Physics:Quantum optics beam splitter experiments



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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.

Beam splitter

Other names	BS, directional coupler (in waveguide form)
Primary uses	Quantum superposition · Entanglement generation · Photon statistics · Hong–Ou–Mandel interferometry · Linear-optical quantum computing · Quantum metrology · Quantum communication

Quantum optics beam splitter experiments.

Introduction to the beam splitter in quantum optics

Historical developments in beam splitting range from Fizeau’s 1851 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. The 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.

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 $0.97$), even when utilizing independent molecular single-photon sources important for the scalability of the quantum internet.

Keywords:

Beam splitter, Integrated photonics, Quantum information,waveguide beam splitter, quantum entanglement, photons, reflection coefficient, phase shift, photon statistics, Hong-Ou-Mandel effect.

Quantum optical classifier

Superexponential speedup classification is a central task in deep learning algorithms. Usually, images are first captured and then processed by a sequence of operations, of which the 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 quantum optical pattern recognition method for binary classification tasks. It classifies objects without reconstructing their images, using the rate of two-photon coincidences at the output of a Hong-Ou-Mandel 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 $𝒪(1)$ 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 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 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 . 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 and satellite quantum communications .
Indistinguishable single photons are a fundamental resource for optical quantum technologies, 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, deterministic sources offer greater future promise. Solid-state single-quantum emitters, such as quantum dots, colour centres and organic molecules, could serve as a versatile platform

Overview

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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 quantum teleportation 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 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 and quantum communication via satellites

An essential enabling technology in these advances is the coupling of photonic device components supporting the generation, manipulation, and detection of quantum states . 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 , 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.
An interactive simulation of quantum teleportation in the Virtual Lab by Quantum Flytrap,

History

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.
  
• 1965: Angular momentum theory applied to optical fields, foundational for BS symmetries.
  
• 1966: Density operators for coherent fields at BS, enabling statistical analysis.
  
• 1981: General properties of lossless BS in interferometry.
  
• 1987: Experimental observation of HOM effect, demonstrating two-photon bunching and quantum interference.
  
• 1989: SU(2) symmetry and photon statistics for lossless BS.
  
• 1995: Unitary quantum description of BS.
  
• 2001: KLM protocol for efficient quantum computation with linear optics, establishing scalability using beam splitters, single-photon sources, and detectors.
  
• 2002: Demonstration that nonclassical inputs are required for BS-generated entanglement.
  
• 2008: Silica-on-silicon waveguide quantum circuits, advancing integrated photonic implementations of BS.
  
• 2018–2020: Theoretical models of frequency-dependent effects in waveguide BS, including fluctuations in HOM detection.
  
• 2020–2021: Quantum entanglement and reflection coefficients in coupled waveguide BS models; frequency-dependent theory for waveguide BS.
  
• 2022: Quantum entanglement for monochromatic and non-monochromatic photons on waveguide BS; comprehensive review systematizing conventional vs. frequency-dependent BS.
  

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.

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 ${\hat{a}}_1$ and ${\hat{a}}_2$ are transformed into the outgoing operators ${\hat{b}}_1$ and ${\hat{b}}_2$ by means of a unitary matrix. For a traditional beam splitter, this transformation can be written as

$\left(\begin{array}{c}{\hat{b}}_1\\ {\hat{b}}_2\end{array}\right)=U_{\text{BS}}\left(\begin{array}{c}{\hat{a}}_1\\ {\hat{a}}_2\end{array}\right),$

where the unitary matrix is given by

$U_{\text{BS}}=\left(\begin{array}{cc}\sqrt{T}{e}^{i\varphi }& \sqrt{R}\\ -\sqrt{R}{e}^{-i\varphi }& \sqrt{T}\end{array}\right).$

In these expressions, $T$, $R$, and $\varphi$ represent the transmission coefficient, reflection coefficient, and relative phase, respectively. The unitary nature of $U_{\text{BS}}$ 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 ${\hat{L}}_i$. The associated rotation angles are determined by the reflectivity $R$ and the phase $\varphi$.

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 $\phi ({\omega }_1,{\omega }_2)$, 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 $R$ and the phase $\varphi$ depend explicitly on the frequencies ${\omega }_1$ and ${\omega }_2$. A representative expression for the reflection coefficient is

$R={\sin }^2\left(\frac{\Omega t_{\text{BS}}}{2\sqrt{1+{\epsilon }^2}}\right)(1+{\epsilon }^2),$

where

$\epsilon =\frac{{\omega }_2-{\omega }_1}{\Omega },$

$\Omega$ characterizes the coupling strength between the modes, and $t_{\text{BS}}$ 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., Michelson interferometer). Quantum applications emerged mid-20th century with quantum electrodynamics and lasers, The Hong-Ou-Mandel effect first demonstrated in 1987 . Entanglement by a beam splitter (2002) . Quantum entanglement and reflection coefficient for coupled harmonic oscillators (2020). Quantum entanglement and statistics of photons on a beam splitter in the form of coupled waveguides (2022) .
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.

Beam splitters vary by design and frequency dependence.

Waveguide BS (directional couplers):
Evanescent coupling between waveguides, R(ω) = sin²(κ(ω)L).

Waveguide BS:
enable integration in photonic chips for quantum technologies.

Conventional beam splitters:
Cube, plate, or pellicle BS with nearly constant R, T, φ over bandwidths. Used in free-space experiments.

Frequency-dependent beam splitters:
Coupled-mode theory: dâ₁/dz = -i δ â₁ - i κ â₂, yielding frequency-dependent U_ij(ω).

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:
$\left(\begin{array}{c}{\hat{a}}_{\text{out},1}\\ {\hat{a}}_{\text{out},2}\end{array}\right)=\left(\begin{array}{cc}R(\omega )& T(\omega )\\ -T^{*}(\omega )& R(\omega )\end{array}\right)\left(\begin{array}{c}{\hat{a}}_{\text{in},1}\\ {\hat{a}}_{\text{in},2}\end{array}\right)$
Here, the reflection $R(\omega )$ and transmission $T(\omega )$ 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.

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. BS can be classified in different ways, including their characteristics, such as polarizing BS and non-polarizing BS, and other distinct characteristics. 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). In quantum optics, aluminium-coated beam splitters Figure 1(c) 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.

A Beam Splitter (BS) scheme with two input ports and two output ports

The BS with free space optics, i.e., cubic BS (top) and fiber optics, i.e., waveguide BS (bottom).

Figure 1(c). Aluminium-coated beam splitter.

Two main parameters characterizing a BS in quantum optics are the reflection coefficient $R$ (or transmission coefficient $T$, which satisfies $R+T=1$) and the phase angle $\varphi$. 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 $R$ and $\varphi$ are considered to be definite numbers.

For instance, in the Hong–Ou–Mandel (HOM) effect, an equal splitter with $R=T=\frac{1}{2}$ is used, which is independent of $\varphi$. $R$ and $\varphi$ are functions of the wavelength or frequency of the incoming light, and $R=R(\lambda )$ and $\varphi =\varphi (\lambda )$ in both cases, regardless of which BS is used. 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.

However, in some cases, $R$ and $\varphi$ 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.

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 $R$ and $T$ become frequency-dependent for the photons entering the BS. Accounting for this frequency dependence requires corrections to established theories, such as HOM interferometer fringe analysis and BS-generated entanglement of photons. This pronounced frequency dependence of $R$ and $T$ is a distinct characteristic of waveguide BSs.

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.

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. Photonics-based implementations further integrate nonlinear activations and diffractive networks for all-optical machine learning. Recent QML models address barren plateaus and demonstrate quantum verification of NP problems."

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

$|\mathrm{o}\mathrm{u}\mathrm{t}⟩=e^{i\hat{H}{t}_{\mathrm{B}\mathrm{S}}}|\mathrm{i}\mathrm{n}⟩,$

where $\hat{H}$ is the Hamiltonian of the quantized electromagnetic field interacting with matter, $t_{\mathrm{B}\mathrm{S}}$ is the interaction time, and $|\mathrm{i}\mathrm{n}⟩$ is the initial state of the electromagnetic field. $\hat{H}$ can be quite complex, depending on the type of beam splitter.

In general, the initial state can be represented as

$|\mathrm{i}\mathrm{n}⟩=\sum _{s_1,s_2}{C}_{s_1,s_2}\frac{1}{\sqrt{s_1!s_2!}}({\hat{a}}_1^{†}{)}^{s_1}({\hat{a}}_2^{†}{)}^{s_2}|0{⟩}_1|0{⟩}_2,$ _{(Eq. (1))}

where the creation operators of the first and second modes are ${\hat{a}}_1^{†}$ and $\hat{a}{2}^{†}$, respectively. The integers $s_1$ and $s_2$ are the quantum numbers of the first and second modes (i.e. the number of photons in each mode). The coefficients $C{s}_1,s_2$ define the initial state, and $|0{⟩}_1|0{⟩}_2$ are the vacuum states of modes 1 and 2. For convenience, it is $|0{⟩}_1|0{⟩}_2\equiv |0⟩$.

If the initial states are Fock states, then the coefficients satisfy $C_{s_1,s_2}=1$. In this case, it is straightforward to show (up to an insignificant phase) that

$|\mathrm{o}\mathrm{u}\mathrm{t}⟩=\sum _{s_1,s_2}{C}_{s_1,s_2}\frac{1}{\sqrt{s_1!s_2!}}({\hat{b}}_1^{†}{)}^{s_1}({\hat{b}}_2^{†}{)}^{s_2}|0⟩,$

with

${\hat{b}}_k^{†}=e^{i\hat{H}{t}_{\mathrm{B}\mathrm{S}}}{\hat{a}}_k^{†}{e}^{-i\hat{H}{t}_{\mathrm{B}\mathrm{S}}},k=1,2.$ _{(Eq. (2))}

Here ${\hat{b}}_1^{†}$ and ${\hat{b}}_2^{†}$ 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, ${𝐔}_{BS}$, constrained by the conservation of energy and bosonic commutation relations:

$\left(\begin{array}{c}{\hat{b}}_1\hat{b}2\end{array}\right)=𝐔BS\left(\begin{array}{c}{\hat{a}}_1{\hat{a}}_2\end{array}\right)=\left(\begin{array}{ccc}t& r{r}^{\prime }& t^{\prime }\end{array}\right)\left(\begin{array}{c}{\hat{a}}_1{\hat{a}}_2\end{array}\right)$ _{(Eq. (3)}}

The requirement for unitarity (${𝐔}^{†}𝐔=𝐈$) implies that the transmission and reflection coefficients satisfy:

$|r{|}^2+|t{|}^2=1$ (Energy conservation)

$r^{*}{t}^{\prime }+t^{*}{r}^{\prime }=0$ (Phase relationship between ports)

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

${𝐔}_{BS}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}1& ii& 1\end{array}\right)$ In the literature, one often encounters different representations of the beam splitter matrix. The most commonly used case corresponds to a phase shift $\varphi =\pi /2$, while another frequently used representation sets $\varphi =0$. In these cases the matrix $U_{\mathrm{B}\mathrm{S}}$ takes the forms

$U_{\mathrm{B}\mathrm{S}}=\left(\begin{array}{cc}\sqrt{T}& i\sqrt{R}\\ i\sqrt{R}& \sqrt{T}\end{array}\right),U_{\mathrm{B}\mathrm{S}}=\left(\begin{array}{cc}\sqrt{T}& \sqrt{R}\\ -\sqrt{R}& \sqrt{T}\end{array}\right).$ _{(Eq. (4)}

Both representations are valid only when the final result is independent of the phase shift $\varphi$. 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 In this case, the initial wave function of the photons is

$|\mathrm{i}\mathrm{n}⟩=\sum _{s_1,s_2}{C}_{s_1,s_2}\frac{1}{\sqrt{s_1!s_2!}}\int \Phi ({\omega }_1,{\omega }_2)({\hat{a}}_1^{†}{)}^{s_1}({\hat{a}}_2^{†}{)}^{s_2}|0⟩d{\omega }_{1}d{\omega }_2,$ _{(Eq. (5))}

where $\Phi ({\omega }_1,{\omega }_2)$ is the joint spectral amplitude (JSA) of the two-mode wave function. Assuming normalization,

$\int |\Phi ({\omega }_1,{\omega }_2){|}^{2}d{\omega }_{1}d{\omega }_2=1,$

the output state is

$|\mathrm{o}\mathrm{u}\mathrm{t}⟩=\sum _{s_1,s_2}{C}_{s_1,s_2}\frac{1}{\sqrt{s_1!s_2!}}\int \Phi ({\omega }_1,{\omega }_2)({\hat{b}}_1^{†}{)}^{s_1}({\hat{b}}_2^{†}{)}^{s_2}|0⟩d{\omega }_{1}d{\omega }_2.$ _{(Eq. (6))}

Quantum mechanical description

The output state is |Ψ_out⟩ = e^{iĤ t}_BS |Ψ_in⟩, where Ĥ is the Hamiltonian and t_BS the interaction time. For monochromatic light, the input is |Ψ_in⟩ = ∑_s1,s2 C_s1,s2 (â^†₁^s₁ â^†₂^s₂ / √(s₁! s₂!)) |0⟩₁ |0⟩₂. The unitary matrix U_BS is:

$\left(\begin{array}{c}{\hat{b}}_1{\hat{b}}_2\end{array}\right)=\left(\begin{array}{ccc}\sqrt{T}& e^{i\varphi }\sqrt{R}-e^{-i\varphi }\sqrt{R}& \sqrt{T}\end{array}\right)\left(\begin{array}{c}{\hat{a}}_1{\hat{a}}_2\end{array}\right)$

For non-monochromatic light, integrate over joint spectral amplitude φ(ω₁, ω₂).

Angular momentum representation

BS transformation as SU(2) rotation: L̂_i operators, with L̂² = l̂(l̂ + 1), where l̂ = (n̂₁ + n̂₂)/2. Unitary: Û = e^{-i Φ L̂₃} e^{-i Θ L̂₂} e^{-i Ψ L̂₃}, with Θ = 2θ, θ = arcsin√R.

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.

Quantum entanglement of photons and their statistics

As described by D.N. Makarov in 2022 . in the Theory for the quantum optics beam splitter. Recent advancements in hybrid integrated circuits 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. 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 quantum optics, the quantum beam splitter is a fundamental component far beyond classical beam division: it generates quantum superposition and quantum entanglement from non-entangled inputs, reveals non-classical photon statistics, and enables key phenomena like the Hong–Ou–Mandel effect (HOM effect). While conventional beam splitters are often bulk components, recent progress in integrated photonics has allowed for on-chip implementations. For example, independent dibenzoterrylene ${\mathrm{C}}_{30}{\mathrm{H}}_{18}$ (DBT) molecules integrated with silicon nitride (${\mathrm{S}\mathrm{i}}_3{\mathrm{N}}_4$) photonic elements, a single-crystalline anthracene nanosheet doped with dibenzoterrylene (DBT) molecules, and gold electrodes for Stark tuning (Methods). Waveguides have achieved stable, lifetime-limited transitions suitable for scalable quantum networks. 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 interferometry to scalable computing. 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 non-monochromatic light. . 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. In frequency-dependent BS, statistics vary by mode, leading to selective bunching/antibunching.

Hong–Ou–Mandel effect

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. This equivalence to the SWAP test further extends HOM to high-dimensional interference in spatial modes. Classical analogs achieving 97% visibility dips confirm the role of complementarity in such systems. Identical photons on 50:50 BS bunch, suppressing coincidence counts (HOM dip).
For frequency-dependent BS, dip visibility depends on spectral overlap; fluctuations affect detection.
Generalizations: bosons/fermions, wavepackets.
On a molecular quantum photonic chip, on-chip HOM interference was realized with a visibility of over 0.97. 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 $S{i}_3{N}_4$ waveguides, researchers observed on-chip quantum interference with a visibility of $0.97\pm 0.02$. 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 .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 $𝒪(1)$ 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. 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.

The Hong-Ou-Mandel effect has been successfully applied to quantum kernel evaluation, 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. 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 $𝒪(1)$ computational operations and injected photons, whereas the classical methods are at least linear in the image resolution: a superexponential advantage.

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. Quantum mechanical effects, like superposition and entanglement, can provide a significant speedup in such tasks, e.g. by building quantum analogues of the perceptron, by employing variational methods or quantum-inspired approaches. Quantum optical neural networks harness the best of both worlds, i.e. deep learning capabilities from quantum optics.

Mathematical description

Two optical input modes a and b that carry annihilation and creation operators $\hat{a}$, ${\hat{a}}^{†}$, and $\hat{b}$, ${\hat{b}}^{†}$. Identical photons in different modes can be described by the Fock states, so, for example $|0{⟩}_a$ corresponds to mode a empty (the vacuum state), and inserting one photon into a corresponds to $|1{⟩}_a={\hat{a}}^{†}|0{⟩}_a$, etc. A photon in each input mode is therefore

$|1,1{⟩}_{ab}={\hat{a}}^{†}{\hat{b}}^{†}|0,0{⟩}_{ab}.$

When the two modes a and b are mixed in a 1:1 beam splitter, they produce output modes c and d. Inserting a photon in a produces a superposition state of the outputs: if the beam splitter is 50:50 then the probabilities of each output are equal, i.e. ${\hat{a}}^{†}|0{⟩}_a\to \frac{1}{\sqrt{2}}({\hat{c}}^{†}+{\hat{d}}^{†})|00{⟩}_{cd}$, and similarly for inserting a photon in b. Therefore

${\hat{a}}^{†}\to \frac{{\hat{c}}^{†}+{\hat{d}}^{†}}{\sqrt{2}}\text{and}{\hat{b}}^{†}\to \frac{{\hat{c}}^{†}-{\hat{d}}^{†}}{\sqrt{2}}.$

The relative minus sign appears because the classical lossless beam splitter produces a unitary transformation. This can be seen most clearly when wr the two-mode beam splitter transformation in matrix form:

$\left(\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right)\to \frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{c}\hat{c}\\ \hat{d}\end{array}\right).$

Similar transformations hold for the creation operators. Unitarity of the transformation implies unitarity of the matrix. Physically, this beam splitter transformation means that reflection from one surface induces a relative phase shift of π, corresponding to a factor of −1, with respect to reflection from the other side of the beam splitter (see the Physical description above).

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

$|1,1{⟩}_{ab}={\hat{a}}^{†}{\hat{b}}^{†}|0,0{⟩}_{ab}\to \frac{1}{2}({\hat{c}}^{†}+{\hat{d}}^{†})({\hat{c}}^{†}-{\hat{d}}^{†})|0,0{⟩}_{cd}$

$=\frac{1}{2}({\hat{c}}^{†2}-{\hat{d}}^{†2})|0,0{⟩}_{cd}=\frac{|2,0{⟩}_{cd}-|0,2{⟩}_{cd}}{\sqrt{2}},$

where used ${\hat{c}}^{†2}|0,0{⟩}_{cd}={\hat{c}}^{†}|1,0{⟩}_{cd}=\sqrt{2}|2,0{⟩}_{cd}$ etc. Since the commutator of the two creation operators ${\hat{c}}^{†}$ and ${\hat{d}}^{†}$ is zero because they operate on different spaces, the product term vanishes. The surviving terms in the superposition are only the ${\hat{c}}^{†2}$ and ${\hat{d}}^{†2}$ 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.

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.

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. 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).

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. In tests of photon statistics or entanglement generation, post-selected coincidence measurements between SPDs enable quantification of antibunching ($g^{(2)}(0)<1$) or violation of Bell inequalities.

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. Such integrations facilitate advanced HOM-based classifiers, where SLMs encode trainable parameters for pattern recognition, with software tools enabling simulation and optimization. Training challenges, including initialization and convergence, mirror those in deep neural networks.

Key technologies for quantum photonic chips

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, quantum photonic components such as quantum light sources, high-speed modulators and highly efficient photodetectors, and representative applications including QKD and quantum teleportation. 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 and interferometric circuits realized using silica-on-silicon planar lightwave circuits (PLCs). 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 (${\mathrm{S}\mathrm{i}}_3{\mathrm{N}}_4$), lithium niobate (LN), gallium arsenide (GaAs), indium phosphide (InP), and silicon oxynitride (${\mathrm{S}\mathrm{i}\mathrm{O}}_x{\mathrm{N}}_y$). 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 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.

Notable achievements along these lines include heterogeneous quantum photonic devices such as integrated superconducting nanowire single-photon detectors (SNSPDs) and on-chip lasers for weak coherent pulse generation. Additional important technologies, including semiconductor quantum dots (QDs) coupled to photonic nanostructures and diamond-on-insulator platforms, 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, quantum teleportation realized on a photonic chip, chip-based DV-QKD, CV-QKD, and MDI-QKD, as well as chip-to-chip quantum teleportation.

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 . There are several examples of programmable integrated devices in SoS , Si , SiN silica laser written waveguides with FLW and UV-writing. 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 and Si-chips . 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.

Femtosecond-laser-writing (FLW)

Femtosecond laser direct-writing setup for waveguides in BK7 glass: processing and characterization configurations with beam profile inset (720 nJ pulse energy).

FLW schematic (A) with 3D stage and 40× objective; (B,C) circular waveguide cross-sections (20 µm and 50 µm scales).

The femtosecond laser writing, using Mode locking is a further method for silica waveguide fabrication. 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 as well as devices able to manipulate polarization as waveplates or partially polarizing beamsplitter . 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 and 3D geometries and continuously coupled waveguide lattices . There are instances of re-programmable circuits in small and large scale realization of integrated devices. The integration of single-photon sources exploiting nonlinear effects is still challenging due to the low birefringence ($\Delta n\approx 0$) and the null third-order nonlinear susceptibility (${\chi }^{(3)}=0$) 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. 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.

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: specific quantum computing models (e.g., boson sampling), and universal quantum computing models (e.g., one-way or measurement-based). For specific quantum computation, a variety of photonic systems were demonstrated using quantum photonic chips, enabling a natural and effective implementation of boson sampling. Gaussian boson sampling, 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 $Si$ chip (up to 8 photons) and a $SiN$ chip (up to 18 photons). Recently, quantum computational advantage has been delivered by photonic Gaussian boson sampling processors, paving the path for further development of integrated specific quantum computers with potential applications including graph optimization, complex molecular spectra, molecular docking, quantum chemistry, 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, and compiled Shor’s factorization. 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 and can be greatly enhanced by implementing resource state generation and fusion operation natively. The relevant circuit implementations include programmable fourphoton graph states on a Si chip, path-polarization hyperentangled and cluster states on a $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$ chip and programmable eight-qubit graph states on a Si chip. 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, 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. 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. 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). 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, such as inverted-taper edge couplers interfaced with lensed SMF fibers or ultrahigh numerical aperture fibers, and grating couplers interfaced with SMF fibers. 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. Additionally, efficiencies over 90% have been experimentally demonstrated using edge couplers fabricated on 200-mm SOI wafers. 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. 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). 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 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. 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. Additionally, liquid cooling can be installed to further increase the cooling capacity of the system

On-Chip Quantum Interference

a, Photograph of the quantum photonic chip.b, Optical micrograph of all 24 independent devices integrated on the chip.c, 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). d, SEM images of the waveguides W1 and W2 (with gold electrodes flanking them), the MMI, and one of the output gratings G3. e, DBT molecular structure and energy-level scheme. em., emission; exc., excitation. f, 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 (a), 300 μm (b), 50 μm (c) and 10 μm (d). Recent experimental breakthroughs have successfully implemented these waveguide principles using molecular quantum photonic chips. By integrating independent dibenzoterrylene (DBT) molecules into $S{i}_3{N}_4$ waveguides, researchers have achieved on-chip Hong–Ou–Mandel (HOM) interference with a visibility of $0.97\pm 0.02$.

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.

Evaluating photon indistinguishability from the TPQI experiment under CW excitation

A recent report 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 $\tilde{V}(S)=\frac{\int d\tau [1-g_{\mathrm{H}\mathrm{O}\mathrm{M}}^{(2)}(\tau )]-\int d\tau [1-g_{\mathrm{H}\mathrm{O}\mathrm{M},d}^{(2)}(\tau )]}{\int d\tau [1-g_{\mathrm{H}\mathrm{O}\mathrm{M},d}^{(2)}(\tau )]}$.
Substituting the theoretical expressions for $g_{\mathrm{H}\mathrm{O}\mathrm{M}}^{(2)}(\tau )$ and $g_{\mathrm{H}\mathrm{O}\mathrm{M},d}^{(2)}(\tau )$ from equations respectively,

$\tilde{V}(S)$ is

$\tilde{V}(S)=\frac{ℳ(2ℳ+1)}{ℳ+(ℳ+1)/(1+S)}$ .

where $ℳ=\frac{{\tau }_2}{2{\tau }_1}$. Equation (26) expresses $\tilde{V}$ as a function of $S$.

In the weak excitation limit ($S\to 0$), $\tilde{V}$ reduces to $\frac{{\tau }_2}{2{\tau }_1}$, thereby yieldingn the true photon indistinguishability, consistent with TPQI experiments under pulsed excitation.

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. The artificial neuron, also called perceptron, 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. 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. Adopting the 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. 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. Photonic qubits 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.

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. Later, the distance record for free-space teleportation was pushed beyond 1,400 km between the Micius satellite and a ground station, 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.

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, 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.

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), polarization entanglement created in birefringent media, and time-bin entanglement in Franson interferometers.

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. 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. 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. 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 ($S{i}_3{N}_4$) 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.

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 $\frac{1}{16}$.

• Waveguide Lattices: Integrated arrays of beam splitters allow for the simulation of quantum walks and complex multi-photon interference patterns.

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, where devices are miniaturized for applications like photonic computing, optical communications, and quantum technologies. These are far smaller than commercial or conventional beam splitters (which often measure millimeters to centimeters).

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. This device splits incoming light into two separate polarized channels and was designed to enable light-speed computing by replacing electrons with photons. It was published in 2015 and claimed as the world's smallest at the time.

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. 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. 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).

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. A 2018 example uses gradient metasurfaces for nanoscale thickness, though specific lateral sizes vary by design (typically 5–10 μm across for efficient operation).

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

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). 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. 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. For example, fiber-based and satellite-to-ground QKD experiments have been demonstrated over 800 km in ultra-low-loss optical fiber and 2000 km in free space, respectively. The maximal secure key rate for a single channel has been pushed to more than 110 Mbit/s. A number of field-test QKD networks have been established in Europe, Japan, China, UK, and so forth. Furthermore, the security of practical QKD systems was intensively studied to overcome the current technical limitations. Post-quantum cryptography has been combined with QKD to achieve short-term security of authentication and long-term security of keys.

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.

• Quantum Repeaters: Used in Bell-state measurements (BSM) to perform entanglement swapping, extending the range of quantum communication.

• Teleportation: A beam splitter is used to perform the joint measurement required to transfer a quantum state $|\psi ⟩$ between distant nodes.

Quantum Metrology and Sensing

By creating path-entangled states, such as N00N states of the form $(|N,0⟩+|0,N⟩)/\sqrt{2}$, beam splitters allow sensors to surpass the Standard Quantum Limit.

• Heisenberg-Limit Sensing: Utilizing quantum interference to achieve a phase sensitivity $\Delta \varphi$ that scales with $1/N$ rather than the classical $1/\sqrt{N}$.
• Beam splitter interference in HOM setups enhances metrological precision in quantum kernel methods for feature space analysis.

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.

• Foundational Tests: Beam splitters provide the platform for Bell test violations and studies of decoherence in open quantum systems.

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.

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (21) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Aharonov-Casher effect

77. Physics:Quantum Scattering theory

78. Physics:Quantum Scattering matrix

79. Physics:Quantum S-matrix

80. Physics:Quantum tunnelling

81. Physics:Quantum speed limit

82. Physics:Quantum revival

83. Physics:Quantum reflection

84. Physics:Quantum oscillations

85. Physics:Quantum jump

86. Physics:Quantum boomerang effect

87. Physics:Quantum chaos

7. Measurement and information (9) Back to index

88. Physics:Quantum Measurement theory

89. Physics:Quantum Measurement operators

90. Physics:Quantum Projective measurement

91. Physics:Quantum POVM

92. Physics:Quantum Weak measurement

93. Physics:Quantum Measurement collapse

94. Physics:Quantum entanglement

95. Physics:Quantum Zeno effect

96. Physics:Quantum limit

8. Quantum information and computing (15) Back to index

97. Physics:Quantum information theory

98. Physics:Quantum Qubit

99. Physics:Quantum Entanglement

100. Physics:Quantum Bell state

101. Physics:Quantum Gates and circuits

102. Physics:Quantum BB84

103. Physics:Quantum No-cloning theorem

104. Physics:Quantum Computing Algorithms in the NISQ Era

105. Physics:Quantum Noisy Qubits

106. Physics:Quantum error correction

107. Physics:Quantum Boson sampling

108. Physics:Quantum random access code

109. Physics:Quantum pseudo-telepathy

110. Physics:Quantum network

111. Physics:Quantum money

9. Quantum optics and experiments (10) Back to index

112. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

113. Physics:Quantum Stern-Gerlach experiment

114. Physics:Quantum Experimental quantum physics

115. Physics:Quantum optics

116. Physics:Quantum optics beam splitter experiments

117. Physics:Quantum Mach-Zehnder interferometer

118. Physics:Quantum Hong-Ou-Mandel effect

119. Physics:Quantum eraser experiment

120. Physics:Quantum delayed-choice quantum eraser

121. Physics:Quantum Ultra fast lasers

122. Template:Quantum optics operators

10. Open quantum systems (15) Back to index

123. Physics:Quantum Open systems

124. Physics:Quantum channel

125. Physics:Quantum Kraus operators

126. Physics:Quantum Amplitude damping

127. Physics:Quantum Phase damping

128. Physics:Quantum Depolarizing channel

129. Physics:Quantum Master equation

130. Physics:Quantum Lindblad equation

131. Physics:Quantum Decoherence

132. Physics:Quantum Dynamical decoupling

133. Physics:Quantum dissipation

134. Physics:Quantum Markov semigroup

135. Physics:Quantum Markovian dynamics

136. Physics:Quantum Non-Markovian dynamics

137. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

138. Physics:Quantum field theory (QFT) basics

139. Physics:Quantum field theory (QFT) core

140. Physics:Quantum Fields and Particles

141. Physics:Quantum Second quantization

142. Physics:Quantum Fock space

143. Physics:Quantum Harmonic Oscillator field modes

144. Physics:Quantum Creation and annihilation operators

145. Physics:Quantum vacuum fluctuations

146. Physics:Quantum Casimir effect

147. Physics:Quantum Propagators in quantum field theory

148. Physics:Quantum Feynman diagrams

149. Physics:Quantum Path integral formulation

150. Physics:Quantum Renormalization in field theory

151. Physics:Quantum Renormalization group

152. Physics:Quantum Field Theory Gauge symmetry

153. Physics:Quantum Spontaneous symmetry breaking

154. Physics:Quantum Non-Abelian gauge theory

155. Physics:Quantum Electrodynamics (QED)

156. Physics:Quantum chromodynamics (QCD)

157. Physics:Quantum Electroweak theory

158. Physics:Quantum Standard Model

159. Physics:Quantum triviality

160. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

161. Physics:Quantum Statistical mechanics

162. Physics:Quantum Partition function

163. Physics:Quantum Distribution functions

164. Physics:Quantum Liouville equation

165. Physics:Quantum Kinetic theory

166. Physics:Quantum Boltzmann equation

167. Physics:Quantum BBGKY hierarchy

168. Physics:Quantum Relaxation and thermalization

169. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (17) Back to index

170. Physics:Quantum Band structure

171. Physics:Quantum Fermi surfaces

172. Physics:Quantum Landau levels

173. Physics:Quantum fractional Hall effect

174. Physics:Quantum Semiconductor physics

175. Physics:Quantum Phonons

176. Physics:Quantum Electron-phonon interaction

177. Physics:Quantum Exchange interaction

178. Physics:Quantum Superconductivity

179. Physics:Quantum Topological phases of matter

180. Physics:Quantum anyon

181. Physics:Quantum well

182. Physics:Quantum spin liquid

183. Physics:Quantum spin Hall effect

184. Physics:Quantum phase transition

185. Physics:Quantum critical point

186. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

187. Physics:Quantum Fusion reactions and Lawson criterion

188. Physics:Quantum Plasma (fusion context)

189. Physics:Quantum Magnetic confinement fusion

190. Physics:Quantum Inertial confinement fusion

191. Physics:Quantum Plasma instabilities and turbulence

192. Physics:Quantum Tokamak core plasma

193. Physics:Quantum Tokamak edge physics and recycling asymmetries

194. Physics:Quantum Stellarator

15. Timeline (8) Back to index

195. Physics:Quantum mechanics/Timeline

196. Physics:Quantum mechanics/Timeline/Pre-quantum era

197. Physics:Quantum mechanics/Timeline/Old quantum theory

198. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

199. Physics:Quantum mechanics/Timeline/Quantum field theory era

200. Physics:Quantum mechanics/Timeline/Quantum information era

201. Physics:Quantum mechanics/Timeline/Quantum technology era

202. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

203. Physics:Quantum topology

204. Physics:Quantum battery

205. Physics:Quantum Supersymmetry

206. Physics:Quantum Black hole thermodynamics

207. Physics:Quantum Holographic principle

208. Physics:Quantum gravity

209. Physics:Quantum De Sitter invariant special relativity

210. Physics:Quantum Doubly special relativity

211. Physics:Quantum arithmetic geometry

212. Physics:Quantum unsolved problems

213. Physics:Quantum Yang-Mills mass gap

214. Physics:Quantum gravity problem

215. Physics:Quantum black hole information paradox

216. Physics:Quantum dark matter problem

217. Physics:Quantum neutrino mass problem

218. Physics:Quantum matter-antimatter asymmetry problem

See also

• Hong–Ou–Mandel effect
• Waveguide (optics)
• Integrated quantum photonics
• Linear optical quantum computing

Index

Core theory Foundations Conceptual and interpretations Mathematical structure and systems Atomic and spectroscopy Wavefunctions and modes Quantum dynamics and evolution Measurement and information Quantum information and computing

Applications and extensions Quantum optics and experiments Open quantum systems Quantum field theory Statistical mechanics and kinetic theory Condensed matter and solid-state physics Plasma and fusion physics Timeline Advanced and frontier topics

Quantum Book II

Matter by scale

Quantum Book III

Methods and tools

Quantum Book IV

Data Analysis Techniques

Full contents

Foundations



1. Physics:Quantum basics
2. Physics:Quantum photoelectric effect
3. Physics:Quantum black-body radiation
4. Physics:Quantum Planck constant
5. Physics:Quantum Postulates
6. Physics:Quantum Hilbert space
7. Physics:Quantum Observables and operators
8. Physics:Quantum mechanics
9. Physics:Quantum mechanics measurements
10. Physics:Quantum state
11. Physics:Quantum system
12. Physics:Quantum superposition
13. Physics:Quantum probability
14. Physics:Quantum Mathematical Foundations of Quantum Theory

Conceptual and interpretations



15. Physics:Quantum Interpretations of quantum mechanics
16. Physics:Quantum Wave–particle duality
17. Physics:Quantum Complementarity principle
18. Physics:Quantum Uncertainty principle
19. Physics:Quantum Measurement problem
20. Physics:Quantum Bell's theorem
21. Physics:Quantum Hidden variable theory
22. Physics:Quantum nonlocality
23. Physics:Quantum contextuality
24. Physics:Quantum Darwinism
25. Physics:Quantum A Spooky Action at a Distance
26. Physics:Quantum A Walk Through the Universe
27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together
28. Physics:Quantum measurement problem

Mathematical structure and systems



29. Physics:Quantum Density matrix
30. Physics:Quantum Exactly solvable quantum systems
31. Physics:Quantum many-body problem
32. Physics:Quantum Formulas Collection
33. Physics:Quantum A Matter Of Size
34. Physics:Quantum Symmetry in quantum mechanics
35. Physics:Quantum Noether theorem
36. Physics:Quantum Angular momentum operator
37. Physics:Quantum Runge–Lenz vector
38. Physics:Quantum Approximation Methods
39. Physics:Quantum Matter Elements and Particles
40. Physics:Quantum Dirac equation
41. Physics:Quantum Klein–Gordon equation
42. Physics:Quantum pendulum
43. Physics:Quantum configuration space

Atomic and spectroscopy



Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

44. Physics:Quantum Atomic structure and spectroscopy
45. Physics:Quantum Hydrogen atom
46. Physics:Quantum number
47. Physics:Quantum Multi-electron atoms
48. Physics:Quantum Fine structure
49. Physics:Quantum Hyperfine structure
50. Physics:Quantum Isotopic shift
51. Physics:Quantum defect
52. Physics:Quantum Zeeman effect
53. Physics:Quantum Stark effect
54. Physics:Quantum Spectral lines and series
55. Physics:Quantum Selection rules
56. Physics:Quantum Fermi's golden rule
57. Physics:Quantum beats

Wavefunctions and modes



A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

58. Physics:Quantum Wavefunction
59. Physics:Quantum Superposition principle
60. Physics:Quantum Eigenstates and eigenvalues
61. Physics:Quantum Boundary conditions and quantization
62. Physics:Quantum Standing waves and modes
63. Physics:Quantum Normal modes and field quantization
64. Physics:Number of independent spatial modes in a spherical volume
65. Physics:Quantum Density of states
66. Physics:Quantum carpet

Quantum dynamics and evolution



67. Physics:Quantum Time evolution
68. Physics:Quantum Schrödinger equation
69. Physics:Quantum Time-dependent Schrödinger equation
70. Physics:Quantum Stationary states
71. Physics:Quantum Perturbation theory
72. Physics:Quantum Time-dependent perturbation theory
73. Physics:Quantum Adiabatic theorem
74. Physics:Quantum Berry phase
75. Physics:Quantum Aharonov-Bohm effect
76. Physics:Quantum Aharonov-Casher effect
77. Physics:Quantum Scattering theory
78. Physics:Quantum Scattering matrix
79. Physics:Quantum S-matrix
80. Physics:Quantum tunnelling
81. Physics:Quantum speed limit
82. Physics:Quantum revival
83. Physics:Quantum reflection
84. Physics:Quantum oscillations
85. Physics:Quantum jump
86. Physics:Quantum boomerang effect
87. Physics:Quantum chaos

Measurement and information



88. Physics:Quantum Measurement theory
89. Physics:Quantum Measurement operators
90. Physics:Quantum Projective measurement
91. Physics:Quantum POVM
92. Physics:Quantum Weak measurement
93. Physics:Quantum Measurement collapse
94. Physics:Quantum entanglement
95. Physics:Quantum Zeno effect
96. Physics:Quantum limit

Quantum information and computing



97. Physics:Quantum information theory
98. Physics:Quantum Qubit
99. Physics:Quantum Entanglement
100. Physics:Quantum Bell state
101. Physics:Quantum Gates and circuits
102. Physics:Quantum BB84
103. Physics:Quantum No-cloning theorem
104. Physics:Quantum Computing Algorithms in the NISQ Era
105. Physics:Quantum Noisy Qubits
106. Physics:Quantum error correction
107. Physics:Quantum Boson sampling
108. Physics:Quantum random access code
109. Physics:Quantum pseudo-telepathy
110. Physics:Quantum network
111. Physics:Quantum money

Quantum optics and experiments



Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

112. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
113. Physics:Quantum Stern-Gerlach experiment
114. Physics:Quantum Experimental quantum physics
115. Physics:Quantum optics
116. Physics:Quantum optics beam splitter experiments
117. Physics:Quantum Mach-Zehnder interferometer
118. Physics:Quantum Hong-Ou-Mandel effect
119. Physics:Quantum eraser experiment
120. Physics:Quantum delayed-choice quantum eraser
121. Physics:Quantum Ultra fast lasers
122. Template:Quantum optics operators

Open quantum systems



123. Physics:Quantum Open systems
124. Physics:Quantum channel
125. Physics:Quantum Kraus operators
126. Physics:Quantum Amplitude damping
127. Physics:Quantum Phase damping
128. Physics:Quantum Depolarizing channel
129. Physics:Quantum Master equation
130. Physics:Quantum Lindblad equation
131. Physics:Quantum Decoherence
132. Physics:Quantum Dynamical decoupling
133. Physics:Quantum dissipation
134. Physics:Quantum Markov semigroup
135. Physics:Quantum Markovian dynamics
136. Physics:Quantum Non-Markovian dynamics
137. Physics:Quantum Trajectories

Quantum field theory



Quantum field theory: fields, particles, interactions, diagrams, and renormalization.

138. Physics:Quantum field theory (QFT) basics
139. Physics:Quantum field theory (QFT) core
140. Physics:Quantum Fields and Particles
141. Physics:Quantum Second quantization
142. Physics:Quantum Fock space
143. Physics:Quantum Harmonic Oscillator field modes
144. Physics:Quantum Creation and annihilation operators
145. Physics:Quantum vacuum fluctuations
146. Physics:Quantum Casimir effect
147. Physics:Quantum Propagators in quantum field theory
148. Physics:Quantum Feynman diagrams
149. Physics:Quantum Path integral formulation
150. Physics:Quantum Renormalization in field theory
151. Physics:Quantum Renormalization group
152. Physics:Quantum Field Theory Gauge symmetry
153. Physics:Quantum Spontaneous symmetry breaking
154. Physics:Quantum Non-Abelian gauge theory
155. Physics:Quantum Electrodynamics (QED)
156. Physics:Quantum chromodynamics (QCD)
157. Physics:Quantum Electroweak theory
158. Physics:Quantum Standard Model
159. Physics:Quantum triviality
160. Physics:Quantum confinement problem

Statistical mechanics and kinetic theory



Statistical mechanics and kinetic theory: distributions, ensembles, transport, and thermalization.

161. Physics:Quantum Statistical mechanics
162. Physics:Quantum Partition function
163. Physics:Quantum Distribution functions
164. Physics:Quantum Liouville equation
165. Physics:Quantum Kinetic theory
166. Physics:Quantum Boltzmann equation
167. Physics:Quantum BBGKY hierarchy
168. Physics:Quantum Relaxation and thermalization
169. Physics:Quantum Thermodynamics

Condensed matter and solid-state physics



170. Physics:Quantum Band structure
171. Physics:Quantum Fermi surfaces
172. Physics:Quantum Landau levels
173. Physics:Quantum fractional Hall effect
174. Physics:Quantum Semiconductor physics
175. Physics:Quantum Phonons
176. Physics:Quantum Electron-phonon interaction
177. Physics:Quantum Exchange interaction
178. Physics:Quantum Superconductivity
179. Physics:Quantum Topological phases of matter
180. Physics:Quantum anyon
181. Physics:Quantum well
182. Physics:Quantum spin liquid
183. Physics:Quantum spin Hall effect
184. Physics:Quantum phase transition
185. Physics:Quantum critical point
186. Physics:Quantum dot

Plasma and fusion physics



Plasma and fusion physics: magnetic confinement, plasma flow, turbulence, and collective field dynamics.

187. Physics:Quantum Fusion reactions and Lawson criterion
188. Physics:Quantum Plasma (fusion context)
189. Physics:Quantum Magnetic confinement fusion
190. Physics:Quantum Inertial confinement fusion
191. Physics:Quantum Plasma instabilities and turbulence
192. Physics:Quantum Tokamak core plasma
193. Physics:Quantum Tokamak edge physics and recycling asymmetries
194. Physics:Quantum Stellarator

Timeline



195. Physics:Quantum mechanics/Timeline
196. Physics:Quantum mechanics/Timeline/Pre-quantum era
197. Physics:Quantum mechanics/Timeline/Old quantum theory
198. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
199. Physics:Quantum mechanics/Timeline/Quantum field theory era
200. Physics:Quantum mechanics/Timeline/Quantum information era
201. Physics:Quantum mechanics/Timeline/Quantum technology era
202. Physics:Quantum mechanics/Timeline/Quiz

Advanced and frontier topics



203. Physics:Quantum topology
204. Physics:Quantum battery
205. Physics:Quantum Supersymmetry
206. Physics:Quantum Black hole thermodynamics
207. Physics:Quantum Holographic principle
208. Physics:Quantum gravity
209. Physics:Quantum De Sitter invariant special relativity
210. Physics:Quantum Doubly special relativity
211. Physics:Quantum arithmetic geometry
212. Physics:Quantum unsolved problems
213. Physics:Quantum Yang-Mills mass gap
214. Physics:Quantum gravity problem
215. Physics:Quantum black hole information paradox
216. Physics:Quantum dark matter problem
217. Physics:Quantum neutrino mass problem
218. Physics:Quantum matter-antimatter asymmetry problem

Notes

Sources

• Theory for the beam splitter in quantum optics: quantum entanglement of photons and their statistics, HOM effect. 2022
  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.
  
• Recent progress in quantum photonic chips for quantum communication and internet 2023
  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.
• Integrated photonics in quantum technologies 2023
  This paper explores the transition from bulk optics to integrated circuits, highlighting how waveguide-based miniaturization is essential for scaling quantum technologies
• On-chip quantum interference of indistinguishable single photons from integrated independent molecules.
  This study demonstrates a molecular quantum photonic chip using dibenzoterrylene (DBT) molecules. By integrating these into organic nanosheets and $S{i}_3{N}_4$ waveguides, stable transitions are achieved. This approach builds on the fundamental principles of waveguide-coupled emitters and quantum networks.
• Quantum optical classifier with superexponential speedup 2025
  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 $𝒪(1)$ 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



Source attribution: Template Quantum optics operators

Author: Harold Foppele

Source attribution: Physics:Quantum optics beam splitter experiments