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:$$N_{\text{Free}}=2N^{2}-\left(N+2\left(N(N-1)/2\right)\right)=N^{2}.$$Since ${\textstyle {\hat {U}}^{\dagger }{\hat {U}}}$ has the same dimension as ${\textstyle {\hat {U}}}$, the number of free parameters describing ${\textstyle {\hat {U}}^{\dagger }{\hat {U}}}$ is the same number of free parameters describing ${\textstyle {\hat {U}}}$.

For example, suppose ${\textstyle {\hat {U}}}$ is two-dimensional.$${\hat {U}}^{\dagger }{\hat {U}}\Rightarrow {\begin{bmatrix}u_{11}^{*}&u_{21}^{*}\\u_{12}^{*}&u_{22}^{*}\end{bmatrix}}\cdot {\begin{bmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{bmatrix}}={\begin{bmatrix}|u_{11}|^{2}+|u_{21}|^{2}&u_{11}^{*}u_{12}+u_{21}^{*}u_{22}\\u_{12}^{*}u_{11}+u_{22}^{*}u_{21}&|u_{12}|^{2}+|u_{22}|^{2}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$$$${\hat {U}}{\hat {U}}^{\dagger }\Rightarrow {\begin{bmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{bmatrix}}\cdot {\begin{bmatrix}u_{11}^{*}&u_{21}^{*}\\u_{12}^{*}&u_{22}^{*}\end{bmatrix}}={\begin{bmatrix}|u_{11}|^{2}+|u_{12}|^{2}&u_{11}u_{21}^{*}+u_{12}u_{22}^{*}\\u_{21}u_{11}^{*}+u_{22}u_{12}^{*}&|u_{21}|^{2}+|u_{22}|^{2}\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$$The Hermiticity condition gives, for the upper and lower triangular elements:$${\begin{cases}u_{11}^{*}u_{12}+u_{21}^{*}u_{22}=u_{11}u_{21}^{*}+u_{12}u_{22}^{*}\\u_{12}^{*}u_{11}+u_{22}^{*}u_{21}=u_{21}u_{11}^{*}+u_{22}u_{12}^{*},\end{cases}}$$which are equivalent equations up to complex conjugation. By setting the right-hand side to the matrix representation of ${\textstyle {\hat {I}}}$, one arrives at the following system of three equations:$${\begin{cases}|u_{11}|^{2}+|u_{21}|^{2}=1\\|u_{12}|^{2}+|u_{22}|^{2}=1\\u_{11}^{*}u_{12}+u_{21}^{*}u_{22}=0.\end{cases}}$$As there are only three equations, one of the parameters is free. Using the formula derived above for ${\textstyle N_{\text{free}}}$, one finds the expected number of free parameters for a ${\textstyle 2{\text{x}}2}$ unitary is one, which is in agreement with the example.

Photon states

In quantum mechanics, the electromagnetic (EM) field is quantized and this allows one to define a photon. In this sense, a photon is an excitation in a specific mode of the quantized EM field. For example, the mode of a plane wave is characterized by its wavevector ${\textstyle {\vec {k}}}$ and a photon associated with a given plane wave mode must be labelled by such wavevector in order to differentiate it. In addition, two frequently studied mode sets are the set of Hermite-Gaussian (HG) modes and the set of Laguerre-Gaussian (LG) modes. These are two transverse spatial bases representing the eigenfunctions of the paraxial wave equation in Cartesian and cylindrical coordinates, respectively.

To analyze a given set of modes, the electromagnetic field is treated as a set of quantum harmonic oscillators. The states are then built-up using a creation operator, ${\textstyle {\hat {a}}_{j}^{\dagger }}$, and an annihilation operator, ${\textstyle {\hat {a}}_{j}}$, both of which are defined for the specific mode, ${\textstyle j}$, being studied. These operators obey bosonic commutation relations:$$\left[{\hat {a}}_{j},{\hat {a}}_{j}\right]=0=\left[{\hat {a}}_{j}^{\dagger },{\hat {a}}_{j}^{\dagger }\right],\left[{\hat {a}}_{j},{\hat {a}}_{j}^{\dagger }\right]=1$$and they transform the photon number state ${\textstyle |n\rangle }$, which contains ${\textstyle n}$ photons (again, existing in the mode being studied), according to the following relations:$${\begin{array}{lcl}{\hat {a}}_{j}|n\rangle &=&{\sqrt {n}}|n-1\rangle \\{\hat {a}}_{j}^{\dagger }|n\rangle &=&{\sqrt {n+1}}|n+1\rangle .\end{array}}$$The photon number states ${\textstyle |n\rangle }$ are orthonormal, i.e., ${\textstyle \langle n|m\rangle =\delta _{n,m}}$, and form a basis for the Hilbert space of such ${\textstyle n}$ photons. In general, an ${\textstyle n}$ photon state ${\textstyle |\psi _{n}\rangle }$ can be expressed as:$$|\psi _{n}\rangle =\sum _{n}c_{n}|n\rangle$$where ${\textstyle \{c_{n}\}}$​ are complex coefficients satisfying the normalization condition ${\textstyle \sum _{n}|c_{n}|^{2}=1}$. In other words, the general photon state ${\textstyle |\psi _{n}\rangle }$ can be decomposed into a superposition of photon number states with weights given by ${\textstyle c_{n}=\langle \psi |n\rangle }$.

Unitary spatial mode transformations

In the previous section, the creation and annihilation operators were defined for a given basis mode basis ${\textstyle \{{\hat {a}}_{j}\}}$. The set of all modes ${\textstyle {\ce {j=\{0,1,...,J\}}}}$ forms the basis, with ${\textstyle J}$ being the maximum number of modes in the system. Using a unitary transformation, the creation operators of can be expressed in a rotated mode basis ${\textstyle \{{\hat {b}}_{j}\}}$ ​related to the original basis as:$$b_{k}^{\dagger }=\sum _{j=0}^{J}U_{k,j}a_{j}^{\dagger },$$where ${\textstyle U_{k,j}}$ are elements of the unitary matrix ${\textstyle U}$. This transformation allows photon states to be expressed in a new basis, facilitating tasks such as simplifying descriptions or enabling experimental realizations in specific bases.

The vacuum state ${\textstyle |0\rangle }$, which contains no photons, remains invariant under any unitary transformation. States with ${\textstyle n}$-photons in a mode ${\textstyle k}$ are generated by applying the operator ${\textstyle \left(b_{k}^{\dagger }\right)^{n}}$ to the vacuum state:$$|n\rangle _{k}={\frac {1}{\sqrt {n!}}}\left({\hat {b}}_{k}^{\dagger }\right)^{n}|0\rangle .$$Substituting the expression for ${\textstyle \left({\hat {b}}_{k}^{\dagger }\right)^{n}}$​, this can be written as:$$|n\rangle _{k}={\frac {1}{\sqrt {n!}}}\left(\sum _{j=0}^{J}U_{k,j}{\hat {a}}_{j}^{\dagger }\right)^{n}|0\rangle .$$Examples of Spatial Mode Transformations

1. Hermite-Gauss (HG) to Laguerre-Gauss (LG) Modes: The transition between HG and LG modes is a common application of unitary transformations. These transformations are realized using mode converters, such as cylindrical lens setups, that act as unitary operators: ∣HGm,n​⟩U​∣LGp,l​⟩, where p and l are the radial and azimuthal indices of the LG modes.
2. Orbital Angular Momentum (OAM) Modes: Photons in LG modes carry orbital angular momentum, with a unitary transformation enabling changes in their OAM states: Ul,l′​:∣OAMl​⟩→∣OAMl′​⟩.

Motivation

Givens or Jacobi rotation

A Givens rotation, also known as a Jacobi rotation, is a linear algebraic operation that performs a rotation in a two-dimensional subspace of a higher-dimensional space. Mathematically, it is represented by the rotation matrix:$${\begin{bmatrix}\cos \phi &\sin \phi \\-\sin \phi &\cos \phi \end{bmatrix}}$$This operation transforms input amplitudes a and b into new amplitudes c and d as follows:$${\begin{bmatrix}c\\d\end{bmatrix}}={\begin{bmatrix}\cos \phi &\sin \phi \\-\sin \phi &\cos \phi \end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}$$The Givens rotation can zero out a specific element of a vector or systematically triangularize a matrix, making it essential for linear algebra algorithms like matrix factorization and solving systems of equations.

Implementing Givens rotations

The paper presents an integrated-optical device for implementing the Givens rotation. This device uses optical components to achieve high-speed and lossless computations. Key elements include:

Electro-Optic Gratings:

• A thick diffraction grating modulated by a voltage generates the sine and cosine multiplications naturally. The input wave amplitudes a and b are processed through the grating to produce transmitted (cosϕ) and diffracted (sinϕ) components.

Phase Shifters:

• Electro-optic phase shifters adjust the phases of the optical waves to ensure coherent addition and subtraction, corresponding to the operations required for the rotation matrix.

Channel Waveguides:

• The input and output light signals are guided in waveguides. The use of z-cut lithium niobate waveguides ensures low-loss and high-speed operation.

Device Configuration:

• The device operates with two coherent, monochromatic input waves representing amplitudes a and b. The phase shifters adjust the relative phase of these inputs, while the diffraction grating computes the sine and cosine components. The outputs c and d are coherently combined to produce the desired rotation.

External Adjustments:

• External phase shifter voltages Vp,1​ and Vp,2​ allow fine-tuning of the rotation parameters.

The final implementation achieves the desired outputs:$$c=a\cos \phi +b\sin \phi ,\quad d=-a\sin \phi +b\cos \phi .$$

Significance and applications

Givens rotations are pivotal in linear algebra and signal processing for the following reasons:Matrix Triangularization:

• Successive Givens rotations can zero out specific matrix elements, transforming a general matrix into triangular form. This is a foundational step in algorithms for solving linear systems, eigenvalue problems, and singular value decomposition.

Signal Processing:

• The operation is widely used in lattice filters, which are critical for applications such as speech processing, channel equalization, and seismic data analysis.

Optical Implementation Advantages:

• The integrated-optical implementation described in the paper offers significant advantages over digital implementations:
  • Speed: Optical devices operate at the speed of light, enabling high-throughput computations.
  • Parallelism: Arrays of Givens rotation devices can compute multiple operations simultaneously.
  • Efficiency: Natural sine and cosine multiplication via diffraction reduces the need for complex digital calculations.

Applications:

• The proposed device is suitable for tasks requiring real-time computations, such as radar imaging, adaptive antenna beamforming, and high-resolution image processing.