One hundred years ago this month, in Feb. 1924, a hereditary member of the French nobility, Louis Victor Pierre Raymond, the 7th Duc de Broglie, published a landmark paper in the Philosophical Magazine of London [1] that revolutionized the nascent quantum theory of the day.
Prior to de Broglie’s theory of quantum matter waves, quantum physics had been mired in ad hoc phenomenological prescriptions like Bohr’s theory of the hydrogen atom and Sommerfeld’s theory of adiabatic invariants. After de Broglie, Erwin Schrödinger would turn the concept of matter waves into the theory of wave mechanics that we still practice today.
Fig. 1 The 1924 paper by de Broglie in the Philosophical Magazine.
The story of how de Broglie came to his seminal idea had an odd twist, based on an initial misconception that helped him get the right answer ahead of everyone else, for which he was rewarded with the Nobel Prize in Physics.
de Broglie’s Early Days
When Louis de Broglie was a student, his older brother Maurice (the 6th Duc de Broglie) was already a practicing physicist making important discoveries in x-ray physics. Although Louis initially studied history in preparation for a career in law, and he graduated from the Sorbonne with a degree in history, his brother’s profession drew him like a magnet. He also read Poincaré at this critical juncture in his career, and he was hooked. He enrolled in the Faculty of Sciences for his advanced degree, but World War I side-tracked him into the signal corps, where he was assigned to the wireless station on top of the Eiffel Tower. He may have participated in the famous interception of a coded German transmission in 1918 that helped turn the tide of the war.
Beginning in 1919, Louis began assisting his brother in the well-equiped private laboratory that Maurice had outfitted in the de Broglie ancestral home. At that time Maurice was performing x-ray spectroscopy of the inner quantum states of atoms, and he was struck by the duality of x-ray properties that made them behave like particles under some conditions and like waves in others.
Fig. 2 Maurice de Broglie in his private laboratory (Figure credit).
Through his close work with his brother, Louis also came to subscribe to the wave-particle duality of x-rays and chose the topic for his PhD thesis—and hence the twist that launched de Broglie backwards towards his epic theory.
de Broglie’s Massive Photons
Today, we say that photons have energy and momentum although they are massless. The momentum is a simple consequence of Einstein’s special relativity
And if m = 0, then
and momentum requires energy but not necessarily mass.
But de Broglie started out backwards. He was so convinced of the particle-like nature of the x-ray photons, that he first considered what would happen if the photons actually did have mass. He constructed a massive photon and compared its proper frequency with a Lorentz-boosted frequency observed in a laboratory. The frequency he set for the photon was like an internal clock, set by its rest-mass energy and by Bohr’s quantization condition
He then boosted it into the lab frame by time dilation
But the energy would be transformed according to
with a corresponding frequency
which is in direct contradiction with Bohr’s quantization condition. What is the resolution of this seeming paradox?
de Broglie’s Matter Wave
de Broglie realized that his “massive photon” must satisfy a condition relating the observed lab frequency to the transformed frequency, such that
This only made sense if his “massive photon” could be represented as a wave with a frequency
that propagated with a phase velocity given by c/β. (Note that β < 1 so that the phase velocity is greater than the speed of light, which is allowed as long as it does not transmit any energy.)
To a modern reader, this all sounds alien, but only because this work in early 1924 represented his first pass at his theory. As he worked on this thesis through 1924, finally defending it in November of that year, he refined his arguments, recognizing that when he combined his frequency with his phase velocity,
it yielded the wavelength for a matter wave to be
where p was the relativistic mechanical momentum of a massive particle.
Using this wavelength, he explained Bohr’s quantization condition as a simple standing wave of the matter wave. In the light of this derivation, de Broglie wrote
We are then inclined to admit that any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave.
pg. 450, Philosophical Magazine of London (1924)
Here was the strongest statement yet of the wave-particle duality of quantum particles. de Broglie went even further and connected the ideas of waves and rays through the Hamilton-Jacobi formalism, an approach that Dirac would extend several years later, establishing the formal connection between Hamiltonian physics and wave mechanics. Furthermore, de Broglie conceived of a “pilot wave” interpretation that removed some of Einstein’s discomfort with the random character of quantum measurement that ultimately led Einstein to battle Bohr in their famous debates, culminating in the iconic EPR paper that has become a cornerstone for modern quantum information science. After the wave-like nature of particles was confirmed in the Davisson-Germer experiments, de Broglie received the Nobel Prize in Physics in 1929.
Fig. 4 A standing matter wave is a stationary state of constructive interference. This wavefunction is in the L = 5 quantum manifold of the hydrogen atom.
Louis de Broglie was clearly ahead of his times. His success was partly due to his isolation from the dogma of the day. He was able to think without the constraints of preconceived ideas. But as soon as he became a regular participant in the theoretical discussions of his day, and bowed under the pressure from Copenhagen, his creativity essentially ceased. The subsequent development of quantum mechanics would be dominated by Heisenberg, Born, Pauli, Bohr and Schrödinger, beginning at the 1927 Solvay Congress held in Brussels.
Interference: The History of Optical Interferometry and the Scientists who Tamed Light, is published! It is available now at Oxford University Press and can be pre-ordered at Amazon and Barnes&Nobles to ship on Sept. 6.
The synopses of the first chapters can be found in my previous blog. Here are previews of the final chapters.
Chapter 6. Across the Universe: Exoplanets, Black Holes and Gravitational Waves
Stellar interferometry is opening new vistas of astronomy, exploring the wildest occupants of our universe, from colliding black holes half-way across the universe (LIGO) to images of neighboring black holes (EHT) to exoplanets near Earth that may harbor life.
Image of the supermassive black hole in M87 from Event Horizon Telescope.
Across the Universe: Gravitational Waves, Black Holes and the Search for Exoplanets describes the latest discoveries of interferometry in astronomy including the use of nulling interferometry in the Very Large Telescope Interferometer (VLTI) to detect exoplanets orbiting distant stars. The much larger Event Horizon Telescope (EHT) used long baseline interferometry and closure phase advanced by Roger Jenison to make the first image of a black hole. The Laser Interferometric Gravitational Observatory (LIGO) represented a several-decade-long drive to detect the first gravitational waves first predicted by Albert Einstein a hundred years ago.
Chapter 7. Two Faces of Microscopy: Diffraction and Interference
From the astronomically large dimensions of outer space to the microscopically small dimensions of inner space, optical interference pushes the resolution limits of imaging.
Two Faces of Microscopy: Diffraction and Interference describes the development of microscopic principles starting with Joseph Fraunhofer and the principle of diffraction gratings that was later perfected by Henry Rowland for high-resolution spectroscopy. The company of Carl Zeiss advanced microscope technology after enlisting the help of Ernst Abbe who formed a new theory of image formation based on light interference. These ideas were extended by Fritz Zernike in the development of phase-contrast microscopy. The ultimate resolution of microscopes, defined by Abbe and known as the Abbe resolution limit, turned out not to be a fundamental limit, but was surpassed by super-resolution microscopy using concepts of interference microscopy and structured illumination.
Chapter 8. Holographic Dreams of Princess Leia: Crossing Beams
The coherence of laser light is like a brilliant jewel that sparkles in the darkness, illuminating life, probing science and projecting holograms in virtual worlds.
Ted Maiman
Holographic Dreams of Princess Leia: Crossing Beams presents the history of holography, beginning with the original ideas of Denis Gabor who invented optical holography as a means to improve the resolution of electron microscopes. Holography became mainstream after the demonstrations by Emmett Leith and Juris Upatnieks using lasers that were first demonstrated by Ted Maiman at Hughes Research Lab after suggestions by Charles Townes on the operating principles of the optical maser. Dynamic holography takes place in crystals that exhibit the photorefractive effect that are useful for adaptive interferometry. Holographic display technology is under development, using ideas of holography merged with light-field displays that were first developed by Gabriel Lippmann.
Chapter 9. Photon Interference: The Foundations of Quantum Communication and Computing
What is the image of one photon interfering? Better yet, what is the image of two photons interfering? The answer to this crucial question laid the foundation for quantum communication.
Photon Interference: The Foundations of Quantum Communication moves the story of interferometry into the quantum realm, beginning with the Einstein-Podolski-Rosen paradox and the principle of quantum entanglement that was refined by David Bohm who tried to banish uncertainty from quantum theory. John Bell and John Clauser pushed the limits of what can be known from quantum measurement as Clauser tested Bell’s inequalities, confirming the fundamental nonlocal character of quantum systems. Leonard Mandel pushed quantum interference into the single-photon regime, discovering two-photon interference fringes that illustrated deep concepts of quantum coherence. Quantum communication began with quantum cryptography and developed into quantum teleportation that can provide the data bus of future quantum computers.
Chapter 10. The Quantum Advantage: Interferometric Computing
There is almost no technical advantage better than having exponential resources at hand. The exponential resources of quantum interference provide that advantage to quantum computing which is poised to usher in a new era of quantum information science and technology.
David Deutsch.
The Quantum Advantage: Interferometric Computing describes the development of quantum algorithms and quantum computing beginning with the first quantum algorithm invented by David Deutsch as a side effect of his attempt to prove the multiple world interpretation of quantum theory. Peter Shor found a quantum algorithm that could factor the product of primes and that threatened all secure communications in the world. Once the usefulness of quantum algorithms was recognized, quantum computing hardware ideas developed rapidly into quantum circuits supported by quantum logic gates. The limitation of optical interactions, that hampered the development of controlled quantum gates, led to the proposal of linear optical quantum computing and boson sampling in a complex cascade of single-photon interferometers that has been used to demonstrate quantum supremacy, also known as quantum computational advantage, using photonic integrated circuits.
New from Oxford Press: Interference
A popular account of the trials and toils of the scientists and engineers who tamed light and used it to probe the universe.
Despite the many apparent paradoxes posed in physics—the twin and ladder paradoxes of relativity theory, Olber’s paradox of the bright night sky, Loschmitt’s paradox of irreversible statistical fluctuations—these are resolved by a deeper look at the underlying assumptions—the twin paradox is resolved by considering shifts in reference frames, the ladder paradox is resolved by the loss of simultaneity, Olber’s paradox is resolved by a finite age to the universe, and Loschmitt’s paradox is resolved by fluctuation theorems. In each case, no physical principle is violated, and each paradox is fully explained.
However, there is at least one “true” paradox in physics that defies consistent explanation—quantum entanglement. Quantum entanglement was first described by Einstein with colleagues Podolsky and Rosen in the famous EPR paper of 1935 as an argument against the completeness of quantum mechanics, and it was given its name by Schrödinger the same year in the paper where he introduced his “cat” as a burlesque consequence of entanglement.
Here is a short history of quantum entanglement [1], from its beginnings in 1935 to the recent 2022 Nobel prize in Physics awarded to John Clauser, Alain Aspect and Anton Zeilinger.
The EPR Papers of 1935
Einstein can be considered as the father of quantum mechanics, even over Planck, because of his 1905 derivation of the existence of the photon as a discrete carrier of a quantum of energy (see Einstein versus Planck). Even so, as Heisenberg and Bohr advanced quantum mechanics in the mid 1920’s, emphasizing the underlying non-deterministic outcomes of measurements, and in particular the notion of instantaneous wavefunction collapse, they pushed the theory in directions that Einstein found increasingly disturbing and unacceptable.
This feature is an excerpt from an upcoming book, Interference: The History of Optical Interferometry and the Scientists Who Tamed Light (Oxford University Press, July 2023), by David D. Nolte.
At the invitation-only Solvay Congresses of 1927 and 1930, where all the top physicists met to debate the latest advances, Einstein and Bohr began a running debate that was epic in the history of physics as the two top minds went head-to-head as the onlookers looked on in awe. Ultimately, Einstein was on the losing end. Although he was convinced that something was missing in quantum theory, he could not counter all of Bohr’s rejoinders, even as Einstein’s assaults became ever more sophisticated, and he left the field of battle beaten but not convinced. Several years later he launched his last and ultimate salvo.
Fig. 1 Niels Bohr and Albert Einstein
At the Institute for Advanced Study in Princeton, New Jersey, in the 1930’s Einstein was working with Nathan Rosen and Boris Podolsky when he envisioned a fundamental paradox in quantum theory that occurred when two widely-separated quantum particles were required to share specific physical properties because of simple conservation theorems like energy and momentum. Even Bohr and Heisenberg could not deny the principle of conservation of energy and momentum, and Einstein devised a two-particle system for which these conservation principles led to an apparent violation of Heisenberg’s own uncertainty principle. He left the details to his colleagues, with Podolsky writing up the main arguments. They published the paper in the Physical Review in March of 1935 with the title “Can Quantum-Mechanical Description of Physical Reality be Considered Complete” [2]. Because of the three names on the paper (Einstein, Podolsky, Rosen), it became known as the EPR paper, and the paradox they presented became known as the EPR paradox.
When Bohr read the paper, he was initially stumped and aghast. He felt that EPR had shaken the very foundations of the quantum theory that he and his institute had fought so hard to establish. He also suspected that EPR had made a mistake in their arguments, and he halted all work at his institute in Copenhagen until they could construct a definitive answer. A few months later, Bohr published a paper in the Physical Review in July of 1935, using the identical title that EPR had used, in which he refuted the EPR paradox [3]. There is not a single equation or figure in the paper, but he used his “awful incantation terminology” to maximum effect, showing that one of the EPR assumptions on the assessment of uncertainties to position and momentum was in error, and he was right.
Einstein was disgusted. He had hoped that this ultimate argument against the completeness of quantum mechanics would stand the test of time, but Bohr had shot it down within mere months. Einstein was particularly disappointed with Podolsky, because Podolsky had tried too hard to make the argument specific to position and momentum, leaving a loophole for Bohr to wiggle through, where Einstein had wanted the argument to rest on deeper and more general principles.
Despite Bohr’s victory, Einstein had been correct in his initial formulation of the EPR paradox that showed quantum mechanics did not jibe with common notions of reality. He and Schrödinger exchanged letters commiserating with each other and encouraging each other in their counter beliefs against Bohr and Heisenberg. In November of 1935, Schrödinger published a broad, mostly philosophical, paper in Naturwissenschaften [4] in which he amplified the EPR paradox with the use of an absurd—what he called burlesque—consequence of wavefunction collapse that became known as Schrödinger’s Cat. He also gave the central property of the EPR paradox its name: entanglement.
Ironically, both Einstein’s entanglement paradox and Schrödinger’s Cat, which were formulated originally to be arguments against the validity of quantum theory, have become established quantum tools. Today, entangled particles are the core workhorses of quantum information systems, and physicists are building larger and larger versions of Schrödinger’s Cat that may eventually merge with the physics of the macroscopic world.
Bohm and Ahronov Tackle EPR
The physicist David Bohm was a rare political exile from the United States. He was born in the heart of Pennsylvania in the town of Wilkes-Barre, attended Penn State and then the University of California at Berkeley, where he joined Robert Oppenheimer’s research group. While there, he became deeply involved in the fight for unions and socialism, activities for which he was called before McCarthy’s Committee on Un-American Activities. He invoked his right to the fifth amendment for which he was arrested. Although he was later acquitted, Princeton University fired him from his faculty position, and fearing another arrest, he fled to Brazil where his US passport was confiscated by American authorities. He had become a physicist without a country.
Fig. 2 David Bohm
Despite his personal trials, Bohm remained scientifically productive. He published his influential textbook on quantum mechanics in the midst of his Senate hearings, and after a particularly stimulating discussion with Einstein shortly before he fled the US, he developed and published an alternative version of quantum theory in 1952 that was fully deterministic—removing Einstein’s “God playing dice”—by creating a hidden-variable theory [5].
Hidden-variable theories of quantum mechanics seek to remove the randomness of quantum measurement by assuming that some deeper element of quantum phenomena—a hidden variable—explains each outcome. But it is also assumed that these hidden variables are not directly accessible to experiment. In this sense, the quantum theory of Bohr and Heisenberg was “correct” but not “complete”, because there were things that the theory could not predict or explain.
Bohm’s hidden variable theory, based on a quantum potential, was able to reproduce all the known results of standard quantum theory without invoking the random experimental outcomes that Einstein abhorred. However, it still contained one crucial element that could not sweep away the EPR paradox—it was nonlocal.
Nonlocality lies at the heart of quantum theory. In its simplest form, the nonlocal nature of quantum phenomenon says that quantum states span spacetime with space-like separations, meaning that parts of the wavefunction are non-causally connected to other parts of the wavefunction. Because Einstein was fundamentally committed to causality, the nonlocality of quantum theory was what he found most objectionable, and Bohm’s elegant hidden-variable theory, that removed Einstein’s dreaded randomness, could not remove that last objection of non-causality.
After working in Brazil for several years, Bohm moved to the Technion University in Israel where he began a fruitful collaboration with Yakir Ahronov. In addition to proposing the Ahronov-Bohm effect, in 1957 they reformulated Podolsky’s version of the EPR paradox that relied on continuous values of position and momentum and replaced it with a much simpler model based on the Stern-Gerlach effect on spins and further to the case of positronium decay into two photons with correlated polarizations. Bohm and Ahronov reassessed experimental results of positronium decay that had been made by Madame Wu in 1950 at Columbia University and found it in full agreement with standard quantum theory.
John Bell’s Inequalities
John Stuart Bell had an unusual start for a physicist. His family was too poor to give him an education appropriate to his skills, so he enrolled in vocational school where he took practical classes that included brick laying. Working later as a technician in a university lab, he caught the attention of his professors who sponsored him to attend the university. With a degree in physics, he began working at CERN as an accelerator designer when he again caught the attention of his supervisors who sponsored him to attend graduate school. He graduated with a PhD and returned to CERN as a card-carrying physicist with all the rights and privileges that entailed.
Fig. 3 John Bell
During his university days, he had been fascinated by the EPR paradox, and he continued thinking about the fundamentals of quantum theory. On a sabbatical to the Stanford accelerator in 1960 he began putting mathematics to the EPR paradox to see whether any local hidden variable theory could be compatible with quantum mechanics. His analysis was fully general, so that it could rule out as-yet-unthought-of hidden-variable theories. The result of this work was a set of inequalities that must be obeyed by any local hidden-variable theory. Then he made a simple check using the known results of quantum measurement and showed that his inequalities are violated by quantum systems. This ruled out the possibility of any local hidden variable theory (but not Bohm’s nonlocal hidden-variable theory). Bell published his analysis in 1964 [6] in an obscure journal that almost no one read…except for a curious graduate student at Columbia University who began digging into the fundamental underpinnings of quantum theory against his supervisor’s advice.
Fig. 4 Polarization measurements on entangled photons violate Bell’s inequality.
John Clauser’s Tenacious Pursuit
As a graduate student in astrophysics at Columbia University, John Clauser was supposed to be doing astrophysics. Instead, he spent his time musing over the fundamentals of quantum theory. In 1967 Clauser stumbled across Bell’s paper while he was in the library. The paper caught his imagination, but he also recognized that the inequalities were not experimentally testable, because they required measurements that depended directly on hidden variables, which are not accessible. He began thinking of ways to construct similar inequalities that could be put to an experimental test, and he wrote about his ideas to Bell, who responded with encouragement. Clauser wrote up his ideas in an abstract for an upcoming meeting of the American Physical Society, where one of the abstract reviewers was Abner Shimony of Boston University. Clauser was surprised weeks later when he received a telephone call from Shimony. Shimony and his graduate student Micheal Horne had been thinking along similar lines, and Shimony proposed to Clauser that they join forces. They met in Boston where they were met Richard Holt, a graudate student at Harvard who was working on experimental tests of quantum mechanics. Collectively, they devised a new type of Bell inequality that could be put to experimental test [7]. The result has become known as the CHSH Bell inequality (after Clauser, Horne, Shimony and Holt).
Fig. 5 John Clauser
When Clauser took a post-doc position in Berkeley, he began searching for a way to do the experiments to test the CHSH inequality, even though Holt had a head start at Harvard. Clauser enlisted the help of Charles Townes, who convinced one of the Berkeley faculty to loan Clauser his graduate student, Stuart Freedman, to help. Clauser and Freedman performed the experiments, using a two-photon optical decay of calcium ions and found a violation of the CHSH inequality by 5 standard deviations, publishing their result in 1972 [8].
Fig. 6 CHSH inequality violated by entangled photons.
Alain Aspect’s Non-locality
Just as Clauser’s life was changed when he stumbled on Bell’s obscure paper in 1967, the paper had the same effect on the life of French physicist Alain Aspect who stumbled on it in 1975. Like Clauser, he also sought out Bell for his opinion, meeting with him in Geneva, and Aspect similarly received Bell’s encouragement, this time with the hope to build upon Clauser’s work.
Fig. 7 Alain Aspect
In some respects, the conceptual breakthrough achieved by Clauser had been the CHSH inequality that could be tested experimentally. The subsequent Clauser Freedman experiments were not a conclusion, but were just the beginning, opening the door to deeper tests. For instance, in the Clauser-Freedman experiments, the polarizers were static, and the detectors were not widely separated, which allowed the measurements to be time-like separated in spacetime. Therefore, the fundamental non-local nature of quantum physics had not been tested.
Aspect began a thorough and systematic program, that would take him nearly a decade to complete, to test the CHSH inequality under conditions of non-locality. He began with a much brighter source of photons produced using laser excitation of the calcium ions. This allowed him to perform the experiment in 100’s of seconds instead of the hundreds of hours by Clauser. With such a high data rate, Aspect was able to verify violation of the Bell inequality to 10 standard deviations, published in 1981 [9].
However, the real goal was to change the orientations of the polarizers while the photons were in flight to widely separated detectors [10]. This experiment would allow the detection to be space-like separated in spacetime. The experiments were performed using fast-switching acoustic-optic modulators, and the Bell inequality was violated to 5 standard deviations [11]. This was the most stringent test yet performed and the first to fully demonstrate the non-local nature of quantum physics.
Anton Zeilinger: Master of Entanglement
If there is one physicist today whose work encompasses the broadest range of entangled phenomena, it would be the Austrian physicist, Anton Zeilinger. He began his career in neutron interferometery, but when he was bitten by the entanglement bug in 1976, he switched to quantum photonics because of the superior control that can be exercised using optics over sources and receivers and all the optical manipulations in between.
Fig. 8 Anton Zeilinger
Working with Daniel Greenberger and Micheal Horne, they took the essential next step past the Bohm two-particle entanglement to consider a 3-particle entangled state that had surprising properties. While the violation of locality by the two-particle entanglement was observed through the statistical properties of many measurements, the new 3-particle entanglement could show violations on single measurements, further strengthening the arguments for quantum non-locality. This new state is called the GHZ state (after Greenberger, Horne and Zeilinger) [12].
As the Zeilinger group in Vienna was working towards experimental demonstrations of the GHZ state, Charles Bennett of IBM proposed the possibility for quantum teleportation, using entanglement as a core quantum information resource [13]. Zeilinger realized that his experimental set-up could perform an experimental demonstration of the effect, and in a rapid re-tooling of the experimental apparatus [14], the Zeilinger group was the first to demonstrate quantum teleportation that satisfied the conditions of the Bennett teleportation proposal [15]. An Italian-UK collaboration also made an early demonstration of a related form of teleportation in a paper that was submitted first, but published after Zeilinger’s, due to delays in review [16]. But teleportation was just one of a widening array of quantum applications for entanglement that was pursued by the Zeilinger group over the succeeding 30 years [17], including entanglement swapping, quantum repeaters, and entanglement-based quantum cryptography. Perhaps most striking, he has worked on projects at astronomical observatories that entangle photons coming from cosmic sources.
By David D. Nolte Nov. 26, 2022
Read more about the history of quantum entanglement in Interference (New From Oxford University Press, 2023)
A popular account of the trials and toils of the scientists and engineers who tamed light and used it to probe the universe.
[2] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Physical Review47, 0777-0780 (1935).
[3] N. Bohr, Can quantum-mechanical description of physical reality be considered complete? Physical Review48, 696-702 (1935).
[4] E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften 23, 807-12; 823-28; 844-49 (1935).
[5] D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .1. Physical Review85, 166-179 (1952); D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .2. Physical Review85, 180-193 (1952).
[6] J. Bell, On the Einstein-Podolsky-Rosen paradox. Physics1, 195 (1964).
[7] 1. J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Proposed experiment to test local hidden-variable theories. Physical Review Letters23, 880-& (1969).
[8] S. J. Freedman, J. F. Clauser, Experimental test of local hidden-variable theories. Physical Review Letters28, 938-& (1972).
[9] A. Aspect, P. Grangier, G. Roger, EXPERIMENTAL TESTS OF REALISTIC LOCAL THEORIES VIA BELLS THEOREM. Physical Review Letters47, 460-463 (1981).
[10] Alain Aspect, Bell’s Theorem: The Naïve Veiw of an Experimentalit. (2004), hal- 00001079
[11] A. Aspect, J. Dalibard, G. Roger, EXPERIMENTAL TEST OF BELL INEQUALITIES USING TIME-VARYING ANALYZERS. Physical Review Letters49, 1804-1807 (1982).
[12] D. M. Greenberger, M. A. Horne, A. Zeilinger, in 1988 Fall Workshop on Bells Theorem, Quantum Theory and Conceptions of the Universe. (George Mason Univ, Fairfax, Va, 1988), vol. 37, pp. 69-72.
[13] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical Review Letters70, 1895-1899 (1993).
[14] J. Gea-Banacloche, Optical realizations of quantum teleportation, in Progress in Optics, Vol 46, E. Wolf, Ed. (2004), vol. 46, pp. 311-353.
[15] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Experimental quantum teleportation. Nature390, 575-579 (1997).
[16] D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-podolsky-Rosen Channels. Phys. Rev. Lett.80, 1121-1125 (1998).
[17] A. Zeilinger, Light for the quantum. Entangled photons and their applications: a very personal perspective. Physica Scripta92, 1-33 (2017).
Quantum physics is often called “weird” because it does things that are not allowed in classical physics and hence is viewed as non-intuitive or strange. Perhaps the two “weirdest” aspects of quantum physics are quantum entanglement and quantum tunneling. Entanglement allows a particle state to extend across wide expanses of space, while tunneling allows a particle to have negative kinetic energy. Neither of these effects has a classical analog.
Quantum entanglement arose out of the Bohr-Einstein debates at the Solvay Conferences in the 1920’s and 30’s, and it was the subject of a recent Nobel Prize in Physics (2022). The quantum tunneling story is just as old, but it was recognized much earlier by the Nobel Prize in 1972 when it was awarded to Brian Josephson, Ivar Giaever and Leo Esaki—each of whom was a graduate student when they discovered their respective effects and two of whom got their big idea while attending a lecture class.
Always go to class, you never know what you might miss, and the payoff is sometimes BIG
Ivar Giaever
Of the two effects, tunneling is the more common and the more useful in modern electronic devices (although entanglement is coming up fast with the advent of quantum information science). Here is a short history of quantum tunneling, told through a series of publications that advanced theory and experiments.
Double-Well Potential: Friedrich Hund (1927)
The first analysis of quantum tunneling was performed by Friedrich Hund (1896 – 1997), a German physicist who studied early in his career with Born in Göttingen and Bohr in Copenhagen. He published a series of papers in 1927 in Zeitschrift für Physik [1] that solved the newly-proposed Schrödinger equation for the case of the double well potential. He was particularly interested in the formation of symmetric and anti-symmetric states of the double well that contributed to the binding energy of atoms in molecules. He derived the first tunneling-frequency expression for a quantum superposition of the symmetric and anti-symmetric states
where f is the coherent oscillation frequency, V is the height of the potential and hν is the quantum energy of the isolated states when the atoms are far apart. The exponential dependence on the potential height V made the tunnel effect extremely sensitive to the details of the tunnel barrier.
Fig. 1 Friedrich Hund
Electron Emission: Lothar Nordheim and Ralph Fowler (1927 – 1928)
The first to consider quantum tunneling from a bound state to a continuum state was Lothar Nordheim (1899 – 1985), a German physicist who studied under David Hilbert and Max Born at Göttingen and worked with John von Neumann and Eugene Wigner and later with Hans Bethe. In 1927 he solved the problem of a particle in a well that is separated from continuum states by a thin finite barrier [2]. Using the new Schrödinger theory, he found transmission coefficients that were finite valued, caused by quantum tunneling of the particle through the barrier. Nordheim’s use of square potential wells and barriers are now, literally, textbook examples that every student of quantum mechanics solves. (For a quantum simulation of wavefunction tunneling through a square barrier see the companion Quantum Tunneling YouTube video.) Nordheim later escaped the growing nationalism and anti-semitism in Germany in the mid 1930’s to become a visiting professor of physics at Purdue University in the United States, moving to a permanent position at Duke University.
Fig. 2 Nordheim square tunnel barrier and Fowler-Nordheim triangular tunnel barrier for electron tunneling from bound states into the continuum.
One of the giants of mathematical physics in the UK from the 1920s through the 1930’s was Ralph Fowler (1889 – 1944). Three of his doctoral students went on to win Nobel Prizes (Chandrasekhar, Dirac and Mott) and others came close (Bhabha, Hartree, Lennard-Jones). In 1928 Fowler worked with Nordheim on a more realistic version of Nordheim’s surface electron tunneling that could explain thermionic emission of electrons from metals under strong electric fields. The electric field modified Nordheim’s square potential barrier into a triangular barrier (which they treated using WKB theory) to obtain the tunneling rate [3]. This type of tunnel effect is now known as Fowler-Nordheim tunneling.
Nuclear Alpha Decay: George Gamow (1928)
George Gamov (1904 – 1968) is one of the icons of mid-twentieth-century physics. He was a substantial physicist who also had a solid sense of humor that allowed him to achieve a level of cultural popularity shared by a few of the larger-than-life physicists of his time, like Richard Feynman and Stephen Hawking. His popular books included One Two Three … Infinity as well as a favorite series of books under the rubric of Mr. Tompkins (Mr. Tompkins in Wonderland and Mr. Tompkins Explores the Atom, among others). He also wrote a history of the early years of quantum theory (Thirty Years that Shook Physics).
In 1928 Gamow was in Göttingen (the Mecca of early quantum theory) with Max Born when he realized that the radioactive decay of Uranium by alpha decay might be explained by quantum tunneling. It was known that nucleons were bound together by some unknown force in what would be an effective binding potential, but that charged alpha particles would also feel a strong electrostatic repulsive potential from a nucleus. Gamow combined these two potentials to create a potential landscape that was qualitatively similar to Nordheim’s original system of 1927, but with a potential barrier that was neither square nor triangular (like the Fowler-Nordheim situation).
Fig. 3 George Gamow
Gamow was able to make an accurate approximation that allowed him to express the decay rate in terms of an exponential term
where Zα is the atomic charge of the alpha particle, Z is the nuclear charge of the Uranium decay product and v is the speed of the alpha particle detected in external measurements [4].
The very next day after Gamow submitted his paper, Ronald Gurney and Edward Condon of Princeton University submitted a paper [5] that solved the same problem using virtually the same approach … except missing Gamow’s surprisingly concise analytic expression for the decay rate.
Molecular Tunneling: George Uhlenbeck (1932)
Because tunneling rates depend inversely on the mass of the particle tunneling through the barrier, electrons are more likely to tunnel through potential barriers than atoms. However, hydrogen is a particularly small atom and is therefore the most amenable to experiencing tunneling.
The first example of atom tunneling is associated with hydrogen in the ammonia molecule NH3. The molecule has a pyramidal structure with the Nitrogen hovering above the plane defined by the three hydrogens. However, an equivalent configuration has the Nitrogen hanging below the hydrogen plane. The energies of these two configurations are the same, but the Nitrogen must tunnel from one side of the hydrogen plane to the other through a barrier. The presence of light-weight hydrogen that can “move out of the way” for the nitrogen makes this barrier very small (infrared energies). When the ammonia is excited into its first vibrational excited state, the molecular wavefunction tunnels through the barrier, splitting the excited level by an energy associated with a wavelength of 1.2 cm which is in the microwave. This tunnel splitting was the first microwave transition observed in spectroscopy and is used in ammonia masers.
Fig. 4 Nitrogen inversion in the ammonia molecule is achieved by excitation to a vibrational excited state followed by tunneling through the barrier, proposed by George Uhlenbeck in 1932.
One of the earliest papers [6] written on the tunneling of nitrogen in ammonia was published by George Uhlenbeck in 1932. George Uhlenbeck (1900 – 1988) was a Dutch-American theoretical physicist. He played a critical role, with Samuel Goudsmit, in establishing the spin of the electron in 1925. Both Uhlenbeck and Goudsmit were close associates of Paul Ehrenfest at Leiden in the Netherlands. Uhlenbeck is also famous for the Ornstein-Uhlenbeck process which is a generalization of Einstein’s theory of Brownian motion that can treat active transport such as intracellular transport in living cells.
Solid-State Electron Tunneling: Leo Esaki (1957)
Although the tunneling of electrons in molecular bonds and in the field emission from metals had been established early in the century, direct use of electron tunneling in solid state devices had remained elusive until Leo Esaki (1925 – ) observed electron tunneling in heavily doped Germanium and Silicon semiconductors. Esaki joined an early precursor of Sony electronics in 1956 and was supported to obtain a PhD from the University of Tokyo. In 1957 he was working with heavily-doped p-n junction diodes and discovered a phenomenon known as negative differential resistance where the current through an electronic device actually decreases as the voltage increases.
Because the junction thickness was only about 100 atoms, or about 10 nanometers, he suspected and then proved that the electronic current was tunneling quantum mechanically through the junction. The negative differential resistance was caused by a decrease in available states to the tunneling current as the voltage increased.
Fig. 5 Esaki tunnel diode with heavily doped p- and n-type semiconductors. At small voltages, electrons and holes tunnel through the semiconductor bandgap across a junction that is only about 10 nm wide. Ht higher voltage, the electrons and hole have no accessible states to tunnel into, producing negative differential resistance where the current decreases with increasing voltage.
Esaki tunnel diodes were the fastest semiconductor devices of the time, and the negative differential resistance of the diode in an external circuit produced high-frequency oscillations. They were used in high-frequency communication systems. They were also radiation hard and hence ideal for the early communications satellites. Esaki was awarded the 1973 Nobel Prize in Physics jointly with Ivar Giaever and Brian Josephson.
Superconducting Tunneling: Ivar Giaever (1960)
Ivar Giaever (1929 – ) is a Norwegian-American physicist who had just joined the GE research lab in Schenectady New York in 1958 when he read about Esaki’s tunneling experiments. He was enrolled at that time as a graduate student in physics at Rensselaer Polytechnic Institute (RPI) where he was taking a course in solid state physics and learning about superconductivity. Superconductivity is carried by pairs of electrons known as Cooper pairs that spontaneously bind together with a binding energy that produced an “energy gap” in the electron energies of the metal, but no one had ever found a way to directly measure it. The Esaki experiment made him immediately think of the equivalent experiment in which Cooper pairs might tunnel between two superconductors (through a thin oxide layer) and yield a measurement of the energy gap. The idea actually came to him during the class lecture.
The experiments used a junction between aluminum and lead (Al—Al2O3—Pb). At first, the temperature of the system was adjusted so that Al remained a normal metal and Pb was superconducting, and Giaever observed a tunnel current with a threshold related to the gap in Pb. Then the temperature was lowered so that both Al and Pb were superconducting, and a peak in the tunnel current appeared at the voltage associated with the difference in the energy gaps (predicted by Harrison and Bardeen).
In Giaever’s experiments, the external circuits had been designed to pick up “ordinary” tunnel currents in which individual electrons tunneled through the oxide rather than the Cooper pairs themselves. However, in 1962, Brian Josephson (1940 – ), a physics graduate student at Cambridge, was sitting in a lecture (just like Giaever) on solid state physics given by Phil Anderson (who was on sabbatical there from Bell Labs). During lecture he had the idea to calculate whether it was possible for the Cooper pairs themselves to tunnel through the oxide barrier. Building on theoretical work by Leo Falicov who was at the University of Chicago and later at Berkeley (years later I was lucky to have Leo as my PhD thesis advisor at Berkeley), Josephson found a surprising result that even when the voltage was zero, there would be a supercurrent that tunneled through the junction (now known as the DC Josephson Effect). Furthermore, once a voltage was applied, the supercurrent would oscillate (now known as the AC Josephson Effect). These were strange and non-intuitive results, so he showed Anderson his calculations to see what he thought. By this time Anderson had already been extremely impressed by Josephson (who would often come to the board after one of Anderson’s lectures to show where he had made a mistake). Anderson checked over the theory and agreed with Josephson’s conclusions. Bolstered by this reception, Josephson submitted the theoretical prediction for publication [9].
As soon as Anderson returned to Bell Labs after his sabbatical, he connected with John Rowell who was making tunnel junction experiments, and they revised the external circuit configuration to be most sensitive to the tunneling supercurrent, which they observed in short time and submitted a paper for publication. Since then, the Josephson Effect has become a standard element of ultra-sensitive magnetometers, measurement standards for charge and voltage, far-infrared detectors, and have been used to construct rudimentary qubits and quantum computers.
[5] R. W. Gurney, E. U. Condon, Nature 122, 439 (1928). R. W. Gurney, E. U. Condon, Phys. Rev. 33, 127 (1929).
[6] Dennison, D. M. and G. E. Uhlenbeck. “The two-minima problem and the ammonia molecule.” Physical Review 41(3): 313-321. (1932)
[7] L. Esaki, New Phenomenon in Narrow Germanium Para-Normal-Junctions, Phys. Rev., 109, 603-604 (1958); L. Esaki, (1974). Long journey into tunneling, disintegration, Proc. of the Nature 123, IEEE, 62, 825.
[8] I. Giaever, Energy Gap in Superconductors Measured by Electron Tunneling, Phys. Rev. Letters, 5, 147-148 (1960); I. Giaever, Electron tunneling and superconductivity, Science, 183, 1253 (1974)
[9] B. D. Josephson, Phys. Lett. 1, 251 (1962); B.D. Josephson, The discovery of tunneling supercurrent, Science, 184, 527 (1974).
[10] P. W. Anderson, J. M. Rowell, Phys. Rev. Lett. 10, 230 (1963); Philip W. Anderson, How Josephson discovered his effect, Physics Today 23, 11, 23 (1970)
[11] Eugen Merzbacher, The Early History of Quantum Tunneling, Physics Today 55, 8, 44 (2002)
[12] Razavy, Mohsen. Quantum Theory Of Tunneling, World Scientific Publishing Company, 2003.
Interference (New from Oxford University Press, 2023)
Read the stories of the scientists and engineers who tamed light and used it to probe the universe.
At the dawn of quantum theory, Heisenberg, Schrödinger, Bohr and Pauli were embroiled in a dispute over whether trajectories of particles, defined by their positions over time, could exist. The argument against trajectories was based on an apparent paradox: To draw a “line” depicting a trajectory of a particle along a path implies that there is a momentum vector that carries the particle along that path. But a line is a one-dimensional curve through space, and since at any point in time the particle’s position is perfectly localized, then by Heisenberg’s uncertainty principle, it can have no definable momentum to carry it along.
My previous blog shows the way out of this paradox, by assembling wavepackets that are spread in both space and momentum, explicitly obeying the uncertainty principle. This is nothing new to anyone who has taken a quantum course. But the surprising thing is that in some potentials, like a harmonic potential, the wavepacket travels without broadening, just like classical particles on a trajectory. A dramatic demonstration of this can be seen in this YouTube video. But other potentials “break up” the wavepacket, especially potentials that display classical chaos. Because phase space is one of the best tools for studying classical chaos, especially Hamiltonian chaos, it can be enlisted to dig deeper into the question of the quantum trajectory—not just about the existence of a quantum trajectory, but why quantum systems retain a shadow of their classical counterparts.
Phase Space
Phase space is the state space of Hamiltonian systems. Concepts of phase space were first developed by Boltzmann as he worked on the problem of statistical mechanics. Phase space was later codified by Gibbs for statistical mechanics and by Poincare for orbital mechanics, and it was finally given its name by Paul and Tatiana Ehrenfest (a husband-wife team) in correspondence with the German physicist Paul Hertz (See Chapter 6, “The Tangled Tale of Phase Space”, in Galileo Unbound by D. D. Nolte (Oxford, 2018)).
The stretched-out phase-space functions … are very similar to the stochastic layer that forms in separatrix chaos in classical systems.
The idea of phase space is very simple for classical systems: it is just a plot of the momentum of a particle as a function of its position. For a given initial condition, the trajectory of a particle through its natural configuration space (for instance our 3D world) is traced out as a path through phase space. Because there is one momentum variable per degree of freedom, then the dimensionality of phase space for a particle in 3D is 6D, which is difficult to visualize. But for a one-dimensional dynamical system, like a simple harmonic oscillator (SHO) oscillating in a line, the phase space is just two-dimensional, which is easy to see. The phase-space trajectories of an SHO are simply ellipses, and if the momentum axis is scaled appropriately, the trajectories are circles. The particle trajectory in phase space can be animated just like a trajectory through configuration space as the position and momentum change in time p(x(t)). For the SHO, the point follows the path of a circle going clockwise.
Fig. 1 Phase space of the simple harmonic oscillator. The “orbits” have constant energy.
A more interesting phase space is for the simple pendulum, shown in Fig. 2. There are two types of orbits: open and closed. The closed orbits near the origin are like those of a SHO. The open orbits are when the pendulum is spinning around. The dividing line between the open and closed orbits is called a separatrix. Where the separatrix intersects itself is a saddle point. This saddle point is the most important part of the phase space portrait: it is where chaos emerges when perturbations are added.
Fig. 2 Phase space for a simple pendulum. For small amplitudes the orbits are closed like those of a SHO. For large amplitudes the orbits become open as the pendulum spins about its axis. (Reproduced from Introduction to Modern Dynamics, 2nd Ed., pg. )
One route to classical chaos is through what is known as “separatrix chaos”. It is easy to see why saddle points (also known as hyperbolic points) are the source of chaos: as the system trajectory approaches the saddle, it has two options of which directions to go. Any additional degree of freedom in the system (like a harmonic drive) can make the system go one way on one approach, and the other way on another approach, mixing up the trajectories. An example of the stochastic layer of separatrix chaos is shown in Fig. 3 for a damped driven pendulum. The chaotic behavior that originates at the saddle point extends out along the entire separatrix.
Fig. 3 The stochastic layer of separatrix chaos for a damped driven pendulum. (Reproduced from Introduction to Modern Dynamics, 2nd Ed., pg. )
The main question about whether or not there is a quantum trajectory depends on how quantum packets behave as they approach a saddle point in phase space. Since packets are spread out, it would be reasonable to assume that parts of the packet will go one way, and parts of the packet will go another. But first, one has to ask: Is a phase-space description of quantum systems even possible?
Quantum Phase Space: The Wigner Distribution Function
Phase-space portraits are arguably the most powerful tool in the toolbox of classical dynamics, and one would like to retain its uses for quantum systems. However, there is that pesky paradox about quantum trajectories that cannot admit the existence of one-dimensional curves through such a phase space. Furthermore, there is no direct way of taking a wavefunction and simply “finding” its position or momentum to plot points on such a quantum phase space.
The answer was found in 1932 by Eugene Wigner (1902 – 1905), an Hungarian physicist working at Princeton. He realized that it was impossible to construct a quantum probability distribution in phase space that had positive values everywhere. This is a problem, because negative probabilities have no direct interpretation. But Wigner showed that if one relaxed the requirements a bit, so that expectation values computed over some distribution function (that had positive and negative values) gave correct answers that matched experiments, then this distribution function would “stand in” for an actual probability distribution.
The distribution function that Wigner found is called the Wigner distribution function. Given a wavefunction ψ(x), the Wigner distribution is defined as
Fig. 4 Wigner distribution function in (x, p) phase space.
The Wigner distribution function is the Fourier transform of the convolution of the wavefunction. The pure position dependence of the wavefunction is converted into a spread-out position-momentum function in phase space. For a Gaussian wavefunction ψ(x) with a finite width in space, the W-function in phase space is a two-dimensional Gaussian with finite widths in both space and momentum. In fact, the Δx-Δp product of the W-function is precisely the uncertainty production of the Heisenberg uncertainty relation.
The question of the quantum trajectory from the phase-space perspective becomes whether a Wigner function behaves like a localized “packet” that evolves in phase space in a way analogous to a classical particle, and whether classical chaos is reflected in the behavior of quantum systems.
The Harmonic Oscillator
The quantum harmonic oscillator is a rare and special case among quantum potentials, because the energy spacings between all successive states are all the same. This makes it possible for a Gaussian wavefunction, which is a superposition of the eigenstates of the harmonic oscillator, to propagate through the potential without broadening. To see an example of this, watch the first example in this YouTube video for a Schrödinger cat state in a two-dimensional harmonic potential. For this very special potential, the Wigner distribution behaves just like a (broadened) particle on an orbit in phase space, executing nice circular orbits.
A comparison of the classical phase-space portrait versus the quantum phase-space portrait is shown in Fig. 5. Where the classical particle is a point on an orbit, the quantum particle is spread out, obeying the Δx-Δp Heisenberg product, but following the same orbit as the classical particle.
Fig. 5 Classical versus quantum phase-space portraits for a harmonic oscillator. For a classical particle, the trajectory is a point executing an orbit. For a quantum particle, the trajectory is a Wigner distribution that follows the same orbit as the classical particle.
However, a significant new feature appears in the Wigner representation in phase space when there is a coherent superposition of two states, known as a “cat” state, after Schrödinger’s cat. This new feature has no classical analog. It is the coherent interference pattern that appears at the zero-point of the harmonic oscillator for the Schrödinger cat state. There is no such thing as “classical” coherence, so this feature is absent in classical phase space portraits.
Two examples of Wigner distributions are shown in Fig. 6 for a statistical (incoherent) mixture of packets and a coherent superposition of packets. The quantum coherence signature is present in the coherent case but not the statistical mixture case. The coherence in the Wigner distribution represents “off-diagonal” terms in the density matrix that leads to interference effects in quantum systems. Quantum computing algorithms depend critically on such coherences that tend to decay rapidly in real-world physical systems, known as decoherence, and it is possible to make statements about decoherence by watching the zero-point interference.
Fig. 6 Quantum phase-space portraits of double wave packets. On the left, the wave packets have no coherence, being a statistical mixture. On the right is the case for a coherent superposition, or “cat state” for two wave packets in a one-dimensional harmonic oscillator.
Whereas Gaussian wave packets in the quantum harmonic potential behave nearly like classical systems, and their phase-space portraits are almost identical to the classical phase-space view (except for the quantum coherence), most quantum potentials cause wave packets to disperse. And when saddle points are present in the classical case, then we are back to the question about how quantum packets behave as they approach a saddle point in phase space.
Quantum Pendulum and Separatrix Chaos
One of the simplest anharmonic oscillators is the simple pendulum. In the classical case, the period diverges if the pendulum gets very close to going vertical. A similar thing happens in the quantum case, but because the motion has strong anharmonicity, an initial wave packet tends to spread dramatically as parts of the wavefunction less vertical stretch away from the part of the wave function that is more nearly vertical. Fig. 7 is a snap-shot about a eighth of a period after the wave packet was launched. The packet has already stretched out along the separatrix. A double-cat-state was used, so there is a second packet that has coherent interference with the first. To see a movie of the time evolution of the wave packet and the orbit in quantum phase space, see the YouTube video.
Fig. 7 Wavefunction of a quantum pendulum released near vertical. The phase-space portrait is very similar to the classical case, except that the phase-space distribution is stretched out along the separatrix. The initial state for the phase-space portrait was a cat state.
The simple pendulum does have a saddle point, but it is degenerate because the angle is modulo -2-pi. A simple potential that has a non-degenerate saddle point is a double-well potential.
Quantum Double-Well and Separatrix Chaos
The symmetric double-well potential has a saddle point at the mid-point between the two well minima. A wave packet approaching the saddle will split into to packets that will follow the individual separatrixes that emerge from the saddle point (the unstable manifolds). This effect is seen most dramatically in the middle pane of Fig. 8. For the full video of the quantum phase-space evolution, see this YouTube video. The stretched-out distribution in phase space is highly analogous to the separatrix chaos seen for the classical system.
Fig. 8 Phase-space portraits of the Wigner distribution for a wavepacket in a double-well potential. The packet approaches the central saddle point, where the probability density splits along the unstable manifolds.
Conclusion
A common statement often made about quantum chaos is that quantum systems tend to suppress chaos, only exhibiting chaos for special types of orbits that produce quantum scars. However, from the phase-space perspective, the opposite may be true. The stretched-out Wigner distribution functions, for critical wave packets that interact with a saddle point, are very similar to the stochastic layer that forms in separatrix chaos in classical systems. In this sense, the phase-space description brings out the similarity between classical chaos and quantum chaos.
By David D. Nolte Sept. 25, 2022
YouTube Video
For more on the history of quantum trajectories, see Galileo Unbound from Oxford Press:
Heisenberg’s uncertainty principle is a law of physics – it cannot be violated under any circumstances, no matter how much we may want it to yield or how hard we try to bend it. Heisenberg, as he developed his ideas after his lone epiphany like a monk on the isolated island of Helgoland off the north coast of Germany in 1925, became a bit of a zealot, like a religious convert, convinced that all we can say about reality is a measurement outcome. In his view, there was no independent existence of an electron other than what emerged from a measuring apparatus. Reality, to Heisenberg, was just a list of numbers in a spread sheet—matrix elements. He took this line of reasoning so far that he stated without exception that there could be no such thing as a trajectory in a quantum system. When the great battle commenced between Heisenberg’s matrix mechanics against Schrödinger’s wave mechanics, Heisenberg was relentless, denying any reality to Schrödinger’s wavefunction other than as a calculation tool. He was so strident that even Bohr, who was on Heisenberg’s side in the argument, advised Heisenberg to relent [1]. Eventually a compromise was struck, as Heisenberg’s uncertainty principle allowed Schrödinger’s wave functions to exist within limits—his uncertainty limits.
Disaster in the Poconos
Yet the idea of an actual trajectory of a quantum particle remained a type of heresy within the close quantum circles. Years later in 1948, when a young Richard Feynman took the stage at a conference in the Poconos, he almost sabotaged his career in front of Bohr and Dirac—two of the giants who had invented quantum mechanics—by having the audacity to talk about particle trajectories in spacetime diagrams.
Feynman was making his first presentation of a new approach to quantum mechanics that he had developed based on path integrals. The challenge was that his method relied on space-time graphs in which “unphysical” things were allowed to occur. In fact, unphysical things were required to occur, as part of the sum over many histories of his path integrals. For instance, a key element in the approach was allowing electrons to travel backwards in time as positrons, or a process in which the electron and positron annihilate into a single photon, and then the photon decays back into an electron-positron pair—a process that is not allowed by mass and energy conservation. But this is a possible history that must be added to Feynman’s sum.
It all looked like nonsense to the audience, and the talk quickly derailed. Dirac pestered him with questions that he tried to deflect, but Dirac persisted like a raven. A question was raised about the Pauli exclusion principle, about whether an orbital could have three electrons instead of the required two, and Feynman said that it could—all histories were possible and had to be summed over—an answer that dismayed the audience. Finally, as Feynman was drawing another of his space-time graphs showing electrons as lines, Bohr rose to his feet and asked derisively whether Feynman had forgotten Heisenberg’s uncertainty principle that made it impossible to even talk about an electron trajectory.
It was hopeless. The audience gave up and so did Feynman as the talk just fizzled out. It was a disaster. What had been meant to be Feynman’s crowning achievement and his entry to the highest levels of theoretical physics, had been a terrible embarrassment. He slunk home to Cornell where he sank into one of his depressions. At the close of the Pocono conference, Oppenheimer, the reigning king of physics, former head of the successful Manhattan Project and newly selected to head the prestigious Institute for Advanced Study at Princeton, had been thoroughly disappointed by Feynman.
But what Bohr and Dirac and Oppenheimer had failed to understand was that as long as the duration of unphysical processes was shorter than the energy differences involved, then it was literally obeying Heisenberg’s uncertainty principle. Furthermore, Feynman’s trajectories—what became his famous “Feynman Diagrams”—were meant to be merely cartoons—a shorthand way to keep track of lots of different contributions to a scattering process. The quantum processes certainly took place in space and time, conceptually like a trajectory, but only so far as time durations, and energy differences and locations and momentum changes were all within the bounds of the uncertainty principle. Feynman had invented a bold new tool for quantum field theory, able to supply deep results quickly. But no one at the Poconos could see it.
Fig. 1 The first Feynman diagram.
Coherent States
When Feynman had failed so miserably at the Pocono conference, he had taken the stage after Julian Schwinger, who had dazzled everyone with his perfectly scripted presentation of quantum field theory—the competing theory to Feynman’s. Schwinger emerged the clear winner of the contest. At that time, Roy Glauber (1925 – 2018) was a young physicist just taking his PhD from Schwinger at Harvard, and he later received a post-doc position at Princeton’s Institute for Advanced Study where he became part of a miniature revolution in quantum field theory that revolved around—not Schwinger’s difficult mathematics—but Feynman’s diagrammatic method. So Feynman won in the end. Glauber then went on to Caltech, where he filled in for Feynman’s lectures when Feynman was off in Brazil playing the bongos. Glauber eventually returned to Harvard where he was already thinking about the quantum aspects of photons in 1956 when news of the photon correlations in the Hanbury-Brown Twiss (HBT) experiment were published. Three years later, when the laser was invented, he began developing a theory of photon correlations in laser light that he suspected would be fundamentally different than in natural chaotic light.
Because of his background in quantum field theory, and especially quantum electrodynamics, it was fairly easy to couch the quantum optical properties of coherent light in terms of Dirac’s creation and annihilation operators of the electromagnetic field. Glauber developed a “coherent state” operator that was a minimum uncertainty state of the quantized electromagnetic field, related to the minimum-uncertainty wave functions derived initially by Schrödinger in the late 1920’s. The coherent state represents a laser operating well above the lasing threshold and behaved as “the most classical” wavepacket that can be constructed. Glauber was awarded the Nobel Prize in Physics in 2005 for his work on such “Glauber states” in quantum optics.
Fig. 2 Roy Glauber
Quantum Trajectories
Glauber’s coherent states are built up from the natural modes of a harmonic oscillator. Therefore, it should come as no surprise that these coherent-state wavefunctions in a harmonic potential behave just like classical particles with well-defined trajectories. The quadratic potential matches the quadratic argument of the the Gaussian wavepacket, and the pulses propagate within the potential without broadening, as in Fig. 3, showing a snapshot of two wavepackets propagating in a two-dimensional harmonic potential. This is a somewhat radical situation, because most wavepackets in most potentials (or even in free space) broaden as they propagate. The quadratic potential is a special case that is generally not representative of how quantum systems behave.
Fig. 3 Harmonic potential in 2D and two examples of pairs of pulses propagating without broadening. The wavepackets in the center are oscillating in line, and the wavepackets on the right are orbiting the center of the potential in opposite directions. (Movies of the quantum trajectories can be viewed at Physics Unbound.)
To illustrate this special status for the quadratic potential, the wavepackets can be launched in a potential with a quartic perturbation. The quartic potential is anharmonic—the frequency of oscillation depends on the amplitude of oscillation unlike for the harmonic oscillator, where amplitude and frequency are independent. The quartic potential is integrable, like the harmonic oscillator, and there is no avenue for chaos in the classical analog. Nonetheless, wavepackets broaden as they propagate in the quartic potential, eventually spread out into a ring in the configuration space, as in Fig. 4.
Fig. 4 Potential with a quartic corrections. The initial gaussian pulses spread into a “ring” orbiting the center of the potential.
A potential with integrability has as many conserved quantities to the motion as there are degrees of freedom. Because the quartic potential is integrable, the quantum wavefunction may spread, but it remains highly regular, as in the “ring” that eventually forms over time. However, integrable potentials are the exception rather than the rule. Most potentials lead to nonintegrable motion that opens the door to chaos.
A classic (and classical) potential that exhibits chaos in a two-dimensional configuration space is the famous Henon-Heiles potential. This has a four-dimensional phase space which admits classical chaos. The potential has a three-fold symmetry which is one reason it is non-integral, since a particle must “decide” which way to go when it approaches a saddle point. In the quantum regime, wavepackets face the same decision, leading to a breakup of the wavepacket on top of a general broadening. This allows the wavefunction eventually to distribute across the entire configuration space, as in Fig. 5.
Fig. 5 The Henon-Heiles two-dimensional potential supports Hamiltonian chaos in the classical regime. In the quantum regime, the wavefunction spreads to eventually fill the accessible configuration space (for constant energy).
Youtube Video
Movies of quantum trajectories can be viewed at my Youtube Channel, Physics Unbound. The answer to the question “Is there a quantum trajectory?” can be seen visually as the movies run—they do exist in a very clear sense under special conditions, especially coherent states in a harmonic oscillator. And the concept of a quantum trajectory also carries over from a classical trajectory in cases when the classical motion is integrable, even in cases when the wavefunction spreads over time. However, for classical systems that display chaotic motion, wavefunctions that begin as coherent states break up into chaotic wavefunctions that fill the accessible configuration space for a given energy. The character of quantum evolution of coherent states—the most classical of quantum wavefunctions—in these cases reflects the underlying character of chaotic motion in the classical analogs. This process can be seen directly watching the movies as a wavepacket approaches a saddle point in the potential and is split. Successive splits of the multiple wavepackets as they interact with the saddle points is what eventually distributes the full wavefunction into its chaotic form.
Therefore, the idea of a “quantum trajectory”, so thoroughly dismissed by Heisenberg, remains a phenomenological guide that can help give insight into the behavior of quantum systems—both integrable and chaotic.
As a side note, the laws of quantum physics obey time-reversal symmetry just as the classical equations do. In the third movie of “A Quantum Ballet“, wavefunctions in a double-well potential are tracked in time as they start from coherent states that break up into chaotic wavefunctions. It is like watching entropy in action as an ordered state devolves into a disordered state. But at the half-way point of the movie, the imaginary part of the wavefunction has its sign flipped, and the dynamics continue. But now the wavefunctions move from disorder into an ordered state, seemingly going against the second law of thermodynamics. Flipping the sign of the imaginary part of the wavefunction at just one instant in time plays the role of a time-reversal operation, and there is no violation of the second law.
Alice’s disturbing adventures in Wonderland tumbled upon her like a string of accidents as she wandered a world of chaos. Rules were never what they seemed and shifted whenever they wanted. She even met a cat who grinned ear-to-ear and could disappear entirely, or almost entirely, leaving only its grin.
The vanishing Cheshire Cat reminds us of another famous cat—Arnold’s Cat—that introduced the ideas of stretching and folding of phase-space volumes in non-integrable Hamiltonian systems. But when Arnold’s Cat becomes a Quantum Cat, a central question remains: What happens to the chaotic behavior of the classical system … does it survive the transition to quantum mechanics? The answer is surprisingly like the grin of the Cheshire Cat—the cat vanishes, but the grin remains. In the quantum world of the Cheshire Cat, the grin of the classical cat remains even after the rest of the cat vanished.
The Cheshire Cat fades away, leaving only its grin, like a fine filament, as classical chaos fades into quantum, leaving behind a quantum scar.
The Quantum Mechanics of Classically Chaotic Systems
The simplest Hamiltonian systems are integrable—they have as many constants of the motion as degrees of freedom. This holds for quantum systems as well as for classical. There is also a strong correspondence between classical and quantum systems for the integrable cases—literally the Correspondence Principle—that states that quantum systems at high quantum number approach classical behavior. Even at low quantum numbers, classical resonances are mirrored by quantum eigenfrequencies that can show highly regular spectra.
But integrable systems are rare—surprisingly rare. Almost no real-world Hamiltonian system is integrable, because the real world warps the ideal. No spring can displace indefinitely, and no potential is perfectly quadratic. There are always real-world non-idealities that destroy one constant of the motion or another, opening the door to chaos.
When classical Hamiltonian systems become chaotic, they don’t do it suddenly. Almost all transitions to chaos in Hamiltonian systems are gradual. One of the best examples of this is the KAM theory that starts with invariant action integrals that generate invariant tori in phase space. As nonintegrable perturbations increase, the tori break up slowly into island chains of stability as chaos infiltrates the separatrixes—first as thin filaments of chaos surrounding the islands—then growing in width to take up more and more of phase space. Even when chaos is fully developed, small islands of stability can remain—the remnants of stable orbits of the unperturbed system.
When the classical becomes quantum, chaos softens. Quantum wave functions don’t like to be confined—they spread and they tunnel. The separatrix of classical chaos—that barrier between regions of phase space—cannot constrain the exponential tails of wave functions. And the origin of chaos itself—the homoclinic point of the separatrix—gets washed out. Then the regular orbits of the classical system reassert themselves, and they appear, like the vestige of the Cheshire Cat, as a grin.
The Quantum Circus
The empty stadium is a surprisingly rich dynamical system that has unexpected structure in both the classical and the quantum domain. Its importance in classical dynamics comes from the fact that its periodic orbits are unstable and its non-periodic orbits are ergodic (filling all available space if given long enough). The stadium itself is empty so that particles (classical or quantum) are free to propagate between reflections from the perfectly-reflecting walls of the stadium. The ergodicity comes from the fact that the stadium—like a classic Roman chariot-race stadium, also known as a circus—is not a circle, but has a straight stretch between two half circles. This simple modification takes the stable orbits of the circle into the unstable orbits of the stadium.
A single classical orbit in a stadium is shown in Fig 1. This is an ergodic orbit that is non-periodic and eventually would fill the entire stadium space. There are other orbits that are nearly periodic, such as one that bounces back and forth vertically between the linear portions, but even this orbit will eventually wander into the circular part of the stadium and then become ergodic. The big quantum-classical question is what happens to these classical orbits when the stadium is shrunk to the nanoscale?
Fig. 1 A classical trajectory in a stadium. It will eventually visit every point, a property known as ergodicity.
Simulating an evolving quantum wavefunction in free space is surprisingly simple. Given a beginning quantum wavefunction A(x,y,t0), the discrete update equation is
Perfect reflection from the boundaries of the stadium are incorporated through imposing a boundary condition that sends the wavefunction to zero. Simple!
A snap-shot of a wavefunction evolving in the stadium is shown in Fig. 2. To see a movie of the time evolution, see my YouTube episode.
Fig. 2 Snapshot of a quantum wavefunction in the stadium. (From YouTube)
The time average of the wavefunction after a long time has passed is shown in Fig. 3. Other than the horizontal nodal line down the center of the stadium, there is little discernible structure or symmetry. This is also true for the mean squared wavefunction shown in Fig. 4, although there is some structure that may be emerging in the semi-circular regions.
Fig. 3 Time-average wavefunction after a long time.
Fig. 4 Time-average of the squared wavefunction after a long time.
On the other hand, for special initial conditions that have a lot of symmetry, something remarkable happens. Fig. 5 shows several mean-squared results for special initial conditions. There is definite structure in these cases that were given the somewhat ugly name “quantum scars” in the 1980’s by Eric Heller who was one of the first to study this phenomenon [1].
Quantum ScarsQuantum ScarsQuantum Scars
Fig. 5 Quantum scars reflect periodic (but unstable) orbits of the classical system. Quantum effects tend to quench chaos and favor regular motion.
One can superpose highly-symmetric classical trajectories onto the figures, as shown in the bottom row. All of these classical orbits go through a high-symmetry point, such as the center of the stadium (on the left image) and through the focal point of the circular mirrors (in the other two images). The astonishing conclusion of this exercise is that the highly-symmetric periodic classical orbits remain behind as quantum scars—like the Cheshire Cat’s grin—when the system is in the quantum realm. The classical orbits that produce quantum scars have the important property of being periodic but unstable. A slight perturbation from the symmetric trajectory causes it to eventually become ergodic (chaotic). These scars are regions with enhanced probability density, what might be termed “quantum trajectories”, but do not show strong interference patterns.
It is important to make the distinction that it is also possible to construct special wavefunctions that are strictly periodic, such as a wave bouncing perfectly vertically between the straight portions. This leads to large-scale interference patterns that are not the same as the quantum scars.
Quantum Chaos versus Laser Speckle
In addition to the bouncing-wave cases that do not strictly produce quantum scars, there is another “neutral” phenomenon that produces interference patterns that look a lot like scars, but are simply the random addition of lots of plane waves with the same wavelength [2]. A snapshot in time of one of these superpositions is shown in Fig. 6. To see how the waves add together, see the YouTube channel episode.
Fig. 6 The sum of 100 randomly oriented plane waves of constant wavelength. (A snapshot from YouTube.)
By David D. Nolte, Aug. 14, 2022
YouTube Video
Read more about the history of chaos theory in Galileo Unbound from Oxford Press:
In one interpretation of quantum physics, when you snap your fingers, the trajectory you are riding through reality fragments into a cascade of alternative universes—one for each possible quantum outcome among all the different quantum states composing the molecules of your fingers.
This is the Many-Worlds Interpretation (MWI) of quantum physics first proposed rigorously by Hugh Everett in his doctoral thesis in 1957 under the supervision of John Wheeler at Princeton University. Everett had been drawn to this interpretation when he found inconsistencies between quantum physics and gravitation—topics which were supposed to have been his actual thesis topic. But his side-trip into quantum philosophy turned out to be a one-way trip. The reception of his theory was so hostile, no less than from Copenhagen and Bohr himself, that Everett left physics and spent a career at the Pentagon.
Resurrecting MWI in the Name of Quantum Information
Fast forward by 20 years, after Wheeler had left Princeton for the University of Texas at Austin, and once again a young physicist was struggling to reconcile quantum physics with gravity. Once again the many worlds interpretation of quantum physics seemed the only sane way out of the dilemma, and once again a side-trip became a life-long obsession.
David Deutsch, visiting Wheeler in the early 1980’s, became convinced that the many worlds interpretation of quantum physics held the key to paradoxes in the theory of quantum information (For the full story of Wheeler, Everett and Deutsch, see Ref [1]). He was so convinced, that he began a quest to find a physical system that operated on more information than could be present in one universe at a time. If such a physical system existed, it would be because streams of information from more than one universe were coming together and combining in a way that allowed one of the universes to “borrow” the information from the other.
It took only a year or two before Deutsch found what he was looking for—a simple quantum algorithm that yielded twice as much information as would be possible if there were no parallel universes. This is the now-famous Deutsch algorithm—the first quantum algorithm [2]. At the heart of the Deutsch algorithm is a simple quantum interference. The algorithm did nothing useful—but it convinced Deutsch that two universes were interfering coherently in the measurement process, giving that extra bit of information that should not have been there otherwise. A few years later, the Deutsch-Josza algorithm [2] expanded the argument to interfere an exponentially larger amount of information streams from an exponentially larger number of universes to create a result that was exponentially larger than any classical computer could produce. This marked the beginning of the quest for the quantum computer that is running red-hot today.
Deutsch’s “proof” of the many-worlds interpretation of quantum mechanics is not a mathematical proof but is rather a philosophical proof. It holds no sway over how physicists do the math to make their predictions. The Copenhagen interpretation, with its “spooky” instantaneous wavefunction collapse, works just fine predicting the outcome of quantum algorithms and the exponential quantum advantage of quantum computing. Therefore, the story of David Deutsch and the MWI may seem like a chimera—except for one fact—it inspired him to generate the first quantum algorithm that launched what may be the next revolution in the information revolution of modern society. Inspiration is important in science, because it lets scientists create things that had been impossible before.
But if quantum interference is the heart of quantum computing, then there is one physical system that has the ultimate simplicity that may yet inspire future generations of physicists to invent future impossible things—the quantum beam splitter. Nothing in the study of quantum interference can be simpler than a sliver of dielectric material sending single photons one way or another. Yet the outcome of this simple system challenges the mind and reminds us of why Everett and Deutsch embraced the MWI in the first place.
The Classical Beam Splitter
The so-called “beam splitter” is actually a misnomer. Its name implies that it takes a light beam and splits it into two, as if there is only one input. But every “beam splitter” has two inputs, which is clear by looking at the classical 50/50 beam splitter. The actual action of the optical element is the combination of beams into superpositions in each of the outputs. It is only when one of the input fields is zero, a special case, that the optical element acts as a beam splitter. In general, it is a beam combiner.
Given two input fields, the output fields are superpositions of the inputs
The square-root of two factor ensures that energy is conserved, because optical fluence is the square of the fields. This relation is expressed more succinctly as a matrix input-output relation
The phase factors in these equations ensure that the matrix is unitary
reflecting energy conservation.
The Quantum Beam Splitter
A quantum beam splitter is just a classical beam splitter operating at the level of individual photons. Rather than describing single photons entering or leaving the beam splitter, it is more practical to describe the properties of the fields through single-photon quantum operators
where the unitary matrix is the same as the classical case, but with fields replaced by the famous “a” operators. The photon operators operate on single photon modes. For instance, the two one-photon input cases are
where the creation operators operate on the vacuum state in each of the input modes.
The fundamental combinational properties of the beam splitter are even more evident in the quantum case, because there is no such thing as a single input to a quantum beam splitter. Even if no photons are directed into one of the input ports, that port still receives a “vacuum” input, and this vacuum input contributes to the fluctuations observed in the outputs.
The input-output relations for the quantum beam splitter are
The beam splitter operating on a one-photon input converts the input-mode creation operator into a superposition of out-mode creation operators that generates
The resulting output is entangled: either the single photon exits one port, or it exits the other. In the many worlds interpretation, the photon exits from one port in one universe, and it exits from the other port in a different universe. On the other hand, in the Copenhagen interpretation, the two output ports of the beam splitter are perfectly anti-correlated.
Fig. 1 Quantum Operations of a Beam Splitter. A beam splitter creates a quantum superposition of the input modes. The a-symbols are quantum number operators that create and annihilate photons. A single-photon input produces an entangled output that is a quantum superposition of the photon coming out of one output or the other.
The Hong-Ou-Mandel (HOM) Interferometer
When more than one photon is incident on a beam splitter, the fascinating effects of quantum interference come into play, creating unexpected outputs for simple inputs. For instance, the simplest example is a two photon input where a single photon is present in each input port of the beam splitter. The input state is represented with single creation operators operating on each vacuum state of each input port
creating a single photon in each of the input ports. The beam splitter operates on this input state by converting the input-mode creation operators into out-put mode creation operators to give
The important step in this process is the middle line of the equations: There is perfect destructive interference between the two single-photon operations. Therefore, both photons always exit the beam splitter from the same port—never split. Furthermore, the output is an entangled two-photon state, once more splitting universes.
Fig. 2 The HOM interferometer. A two-photon input on a beam splitter generates an entangled superposition of the two photons exiting the beam splitter always together.
The two-photon interference experiment was performed in 1987 by Chung Ki Hong and Jeff Ou, students of Leonard Mandel at the Optics Institute at the University of Rochester [4], and this two-photon operation of the beam splitter is now called the HOM interferometer. The HOM interferometer has become a center-piece for optical and photonic implementations of quantum information processing and quantum computers.
N-Photons on a Beam Splitter
Of course, any number of photons can be input into a beam splitter. For example, take the N-photon input state
The beam splitter acting on this state produces
The quantity on the right hand side can be re-expressed using the binomial theorem
where the permutations are defined by the binomial coefficient
The output state is given by
which is a “super” entangled state composed of N multi-photon states, involving N different universes.
Coherent States on a Quantum Beam Splitter
Surprisingly, there is a multi-photon input state that generates a non-entangled output—as if the input states were simply classical fields. These are the so-called coherent states, introduced by Glauber and Sudarshan [5, 6]. Coherent states can be described as superpositions of multi-photon states, but when a beam splitter operates on these superpositions, the outputs are simply 50/50 mixtures of the states. For instance, if the input scoherent tates are denoted by α and β, then the output states after the beam splitter are
This output is factorized and hence is NOT entangled. This is one of the many reasons why coherent states in quantum optics are considered the “most classical” of quantum states. In this case, a quantum beam splitter operates on the inputs just as if they were classical fields.
By David D. Nolte, May 8, 2022
Read more in “Interference” (New from Oxford University Press, 2023)
A popular account of the trials and toils of the scientists and engineers who tamed light and used it to probe the universe.
[2] D. Deutsch, “Quantum-theory, the church-turing principle and the universal quantum computer,” Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 400, no. 1818, pp. 97-117, (1985)
[3] D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation,” Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 439, no. 1907, pp. 553-558, Dec (1992)
[4] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Physical Review Letters, vol. 59, no. 18, pp. 2044-2046, Nov (1987)
[5] Glauber, R. J. (1963). “Photon Correlations.” Physical Review Letters 10(3): 84.
[6] Sudarshan, E. C. G. (1963). “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams.” Physical Review Letters 10(7): 277-&.; Mehta, C. L. and E. C. Sudarshan (1965). “Relation between quantum and semiclassical description of optical coherence.” Physical Review 138(1B): B274.
Now is exactly the wrong moment to be reviewing the state of photonic quantum computing — the field is moving so rapidly, at just this moment, that everything I say here now will probably be out of date in just a few years. On the other hand, now is exactly the right time to be doing this review, because so much has happened in just the past few years, that it is important to take a moment and look at where this field is today and where it will be going.
At the 20-year anniversary of the publication of my book Mind at Light Speed (Free Press, 2001), this blog is the third in a series reviewing progress in three generations of Machines of Light over the past 20 years (see my previous blogs on the future of the photonic internet and on all-optical computers). This third and final update reviews progress on the third generation of the Machines of Light: the Quantum Optical Generation. Of the three generations, this is the one that is changing the fastest.
Quantum computing is almost here … and it will be at room temperature, using light, in photonic integrated circuits!
Quantum Computing with Linear Optics
Twenty years ago in 2001, Emanuel Knill and Raymond LaFlamme at Los Alamos National Lab, with Gerald Mulburn at the University of Queensland, Australia, published a revolutionary theoretical paper (known as KLM) in Nature on quantum computing with linear optics: “A scheme for efficient quantum computation with linear optics” [1]. Up until that time, it was believed that a quantum computer — if it was going to have the property of a universal Turing machine — needed to have at least some nonlinear interactions among qubits in a quantum gate. For instance, an example of a two-qubit gate is a controlled-NOT, or CNOT, gate shown in Fig. 1 with the Truth Table and the equivalent unitary matrix. It clear that one qubit is controlling the other, telling it what to do.
The quantum CNOT gate gets interesting when the control line has a quantum superposition, then the two outputs become entangled.
Entanglement is a strange process that is unique to quantum systems and has no classical analog. It also has no simple intuitive explanation. By any normal logic, if the control line passes through the gate unaltered, then absolutely nothing interesting should be happening on the Control-Out line. But that’s not the case. The control line going in was a separate state. If some measurement were made on it, either a 1 or 0 would be seen with equal probability. But coming out of the CNOT, the signal has somehow become perfectly correlated with whatever value is on the Signal-Out line. If the Signal-Out is measured, the measurement process collapses the state of the Control-Out to a value equal to the measured signal. The outcome of the control line becomes 100% certain even though nothing was ever done to it! This entanglement generation is one reason the CNOT is often the gate of choice when constructing quantum circuits to perform interesting quantum algorithms.
However, optical implementation of a CNOT is a problem, because light beams and photons really do not like to interact with each other. This is the problem with all-optical classical computers too (see my previous blog). There are ways of getting light to interact with light, for instance inside nonlinear optical materials. And in the case of quantum optics, a single atom in an optical cavity can interact with single photons in ways that can act like a CNOT or related gates. But the efficiencies are very low and the costs to implement it are very high, making it difficult or impossible to scale such systems up into whole networks needed to make a universal quantum computer.
Therefore, when KLM published their idea for quantum computing with linear optics, it caused a shift in the way people were thinking about optical quantum computing. A universal optical quantum computer could be built using just light sources, beam splitters and photon detectors.
The way that KLM gets around the need for a direct nonlinear interaction between two photons is to use postselection. They run a set of photons — signal photons and ancilla (test) photons — through their linear optical system and they detect (i.e., theoretically…the paper is purely a theoretical proposal) the ancilla photons. If these photons are not detected where they are wanted, then that iteration of the computation is thrown out, and it is tried again and again, until the photons end up where they need to be. When the ancilla outcomes are finally what they need to be, this run is selected because the signal state are known to have undergone a known transformation. The signal photons are still unmeasured at this point and are therefore in quantum superpositions that are useful for quantum computation. Postselection uses entanglement and measurement collapse to put the signal photons into desired quantum states. Postselection provides an effective nonlinearity that is induced by the wavefunction collapse of the entangled state. Of course, the down side of this approach is that many iterations are thrown out — the computation becomes non-deterministic.
KLM could get around most of the non-determinism by using more and more ancilla photons, but this has the cost of blowing up the size and cost of the implementation, so their scheme was not imminently practical. But the important point was that it introduced the idea of linear quantum computing. (For this, Milburn and his collaborators have my vote for a future Nobel Prize.) Once that idea was out, others refined it, and improved upon it, and found clever ways to make it more efficient and more scalable. Many of these ideas relied on a technology that was co-evolving with quantum computing — photonic integrated circuits (PICs).
Quantum Photonic Integrated Circuits (QPICs)
Never underestimate the power of silicon. The amount of time and energy and resources that have now been invested in silicon device fabrication is so astronomical that almost nothing in this world can displace it as the dominant technology of the present day and the future. Therefore, when a photon can do something better than an electron, you can guess that eventually that photon will be encased in a silicon chip–on a photonic integrated circuit (PIC).
The dream of integrated optics (the optical analog of integrated electronics) has been around for decades, where waveguides take the place of conducting wires, and interferometers take the place of transistors — all miniaturized and fabricated in the thousands on silicon wafers. The advantages of PICs are obvious, but it has taken a long time to develop. When I was a post-doc at Bell Labs in the late 1980’s, everyone was talking about PICs, but they had terrible fabrication challenges and terrible attenuation losses. Fortunately, these are just technical problems, not limited by any fundamental laws of physics, so time (and an army of researchers) has chipped away at them.
One of the driving forces behind the maturation of PIC technology is photonic fiber optic communications (as discussed in a previous blog). Photons are clear winners when it comes to long-distance communications. In that sense, photonic information technology is a close cousin to silicon — photons are no less likely to be replaced by a future technology than silicon is. Therefore, it made sense to bring the photons onto the silicon chips, tapping into the full array of silicon fab resources so that there could be seamless integration between fiber optics doing the communications and the photonic chips directing the information. Admittedly, photonic chips are not yet all-optical. They still use electronics to control the optical devices on the chip, but this niche for photonics has provided a driving force for advancements in PIC fabrication.
Fig. 2 Schematic of a silicon photonic integrated circuit (PIC). The waveguides can be silica or nitride deposited on the silicon chip. From the Comsol WebSite.
One side-effect of improved PIC fabrication is low light losses. In telecommunications, this loss is not so critical because the systems use OEO regeneration. But less loss is always good, and the PICs can now safeguard almost every photon that comes on chip — exactly what is needed for a quantum PIC. In a quantum photonic circuit, every photon is valuable and informative and needs to be protected. The new PIC fabrication can do this. In addition, light switches for telecom applications are built from integrated interferometers on the chip. It turns out that interferometers at the single-photon level are unitary quantum gates that can be used to build universal photonic quantum computers. So the same technology and control that was used for telecom is just what is needed for photonic quantum computers. In addition, integrated optical cavities on the PICs, which look just like wavelength filters when used for classical optics, are perfect for producing quantum states of light known as squeezed light that turn out to be valuable for certain specialty types of quantum computing.
Therefore, as the concepts of linear optical quantum computing advanced through that last 20 years, the hardware to implement those concepts also advanced, driven by a highly lucrative market segment that provided the resources to tap into the vast miniaturization capabilities of silicon chip fabrication. Very fortuitous!
Room-Temperature Quantum Computers
There are many radically different ways to make a quantum computer. Some are built of superconducting circuits, others are made from semiconductors, or arrays of trapped ions, or nuclear spins on nuclei on atoms in molecules, and of course with photons. Up until about 5 years ago, optical quantum computers seemed like long shots. Perhaps the most advanced technology was the superconducting approach. Superconducting quantum interference devices (SQUIDS) have exquisite sensitivity that makes them robust quantum information devices. But the drawback is the cold temperatures that are needed for them to work. Many of the other approaches likewise need cold temperature–sometimes astronomically cold temperatures that are only a few thousandths of a degree above absolute zero Kelvin.
Cold temperatures and quantum computing seemed a foregone conclusion — you weren’t ever going to separate them — and for good reason. The single greatest threat to quantum information is decoherence — the draining away of the kind of quantum coherence that allows interferences and quantum algorithms to work. In this way, entanglement is a two-edged sword. On the one hand, entanglement provides one of the essential resources for the exponential speed-up of quantum algorithms. But on the other hand, if a qubit “sees” any environmental disturbance, then it becomes entangled with that environment. The entangling of quantum information with the environment causes the coherence to drain away — hence decoherence. Hot environments disturb quantum systems much more than cold environments, so there is a premium for cooling the environment of quantum computers to as low a temperature as they can. Even so, decoherence times can be microseconds to milliseconds under even the best conditions — quantum information dissipates almost as fast as you can make it.
Enter the photon! The bottom line is that photons don’t interact. They are blind to their environment. This is what makes them perfect information carriers down fiber optics. It is also what makes them such good qubits for carrying quantum information. You can prepare a photon in a quantum superposition just by sending it through a lossless polarizing crystal, and then the superposition will last for as long as you can let the photon travel (at the speed of light). Sometimes this means putting the photon into a coil of fiber many kilometers long to store it, but that is OK — a kilometer of coiled fiber in the lab is no bigger than a few tens of centimeters. So the same properties that make photons excellent at carrying information also gives them very small decoherence. And after the KLM schemes began to be developed, the non-interacting properties of photons were no longer a handicap.
In the past 5 years there has been an explosion, as well as an implosion, of quantum photonic computing advances. The implosion is the level of integration which puts more and more optical elements into smaller and smaller footprints on silicon PICs. The explosion is the number of first-of-a-kind demonstrations: the first universal optical quantum computer [2], the first programmable photonic quantum computer [3], and the first (true) quantum computational advantage [4].
All of these “firsts” operate at room temperature. (There is a slight caveat: The photon-number detectors are actually superconducting wire detectors that do need to be cooled. But these can be housed off-chip and off-rack in a separate cooled system that is coupled to the quantum computer by — no surprise — fiber optics.) These are the advantages of photonic quantum computers: hundreds of qubits integrated onto chips, room-temperature operation, long decoherence times, compatibility with telecom light sources and PICs, compatibility with silicon chip fabrication, universal gates using postselection, and more. Despite the head start of some of the other quantum computing systems, photonics looks like it will be overtaking the others within only a few years to become the dominant technology for the future of quantum computing. And part of that future is being helped along by a new kind of quantum algorithm that is perfectly suited to optics.
Fig. 3 Superconducting photon counting detector. From WebSite
A New Kind of Quantum Algorithm: Boson Sampling
In 2011, Scott Aaronson (then at at MIT) published a landmark paper titled “The Computational Complexity of Linear Optics” with his post-doc, Anton Arkhipov [5]. The authors speculated on whether there could be an application of linear optics, not requiring the costly step of post-selection, that was still useful for applications, while simultaneously demonstrating quantum computational advantage. In other words, could one find a linear optical system working with photons that could solve problems intractable to a classical computer? To their own amazement, they did! The answer was something they called “boson sampling”.
To get an idea of what boson sampling is, and why it is very hard to do on a classical computer, think of the classic demonstration of the normal probability distribution found at almost every science museum you visit, illustrated in Fig. 2. A large number of ping-pong balls are dropped one at a time through a forest of regularly-spaced posts, bouncing randomly this way and that until they are collected into bins at the bottom. Bins near the center collect many balls, while bins farther to the side have fewer. If there are many balls, then the stacked heights of the balls in the bins map out a Gaussian probability distribution. The path of a single ping-pong ball represents a series of “decisions” as it hits each post and goes left or right, and the number of permutations of all the possible decisions among all the other ping-pong balls grows exponentially—a hard problem to tackle on a classical computer.
Fig. 4 Ping-pont ball normal distribution. Watch the YouTube video.
In the paper, Aaronson considered a quantum analog to the ping-pong problem in which the ping-pong balls are replaced by photons, and the posts are replaced by beam splitters. As its simplest possible implementation, it could have two photon channels incident on a single beam splitter. The well-known result in this case is the “HOM dip” [6] which is a consequence of the boson statistics of the photon. Now scale this system up to many channels and a cascade of beam splitters, and one has an N-channel multi-photon HOM cascade. The output of this photonic “circuit” is a sampling of the vast number of permutations allowed by bose statistics—boson sampling.
To make the problem more interesting, Aaronson allowed the photons to be launched from any channel at the top (as opposed to dropping all the ping-pong balls at the same spot), and they allowed each beam splitter to have adjustable phases (photons and phases are the key elements of an interferometer). By adjusting the locations of the photon channels and the phases of the beam splitters, it would be possible to “program” this boson cascade to mimic interesting quantum systems or even to solve specific problems, although they were not thinking that far ahead. The main point of the paper was the proposal that implementing boson sampling in a photonic circuit used resources that scaled linearly in the number of photon channels, while the problems that could be solved grew exponentially—a clear quantum computational advantage [4].
On the other hand, it turned out that boson sampling is not universal—one cannot construct a universal quantum computer out of boson sampling. The first proposal was a specialty algorithm whose main function was to demonstrate quantum computational advantage rather than do something specifically useful—just like Deutsch’s first algorithm. But just like Deutsch’s algorithm, which led ultimately to Shor’s very useful prime factoring algorithm, boson sampling turned out to be the start of a new wave of quantum applications.
Shortly after the publication of Aaronson’s and Arkhipov’s paper in 2011, there was a flurry of experimental papers demonstrating boson sampling in the laboratory [7, 8]. And it was discovered that boson sampling could solve important and useful problems, such as the energy levels of quantum systems, and network similarity, as well as quantum random-walk problems. Therefore, even though boson sampling is not strictly universal, it solves a broad class of problems. It can be viewed more like a specialty chip than a universal computer, like the now-ubiquitous GPU’s are specialty chips in virtually every desktop and laptop computer today. And the room-temperature operation significantly reduces cost, so you don’t need a whole government agency to afford one. Just like CPU costs followed Moore’s Law to the point where a Raspberry Pi computer costs $40 today, the photonic chips may get onto their own Moore’s Law that will reduce costs over the next several decades until they are common (but still specialty and probably not cheap) computers in academia and industry. A first step along that path was a recently-demonstrated general programmable room-temperature photonic quantum computer.
Fig. 5 A classical Galton board on the left, and a photon-based boson sampling on the right. From the Walmsley (Oxford) WebSite.
A Programmable Photonic Quantum Computer: Xanadu’s X8 Chip
I don’t usually talk about specific companies, but the new photonic quantum computer chip from Xanadu, based in Toronto, Canada, feels to me like the start of something big. In the March 4, 2021 issue of Nature magazine, researchers at the company published the experimental results of their X8 photonic chip [3]. The chip uses boson sampling of strongly non-classical light. This was the first generally programmable photonic quantum computing chip, programmed using a quantum programming language they developed called Strawberry Fields. By simply changing the quantum code (using a simple conventional computer interface), they switched the computer output among three different quantum applications: transitions among states (spectra of molecular states), quantum docking, and similarity between graphs that represent two different molecules. These are radically different physics and math problems, yet the single chip can be programmed on the fly to solve each one.
The chip is constructed of nitride waveguides on silicon, shown in Fig. 6. The input lasers drive ring oscillators that produce squeezed states through four-wave mixing. The key to the reprogrammability of the chip is the set of phase modulators that use simple thermal changes on the waveguides. These phase modulators are changed in response to commands from the software to reconfigure the application. Although they switch slowly, once they are set to their new configuration, the computations take place “at the speed of light”. The photonic chip is at room temperature, but the outputs of the four channels are sent by fiber optic to a cooled unit containing the superconductor nanowire photon counters.
Fig. 6 The Xanadu X8 photonic quantum computing chip. From Ref.Fig. 7 To see the chip in operation, see the YouTube video.
Admittedly, the four channels of the X8 chip are not large enough to solve the kinds of problems that would require a quantum computer, but the company has plans to scale the chip up to 100 channels. One of the challenges is to reduce the amount of photon loss in a multiplexed chip, but standard silicon fabrication approaches are expected to reduce loss in the next generation chips by an order of magnitude.
Additional companies are also in the process of entering the photonic quantum computing business, such as PsiQuantum, which recently closed a $450M funding round to produce photonic quantum chips with a million qubits. The company is led by Jeremy O’Brien from Bristol University who has been a leader in photonic quantum computing for over a decade.
[1] E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature, vol. 409, no. 6816, pp. 46-52, Jan (2001)
[5] S. Aaronson and A. Arkhipov, “The Computational Complexity of Linear Optics,” in 43rd ACM Symposium on Theory of Computing, San Jose, CA, Jun 06-08 2011, NEW YORK: Assoc Computing Machinery, in Annual ACM Symposium on Theory of Computing, 2011, pp. 333-342
[8] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic Boson Sampling in a Tunable Circuit,” Science, vol. 339, no. 6121, pp. 794-798, Feb (2013)
Interference (New from Oxford University Press, 2023)
Read the stories of the scientists and engineers who tamed light and used it to probe the universe.
The idea of parallel dimensions in physics has a long history dating back to Bernhard Riemann’s famous 1954 lecture on the foundations of geometry that he gave as a requirement to attain a teaching position at the University of Göttingen. Riemann laid out a program of study that included physics problems solved in multiple dimensions, but it was Rudolph Lipschitz twenty years later who first composed a rigorous view of physics as trajectories in many dimensions. Nonetheless, the three spatial dimensions we enjoy in our daily lives remained the only true physical space until Hermann Minkowski re-expressed Einstein’s theory of relativity in 4-dimensional space time. Even so, Minkowski’s time dimension was not on an equal footing with the three spatial dimensions—the four dimensions were entwined, but time had a different characteristic, what is known as pseudo-Riemannian metric. It is this pseudo-metric that allows space-time distances to be negative as easily as positive.
In 1919 Theodore Kaluza of the University of Königsberg in Prussia extended Einstein’s theory of gravitation to a fifth spatial dimension, and physics had its first true parallel dimension. It was more than just an exercise in mathematics—adding a fifth dimension to relativistic dynamics adds new degrees of freedom that allow the dynamical 5-dimensional theory to include more than merely relativistic massive particles and the electric field they generate. In addition to electro-magnetism, something akin to Einstein’s field equation of gravitation emerges. Here was a five-dimensional theory that seemed to unify E&M with gravity—a first unified theory of physics. Einstein, to whom Kaluza communicated his theory, was intrigued but hesitant to forward Kaluza’s paper for publication. It seemed too good to be true. But Einstein finally sent it to be published in the proceedings of the Prussian Academy of Sciences [Kaluza, 1921]. He later launched his own effort to explore such unified field theories more deeply.
Yet Kaluza’s theory was fully classical—if a fifth dimension can be called that—because it made no connection to the rapidly developing field of quantum mechanics. The person who took the step to make five-dimensional space-time into a quantum field theory was Oskar Klein.
Oskar Klein (1894 – 1977)
Oskar Klein was a Swedish physicist who was in the “second wave” of quantum physicists just a few years behind the titans Heisenberg and Schrödinger and Pauli. He began as a student in physical chemistry working in Stockholm under the famous Arrhenius. It was arranged for him to work in France and Germany in 1914, but he was caught in Paris at the onset of World War I. Returning to Sweden, he enlisted in military service from 1915 to 1916 and then joined Arrhenius’ group at the Nobel Institute where he met Hendrick Kramers—Bohr’s direct assistant at Copenhagen at that time. At Kramer’s invitation, Klein traveled to Copenhagen and worked for a year with Kramers and Bohr before returning to defend his doctoral thesis in 1921 in the field of physical chemistry. Klein’s work with Bohr had opened his eyes to the possibilities of quantum theory, and he shifted his research interest away from physical chemistry. Unfortunately, there were no positions at that time in such a new field, so Klein accepted a position as assistant professor at the University of Michigan in Ann Arbor where he stayed from 1923 to 1925.
Oskar Klein in the late 1920’s
The Fifth Dimension
In an odd twist of fate, this isolation of Klein from the mainstream quantum theory being pursued in Europe freed him of the bandwagon effect and allowed him to range freely on topics of his own devising and certainly in directions all his own. Unaware of Kaluza’s previous work, Klein expanded Minkowski’s space-time from four to five spatial dimensions, just as Kaluza had done, but now with a quantum interpretation. This was not just an incremental step but had far-ranging consequences in the history of physics.
Klein found a way to keep the fifth dimension Euclidean in its metric properties while rolling itself up compactly into a cylinder with the radius of the Planck length—something inconceivably small. This compact fifth dimension made the manifold into something akin to an infinitesimal string. He published a short note in Nature magazine in 1926 on the possibility of identifying the electric charge within the 5-dimensional theory [Klein, 2916a]. He then returned to Sweden to take up a position at the University of Lund. This odd string-like feature of 5-dimensional space-time was picked up by Einstein and others in their search for unified field theories of physics, but the topic soon drifted from the lime light where it lay dormant for nearly fifty years until the first forays were made into string theory. String theory resurrected the Kaluza-Klein theory which has bourgeoned into the vast topic of String Theory today, including Superstrings that occur in 10+1 dimensionsat the frontiers of physics.
Dirac Electrons without the Spin: Klein-Gordon Equation
Once back in Europe, Klein reengaged with the mainstream trends in the rapidly developing quantum theory and in 1926 developed a relativistic quantum theory of the electron [Klein, 1926b]. Around the same time Walter Gordon also proposed this equation, which is now called the “Klein-Gordon Equation”. The equation was a classic wave equation that was second order in both space and time. This was the most natural form for a wave equation for quantum particles and Schrödinger himself had started with this form. But Schrödinger had quickly realized that the second-order time term in the equation did not capture the correct structure of the hydrogen atom, which led him to express the time-dependent term in first order and non-relativistically—which is today’s “Schrödinger Equation”. The problem was in the spin of the electron. The electron is a spin-1/2 particle, a Fermion, which has special transformation properties. It was Dirac a few years later who discovered how to express the relativistic wave equation for the electron—not by promoting the time-dependent term to second order, but by demoting the space-dependent term to first order. The first-order expression for both the space and time derivatives goes hand in hand with the Pauli spin matrices for the electron, and the Dirac Equation is the appropriate relativistically-correct wave equation for the electron.
Klein’s relativistic quantum wave equation does turn out to be the relevant form for a spin-less particle like the pion, but the pion decays by the strong nuclear force and the Klein-Gordon equation is not a practical description. However, the Higgs boson also is a spin-zero particle, and the Klein-Gordon expression does have relevance for this fundamental exchange particle.
Klein Tunneling
In those early days of the late 1920’s, the nature of the nucleus was still a mystery, especially the problem of nuclear radioactivity where a neutron could convert to a proton with the emission of an electron. Some suggested that the neutron was somehow a proton that had captured an electron in a potential barrier. Klein showed that this was impossible, that the electrons would be highly relativistic—something known as a Dirac electron—and they would tunnel with perfect probability through any potential barrier [Klein, 1929]. Therefore, Klein concluded, no nucleon or nucleus could bind an electron.
This phenomenon of unity transmission through a barrier became known as Klein tunneling. The relativistic electron transmits perfectly through an arbitrary potential barrier—independent of its width or height. This is unlike light that transmits through a dielectric slab in resonances that depend on the thickness of the slab—also known as a Fabry-Perot interferometer. The Dirac electron can have any energy, and the potential barrier can have any width, yet the electron will tunnel with 100% probability. How can this happen?
The answer has to do with the dispersion (velocity versus momentum) of the Dirac electron. As the momentum changes in a potential the speed of the Dirac electron stays constant. In the potential barrier, the moment flips sign, but the speed remains unchanged. This is equivalent to the effects of negative refractive index in optics. If a photon travels through a material with negative refractive index, its momentum is flipped, but its speed remains unchanged. From Fermat’s principle, it is speed which determines how a particle like a photon refracts, so if there is no speed change, then there is no reflection.
For the case of Dirac electrons in a potential with field F, speed v and transverse momentum py, the transmission coefficient is given by
If the transverse momentum is zero, then the transmission is perfect. A visual schematic of the role of dispersion and potentials for Dirac electrons undergoing Klein tunneling is shown in the next figure.
In this case, even if the transverse momentum is not strictly zero, there can still be perfect transmission. It is simply a matter of matching speeds.
Graphene became famous over the past decade because its electron dispersion relation is just like a relativistic Dirac electron with a Dirac point between conduction and valence bands. Evidence for Klein tunneling in graphene systems has been growing, but clean demonstrations have remained difficult to observe.
Now, published in the Dec. 2020 issue of Science magazine—almost a century after Klein first proposed it—an experimental group at the University of California at Berkeley reports a beautiful experimental demonstration of Klein tunneling—not from a nucleus, but in an acoustic honeycomb sounding board the size of a small table—making an experimental analogy between acoustics and Dirac electrons that bears out Klein’s theory.
In this special sounding board, it is not electrons but phonons—acoustic vibrations—that have a Dirac point. Furthermore, by changing the honeycomb pattern, the bands can be shifted, just like in a p-n-p junction, to produce a potential barrier. The Berkeley group, led by Xiang Zhang (now president of Hong Kong University), fabricated the sounding board that is about a half-meter in length, and demonstrated dramatic Klein tunneling.
It is amazing how long it can take between the time a theory is first proposed and the time a clean experimental demonstration is first performed. Nearly 90 years has elapsed since Klein first derived the phenomenon. Performing the experiment with actual relativistic electrons was prohibitive, but bringing the Dirac electron analog into the solid state has allowed the effect to be demonstrated easily.
References
[1921] Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972
[1926a] Klein, O. (1926). “The Atomicity of Electricity as a Quantum Theory Law”. Nature118: 516-516.
[1926b] Klein, O. (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie”. Zeitschrift für Physik. 37 (12): 895
[1929] Klein, O. (1929). “Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac”. Zeitschrift für Physik. 53 (3–4): 157
Galileo Unbound 
