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.  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 [1].  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 shown in Fig. 1.  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 [3], 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.

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 [4, 5].  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 a and b, 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.


References

[1] 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)

[2] 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)

[3] 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)

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

[5] 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.

Who Invented the Quantum? Einstein vs. Planck

Albert Einstein defies condensation—it is impossible to condense his approach, his insight, his motivation—into a single word like “genius”.  He was complex, multifaceted, contradictory, revolutionary as well as conservative.  Some of his work was so simple that it is hard to understand why no-one else did it first, even when they were right in the middle of it.  Lorentz and Poincaré spring to mind—they had been circling the ideas of spacetime for decades—but never stepped back to see what the simplest explanation could be.  Einstein did, and his special relativity was simple and beautiful, and the math is just high-school algebra.  On the other hand, parts of his work—like gravitation—are so embroiled in mathematics and the religion of general covariance that it remains opaque to physics neophytes 100 years later and is usually reserved for graduate study. 

            Yet there is a third thread in Einstein’s work that relies on pure intuition—neither simple nor complicated—but almost impossible to grasp how he made his leap.  This is the case when he proposed the real existence of the photon—the quantum particle of light.  For ten years after this proposal, it was considered by almost everyone to be his greatest blunder. It even came up when Planck was nominating Einstein for membership in the German Academy of Science. Planck said

That he may sometimes have missed the target of his speculations, as for example, in his hypothesis of light quanta, cannot really be held against him.

In this single statement, we have the father of the quantum being criticized by the father of the quantum discontinuity.

Max Planck’s Discontinuity

In histories of the development of quantum theory, the German physicist Max Planck (1858—1947) is characterized as an unlikely revolutionary.  He was an establishment man, in the stolid German tradition, who was already embedded in his career, in his forties, holding a coveted faculty position at the University of Berlin.  In his research, he was responding to a theoretical challenge issued by Kirchhoff many years ago in 1860 to find the function of temperature and wavelength that described and explained the observed spectrum of radiating bodies.  Planck was not looking for a revolution.  In fact, he was looking for the opposite.  One of his motivations in studying the thermodynamics of electromagnetic radiation was to rebut the statistical theories of Boltzmann.  Planck had never been convinced by the atomistic and discrete approach Boltzmann had used to explain entropy and the second law of thermodynamics.  With the continuum of light radiation he thought he had the perfect system that would show how entropy behaved in a continuous manner, without the need for discrete quantities. 

Therefore, Planck’s original intentions were to use blackbody radiation to argue against Boltzmann—to set back the clock.  For this reason, not only was Planck an unlikely revolutionary, he was a counter-revolutionary.  But Planck was a revolutionary because that is what he did, whatever his original intentions were, and he accepted his role as a revolutionary when he had the courage to stand in front of his scientific peers and propose a quantum hypothesis that lay at the heart of physics.

            Blackbody radiation, at the end of the nineteenth century, was a topic of keen interest and had been measured with high precision.  This was in part because it was such a “clean” system, having fundamental thermodynamic properties independent of any of the material properties of the black body, unlike the so-called ideal gases, which always showed some dependence on the molecular properties of the gas. The high-precision measurements of blackbody radiation were made possible by new developments in spectrometers at the end of the century, as well as infrared detectors that allowed very precise and repeatable measurements to be made of the spectrum across broad ranges of wavelengths. 

In 1893 the German physicist Wilhelm Wien (1864—1928) had used adiabatic expansion arguments to derive what became known as Wien’s Displacement Law that showed a simple linear relationship between the temperature of the blackbody and the peak wavelength.  Later, in 1896, he showed that the high-frequency behavior could be described by an exponential function of temperature and wavelength that required no other properties of the blackbody.  This was approaching the solution of Kirchhoff’s challenge of 1860 seeking a universal function.  However, at lower frequencies Wien’s approximation failed to match the measured spectrum.  In mid-year 1900, Planck was able to define a single functional expression that described the experimentally observed spectrum.  Planck had succeeded in describing black-body radiation, but he had not satisfied Kirchhoff’s second condition—to explain it. 

            Therefore, to describe the blackbody spectrum, Planck modeled the emitting body as a set of ideal oscillators.  As an expert in the Second Law, Planck derived the functional form for the radiation spectrum, from which he found the entropy of the oscillators that produced the spectrum.  However, once he had the form for the entropy, he needed to explain why it took that specific form.  In this sense, he was working backwards from a known solution rather than forwards from first principles.  Planck was at an impasse.  He struggled but failed to find any continuum theory that could work. 

Then Planck turned to Boltzmann’s statistical theory of entropy, the same theory that he had previously avoided and had hoped to discredit.  He described this as “an act of despair … I was ready to sacrifice any of my previous convictions about physics.”  In Boltzmann’s expression for entropy, it was necessary to “count” possible configurations of states.  But counting can only be done if the states are discrete.  Therefore, he lumped the energies of the oscillators into discrete ranges, or bins, that he called “quanta”.  The size of the bins was proportional to the frequency of the oscillator, and the proportionality constant had the units of Maupertuis’ quantity of action, so Planck called it the “quantum of action”. Finally, based on this quantum hypothesis, Planck derived the functional form of black-body radiation.

            Planck presented his findings at a meeting of the German Physical Society in Berlin on November 15, 1900, introducing the word quantum (plural quanta) into physics from the Latin word that means quantity [1].  It was a casual meeting, and while the attendees knew they were seeing an intriguing new physical theory, there was no sense of a revolution.  But Planck himself was aware that he had created something fundamentally new.  The radiation law of cavities depended on only two physical properties—the temperature and the wavelength—and on two constants—Boltzmann’s constant kB and a new constant that later became known as Planck’s constant h = ΔE/f = 6.6×10-34 J-sec.  By combining these two constants with other fundamental constants, such as the speed of light, Planck was able to establish accurate values for long-sought constants of nature, like Avogadro’s number and the charge of the electron.

            Although Planck’s quantum hypothesis in 1900 explained the blackbody radiation spectrum, his specific hypothesis was that it was the interaction of the atoms and the light field that was somehow quantized.  He certainly was not thinking in terms of individual quanta of the light field.

Figure. Einstein and Planck at a dinner held by Max von Laue in Berlin on Nov. 11, 1931.

Einstein’s Quantum

When Einstein analyzed the properties of the blackbody radiation in 1905, using his deep insight into statistical mechanics, he was led to the inescapable conclusion that light itself must be quantized in amounts E = hf, where h is Planck’s constant and f is the frequency of the light field.  Although this equation is exactly the same as Planck’s from 1900, the meaning was completely different.  For Planck, this was the discreteness of the interaction of light with matter.  For Einstein, this was the quantum of light energy—whole and indivisible—just as if the light quantum were a particle with particle properties.  For this reason, we can answer the question posed in the title of this Blog—Einstein takes the honor of being the inventor of the quantum.

            Einstein’s clarity of vision is a marvel to behold even to this day.  His special talent was to take simple principles, ones that are almost trivial and beyond reproach, and to derive something profound.  In Special Relativity, he simply assumed the constancy of the speed of light and derived Lorentz’s transformations that had originally been based on obtuse electromagnetic arguments about the electron.  In General Relativity, he assumed that free fall represented an inertial frame, and he concluded that gravity must bend light.  In quantum theory, he assumed that the low-density limit of Planck’s theory had to be consistent with light in thermal equilibrium in thermal equilibrium with the black body container, and he concluded that light itself must be quantized into packets of indivisible energy quanta [2].  One immediate consequence of this conclusion was his simple explanation of the photoelectric effect for which the energy of an electron ejected from a metal by ultraviolet irradiation is a linear function of the frequency of the radiation.  Einstein published his theory of the quanta of light [3] as one of his four famous 1905 articles in Annalen der Physik in his Annus Mirabilis

Figure. In the photoelectric effect a photon is absorbed by an electron state in a metal promoting the electron to a free electron that moves with a maximum kinetic energy given by the difference between the photon energy and the work function W of the metal. The energy of the photon is absorbed as a whole quantum, proving that light is composed of quantized corpuscles that are today called photons.

            Einstein’s theory of light quanta was controversial and was slow to be accepted.  It is ironic that in 1914 when Einstein was being considered for a position at the University in Berlin, Planck himself, as he championed Einstein’s case to the faculty, implored his colleagues to accept Einstein despite his ill-conceived theory of light quanta [4].  This comment by Planck goes far to show how Planck, father of the quantum revolution, did not fully grasp, even by 1914, the fundamental nature and consequences of his original quantum hypothesis.  That same year, the American physicist Robert Millikan (1868—1953) performed a precise experimental measurement of the photoelectric effect, with the ostensible intention of proving Einstein wrong, but he accomplished just the opposite—providing clean experimental evidence confirming Einstein’s theory of the photoelectric effect. 

The Stimulated Emission of Light

About a year after Millikan proved that the quantum of energy associated with light absorption was absorbed as a whole quantum of energy that was not divisible, Einstein took a step further in his theory of the light quantum. In 1916 he published a paper in the proceedings of the German Physical Society that explored how light would be in a state of thermodynamic equilibrium when interacting with atoms that had discrete energy levels. Once again he used simple arguments, this time using the principle of detailed balance, to derive a new and unanticipated property of light—stimulated emission!

Figure. The stimulated emission of light. An excited state is stimulated to emit an identical photon when the electron transitions to its ground state.

The stimulated emission of light occurs when an electron is in an excited state of a quantum system, like an atom, and an incident photon stimulates the emission of a second photon that has the same energy and phase as the first photon. If there are many atoms in the excited state, then this process leads to a chain reaction as 1 photon produces 2, and 2 produce 4, and 4 produce 8, etc. This exponential gain in photons with the same energy and phase is the origin of laser radiation. At the time that Einstein proposed this mechanism, lasers were half a century in the future, but he was led to this conclusion by extremely simple arguments about transition rates.

Figure. Section of Einstein’s 1916 paper that describes the absorption and emission of light by atoms with discrete energy levels [5].

Detailed balance is a principle that states that in thermal equilibrium all fluxes are balanced. In the case of atoms with ground states and excited states, this principle requires that as many transitions occur from the ground state to the excited state as from the excited state to the ground state. The crucial new element that Einstein introduced was to distinguish spontaneous emission from stimulated emission. Just as the probability to absorb a photon must be proportional to the photon density, there must be an equivalent process that de-excites the atom that also must be proportional the photon density. In addition, an electron must be able to spontaneously emit a photon with a rate that is independent of photon density. This leads to distinct coefficients in the transition rate equations that are today called the “Einstein A and B coefficients”. The B coefficients relate to the photon density, while the A coefficient relates to spontaneous emission.

Figure. Section of Einstein’s 1917 paper that derives the equilibrium properties of light interacting with matter. The “B”-coefficient for transition from state m to state n describes stimulated emission. [6]

Using the principle of detailed balance together with his A and B coefficients as well as Boltzmann factors describing the number of excited states relative to ground state atoms in equilibrium at a given temperature, Einstein was able to derive an early form of what is today called the Bose-Einstein occupancy function for photons.

Derivation of the Einstein A and B Coefficients

Detailed balance requires the rate from m to n to be the same as the rate from n to m

where the first term is the spontaneous emission rate from the excited state m to the ground state n, the second term is the stimulated emission rate, and the third term (on the right) is the absorption rate from n to m. The numbers in each state are Nm and Nn, and the density of photons is ρ. The relative numbers in the excited state relative to the ground state is given by the Boltzmann factor

By assuming that the stimulated transition coefficient from n to m is the same as m to n, and inserting the Boltzmann factor yields

The Planck density of photons for ΔE = hf is

which yields the final relation between the spontaneous emission coefficient and the stimulated emission coefficient

The total emission rate is

where the p-bar is the average photon number in the cavity. One of the striking aspects of this derivation is that no assumptions are made about the physical mechanisms that determine the coefficient B. Only arguments of detailed balance are required to arrive at these results.

Einstein’s Quantum Legacy

Einstein was awarded the Nobel Prize in 1921 for the photoelectric effect, not for the photon nor for any of Einstein’s other theoretical accomplishments.  Even in 1921, the quantum nature of light remained controversial.  It was only in 1923, after the American physicist Arthur Compton (1892—1962) showed that energy and momentum were conserved in the scattering of photons from electrons, that the quantum nature of light began to be accepted.  The very next year, in 1924, the quantum of light was named the “photon” by the American American chemical physicist Gilbert Lewis (1875—1946). 

            A blog article like this, that attributes the invention of the quantum to Einstein rather than Planck, must say something about the irony of this attribution.  If Einstein is the father of the quantum, he ultimately was led to disinherit his own brain child.  His final and strongest argument against the quantum properties inherent in the Copenhagen Interpretation was his famous EPR paper which, against his expectations, launched the concept of entanglement that underlies the coming generation of quantum computers.


Einstein’s Quantum Timeline

1900 – Planck’s quantum discontinuity for the calculation of the entropy of blackbody radiation.

1905 – Einstein’s “Miracle Year”. Proposes the light quantum.

1911 – First Solvay Conference on the theory of radiation and quanta.

1913 – Bohr’s quantum theory of hydrogen.

1914 – Einstein becomes a member of the German Academy of Science.

1915 – Millikan measurement of the photoelectric effect.

1916 – Einstein proposes stimulated emission.

1921 – Einstein receives Nobel Prize for photoelectric effect and the light quantum. Third Solvay Conference on atoms and electrons.

1927 – Heisenberg’s uncertainty relation. Fifth Solvay International Conference on Electrons and Photons in Brussels. “First” Bohr-Einstein debate on indeterminancy in quantum theory.

1930 – Sixth Solvay Conference on magnetism. “Second” Bohr-Einstein debate.

1935 – Einstein-Podolsky-Rosen (EPR) paper on the completeness of quantum mechanics.


Selected Einstein Quantum Papers

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

Einstein, A. (1907). “Die Plancksche Theorie der Strahlung und die Theorie der spezifischen W ̈arme.” Annalen der Physik 22: 180–190.

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

Einstein, A. and O. Stern (1913). “An argument for the acceptance of molecular agitation at absolute zero.” Annalen Der Physik 40(3): 551-560.

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.

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


Notes

[1] M. Planck, “Elementary quanta of matter and electricity,” Annalen Der Physik, vol. 4, pp. 564-566, Mar 1901.

[2] Klein, M. J. (1964). Einstein’s First Paper on Quanta. The natural philosopher. D. A. Greenberg and D. E. Gershenson. New York, Blaidsdell. 3.

[3] A. Einstein, “Generation and conversion of light with regard to a heuristic point of view,” Annalen Der Physik, vol. 17, pp. 132-148, Jun 1905.

[4] Chap. 2 in “Mind at Light Speed”, by David Nolte (Free Press, 2001)

[5] Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.

[6] Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.