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Photonics Research
Vol. 10, Issue 7, 1763 (2022)

Nonlinear quantum spectroscopy with parity–time-symmetric integrated circuits

Pawan Kumar^1、2、*, Sina Saravi¹, Thomas Pertsch^1、2, Frank Setzpfandt¹, and Andrey A. Sukhorukov^3、4

Author Affiliations

¹Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena, 07745 Jena, Germany

²Fraunhofer Institute for Applied Optics and Precision Engineering, 07745 Jena, Germany

³Research School of Physics, Australian National University, Canberra, ACT 2601, Australia

⁴ARC Centre of Excellence for Transformative Meta-Optical Systems (TMOS), Australia

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DOI: 10.1364/PRJ.450410 Cite this Article Set citation alerts

Pawan Kumar, Sina Saravi, Thomas Pertsch, Frank Setzpfandt, Andrey A. Sukhorukov. Nonlinear quantum spectroscopy with parity–time-symmetric integrated circuits[J]. Photonics Research, 2022, 10(7): 1763 Copy Citation Text

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Abstract

We propose a novel quantum nonlinear interferometer design that incorporates a passive parity–time (PT)-symmetric coupler sandwiched between two nonlinear sections where signal–idler photon pairs are generated. The PT symmetry enables efficient coupling of the longer-wavelength idler photons and facilitates the sensing of losses in the second waveguide exposed to analyte under investigation, whose absorption can be inferred by measuring only the signal intensity at a shorter wavelength where efficient detectors are readily available. Remarkably, we identify a new phenomenon of sharp signal intensity fringe shift at critical idler loss values, which is distinct from the previously studied PT symmetry breaking. We discuss how such unconventional properties arising from quantum interference can provide a route to enhancing the sensing of analytes and facilitate broadband spectroscopy applications in integrated photonic platforms.

\frac{\partial A_{i} (z)}{\partial z} = M_{i} A_{i} (z),

(1)

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M_{i} = [\begin{matrix} - i ({\bar{β}}_{i} + Δ β_{i}) & - i C_{i} \\ - i C_{i} & - i ({\bar{β}}_{i} - Δ β_{i}) - γ_{i} \end{matrix}] .

(2)

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{\tilde{λ}}_{\pm} = \frac{\pm ξ_{i} - γ_{i}}{2}, V_{+} = {[\begin{matrix} γ_{i} + ξ_{i} - 2 i Δ β_{i} & - 2 i C_{i} \end{matrix}]}^{T}, V_{-} = {[\begin{matrix} 2 i C_{i} & γ_{i} + ξ_{i} - 2 i Δ β_{i} \end{matrix}]}^{T},

(3)

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A_{i} (z) = α_{+} e^{λ_{+} z} V_{+} + α_{-} e^{λ_{-} z} V_{-} .

(4)

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\frac{\partial ϕ_{1, n_{i}} (z)}{\partial z} = - i (β_{s} + β_{i}^{(n_{i})}) ϕ_{1, n_{i}} (z) + δ_{1, n_{i}} A_{p} (z),

(5)

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ϕ (l) = {[\begin{matrix} 1 & 0 \end{matrix}]}^{T} ϕ_{0} (l)

(6)

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ϕ_{0} (l) = [A_{p} (0) l / 2] \cdot e^{- i (β_{s} + β_{i}^{(1)} + \frac{Δ β_{NL}}{2}) l} sinc (\frac{Δ β_{NL} l}{2}),

(7)

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\frac{\partial ϕ (z)}{\partial z} = (- i β_{s} 1 + M_{i}) ϕ (z),

(8)

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ϕ (l + L) = e^{- i (β_{s} + {\bar{β}}_{i}) L} e^{- γ_{i} L / 2} \times [\begin{matrix} \cosh^{2} Θ e^{ξ_{i} L / 2} - \sinh^{2} Θ e^{- ξ_{i} L / 2} \\ i \sinh Θ \cosh Θ (e^{- ξ_{i} L / 2} - e^{ξ_{i} L / 2}) \end{matrix}] ϕ_{0} (l),

(9)

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ϕ (2 l + L) = [\begin{matrix} e^{- i β_{i}^{(1)} l} & 0 \\ 0 & e^{- i β_{i}^{(2)} l} \end{matrix}] e^{- i β_{s} l} ϕ (l + L) + [\begin{matrix} 1 \\ 0 \end{matrix}] ϕ_{0} e^{- i β_{p} (l + L)},

(10)

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I_{1}^{s} = I_{11} + I_{12} + I_{12}^{(s)} .

(11)

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\frac{\partial ϕ_{12}^{(s)} (z, z^{'})}{\partial z} = {\begin{cases} - i β_{s} ϕ_{12}^{(s)} (z, z^{'}) & for z^{'} \leq z \\ 0 & otherwise \end{cases},

(12)

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I_{12}^{(s)} (z) = \int_{l}^{z} {| ϕ_{12}^{(s)} (z, z^{'}) |}^{2} d z^{'} = 2 γ_{i} \int_{l}^{z} {| ϕ_{12} (z^{'}) |}^{2} d z^{'}, for z > l,

(13)

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I_{11} (2 l + L) = {1 + {| V_{γ_{i}} |}^{2} + 2 Re [V_{γ_{i}} \times e^{i (Δ β_{NL} + K) (L + l)}]} {| ϕ_{0} |}^{2},

(14)

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V_{γ_{i}} = e^{i Δ β_{i} L - γ_{i} L / 2} [\cosh \frac{ξ_{i} L}{2} + (1 + 2 \sinh^{2} Θ) \sinh \frac{ξ_{i} L}{2}] .

(15)

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V_{γ_{i}} = e^{- \frac{γ_{i} L}{2}} (\cos \frac{σ_{i} L}{2} + \frac{γ_{i}}{σ_{i}} \sin \frac{σ_{i} L}{2}) with σ_{i} = \sqrt{{(2 C_{i})}^{2} - γ_{i}^{2}}

(16)

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V_{γ_{i}} = e^{- \frac{γ_{i} L}{2}} (\cosh \frac{η_{i} L}{2} + \frac{γ_{i}}{η_{i}} \sinh \frac{η_{i} L}{2}) with η_{i} = \sqrt{γ_{i}^{2} - {(2 C_{i})}^{2}} .

(17)

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V_{γ_{cr}} = 0 .

(18)

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Δ β_{NL} = Δ (\frac{1}{v_{g}}) \times \frac{Δ ω_{s}}{ω_{p}},

(19)

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\frac{\partial ϕ_{11} (z)}{\partial z} = - i (β_{s} + β_{i}^{(1)}) ϕ_{11} (z) + A_{p} (z), with ϕ_{11} (0) = 0 .

(A1)

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A_{p} (z) = A_{p} (0) \times \cos K z \times e^{- i β_{p} z} = \frac{A_{p} (0)}{2} (e^{i K z} + e^{- i K z}) e^{- i β_{p} z} = \frac{A_{p} (0)}{2} [e^{- i (β_{p} - K) z} + e^{- i (β_{p} + K) z}] .

(A2)

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\frac{\partial {\tilde{ϕ}}_{1,1} (z)}{\partial z} = \frac{A_{p} (0)}{2} e^{- i Δ β_{NL} z} .

(A3)

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{\tilde{ϕ}}_{1,1} (z) - {\tilde{ϕ}}_{1,1} (0) = \frac{A_{p} (0)}{2} \frac{e^{- i Δ β_{NL} z} - 1}{- i Δ β_{NL}} = \frac{A_{p} (0) \times z}{2} e^{\frac{- i Δ β_{NL} z}{2}} sinc (\frac{Δ β_{NL} z}{2}),

(A4)

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ϕ_{1,1} (z) = \frac{A_{p} (0) \times z}{2} e^{- i (β_{s} + β_{i}^{(1)} + \frac{Δ β_{NL}}{2}) z} sinc (\frac{Δ β_{NL} z}{2}) .

(A5)

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ϕ (l) = [\begin{matrix} ϕ_{11} (l) \\ ϕ_{12} (l) \end{matrix}] = [\begin{matrix} 1 \\ 0 \end{matrix}] ϕ_{0} (l),

(A6)

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\frac{\partial ϕ (z)}{\partial z} = (- i β_{s} 1 + M_{i}) ϕ (z),

(A7)

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ϕ (z) = α_{+} e^{{\bar{λ}}_{+} (z - l)} V_{+} + α_{-} e^{{\bar{λ}}_{-} (z - l)} V_{-} .

(A8)

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α_{+} = \frac{(γ_{i} + ξ_{i} - 2 i Δ β_{i}) ϕ_{0} (l)}{{(γ_{i} + ξ_{i} - 2 i Δ β_{i})}^{2} + {(2 i C_{i})}^{2}},

(A9)

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α_{-} = \frac{(2 i C_{i}) ϕ_{0} (l)}{{(γ_{i} + ξ_{i} - 2 i Δ β_{i})}^{2} + {(2 i C_{i})}^{2}} .

(A10)

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ϕ (z) = e^{- i (β_{s} + {\bar{β}}_{i}) (z - l)} e^{- γ_{i} (z - l) / 2} \times [\begin{matrix} \cosh^{2} Θ e^{ξ_{i} (z - l) / 2} - \sinh^{2} Θ e^{- ξ_{i} (z - l) / 2} \\ i \sinh Θ \cosh Θ [e^{- ξ_{i} (z - l) / 2} - e^{ξ_{i} (z - l) / 2}] \end{matrix}] ϕ_{0} (l),

(A11)

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ϕ_{12}^{(s)} (z, z^{'}) = [- i \sqrt{2 γ_{i}} \times ϕ_{12} (z^{'})] e^{- i β_{s} (z - l)}, l \leq z^{'} \leq l + L,

(A12)

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I_{11} (2 l + L) = {1 + {| V_{γ_{i}} |}^{2} + 2 Re [V_{γ_{i}} \times e^{i (Δ β_{NL} + K) (L + l)}]} {| ϕ_{0} |}^{2},

(B1)

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V_{γ_{i}} = e^{i Δ β_{i} L - γ_{i} L / 2} [\cosh \frac{ξ_{i} L}{2} + (1 + 2 \sinh^{2} Θ) \sinh \frac{ξ_{i} L}{2}],

(B2)

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I_{12} (2 l + L) = e^{- γ_{i} L} {| \sinh 2 Θ \times \sinh \frac{ξ_{i} L}{2} |}^{2} {| ϕ_{0} (l) |}^{2} .

(B3)

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\tan \frac{σ_{i} L}{2} = - \frac{σ_{i}}{γ_{i}} .

(B4)

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Pawan Kumar, Sina Saravi, Thomas Pertsch, Frank Setzpfandt, Andrey A. Sukhorukov. Nonlinear quantum spectroscopy with parity–time-symmetric integrated circuits[J]. Photonics Research, 2022, 10(7): 1763

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