High-dimension experimental tomography of a path-encoded photon quantum state

D. Curic; L. Giner; J. S. Lundeen

doi:10.1364/PRJ.7.000A27

Journals >Photonics Research >Volume 7 >Issue 7 >Page A27 > Article

Photonics Research
Vol. 7, Issue 7, A27 (2019)

High-dimension experimental tomography of a path-encoded photon quantum state

D. Curic^1、2、*, L. Giner¹, and J. S. Lundeen¹

Author Affiliations

¹Department of Physics and Centre for Research in Photonics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada

²Current address: Department of Physics and Astronomy, University of Calgary, Calgary, Alberta T2N 1N4, Canada

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DOI: 10.1364/PRJ.7.000A27 Cite this Article Set citation alerts

D. Curic, L. Giner, J. S. Lundeen. High-dimension experimental tomography of a path-encoded photon quantum state[J]. Photonics Research, 2019, 7(7): A27 Copy Citation Text

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Abstract

Quantum information protocols often rely on tomographic techniques to determine the state of the system. A popular method of encoding information is on the different paths a photon may take, e.g., parallel waveguides in integrated optics. However, reconstruction of states encoded onto a large number of paths is often prohibitively resource intensive and requires complicated experimental setups. Addressing this, we present a simple method for determining the state of a photon in a superposition of

d

paths using a rotating one-dimensional optical Fourier transform. We establish the theory and experimentally demonstrate the technique by measuring a wide variety of six-dimensional density matrices. The average fidelity of these with the expected state is as high as

0.9852 \pm 0.0008

. This performance is comparable to or exceeds established tomographic methods for other types of systems.

P (k_{x}) = {| \tilde{ψ} (k_{x}) |}^{2} [ρ_{11} + ρ_{22} + 2 | ρ_{12} | \cos (L_{12}^{x} k_{x} + ϕ_{12})],

(1)

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F_{k_{x}} {P (k_{x})} (\bar{x}) = (ρ_{00} + ρ_{11}) δ (\bar{x}) + ρ_{12} δ (\bar{x} - L_{12}^{x}) + ρ_{21} δ (\bar{x} + L_{12}^{x}) .

(2)

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ρ_{x} (k_{y}) = ⟨ k_{y} | ρ | k_{y} ⟩ = \sum_{i j}^{d} ρ_{i j} ⟨ k_{y} {| ψ_{i} ⟩}_{y} ⟨ ψ_{j} | k_{y} ⟩ {| ψ_{i} ⟩}_{x} ⟨ ψ_{j} | .

(3)

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ρ_{x} (k_{y}) = {| \tilde{ψ} (k_{y}) |}^{2} \sum_{i j}^{d} ρ_{i j} e^{i L_{i j}^{y} k_{y}} {| ψ_{i} ⟩}_{x} ⟨ ψ_{j} | .

(4)

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ρ_{x} (k_{y}) = \sum_{i}^{d} ρ_{i i} {| ψ_{i} ⟩}_{x} ⟨ ψ_{i} | + \sum_{i \neq j}^{d} ρ_{i j} e^{i L_{i j}^{y} k_{y}} {| ψ_{i} ⟩}_{x} ⟨ ψ_{j} | .

(5)

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P (x_{m}, k_{y}) = ⟨ x_{m} | ρ_{x} (k_{y}) | x_{m} ⟩ = {| ψ (0) |}^{2} (\sum_{i}^{d} ρ_{i i} δ_{x_{m} x_{i}} + \sum_{i \neq j}^{d} ρ_{i j} e^{i L_{i j}^{y} k_{y}} δ_{x_{m} x_{i}} δ_{x_{m} x_{j}}) .

(6)

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F_{k_{y}} {P (x_{m}, k_{y})} (\bar{y}) = \int d k_{y} P (x_{m}, k_{y}) e^{i k_{y} \bar{y}} = \sum_{i}^{d} ρ_{i i} δ_{x_{m} x_{i}} δ (\bar{y}) + \sum_{i \neq j}^{d} ρ_{i j} δ (\bar{y} - L_{i j}^{y}) δ_{x_{m} x_{i}} δ_{x_{m} x_{j}} .

(7)

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D. Curic, L. Giner, J. S. Lundeen. High-dimension experimental tomography of a path-encoded photon quantum state[J]. Photonics Research, 2019, 7(7): A27

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