arXiv:1912.03707

Linear multi-particle, multi-port interferometers and their matrix representations

Bülent Demirel

Published Dec 8, 2019• 0 citations

Abstract

Interferences in multi-path systems for single and multiple particles are theoretically analyzed. A holistic method is presented, which allows to construct the unitary transition matrix describing interferometers for any port number d and particle number n. This work has two centerpieces. First, a solution algorithm for realizing any desired unitary transformation in an interferometer for the single particle case is demonstrated. Secondly, a simple scheme for calculating the rotation matrices of beam splitters and phase shifters for multiple particles is shown. Such linear optical devices are especially important for quantum circuits and quantum information processing.

DOI: 10.48550/arxiv.1912.03707

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arXiv:1912.03707

Linear multi-particle, multi-port interferometers and their matrix representations

Bülent Demirel

Published Dec 8, 2019• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1912.03707

Related papers

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n_{s}n_{p}\times n_{s}n_{p}

unitary matrix on the state of light in

n_{s}

spatial and

n_{p}

internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU

(n)

basis states, which reduce the canonical subgroup chain, for arbitrary

n

. The boson realizations are employed to construct

\mathcal{D}

-functions, which are the matrix elements of arbitrary irreducible representations, of SU

(n)

in the canonical basis. I show that immanants of principal submatrices of a unitary matrix

T

are a sum of the diagonal

\mathcal{D}(Ω)

-functions of group element

Ω

over

t

determined by the choice of submatrix and over the irrep

(λ)

determined by the immanant under consideration. The algorithm for

\mathrm{SU}(n)

\mathcal{D}

-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.

n_{s}n_{p}\\times n_{s}n_{p}

unitary matrix on the state of light in

n_{s}

spatial and

n_{p}

(n)

basis states, which reduce the canonical subgroup chain, for arbitrary

n

. The boson realizations are employed to construct

\\mathcal{D}

-functions, which are the matrix elements of arbitrary irreducible representations, of SU

(n)

in the canonical basis. I show that immanants of principal submatrices of a unitary matrix

T

are a sum of the diagonal

\\mathcal{D}(\\Omega)

-functions of group element

\\Omega

over

t

determined by the choice of submatrix and over the irrep

(\\lambda)

determined by the immanant under consideration. The algorithm for

\\mathrm{SU}(n)

\\mathcal{D}

-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.

Multiport interferometers with

N

ports are basic devices in both classical and quantum photonics. Ideally, they implement a linear unitary transformation between the input and output electric field vectors with

N

components, each associated with a spatial mode of classical coherent light or a single photon. Standard designs for a fully reconfigurable universal multiport interferometer are given by the Reck or the Clements schemes. In this work, we introduce routing schemes to implement a generic unitary transformation on classical coherent light or single photons using linear or tree geometries via multiple projective measurements on a single detector with the minimum number of components. Then, we generalize this result to the case of any multi-photon state for scattershot boson sampling experiments with multi-routing schemes. Finally, we test the robustness of routing schemes compared to universal schemes with respect to losses and phase noise.