Category / Quantum Explorations

Posts on the quantum physics and its history.

by David D. Nolte

A Short History of Quantum Entanglement

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

Video Lectures

YouTube Lecture on the History of Quantum Entanglement

Physics Colloquium on the Backstory of the 2023 Nobel Prize in Physics

Timeline

1935 – Einstein EPR

1935 – Bohr EPR

1935 – Schrödinger: Entanglement and Cat

1950 – Madam Wu positron decay

1952 – David Bohm and Non-local hidden variables

1957 – Bohm and Ahronov version of EPR

1963 – Bell’s inequalities

1967 – Clauser reads Bell’s paper

1967 – Commins experiment with Calcium

1969 – CHSH inequality: measurable with detection inefficiencies

1972 – Clauser and Freedman experiment

1975 – Aspect reads Bell’s paper

1976 – Zeilinger reads Bell’s paper

1981 – Aspect two-photon generation source

1982 – Aspect time variable analyzers

1988 – Parametric down-conversion of EPR pairs (Shih and Alley, Ou and Mandel)

1989 – GHZ state proposed

1993 – Bennett quantum teleportation proposal

1995 – High-intensity down-conversion source of EPR pairs (Kwiat and Zeilinger)

1997 – Zeilinger quantum teleportation experiment

1999 – Observation of the GHZ state

Bibliography

[1] See the full details in: David D. Nolte, Interference: A History of Interferometry and the Scientists Who Tamed Light (Oxford University Press, July 2023)

[2] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Physical Review 47, 0777-0780 (1935).

[3] N. Bohr, Can quantum-mechanical description of physical reality be considered complete? Physical Review 48, 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 Review 85, 166-179 (1952); D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .2. Physical Review 85, 180-193 (1952).

[6] J. Bell, On the Einstein-Podolsky-Rosen paradox. Physics 1, 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 Letters 23, 880-& (1969).

[8] S. J. Freedman, J. F. Clauser, Experimental test of local hidden-variable theories. Physical Review Letters 28, 938-& (1972).

[9] A. Aspect, P. Grangier, G. Roger, EXPERIMENTAL TESTS OF REALISTIC LOCAL THEORIES VIA BELLS THEOREM. Physical Review Letters 47, 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 Letters 49, 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 Letters 70, 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. Nature 390, 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 Scripta 92, 1-33 (2017).

New from Oxford Press: The History of Optical Interferometry (Late Summer 2023)

Pre-order Today

by David D. Nolte

A Short History of Quantum Tunneling

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).

Fig. 6 Diagram from Giaever “The Discovery of Superconducting Tunneling” at https://conferences.illinois.edu/bcs50/pdf/giaever.pdf

The Josephson Effect: Brian Josephson (1962)

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.

By David D. Nolte: Nov. 6, 2022

YouTube Video

YouTube Video of Quantum Tunneling Systems

References:

[1] F. Hund, Z. Phys. 40, 742 (1927). F. Hund, Z. Phys. 43, 805 (1927).

[2] L. Nordheim, Z. Phys. 46, 833 (1928).

[3] R. H. Fowler, L. Nordheim, Proc. R. Soc. London, Ser. A 119, 173 (1928).

[4] G. Gamow, Z. Phys. 51, 204 (1928).

[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.

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by David D. Nolte

Is There a Quantum Trajectory? The Phase-Space Perspective

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

YouTube Video of Dynamics in Quantum Phase Space

For more on the history of quantum trajectories, see Galileo Unbound from Oxford Press:

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References

1. T. Curtright, D. Fairlie, C. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space.  (World Scientific, New Jersey, 2014).

2. J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

by David D. Nolte

Is There a Quantum Trajectory?

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.

By David D. Nolte, Sept. 4, 2022

YouTube Video

YouTube Video of Quantum Trajectories

For more on the history of quantum trajectories, see Galileo Unbound from Oxford Press:

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References

[1] See Chapter 8 , On the Quantum Footpath, in Galileo Unbound, D. D. Nolte (Oxford University Press, 2018)

[2] J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

by David D. Nolte

Quantum Chaos and the Cheshire Cat

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 Scars
Quantum Scars
Quantum 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

YouTube Video of Quantum Chaos

For more on the history of chaos theory, buy Galileo Unbound from Oxford Press:

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References

[1] Heller E J, Bound-state eigenfunctions of classically chaotic hamiltonian-systems – scars of periodic-orbits, Physical Review Letters 53 ,1515 (1984)

[2] Gutzwiller M C, Chaos in classical and quantum mechanics (New York: New York : Springer-Verlag, 1990)

by David D. Nolte

The Many Worlds of the Quantum Beam Splitter

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

References

[1] David D. Nolte, Interference: The History of Optical Interferometry and the Scientists who Tamed Light, (Oxford, July 2023)

[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.

New from Oxford Press: The History of Optical Interferometry (Late Summer 2023)

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Twenty Years at Light Speed: The Future of Photonic Quantum Computing

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.

Stay tuned!

By David D. Nolte, Dec. 20, 2021

Further Reading

• David D. Nolte, “Interference: A History of Interferometry and the Scientists who Tamed Light” (Oxford University Press, to be published in 2023)

• J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photonics, Review vol. 3, no. 12, pp. 687-695, Dec (2009)

• T. C. Ralph and G. J. Pryde, “Optical Quantum Computation,” in Progress in Optics, Vol 54, vol. 54, E. Wolf Ed.,  (2010), pp. 209-269.

• S. Barz, “Quantum computing with photons: introduction to the circuit model, the one-way quantum computer, and the fundamental principles of photonic experiments,” (in English), Journal of Physics B-Atomic Molecular and Optical Physics, Article vol. 48, no. 8, p. 25, Apr (2015), Art no. 083001

References

[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)

KLM01Download

[2] J. Carolan, J. L. O’Brien et al, “Universal linear optics,” Science, vol. 349, no. 6249, pp. 711-716, Aug (2015)

[3] J. M. Arrazola, et al, “Quantum circuits with many photons on a programmable nanophotonic chip,” Nature, vol. 591, no. 7848, pp. 54-+, Mar (2021)

[4] H.-S. Zhong J.-W. Pan et al, “Quantum computational advantage using photons,” Science, vol. 370, no. 6523, p. 1460, (2020)

[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

[6] 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)

[7] J. B. Spring, I. A. Walmsley et al, “Boson Sampling on a Photonic Chip,” Science, vol. 339, no. 6121, pp. 798-801, Feb (2013)

[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)

New from Oxford Press: The History of Optical Interferometry (Late Summer 2023)

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by David D. Nolte

The Many Dimensions of Oskar Klein

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 dimensions at 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.

Dispersion of Dirac electrons at a potential step. Reprinted from https://www.arxiv-vanity.com/papers/0710.3848/

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.

The accoustic Klein tunneling sounding board at Berkeley. Reprinted from https://science.sciencemag.org/content/370/6523/1447

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”. Nature 118: 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

by David D. Nolte

A Short History of the Photon

The quantum of light—the photon—is a little over 100 years old.  It was born in 1905 when Einstein merged Planck’s blackbody quantum hypothesis with statistical mechanics and concluded that light itself must be quantized.  No one believed him!  Fast forward to today, and the photon is a modern workhorse of modern quantum technology.  Quantum encryption and communication are performed almost exclusively with photons, and many prototype quantum computers are optics based.  Quantum optics also underpins atomic and molecular optics (AMO), which is one of the hottest and most rapidly advancing  frontiers of physics today.

Only after the availability of “quantum” light sources … could photon numbers be manipulated at will, launching the modern era of quantum optics.

This blog tells the story of the early days of the photon and of quantum optics.  It begins with Einstein in 1905 and ends with the demonstration of photon anti-bunching that was the first fundamentally quantum optical phenomenon observed seventy years later in 1977.  Across that stretch of time, the photon went from a nascent idea in Einstein’s fertile brain to the most thoroughly investigated quantum particle in the realm of physics.

The Photon: Albert Einstein (1905)

When Planck presented his quantum hypothesis in 1900 to the German Physical Society [1], his model of black body radiation retained all its classical properties but one—the quantized interaction of light with matter.  He did not think yet in terms of quanta, only in terms of steps in a continuous interaction.

The quantum break came from Einstein when he published his 1905 paper proposing the existence of the photon—an actual quantum of light that carried with it energy and momentum [2].  His reasoning was simple and iron-clad, resting on Planck’s own blackbody relation that Einstein combined with simple reasoning from statistical mechanics.  He was led inexorably to the existence of the photon.  Unfortunately, almost no one believed him (see my blog on Einstein and Planck). 

This was before wave-particle duality in quantum thinking, so the notion that light—so clearly a wave phenomenon—could be a particle was unthinkable.  It had taken half of the 19th century to rid physics of Newton’s corpuscules and emmisionist theories of light, so to bring it back at the beginning of the 20th century seemed like a great blunder.  However, Einstein persisted.

In 1909 he published a paper on the fluctuation properties of light [3] in which he proposed that the fluctuations observed in light intensity had two contributions: one from the discreteness of the photons (what we call “shot noise” today) and one from the fluctuations in the wave properties.  Einstein was proposing that both particle and wave properties contributed to intensity fluctuations, exhibiting simultaneous particle-like and wave-like properties.  This was one of the first expressions of wave-particle duality in modern physics.

In 1916 and 1917 Einstein took another bold step and proposed the existence of stimulated emission [4].  Once again, his arguments were based on simple physics—this time the principle of detailed balance—and he was led to the audacious conclusion that one photon can stimulated the emission of another.  This would become the basis of the laser forty-five years later.

While Einstein was confident in the reality of the photon, others sincerely doubted its existence.  Robert Milliken (1868 – 1953) decided to put Einstein’s theory of photoelectron emission to the most stringent test ever performed.  In 1915 he painstakingly acquired the definitive dataset with the goal to refute Einstein’s hypothesis, only to confirm it in spectacular fashion [5].  Partly based on Milliken’s confirmation of Einstein’s theory of the photon, Einstein was awarded the Nobel Prize in Physics in 1921.

Einstein at a blackboard.

From that point onward, the physical existence of the photon was accepted and was incorporated routinely into other physical theories.  Compton used the energy and the momentum of the photon in 1922 to predict and measure Compton scattering of x-rays off of electrons [6].  The photon was given its modern name by Gilbert Lewis in 1926 [7].

Single-Photon Interference: Geoffry Taylor (1909)

If a light beam is made up of a group of individual light quanta, then in the limit of very dim light, there should just be one photon passing through an optical system at a time.  Therefore, to do optical experiments on single photons, one just needs to reach the ultimate dim limit.  As simple and clear as this argument sounds, it has problems that only were sorted out after the Hanbury Brown and Twiss experiments in the 1950’s and the controversy they launched (see below).  However, in 1909, this thinking seemed like a clear approach for looking for deviations in optical processes in the single-photon limit.

In 1909, Geoffry Ingram Taylor (1886 – 1975) was an undergraduate student at Cambridge University and performed a low-intensity Young’s double-slit experiment (encouraged by J. J. Thomson).  At that time the idea of Einstein’s photon was only 4 years old, and Bohr’s theory of the hydrogen atom was still a year away.  But Thomson believed that if photons were real, then their existence could possibly show up as deviations in experiments involving single photons.  Young’s double-slit experiment is the classic demonstration of the classical wave nature of light, so performing it under conditions when (on average) only a single photon was in transit between a light source and a photographic plate seemed like the best place to look.

G. I. Taylor

The experiment was performed by finding an optimum exposure of photographic plates in a double slit experiment, then reducing the flux while increasing the exposure time, until the single-photon limit was achieved while retaining the same net exposure of the photographic plate.  Under the lowest intensity, when only a single photon was in transit at a time (on average), Taylor performed the exposure for three months.  To his disappointment, when he developed the film, there was no significant difference between high intensity and low intensity interference fringes [8].  If photons existed, then their quantized nature was not showing up in the low-intensity interference experiment.

The reason that there is no single-photon-limit deviation in the behavior of the Young double-slit experiment is because Young’s experiment only measures first-order coherence properties.  The average over many single-photon detection events is described equally well either by classical waves or by quantum mechanics.  Quantized effects in the Young experiment could only appear in fluctuations in the arrivals of photons, but in Taylor’s day there was no way to detect the arrival of single photons. 

Quantum Theory of Radiation : Paul Dirac (1927)

After Paul Dirac (1902 – 1984) was awarded his doctorate from Cambridge in 1926, he received a stipend that sent him to work with Niels Bohr (1885 – 1962) in Copenhagen. His attention focused on the electromagnetic field and how it interacted with the quantized states of atoms.  Although the electromagnetic field was the classical field of light, it was also the quantum field of Einstein’s photon, and he wondered how the quantized harmonic oscillators of the electromagnetic field could be generated by quantum wavefunctions acting as operators.  He decided that, to generate a photon, the wavefunction must operate on a state that had no photons—the ground state of the electromagnetic field known as the vacuum state.

Dirac put these thoughts into their appropriate mathematical form and began work on two manuscripts.  The first manuscript contained the theoretical details of the non-commuting electromagnetic field operators.  He called the process of generating photons out of the vacuum “second quantization”.  In second quantization, the classical field of electromagnetism is converted to an operator that generates quanta of the associated quantum field out of the vacuum (and also annihilates photons back into the vacuum).  The creation operators can be applied again and again to build up an N-photon state containing N photons that obey Bose-Einstein statistics, as they must, as required by their integer spin, and agreeing with Planck’s blackbody radiation. 

Dirac then showed how an interaction of the quantized electromagnetic field with quantized energy levels involved the annihilation and creation of photons as they promoted electrons to higher atomic energy levels, or demoted them through stimulated emission.  Very significantly, Dirac’s new theory explained the spontaneous emission of light from an excited electron level as a direct physical process that creates a photon carrying away the energy as the electron falls to a lower energy level.  Spontaneous emission had been explained first by Einstein more than ten years earlier when he derived the famous A and B coefficients [4], but the physical mechanism for these processes was inferred rather than derived. Dirac, in late 1926, had produced the first direct theory of photon exchange with matter [9]. 

Paul Dirac in his early days.

Einstein-Podolsky-Rosen (EPR) and Bohr (1935)

The famous dialog between Einstein and Bohr at the Solvay Conferences culminated in the now famous “EPR” paradox of 1935 when Einstein published (together with B. Podolsky and N. Rosen) a paper that contained a particularly simple and cunning thought experiment. In this paper, not only was quantum mechanics under attack, but so was the concept of reality itself, as reflected in the paper’s title “Can Quantum Mechanical Description of Physical Reality Be Considered Complete?” [10].

Bohr and Einstein at Paul Ehrenfest’s house in 1925.

Einstein considered an experiment on two quantum particles that had become “entangled” (meaning they interacted) at some time in the past, and then had flown off in opposite directions. By the time their properties are measured, the two particles are widely separated. Two observers each make measurements of certain properties of the particles. For instance, the first observer could choose to measure either the position or the momentum of one particle. The other observer likewise can choose to make either measurement on the second particle. Each measurement is made with perfect accuracy. The two observers then travel back to meet and compare their measurements.   When the two experimentalists compare their data, they find perfect agreement in their values every time that they had chosen (unbeknownst to each other) to make the same measurement. This agreement occurred either when they both chose to measure position or both chose to measure momentum.

It would seem that the state of the particle prior to the second measurement was completely defined by the results of the first measurement. In other words, the state of the second particle is set into a definite state (using quantum-mechanical jargon, the state is said to “collapse”) the instant that the first measurement is made. This implies that there is instantaneous action at a distance −− violating everything that Einstein believed about reality (and violating the law that nothing can travel faster than the speed of light). He therefore had no choice but to consider this conclusion of instantaneous action to be false.  Therefore quantum mechanics could not be a complete theory of physical reality −− some deeper theory, yet undiscovered, was needed to resolve the paradox.

Bohr, on the other hand, did not hold “reality” so sacred. In his rebuttal to the EPR paper, which he published six months later under the identical title [11], he rejected Einstein’s criterion for reality. He had no problem with the two observers making the same measurements and finding identical answers. Although one measurement may affect the conditions of the second despite their great distance, no information could be transmitted by this dual measurement process, and hence there was no violation of causality. Bohr’s mind-boggling viewpoint was that reality was nonlocal, meaning that in the quantum world the measurement at one location does influence what is measured somewhere else, even at great distance. Einstein, on the other hand, could not accept a nonlocal reality.

Entangled versus separable states. When the states are separable, no measurement on photon A has any relation to measurements on photon B. However, in the entangled case, all measurements on A are related to measurements on B (and vice versa) regardless of what decision is made to make what measurement on either photon, or whether the photons are separated by great distance. The entangled wave-function is “nonlocal” in the sense that it encompasses both particles at the same time, no matter how far apart they are.

The Intensity Interferometer:  Hanbury Brown and Twiss (1956)

Optical physics was surprisingly dormant from the 1930’s through the 1940’s. Most of the research during this time was either on physical optics, like lenses and imaging systems, or on spectroscopy, which was more interested in the physical properties of the materials than in light itself. This hiatus from the photon was about to change dramatically, not driven by physicists, but driven by astronomers.

The development of radar technology during World War II enabled the new field of radio astronomy both with high-tech receivers and with a large cohort of scientists and engineers trained in radio technology. In the late 1940’s and early 1950’s radio astronomy was starting to work with long baselines to better resolve radio sources in the sky using interferometery. The first attempts used coherent references between two separated receivers to provide a common mixing signal to perform field-based detection. However, the stability of the reference was limiting, especially for longer baselines.

In 1950, a doctoral student in the radio astronomy department of the University of Manchester, R. Hanbury Brown, was given the task to design baselines that could work at longer distances to resolve smaller radio sources. After struggling with the technical difficulties of providing a coherent “local” oscillator for distant receivers, Hanbury Brown had a sudden epiphany one evening. Instead of trying to reference the field of one receiver to the field of another, what if, instead, one were to reference the intensity of one receiver to the intensity of the other, specifically correlating the noise on the intensity? To measure intensity requires no local oscillator or reference field. The size of an astronomical source would then show up in how well the intensity fluctuations correlated with each other as the distance between the receivers was changed. He did a back of the envelope calculation that gave him hope that his idea might work, but he needed more rigorous proof if he was to ask for money to try out his idea. He tracked down Richard Twiss at a defense research lab and the two working out the theory of intensity correlations for long-baseline radio interferometry. Using facilities at the famous Jodrell Bank Radio Observatory at Manchester, they demonstrated the principle of their intensity interferometer and measured the angular size of Cygnus A and Cassiopeia A, two of the strongest radio sources in the Northern sky.

R. Hanbury Brown

One of the surprising side benefits of the intensity interferometer over field-based interferometry was insensitivity to environmental phase fluctuations. For radio astronomy the biggest source of phase fluctuations was the ionosphere, and the new intensity interferometer was immune to its fluctuations. Phase fluctuations had also been the limiting factor for the Michelson stellar interferometer which had limited its use to only about half a dozen stars, so Hanbury Brown and Twiss decided to revisit visible stellar interferometry using their new concept of intensity interferometry.

To illustrate the principle for visible wavelengths, Hanbury Brown and Twiss performed a laboratory experiment to correlate intensity fluctuations in two receivers illuminated by a common source through a beam splitter. The intensity correlations were detected and measured as a function of path length change, illustrating an excess correlation in noise for short path lengths that decayed as the path length increased. They published their results in Nature magazine in 1956 that immediately ignited a firestorm of protest from physicists [12].

In the 1950’s, many physicists had embraced the discrete properties of the photon and had developed a misleading mental picture of photons as individual and indivisible particles that could only go one way or another from a beam splitter, but not both. Therefore, the argument went, if the photon in an attenuated beam was detected in one detector at the output of a beam splitter, then it cannot be detected at the other. This would produce an anticorrelation in coincidence counts at the two detectors. However, the Hanbury Brown Twiss (HBT) data showed a correlation from the two detectors. This launched an intense controversy in which some of those who accepted the results called for a radical new theory of the photon, while most others dismissed the HBT results as due to systematics in the light source. The heart of this controversy was quickly understood by the Nobel laureate E. M Purcell. He correctly pointed out that photons are bosons and are indistinguishable discrete particles and hence are likely to “bunch” together, according to quantum statistics, even under low light conditions [13]. Therefore, attenuated “chaotic” light would indeed show photodetector correlations, even if the average photon number was less than a single photon at a time, the photons would still bunch.

The bunching of photons in light is a second order effect that moves beyond the first-order interference effects of Young’s double slit, but even here the quantum nature of light is not required. A semiclassical theory of light emission from a spectral line with a natural bandwidth also predicts intensity correlations, and the correlations are precisely what would be observed for photon bunching. Therefore, even the second-order HBT results, when performed with natural light sources, do not distinguish between classical and quantum effects in the experimental results. But this reliance on natural light sources was about to change fundmaentally with the invention of the laser.

Invention of the Laser : Ted Maiman (1959)

One of the great scientific breakthroughs of the 20th century was the nearly simultaneous yet independent realization by several researchers around 1951 (by Charles H. Townes of Columbia University, by Joseph Weber of the University of Maryland, and by Alexander M. Prokhorov and Nikolai G. Basov at the Lebedev Institute in Moscow) that clever techniques and novel apparati could be used to produce collections of atoms that had more electrons in excited states than in ground states. Such a situation is called a population inversion. If this situation could be attained, then according to Einstein’s 1917 theory of photon emission, a single photon would stimulate a second photon, which in turn would stimulate two additional electrons to emit two identical photons to give a total of four photons −− and so on. Clearly this process turns a single photon into a host of photons, all with identical energy and phase.

Theodore Maiman

Charles Townes and his research group were the first to succeed in 1953 in producing a device based on ammonia molecules that could work as an intense source of coherent photons. The initial device did not amplify visible light, but amplified microwave photons that had wavelengths of about 3 centimeters. They called the process microwave amplification by stimulated emission of radiation, hence the acronym “MASER”. Despite the significant breakthrough that this invention represented, the devices were very expensive and difficult to operate. The maser did not revolutionize technology, and some even quipped that the acronym stood for “Means of Acquiring Support for Expensive Research”. The maser did, however, launch a new field of study, called quantum electronics, that was the direct descendant of Einstein’s 1917 paper. Most importantly, the existence and development of the maser became the starting point for a device that could do the same thing for light.

The race to develop an optical maser (later to be called laser, for light amplification by stimulated emission of radiation) was intense. Many groups actively pursued this holy grail of quantum electronics. Most believed that it was possible, which made its invention merely a matter of time and effort. This race was won by Theodore H. Maiman at Hughes Research Laboratory in Malibu California in 1960 [14]. He used a ruby crystal that was excited into a population inversion by an intense flash tube (like a flash bulb) that had originally been invented for flash photography. His approach was amazingly simple −− blast the ruby with a high-intensity pulse of light and see what comes out −− which explains why he was the first. Most other groups had been pursuing much more difficult routes because they believed that laser action would be difficult to achieve.

Perhaps the most important aspect of Maiman’s discovery was that it demonstrated that laser action was actually much simpler than people anticipated, and that laser action is a fairly common phenomenon. His discovery was quickly repeated by other groups, and then additional laser media were discovered such as helium-neon gas mixtures, argon gas, carbon dioxide gas, garnet lasers and others. Within several years, over a dozen different material and gas systems were made to lase, opening up wide new areas of research and development that continues unabated to this day. It also called for new theories of optical coherence to explain how coherent laser light interacted with matter.

Coherent States : Glauber (1963)

The HBT experiment had been performed with attenuated chaotic light that had residual coherence caused by the finite linewidth of the filtered light source. The theory of intensity correlations for this type of light was developed in the 1950’s by Emil Wolf and Leonard Mandel using a semiclassical theory in which the statistical properties of the light was based on electromagnetics without a direct need for quantized photons. The HBT results were fully consistent with this semiclassical theory. However, after the invention of the laser, new “coherent” light sources became available that required a fundamentally quantum depiction.

Roy Glauber was a theoretical physicist who received his PhD working with Julian Schwinger at Harvard. He spent several years as a post-doc at Princeton’s Institute for Advanced Study starting in 1949 at the time when quantum field theory was being developed by Schwinger, Feynman and Dyson. While Feynman was off in Brazil for a year learning to play the bongo drums, Glauber filled in for his lectures at Cal Tech. He returned to Harvard in 1952 in the position of an assistant professor. He was already thinking about the quantum aspects of photons in 1956 when news of the photon correlations in the HBT experiment were published, and when the laser was invented three years later, he began developing a theory of photon correlations in laser light that he suspected would be fundamentally different than in natural chaotic light.

Roy Glauber

Because of his background in quantum field theory, and especially quantum electrodynamics, it was a fairly easy task to couch the quantum optical properties of coherent light in terms of Dirac’s creation and annihilation operators of the electromagnetic field. Related to the minimum-uncertainty wave functions derived initially by Schrödinger in the late 1920’s, Glauber developed a “coherent state” operator that was a minimum uncertainty state of the quantized electromagnetic field [15]. This coherent state represents a laser operating well above the lasing threshold and predicted that the HBT correlations would vanish. Glauber was awarded the Nobel Prize in Physics in 2005 for his work on such “Glauber” states in quantum optics.

Single-Photon Optics: Kimble and Mandel (1977)

Beyond introducing coherent states, Glauber’s new theoretical approach, and parallel work by George Sudarshan around the same time [16], provided a new formalism for exploring quantum optical properties in which fundamentally quantum processes could be explored that could not be predicted using only semiclassical theory. For instance, one could envision producing photon states in which the photon arrivals at a detector could display the kind of anti-bunching that had originally been assumed (in error) by the critics of the HBT experiment. A truly one-photon state, also known as a Fock state or a number state, would be the extreme limit in which the quantum field possessed a single quantum that could be directed at a beam splitter and would emerge either from one side or the other with complete anti-correlation. However, generating such a state in the laboratory remained a challenge.

In 1975 by Carmichel and Walls predicted that resonance fluorescence could produce quantized fields that had lower correlations than coherent states [17]. In 1977 H. J. Kimble, M. Dagenais and L. Mandel demonstrated, for the first time, photon antibunching between two photodetectors at the two ports of a beam splitter [18]. They used a beam of sodium atoms pumped by a dye laser.

This first demonstration of photon antibunching represents a major milestone in the history of quantum optics. Taylor’s first-order experiments in 1909 showed no difference between classical electromagnetic waves and a flux of photons. Similarly the second-order HBT experiment of 1956 using chaotic light could be explained equally well using classical or quantum approaches to explain the observed photon correlations. Even laser light (when the laser is operated far above threshold) produced classic “classical” wave effects with only the shot noise demonstrating the discreteness of photon arrivals. Only after the availability of “quantum” light sources, beginning with the work of Kimble and Mandel, could photon numbers be manipulated at will, launching the modern era of quantum optics. Later experiments by them and others have continually improved the control of photon states.

By David D. Nolte, Jan. 18, 2021

TimeLine:

  • 1900 – Planck (1901). “Law of energy distribution in normal spectra.” Annalen Der Physik 4(3): 553-563.
  • 1905 – A. Einstein (1905). “Generation and conversion of light with regard to a heuristic point of view.” Annalen Der Physik 17(6): 132-148.
  • 1909 – A. Einstein (1909). “On the current state of radiation problems.” Physikalische Zeitschrift 10: 185-193.
  • 1909 – G.I. Taylor: Proc. Cam. Phil. Soc. Math. Phys. Sci. 15 , 114 (1909) Single photon double-slit experiment
  • 1915 – Millikan, R. A. (1916). “A direct photoelectric determination of planck’s “h.”.” Physical Review 7(3): 0355-0388. Photoelectric effect.
  • 1916 – Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.. Einstein predicts stimulated emission
  • 1923 –Compton, Arthur H. (May 1923). “A Quantum Theory of the Scattering of X-Rays by Light Elements”. Physical Review. 21 (5): 483–502.
  • 1926 – Lewis, G. N. (1926). “The conservation of photons.” Nature 118: 874-875.. Gilbert Lewis named “photon”
  • 1927 – D. Dirac, P. A. M. (1927). “The quantum theory of the emission and absorption of radiation.” Proceedings of the Royal Society of London Series a-Containing Papers of a Mathematical and Physical Character 114(767): 243-265.
  • 1932 – E. P. Wigner: Phys. Rev. 40, 749 (1932)
  • 1935 – A. Einstein, B. Podolsky, N. Rosen: Phys. Rev. 47 , 777 (1935). EPR paradox.
  • 1935 – N. Bohr: Phys. Rev. 48 , 696 (1935). Bohr’s response to the EPR paradox.
  • 1956 – R. Hanbury-Brown, R.W. Twiss: Nature 177 , 27 (1956) Photon bunching
  • 1963 – R. J. Glauber: Phys. Rev. 130 , 2529 (1963) Coherent states
  • 1963 – E. C. G. Sudarshan: Phys. Rev. Lett. 10, 277 (1963) Coherent states
  • 1964 – P. L. Kelley, W.H. Kleiner: Phys. Rev. 136 , 316 (1964)
  • 1966 – F. T. Arecchi, E. Gatti, A. Sona: Phys. Rev. Lett. 20 , 27 (1966); F.T. Arecchi, Phys. Lett. 16 , 32 (1966)
  • 1966 – J. S. Bell: Physics 1 , 105 (1964); Rev. Mod. Phys. 38 , 447 (1966) Bell inequalities
  • 1967 – R. F. Pfleegor, L. Mandel: Phys. Rev. 159 , 1084 (1967) Interference at single photon level
  • 1967 – M. O. Scully, W.E. Lamb: Phys. Rev. 159 , 208 (1967).  Quantum theory of laser
  • 1967 – B. R. Mollow, R. J. Glauber: Phys. Rev. 160, 1097 (1967); 162, 1256 (1967) Parametric converter
  • 1969 – M. O. Scully, W.E. Lamb: Phys. Rev. 179 , 368 (1969).  Quantum theory of laser
  • 1969 – M. Lax, W.H. Louisell: Phys. Rev. 185 , 568 (1969).  Quantum theory of laser
  • 1975 – Carmichael, H. J. and D. F. Walls (1975). Journal of Physics B-Atomic Molecular and Optical Physics 8(6): L77-L81. Photon anti-bunching predicted in resonance fluorescence
  • 1977 – H. J. Kimble, M. Dagenais and L. Mandel (1977) Photon antibunching in resonance fluorescence. Phys. Rev. Lett. 39, 691-5:  Kimble, Dagenais and Mandel demonstrate the effect of antibunching

References

• Parts of this blog are excerpted from Mind at Light Speed, D. Nolte (Free Press, 2001) that tells the story of light’s central role in telecommunications and in the future of optical and quantum computers. Further information can be found in Interference: The History of Optical Interferometry and the Scientists who Tamed Light (Oxford, 2023).

[1] Planck (1901). “Law of energy distribution in normal spectra.” Annalen Der Physik 4(3): 553-563.

[2] A. Einstein (1905). “Generation and conversion of light with regard to a heuristic point of view.” Annalen Der Physik 17(6): 132-148

[3] A. Einstein (1909). “On the current state of radiation problems.” Physikalische Zeitschrift 10: 185-193.

[4] Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318; Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.

[5] Millikan, R. A. (1916). “A direct photoelectric determination of planck‘s “h.”.” Physical Review 7(3): 0355-0388.

[6] Compton, A. H. (1923). “A quantum theory of the scattering of x-rays by light elements.” Physical Review 21(5): 0483-0502.

[7] Lewis, G. N. (1926). “The conservation of photons.” Nature 118: 874-875.

[8] Taylor, G. I. (1910). “Interference fringes with feeble light.” Proceedings of the Cambridge Philosophical Society 15: 114-115.

[9] Dirac, P. A. M. (1927). “The quantum theory of the emission and absorption of radiation.” Proceedings of the Royal Society of London Series a-Containing Papers of a Mathematical and Physical Character 114(767): 243-265.

EinsteinPR35Download

[10] Einstein, A., B. Podolsky and N. Rosen (1935). “Can quantum-mechanical description of physical reality be considered complete?” Physical Review 47(10): 0777-0780.

BohrPR35Download

[11] Bohr, N. (1935). “Can quantum-mechanical description of physical reality be considered complete?” Physical Review 48(8): 696-702.

• Hanbury Brown 1956Download

[12] Brown, R. H. and R. Q. Twiss (1956). “Correlation Between Photons in 2 Coherent Beams of Light.” Nature 177(4497): 27-29; [1] R. H. Brown and R. Q. Twiss, “Test of a new type of stellar interferometer on Sirius,” Nature, vol. 178, no. 4541, pp. 1046-1048, (1956).

• Purcell 1956Download

[13] Purcell, E. M. (1956). “Question of Correlation Between Photons in Coherent Light Rays.” Nature 178(4548): 1448-1450.

[14] Maimen, T. H. (1960). “Stimulated optical radiation in ruby.” Nature 187: 493.

• Glauber 1963Download

[15] Glauber, R. J. (1963). “Photon Correlations.” Physical Review Letters 10(3): 84.

[16] 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.

• Carmichel 1975Download

[17] Carmichael, H. J. and D. F. Walls (1975). “Quantum treatment of spontaneous emission from a strongly driven 2-level atom.” Journal of Physics B-Atomic Molecular and Optical Physics 8(6): L77-L81.

[18] Kimble, H. J., M. Dagenais and L. Mandel (1977). “Photon anti bunching in resonance fluorescence.” Physical Review Letters 39(11): 691-695.

New from Oxford Press: The History of Optical Interferometry (Summer 2023)

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by David D. Nolte

Quantum Seeing without Looking? The Strange Physics of Quantum Sensing

Quantum sensors have amazing powers.  They can detect the presence of an obstacle without ever interacting with it.  For instance, consider a bomb that is coated with a light sensitive layer that sets off the bomb if it absorbs just a single photon.  Then put this bomb inside a quantum sensor system and shoot photons at it.  Remarkably, using the weirdness of quantum mechanics, it is possible to design the system in such a way that you can detect the presence of the bomb using photons without ever setting it off.  How can photons see the bomb without illuminating it?  The answer is a bizarre side effect of quantum physics in which quantum wavefunctions are recognized as the root of reality as opposed to the pesky wavefunction collapse at the moment of measurement.

The ability for a quantum system to see an object with light, without exposing it, is uniquely a quantum phenomenon that has no classical analog.

All Paths Lead to Feynman

When Richard Feynman was working on his PhD under John Archibald Wheeler at Princeton in the early 1940’s he came across an obscure paper written by Paul Dirac in 1933 that connected quantum physics with classical Lagrangian physics.  Dirac had recognized that the phase of a quantum wavefunction was analogous to the classical quantity called the “Action” that arises from Lagrangian physics.  Building on this concept, Feynman constructed a new interpretation of quantum physics, known as the “many histories” interpretation, that occupies the middle ground between Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics.  One of the striking consequences of the many histories approach is the emergence of the principle of least action—a classical concept—into interpretations of quantum phenomena.  In this approach, Feynman considered ALL possible histories for the propagation of a quantum particle from one point to another, he tabulated the quantum action in the phase factor, and he then summed all of these histories.

One of the simplest consequences of the sum over histories is a quantum interpretation of Snell’s law of refraction in optics.  When summing over all possible trajectories of a photon from a point above to a point below an interface, there are a subset of paths for which the action integral varies very little from one path in the subset to another.  The consequence of this is that the phases of all these paths add constructively, producing a large amplitude to the quantum wavefunction along the centroid of these trajectories.  Conversely, for paths far away from this subset, the action integral takes on many values and the phases tend to interfere destructively, canceling the wavefunction along these other paths.  Therefore, the most likely path of the photon between the two points is the path of maximum constructive interference and hence the path of stationary action.  It is simple so show that this path is none other than the classical path determined by Snell’s Law and equivalently by Fermat’s principle of least time.  With the many histories approach, we can add the principle of least (or stationary) action to the list of explanations of Snell’s Law.  This argument holds as well for an electron (with mass and a de Broglie wavelength) as it does for a photon, so this not just a coincidence specific to optics but is a fundamental part of quantum physics.

A more subtle consequence of the sum over histories view of quantum phenomena is Young’s double slit experiment for electrons, shown at the top of Fig 1.  The experiment consists of a source that emits only a single electron at a time that passes through a double-slit mask to impinge on an electron detection screen.  The wavefunction for a single electron extends continuously throughout the full spatial extent of the apparatus, passing through both slits.  When the two paths intersect at the screen, the difference in the quantum phases of the two paths causes the combined wavefunction to have regions of total constructive interference and other regions of total destructive interference.  The probability of detecting an electron is proportional to the squared amplitude of the wavefunction, producing a pattern of bright stripes separated by darkness.  At positions of destructive interference, no electrons are detected when both slits are open.  However, if an opaque plate blocks the upper slit, then the interference pattern disappears, and electrons can be detected at those previously dark locations.  Therefore, the presence of the object can be deduced by the detection of electrons at locations that should be dark.

Fig. 1  Demonstration of the sum over histories in a double-slit experiment for electrons. In the upper frame, the electron interference pattern on the phosphorescent screen produces bright and dark stripes.  No electrons hit the screen in a dark stripe.  When the upper slit is blocked (bottom frame), the interference pattern disappears, and an electron can arrive at the location that had previously been dark.

Consider now when the opaque plate is an electron-sensitive detector.  In this case, a single electron emitted by the source can be detected at the screen or at the plate.  If it is detected at the screen, it can appear at the location of a dark fringe, heralding the presence of the opaque plate.  Yet the quantum conundrum is that when the electron arrives at a dark fringe, it must be detected there as a whole, it cannot be detected at the electron-sensitive plate too.  So how does the electron sense the presence of the detector without exposing it, without setting it off? 

In Feynman’s view, the electron does set off the detector as one possible history.  And that history interferes with the other possible history when the electron arrives at the screen.  While that interpretation may seem weird, mathematically it is a simple statement that the plate blocks the wavefunction from passing through the upper slit, so the wavefunction in front of the screen, resulting from all possible paths, has no interference fringes (other than possible diffraction from the lower slit).  From this point of view, the wavefunction samples all of space, including the opaque plate, and the eventual absorption of a photon one place or another has no effect on the wavefunction.  In this sense, it is the wavefunction, prior to any detection event, that samples reality.  If the single electron happens to show up at a dark fringe at the screen, the plate, through its effects on the total wavefunction, has been detected without interacting with the photon. 

This phenomenon is known as an interaction-free measurement, but there are definitely some semantics issues here.  Just because the plate doesn’t absorb a photon, it doesn’t mean that the plate plays no role.  The plate certainly blocks the wavefunction from passing through the upper slit.  This might be called an “interaction”, but that phrase it better reserved for when the photon is actually absorbed, while the role of the plate in shaping the wavefunction is better described as one of the possible histories.

Quantum Seeing in the Dark

Although Feynman was thinking hard (and clearly) about these issues as he presented his famous lectures in physics at Cal Tech during 1961 to 1963, the specific possibility of interaction-free measurement dates more recently to 1993 when Avshalom C. Elitzur and Lev Vaidman at Tel Aviv University suggested a simple Michelson interferometer configuration that could detect an object half of the time without interacting with it [1].  They are the ones who first pressed this point home by thinking of a light-sensitive bomb.  There is no mistaking when a bomb goes off, so it tends to give an exaggerated demonstration of the interaction-free measurement. 

The Michelson interferometer for interaction-free measurement is shown in Fig. 2.  This configuration uses a half-silvered beamsplitter to split the possible photon paths.  When photons hit the beamsplitter, they either continue traveling to the right, or are deflected upwards.  After reflecting off the mirrors, the photons again encounter the beamsplitter, where, in each case, they continue undeflected or are reflected.  The result is that two paths combine at the beamsplitter to travel to the detector, while two other paths combine to travel back along the direction of the incident beam. 

Fig. 2 A quantum-seeing in the dark (QSD) detector with a photo-sensitive bomb. A single photon is sent into the interferometer at a time. If the bomb is NOT present, destructive interference at the detector guarantees that the photon is not detected. However, if the bomb IS present, it destroys the destructive interference and the photon can arrive at the detector. That photon heralds the presence of the bomb without setting it off. (Reprinted from Mind @ Light Speed)

The paths of the light beams can be adjusted so that the beams that combine to travel to the detector experience perfect destructive interference.  In this situation, the detector never detects light, and all the light returns back along the direction of the incident beam.  Quantum mechanically, when only a single photon is present in the interferometer at a time, we would say that the quantum wavefunction of the photon interferes destructively along the path to the detector, and constructively along the path opposite to the incident beam, and the detector would detect no photons.  It is clear that the unobstructed path of both beams results in the detector making no detections.

Now place the light sensitive bomb in the upper path.  Because this path is no longer available to the photon wavefunction, the destructive interference of the wavefunction along the detector path is removed.  Now when a single photon is sent into the interferometer, three possible things can happen.  One, the photon is reflected by the beamsplitter and detonates the bomb.  Two, the photon is transmitted by the beamsplitter, reflects off the right mirror, and is transmitted again by the beamsplitter to travel back down the incident path without being detected by the detector.  Three, the photon is transmitted by the beamsplitter, reflects off the right mirror, and is reflected off the beamsplitter to be detected by the detector. 

In this third case, the photon is detected AND the bomb does NOT go off, which succeeds at quantum seeing in the dark.  The odds are much better than for Young’s experiment.  If the bomb is present, it will detonate a maximum of 50% of the time.  The other 50%, you will either detect a photon (signifying the presence of the bomb), or else you will not detect a photon (giving an ambiguous answer and requiring you to perform the experiment again).  When you perform the experiment again, you again have a 50% chance of detonating the bomb, and a 25% chance of detecting it without it detonating, but again a 25% chance of not detecting it, and so forth.  All in all, every time you send in a photon, you have one chance in four of seeing the bomb without detonating it.  These are much better odds than for the Young’s apparatus where only exact detection of the photon at a forbidden location would signify the presence of the bomb.

It is possible to increase your odds above one chance in four by decreasing the reflectivity of the beamsplitter.  In practice, this is easy to do simply by depositing less and less aluminum on the surface of the glass plate.  When the reflectivity gets very low, let us say at the level of 1%, then most of the time the photon just travels back along the direction it came and you have an ambiguous result.  On the other hand, when the photon does not return, there is an equal probability of detonation as detection.  This means that, though you may send in many photons, your odds for eventually seeing the bomb without detonating it are nearly 50%, which is a factor of two better odds than for the half-silvered beamsplitter.  A version of this experiment was performed by Paul Kwiat in 1995 as a postdoc at Innsbruck with Anton Zeilinger.  It was Kwiat who coined the phrase “quantum seeing in the dark” as a catchier version of “interaction-free measurement” [2].

A 50% chance of detecting the bomb without setting it off sounds amazing, until you think that there is a 50% chance that it will go off and kill you.  Then those odds don’t look so good.  But optical phenomena never fail to surprise, and they never let you down.  A crucial set of missing elements in the simple Michelson experiment was polarization-control using polarizing beamsplitters and polarization rotators.  These are common elements in many optical systems, and when they are added to the Michelson quantum sensor, they can give almost a 100% chance of detecting the bomb without setting it off using the quantum Zeno effect.

The Quantum Zeno Effect

Photons carry polarization as their prime quantum number, with two possible orientations.  These can be defined in different ways, but the two possible polarizations are orthogonal to each other.  For instance, these polarization pairs can be vertical (V)  and horizontal (H), or they can be right circular  and left circular.  One of the principles of quantum state evolution is that a quantum wavefunction can be maintained in a specific state, even if it has a tendency naturally to drift out of that state, by repeatedly making a quantum measurement that seeks to measure deviations from that state.  In practice, the polarization of a photon can be maintained by repeatedly passing it through a polarizing beamsplitter with the polarization direction parallel to the original polarization of the photon.  If there is a deviation in the photon polarization direction by a small angle, then a detector on the side port of the polarizing beamsplitter will fire with a probability equal to the square of the sine of the deviation.  If the deviation angle is very small, say Δθ, then the probability of measuring the deviation is proportional to (Δθ)2, which is an even smaller number.  Furthermore, the probability that the photon will transmit through the polarizing beamsplitter is equal to 1-(Δθ)2 , which is nearly 100%.

This is what happens in Fig. 3 when the photo-sensitive bomb IS present. A single H-polarized photon is injected through a switchable mirror into the interferometer on the right. In the path of the photon is a polarization rotator that rotates the polarization by a small angle Δθ. There is nearly a 100% chance that the photon will transmit through the polarizing beamsplitter with perfect H-polarization reflect from the mirror and return through the polarizing beamsplitter, again with perfect H-polarization to pass through the polarization rotator to the switchable mirror where it reflects, gains another increment to its polarization angle, which is still small, and transmits through the beamsplitter, etc. At each pass, the photon polarization is repeatedly “measured” to be horizontal. After a number of passes N = π/Δθ/2, the photon is switched out of the interferometer and is transmitted through the external polarizing beamsplitter where it is detected at the H-photon detector.

Now consider what happens when the bomb IS NOT present. This time, even though there is a high amplitude for the transmitted photon, there is that Δθ amplitude for reflection out the V port. This small V-amplitude, when it reflects from the mirror, recombines with the H-amplitude at the polarizing beamsplitter to produce a polarization that has the same tilted polarizaton that it started with, sending it back in the direction from which it came. (In this situation, the detector on the “dark” port of the internal beamsplitter never sees the photon because of destructive interference along this path.) The photon is then rotated once more by the polarization rotator, and the photon polarization is rotated again, etc.. Now, after a number of passes N = π/Δθ/2, the photon has acquired a V polarization and is switched out of the interferometer. At the external polarizing beamsplitter it is reflected out of the V-port where it is detected at the V-photon detector.

Fig. 3  Quantum Zeno effect for interaction-free measurement.  If the bomb is present, the H-photon detector detects the output photon without setting it off.  The switchable mirror ejects the photon after it makes π/Δθ/2 round trips in the polarizing interferometer.

The two end results of this thought experiment are absolutely distinct, giving a clear answer to the question whether the bomb is present or not. If the bomb IS present, the H-detector fires. If the bomb IS NOT present, then the V-detector fires. Through all of this, the chance to set off the bomb is almost zero. Therefore, this quantum Zeno interaction-free measurement detects the bomb with nearly 100% efficiency with almost no chance of setting it off. This is the amazing consequence of quantum physics. The wavefunction is affected by the presence of the bomb, altering the interference effects that allow the polarization to rotate. But the likelihood of a photon being detected by the bomb is very low.

On a side note: Although ultrafast switchable mirrors do exist, the experiment was much easier to perform by creating a helix in the optical path through the system so that there is only a finite number of bounces of the photon inside the cavity. See Ref. [2] for details.

In conclusion, the ability for a quantum system to see an object with light, without exposing it, is uniquely a quantum phenomenon that has no classical analog.  No E&M wave description can explain this effect.

Further Reading

I first wrote about quantum seeing the dark in my 2001 book on the future of optical physics and technology: Nolte, D. D. (2001). Mind at Light Speed : A new kind of intelligence. (New York, Free Press)

More on the story of Feynman and Wheeler and what they were trying to accomplish is told in Chapter 8 of Galileo Unbound on the physics and history of dynamics: Nolte, D. D. (2018). Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press).

Paul Kwiat introduced to the world to interaction-free measurements in 1995 in this illuminating Scientific American article: Kwiat, P., H. Weinfurter and A. Zeilinger (1996). “Quantum seeing in the dark – Quantum optics demonstrates the existence of interaction-free measurements: the detection of objects without light-or anything else-ever hitting them.” Scientific American 275(5): 72-78.

References

[1] Elitzur, A. C. and L. Vaidman (1993). “QUANTUM-MECHANICAL INTERACTION-FREE MEASUREMENTS.” Foundations of Physics 23(7): 987-997.

[2] Kwiat, P., H. Weinfurter, T. Herzog, A. Zeilinger and M. A. Kasevich (1995). “INTERACTION-FREE MEASUREMENT.” Physical Review Letters 74(24): 4763-4766.