laser at them. On the wall behind the card, the interference pattern from the slits should be clearly visible.

If you don’t want to go through all that trouble, you can also use Photoshop or similar software to recreate the effect.

First, create a template of evenly spaced concentric circles. Using different layers for each source, as well as a background later, position the center of the concentric rings near to one another. On a 1200 pixel wide canvas, a distance of 100 pixels between the two centers should do nicely.

Then, fill in the color of each concentric ring, alternating light and dark, with an opacity set to about 33%. You may need to hide one of the concentric circle layers while you work on the other. When you’re done, reveal the two overlapping layers of circles and the interference pattern should jump out at you immediately, looking something like this:

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Of course, if you want to dig into the quantum mechanics side of things, you’ll need to work in a pretty advanced physics lab at a university or science institute, since photon detectors aren’t the kind of thing you can pick up at the hobby store.

Still, if you’re compelled to try the heavier stuff out for yourself, you wouldn’t be the first person to get drawn into a career in physics because of the weirdness of quantum mechanics, and there are definitely worse ways to make a living.

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• Published: 10 July 2020

Quantum double-double-slit experiment with momentum entangled photons

• Manpreet Kaur 1 &
• Mandip Singh 1  

Scientific Reports volume  10 , Article number:  11427 ( 2020 ) Cite this article

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• Optics and photonics

Double-double-slit thought experiment provides profound insight on interference of quantum entangled particles. This paper presents a detailed experimental realisation of quantum double-double-slit thought experiment with momentum entangled photons and theoretical analysis of the experiment. Experiment is configured in such a way that photons are path entangled and each photon can reveal the which-slit path information of the other photon. As a consequence, single photon interference is suppressed. However, two-photon interference pattern appears if locations of detection of photons are correlated without revealing the which-slit path information. It is also shown experimentally and theoretically that two-photon quantum interference disappears when the which-slit path of a photon in the double-double-slit is detected.

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Introduction.

Wave nature of light was first experimentally demonstrated by the famous Young’s double-slit experiment 1 , 2 . In quantum physics, light is quantised in the form of energy quanta known as photon. According to the statement of P.A.M. Dirac, “Each photon interferes only with itself” 3 . This self interference of a photon is a consequence of quantum superposition principle. If photons are incident on a double-slit one by one then the interference pattern of a photon gradually emerges. Where detection of each photon corresponds to a point on the screen. Young’s double-slit experiment provides profound insight on the wave-particle duality if it is imagined for individual particles 4 . Interference pattern of a single particle is not formed if the path information of a particle i.e. a slit through which a particle has passed, is known. According to Copenhagen interpretation, an observation on the quantum superposition of paths of a particle corresponds to a measurement that collapses quantum superposition therefore, no interference pattern is formed. On the other hand, what happens if we modify the experiment in such a way that the which-path information of a particle is not available during its passage through a double-slit but can be obtained even after its detection. In this case, the which-path information can be carried out by the quantum state of another particle if total quantum state of particles is an entangled quantum state. By knowing its path by a measurement, the path information of the other particle is immediately determined. Because of path revealing quantum entanglement of particles the single particle interference is suppressed. However, quantum interference can be recovered even after completion of experiment by making correlated selection of measurement outcomes.

The first experiment to show the interference of light with very low intensity in the Young’s double-slit experiment was performed in 1909 by G.I. Taylor 5 . Interesting experiments showing the Young’s double-slit interference are performed with neutrons from the foundational perspective of quantum mechanics 6 , with electron beams 7 and with a single electron passing through a double-slit 8 , 9 , 10 . Recently, a first experimental demonstration of interference of antiparticles with a double-slit is reported 11 . Interference of macromolecules is the subject of great interest in the quest to realise quantum superposition of mesoscopic and macroscopic objects 12 , 13 . In this context, number of interesting experiments have been performed to produce a path superposition of large molecules similar to the double-slit type interference experiments 14 , 15 , 16 .

The main concept of a quantum single double-slit experiment was extended to a quantum double-double-slit thought experiment by Greenberger, Horne and Zeilinger 17 to provide foundational insight on the multiparticle quantum interference. In their paper. they have considered two double-slits and a source of particles placed in the middle of double-slits. Each particle is detected individually after it traverses a double-slit. Quantum entanglement of particles appears naturally in their considerations 18 , 19 and it is shown, when single particle interference disappears and two-particle interference appears. An experimental realisation of quantum double-double-slit thought experiment showing a two-photon interference has been demonstrated with quantum correlated photons produced by spontaneous parametric down conversion (SPDC) process 20 . However, in this paper, we present a detailed experimental realisation of the quantum double-double-slit thought experiment with momentum entangled photon pairs, where a virtual double-double-slit configuration is realised with two Fresnel biprisms. This paper provides a detailed conceptual, theoretical and experimental analysis of the quantum double-double-slit experiment. In addition, an experiment of detection of a which-slit path of a photon is presented where it is shown that the two-photon interference disappears when a which-slit path of a photon is detected.

In this paper, experiments are presented in the context of a quantum double-double-slit thought experiment. However, experiments of foundational significance with polarization entangled photons 21 , 22 , 23 and momentum entangled photons 24 have been intensively studied. In addition, interesting experiments on delayed choice path erasure 25 , 26 , 27 , 28 , 29 , 30 and two-photon interference 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 are performed. Similar experiments have been proposed with Einstein–Podolsky–Rosen (EPR) entangled pair of atoms 41 .

Quantum double-double-slit experiment

Quantum double-double-slit experiment consists of two double-slits and a source of photon pairs. In this experimental situation, a single photon passes through each double-slit and detected individually on screens positioned behind the double-slits as shown in Fig.  1 . However, interference of photons depends on the quantum state of two photons. To understand quantum interference of two photons in a double-double-slit experiment, consider a source is producing photons in pairs and both the photons have same linear polarisation. Double-slit 1 and double-slit 2 are aligned parallel to y -axis and positioned at distances \(l_{1}\) and \(l_{2}\) , respectively along the x -axis from the source. Single slits \(a_{1}\) and \(b_{1}\) of double-slit 1 are separated by a distance \(d_{1}\) and single slits \(a_{2}\) and \(b_{2}\) of double-slit 2 are separated by a distance \(d_{2}\) as shown in Fig.  1 where each slit width is considered to be infinitesimally small. A single photon of a photon pair is detected on screen 1, which is positioned at a distance \(s_{1}\) from double-slit 1 and a second photon is detected on screen 2, which is positioned at a distance \(s_{2}\) from double-slit 2. There are four different possible paths by which photons can arrive at the respective screens i.e. a photon can arrive at a point \(o_{1}\) on screen 1 via double-slit 1 and the other photon can arrive at a point \(o_{2}\) on screen 2 via double-slit 2. Therefore, possible paths of photons are (i) a first photon can pass through slit \(a_{1}\) and the second photon can pass through slit \(a_{2}\) , or (ii) a first photon can pass through slit \(b_{1}\) and the second photon can pass through slit \(b_{2}\) , or (iii) a first photon can pass through slit \(a_{1}\) and the second photon can pass through slit \(b_{2}\) , or (iv) a first photon can pass through slit \(b_{1}\) and the second photon can pass through slit \(a_{2}\) . Since all the possible paths are indistinguishable and not revealing any which-path information therefore, total amplitude \(A_{12}\) to find a photon at \(o_{1}\) and a photon at \(o_{2}\) together is a quantum superposition of all the possible paths, which can be successively written as

where \(t_{a1}\) , \(t_{b1}\) , \(t_{a2}\) , \(t_{b2}\) are amplitudes of transmission of slits \(a_{1}\) , \(b_{1}\) , \(a_{2}\) , \(b_{2}\) , respectively. Quantum states \(|a_{1}\rangle\) , \(|b_{1}\rangle\) \(|a_{2}\rangle\) , \(|b_{2}\rangle\) are position space basis states of locations on the slits on double-slit 1 and double-slit 2, respectively where a photon can be found. Similarly, \(|o_{1}\rangle\) and \(|o_{2}\rangle\) are the position space basis states of locations on the screens. However, position basis states corresponding to points on each double-slit and a screen form a different basis set such that \(\langle o_{1}|a_{1}\rangle\) represents the amplitude of transmitted photon to go from slit \(a_{1}\) to a location \(o_{1}\) on screen 1. Same terminology is applied for other amplitudes in Eq.  1 .

A schematic diagram of a double-double-slit experiment. Photons are individually detected on screens after they pass through the double-slits separately. Which-slit path information of photons can be detected by blocking any single slit by closing the shutter.

Further, consider photon pairs produced by a source of finite size are emitted in opposite directions w.r.t. each other such that they are momentum entangled, their net momentum is zero and momentum of each photon is definitely unknown. Consider the spatial extension of source is much smaller than the slit separation but large to produce momentum entanglement. As a consequence of momentum entanglement, if a photon passes through slit \(a_{1}\) then the other photon passes through slit \(b_{2}\) and if a photon passes through slit \(b_{1}\) then the other photon passes through slit \(a_{2}\) after their transmission through the slits. For momentum entangled photons, both these possibilities are quantum superimposed, as result of it both the photons are path entangled via the slits and first two terms in the summation of Eq.  1 become zero. The last two terms in the summation are due to path entanglement via the slits, these two amplitudes interfere with each other and produce a two-photon interference of momentum entangled photons. When all four slits are opened, a two-photon path information is not revealed and a two-photon interference can be observed by recording detection locations of a photon corresponding to a particular location of detection of other photon on the other screen during each repetition of the experiment.

On the other hand, if one measures the direction of momentum of any single photon prior to its passage through double-slits then the momentum entangled state is collapsed. This measurement outcome reveals momentum direction of a photon on which a measurement is performed and the direction of momentum of the other photon is also revealed instantly after the collapse even without making any measurement on it. This measurement reveals which-slit path information of photons. On the other hand, which-slit path of photons in the double-double-slit can be detected by closing any single slit with a shutter. A shutter shown in Fig.  1 is considered as a photon measuring detector, if shutter is closed to block a slit \(a_{2}\) and a photon is detected on screen 2 then it reveals that a photon is passed through a slit \(b_{2}\) due to collapse of quantum entangled state caused by the shutter detector. As a consequence, one can find out that the other photon is passed through a slit \(a_{1}\) if it is detected at \(o_{1}\) . Since a path of both photons is known therefore, two-photon interference is suppressed. Interesting situation appears when double-slit 2 and screen 2 are removed to allow a photon to propagate in space while other photon is passed through double-slit 1 and detected on screen 1. A single-photon interference not produced on screen 1 because of path entanglement the which-slit path information of photons can be obtained by measuring momentum of the propagating photon even after the detection of a photon on screen 1.

Furthermore, when all slits are opened and which-slit path is not detected, a photon can be detected at any location on a screen randomly during each repetition of the experiment and its detection location is not known prior to a measurement on screen. Once a photon is detected on a screen, its detection location instantly determines the amplitude to find other photon on other screen if it is not reached there. Individual photons show no interference on a screen because a well defined phase coherent amplitude to find a photon on a screen depends on a particular detection location of other photon. In this case, a single photon amplitude is completely incoherent. The information of detection location of a photon determines a particular two-photon interference pattern. In other words, in this type of joint and correlated registration of detection locations of photons, if a different detection location of a photon is selected the two-photon interference pattern exhibits a shift. If only single photons are registered on each screen without making any correlation between their detection locations then the interference pattern is not formed on each screen.

Two-photon interference

To find out a two-photon interference in the double-double-slit experiment for a finite width of each slit, consider a source of photons located at origin is producing a two-photon quantum state \(|\Psi \rangle\) as shown in Fig.  1 . Double-slits can be defined by amplitude transmission functions \(t_{1}(y')\) and \(t_{2}(y'')\) of double-slit 1 and double-slit 2 respectively. Where \(y'\) and \(y''\) are the arbitrary points on double-slit 1 and double-slit 2, respectively such that the position basis states corresponding to these points located on the double-slits where a photon can be found are \(|l_{1}, y'\rangle\) and \(|l_{2}, y''\rangle\) . Therefore, the amplitude \(A_{12}\) to find photons at points \(o_{1}\) and \(o_{2}\) together on screens can be written as

Consider photon source has finite size and two-photon quantum state \(|\Psi \rangle\) is a momentum entangled quantum state, where both the photons have same linear polarisation and frequency. Such a two-photon quantum entangled state can be produced by degenerate noncollinear SPDC with type-I phase matching in a beta-barium-borate (BBO) crystal which is pumped by a laser beam propagating along the z -axis (longitudinal direction), where the z -axis (not shown in Fig.  1 ) is perpendicular to the xy -plane (transverse plane). Photons known as the signal and the idler photons are emitted from the source with opposite momenta with nearly equal in magnitude in the transverse plane such that their two-photon momentum entangled state in the transverse momentum space is 42 , 43 , 44 , 45 , 46

where \(|\mathbf {q_{s}}\rangle\) , \(|\mathbf {q_{i}}\rangle\) are the transverse momentum quantum states of the signal and the idler photons of momentum \(\mathbf {q_{s}}\) and \(\mathbf {q_{i}}\) , respectively and N is a normalisation constant. Two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) represents the amplitude to find a signal photon in momentum state \(|\mathbf {q_{s}}\rangle\) and an idler photon in momentum state \(|\mathbf {q_{i}}\rangle\) . Quantum entanglement is manifested by non separability of \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) . For the pump laser beam with gaussian intensity profile of finite width in the transverse plane, the two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) is prominent only for momentum states of photons with opposite transverse momenta. Since source size is finite therefore, if momentum of a photon is measured precisely then the quantum state of the other photon corresponds to a momentum state of opposite momentum with finite uncertainty. Further detail on momentum entanglement of photons produced by degenerate noncollinear SPDC in the BBO crystal is given in methods.

There are two possibilities that can result in a joint detection of photons on screen 1 and screen 2. These indistinguishable possibilities are (i) the signal photon is passed through double-slit 1 and detected on screen 1 and the idler photon is passed through double-slit 2 and detected on screen 2 and (ii) the idler photon is passed through double-slit 1 and detected on screen 1 and the signal photon is passed through double-slit 2 and detected on screen 2. Any single photon (signal or idler) that can be detected after passing through the double-slit 1 is labeled as photon 1 and any single photon that can be detected after passing through the double-slit 2 is labeled as photon 2. Photon 1 and photon 2 are indistinguishable as they have same frequency and polarisation. Consider, transmission function of double-slit 1 is \(t_{1}(y')= a'_{t}\left( \frac{e^{-(y'-d_{1}/2)^2/2\sigma _{1}^2}}{(2\pi )^{1/2}\sigma _{1}}+\frac{e^{-(y'+d_{1}/2)^2/2\sigma _{1}^2}}{(2\pi )^{1/2}\sigma _{1}}\right)\) , which represents two gaussian slits with separation between them \(d_{1}\) and slit width \(\sigma _{1}\) of each slit is such that \(d_{1}\) is considerably larger than \(\sigma _{1}\) . Similarly, transmission function of double-slit 2 is \(t_{2}(y'')= a''_{t}\left( \frac{e^{-(y''-d_{2}/2)^2/2\sigma _{2}^2}}{(2\pi )^{1/2}\sigma _{2}}+\frac{e^{-(y''+d_{2}/2)^2/2\sigma _{2}^2}}{(2\pi )^{1/2}\sigma _{2}}\right)\) , which represents two gaussian slits with separation between them \(d_{2}\) and slit width \(\sigma _{2}\) of each slit is such that \(d_{2}\) is considerably larger than \(\sigma _{2}\) . Where, \(a'_{t}\) and \(a''_{t}\) are the complex multipliers of transmission functions, they include the phase shift introduced by the slits and limit the maximum transmission to one. For \(a'_{t}= a''_{t}=0\) , the transmission of slits is zero. Each double-slit is positioned far away from the source as compared to its slit separation. Therefore, slits are located at close inclination with the x -axis such that photons coming from source are incident on slits almost close to the normal incidence. To have two-photon path entanglement via the slits the double-slits are positioned such that \(d_{1}/l_{1}=d_{2}/l_{2}\) and \(\sigma _{1}/l_{1}=\sigma _{2}/l_{2}\) . In addition, uncertainty \(\Delta q_{\parallel }\) of momentum component of each photon parallel to the double-slits, provided momentum of other photon is precisely determined, is small such that \(\Delta q_{\parallel }/q\ll d_{1}/l_{1}= d_{2}/l_{2}\) to suppress single photon interference by each double-slit, where q is the magnitude of momentum of a photon 41 . However, \(\Delta q_{\parallel }/q\approx \sigma _{1}/l_{1}=\sigma _{2}/l_{2}\) . These conditions implies, if a photon is passed through slit \(a_{1}\) then the other photon is most likely passed through slit \(b_{2}\) and if a photon is passed through slit \(b_{1}\) the other photon is most likely passed through slit \(a_{2}\) . Therefore, photons contributing to the joint detection on screens are path entangled via the slits. However, if a photon is absorbed far away from slits at an arbitrary location \(y'\) on double-slit 1 then the other photon is most probably absorbed at \(y''=-y'l_{2}/l_{1}\) far away from slits of double-slit 2. Transmission of each slit is considered to be gaussian with very small width that allows a photon to pass through it. Under these considerations, the amplitude \(A_{12}\) of joint detection of photons on screens gets a major contribution from a small range of momentum states of quantum state \(|\Psi \rangle\) . Remaining momentum states in \(|\Psi \rangle\) are absorbed at double-slits. Therefore, to evaluate \(A_{12}\) by using Eq.  2 , a following approximation can be applied

where \(c_{w}\) is a constant of proportionality that depends on the two-photon wavefunction. Since photons are incident on each slit close to the normal incidence therefore, \(e^{i q (r_{a1}+ r_{b2})/\hslash }\) is the two-photon amplitude of a photon to go from source to slit \(a_{1}\) located at a distance \(r_{a1}\) and other photon to go from source to slit \(b_{2}\) located at a distance \(r_{b2}\) . Similarly, \(e^{i q (r_{b1}+ r_{a2})/\hslash }\) is the two-photon amplitude of a photon to go from source to slit \(b_{1}\) located at a distance \(r_{b1}\) and other photon to go from source to slit \(a_{2}\) located at a distance \(r_{a2}\) . The transmitted amplitude of photons via the slits \(a_{1}\) and \(a_{2}\) or via the slits \(b_{1}\) and \(b_{2}\) is negligible because \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) is very small for these paths. Photons are path entangled via the slits and Eq.  4 represents the amplitude of transmitted photons on the double-slits that leads to the joint detection of photons.

Transmitted photon amplitude of a photon further emanates from a point on a double-slit such that it corresponds to an uniform probability distribution of the photon to be found on the screen. The amplitudes of transmitted photons to go from a point location on a double-slit to a point location on the nearest screen are \(\langle o_{1}| l_{1},y'\rangle \propto e^{iq|R'|/\hslash }/|R'|^{1/2}\) and \(\langle o_{2}| l_{2},y''\rangle \propto e^{iq|R''|/\hslash }/|R''|^{1/2}\) for photon 1 and photon 2, respectively. Where \(R'\) and \(R''\) are the distances of \(o_{1}\) and \(o_{2}\) from arbitrary points \(y'\) and \(y''\) located on double-slit 1 and double-slit 2, respectively. Since distances \(s_{1}\) and \(s_{2}\) of the screens from the nearest double-slits are much larger than the slit separations therefore, \(\langle o_{1}| l_{1},y'\rangle \propto e^{iq(r_{1}-y'\sin (\theta _{1}))/\hslash }/r^{1/2}_{1}\) and \(\langle o_{2}| l_{2},y''\rangle \propto e^{iq(r_{2}-y''\sin (\theta _{2}))/\hslash }/r^{1/2}_{2}\) , where \(r_{1}\) and \(r_{2}\) are the distances of \(o_{1}\) and \(o_{2}\) from the middle points of double-slit 1 and double-slit 2, respectively as shown in Fig.  1 . After solving Eq.  2 by using Eq.  4 the amplitude of joint detection of photons can be written as

where \(\delta =q (r_{a1}+r_{b2}-r_{a2}-r_{b1})/2\hslash\) and \(c_{n}\) is a proportionality constant. Therefore, probability of coincidence detection \(p_{12}=|A_{12}|^{2}\) of photons is

Probability of coincidence detection of photons is a product of two functions, where the exponential functions corresponds to a single-photon diffraction of photons from single slits and a cosine function corresponds to two-photon interference from the double-double-slit. Since photons are path entangled via the slits therefore, Eq.  6 can not be written as a product of two separate functions of variables of photon 1 and photon 2, respectively. If only the single photon detection locations on each screen are recorded without making any correlation among them then no interference pattern is formed. A single photon interference is suppressed due to quantum entanglement of paths of photons in the double-double-slit. Both photons can be detected anywhere randomly on the respective screens however, a two-photon quantum interference pattern appears only in the position correlated measurements. Probability of detection of a single photon on the respective screens can be calculated by integrating all possible paths of a single photon. However, due to quantum entanglement of paths this integral results in an addition of probability of detection of a single photon via each slit of a double-slit. Therefore, probabilities \(p_{1}\) and \(p_{2}\) to find a single photon on screen 1 and screen 2 are

where each probability distribution of a single photon detection is gaussian and single photon interference pattern is not exhibited.

Actual experiment is performed in the three-dimensional position space, where momentum of photons and distances of detectors from double-slits are measured in the three-dimensional position space. Therefore, projection of momentum and distances onto the transverse plane should be considered in order to be consistent with Eqs.  6 and 7 . In actual experiment the slits are located parallel to the transverse plane, detector displacement is parallel to the transverse plane and displacement range is such that \(y_{1}\ll s_{1}\) , \(y_{2}\ll s_{2}\) therefore, \(\sin {\theta _{1}}\sim y_{1}/s_{1}\) and \(\sin \theta _{2}\sim y_{2}/s_{2}\) . Under these considerations, terms in the form of a ratio, of transverse momentum and distance of a screen from a corresponding double slit, appears in Eqs.  6 and 7 . Therefore, photon momentum and distances of detectors from double-slits measured in the three-dimensional position space can be placed in these equations to calculate the patterns.

Double-double-slit experiment presented in this paper is performed with momentum entangled photons produced by type-I degenerate noncollinear SPDC 21 , 24 , 39 , 42 , 43 , 44 , 45 , 47 . A BBO crystal is pumped by an extraordinary linearly polarised laser beam of wavelength 405 nm and down converted photon pairs of wavelength 810 nm with ordinary polarisation are produced in the forward direction in a conical emission pattern according to momentum and energy conservation as shown in Fig.  2 . To produce a virtual double-double-slit configuration, two Fresnel biprisms are placed in the path of photons and photons are detected by single photon avalanche photodetectors \(D_{1}\) and \(D_{2}\) . Optical narrow band pass filters are placed in front of each photon detector to stop the background light. Down converted photons have same frequency and linear polarisation, which is perpendicular to the polarisation of the pump laser beam. Pump laser intensity is such that probability of more than single photon pair production is extremely small. Number of photon counts of each single photon detector and their mutual coincidence photon counts are measured with a two channel single photon counting module. Transverse mode extension of the pump laser beam is reduced to keep the source size much smaller than the slit separation but it is large so that momentum entanglement of photons is preserved.

A schematic diagram of the experimental configuration of the double-double-slit experiment. Momentum entangled photon pairs are produced in a conical emission pattern by a nonlinear crystal. A double-double-slit configuration is realised with two Fresnel biprisms.

An unfolded diagram of the double-double-slit experiment realised with Fresnel biprisms. Virtual sources correspond to virtual slits. To detect which-slit path information of photons, a shutter can be placed in a path of photon 1 in such a way that a virtual slit \(b_{1}\) is blocked.

Two-photon interference pattern obtained by measuring the coincidence photon counts when measurement location \(y_{2}\) of photon 2 is stationary. Experimental measurements are represented by open circles and solid line interference pattern is the two-photon interference calculated from theory. There is no interference exhibited by the individual photons as shown by single photon counts of single photon detectors \(D_{1}\) and \(D_{2}\) . Where ( a ) for \(y_{2}=~0~{\text {mm}}\) and ( b ) for \(y_{2}=~0.07~{\text {mm}}\) . Two-photon interference pattern is shifted as the location \(y_{2}\) of photon detector \(D_{2}\) is displaced.

Two-photon interference pattern obtained by measuring the coincidence photon counts when measurement location \(y_{1}\) of photon 1 is stationary. Experimental measurements are represented by open circles and solid line interference pattern is the two-photon interference calculated from theory. There is no interference exhibited by the individual photons as shown by single photon counts of single photon detectors \(D_{1}\) and \(D_{2}\) . Where ( a ) for \(y_{1}=~0~{\text {mm}}\) and ( b ) for \(y_{1}=~0.07~{\text {mm}}\) . Two-photon interference pattern is shifted as the location \(y_{1}\) of photon detector \(D_{1}\) is displaced.

For type-I phase matching, the BBO emits two degenerate photons with opposite transverse momenta in the transverse plane as shown in Fig.  2 and quantum state of photons of a pair corresponds to a continuous variable momentum entangled quantum state. A two-dimensional unfolded diagram of the experimental schematic given in Fig.  2 is shown in Fig.  3 , where a source S positioned at origin is a BBO crystal that emits momentum entangled photons pairs. Two virtual double-slits are realised with two Fresnel biprisms positioned in the path of both photons. Fresnel biprisms are aligned in such a way that after passing through each Fresnel biprism, paths of a photon can be extrapolated in the backward direction such that it appears as if the photon is originated from two virtual sources which are considered as slits. Each Fresnel biprism produces a virtual double-slit with gaussian slits of finite size. In this way, a double-double-slit configuration is realised with slit separation \(d_{1}\) and \(d_{2}\) of a virtual double-slit 1 and a virtual double-slit-2, respectively as shown in Fig.  3 . The virtual double-double-slit is parallel to the transverse plane and both photon detectors are displaced parallel to the transverse plane. Photon 1 is detected at location \(o_{1}\) and photon 2 is detected at location \(o_{2}\) by single photon detectors. Shortest distance of \(D_{1}\) , \(D_{2}\) are \(L_{1}\) , \(L_{2}\) from double-slit 1 and double-slit 2, respectively as shown in Fig.  3 . Single photon counts of single photon detectors positioned at different locations \(y_{1}\) and \(y_{2}\) and the corresponding coincidence photons counts are recorded. Experimental results on the double-double-slit interference of momentum entangled photons are shown in Fig.  4 , where the coincidence and single photon counts of photons are measured at different \(y_{1}\) positions of single photon detector \(D_{1}\) when single photon detector \(D_{2}\) a kept stationary at a location \(y_{2}\) . Single photon counts of each single photon detector and the coincidence photon counts are presented by open circles in Fig.  4 a for \(y_{2}\) = 0 mm where each data point is the mean of photon counts acquired for 5 s and twenty five repetitions of the experiment. The coincidence photon counts represent a two-photon interference pattern and the corresponding theoretically calculated interference given by Eq.  6 with a consideration of finite size of photon detectors is shown by a solid line. Effect of finite size of detectors raises the minima of the interference pattern. Single photon counts show no interference pattern as presented by the theoretical analysis also. According to the experimental considerations, \(\sin \theta _{1}\sim y_{1}/s_{1}\) and \(\sin \theta _{2}\sim y_{2}/s_{2}\) . The coincidence interference pattern exhibits a shift when measurement location \(y_{2}\) of photon 2 is shifted to another position by displacing single photon detector \(D_{2}\) . A shift in the two-photon interference pattern is shown in Fig.  4 b for \(y_{2}\) = 0.07 mm. In the opposite case, photon 1 is detected at a stationary location \(y_{1}\) and photon 2 is detected at different locations \(y_{2}\) . Results of the coincidence measurements of photon counts and single photon counts are shown in Fig.  5 a for \(y_{1}\) = 0 mm and Fig.  5 b for \(y_{1}\) = 0.07 mm. Solid line in each plot of coincidence measurements is the two-photon interference calculated from Eq.  6 by including the effect of finite size of detectors. A two-photon interference shows a shift with the displacement of position of detection location \(y_{1}\) of photon 1, while single photon counts show no interference as theoretically shown in the previous section. In the experiment, each virtual double-slit has a same slit separation \(d_{1}=d_{2}=~0.67~\hbox {mm}\) and \(L_{1}=L_{2}=528\)  mm.

It is evident from the probability of coincidence photon detection given in Eq.  6 that for \(d_{1}=d_{2}\) , the fringe separation of the two-photon interference pattern will reduce to half if the coincidence photon counts are measured for \(y_{2}=-y_{1}\) i.e. when both single photon detectors are displaced in the opposite direction. For this case, a two-photon interference pattern and a single photon pattern are shown in Fig.  6 , where each measured data point of photon counts is the mean of data acquired for 5 s and twenty five repetitions of the experiment. It is a different experimental set-up than the previous case and in this case \(d_{1}=d_{2}=~0.682~\hbox {mm}\) and \(L_{1}=L_{2}=520\) mm. Solid line represents a theoretically calculated two-photon interference by including the effect of finite size of photon detectors. It is evident that the fringe separation is reduced to half and therefore, the number of fringes are increased within the same gaussian envelop. There is no formation of coincidence interference pattern if both the single photon detectors are displaced in the same direction such that \(y_{2}=y_{1}\) as it is evident from Eq.  6 .

In this experiment both the single photon detectors \(D_{1}\) and \(D_{2}\) are displaced in the opposite direction such that \(y_{2}=-y_{1}\) . Fringe separation of two-photon interference is reduced and individual photons exhibit no interference.

Detection of which-slit path of photons

In the double-double-slit experiment, photons are momentum entangled and they can reveal the which-slit path information of each other if one of them is detected close to any double-slit. If one blocks a single slit of a double-slit then the which-slit path can be detected from the coincidence detection of photons. Consider a slit \(a_{2}\) is blocked by closing a shutter shown in Fig.  1 .

Two-photon coincidence pattern when a virtual slit \(b_{1}\) is blocked. It is evident that two-photon interference is suppressed. Solid line is a theoretically calculated two-photon pattern.

If photons of a single pair are detected on screen 1 and screen 2 together then it is evident that photon 2 has passed through slit \(b_{2}\) . One can consider a path blocking shutter as another single photon detector \(D_{3}\) . If \(D_{3}\) detects a photon 2 then the path entangled state of photons collapses and the which-slit path of photon 1 in the double-slit 1 is also determined. In this case, the which-slit path of photon 1 is through the single slit \(b_{1}\) . Since each photon is passed through a single slit therefore, neither a single photon nor a two-photon interference of joint detections of photons on screen 1 and a path blocking single photon detector \(D_{3}\) will occur. On the other hand, if photon 2 is detected on screen 2 then the path entangled state is collapsed by \(D_{3}\) such that photon 2 is passed though slit \(b_{2}\) and photon 1 is passed through slit \(a_{1}\) . Single photon detection probability of photon 2 on screen 2 will reduce by half in comparison to the case when both slits were open. Probability of a single photon detection of photon 1 on screen 1 will remain unchanged because detection of photon 1 does not reveal any information whether a photon 2 is detected at screen 2 or by \(D_{3}\) . In the experiment, shutter is placed after the Fresnel biprism 1 such that a virtual slit \(b_{1}\) shown in Fig.  3 is blocked. This configuration resembles to a double-double-slit schematic shown in Fig.  1 where the slit \(a_{2}\) can be blocked by a shutter. Photon counts are measured for different locations \(y_{2}\) of single photon detector \(D_{2}\) by keeping single photon detector \(D_{1}\) stationary at \(y_{1}\) . Experimental results of a path detection experiment are shown in Fig.  7 . Experimental parameters in this case are same as for the experiment described in the previous section. It is evident from the experimental results, if a which-slit path information of photons is extracted by blocking any single slit then both single and two-photon interferences are suppressed.

This paper has presented experimental and conceptual insights on the quantum double-double-slit thought experiment first introduced by Greenberger, Horne and Zeilinger 17 . Experiments presented in this paper are performed with momentum entangled photons produced by type-I degenerate noncollinear SPDC process in a BBO crystal. In the experiment, once both photons traverse the respective double-slits, they can be detected anywhere on screens randomly because when a photon strikes a screen its quantum state collapses to one location randomly. Patterns emerge in many repetitions of the same experiment. Since paths of photons in the double-double-slit configuration are quantum entangled, their individual quantum states are phase incoherent therefore, formation of a single photon interference is suppressed. However, if a photon is detected on a screen at a well defined location, the quantum state of other photon, which is not detected, corresponds to a phase coherent amplitude to find it on second screen. Therefore, knowledge of detection locations of a photon labels the different phase coherent amplitudes to find other photon on second screen. However, in subsequent repetitions of the experiment, detection locations of photons can vary randomly. For a given location of detection of a photon the other photon shows interference pattern which corresponds to the conditional interference pattern of two photons. As a detection location of a photon is varied the conditional interference pattern is shifted. On the other hand, if no correlations of detection locations of photons are made then there is no way to select a particular phase coherent amplitude in repeated measurements. Eventually, a single photon interference pattern does not appear. It is also shown experimentally and conceptually, if a which-slit path information of any one of the photons is detected then a single photon interference and a two-photon interference disappear because of random collapse of quantum superposition of paths.

Two-photon momentum entangled state

Two-photon momentum entangled state is produced by a negative uniaxial second order nonlinear BBO crystal by type-I SPDC process. A pump photon of frequency \(\omega _{p}\) is split into two photons known as the signal photon and the idler photon of frequency \(\omega _{s}\) and \(\omega _{i}\) , respectively. A linearly polarised extraordinary pump laser beam propagating along the z -axis is incident on the crystal. A planar surface of the crystal is in the xy -plane with origin at the centre, where \(l_{x}\) , \(l_{y}\) , \(l_{z}\) are the spatial extensions of the crystal along each axis. Ordinary photons produced by SPDC are linearly polarised with propagation vectors in three dimensions \(\mathbf {k_{s}}\) and \(\mathbf {k_{i}}\) . In type-I phase matching, due to dispersion and anisotropy of the crystal, the signal and the idler photons are produced with non zero angle of their propagation vectors with the propagation vector \(\mathbf {k_{p}}\) of the pump laser beam to conserve momentum of photons. For a thin crystal and a narrow pump laser beam, it produces a conical emission pattern of down converted photons. Pump laser beam is considered to be a continuous beam and due to low down conversion efficiency the pump laser beam amplitude is considered to be constant. The amplitude to produce more than one photon pair is extremely small, which is desirable in experiments with a single quantum entangled pair of photons during each cycle of the experiment. Pump laser beam is considered to be monochromatic, frequencies of the signal and the idler photons are same in the experiment and their propagation vectors are making a nonzero angle with the propagation direction of pump photons. Narrow band pass filters are placed after the crystal and prior to the detectors to increase coherence length. Due to sufficiently long interaction time, energy conservation condition is fulfilled such that \(\hslash \omega _{p}=\hslash \omega _{s}+\hslash \omega _{i}\) . Since polarisation of down converted photons is same therefore, two-photon quantum state produced by degenerate type-I noncollinear SPDC process can be written as 42 , 43 , 44 , 45 , 46 , 47

where \(c_{0}\) , \(c_{1}\) are complex coefficients, \(c_{1}\) depends on the pump laser beam intensity and second order nonlinear coefficient of the crystal. The quantum states \(|\mathbf {p_{s}}\rangle\) and \(|\mathbf {p_{i}}\rangle\) represent single photon momentum states of the signal and the idler modes of momentum vectors \(\mathbf {p_{s}}=\hslash \mathbf {k_{s}}\) and \(\mathbf {p_{i}}=\hslash \mathbf {k_{i}}\) , respectively. The quantum state \(|0\rangle\) is a vacuum state of the signal and the idler modes without any photon. A two-photon wavefunction \(\Phi (\mathbf {p_{s},\mathbf {p_{i}}})\) in the momentum space can be written as

where \(c_{p}\) is a constant and the integration is carried out in transverse momentum space which is a projection of three-dimensional momenta onto the transverse two-dimensional xy -plane, \(\Delta p_{j}\) = \(p_{sj}+p_{ij}-p_{pj}\) for \(j\in \{x,y,z\}\) and \(p_{sj}\) , \(p_{ij}\) , \(p_{pj}\) represent components of momentum of the signal, the idler and pump photons along the j -axis, respectively. A function \(\mathbf {\nu }(\mathbf {q_{p}})\) is the normalised amplitude of pump laser beam corresponding to momentum projection \(\mathbf {q_{p}}\) in the transverse plane. For a plane wave, \(\mathbf {\nu }(\mathbf {q_{p}})\) is a Dirac delta function. If crystal extensions \(l_{x}\) and \(l_{y}\) are much larger than the wavelength of pump laser beam then \(\Delta p_{x}\) and \(\Delta p_{y}\) should be very small otherwise \(\Phi (\mathbf {p_{s},\mathbf {p_{i}}})\) diminishes. Therefore, \(\mathbf {q_{p}}\) = \(\mathbf {q_{s}}+\mathbf {q_{i}}\) for transverse momentum \(\mathbf {q_{s}}\) of the signal photon and transverse momentum \(\mathbf {q_{i}}\) of the idler photon. It corresponds to conservation of transverse momentum of photons. For a gaussian transverse momentum profile of the pump laser beam with radius \(\sigma _{p}\) in the position-space and for a very small angle between the pump photon momentum and the signal photon or the idler photon momentum, the two-photon wavefunction is given in Ref. 44 ,

where, \(c_{\Phi }\) is a constant of proportionality. Two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) is prominent if transverse momenta of photons are equal and opposite to each other.

Two-photon quantum entangled state in the transverse momentum space can be written as

where N is a normalisation constant and the vacuum state is not relevant in the context of present experiment. In general the momentum entanglement is manifested by non separability of two-photon wavefunction \(\Phi (\mathbf {q_{s},\mathbf {q_{i}}})\) .

Experimental details

A linearly polarised pump laser light of wavelength 405 nm of gaussian beam profile is incident on a BBO crystal at room temperature. Two orthogonally polarised momentum entangled photons of wavelength 810 nm are emitted in a conical emission pattern and single photons are detected by avalanche single photon detectors. Each photon pair is passed through two Fresnel biprisms to realise a virtual double-double-slit configuration. After the crystal, pump light at 405 nm is blocked by an optical band pass filter with transmission window peak at 810 nm where the full-width-half-maximum of the transmission window is about 10 nm. Two multimode optical fibers carry photons from points \(o_{1}\) and \(o_{2}\) to each single photon detector. The other end of each optical fiber is mounted on separate three-dimensional precision displacement stages and photons are coupled to each optical fiber with an objective lens. Narrow apertures are positioned at \(o_{1}\) and \(o_{2}\) prior to the objective lens to allow photons to be detected at these two points only. Prior to each fiber coupler two optical band pass filters (filter 1 and filter 2) are placed in the path of each photon to block scattered photon of wavelength 405 nm and background photons reaching each single photon detector. Photon correlations are measured by counting electrical pulses produced by each single photon detector. Experimentally measured coincidence and single photon counts are shown by open circles data points in the figures. Each data point is acquired for 5 s with twenty five repetitions of the same experiment. Experimental results are compared with theoretical calculations considering the effect of finite size of photon detectors.

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Acknowledgements

Mandip Singh acknowledges research funding by the Department of Science and Technology, Quantum Enabled Science and Technology grant for project No. Q.101 of theme title “Quantum Information Technologies with Photonic Devices”. DST/ICPS/QuST/Theme-1/2019 (General).

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M.S. designed and setup the experiment, M.K. took data and also aligned the experiment. M.S. did theory, analysed data and wrote the manuscript, both authors discussed the experiment.

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Kaur, M., Singh, M. Quantum double-double-slit experiment with momentum entangled photons. Sci Rep 10 , 11427 (2020). https://doi.org/10.1038/s41598-020-68181-1

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Classic Physics 'Thought' Experiment Finally Recreated

Scientists have finally performed a famous experiment described by physicist Richard Feynman that captures the mysterious quantum properties that allow electrons to behave like either waves or particles.

The findings, published Wednesday (March 13) in the New Journal of Physics, don't upend any fundamental laws, but confirm the results of a classic  double-slit experiment  once thought too challenging to perform.

"It confirms quantum mechanics," said study co-author Herman Batelaan, a physicist at the University of Nebraska. "We all know it's supposed to work that way, but to actually go out there and do it is a valuable thing."

From 1961 to 1963, Nobel prize -winning physicist Richard Feynman gave a series of electrifying lectures on quantum mechanics, revealing the strange laws that govern the world of the very small.

In one, he described a thought experiment to illustrate how electrons can behave like waves or particles.

The setup was simple: A gun shoots single electrons at a wall with two tiny slits that can be opened or closed. Once they pass through the slit, or slits, the electrons hit a detector. [ Twisted Physics: 7 Mind-Blowing Findings ]

Feynman predicted that when just one slit was open, the electrons would behave as particles. But when both slits were open, the electrons would create a characteristic pattern of light and dark fringes on the detector formed by wave interference . In essence, the peaks of one wave would sometimes add to the peaks of others, creating bright patches, while at other places the peaks and troughs would overlap, creating dark areas.

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In the lecture, Feynman said the setup, with tiny gaps opening and closing at will, would be too hard to create.

"You have to control this at a small scale," Batelaan told LiveScience.

But over the last 20 years, the technology to create the setup has emerged, and the research team realized all the elements were in place to recreate the famous experiment .

To do so, the team fired an electron beam that could shoot individual electrons at a gold-silicon membrane wall with two 62 nanometer-wide slits. To open and close the slits, the team used an piezo-electric motor to move a 10 micrometer-tall mask behind the slits up and down.

It's no surprise the team's results match those predicted by the brilliant physicist.

But while the new experiment may not raise questions about quantum mechanics, it's important to check that these fundamental laws are behaving as physicists expect in many different setups, Batelaan said.

"Something that's so fundamental, you want to overturn every stone," Batelaan said.

Follow Tia Ghose on Twitter @tiaghose .   Follow   LiveScience @livescience, Facebook   & Google+ . Original article on  Live Science.

Tia is the managing editor and was previously a senior writer for Live Science. Her work has appeared in Scientific American, Wired.com and other outlets. She holds a master's degree in bioengineering from the University of Washington, a graduate certificate in science writing from UC Santa Cruz and a bachelor's degree in mechanical engineering from the University of Texas at Austin. Tia was part of a team at the Milwaukee Journal Sentinel that published the Empty Cradles series on preterm births, which won multiple awards, including the 2012 Casey Medal for Meritorious Journalism.

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What is the *DETECTOR* in the double slit experiment and how does it work?

Is the detector a passive device or is it just a fictional mathematical probe?

I think the detector is somehow consuming the energy responsible for the wave nature of the photons, electrons or atoms, but I can't find any information about the detector and how it works.

Any help is appreciated since all videos and articles are suspiciously skipping the detector or simplifying it as a 3d cat or fictional cartoon eye. I know about the quantum eraser experiment but before moving to it, I need to know about the detector and how it exactly measures.

I'm a software programmer trying to understand how quantum computers work.

• quantum-mechanics
• experimental-physics
• double-slit-experiment
• particle-detectors

• 4 $\begingroup$ You can use a wall as a detector. $\endgroup$ –  rob ♦ Commented Mar 23, 2019 at 1:11
• 5 $\begingroup$ @rob I mean the detector that is supposedly causing the photones to change their mind and act as particles withot wave characteristics and not the surface $\endgroup$ –  USER249 Commented Mar 23, 2019 at 1:21
• $\begingroup$ @rob Its somthing scientists said or imply they placed behind the slits to determine which slit a particular photon or atom went through before hitting the surface. Th surface hier is the wall you are talking about. $\endgroup$ –  USER249 Commented Mar 23, 2019 at 1:59
• $\begingroup$ @AmrBerag I’m glad to see you’re still interested in this considering you never did get an actual physical answer. In experiments involving photons there’s only one way to detect them. Place matter in the path of the photon and the photon will be absorbed. From this you will know what photons are missing or where they landed. Like spectral lines you can have emission lines or absorption lines. It’s all physical and no waves involved. $\endgroup$ –  Bill Alsept Commented Aug 21, 2020 at 15:40
• 1 $\begingroup$ @USER249 did you ever get a good answer or explanation to this? I feel as though it would be impossible to build such a detector withour perturbing the whole experiment in the first place. I'm not a physicist though so I don't know what I don't know. $\endgroup$ –  rollsch Commented Aug 7, 2023 at 0:11

3 Answers 3

I misunderstood your question at first. I thought you were asking about the detector downstream of the double slit, where the interference pattern is visible; every practical double-slit experiment includes such a detector. But instead, you are asking about a hypothetical detector which could "tag" a particle as having gone through one slit or the other. Most interference experiments do not have such a detector.

The idea of "tagging" a particle as having gone through one slit or the other, and the realization that such tagging would destroy the double-slit interference pattern, was hashed out in a long series of debates between Bohr and Einstein . Most introductory quantum mechanics textbooks will have at least some summary of the history of these discussions, which include many possible "detectors" with varying degrees of fancifulness.

A practical way to tag photons as having gone through one slit or another is to cover both slits with polarizing films. If the light polarizations are parallel, it's not possible to use this technique to tell which slit a given photon came through, and the interference pattern survives. If the light polarizations are perpendicular, it would be possible in principle to detect whether a given photon went through one slit or the other; in this case, the interference pattern is also absent. If the polarizers are at some other angle, it's a good homework problem to predict the intensity of the interference pattern.

• 5 $\begingroup$ Wouldn't perpendicular ploarizers also explain the lack of an interference pattern because perpendicular waves wouldn't interfere with each other? $\endgroup$ –  JohnFx Commented Jun 28, 2020 at 1:49
• 1 $\begingroup$ @JohnFx I thought that's what I wrote? $\endgroup$ –  rob ♦ Commented Jun 28, 2020 at 6:05
• $\begingroup$ When they go through the polarizers, are they in a superposition of both polarities, or just in one of either? $\endgroup$ –  Juan Perez Commented May 20, 2023 at 13:50
• $\begingroup$ @JuanPerez I'm not sure that question has a simple answer like you want it to. It might be possible to use the ambiguity to make a variation of the "quantum eraser" experiment, but I don't think I can explain that in a comment. $\endgroup$ –  rob ♦ Commented May 20, 2023 at 16:20
• $\begingroup$ I think that polarizers at the slits are a bad example because even classical electrodynamics predicts the disappearance of the interference pattern in that case, so it can't be said to provide evidence for quantum mechanics. $\endgroup$ –  benrg Commented May 20, 2023 at 19:22

I watched a couple of videos on the double slit experiment and I had exactly the same question as the original poster (Amr Berag) and stumbled upon this post and was just wondering how come everyone else isn't wondering the same thing.

There are so many videos showing the actual double slit experiment but none show the actual wave function collapse in reality when the particles are "observed".

It turns out that it was merely a thought experiment when it was first proposed and it's not super trivial to put an actual detector, but in 1987 an experiment was performed and subsequent experiments were performed but none shown on video.

This link explains that a bit

Please look at the "Which-way" section here

I'm just surprised that they don't mention this in any of the videos and how come no one else asks for proof of this. When I saw the original video I was just waiting till the end to see them put a "detector" and see the interference pattern disappear, but nope.

• 3 $\begingroup$ It seems nuts that this is taken as a given when it was just a thought experiment. I read the link but I don't see how it demonstrates the thought experiment practically... $\endgroup$ –  Cloudyman Commented Dec 3, 2021 at 23:13
• $\begingroup$ The reason why no serious physicists does these experiments is trivial: one can't learn anything from them. We know what "a detector" does to a system: it either removes energy from it or it adds energy to it. The consequences of that are trivial to calculate, both in quantum mechanics and in classical physics. So what's the point of wasting time and money on experiments that we can predicts at the introductory textbook level? In practice science is about what we don't know. It's not an endless repetition of stuff that we do know. We owe that to the taxpayer who funds us. $\endgroup$ –  FlatterMann Commented May 20, 2023 at 19:49
• 2 $\begingroup$ For such a big result, experimental verification seems like a good idea! The failure of the thought experiment to happen in practice can lead to new results in themselves, for example. $\endgroup$ –  apg Commented Dec 23, 2023 at 20:42
• 1 $\begingroup$ It almost seems like a conspiracy that noone ever filmed this famous experiment, while at the same time we have hundreds of thousands of the double-slit experiment adaptations. Good to see that I'm not the only one confused here. This would be one of the greatest resources for introducing newbies to the uncertainity in the world of quantum physics. The description of the experiment is so mind-boggling that it is one of not very few things that I remember from high school. It would be so live-changing to be able to see it then, even on film. $\endgroup$ –  jannis Commented Feb 10 at 11:56
• 1 $\begingroup$ And you know @FlatterMann "seeing is believing". When I first learned about the uncertainty and wave-particle dualism it was like "meh, this cannot be true, just some random theoretical stuff they are telling us so that they can later do an exam about it". Having this on film would have great educational consequences. IMO putting such content in the Internet would give the author eternal fame (not mentioning likes/views and, therefore, money). And when it comes to cost Mr Beast puts millions of dollars in his movies, maybe he'd get interested:) Anyway I would surely pay to see it. $\endgroup$ –  jannis Commented Feb 10 at 12:03

It's anything that gives you information about where the particle passed by. The problem in measurement in QM is that to measure anything you need to interact with the "thing" you want to measure. If you want to measure temperature in a drop of water with a large thermometer, the heat of the thermometer will affect the drop temperature. If you want to measure the distance to the moon you may shoot a laser (knowing c=the speed of light in vacuum) and wait till it returns. But if the smallest thing you have is a rock, you can throw it and do the same process knowing the rock's speed (ignoring gravity and air resistance), but if the smallest thing you have, to measure it, is another moon, you will affect the position of the "original" moon and so affecting the whole state of the thing you want to measure. Well, the quantum world is so small that to measure things you have to destroy the original state or perturb it. The device in the double slit is just to block or interact with the particle passing through that slit.

• 3 $\begingroup$ You just rephrased my question bro. Are you telling me that the detector in the double slit experiment is not passive enough to detect without afecting the outcome? Then why all these credited sientists jumping to the assumption that photones are conscious instead of saying that the very act of trying to detect is affecting these photones and particles in a way that they lose their wave characteristics by withdrawing their enrgy for example. Or do you mean the detector is nothing but another surface moving toward the slits but then it will be too close and prevent the interference! $\endgroup$ –  USER249 Commented Mar 23, 2019 at 1:47
• 1 $\begingroup$ I din't rephrase your question. There's no single question mark on my reply. The math is a model to represent a behavior that is very consistent. It is not passive enough $\endgroup$ –  Gndk Commented Mar 23, 2019 at 1:59
• 1 $\begingroup$ Do you mean there is no detector? $\endgroup$ –  USER249 Commented Mar 23, 2019 at 2:04
• 1 $\begingroup$ The behavior Im asking about is that these things behave like waves when it is unknown which path they take but behave like sand partjcles or bullets when sientists attempt to detect their pass. I would like to know what device they used to detect the path and how it works. This device is referred to as "The Detector" that is causing the mind puzzling behavior implying that such device exists and is good enough to make scientists and univesity professors believe that atoms not only conscious but can communicate and tell their friends that some curious humans are trying to spy on them. $\endgroup$ –  USER249 Commented Mar 23, 2019 at 2:37
• 1 $\begingroup$ @Esther You can start here if you are coming from that cartoon robotic eye detector video youtu.be/yotBpxXiivA $\endgroup$ –  USER249 Commented Dec 5, 2020 at 13:25

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The Physicist Who’s Challenging the Quantum Orthodoxy

July 10, 2023

Jonathan Oppenheim, a physicist at University College London, is developing hybrid theories that could unify classical gravity and quantum mechanics.

Philipp Ammon for Quanta Magazine

Introduction

Most physicists expect that when we zoom in on the fabric of reality, the unintuitive weirdness of quantum mechanics persists down to the very smallest scales. But in those settings, quantum mechanics collides with classical gravity in a resolutely incompatible way.

So for almost a century, theorists have tried to create a unified theory by quantizing gravity, or sculpting it according to the rules of quantum mechanics. They still haven’t succeeded.

Jonathan Oppenheim , who runs a program exploring post-quantum alternatives at University College London, suspects that’s because gravity simply can’t be squeezed into a quantum box. Maybe, he argues, our presumption that it must be quantized is wrong. “That view is ingrained,” he said. “But no one knows what the truth is.”

Quantum theories are based on probabilities rather than certainties. For example, when you measure a quantum particle, you can’t predict exactly where you will find it, but you can predict the likelihood that it will be found in a particular place. What’s more, the more certain you are about a particle’s location, the less certain you are about its momentum. Over the 20th century, physicists gradually made sense of electromagnetism and other forces using this framework.  

But when they tried to quantize gravity, they ran into unnatural infinities that had to be sidestepped with clumsy mathematical tricks.

  The problems arise because gravity is a result of space-time itself, rather than something that acts on top of it. So if gravity is quantized, that means space-time is also quantized. But that doesn’t work, because quantum theory only makes sense against a classical space-time background — you can’t add and then evolve quantum states on top of an uncertain foundation.  

Video : Oppenheim describes why he thinks gravity can’t be squeezed into the same quantum box as the other fundamental forces — and what he’s proposing as an alternative.

Christopher Webb Young/ Quanta Magazine ; Noah Hutton for Quanta Magazine

To deal with this deep conceptual conflict, most theorists turned to string theory, which imagines that matter and space-time emerge from tiny, vibrating strings. A smaller faction looked to loop quantum gravity, which replaces the smooth space-time of Einstein’s general relativity with a network of interlocked loops. In both theories, our familiar, classical world somehow emerges from these fundamentally quantum building blocks.  

Oppenheim was originally a string theorist, and string theorists believe in the primacy of quantum mechanics. But he soon became uncomfortable with the elaborate mathematical acrobatics his peers performed to tackle one of the most notorious problems in modern physics: the black hole information paradox .  

In 2017, Oppenheim started searching for alternatives that avoided the information paradox by taking both the quantum and the classical worlds as bedrocks. He stumbled across some overlooked research on quantum-classical hybrid theories from the 1990s, which he’s been extending and exploring ever since. By studying how the classical and quantum worlds interrelate, Oppenheim hopes to find a deeper theory that is neither quantum nor classical, but some kind of hybrid. “Often we put all our eggs in a few baskets, when there are lots of possibilities,” he said.  

To make his point, Oppenheim recently made a bet with Geoff Penington and Carlo Rovelli — leaders in their respective fields of string theory and loop quantum gravity. The odds? 5,000-to-1. If Oppenheim’s hunch is correct and space-time isn’t quantized, he stands to win bucketloads of potato chips, colorful plastic bazinga balls , or shots of olive oil, according to his fancy — as long as each item costs at most 20 pence (about 25 cents).

We met in a north London café lined with books, where he calmly unpacked his concerns about the quantum gravity status quo and extolled the surprising beauty of these hybrid alternatives. “They raise all kinds of remarkably subtle questions,” he said. “I’ve really lost my feet trying to understand these systems.” But he perseveres.  

“I want my 5,000 bazinga balls.”

The interview has been condensed and edited for clarity.

Why are most theorists so sure that space-time is quantized?

It’s become dogma. All the other fields in nature are quantized. There’s a sense that there’s nothing special about gravity — it’s just a field like any other — and therefore we should quantize it.

Oppenheim and his students, seen here in and around the UCL campus, are developing a new class of hybrid quantum-classical theories in which gravity stays classical. Maybe, Oppenheim argues, gravity is special and the quantum consensus is wrong.

Is gravity special in your view?

Yes. Physicists define all the other forces in terms of fields evolving in space-time. Gravity alone tells us about the geometry and curvature of space-time itself. None of the other forces describe the universal background geometry that we live in like gravity does.

At the moment, our best theory of quantum mechanics uses this background structure of space-time — which gravity defines. And if you really believe that gravity is quantized, then we lose that background structure.

What sorts of problems do you run into if gravity is classical and not quantized?

For a long time, the community believed it was logically impossible for gravity to be classical because coupling a quantum system with a classical system would lead to inconsistencies. In the 1950s, Richard Feynman imagined a situation that illuminated the problem: He began with a massive particle that is in a superposition of two different locations. These locations could be two holes in a metal sheet, as in the famous double-slit experiment. Here, the particle also behaves like a wave. It creates an interference pattern of light and dark stripes on the other side of the slits, which makes it impossible to know which slit it went through. In popular accounts, the particle is sometimes described as going through both slits at once.

But since the particle has mass, it creates a gravitational field that we can measure. And that gravitational field tells us its location. If the gravitational field is classical, we can measure it to infinite precision, infer the particle’s location, and determine which slit it went through. So we then have a paradoxical situation — the interference pattern tells us that we can’t determine which slit the particle went through, but the classical gravitational field lets us do just that.

But if the gravitational field is quantum, there is no paradox — uncertainty creeps in when measuring the gravitational field, and so we still have uncertainty in determining the particle’s location.

So if gravity behaves classically, you end up knowing too much. And that means that cherished ideas from quantum mechanics, like superposition, break down?

Yes, the gravitational field knows too much. But there’s a loophole in Feynman’s argument that could allow classical gravity to work.

What is that loophole?

As it stands, we only know which path the particle took because it produces a definite gravitational field that bends space-time and allows us to determine the particle’s location.  

But if that interaction between the particle and space-time is random — or unpredictable — then the particle itself doesn’t completely dictate the gravitational field. Which means that measuring the gravitational field will not always determine which slit the particle went through because the gravitational field could be in one of many states. Randomness creeps in, and you no longer have a paradox.

So why don’t more physicists think gravity is classical?

Well, it is logically possible to have a theory in which we don’t quantize all the fields. But for a classical theory of gravity to be consistent with everything else being quantized, then gravity has to be fundamentally random. To a lot of physicists that’s unacceptable.

Oppenheim started out as a string theorist, but he eventually grew frustrated with the clumsy mathematical tricks his colleagues employed to get around one of the most notorious conundrums in physics: the black hole information paradox.

Physicists spend a lot of time trying to figure out how nature works. So the idea that there is, on a very deep level, something inherently unpredictable is troubling to many.

The outcome of measurements within quantum theory appears to be probabilistic. But many physicists prefer to think that what appears as randomness is just the quantum system and the measuring apparatus interacting with the environment. They don’t see it as some fundamental feature of reality.

What are you proposing instead?

My best guess is that the next theory of gravity will be something that is neither completely classical nor completely quantum, but something else entirely.

Physicists are only ever coming up with models that approximate nature. But as an attempt at a closer approximation, my students and I constructed a fully consistent theory in which quantum systems and classical space-time interact. We just had to modify quantum theory slightly and modify classical general relativity slightly to allow for the breakdown of predictability that is required.

Why did you start working on these hybrid theories?

I was motivated by the black hole information paradox. When you throw a quantum particle into a black hole and then let that black hole evaporate, you encounter a paradox if you believe that black holes preserve information. Standard quantum theory demands that whatever object you throw into the black hole is radiated back out in some scrambled but recognizable way. But that violates general relativity, which tells us that you can never know about objects that cross the black hole’s event horizon.

But if the black hole evaporation process is indeterministic then there’s no paradox. We never learn what was thrown into the black hole because predictability breaks down. General relativity is safe.

Recently, Oppenheim made a 5,000-to-1 bet with two colleagues that gravity can’t be quantized. If he wins, he gets to stuff his pockets with 5,000 bags of potato chips or bazinga balls or anything else that suits his fancy — as long as each item costs at most 20 pence (about 25 cents). “I feel I’ve made a pretty safe bet, even if I lose,” Oppenheim said.

So the noisiness in these quantum-classical hybrid theories allows information to be lost?

But information conservation is a key principle in quantum mechanics. losing this can’t sit easily with many theorists..

That’s true. There were huge debates about this in recent decades, and almost everybody came to believe that black hole evaporation is deterministic. I’m always puzzled by that.

Will experiments ever resolve if gravity is quantized or not?

At some point. We still know almost nothing about gravity on the smallest scales. It hasn’t even been tested to the millimeter scale, let alone to the scale of a proton. But there are some exciting experiments coming online which will do that.

One is a modern-day version of the “Cavendish experiment,” which calculates the strength of the gravitational attraction between two lead spheres. If there is randomness in the gravitational field, as in these quantum-classical hybrids, then when we try and measure its strength we won’t always get the same answer. The gravitational field will jiggle around. Any theory in which gravity is fundamentally classical has a certain level of gravitational noise.

How do you know this randomness is intrinsic to the gravitational field and not some noise from the environment?

You don’t. Gravity is such a weak force that even the best experiments already have a lot of jiggle in them. So you have to eliminate all these other sources of noise as much as possible. What’s exciting is that my students and I showed that if these hybrid theories are true, there must be some minimal amount of gravitational noise. This can be measured by studying gold atoms in a double-slit experiment. These experiments already place bounds on whether gravity is fundamentally classical. We are gradually closing in on the amount of indeterminacy allowed.

On the flip side of the bet, are there any experiments that would prove that gravity is quantized?

There are proposed experiments that look for entanglement mediated by the gravitational field. As entanglement is a quantum phenomenon, that would be a direct test of the quantum nature of gravity. These experiments are very exciting, but probably decades away.

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