Hong–Ou–Mandel Effect
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The Hong–Ou–Mandel effect is a two-
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...
effect in
quantum optics Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction ...
that was demonstrated in 1987 by Chung Ki Hong (), Zheyu Jeff Ou () and
Leonard Mandel Leonard Mandel (May 9, 1927 – February 9, 2001) was an American physicist who contributed to the development of theoretical and experimental modern optics and is widely considered one of the founding fathers of the field of quantum optics. With ...
at the
University of Rochester The University of Rochester is a private university, private research university in Rochester, New York, United States. It was founded in 1850 and moved into its current campus, next to the Genesee River in 1930. With approximately 30,000 full ...
:. The effect occurs when two identical single photons enter a 1:1
beam splitter A beam splitter or beamsplitter is an optical instrument, optical device that splits a beam of light into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as Interferometry, int ...
, one in each input port. When the temporal overlap of the photons on the beam splitter is perfect, the two photons will always exit the beam splitter together in the same output mode, meaning that there is zero chance that they will exit separately with one photon in each of the two outputs giving a coincidence event. The photons have a 50:50 chance of exiting (together) in either output mode. If they become more distinguishable (e.g. because they arrive at different times or with different wavelength), the probability of them each going to a different detector will increase. In this way, the interferometer coincidence signal can accurately measure bandwidth, path lengths, and timing. Since this effect relies on the existence of photons and the
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
, it can not be fully explained by classical optics. The effect provides one of the underlying physical mechanisms for logic gates in
linear optical quantum computing Linear optical quantum computing or linear optics quantum computation (LOQC), also photonic quantum computing (PQC), is a paradigm of Quantum computing, quantum computation, allowing (under certain conditions, described below) universal quantum com ...
(the other mechanism being the action of measurement).


Quantum-mechanical description


Physical description

When a photon enters a beam splitter, there are two possibilities: it will either be reflected or transmitted. The relative probabilities of transmission and reflection are determined by the
reflectivity The reflectance of the surface of a material is its effectiveness in Reflection (physics), reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the respon ...
of the beam splitter. Here, we assume a 1:1 beam splitter, in which a photon has equal
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
of being reflected and transmitted. Next, consider two photons, one in each input mode of a 1:1 beam splitter. There are four possibilities regarding how the photons will behave: # The photon coming in from above is reflected and the photon coming in from below is transmitted. # Both photons are transmitted. # Both photons are reflected. # The photon coming in from above is transmitted and the photon coming in from below is reflected. We assume now that the two photons are identical in their physical properties (i.e.,
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
, spatio-temporal mode structure, and
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
). Since the state of the beam splitter does not "record" which of the four possibilities actually happens,
Feynman rules In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
dictates that we have to add all four possibilities at the
probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity at a point in space represents a probability density at that point. Probability amplitu ...
level. In addition, reflection from the bottom side of the beam splitter introduces a relative
phase shift In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
of π, corresponding to a factor of −1 in the associated term in the superposition. This sign is required by the reversibility (or unitarity of the quantum evolution) of the beam splitter. Since the two photons are identical, we cannot distinguish between the output states of possibilities 2 and 3, and their relative minus sign ensures that these two terms cancel. This cancelation can be interpreted as
destructive interference In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater amplitude (constructive in ...
of the transmission/transmission and reflection/reflection possibilities. If a detector is set up on each of the outputs then coincidences can never be observed, while both photons can appear together in either one of the two detectors with equal probability. A classical prediction of the intensities of the output beams for the same beam splitter and identical coherent input beams would suggest that all of the light should go to one of the outputs (the one with the positive phase).


Mathematical description

Consider two optical input modes ''a'' and ''b'' that carry annihilation and creation operators \hat, \hat^\dagger, and \hat, \hat^\dagger. Identical photons in different modes can be described by the
Fock state In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an im ...
s, so, for example , 0\rangle_a corresponds to mode ''a'' empty (the vacuum state), and inserting one photon into ''a'' corresponds to , 1\rangle_a=\hat^\dagger, 0\rangle_a, etc. A photon in each input mode is therefore : , 1, 1\rangle_ = \hat^\dagger \hat^\dagger , 0, 0\rangle_. When the two modes ''a'' and ''b'' are mixed in a 1:1 beam splitter, they produce output modes ''c'' and ''d''. Inserting a photon in ''a'' produces a superposition state of the outputs: if the beam splitter is 50:50 then the probabilities of each output are equal, i.e. \hat^\dagger , 0\rangle_a \to \frac\left( \hat^\dagger + \hat^\dagger\right), 00\rangle_, and similarly for inserting a photon in ''b''. Therefore : \hat^\dagger \to \frac \quad\text\quad \hat^\dagger \to \frac. The relative minus sign appears because the classical lossless beam splitter produces a unitary transformation. This can be seen most clearly when we write the two-mode beam splitter transformation in
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
form: : \begin \hat \\ \hat \end \to \frac \begin 1 & 1 \\ 1 & -1 \end \begin \hat \\ \hat \end. Similar transformations hold for the creation operators. Unitarity of the transformation implies unitarity of the matrix. Physically, this beam splitter transformation means that reflection from one surface induces a relative phase shift of π, corresponding to a factor of −1, with respect to reflection from the other side of the beam splitter (see the Physical description above). When two photons enter the beam splitter, one on each side, the state of the two modes becomes : , 1, 1\rangle_ = \hat^\dagger \hat^\dagger , 0, 0\rangle_ \to \frac \left( \hat^\dagger + \hat^\dagger \right) \left( \hat^\dagger - \hat^\dagger \right) , 0, 0\rangle_ : = \frac \left( \hat^ - \hat^ \right) , 0, 0\rangle_ = \frac, where we used \hat^, 0, 0\rangle_=\hat^\dagger, 1, 0\rangle_=\sqrt, 2, 0\rangle_ etc. Since the commutator of the two creation operators \hat^\dagger and \hat^\dagger is zero because they operate on different spaces, the product term vanishes. The surviving terms in the superposition are only the \hat^ and \hat^ terms. Therefore, when two identical photons enter a 1:1 beam splitter, they will always exit the beam splitter in the same (but random) output mode. The result is non-classical: a classical light wave entering a classical beam splitter with the same transfer matrix would always exit in arm ''c'' due to destructive interference in arm ''d'', whereas the quantum result is random. Changing the beam splitter phases can change the classical result to arm ''d'' or a mixture of both, but the quantum result is independent of these phases. For a more general treatment of the beam splitter with arbitrary reflection/transmission coefficients, and arbitrary numbers of input photons, see the general quantum mechanical treatment of a beamsplitter for the resulting output Fock state.


Experimental signature

Customarily the Hong–Ou–Mandel effect is observed using two
photodetector Photodetectors, also called photosensors, are devices that detect light or other forms of electromagnetic radiation and convert it into an electrical signal. They are essential in a wide range of applications, from digital imaging and optical ...
s monitoring the output modes of the beam splitter. The coincidence rate of the detectors will drop to zero when the identical input photons overlap perfectly in time. This is called the ''Hong–Ou–Mandel dip'', or HOM dip. The coincidence count reaches a minimum, indicated by the dotted line. The minimum drops to zero when the two photons are perfectly identical in all properties. When the two photons are perfectly distinguishable, the dip completely disappears. The precise shape of the dip is directly related to the
power spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...
of the single-photon
wave packet In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an Envelope (waves), envelope. A wave packet can be analyzed into, or can be synthesized from, a ...
and is therefore determined by the physical process of the source. Common shapes of the HOM dip are
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
and Lorentzian. A classical analogue to the HOM effect occurs when two
coherent state In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
s (e.g. laser beams) interfere at the beamsplitter. If the states have a rapidly varying phase difference (i.e. faster than the integration time of the detectors) then a dip will be observed in the coincidence rate equal to one half the average coincidence count at long delays. (Nevertheless, it can be further reduced with a proper discriminating trigger level applied to the signal.) Consequently, to prove that destructive interference is two-photon quantum interference rather than a classical effect, the HOM dip must be lower than one half. The Hong–Ou–Mandel effect can be directly observed using single-photon-sensitive
intensified ''Intensified'' is an album by Desmond Dekker & the Aces released in 1970. Track listing All tracks composed by Desmond Dekker; except where indicated #"It Mek" (Dekker, Leslie Kong Leslie Kong (20 December 1933 – 9 August 1971) was a Jam ...
cameras. Such cameras have the ability to register single photons as bright spots clearly distinguished from the low-noise background. In the figure above, the pairs of photons are registered in the middle of the Hong–Ou–Mandel dip. In most cases, they appear grouped in pairs either on the left or right side, corresponding to two output ports of a beam splitter. Occasionally a coincidence event occurs, manifesting a residual distinguishability between the photons.


Applications and experiments

The Hong–Ou–Mandel effect can be used to test the degree of indistinguishability of the two incoming photons. When the HOM dip reaches all the way down to zero coincident counts, the incoming photons are perfectly indistinguishable, whereas if there is no dip, the photons are distinguishable. In 2002, the Hong–Ou–Mandel effect was used to demonstrate the
purity Purity may refer to: Books * ''Pureza'' (novel), a 1937 Brazilian novel by José Lins do Rego * ''Purity'' (novel), a 2015 novel by Jonathan Franzen ** ''Purity'' (TV series), a TV series based on the novel *''Purity'', a 2012 novel by Jackson ...
of a solid-state single-photon source by feeding two successive photons from the source into a 1:1 beam splitter. The
interference visibility The interferometric visibility (also known as interference visibility and fringe visibility, or just visibility when in context) is a measure of the contrast of ''interference'' in any system subject to wave superposition. Examples include as opt ...
''V'' of the dip is related to the states of the two photons \rho_a and \rho_b as : V = \operatorname(\rho_ \rho_). If \rho_a = \rho_b = \rho, then the visibility is equal to the purity P = \operatorname(\rho^2) of the photons. In 2006, an experiment was performed in which two atoms independently emitted a single photon each. These photons subsequently produced the Hong–Ou–Mandel effect. Multimode Hong–Ou–Mandel interference was studied in 2003. The Hong–Ou–Mandel effect also underlies the basic entangling mechanism in linear optical
quantum computing A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using s ...
, and the two-photon quantum state , 2,0\rangle + , 0,2\rangle that leads to the HOM dip is the simplest non-trivial state in a class called
NOON state In quantum optics, a NOON state or N00N state is a quantum-mechanical many-body entangled state: : , \text \rangle = \frac, \, which represents a superposition of ''N'' particles in mode ''a'' with zero particles in mode ''b'', and vice ver ...
s. In 2015 the Hong–Ou–Mandel effect for photons was directly observed with spatial resolution using an sCMOS camera with an image intensifier. Also in 2015 the effect was observed with helium-4 atoms. The HOM effect can be used to measure the biphoton wave function from a spontaneous
four-wave mixing Four-wave mixing (FWM) is an intermodulation phenomenon in nonlinear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave ...
process. In 2016 a frequency converter for photons demonstrated the Hong–Ou–Mandel effect with different-color photons. In 2018, HOM interference was used to demonstrate high-fidelity quantum interference between topologically protected states on a photonic chip. Topological photonics have intrinsically high-coherence, and unlike other quantum processor approaches, do not require strong magnetic fields and operate at room temperature.


Three-photon interference

Three-photon interference effect has been identified in experiments.


See also

*
Degree of coherence In quantum optics, correlation functions are used to characterize the statistical and Coherence (physics), coherence properties – the ability of waves to interfere – of electromagnetic radiation, like optical light. Higher order coherence or ...
*
Photon antibunching Photon antibunching generally refers to a light field with photons more equally spaced than a coherent laser field, a signature being a measured two-time correlation suppressed below that of a coherent laser field. More specifically, it can refe ...
*
Photon bunching In physics, the Hanbury Brown and Twiss (HBT) effect is any of a variety of correlation and anti-correlation effects in the intensities received by two detectors from a beam of particles. HBT effects can generally be attributed to the wave–par ...


References


External links


Lectures on Quantum Computing: Interference (2 of 6)
-
David Deutsch David Elieser Deutsch ( ; ; born 18 May 1953) is a British physicist at the University of Oxford, often described as the "father of quantum computing". He is a visiting professor in the Department of Atomic and Laser Physics at the Centre for ...
lecture video, video of related experiment (a single photon in a sharp direction is split, mirrored and rejoined in a second splitter (joiner) output in the sharp direction).
Can Two-Photon Interference be Considered the Interference of Two Photons?
- Discussion of the interpretation of the HOM interferometer results.
YouTube animation showing HOM effect in a semiconductor device.

YouTube movie showing experimental results of HOM effect observed on a camera.

Hong-Ou-Mandel in the Virtual Lab by Quantum Flytrap, an interactive simulation
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{{DEFAULTSORT:Hong-Ou-Mandel effect Quantum optics Interferometry