View source for Physics:Quantum optics beam splitter experiments

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<ref name="29E">Thorlabs 50:50 (R:T) non‑polarizing beamsplitter cube. Example product: BS005, 700‑1100&nbsp;nm. Manufacturer page: <a href="https://www.thorlabs.com/thorproduct.cfm?partnumber=BS005">https://www.thorlabs.com/thorproduct.cfm?partnumber=BS005</a>.</ref>

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Theory for the beam splitter (BS)  in quantum optics, quantum entanglement of photons and their statistics, the HOM effect, is well developed and based on fairly simple mathematical and physical foundations. This theory has been developed for any type of BS and is based on the constancy of the reflection coefficients R (or the transmission coefficient, where R + T = 1) and the phase shift ϕ. The constancy of these coefficients cannot always be satisfied for a waveguide BS, where R and ϕ depend in a special way on photon frequencies. Based on this, the concept of BS systematizes in quantum optics into "Conventional" and frequency-dependent BS, and also confirms the theory of such BS.The quantum entanglement, photon statistics at the output ports, and the Hong-Ou-Mandel (HOM) effect for such BS can be very different. Taking into account the fact that the waveguide BS is currently acquiring an important role in quantum technologies due to the possibility of its miniaturization, this article will be useful not only for theoreticians, but also for experimenters.
[[File:Quantum_beam_splitter_interference_yellow1.jpg|thumb|400px|Beam splitter experiment: an incoming photon is placed into a superposition of two paths, which can interfere when recombined.]]
{{Infobox
| title        = Beam splitter
| name         = Beam splitter
| image        = [[File:Entangled photons from type-I SPDC with a Beam Splitter.svg|center|320px]]
| caption      = Diagram of entangled photon generation: A pump beam induces type-I spontaneous parametric down-conversion (SPDC) in a nonlinear crystal, producing a polarization-entangled photon pair (signal and idler modes). The pair is input to a 50:50 beam splitter,[https://www.thorlabs.com/non-polarizing-cube-beamsplitters-700---1100-nm?tabName=Overview 700-1100nm]creating path-entangled output modes for quantum experiments like Bell tests.
| headerstyle  = background:#e0f0ff;
| labelstyle   = background:#f0f8ff; width: 43%;
| bodystyle    = width: 24em;

| label1       = Other names
| data1        = BS, directional coupler (in waveguide form)
| label2       = Primary uses
| data2        = [[Wikipedia:Quantum superposition|Quantum superposition]] · [[Wikipedia:Optical cluster state#Polarization encoding|Entanglement generation]] · [[Wikipedia:Photon statistics|Photon statistics]] · [[Wikipedia:Hong–Ou–Mandel effect|Hong–Ou–Mandel interferometry]] · [[Wikipedia:Linear optical quantum computing|Linear-optical quantum computing]] · [[Wikipedia:Quantum metrology|Quantum metrology]] · [[Wikipedia:Quantum channel|Quantum communication]]
}}
	
[[File:InfraTec Beamsplitter Principle.jpg|thumb|Schematic diagram of a reflection beam splitter in a pyroelectric infrared detector (four optical channels)]]
[[File:Hong-Ou-Mandel interference schematic.jpg|thumb|Schematic of the Hong–Ou–Mandel effect: Two indistinguishable photons incident on a 50:50 beam splitter exhibit quantum interference, leading to photon bunching (both photons exit the same port).On-chip HOM interference visibilities exceeding 0.97 were observed using independent molecular sources. <ref name="03K" /> Additionally, quantum beating was observed when a frequency detuning was applied between two molecules, persisting for over 100 µs. <ref name="04K" />]]
===Introduction to the beam splitter in quantum optics===

Historical developments in beam splitting range from '''Fizeau’s 1851'''<ref name="Fizeau1851" /> interference measurements to the development of the '''Michelson interferometer'''. The transition to the quantum regime occurred in 1987 with the first experimental demonstration of the HOM effect.<ref name="04E" /> The [[Wikipedia:KLM protocol|KLM protocol]](2001) demonstrated that universal linear optical quantum computing is possible by using only beam splitters, phase shifters, and single-photon detectors.It uses a process called Measurement-Induced Nonlinearity.<ref name="ML03" /><ref name="06E" />

== Recent years ==
Have witnessed significant progress in quantum communication and quantum internet with the emerging quantum photonic chips, whose characteristics of scalability, stability, and low cost, open up new possibilities in miniaturized essentials. This provides an overview of the advances in quantum photonic chips for quantum communication, beginning with a summary of the prevalent photonic integrated fabrication platforms and key components for integrated quantum communication systems. Then discusses a range of quantum communication applications, such as quantum key distribution and quantum teleportation. Finally, the review culminates with a perspective on challenges towards high-performance chip-based quantum communication, as well as a glimpse into future opportunities for integrated quantum networks. Recent advancements in '''integrated quantum photonics''' focus on on-chip beam splitters fabricated on silicon, silicon nitride, and femtosecond-laser-written waveguides. These platforms enable high-fidelity interference (visibilities <math>0.97</math>), even when utilizing independent molecular single-photon sources<ref name="01K" /><ref name="02K" /><ref name="64K" /> important for the scalability of the [[Wikipedia:Quantum network|quantum internet]].<ref name="22G" /><ref name="23G" />

===='''Keywords''':====
[[Wikipedia:Beam splitter|Beam splitter]], [[Wikipedia:Photonic integrated circuit|Integrated photonics]], [[Physics:Quantum information theory|Quantum information]],[[Wikipedia:waveguide beam splitter|waveguide beam splitter]], [[Wikipedia::quantum entanglement|quantum entanglement]], [[Wikipedia:photons|photons]], [[Wikipedia:reflection coefficient|reflection coefficient]], [[Wikipedia:phase shift|phase shift]], [[Wikipedia:photon statistics|photon statistics]], [[Wikipedia:Hong-Ou-Mandel effect|Hong-Ou-Mandel effect]].

===Quantum optical classifier===
Superexponential speedup classification is a central task in [[Wikipedia:Neural network (machine learning)|deep learning]] algorithms. Usually, images are first captured and then processed by a sequence of operations, of which the [[Wikipedia:Artificial neuron|artificial neuron]] represents one of the fundamental units. This paradigm requires significant resources that scale (at least) linearly in the image resolution, both in terms of photons and computational operations. Present is a [[Wikipedia:Quantum neural network|quantum optical pattern recognition]] method for [[Wikipedia:Binary classification|binary classification]] tasks. It classifies objects without reconstructing their images, using the rate of [[Wikipedia:Hong–Ou–Mandel effect|two-photon coincidences]] at the output of a Hong-Ou-Mandel [[Wikipedia:Interferometry|interferometer]], where both the input and the classifier parameters are encoded into single-photon states. This method exhibits the behaviour of a classical neuron of unit depth. Once trained, it shows a constant <math>\mathcal{O}(1)</math> complexity in the number of computational operations and photons required by a single classification. This is a superexponential advantage over a classical artificial neuron.

===On-chip integration===
Of independent channels of indistinguishable single photons is a prerequisite for scalable [[Wikipedia:Linear optical quantum computing|optical quantum information processing]]. This requires separate solid-state single-photon emitters to exhibit identical lifetime-limited transitions. This challenging task is usually further exacerbated by spectral diffusion due to complex charge noise near material surfaces made by nanofabrication processes. A molecular [[Wikipedia:Integrated quantum photonics|quantum photonic chip]] that demonstrate on-chip Hong–Ou–Mandel quantum interference of indistinguishable single photons from independent molecules is developed. The molecules are embedded in a single-crystalline organic nanosheet and integrated with single-mode waveguides without nanofabrication, thereby ensuring stable, lifetime-limited transitions. With the aid of Stark tuning, 100 waveguide-coupled molecules can be tuned to the same frequency and achieve on-chip Hong–Ou–Mandel interference visibilities exceeding 0.97 for 2 molecules separately coupled to 2 waveguides. For two molecules with a controlled frequency difference, over 100-µs-long quantum beating in the interference, showing both excellent single-photon purity (particle nature) and long coherence (wave nature) of the emission.The results show a possible strategy towards constructing scalable optical universal quantum processors and a promising platform for studying waveguide quantum electrodynamics with identical single emitters wired via photonic circuits.

===Integrated Photonics in Quantum Technologies===
Integrated photonics in quantum technologies <ref name="01H" /><ref name="02H" />. The advantages of single-photon state encoding are several and include the lack of decoherence phenomena, the possibility to realize information processing at room temperature and to send photons through fibers and free space channels. In the last ten years, improvements in photonic quantum technologies enabled an increase in the complexity of the implemented system, supporting relevant advances in various branches of quantum information, including the demonstration of quantum advantage <ref name="03H" /><ref name="04H" /><ref name="05H" /> and satellite quantum communications <ref name="06H" /><ref name="07H" />.<br>
Indistinguishable single photons are a fundamental resource for optical quantum technologies<ref name="01K"/><ref name="02K"/><ref name="03K"/>, underpinning universal quantum computing, quantum simulation and quantum networks. Although recent demonstrations of some preliminary quantum photonic applications primarily rely on parametric-process-based single-photon sources<ref name="04K"/><ref name="05K"/>, deterministic sources offer greater future promise<ref name="02K"/><ref name="06K"/><ref name="07K"/><ref name="08K"/><ref name="09K"/>. Solid-state single-quantum emitters, such as quantum dots<ref name="02K"/><ref name="09K"/>, colour centres<ref name="10K"/><ref name="11K"/><ref name="12K"/><ref name="13K"/> and organic molecules<ref name="14K"/><ref name="15K"/>, could serve as a versatile platform

==Overview==
[[File:Quantum teleportation video.ogg|thumb|upright=1.0|Video depicting the quantum teleportation protocol. The goal is to send a quantum state Q from one station, A, to another station, B. At first, a pair of entangled particles is distributed to A and B, which pair is shown as two particles connected by a wavy line and produced by source S. Once this preparation step is finished, the quantum teleportation itself begins. Station A measures its entangled particle together with the particle in state Q and obtains one of four possible results. These results are represented by different positions of an arrow in a "clock". The result is communicated to station B via the classical channel, represented as "radio waves". Based on the received message, station B chooses an appropriate device and applies it to its particle. In the video, the specific result measured by A is represented by an arrow pointing to the bottom right corner and so station B applies the bottom-right device. After the particle leaves the device, its state is Q, which is equal to the original state of the particle at station A. This way, the quantum teleportation of state Q is successfully completed.]]
Quantum states of light are basic resources for the realization of quantum information processing tasks, starting from pioneering experiments of quantum non-locality and [[Wikipedia:Quantum teleportation|quantum teleportation]]<ref name="20G" /><ref name="21G" /> and extending to modern quantum communication and computation efforts. The transition from bulk optics to integrated photonic circuits has been essential for scaling these technologies, enabling the miniaturization of complex [[Wikipedia:Interferometry|interferometric networks]] on a single chip. The advantages of single-photon encoding include resistance to decoherence effects, the possibility of operation in an ambient temperature environment, and the ability to transfer photons via an optical fiber as well as free space communication links. The last decade has marked a growing complexity of photonic quantum technology efforts that have made possible the enhancement of quantum advantage experiments<ref name="02H" /><ref name="03H" /><ref name="04H" /> and quantum communication via satellites <ref name="05H" /><ref name="06H" />

An essential enabling technology in these advances is the coupling of photonic device components supporting the generation, manipulation, and detection of quantum states <ref name="07H" /><ref name="08H" /><ref name="09H" />. On-chip integration of independent channels of indistinguishable single photons is a prerequisite for scalable optical quantum information processing. Integrated photonics enables the realization of waveguides and reconfigurable optical components, which in turn make possible multi-port reprogrammable optical networks, and most recently, integrated processors merging both quantum state preparation and quantum processing.  A molecular quantum photonic chip has been developed, demonstrating on-chip Hong–Ou–Mandel quantum interference of indistinguishable single photons emitted from independent molecules..This requires separate solid-state single-photon emitters to exhibit identical lifetime-limited transitions.While the integration of single-photon detectors is still a challenge, some very promising advances have been made in recent years towards fully integrated photonic platforms. Compared to conventional discrete optical platforms, which demand a very careful alignment of discrete components, experience stability problems, and face cost scalability, quantum photonic chips on a microchip offer advantages in miniaturization, scalability, stability, and potentially low cost mass production. In this sense, quantum photonic chips constitute a highly promising platform for applied quantum communication, specifically in quantum key distribution (QKD), quantum secure direct communication, quantum teleportation<ref name="89H" /><ref name="50H" /> , and, in general, in quantum networks.

Of all the necessary components of an integrated photonic circuit, beam splitter (BS) is an integral part of it. The theoretical foundations of BS in quantum optics and its relation to photon statistics, entanglement, and other phenomena like Hong-Ou-Mandel effect have long been established. The recent theoretical interest has particularly underscored how waveguide BSs can differ in terms of reflection and transmission coefficients for different frequencies, going against the conventional way of designing a beam splitter. As waveguide BSs play a vital role in designing scaled-down and scalable quantum optical components, a thorough understanding of both conventional and frequency-dependent beam splitters is necessary for carrying out experiments in integrated quantum communication.<br>[https://lab.quantumflytrap.com/lab/quantum-teleportation?mode=waves An interactive simulation of quantum teleportation in the Virtual Lab by Quantum Flytrap,]

== History ==
[[File:Table of Opticks, Cyclopaedia, Volume 2.jpg|thumb|1728 Cyclopeadia. Drawings of optical equipment]]
====Key milestones:====
* 1851: The Fizeau experiment to measure the speeds of light in water. The Fizeau experiment, conducted by French physicist Hippolyte Fizeau (1819–1896), was a test to determine how the motion of a medium (water) affects the speed of light propagating through it. This was not a direct measurement of the absolute speed of light in stationary water (that had been approximated earlier), but rather an investigation into the relative speeds of light traveling with and against the flow of moving water.<ref name="Fizeau1851" /><br>
* '''1965''': Angular momentum theory applied to optical fields, foundational for BS symmetries.<ref name="28E" /><br>
* '''1966''': Density operators for coherent fields at BS, enabling statistical analysis.<ref name="26E" /><br>
* '''1981''': General properties of lossless BS in interferometry.<ref name="18E" /><br>
* '''1987''': Experimental observation of HOM effect, demonstrating two-photon bunching and quantum interference.<ref name="04E" /><br>
* '''1989''': SU(2) symmetry and photon statistics for lossless BS.<ref name="19E" /><br>
* '''1995''': Unitary quantum description of BS.<ref name="27E" /><br>
* '''2001''': KLM protocol for efficient quantum computation with linear optics, establishing scalability using beam splitters, single-photon sources, and detectors.<ref name="06E" /><ref name="ML03" /><br>
* '''2002''': Demonstration that nonclassical inputs are required for BS-generated entanglement.<ref name="20E" /><br>
* '''2008''': Silica-on-silicon waveguide quantum circuits, advancing integrated photonic implementations of BS.<ref name="15E" /><br>
* '''2018–2020''': Theoretical models of frequency-dependent effects in waveguide BS, including fluctuations in HOM detection.<ref name="24E" /><ref name="25E" /><br>
* '''2020–2021''': Quantum entanglement and reflection coefficients in coupled waveguide BS models; frequency-dependent theory for waveguide BS.<ref name="17E" /><ref name="21E" /><ref name="22E" /><br>
* '''2022''': Quantum entanglement for monochromatic and non-monochromatic photons on waveguide BS; comprehensive review systematizing conventional vs. frequency-dependent BS.<ref name="23E" /><br>
This timeline highlights the historical development from foundational quantum formulas to the recognition of frequency-dependent effects in waveguide implementations, which are important for scalable quantum technologies.
[[File:Timeline Integration Quantum Optics Technology.png|center|Summary of the features of the principal fabrication technologies for what concerns the operating wavelengths, circuits geometry, integration of sources and detectors, and the interface with external fibers.]]
== Theoretical Framework ==
In quantum optics, the mathematical description of a beam splitter describes how the incoming annihilation operators <math>\hat{a}_1</math> and <math>\hat{a}_2</math> are transformed into the outgoing operators <math>\hat{b}_1</math> and <math>\hat{b}_2</math> by means of a unitary matrix. For a traditional beam splitter, this transformation can be written as

<math> \begin{pmatrix} \hat{b}_1 \\ \hat{b}_2 \end{pmatrix} = U_{\text{BS}} \begin{pmatrix} \hat{a}_1 \\ \hat{a}_2 \end{pmatrix}, </math>

where the unitary matrix is given by

<math> U_{\text{BS}} = \begin{pmatrix} \sqrt{T}\, e^{i\phi} & \sqrt{R} \\ -\sqrt{R}\, e^{-i\phi} & \sqrt{T} \end{pmatrix}. </math>

In these expressions, <math>T</math>, <math>R</math>, and <math>\phi</math> represent the transmission coefficient, reflection coefficient, and relative phase, respectively. The unitary nature of <math>U_{\text{BS}}</math> guarantees that bosonic commutation relations are preserved.

In the angular-momentum representation, the action of the beam splitter corresponds to an SU(2) rotation generated by angular momentum operators <math>\hat{L}_i</math>. The associated rotation angles are determined by the reflectivity <math>R</math> and the phase <math>\phi</math>.

For non-monochromatic light, the spectral degrees of freedom must also be taken into account. In this case, the output quantum state depends on the joint spectral amplitude function <math>\varphi(\omega_1,\omega_2)</math>, which must be integrated over the relevant frequency variables.

Frequency-dependent beam splitters, commonly encountered in waveguide couplers, can be derived using coupled-mode theory. Within this framework, both the reflection coefficient <math>R</math> and the phase <math>\phi</math> depend explicitly on the frequencies <math>\omega_1</math> and <math>\omega_2</math>. A representative expression for the reflection coefficient is

<math> R = \sin^2\!\left( \frac{\Omega\, t_{\text{BS}}}{2\sqrt{1+\varepsilon^2}} \right)\,(1+\varepsilon^2), </math>

where

<math> \varepsilon = \frac{\omega_2 - \omega_1}{\Omega}, </math>

<math>\Omega</math> characterizes the coupling strength between the modes, and <math>t_{\text{BS}}</math> denotes the effective interaction time.

This spectral dependence significantly influences quantum interference and entanglement properties. To observe genuinely quantum effects, non-classical input states such as Fock states or squeezed states are required. Measures of entanglement, including concurrence, decrease when the spectral overlap between modes is limited.

Photon-number statistics also depend on both the input state and the spectral structure. Coherent states exhibit Poissonian statistics, whereas non-classical states can display sub-Poissonian or super-Poissonian behavior. In multimode fields, frequency selectivity can lead to partial photon bunching.

A prominent example of such interference phenomena is the Hong–Ou–Mandel (HOM) effect, in which two identical photons incident on a beam splitter tend to bunch together, resulting in suppressed coincidence counts. When the beam splitter is frequency dependent, spectral variations reduce the visibility of the Hong–Ou–Mandel dip. Generalizations of this effect include formulations based on wave packets as well as analogous interference phenomena involving fermions.

==The beam splitter==
Dates to classical interferometry in the 19th century (e.g., [[Wikipedia:Michelson interferometer|Michelson interferometer]]). Quantum applications emerged mid-20th century with quantum electrodynamics and lasers, The Hong-Ou-Mandel effect first demonstrated in 1987<ref name="04E" /> <ref name="55E" /><ref name="58E" /><ref name="63E" />. Entanglement by a beam splitter (2002) <ref name="20E" /> . Quantum entanglement and reflection coefficient for coupled harmonic oscillators (2020)<ref name="21E" />. Quantum entanglement and statistics of photons on a beam splitter in the form of coupled waveguides (2022) <ref name="22E" />.<br>Beam Splitters (BS) have a variety of forms, such as a glass plate with a coat of silver or a thin dielectric film, a glass prism with a coat along its diagonal, two parallel glass plates with a coat in between, or a thin film with a deposited coat. Waveguide BSs are formed by bringing two waveguides side by side so that their electromagnetic fields interact with each other<ref name="06E"/>.

====Beam splitters vary by design and frequency dependence.<ref name="34E" /><ref name="35E" />====

'''Waveguide BS (directional couplers)''': <br>
[[Wikipedia:Evanescent field|Evanescent coupling]] between waveguides, ''R(ω) = sin²(κ(ω)L)''.<ref name="41E" /><ref name="42E" /><ref name="43E" /><ref name="44E" /><ref name="45E" /><ref name="46E" /><ref name="47E" />

'''Waveguide BS''':<br> enable integration in [[Wikipedia:Photonic integrated circuit|photonic chips]] for quantum technologies.<ref name="48E" /><ref name="49E" /><ref name="50E" />

'''Conventional beam splitters:'''<br>
Cube, plate, or pellicle BS with nearly constant ''R'', ''T'', ''φ'' over bandwidths. Used in free-space experiments.<ref name="36E" /><ref name="37E" /><ref name="38E" /><ref name="39E" /><ref name="40E" />

'''Frequency-dependent beam splitters:'''<br>
Coupled-mode theory: dâ<sub>1</sub>/dz = -i δ â<sub>1</sub> - i κ â<sub>2</sub>, yielding frequency-dependent U<sub>ij</sub>(ω).<ref name="64E" /><ref name="17E" /><ref name="22E" /><ref name="23E" />

== Theory of Waveguide Beam Splitters == 
While classical beam splitters are often treated as constant, the scattering matrix for a waveguide beam splitter is explicitly frequency-dependent. The transformation of input modes into output modes is represented as:<br>
<math>
\begin{pmatrix}
\hat{a}_{\text{out},1} \\
\hat{a}_{\text{out},2}
\end{pmatrix}
=
\begin{pmatrix}
R(\omega) & T(\omega) \\
-T^*(\omega) & R(\omega)
\end{pmatrix}
\begin{pmatrix}
\hat{a}_{\text{in},1} \\
\hat{a}_{\text{in},2}
\end{pmatrix}
</math><br>
Here, the reflection <math>R(\omega)</math> and transmission <math>T(\omega)</math> coefficients are determined by the coupling constant and the interaction length within the waveguide. This frequency dependence is crucial for accurately describing the interference of non-monochromatic single photons. <ref name="23E" />

====Waveguide BS====
Has some important advantages over a conventional BS: they are significantly more compact and have other advantages in terms of performance and integration<ref name="15E"/>. BS can be classified in different ways, including their characteristics, such as polarizing BS and non-polarizing BS, and other distinct characteristics<ref name="18E"/>. A BS in quantum optics can be described regardless of its physical implementation, as shown in Figure 1(a); BS illustrations vary based on BS type, as in Figure 1(b)<ref name="03E"/>.
In quantum optics, aluminium-coated beam splitters Figure 1(c) <ref name="AluSplit" /> are often modeled as ideal two-port devices characterized solely by 
𝑅
R, 
𝑇
T, and a relative phase shift 
𝜙
ϕ between reflected and transmitted fields. The metallic coating introduces a well-defined phase relation between the output modes, allowing such beam splitters to be used in interference experiments, including Hong–Ou–Mandel–type configurations, despite their intrinsic losses.
<div style="column-count:3; column-gap:2em; margin-left:0; padding-left:0;">
[[File:Beam Splitter (a).png|thumb|left|A Beam Splitter (BS) scheme with two input ports and two output ports]]
[[File:Beam Splitter (b).png|thumb|The BS with free space optics, i.e., cubic BS (top) and fiber optics, i.e., waveguide BS (bottom).]]
[[File:Flat metal-coated beamsplitter.png|thumb|Figure 1(c). Aluminium-coated beam splitter.]]
  </div>

Two main parameters characterizing a BS in quantum optics are the reflection coefficient <math>R</math> (or transmission coefficient <math>T</math>, which satisfies <math>R + T = 1</math>) and the phase angle <math>\phi</math><ref name="19E"/>. In conventional reviews of quantum optics, during calculations concerning the behavior of photons (or electromagnetic waves) in the output ports or in devices incorporating a BS, parameters <math>R</math> and <math>\phi</math> are considered to be definite numbers<ref name="07E"/>.

For instance, in the Hong–Ou–Mandel (HOM) effect, an equal splitter with <math>R = T = \tfrac{1}{2}</math> is used, which is independent of <math>\phi</math><ref name="04E"/>.  <math>R</math> and <math>\phi</math> are functions of the wavelength or frequency of the incoming light, and <math>R = R(\lambda)</math> and <math>\phi = \phi(\lambda)</math> in both cases, regardless of which BS is used<ref name="17E"/>. If a fixed wavelength or a small frequency band is used in the experiment, this dependency can be ignored, and the output photons can be considered constant. This has long been considered self-evident<ref name="07E"/>.

However, in some cases, <math>R</math> and <math>\phi</math> are not constant by definition, and their frequency dependence is strong enough to affect, in a substantial way, the quantum state of light waves distributed in the output ports. Phenomena related to entanglement of light waves in a BS, unlike in the constant-parameter setting, behave differently if waveguide BSs (further referred to as fiber-optic BSs) are considered, since they differ from other BSs in this respect<ref name="17E"/><ref name="22E"/>.

There is a theoretical basis for frequency-dependent waveguide BS. It shows that for a waveguide model interpreted as two coupled waveguides, the amplitude reflection and transmission coefficients <math>R</math> and <math>T</math> become frequency-dependent for the photons entering the BS<ref name="17E"/>. Accounting for this frequency dependence requires corrections to established theories, such as HOM interferometer fringe analysis<ref name="24E"/><ref name="25E"/> and BS-generated entanglement of photons<ref name="22E"/><ref name="23E"/>. This pronounced frequency dependence of <math>R</math> and <math>T</math> is a distinct characteristic of waveguide BSs<ref name="17E"/>.

There is a need for a comprehensive analysis that classifies BS in quantum optics into two types: conventional (frequency-independent) and frequency-dependent. Based on this classification, researchers can examine differences in photon entanglement at the output ports. The present analysis performs exactly this, considering entanglement, photon statistics at the outputs, and the HOM effect<ref name="22E"/><ref name="24E"/>.

== Beam splitter in quantum optics ==
* Beam splitters also enable quantum optical neural networks for tasks like image classification and optimal quantum cloning, offering variational quantum algorithms and perceptron models that exploit entanglement for supervised learning.<ref name="15L"/><ref name="16L"/><ref name="17L"/><ref name="22L"/><ref name="23L"/><ref name="27L"/> Photonics-based implementations further integrate nonlinear activations and diffractive networks for all-optical machine learning.<ref name="07L"/><ref name="08L"/><ref name="09L"/><ref name="13L"/> Recent QML models address barren plateaus and demonstrate quantum verification of NP problems.<ref name="19L"/><ref name="20L"/><ref name="26L"/>"

Since a beam splitter (BS) separates incoming beams, the quantum state of photons at the BS output ports is given by

<math> | \mathrm{out} \rangle = e^{i \hat{H} t_{\mathrm{BS}}} | \mathrm{in} \rangle , </math>

where <math>\hat{H}</math> is the Hamiltonian of the quantized electromagnetic field interacting with matter, <math>t_{\mathrm{BS}}</math> is the interaction time, and <math>| \mathrm{in} \rangle</math> is the initial state of the electromagnetic field.  <math>\hat{H}</math> can be quite complex, depending on the type of beam splitter.

In general, the initial state can be represented as <ref name="13E" /><ref name="26E" />

<math> | \mathrm{in} \rangle = \sum_{s_1,s_2} C_{s_1,s_2} \frac{1}{\sqrt{s_1! s_2!}} (\hat{a}_1^\dagger)^{s_1} (\hat{a}_2^\dagger)^{s_2} |0\rangle_1 |0\rangle_2 , </math>  <sub>(Eq. (1))</sub>

where the creation operators of the first and second modes are <math>\hat{a}_1^\dagger</math> and <math>\hat{a}2^\dagger</math>, respectively. The integers <math>s_1</math> and <math>s_2</math> are the quantum numbers of the first and second modes (i.e. the number of photons in each mode). The coefficients <math>C{s_1,s_2}</math> define the initial state, and <math>|0\rangle_1 |0\rangle_2</math> are the vacuum states of modes 1 and 2. For convenience, it is <math>|0\rangle_1 |0\rangle_2 \equiv |0\rangle</math>.

If the initial states are Fock states, then the coefficients satisfy <math>C_{s_1,s_2} = 1</math>. In this case, it is straightforward to show (up to an insignificant phase) that <ref name="01E" /><ref name="13E" /><ref name="26E" />

<math> | \mathrm{out} \rangle = \sum_{s_1,s_2} C_{s_1,s_2} \frac{1}{\sqrt{s_1! s_2!}} (\hat{b}_1^\dagger)^{s_1} (\hat{b}_2^\dagger)^{s_2} |0\rangle , </math>

with

<math> \hat{b}_k^\dagger = e^{i \hat{H} t_{\mathrm{BS}}} \hat{a}_k^\dagger e^{-i \hat{H} t_{\mathrm{BS}}}, \qquad k = 1,2 . </math>  <sub>(Eq. (2))</sub>

Here <math>\hat{b}_1^\dagger</math> and <math>\hat{b}_2^\dagger</math> are the creation operators at the output ports of the beam splitter for modes 1 and 2, respectively.

For any lossless two-mode beam splitter, the transformation between input and output operators is governed by a unitary matrix, <math>\mathbf{U}_{BS}</math>, constrained by the conservation of energy and bosonic commutation relations:<ref name="18E" /><ref name="19E" /><ref name="27E" />

<math> \begin{pmatrix} \hat{b}_1 \ \hat{b}2 \end{pmatrix} = \mathbf{U}{BS} \begin{pmatrix} \hat{a}_1 \ \hat{a}_2 \end{pmatrix} = \begin{pmatrix} t & r \ r' & t' \end{pmatrix} \begin{pmatrix} \hat{a}_1 \ \hat{a}_2 \end{pmatrix} </math>  <sub>(Eq. (3)}</sub>

The requirement for unitarity (<math>\mathbf{U}^\dagger \mathbf{U} = \mathbf{I}</math>) implies that the transmission and reflection coefficients satisfy:

<math>|r|^2 + |t|^2 = 1</math> (Energy conservation)

<math>r^* t' + t^* r' = 0</math> (Phase relationship between ports)

For a symmetric 50:50 beam splitter, this is commonly expressed as:

<math> \mathbf{U}_{BS} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \ i & 1 \end{pmatrix} </math>
In the literature, one often encounters different representations of the beam splitter matrix. The most commonly used case corresponds to a phase shift <math>\phi = \pi/2</math>, while another frequently used representation sets <math>\phi = 0</math>. In these cases the matrix <math>U_{\mathrm{BS}}</math> takes the forms

<math> U_{\mathrm{BS}} = \begin{pmatrix} \sqrt{T} & i \sqrt{R} \\ i \sqrt{R} & \sqrt{T} \end{pmatrix}, \qquad U_{\mathrm{BS}} = \begin{pmatrix} \sqrt{T} & \sqrt{R} \\ - \sqrt{R} & \sqrt{T} \end{pmatrix}. </math>   <sub>(Eq. (4)</sub>

Both representations are valid only when the final result is independent of the phase shift <math>\phi</math>. As shown below, many quantities of interest in quantum optics, such as quantum entanglement at the output ports of the beam splitter, do not depend on this phase. Nevertheless, using the general form given in Eq. (3).

In reality, photons are not monochromatic and their frequency distribution must be taken into account <ref name="01G" /><ref name="05G" /> In this case, the initial wave function of the photons is

<math> | \mathrm{in} \rangle = \sum_{s_1,s_2} C_{s_1,s_2} \frac{1}{\sqrt{s_1! s_2!}} \int \Phi(\omega_1,\omega_2) (\hat{a}_1^\dagger)^{s_1} (\hat{a}_2^\dagger)^{s_2} |0\rangle \, d\omega_1 d\omega_2 , </math>  <sub>(Eq. (5))</sub>

where <math>\Phi(\omega_1,\omega_2)</math> is the joint spectral amplitude (JSA) of the two-mode wave function. Assuming normalization,

<math> \int |\Phi(\omega_1,\omega_2)|^2 \, d\omega_1 d\omega_2 = 1 , </math>

the output state is

<math> | \mathrm{out} \rangle = \sum_{s_1,s_2} C_{s_1,s_2} \frac{1}{\sqrt{s_1! s_2!}} \int \Phi(\omega_1,\omega_2) (\hat{b}_1^\dagger)^{s_1} (\hat{b}_2^\dagger)^{s_2} |0\rangle \, d\omega_1 d\omega_2 . </math>  <sub>(Eq. (6))</sub>

== Quantum mechanical description ==
The output state is ''|Ψ<sub>out</sub>⟩ = e<sup>iĤ t</sup><sub>BS</sub> |Ψ<sub>in</sub>⟩'', where Ĥ is the Hamiltonian and ''t''<sub>BS</sub> the interaction time.<ref name="51E" /><ref name="52E" /><ref name="53E" />
For monochromatic light, the input is |Ψ<sub>in</sub>⟩ = ∑<sub>s<sub>1</sub>,s<sub>2</sub></sub> C<sub>s<sub>1</sub>,s<sub>2</sub></sub> (â<sup>†</sup><sub>1</sub><sup>s<sub>1</sub></sup> â<sup>†</sup><sub>2</sub><sup>s<sub>2</sub></sup> / √(s<sub>1</sub>! s<sub>2</sub>!)) |0⟩<sub>1</sub> |0⟩<sub>2</sub>.<ref name="54E" /><ref name="55E" />
The unitary matrix U<sub>BS</sub> is:
: <math>\begin{pmatrix} \hat{b}_1 \ \hat{b}_2 \end{pmatrix} = \begin{pmatrix} \sqrt{T} & e^{i\phi} \sqrt{R} \ -e^{-i\phi} \sqrt{R} & \sqrt{T} \end{pmatrix} \begin{pmatrix} \hat{a}_1 \ \hat{a}_2 \end{pmatrix}</math><ref name="56E" /><ref name="57E" /><ref name="58E" />

For non-monochromatic light, integrate over joint spectral amplitude φ(ω<sub>1</sub>, ω<sub>2</sub>).<ref name="59E" /><ref name="34E" /><ref name="60E" />

=== Angular momentum representation ===
BS transformation as SU(2) rotation: L̂<sub>i</sub> operators, with L̂<sup>2</sup> = l̂(l̂ + 1), where l̂ = (n̂<sub>1</sub> + n̂<sub>2</sub>)/2.<ref name="61E" /><ref name="34E" />
Unitary: Û = e<sup>-i Φ L̂<sub>3</sub></sup> e<sup>-i Θ L̂<sub>2</sub></sup> e<sup>-i Ψ L̂<sub>3</sub></sup>, with Θ = 2θ, θ = arcsin√R.<ref name="62E" /><ref name="63E" /><ref name="64E" />

== Quantum entanglement ==
BS generates entanglement if at least one input is non-classical (e.g., squeezed or Fock state).
In frequency-dependent BS, entanglement varies with spectral overlap; broadband photons may reduce concurrence.<ref name="20E" /><ref name="21E" /><ref name="43E" /><ref name="44E" />

===='''Quantum entanglement of photons and their statistics'''==== 
As described by D.N. Makarov in 2022 <ref name="23E" />{{efn-ua|The references in this article have been adjusted. Some where damaged/misspeld in the original article.}}. in the Theory for the '''quantum optics beam splitter'''. Recent advancements in hybrid integrated circuits <ref name="106G" /><ref name="48H" /> have transitioned these theories from bulk optics to scalable chip-based platforms. Quantum states of light are fundamental resources for the implementation of quantum information protocols since the pioneering tests on nonlocality and quantum teleportation.<ref name="01H" /><ref name="02H" /> The optical device that divides an incident beam of light into two or more output beams, typically a transmitted beam and a reflected beam. In [[Wikipedia:Quantum optics|quantum optics]], the quantum beam splitter is a fundamental component far beyond classical beam division: it generates quantum [[Wikipedia:superposition|superposition]] and [[Wikipedia:quantum entanglement|quantum entanglement]] from non-entangled inputs, reveals non-classical [[Wikipedia:photon statistics|photon statistics]], and enables key phenomena like the [[Wikipedia:Hong–Ou–Mandel effect|Hong–Ou–Mandel effect]] (HOM effect).<ref name="01E" /><ref name="02E" /><ref name="03E" /><ref name="04E" /><ref name="05E" /><ref name="06E" /> While conventional beam splitters are often bulk components, recent progress in [[Wikipedia:Photonic integrated circuit|integrated photonics]] <ref name="01G" /> has allowed for on-chip implementations. For example, independent dibenzoterrylene <math>\mathrm{C}_{30}\mathrm{H}_{18}</math> (DBT) molecules integrated with silicon nitride (<math>\mathrm{Si}_{3}\mathrm{N}_{4}</math>) photonic elements, a single-crystalline [[Wikipedia:Anthracene|anthracene]] nanosheet doped with dibenzoterrylene (DBT) molecules<ref name="34K" />, and gold electrodes for Stark tuning (Methods).<ref name="25K" /><ref name="27K" /><ref name="29K" /><ref name="39K" /> Waveguides have achieved stable, lifetime-limited transitions suitable for scalable quantum networks. <ref name="01K" />  [[Wikipedia:Atomic line filter|Stark tuning]] experiments show how 100 waveguide-coupled molecules can be tuned to the same frequency and achieve on-chip Hong–Ou–Mandel interference. The quantum theory of the beam splitter is remarkably simple, parameterized by the reflection coefficient ''R'' (or transmission ''T'', with ''R'' + ''T'' = 1) and a relative phase shift ''φ''. This minimal description underpins linear-optical quantum protocols, from [[Wikipedia:Interferometry|interferometry]] to [[Wikipedia:Quantum computing scaling laws|scalable computing]].<ref name="07E" /><ref name="08E" /> Beam splitters are systematized into "conventional" (frequency-independent ''R'' and ''φ'') and frequency-dependent types (e.g., waveguide beam splitters), with the latter affecting entanglement and interference for [[Wikipedia:Fraunhofer diffraction equation#Non-monochromatic illumination|non-monochromatic light]].<ref name="09E" /><ref name="10E" /><ref name="11E" />
<ref name="12E" /><ref name="13E" />
<ref name="14E" /><ref name="15E" />
<ref name="16E" /><ref name="17E" />
<ref name="18E" /><ref name="19E" />. This article aims at providing
an exhaustive framework of the advances of integrated quantum photonic platforms,
for what concerns the integration of sources, manipulation, and detectors, as well as the contributions in quantum computing, cryptography and simulations.

== Photon statistics ==
Output distributions depend on input states and BS parameters. Coherent inputs yield coherent outputs; Fock inputs show sub/super-Poissonian statistics.<ref name="01E" /><ref name="03E" /><ref name="05E" />
In frequency-dependent BS, statistics vary by mode, leading to selective bunching/antibunching.<ref name="22E" /><ref name="21E" />

== Hong–Ou–Mandel effect ==
[[File:Hong-Ou-Mandel mathematics.png|thumb|Mathematical equivalence between the Hong-Ou-Mandel and a classical artificial neuron.
The left branch of the interferometer corresponds to the input layer, while the probe parameters are related to the trainable neuron weights. The rate of coincidences encodes the square absolute value of their scalar product, further post-processed by adding a bias and a sigmoid activation function]]
Recent advancements have leveraged the HOM effect for quantum kernel evaluation, enabling distance computations in feature spaces for machine learning tasks.<ref name="33L"/> This equivalence to the SWAP test further extends HOM to high-dimensional interference in spatial modes.<ref name="30L"/><ref name="32L"/> Classical analogs achieving 97% visibility dips confirm the role of complementarity in such systems.<ref name="31L"/> Identical photons on 50:50 BS bunch<ref name="29L" />, suppressing coincidence counts (HOM dip).<ref name="56E" /><ref name="57E" /><ref name="58E" /><ref name="59E" /><ref name="60E" /><br>For frequency-dependent BS, dip visibility depends on spectral overlap; fluctuations affect detection.<ref name="17E" /><ref name="22E" /><ref name="23E" /><ref name="21E" /><br>Generalizations: bosons/fermions, wavepackets.<ref name="61E" /><ref name="62E" /><ref name="63E" /><ref name="64E" /><br>On a molecular quantum photonic chip, on-chip HOM interference was realized with a visibility of over 0.97. <ref name="02K" />The high visibility confirms the excellent indistinguishability of photons originating from independent sources on the same chip.Recent experiments have successfully implemented these waveguide principles on-chip. Using independent dibenzoterrylene (DBT) molecules integrated into <math>Si_{3}N_{4}</math> waveguides, researchers observed on-chip quantum interference with a visibility of <math>0.97 \pm 0.02</math>. <ref name="01K" /> This provides experimental proof that integrated molecular emitters can achieve the high level of indistinguishability required for scalable quantum circuits.In 2025 a novel quantum optical pattern recognition method leveraging the Hong-Ou-Mandel (HOM) effect for binary classification tasks was introduced by Simone Roncallo et all
<ref name="39L" />.This encodes input objects and trainable parameters into single-photon states, measures two-photon coincidence rates at the output of a HOM interferometer, demonstrating a superexponential resource advantage (constant <math>\mathcal{O}(1)</math> complexity in photons and operations versus at least linear scaling in classical artificial neurons).
==== Quantum optical setup====
The quantum optical setup classifies objects without reconstructing their images. The approach relies on the Hong-Ou-Mandel effect, for which the probability that two photons exit a beam splitter in different modes, depends on their distinguishability<ref name="29L"/><ref name="30L"/><ref name="31L"/><ref name="32L"/>. In the
implementation, an input object is targeted by a single-photon source, and eventually followed by an arbitrary lens system. The single-photon state interferes with another one, which encodes a set of trainable parameters, e.g. through a spatial light modulator. After the Hong-Ou-Mandel interferometer, the photons are collected by two bucket detectors without spatial sensitivity, one for each output mode. Classification occurs by measuring the rate of two-photon coincidences at the output.<br>

The Hong-Ou-Mandel effect has been successfully applied to quantum kernel evaluation<ref name="33L"/>, which can compute distances between pairs of data points in the feature space. In this case, each point is sent to one branch of the interferometer, encoded in the temporal modes of a single-photon state. In our method, the interferometer has only one independent branch, which takes the spatial modes of a single-photon state reflected off the target object. The other branch remains fixed after training, and contains the layer of parameters.<ref name="25L"/><ref name="38L"/> After the measurement, the response function of our apparatus mathematically resembles that of a classical neuron. For this reason, we refer to our setup as quantum optical neuron. By analytically comparing the resource cost of the classical and quantum neurons, this  method requires constant <math>\mathcal{O}(1)</math>
computational operations and injected photons, whereas the classical methods are at least linear in the image resolution: a '''superexponential advantage'''. <br>

When combining multiple neurons, the large number of parameters involved motivates a consistent effort in reducing the cost of deep learning algorithms, e.g. by leveraging classical implementations that bypass hardware in an all-optical way<ref name="07L"/><ref name="08L"/><ref name="09L"/><ref name="10L"/><ref name="11L"/><ref name="12L"/><ref name="13L"/>. Quantum mechanical effects, like superposition and entanglement, can provide a significant speedup in such tasks<ref name="14L"/><ref name="15L"/>, e.g. by building quantum analogues of the perceptron<ref name="16L"/><ref name="17L"/><ref name="18L"/>, by employing variational methods<ref name="19L"/><ref name="20L"/> or quantum-inspired approaches<ref name="21L"/>. Quantum optical neural networks harness the best of both worlds, i.e. deep learning capabilities from quantum optics<ref name="22L"/><ref name="23L"/><ref name="24L"/><ref name="25L"/><ref name="26L"/><ref name="27L"/><ref name="28L"/>.
======Mathematical description======
Two optical input modes ''a'' and ''b'' that carry [[Wikipedia:annihilation and creation operators|annihilation and creation operators]] <math>\hat{a}</math>, <math>\hat{a}^\dagger</math>, and <math>\hat{b}</math>, <math>\hat{b}^\dagger</math>. Identical photons in different modes can be described by the [[Wikipedia:Fock state|Fock state]]s<ref name="03E" />, so, for example <math>|0\rangle_a</math> corresponds to mode ''a'' empty (the vacuum state), and inserting one photon into ''a'' corresponds to <math>|1\rangle_a=\hat{a}^\dagger|0\rangle_a</math>, etc. A photon in each input mode is therefore

: <math>|1, 1\rangle_{ab} = \hat{a}^\dagger \hat{b}^\dagger |0, 0\rangle_{ab}.</math>

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. <math>\hat{a}^\dagger |0\rangle_a \to \frac{1}{\sqrt{2}}\left( \hat{c}^\dagger + \hat{d}^\dagger\right)|00\rangle_{cd}</math>, and similarly for inserting a photon in ''b''. Therefore

: <math>\hat{a}^\dagger \to \frac{\hat{c}^\dagger + \hat{d}^\dagger}{\sqrt{2}} \quad\text{and}\quad \hat{b}^\dagger \to \frac{\hat{c}^\dagger - \hat{d}^\dagger}{\sqrt{2}}.</math>

The relative minus sign appears because the [[Wikipedia:beam splitter#Classical lossless beam splitter|classical lossless beam splitter produces a unitary transformation]]<ref name="19E" />. This can be seen most clearly when wr the two-mode beam splitter transformation in [[Wikipedia:matrix (mathematics)|matrix]] form:

: <math>\begin{pmatrix}
 \hat{a} \\
 \hat{b}
\end{pmatrix} \to \frac{1}{\sqrt{2}} \begin{pmatrix}
 1 & 1 \\
 1 & -1
\end{pmatrix} \begin{pmatrix}
 \hat{c} \\
 \hat{d}
\end{pmatrix}.</math><ref name="18E" />

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|Physical description]] above)<ref name="27E" />.

When two photons enter the beam splitter, one on each side, the state of the two modes becomes

: <math>|1, 1\rangle_{ab} = \hat{a}^\dagger \hat{b}^\dagger |0, 0\rangle_{ab} \to \frac{1}{2} \left( \hat{c}^\dagger + \hat{d}^\dagger \right) \left( \hat{c}^\dagger - \hat{d}^\dagger \right) |0, 0\rangle_{cd} </math>
: <math> = \frac{1}{2} \left( \hat{c}^{\dagger 2} - \hat{d}^{\dagger 2} \right) |0, 0\rangle_{cd}
 = \frac{|2, 0\rangle_{cd} - |0, 2\rangle_{cd}}{\sqrt{2}},
</math>

where used <math>\hat{c}^{\dagger 2}|0, 0\rangle_{cd}=\hat{c}^\dagger|1, 0\rangle_{cd}=\sqrt{2}|2, 0\rangle_{cd}</math> etc.<ref name="04E" />
Since the commutator of the two creation operators <math>\hat{c}^\dagger</math> and <math>\hat{d}^\dagger</math> is zero because they operate on different spaces, the product term vanishes. The surviving terms in the superposition are only the <math>\hat{c}^{\dagger 2}</math> and <math>\hat{d}^{\dagger 2}</math> 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 [[Wikipedia:Beam splitter#Quantum mechanical description|the general quantum mechanical treatment of a beamsplitter]] for the resulting output Fock state.<ref name="01E" /><ref name="02E" /><ref name="05E" />

==Single-Photon Detection in Beam Splitter Experiments==
In experiments in quantum optics with beam splitters, an individual-photon-catching detector network is obviously decisive to glimpse those striking non-classical effects: antibunching, Hong-Ou-Mandel interference, and entanglement that the beam splitter itself can generate.
[[File:Beam Splitter with Ultra Fast SPDs.png|center|Schematic (left) and scanning electron microscope image (right, scale bar 5 μm) of waveguide-integrated ultra-fast superconducting nanowire single-photon detectors (SNSPDs) coupled to a beam splitter on a photonic chip.]]
Single-photon detectors (SPDs), such as superconducting nanowire single-photon detectors (SNSPDs) or single-photon avalanche diodes (SPADs) operated in Geiger mode, provide the necessary time-resolved, high-efficiency detection at the single-photon level.<ref name="01H" /><ref name="149H" /><ref name="150H" /> In foundational experiments, SPDs are placed at the two output ports of a beam splitter. For a single photon incident on 50:50 beam splitter, the absence of simultaneous detections (zero coincidence counts above vacuum noise) demonstrates the particle-like indivisibility of the photon, while interference effects reveal its wave nature (e.g., in Mach-Zehnder configurations built with beam splitters).<ref name="04E" /><ref name="37E" />

In the Hong–Ou–Mandel effect, two indistinguishable photons entering separate input ports bunch at the outputs, leading to a near-complete suppression of coincidence detections between SPDs at the two ports, a hallmark of quantum interference.<ref name="04E" /><ref name="52E" /><ref name="53E" /> In tests of photon statistics or entanglement generation, post-selected coincidence measurements between SPDs enable quantification of antibunching (<math>g^{(2)}(0) < 1</math>) or violation of Bell inequalities.<ref name="20E" /><ref name="37E" />

Fully integrating SPDs onto photonic chips are still a big challenge. There are some promising developments about waveguide-coupled superconducting detectors. These latest developments open up the possibility that future quantum systems will have detection totally on a chip.<ref name="15E" /><ref name="102G" /><ref name="17H" />
Such integrations facilitate advanced HOM-based classifiers, where SLMs encode trainable parameters for pattern recognition, with software tools enabling simulation and optimization.<ref name="35L"/><ref name="36L"/><ref name="37L"/><ref name="39L"/> Training challenges, including initialization and convergence, mirror those in deep neural networks.<ref name="34L"/><ref name="38L"/>

====Key technologies for quantum photonic chips====
[[File:Glowing waveguides on quantum photonic chip (AI generated).jpg|thumb|Artistic illustration of glowing optical waveguides in a silicon nitride quantum photonic integrated circuit, highlighting on-chip light propagation for quantum interference experiments.]]
Photonic integration provides a clear route toward compact quantum communication systems with growing complexity and improved functionality. Integrated quantum communication can be broadly categorized into three main aspects: photonic material platforms enabling large-scale integration<ref name="36G" /><ref name="37G" /><ref name="38G" />, quantum photonic components such as quantum light sources<ref name="39G" />, high-speed modulators<ref name="40G" /> and highly efficient photodetectors<ref name="41G" />, and representative applications including QKD<ref name="42G" /><ref name="43G" /> and quantum teleportation<ref name="44G" />. Because the materials, fabrication processes, and structural designs used in photonic integration differ substantially from those of discrete optical systems, essential chip-level photonic components must be redesigned and optimized for specific quantum information tasks.

This section summarizes key technical developments covering quantum light sources, encoding and decoding elements, quantum detectors, and packaging techniques for integrated photonic systems. These advances constitute critical points in the evolution of integrated quantum communication. Early work in this area can be traced to the integration of photon sources based on periodically poled lithium niobate waveguides<ref name="45G" /> and interferometric circuits realized using silica-on-silicon planar lightwave circuits (PLCs)<ref name="46G" /><ref name="47G" /><ref name="48G" /><ref name="49G" />. The high efficiency and thermally stable operation of these integrated devices highlighted their intrinsic advantages over discrete and bulky optical components.

Subsequently, a wide range of material platforms was explored, leading to substantial progress in the on-chip generation, manipulation, and detection of quantum states of light for quantum communication and other quantum information applications. Prominent monolithic platforms for chip-based quantum communication include silica waveguides (silica-on-silicon and laser-written silica waveguides), silicon-on-insulator (SOI), silicon nitride (<math>\mathrm{Si}_{3}\mathrm{N}_{4}</math>), lithium niobate (LN), gallium arsenide (GaAs)<ref name="98H" /><ref name="108H" /><ref name="119H" /><ref name="23K" />, indium phosphide (InP), and silicon oxynitride (<math>\mathrm{SiO}_{x}\mathrm{N}_{y}</math>)<ref name="34G" /><ref name="35G" /><ref name="50G" />. The state of the art of these platforms reveals distinct advantages and limitations in terms of waveguiding performance, availability of active components, and compatibility with related technologies.

SOI offers high refractive-index contrast for dense integration, strong optical nonlinearity for nonclassical state generation, and excellent compatibility with advanced CMOS (complementary metal–oxide–semiconductor) fabrication processes widely used in the semiconductor industry. However, the absence of native lasing capability complicates the full monolithic integration of all components required for a complete quantum communication system. Semiconductor platforms such as GaAs<ref name="91G" /><ref name="50H" /><ref name="98H" /><ref name="108H" /> and InP enable full system integration but generally involve higher costs and reduced scalability. These inherent trade-offs indicate that no single material platform can simultaneously satisfy all requirements for quantum communication. As a result, hybrid integration has emerged as a promising strategy to combine the strengths of different platforms<ref name="50G" />.

Notable achievements along these lines include heterogeneous quantum photonic devices such as integrated superconducting nanowire single-photon detectors (SNSPDs)<ref name="41G" /> and on-chip lasers for weak coherent pulse generation<ref name="51G" />. Additional important technologies, including semiconductor quantum dots (QDs) coupled to photonic nanostructures<ref name="52G" /> and diamond-on-insulator platforms<ref name="53G" /><ref name="54G" />, have also emerged as competitive solutions for integrated quantum communication systems.

A timeline of advances in quantum photonic chips for quantum communication highlights several key items, including the first on-chip quantum interferometer for quantum cryptography<ref name="46G" />, quantum teleportation<ref name="05K" /><ref name="108H" /><ref name="31K" /> realized on a photonic chip<ref name="90G" />, chip-based DV-QKD<ref name="42G" />, CV-QKD<ref name="43G" />, and MDI-QKD<ref name="81G" /><ref name="94G" /><ref name="96G" />, as well as chip-to-chip quantum teleportation<ref name="44G" />.

===Encoding schemes:===
====Path encoding====
Photon states distributed among multiple waveguides are employed to encode qubit/qudit states and to observe quantum interference effects due to bosonic statistics. Such information encoding has been one of the most investigated in quantum integrated photonics. Path-encoded single- and multi-photon states can be arbitrarily prepared, manipulated and measured using re-programmable Mach–Zehnder interferometers (MZIs). The effect of the MZI is equivalent to the operations made by a beamsplitter with a tunable splitting ratio and by a phase shift (Fig. 3a). This unit for path encoding processing envisages two unbiased directional couplers and two integrated tunable phase shifts. Relative phases between paths in integrated devices are the results of the geometric deformation of waveguides. The recent developments in the field have demonstrated the capability to realize reconfigurable phase shifters, thus allowing the implementation of several unitary transformations on the same device <ref name="11H" /><ref name="21H" /><ref name="101H" />. There are several examples of programmable integrated devices in SoS<ref name="11H" /><ref name="53H" /> , Si <ref name=" 13H" /> <ref name="15H" /> <ref name="27H" /> <ref name="90H" /> <ref name="92H" /> <ref name="94H" />, SiN<ref name="12H" /><ref name ="14H" /><ref name="101H" /> silica laser written waveguides with FLW<ref name="83H" /> <ref name="84H" /> <ref name="85H" /> and UV-writing<ref name="56H" /><ref name="57H" />. The re-programmable elements inside the MZI are the two phase shifts that are controlled generally through the thermo-optic effect. Heaters placed nearby the location of the waveguide allow for local changes in the refractive index of the material. Such reconfigurable units are sufficient to encode, process and measure any qubit in two optical paths. The complexity reached from integrated devices is nowadays very remarkable allowing for full integration of qubit- and qudit-based quantum gates and algorithms in SoS <ref name="53H" />  and Si-chips <ref name="15H" /><ref name="27H" /><ref name="93H"/><ref name="94H" />. In particular, the most recent silicon quantum processors count up to 16 integrated single-photon sources, more than 100 heaters and likewise integrated optical elements<ref name="13H" /><ref name="15H" /><ref name="27H" />.
{{Annotated image
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| image-width = {{{image-width|1000}}}  
| width = {{{width|1000}}}             
| 
| float = none
| caption =  
| annotations =
  {{Annotation|900|300|<ref name="134H"/><ref name="135H" /> | font-size=14 | color=blue}}
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====Femtosecond-laser-writing (FLW)==== 
[[File:Femtosecond laser waveguide writing setup in BK7 glass 2018.jpg|thumb|Femtosecond laser direct-writing setup for waveguides in BK7 glass: processing and characterization configurations with beam profile inset (720 nJ pulse energy).]]
[[File:Femtosecond laser writing 3D stage setup and circular waveguide cross-sections.png|thumb|FLW schematic (A) with 3D stage and 40× objective; (B,C) circular waveguide cross-sections (20 µm and 50 µm scales).]]
The [[Wikipedia:femtosecond|femtosecond]] laser writing, using [[Wikipedia:Mode locking|Mode locking]]<ref name="Mayer" /> is a further method for silica waveguide fabrication<ref name="45H" /><ref name="65H" />. The mechanism of the process is the non-linear absorption of strong laser pulses tightly focused in the silica sample. Such absorption results in a permanent and localized modification in the refractive index. Waveguides are directly written by translating the silica sample at a constant speed with respect to the laser beam, without needing any preliminary preparation of substrate or layers of different materials as in the previous methods. The cross-section is circular and presents a very low birefringence. Such characteristics together with the 3D geometry capability have allowed the realization of devices insensitive from polarization <ref name="66H" /><ref name="68H" /> as well as devices able to manipulate polarization as waveplates or partially polarizing beamsplitter <ref name="69H" /><ref name="70H" />. The 3D geometry has other advantages regarding the range of possible schemes for optical circuit decomposition. FLW devices demonstrated to realize circuits according to the traditional networks of integrated beam-splitter in planar <ref name="67H" /><ref name="71H" /><ref name="72H" /><ref name="73H" /><ref name="74H" /> and 3D geometries<ref name="75H" /><ref name="76H" /><ref name="77H" /> and continuously coupled waveguide lattices <ref name="78H" /><ref name="79H" /><ref name="80H" /><ref name="81H" /><ref name="82H" />. There are instances of re-programmable circuits in small<ref name="83H" /><ref name="84H" /><ref name="85H" /><ref name="86H" /> and large scale <ref name="87H" /> realization of integrated devices. The integration of single-photon sources exploiting nonlinear effects is still challenging due to the low birefringence (<math>\Delta n \approx 0</math>) and the null third-order nonlinear susceptibility (<math>\chi^{(3)} = 0</math>) of these waveguides. Notwithstanding, femtosecond laser writing (FLW) can be exploited to write waveguides in a nonlinear material to generate pairs of photons through parametric processes. These kinds of sources have been interfaced successfully with FLW chips in<ref name="88H" />. The FLW waveguides display also a good coupling with external fibers, enabling the interface of the optical circuit with remote users or solid-state sources<ref name="85H" /><ref name="86H" />.

===Compact integration of optical components===

A factor that drives the compact integration of
optical components, quantum computing on integrated
photonic chips has attracted much attention in recent
years. There are two types of optical models<ref name="201G" />: specific
quantum computing models<ref name="202G" /><ref name="203G" /> (e.g., boson sampling),
and universal quantum computing models<ref name="204G" /> <ref name="205G" /><ref name="206G" /><ref name="207G" /><ref name="208G" /><ref name="209G" /> (e.g.,
one-way or measurement-based). For specific quantum
computation, a variety of photonic systems were
demonstrated using quantum photonic chips<ref name="210G" /><ref name="211G" /><ref name="212G" /><ref name="213G" /><ref name="214G" /><ref name="215G" /><ref name="215G" /><ref name="217G" />,
enabling a natural and effective implementation of boson
sampling. Gaussian boson sampling<ref name="218G" /><ref name="219G" />, which can
dramatically enhance the sampling rate with the adoption
of squeezed light sources, was performed for the calculation
of molecular vibronic spectra on a <math>Si</math> chip<ref name="217G" /> (up to
8 photons) and a <math>SiN</math> chip<ref name="216G" /> (up to 18 photons). Recently,
quantum computational advantage has been delivered by
photonic Gaussian boson sampling processors<ref name="220G" /><ref name="221G" />,
paving the path for further development of integrated
specific quantum computers with potential applications
including graph optimization<ref name="222G" />, complex molecular
spectra<ref name="223G" />, molecular docking<ref name="224G" />, quantum chemistry<ref name="225G" />,
etc. For universal quantum computation, a number of
major functionalities have been demonstrated with onchip
photonic components, such as controlled-NOT gate
and its heralding version<ref name="92G" /><ref name="226G" />, and compiled Shor’s
factorization<ref name="227G" />. Moreover, both architectural and technological
efforts have been dedicated to photonic one-way
quantum computation. This approach employs cluster
states and sequential single-qubit measurement to perform
universal quantum algorithms<ref name="205G" /><ref name="207G" /><ref name="228G" /> and can be
greatly enhanced by implementing resource state generation
and fusion operation natively<ref name="229G" /><ref name="230G" /><ref name="231G" />. The relevant
circuit implementations include programmable fourphoton
graph states on a Si chip<ref name="232G" />, path-polarization
hyperentangled and cluster states on a <math>\mathrm{SiO_2}</math> chip<ref name="233G" /> and
programmable eight-qubit graph states on a Si chip<ref name="234G" />.
In conclusion, quantum photonic chips have rapidly
matured to become a versatile platform that proves to be
invaluable in the development of cutting-edge quantum
communication technologies. This review delves into the
advancements achieved in this particular field. Considering
the remarkable outcomes, it is anticipated that photonic
integration will eventually assume a crucial role in
building various quantum networks and potentially a
global quantum internet<ref name="22G" /><ref name="23G" /><ref name="33G" /><ref name="19K" />, reshaping the landscape of
future communication methodologies.

==Chip packaging and system integration==
While bare quantum photonic chips can be characterized
using a probe station, they must be packaged into
durable modules to develop working prototype devices<ref name="115G" />.
To this end, numerous processes have been proposed to
package quantum photonic chips into compact systems
for real-world applications.
Generally, photonic packaging involves a range of
techniques and technical competencies needed to make
the optical, electrical, mechanical, and thermal connections
between a photonic chip and the off-chip components
in a photonic module<ref name="116G" /><ref name="117G" /><ref name="118G" />. Fiber-to-chip coupling
is one of the best-known aspects. The main challenge
associated with coupling between an optical fiber and a
typical waveguide on the chip is the large difference
between their mode‐field diameters (MFDs)<ref name="119G" />. For
example, the MFD at 1550 nm is ~10 μm in telecom
single‐mode fiber (SMF), while the cross-section of the
corresponding strip silicon waveguide is usually only
220 × 450 nm. This mismatch can be mitigated by using
configurations that efficiently extract the mode from
waveguide<ref name="97G" />, such as inverted-taper edge couplers
interfaced with lensed SMF fibers<ref name="120G" /><ref name="121G" /> or ultrahigh
numerical aperture fibers<ref name="122G" />, and grating couplers
interfaced with SMF fibers<ref name="119G" /><ref name="123G" />. For the
approach harnessing grating couplers, coupling efficiency
up to 81.3% (−0.9 dB) can be achieved in a 260-nm-thick
SOI platform without the need for a back reflector or
overlayer<ref name="124G" />. Additionally, efficiencies over 90% have been
experimentally demonstrated using edge couplers fabricated
on 200-mm SOI wafers<ref name="125G" />. An alternative approach
for cost-effective and panel-level packaging is the evanescent
coupling scheme, which has been reported to
have a coupling loss of approximately 1 dB at a wavelength
of 1550 nm<ref name="126G" />.
To access the electrical components on quantum photonic
chips, electronic packaging is required to route signals
from electronic drivers, amplifiers, and other control
circuitry. This is often achieved by interfacing with dedicated
printed circuit boards (PCBs)<ref name="127G" />. The connection
between PCBs and the bond-pads on the chip is
usually made using wire-bonds.When a very large number
of electrical connections or precise sub-nanosecond control
on multiple channels is needed, 2.5-dimensional or
3-dimensional integration with customized electronic
integrated circuits (EICs) may be utilized <ref name="115G" /><ref name="128G" />
This integration can be achieved using either solder-ballbump
or copper-pillar-bump interconnects, providing a
robust electrical, mechanical, and thermal interface for the
photonic chips<ref name="129G" /><ref name="130G" />.
Global thermal stabilization of quantumphotonic devices
is essential for prototypes that require high accuracy and
repeatability or for field tests where seasonal temperature
swings are common. This can be achieved using passive
cooling techniques or a thermoelectric cooler (TEC). The
added global stability from the TEC allows for more efficient
and better reproducibility in the local temperature
tuning of individual photonic elements (e.g., micro-ring
resonators, thermo-optic phase shifters, etc.) on the
chip<ref name="115G" />. Additionally, liquid cooling can be installed to
further increase the cooling capacity of the system<ref name="127G" />

==Experiments==
== On-Chip Quantum Interference ==
[[File:Molecular quantum photonic chip and an illustration of on-chip interference of indistinguishable single photons from independent quantum emitters (molecules).jpg]]
 <div style="column-count:2;">'''<big>a</big>''', Photograph of the quantum photonic chip.'''<big>b</big>''', Optical micrograph of all 24 independent devices integrated on the chip.'''<big>c</big>''', Zoomed-in view of one device with hybrid integration of Si3N4 photonic elements (waveguides W1–W4, a 2 × 2 MMI and grating couplers G1–G4), an anthracene nanosheet (light green) doped with DBT molecules, and metal electrodes (yellow). '''<big>d</big>''', SEM images of the waveguides W1 and W2 (with gold electrodes flanking them), the MMI, and one of the output gratings G3. '''<big>e</big>''', DBT molecular structure and energy-level scheme. em., emission; exc., excitation. '''<big>f</big>''', Illustration of the on-chip two-photon quantum interference experiment: two streams of single photons originating from resonantly driven DBT molecules couple to the waveguides, interfere through the MMI, then propagate through the waveguide circuits and out-couple to free space via the gratings for timecorrelated
single-photon detection. The transition frequencies of the molecules can be tuned by the electrode via the Stark effect. Scale bars, 1 mm ('''<big>a</big>'''), 300 μm ('''<big>b</big>'''), 50 μm ('''<big>c</big>''') and 10 μm ('''<big>d</big>'''). Recent experimental breakthroughs have successfully implemented these waveguide principles using molecular quantum photonic chips. By integrating independent dibenzoterrylene (DBT) molecules into <math>Si_{3}N_{4}</math> waveguides, researchers have achieved on-chip Hong–Ou–Mandel (HOM) interference with a visibility of <math>0.97 \pm 0.02</math>. <ref name="01K" /></div>

These integrated systems allow for the observation of quantum beating when a frequency detuning (e.g., 400 MHz) is applied between two emitters. These beats have been shown to persist for over 100 µs, demonstrating the high spectral stability and single-photon purity required for scalable quantum information processing. <ref name="04K" />

====Evaluating photon indistinguishability from the TPQI experiment under CW excitation====
A recent report<ref name="40K" /> establishes a method to evaluate full wave-packet photon
indistinguishability from TPQI experiments under non-resonant CW excitation. Here, this method extends to resonant CW excitation, enabling direct assessment of photon indistinguishability from our
TPQI data. The metric used in this approach is
<math>
\tilde{V}(S) =
\frac{
\int d\tau \, \bigl[1 - g^{(2)}_{\rm HOM}(\tau)\bigr]
-
\int d\tau \, \bigl[1 - g^{(2)}_{{\rm HOM},d}(\tau)\bigr]
}{
\int d\tau \, \bigl[1 - g^{(2)}_{{\rm HOM},d}(\tau)\bigr]
}</math>.<br>Substituting the theoretical expressions for 
<math>g^{(2)}_{\rm HOM}(\tau)</math> and 
<math>g^{(2)}_{{\rm HOM},d}(\tau)</math> from equations respectively, 

<math>\tilde{V}(S)</math> is

<math>
\tilde{V}(S) = \frac{\mathcal{M} \,(2\mathcal{M}+1)}{\mathcal{M}+ (\mathcal{M}+1)/(1+S)}
</math>
. 

where <math>\mathcal{M} = \frac{\tau_2}{2 \tau_1}</math>. Equation (26) expresses <math>\tilde{V}</math> as a function of <math>S</math>. 

In the weak excitation limit (<math>S \to 0</math>), <math>\tilde{V}</math> reduces to <math>\frac{\tau_2}{2 \tau_1}</math>, thereby yieldingn the true photon indistinguishability, consistent with TPQI experiments
under pulsed excitation<ref name="40K" />.

== Applications ==
==Image classification==
Has been significantly improved by the introduction of deep learning methods, which provide several algorithms that can learn and extract image features. Examples include feedforward neural networks, convolutional neural networks and vision transformers<ref name="01L"/><ref name="02L"/><ref name="03L"/><ref name="04L"/>. The artificial neuron, also called perceptron<ref name="05L"/>, represents the fundamental unit of such architectures. In this model, encoded data are processed through a set of weighted trainable connections, by taking the scalar product between the input and the vector of weights. The output is further post-processed, including a bias and an activation function, which is usually non-linear<ref name="06L"/>. Image classification implies a two-fold cost. Computational processing requires a number of operations that scales, at least, linearly in the image resolution. Similarly, the optical cost of image capturing undergoes the same scaling in the number of photons.

This model compared against conventional classifiers, i.e. a single neuron and a convolutional neural network, commonly employed in pattern recognition tasks<ref name="25L" /><ref name="26L" /><ref name="27L" /><ref name="28L" /><ref name="38L" />. Adopting the [[Wikipedia:Tensor (machine learning)|TensorFlow notation]], the convolutional structure is: Conv2D (10, 3 × 3) → Conv2D (4, 2 × 2) → MaxPooling2D (2 × 2). Roughly, all the architectures have ~10³ trainable parameters. The performances are equal in the MNIST dataset, both in terms of trainability and final accuracy. In the CIFAR-10 dataset, our classifier outperforms the conventional ones, showing superior efficiency under a strongly-constrained parameters count. These findings emphasize the competitive accuracy of our method, and also its comparative advantage in pattern recognition tasks with a limited number of parameters.

====Entanglement distribution and quantum teleportation systems====
'''Quantum teleportation''' has been achieved over different types of platforms such as superconducting qubits, trapped atoms, nitrogen-vacancy centers, and continuous-variable states, among others.<ref name="170H" /> Of all the types of quantum teleportation, the photonic qubit is considered to be a very promising candidate for forming the quantum channel of the quantum network due to its stability within noisy environments and the fact that it can be operated at room temperature.<ref name="23H" /> Photonic qubits<ref name="03K" /> are more resistant to long-distance environmental interference. To date, photonic quantum teleportation has been successfully performed experimentally using different methods, including free-space and fiber.<ref name="170H" />

The first experimental validation of quantum teleportation relied on qubits encoded in the polarization of photons produced from a beta-barium borate (BBO) crystal in a free-space setup on an optical table.<ref name="20H" /> Later, the distance record for free-space teleportation was pushed beyond 1,400 km between the Micius satellite and a ground station,<ref name="171H" /> thus providing the basis for a global quantum network. However, due to the issues of beam divergence, pointing, and collection of free-space teleportation, optical-fiber-based teleportation is considered more suitable for the establishment of cost-efficient quantum metropolitan networks. The current distance record for optical-fiber-based teleportation is 102 km.<ref name="172H" />

A major issue related to photonic qubit teleportation involves the efficiency limit of Bell-state measurements (BSMs) using linear optics, with a 50% bound. To go around such a constraint, continuous variable optical modes can be used as a different solution to accomplish full deterministic teleportation. This technique was successfully experimented with on a 6-km fiber link,<ref name="173H" /> but its fidelity needs to be enhanced because of its vulnerability to transmission losses. For non-photonic qubit technology, a distance of 21 m was attained in the case of atom traps.<ref name="174H" />

With increasing momentum in quantum teleportation, another relevant technology is its integration. In future quantum networks, quantum teleportation chips could be integrated into fixed systems (e.g., network relays located in network nodes) or mobile systems (e.g., drones) to create lightweight and compact quantum nodes allowing remote access to quantum equipment for shared information as well as advanced computational power (Luo et al., Light: Science & Applications, 2023, 12:175). All this has become possible due to generation and manipulation of entangled photon pairs in multiple Degrees of Freedom on-chip, including path-encoded entanglement in Mach-Zehnder Interferometers (MZIs),<ref name="93H" /> polarization entanglement created in birefringent media,<ref name="177H" /> and time-bin entanglement in Franson interferometers.<ref name="178H" />

The first telecom-based chip-scale teleportation used an off-chip photon source, showing the feasibility of a fidelity of 0.89 in a single chip system.<ref name="90H" /> The current advancement in integrated quantum photonics has also helped realize entanglement-based quantum communications beyond the chip level. The first entanglement distribution between chips incorporated all necessary components into monolithically integrated silicon photonic chips.<ref name="100H" /> On-chip entangled Bell states were generated, and the qubit was transferred to the other silicon chip by encoding the on-chip path-encoded and in-fiber polarization states using two-dimensional grating couplers. Moreover, more advanced integrated quantum circuits implemented with on-chip sources have implemented inter-chip teleportation, showing a fidelity of 0.88.<ref name="44H" /> The chip-scale realization of photonic qubit creation, processing, and transmission provides one potential promising step toward the realization of the distributed quantum information processing Internet. In addition, entangled photon pairs in the visible and telecom bands have been created on a chip of silicon nitride (<math>Si_3N_4</math>) using a micro-ring resonator, with distribution over more than 20 km, using precisely designed and fabricated micro-ring resonators, entangling photons in the visible range, which can be coupled with quantum memories, and in the telecom range, with lower attenuation in the transmission of the photons over the fibers.<ref name="71H" />

====Quantum Information Processing and Computing====
Beam splitters are the fundamental building blocks for Linear Optical Quantum Computing (LOQC).

* The KLM Protocol: Beam splitters facilitate the probabilistic entangling gates necessary for universal quantum computation using only linear elements. The original CNOT gate in this protocol operates with a success probability of <small><math>\frac{1}{16}</math></small>. <ref name="ML03" />

* Waveguide Lattices: Integrated arrays of beam splitters allow for the simulation of quantum walks and complex multi-photon interference patterns. <ref name="15E" /><ref name="16E" />

===Quantum Photonic Chips for Quantum Communication and Internet===
The smallest optical beam splitters are typically found in advanced research within nanophotonics, plasmonics, and integrated optics<ref name="34G" />, where devices are miniaturized for applications like photonic computing<ref name="09E" />, optical communications, and quantum technologies<ref name="16E" />. These are far smaller than commercial or conventional beam splitters (which often measure millimeters to centimeters)<ref name="29E" />. 

====Photonic Beam Splitters====
An example is a silicon-based photonic polarizing beam splitter developed by researchers at the University of Utah. It measures just 2.4 × 2.4 microns (μm) in footprint, making it one of the smallest low-loss all-dielectric designs<ref name="17E" />. This device splits incoming light into two separate polarized channels and was designed to enable light-speed computing by replacing electrons with photons<ref name="06E" />. It was published in 2015 and claimed as the world's smallest at the time.<ref name="18E" />

====Plasmonic Beam Splitters====
Plasmonic designs, which use surface plasmon polaritons (waves at metal-dielectric interfaces) to manipulate light, can be even smaller due to sub-wavelength confinement, though they often have higher losses<ref name="30E" />.
One ultracompact plasmonic polarizing beam splitter on a silicon-on-insulator (SOI) platform has a coupling region of 1.1 μm in length and 50 nanometers (nm) in width. The overall footprint is approximately 1.1 × 0.95 μm (accounting for the waveguides), resulting in an area of about 1 μm². This was reported in 2013 and leverages silver cylinders sandwiched between silicon waveguides for splitting polarized light.<ref name="82G" />
Other plasmonic variants, such as those based on nanoslits or bent directional couplers, have dimensions ranging from hundreds of nm to a few μm, with some coupling lengths as short as 0.9–8.9 μm in more recent designs (e.g., from 2020–2023 papers on slot waveguides or photonic crystals).<ref name="35G" />

====Metasurface-Based Beam Splitters====
Metasurfaces (ultra-thin engineered arrays of nano-antennas) offer nanoscale thickness, often 50–200 nm, while lateral dimensions can be a few μm to tens of μm to handle the beam. These are among the thinnest possible, enabling flat optics for beam splitting with arbitrary ratios or angles<ref name="52E" />. A 2018 example uses gradient metasurfaces for nanoscale thickness, though specific lateral sizes vary by design (typically 5–10 μm across for efficient operation).<ref name="52G" />

These nanoscale beam splitters are fabricated using techniques like electron-beam lithography and are integrated on chips<ref name="15G" />, making them orders of magnitude smaller than traditional glass cubes or plates<ref name="AluSplit" />. Recent developments (post-2020) focus on reducing losses, broadening bandwidth, and integrating with materials like lithium niobate or silicon nitride<ref name="77G" />, but no widely reported designs have broken below ~1 μm in key dimensions while maintaining functionality<ref name="19E" />. If you're interested in a specific type (e.g., for visible light, IR, or quantum applications), more details could narrow it down further<ref name="27E" />.

===Quantum communication===
which applies the principles of quantum mechanics for quantum information transmission, enables fundamental improvements to security, computing, sensing, and metrology. This realm encapsulates a vast variety of technologies and applications ranging from state-of-the-art laboratory experiments to commercial reality. The best-known example is quantum key distribution (QKD)<ref name="01G"/><ref name="02G"/>. The basic idea of QKD is to use the quantum states of photons to share secret keys between two distant parties. The quantum no-cloning theorem endows the two communicating users with the ability to detect any eavesdropper trying to gain knowledge of the key<ref name="03G"/><ref name="04G"/>. Since security is based on the laws of quantum physics rather than computational complexity, QKD is recognized as a desired solution to address the ever-increasing threat raised by emergent quantum computing hardware and algorithms.

Despite the controversy surrounding its practical security, QKD is leading the way to real-world applications<ref name="05G"/>. For example, fiber-based and satellite-to-ground QKD experiments have been demonstrated over 800 km in ultra-low-loss optical fiber<ref name="06G"/> and 2000 km in free space<ref name="07G"/>, respectively. The maximal secure key rate for a single channel has been pushed to more than 110 Mbit/s<ref name="08G"/>. A number of field-test QKD networks have been established in Europe<ref name="09G"/><ref name="10G"/><ref name="11G"/>, Japan<ref name="12G"/>, China<ref name="13G"/><ref name="14G"/>, UK<ref name="15G"/>, and so forth. Furthermore, the security of practical QKD systems was intensively studied to overcome the current technical limitations<ref name="05G"/><ref name="16G"/><ref name="17G"/>. Post-quantum cryptography has been combined with QKD to achieve short-term security of authentication and long-term security of keys<ref name="18G"/>.
===Quantum Communication and Cryptography===
Beam splitters are used to distribute entanglement across networks, enabling secure information transfer.

* Quantum Key Distribution (QKD): Critical for implementing protocols that detect eavesdropping through signal splitting and interference. <ref name="34E" /><ref name="35E" />

* Quantum Repeaters: Used in Bell-state measurements (BSM) to perform entanglement swapping, extending the range of quantum communication. <ref name="51E" />

* Teleportation: A beam splitter is used to perform the joint measurement required to transfer a quantum state <math>|\psi\rangle</math> between distant nodes. <ref name="38E" /><ref name="39E" />

====Quantum Metrology and Sensing====
By creating path-entangled states, such as N00N states of the form <math>(|N,0\rangle + |0,N\rangle)/\sqrt{2}</math>, beam splitters allow sensors to surpass the Standard Quantum Limit.

* Heisenberg-Limit Sensing: Utilizing quantum interference to achieve a phase sensitivity <math>\Delta\phi</math> that scales with <math>1/N</math> rather than the classical <math>1/\sqrt{N}</math>. <ref name="48E" /><ref name="49E" /><ref name="50E" />
* Beam splitter interference in HOM setups enhances metrological precision in quantum kernel methods for feature space analysis.<ref name="24L"/><ref name="33L"/>

====Characterization and Foundations====
To verify the performance of these applications, measures and tests are employed:

* Entanglement Measures: The quality of the generated states is quantified using Concurrence and Entropy of Formation. <ref name="45E" /><ref name="46E" />

* Foundational Tests: Beam splitters provide the platform for Bell test violations and studies of decoherence in open quantum systems. <ref name="36E" /><ref name="37E" />

====Integrated Photonics====
Implementations focusing at on-chip integration using waveguide architectures. An improvement is the use of dibenzoterrylene (DBT) molecules in an anthracene matrix, which has enabled the on-chip integration of independent channels with high-visibility, indistinguishable single photons. <ref name="46K" />

== See also ==
* [[Wikipedia:Hong–Ou–Mandel effect|Hong–Ou–Mandel effect]] 
* [[Wikipedia:Waveguide (optics)|Waveguide (optics)]]
* [[Wikipedia:Integrated quantum photonics|Integrated quantum photonics]] 
* [[Wikipedia:Linear optical quantum computing|Linear optical quantum computing]]
{{:Physics:Quantum basics/See also}}

==Notes==
{{notelist-ua}}


====Sources====
* [https://arxiv.org/pdf/2211.03359 Theory for the beam splitter in quantum optics: quantum entanglement of photons and their statistics, HOM effect. 2022] <br>This reference provides a comprehensive theoretical framework for conventional and frequency-dependent beam splitters. It systematizes how the reflection coefficient and phase shift impact quantum entanglement and photon statistics at the output ports.<br> 
* [https://www.nature.com/articles/s41377-023-01173-8 Recent progress in quantum photonic chips for quantum communication and internet 2023]<br>A review of the state-of-the-art in integrated quantum photonics, focusing on the development of chip-scale components for quantum communication networks and the challenges of multi-photon interference.
* [https://link.springer.com/article/10.1007/s40766-023-00040-x Integrated photonics in quantum technologies 2023]<br>This paper explores the transition from bulk optics to integrated circuits, highlighting how waveguide-based miniaturization is essential for scaling quantum technologies
* [https://ngdc.cncb.ac.cn/openlb/publication/OLB-PM-41193863 On-chip quantum interference of indistinguishable single photons from integrated independent molecules.] <br>This study demonstrates a molecular quantum photonic chip using dibenzoterrylene (DBT) molecules. By integrating these into organic nanosheets and <math>Si_{3}N_{4}</math> waveguides, stable transitions are achieved. This approach builds on the fundamental principles of waveguide-coupled emitters and quantum networks.
* [https://www.nature.com/articles/s42005-025-02020-5 Quantum optical classifier with superexponential speedup 2025] <br>This paper introduces a quantum optical pattern recognition method for binary classification using the Hong-Ou-Mandel effect in a beam splitter-based interferometer. It achieves superexponential speedup by classifying objects without image reconstruction, requiring constant <math>\mathcal{O}(1)</math> resources in photons and operations, providing an advantage over classical neurons.

🎯 Primary Learning Objective

After working through this resource, students should be able to:

* Define a beam splitter and explain its role in quantum optics. 
* Mathematically describe how a beam splitter transforms quantum modes. 
* Understand and analyze key experiments, especially the Hong–Ou–Mandel (HOM) effect. 
* Connect beam splitter behavior to fundamental quantum concepts such as superposition and entanglement. 
* Relate modern integrated photonics implementations (e.g., waveguide beam splitters) to traditional optics.
==References==
{{Reflist|3}}
{{Quantum optics operators}}
[[Category:Quantum optics]][[Category:Optical components]] [[Category:Interferometry]]

{{Author|Harold Foppele}}
{{Sourceattribution|Quantum optics beam splitter experiments|1}}