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 particle, massless ...
interference effect in
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
that was demonstrated in 1987 by three physicists from the University of Rochester: Chung Ki Hong (홍정기), Zheyu Ou (区泽宇), 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 ...
.
[
] The effect occurs when two identical single-photon waves enter a 1:1
beam splitter
A beam splitter or ''beamsplitter'' is an 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 interferometers, also finding wid ...
, 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
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviol ...
.
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) is a paradigm of quantum computation, allowing (under certain conditions, described below) universal quantum computation. LOQC uses photons as information carriers, main ...
(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 reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electroni ...
of the beam splitter. Here, we assume a 1:1 beam splitter, in which a photon has equal
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
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. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
).

Since the state of the beam splitter does not "record" which of the four possibilities actually happens,
Feynman rules 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 modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the qu ...
level. In addition, reflection from the bottom side of the beam splitter introduces a relative
phase shift
In physics and mathematics, the phase of a 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 denoted \phi(t) and expressed in such a scale that it ...
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 waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
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
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
,
, and
,
. 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 imp ...
s, so, for example
corresponds to mode ''a'' empty (the vacuum state), and inserting one photon into ''a'' corresponds to
, etc. A photon in each input mode is therefore
:
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.
, and similarly for inserting a photon in ''b''. Therefore
:
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 most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
form:
:
Similar transformations hold for the annihilation 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
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally cons ...
above).
When two photons enter the beam splitter, one on each side, the state of the two modes becomes
:
:
where we used
etc.
Since the commutator of the two creation operators
and
is zero because they operate on different spaces, the product term vanishes. The surviving terms in the superposition are only the
and
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 sensors of light or other electromagnetic radiation. There is a wide variety of photodetectors which may be classified by mechanism of detection, such as photoelectric or photochemical effects, or by ...
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
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of the single-photon
wave packet
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
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 states (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 an inf ...
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 of a solid-state single-photon source by feeding two successive photons from the source into a 1:1 beam splitter. The
interference visibility ''V'' of the dip is related to the states of the two photons
and
as
:
If
, then the visibility is equal to the purity
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
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
, and the two-photon quantum state
that leads to the HOM dip is the simplest non-trivial state in a class called
NOON states.
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
*
Photon antibunching
*
Photon bunching
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. He is a Visiting Professor in the Department of Atomic and Laser Physics at the Centre for Quantum Computation (CQC) in the Clarendon Laboratory o ...
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.
{{DEFAULTSORT:Hong-Ou-Mandel effect
Quantum optics
Interferometry