Universal multiport interferometer

Universal unitary (mode transformations)

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, ${\textstyle |\psi \rangle \rightarrow |\psi '\rangle ={\hat {U}}|\psi \rangle }$, 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 triangular network of these two optical elements, it is possible to experimentally realize a discrete universal unitary, specifically for ${\textstyle N{\text{x}}N}$ mode transformations. The resulting device is commonly referred to as a universal multiport interferometer.

In 2016, Clements et al. introduced a variation of Reck and Zeilinger's decomposition, again using beam splitters and phase shifters, but arranged in a symmetrically crossing network as opposed to a triangular network. Importantly, this variation has a smaller optical depth - the longest path through the interferometric mesh - and thus experiences lower propagation losses.

Transformations beyond the subset of linear ${\textstyle N{\text{x}}N}$ mode transformations cannot be realized, in general, using such decompositions. Hence, they are not universal for the complete set of unitary transformations. This distinguishes it from other universal unitary platforms which implement transformations existing within another subset. For example, the universal unitary commonly discussed in quantum computing is the universal gate whereby any ${\textstyle N}$-qubit gate can be realized by a circuit of single qubit gates and CNOT gates. The classical analog of such universality is the idea that an arbitrary Boolean function can be realized using a combination of NOT gates and any one of the two-bit gates (e.g. AND, OR).

Definitions

Unitary operator and transformations

In quantum mechanics, the inverse of an operator ${\textstyle {\hat {O}}}$ is represented by ${\textstyle {\hat {O}}^{-1}}$ , and is defined such that ${\textstyle {\hat {O}}{\hat {O}}^{-1}={\hat {I}}}$ where ${\textstyle {\hat {I}}}$ is the identity operator. Among the complete set of operators, those that satisfy the additional condition that ${\textstyle {\hat {O}}^{-1}={\hat {O}}^{\dagger }}$ are called unitary and are denoted ${\textstyle {\hat {U}}}$. This is where the superscript "$\dagger$" signifies the Hermitian conjugate (or Hermitian adjoint, Hermitian transpose, conjugate transpose) which is the conjugate transpose of the given operator, i.e., ${\textstyle {\hat {O}}^{\dagger }=\left({\hat {O}}^{*}\right)^{T}}$. Therefore, any unitary operator ${\textstyle {\hat {U}}}$ must, by definition, satisfy the following condition:$${\hat {U}}{\hat {U}}^{\dagger }={\hat {I}}={\hat {U}}^{\dagger }{\hat {U}}.$$Unitary operators play an important role in quantum mechanics as most physical processes evolve unitarily. An exception to this is measurement, which is an inherently non-unitary process.

Property 1: Preserving inner product

Suppose there exists a unitary operator ${\textstyle {\hat {U}}}$ which transforms the state ${\textstyle |v\rangle }$ according to ${\textstyle |v\rangle \rightarrow {\hat {U}}|v\rangle =|v'\rangle }$ and which transforms the state ${\textstyle |w\rangle }$ according to ${\textstyle |w\rangle \rightarrow {\hat {U}}|w\rangle =|w'\rangle }$. Then, the inner product of the transformed states, $|v'\rangle$ and $|w'\rangle$, in bra-ket notation is given by,$${\begin{array}{lcl}\langle v'|w'\rangle &=&\left({\hat {U}}|v\rangle \right)^{\dagger }{\hat {U}}|w\rangle \\&=&\left(\langle v|{\hat {U}}^{\dagger }\right){\hat {U}}|w\rangle \\&=&\langle v|{\hat {U}}^{\dagger }{\hat {U}}|w\rangle \\&=&\langle v|{\hat {I}}|w\rangle \\\langle v'|w'\rangle &=&\langle v|w\rangle .\end{array}}$$Hence, the inner product is preserved under the unitary transformation of states. The consequence of this property is information is conserved under unitary transformation.

Property 2: Diagonalization of Hermitian operators

If an operator ${\textstyle {\hat {O}}}$ is Hermitian such that ${\textstyle {\hat {O}}={\hat {O}}^{\dagger }}$, then according to finite-dimensional spectral theorem, ${\textstyle {\hat {O}}}$ can be diagonalized by some unitary operator ${\textstyle {\hat {U}}}$. Mathematically, this property is represented by,$${\hat {O}}={\hat {U}}{\hat {\lambda }}{\hat {U}}^{\dagger },$$where ${\textstyle {\hat {\lambda }}}$ is a diagonal matrix with ${\textstyle \lambda _{i}}$ as elements, i.e.,$${\hat {\lambda }}\Rightarrow \sum _{i}^{d}\lambda _{i}|\lambda _{i}\rangle \langle \lambda _{i}|.$$It follows that,

${\begin{array}{lcl}{\hat {U}}^{\dagger }{\hat {O}}{\hat {U}}&=&{\hat {U}}^{\dagger }\left({\hat {U}}{\hat {\lambda }}{\hat {U}}^{\dagger }\right){\hat {U}}\\&=&{\hat {I}}{\hat {\lambda }}{\hat {I}}\\{\hat {U}}^{\dagger }{\hat {O}}{\hat {U}}&=&{\hat {\lambda }},\end{array}}$

or, in other words, the action of ${\textstyle {\hat {U}}}$ can be described as a mere change-of-basis of ${\textstyle {\hat {O}}}$ to one in which it is diagonal, namely, the set$\{|\lambda _{i}\rangle \langle \lambda _{i}|\}_{d}$. In the physical world, unitary operators are the generalizations of rotations in 3-dimensional Euclidean space.

Property 3: Dimensionality

Suppose there exists a unitary operator ${\textstyle {\hat {U}}}$ of dimension ${\textstyle N}$, then the number of elements contained in its matrix representation is ${\textstyle N^{2}}$. The elements are, in general, complex so this corresponds to a total of ${\textstyle 2N^{2}}$ real independent parameters. However, the conditions of unitarity reduce the number of real independent parameters.

According to the definition of a unitary operator, ${\textstyle {\hat {U}}}$ is required to satisfy ${\textstyle {\hat {U}}^{\dagger }{\hat {U}}={\hat {I}}}$, but ${\textstyle {\hat {I}}}$ is Hermitian. Therefore, the diagonal elements of ${\textstyle {\hat {U}}^{\dagger }{\hat {U}}}$ must be real valued. This corresponds to ${\textstyle N}$ real independent parameters.

The Hermiticity of ${\textstyle {\hat {I}}}$ implies ${\textstyle {\hat {U}}^{\dagger }{\hat {U}}={\hat {U}}{\hat {U}}^{\dagger }}$, which further implies the lower triangular elements of ${\textstyle {\hat {U}}^{\dagger }{\hat {U}}}$ can be expressed in terms of the upper triangular elements such that the off-diagonal elements correspond to only ${\textstyle 2\left(N(N-1)/2\right)}$ real independent parameters instead of ${\textstyle 2\left(N(N-1)\right)}$.

The total number of real independent parameters is thus reduced from ${\textstyle 2N^{2}}$ to:$$N_{\text{ind}}=2N^{2}+N+2\left(N(N-1)/2\right),$$ and hence the number of free parameters is:$$N_{\text{Free}}=2N^{2}-\left(N+2\left(N(N-1)/2\right)\right)=N^{2}.$$For example, suppose ${\textstyle {\hat {U}}}$ is two-dimensional.

$${\hat {U}}^{\dagger }{\hat {U}}\Rightarrow {\begin{bmatrix}u_{11}^{*}&u_{21}^{*}\\u_{12}^{*}&u_{22}^{*}\end{bmatrix}}\cdot {\begin{bmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{bmatrix}}={\begin{bmatrix}|u_{11}|^{2}+|u_{21}|^{2}&u_{11}^{*}u_{12}+u_{21}^{*}u_{22}\\u_{12}^{*}u_{11}+u_{22}^{*}u_{21}&|u_{12}|^{2}+|u_{22}|^{2}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$$$${\hat {U}}{\hat {U}}^{\dagger }\Rightarrow {\begin{bmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{bmatrix}}\cdot {\begin{bmatrix}u_{11}^{*}&u_{21}^{*}\\u_{12}^{*}&u_{22}^{*}\end{bmatrix}}={\begin{bmatrix}|u_{11}|^{2}+|u_{12}|^{2}&u_{11}u_{21}^{*}+u_{21}u_{22}^{*}\\u_{21}u_{11}^{*}+u_{22}u_{12}^{*}&|u_{21}|^{2}+|u_{22}|^{2}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$$

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.

Experimental implementation

Albert Einstein, 1921.

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.

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