Universal multiport interferometer

Universal unitary (Reck and Zeilinger)

In quantum mechanics, an operator ${\textstyle {\hat {U}}}$ is unitary if it and its Hermitian conjugate, ${\textstyle {\hat {U}}^{\dagger }}$, satisfy the condition ${\textstyle {\hat {U}}{\hat {U}}^{\dagger }={\hat {I}}={\hat {U}}^{\dagger }{\hat {U}}}$, where ${\textstyle {\hat {I}}}$ is the identity operator [1]. By extension, a unitary transformation describes the change of a state, ${\textstyle |\psi >\rightarrow |\psi '>={\hat {U}}|\psi >}$, or the change an operator, ${\textstyle {\hat {A}}\rightarrow {\hat {A}}'={\hat {U}}{\hat {A}}{\hat {U}}^{\dagger }}$, under the action of a unitary operator. Hence, a platform capable of imposing completely general unitary transformations is referred to, itself, as a universal unitary. Importantly, any lossless experimental setup can be described by a unitary operator [2]. However, designing an experimental setup that implements any given unitary operator is, unfortunately, a more challenging task.

In general, fully determining any ${\textstyle N}$-dimensional unitary requires specifying ${\textstyle N^{2}}$ independent real parameters [3]. For the simple case of transforming a two-beam array, a universal unitary can be implemented using a variable beam splitter and three phase-shifters [3,4]. In 1994, Michael Reck and Anton Zeilinger generalized this well-known approach by proving that variable beam splitters and phase-shifters, when arranged in an interferometric mesh with ${\textstyle N^{2}}$ arms, can be used to impose any ${\textstyle N{\text{x}}N}$ unitary mode transformation [2]. Using their deterministic algorithm to decompose a given unitary into a network of these two optical elements, it is possible to experimentally realize a discrete universal unitary, specifically for mode transformations.

Distinction between universal unitaries.

Variations include.

Definitions

Unitary transformations

Here is a formal theorem that supports the main topic defined in the previous section. If the main topic of the article is the theorem itself, then this would likely be moved to the previous section.

Rainbow dove surface hologram used as a security feature on most visa cards.

The compression ratio for the three-lens spaceplate is defined as,

$R={\frac {d_{eff}}{d}}={\frac {f_{ext}}{2f_{mid}}}-1$,

where $f_{ext}$ is the focal length of the external lenses and $f_{mid}$ is the focal length of the middle lens.

Photon states

Here is a supporting statement that is easily deduced from the main theorem and may help motivate the main topic of this article.

The ideal spaceplate phase is given by,

$\varphi _{SP}(k_{x},k_{y},d_{eff})=d_{eff}(|{\textbf {k}}|^{2}-k_{x}^{2}-k_{y}^{2})^{1/2}$.

Unitary spatial mode transformations

Here is a classical perspective of the problem and how it contrasts with the quantum perspective. This should highlight the need for a quantum perspective.

Motivation

Almost all scientific Wikipedia articles that cover specific mathematical concepts or formulae provide a properties section. This will likely have links to external sources for proofs.

Beam splitters:

• Definition or equation describing property 1. Example: additivity.
• Example supporting property 1.
  • Additivity:$f(x+y)=f(x)+f(y)$
• Reference to formal proof.

Phase Shifters:

• Definition or equation describing property 2. Example: homogeneity of degree 1.
• Example supporting property 2.
  • Homogeneity of degree 1: $f(\alpha x)=\alpha f(x)$ for all $\alpha$
• Reference to formal proof.

Experimental implementation

Albert Einstein, 1921.

The Reck and Zeilinger (R-Z) scheme

If there is an intriguing story behind the main topic or problem, the article will likely reference it in a history section. If the main topic is experimental in itself, this section may be combined with the following section on experimental evidence.

Albert Einstein is credited with this discovery. Explain the history behind the main topic.

Algorithm

Experiment 1

Overview

Some articles will provide descriptions of famous experimental results that support the main topic.

An interferometer typically used for demonstration purposes in educational settings.

Setup

Here is the setup of experiment 1. Explain the setup.

The interferometer is seen in the image on the right.

Results

Here are the results of experiment 1.

Experiment 2

Overview

Some articles will provide descriptions of famous experimental results that support the main topic.

Setup

Here is the setup of experiment 2. Explain the setup.

$E=mc^{2}$

Results

Here are the results of experiment 2.

Variants and similar decompositions

The interior of LIGO at Hanford, Washington.

Real-world application 1

The interior of LIGO is seen in the image on the right.

Real-world application 2

The Hubble Space Telescope. Explain the application.

See also

• Fourier optics
• Holography
• Quantum information

Further reading

• Savoia, S., Castaldi, G., & Galdi, V. (2013). Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices. Physical Review B - Condensed Matter and Materials Physics, 87(23). https://doi.org/10.1103/PhysRevB.87.235116
• Zhou, Y., Zheng, H., Kravchenko, I. I., & Valentine, J. (2020). Flat optics for image differentiation. Nature Photonics, 14(5), 316–323. https://doi.org/10.1038/s41566-020-0591-3
• Pagé, J. T. R., Reshef, O., Boyd, R. W., & Lundeen, J. S. (2022). Designing high-performance propagation-compressing spaceplates using thin-film multilayer stacks. Optics Express, 30(2), 2197. https://doi.org/10.1364/oe.443067