Thirty years ago, Len Mandel, together with his students Jeff Ou and C. K. Hong, published a description of a remarkably simple experiment (*1*), the consequences of which have had dramatic implications for quantum science and technology. The experiment consisted of sending two elementary particles of light, photons, onto opposite sides of a piece of glass that had been coated with a thin film to give it a reflectivity of 50% (see the figure). They observed that the two photons always left by the same side at the output, though it was not possible to determine beforehand which side that would be. This is completely surprising if one considers the behavior of particles as indivisible, identical entities obeying the laws of classical physics. It is easy to see that in such a circumstance, one might expect there to be four equally likely output configurations, only two of which consist of the particles occupying the same outputs. The phenomenon is a beautiful manifestation of the interference of a quantum field; in this case, the bosonic field associated with photons. The effect would be entirely the opposite with fermionic fields associated with electrons—both particles would always leave by different sides.

In their paper, Hong, Ou, and Mandel (HOM) emphasized the role of distinguishing information in determining the extent to which interference occurred, arguing that the mere presence of such information, whether measured or not, would abrogate the “bunching” effect. Hence, they argued that this would be a good way to measure the time delay between two single photons using a slow-responding detector. If the photons were delayed from each other so that they did not arrive together at the beamsplitter, then they would not exit together, and thus would not result in the two detectors at the output ports both registering a signal simultaneously. The change in the rate at which the detectors register in coincidence as the delay is varied gives the signature HOM “dip” (see the figure), which provides a test of the nonclassical nature of the input light.

Quantum interference between two particles has many subtle effects, including that for entangled states of the photons (states in which the photons are in quantum superpositions of two possible distinguishable configurations; for example, the photon at the upper port is red and that at the lower port blue, or vice versa), it is not even necessary that they arrive at the beamsplitter at the same time. The probability *P* that two photons input at different ports exit at different ports is proportional to the square of the difference between the two-photon wave functions at the input and at the output of the beamsplitter. Whether or not this leads to complete suppression of the probability (that is, *P* = 0) in a 50:50 beamsplitter depends on the symmetry of the two-photon wave function under exchange of their quantum numbers (except for the port number) (*2*). If the two photons are identical in all respects (for example, the same polarization, color, arrival time, spatial shape, and direction), then exchanging any of the properties of the photons leaves the two-photon wave function unchanged. If some of the characteristics are different, then exchanging those can change the wave function. If it does change, then sometimes the two photons will exit at opposite ports, in contrast to the case first demonstrated by Hong, Ou, and Mandel. Entangled states where the photons arrive separately can still satisfy the symmetry constraints.

This property of interference can be generalized to more complicated arrangements of beamsplitters. Imagine three beamsplitters arranged sequentially in a network that has three input and output ports, thereby allowing three photons to interfere. In this case, the probability that each photon exits through a different port depends on the differences between the input and output three-photon wave functions under all possible partial exchanges of the photon quantum numbers. Here, there are three pairwise exchanges and two threefold exchanges. The various exchange permutations accumulate different signs, depending on the exact arrangement of the circuit (*3*).

For circuits with *N* input and *N* output ports, determining the probability amplitudes associated with all of these permutations turns out to be a computationally hard problem. Indeed, it has been shown that even sampling from the output distribution of photons for a network of sufficiently large *N* is beyond the capability of the most powerful classical computers. Further, if it can be shown that a quantum machine could perform this sampling more rapidly than a conventional computer, then it is suggested that this challenges some of the fundamental tenets of computer science (*4*).

Another important feature of HOM interference is that it produces entangled states at the output from unentangled states at the input. For instance, the output state for two input photons, one at each input port, that interfere completely (*P* = 0 at the output of the beamsplitter) is a superposition state consisting of both two photons in one output port and two photons in the other.

This opens the door to some new applications in sensing, communications, simulation, and even computing. For instance, if these two beams are recombined at a second beamsplitter, then, by symmetry, the probability of getting one photon at each output of the second beamsplitter is unity (*P* = 1 at the output of the interferometer). If a phase shift ϕ is applied to one of the beams after the first beamsplitter, the probability of one photon at each output port varies periodically as *P* = ½(1 − cos 2ϕ). The resultant interference fringes (peaks or troughs of this pattern, where *P* = 1 or 0, respectively) occur twice as frequently as ϕ is varied as in a classical interferometer using the same wavelength and phase shifter (*5*, *6*). This feature of two-photon interferometry forms the basis of a sensor with a higher precision for a given amount of light at the input: Quantum states of two photons yield a more precise estimate of the phase shift than does, say, a laser beam attenuated to have on average two photons, because the signal-to-noise ratio is improved. This idea scales to larger numbers of photons, and multiphoton interferometers can attain the so-called Heisenberg limit of precision, the tightest bound known for parameter estimation (*7*, *8*).

Another application arising directly from HOM interference is the possibility of a quantum logic gate for photons. Consider the case where each individual photon can encode a bit of information. For example, a single photon could occupy the two input ports of a beamsplitter. One port would encode a logical “0” and the other a logical “1.” The quantum character of the bit, a qubit, is reflected in the range of possible superpositions of 0 and 1 that are encoded in the photon that occupies the two ports simultaneously. A beamsplitter performs a single qubit logic operation that takes, for instance, a photon in state 0 at the input into a superposition of 0 and 1 at the output.

The other important ingredient for quantum computing is a two-qubit logic gate. This can change the sign of one qubit contingent on the state of a second. So, for instance, if two photons were in state 00 at the input, then they would remain so at the output, whereas if they were in state 11, the state would change sign at the output, to −11. This is exactly what the HOM effect does. If two photons are incident on the beamsplitter at different ports and both exit at different ports, the phase changes upon reflection that lead to the HOM effect also lead to the change of sign in the 11 component of the two-photon wave function when the beamsplitter is unbalanced. That is, the probability that a photon is reflected is not equal to the probability that it is transmitted (*9*).

A feature of all photonic quantum logic gates based on the HOM effect is that they are necessarily probabilistic—you can't predict with certainty whether both photons will exit through different output ports. And this means that sophisticated schemes are needed to ensure that arrays of such logic gates can be used to build a quantum computer, but this is possible in principle (*10*, *11*). HOM opened the door into a realm of new discovery about the nature of quantum particles as well as potential applications in information science that connect with very disparate fields, such as computational complexity and number theory. Its profound beauty lies in the simple way in which it harnesses a purely quantum statistical property to unlock this door.

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