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.