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SPDC-based Entangled Photon Sources Buzzwords: Entanglement, Twin-photons, Quantum State Engineering

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Page 1: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

SPDC-based EntangledPhoton Sources

Buzzwords: Entanglement, Twin-photons, Quantum State Engineering

Page 2: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

This (and next week) on QCommβ€’ Two-photon SPDC state derivation from PDC Hamiltonian (recap)β€’ Basic nonlinear optics, properties of SPDC emission

β€’ Typical nonlinear materialsβ€’ Phase-matchingβ€’ Spectral properties

β€’ Polarization-entanglement generationβ€’ Source realizationsβ€’ Polarization-entanglement by HOM interferenceβ€’ Detour: Space-proof entangled photon source

β€’ Transverse-mode correlationsβ€’ Spatial Mode Entanglementβ€’ Efficient fiber-coupling

β€’ Frequency-mode correlationsβ€’ Frequency Entanglementβ€’ Multi-photon entanglement generation

Page 3: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Spontaneous parametric down conversion

οΏ½H𝐼𝐼 ∝ οΏ½πœ†πœ†πœ‡πœ‡πœ‡πœ‡

∫ 𝑑𝑑𝒓𝒓 πœ’πœ’πœ†πœ†πœ‡πœ‡πœ‡πœ‡πŸπŸ 𝒓𝒓 ΓŠπœ†πœ†,𝑝𝑝

+ 𝒓𝒓, 𝑑𝑑 ΓŠπœ‡πœ‡,π‘ π‘ βˆ’ 𝒓𝒓, 𝑑𝑑 ΓŠπœ‡πœ‡,𝑖𝑖

βˆ’ 𝒓𝒓, 𝑑𝑑 + 𝑐𝑐. 𝑐𝑐

π’Œπ’Œ = π‘˜π‘˜π‘§π‘§ πœ”πœ”,𝒒𝒒 𝒆𝒆𝑧𝑧 + 𝒒𝒒

PDC interaction Hamiltonian

Transverse field operators

𝒒𝒒 = (kx, ky)

𝑓𝑓 πœ”πœ” =β„πœ”πœ”

2πœ–πœ–0𝑐𝑐 𝑛𝑛

Transverse momentum

Ξ¨0 = 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” ⨂ 𝑣𝑣𝑣𝑣𝑐𝑐 ⨂ 𝑣𝑣𝑣𝑣𝑐𝑐

Initial State:

Page 4: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Spontaneous parametric down conversion

οΏ½H𝐼𝐼 ∝ ∫ 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 ΓŠπ‘π‘+ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘ π‘ βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘–π‘–βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 + 𝑐𝑐. 𝑐𝑐

π’Œπ’Œ = π‘˜π‘˜π‘§π‘§ πœ”πœ”,𝒒𝒒 𝒆𝒆𝑧𝑧 + 𝒒𝒒 ,

PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient)

Transverse field operators (narrowband)

𝒒𝒒 = (kx, ky) ,

𝑓𝑓 πœ”πœ”0 =β„πœ”πœ”0

2πœ–πœ–0𝑐𝑐 𝑛𝑛

Transverse momentum

Ξ¨0 = 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” ⨂ 𝑣𝑣𝑣𝑣𝑐𝑐 ⨂ 𝑣𝑣𝑣𝑣𝑐𝑐

Initial State:

πœ”πœ” = πœ”πœ”0 + Ξ©

Page 5: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Ψ𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = exp βˆ’π‘–π‘–β„οΏ½π‘‡π‘‡οΏ½H𝐼𝐼 𝑑𝑑𝑑𝑑 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖

SPDC state in interaction picture

= exp βˆ’π‘–π‘–β„οΏ½π‘‡π‘‡,𝑉𝑉

𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 ΓŠπ‘π‘+ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘ π‘ βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘–π‘–βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 + 𝑐𝑐. 𝑐𝑐 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖

= 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖 βˆ’π‘–π‘–β„οΏ½π‘‡π‘‡,𝑉𝑉

𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 ΓŠπ‘π‘+ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘ π‘ βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘–π‘–βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 + 𝑐𝑐. 𝑐𝑐. 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖

= 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖 βˆ’π‘–π‘–β„οΏ½π‘‡π‘‡,𝑉𝑉

𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓 𝐸𝐸𝑝𝑝 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘ π‘ βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘–π‘–βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖 + β‹―

Two-photon SPDC state

= 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖 βˆ’π‘–π‘–β„οΏ½π‘‡π‘‡,𝑉𝑉

𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 ΓŠπ‘π‘+ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘ π‘ βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘–π‘–βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 𝐸𝐸𝑝𝑝 π‘žπ‘ž , 𝑠𝑠 πœ”πœ” 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖

Page 6: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Ψ𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(2βˆ’photon) ∝ οΏ½

𝑇𝑇,𝑉𝑉𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 𝐸𝐸𝑝𝑝 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘ π‘ βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 ΓŠπ‘–π‘–βˆ’ 𝒓𝒓βŠ₯, 𝑧𝑧, 𝑑𝑑 𝑣𝑣𝑣𝑣𝑐𝑐 𝑠𝑠,𝑖𝑖

Two-photon SPDC state

= �𝑇𝑇,𝑉𝑉

𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓 πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯,𝑧𝑧 ∫ π‘‘π‘‘πœ”πœ”π‘π‘ 𝑑𝑑𝒒𝒒𝑝𝑝 𝐸𝐸𝑝𝑝(𝒒𝒒𝑝𝑝)s πœ”πœ”π‘π‘ e(𝑖𝑖𝑖𝑖𝑝𝑝𝑧𝑧𝑧𝑧 +𝑖𝑖 𝒒𝒒𝒑𝒑⋅𝒓𝒓βŠ₯βˆ’π‘–π‘–πœ‡πœ‡π‘π‘π‘‘π‘‘) Γ—

∫ π‘‘π‘‘πœ”πœ”π‘ π‘  𝑑𝑑𝒒𝒒𝑠𝑠eβˆ’(𝑖𝑖𝑖𝑖𝑠𝑠𝑧𝑧𝑧𝑧 +𝑖𝑖 𝒒𝒒𝒔𝒔⋅𝒓𝒓βŠ₯βˆ’π‘–π‘–πœ‡πœ‡π‘ π‘ π‘‘π‘‘)∫ π‘‘π‘‘πœ”πœ”π‘–π‘– 𝑑𝑑𝒒𝒒𝑖𝑖eβˆ’(𝑖𝑖𝑖𝑖𝑠𝑠𝑧𝑧𝑧𝑧 +𝑖𝑖 𝒒𝒒𝒔𝒔⋅𝒓𝒓βŠ₯βˆ’π‘–π‘–πœ‡πœ‡π‘ π‘ π‘‘π‘‘) πœ”πœ”π‘ π‘ ,π’’π’’π’”π’”βŸ©|πœ”πœ”π‘–π‘– ,π’’π’’π’Šπ’Š

…

Ψ𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(2βˆ’photon) = ∫ π‘‘π‘‘πœ”πœ”π‘ π‘  𝑑𝑑𝒒𝒒𝑠𝑠 π‘‘π‘‘πœ”πœ”π‘–π‘– 𝑑𝑑𝒒𝒒𝑖𝑖Φ(πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖) πœ”πœ”π‘ π‘ ,π’’π’’π’”π’”βŸ©|πœ”πœ”π‘–π‘– ,π’’π’’π’Šπ’Š

SPDC biphoton mode function

Page 7: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Two-photon mode function SPDCΞ¦(πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖) = βˆ«π‘‡π‘‡,𝑉𝑉 𝑑𝑑𝑑𝑑 𝑑𝑑𝒓𝒓βŠ₯𝑑𝑑𝑧𝑧 πœ’πœ’

𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 ∫ π‘‘π‘‘πœ”πœ”π‘π‘ 𝑑𝑑𝒒𝒒𝑝𝑝 𝐸𝐸𝑝𝑝 𝒒𝒒𝑝𝑝 s πœ”πœ”π‘π‘ eβˆ’π‘–π‘– πœ‡πœ‡π‘π‘βˆ’πœ‡πœ‡π‘ π‘ βˆ’πœ‡πœ‡π‘–π‘– 𝑑𝑑e𝑖𝑖 π‘–π‘–π‘π‘π‘§π‘§βˆ’π‘–π‘–π‘§π‘§π‘ π‘ βˆ’π‘–π‘–π‘–π‘–π‘§π‘§ 𝑧𝑧 e𝑖𝑖 π’’π’’π’‘π’‘βˆ’π’’π’’π’”π’”βˆ’π’’π’’π’Šπ’Š ⋅𝒓𝒓βŠ₯

Crystal transversally homogeneous with large aperture:

∫ 𝑑𝑑𝒓𝒓βŠ₯e𝑖𝑖 π’’π’’π’‘π’‘βˆ’π’’π’’π’”π’”βˆ’π’’π’’π’Šπ’Š ⋅𝒓𝒓βŠ₯ β†’ 𝛿𝛿(𝒒𝒒𝑝𝑝 βˆ’ 𝒒𝒒𝑠𝑠 βˆ’ 𝒒𝒒𝑖𝑖)πœ’πœ’ 𝟐𝟐 𝒓𝒓βŠ₯, 𝑧𝑧 β†’ πœ’πœ’ 𝟐𝟐 𝑧𝑧

Transverse momentum conservation

Long interaction time:

∫ 𝑑𝑑𝑑𝑑 eβˆ’π‘–π‘– πœ‡πœ‡π‘π‘βˆ’πœ‡πœ‡π‘ π‘ βˆ’πœ‡πœ‡π‘–π‘– 𝑑𝑑 β†’ 𝛿𝛿(πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  βˆ’ πœ”πœ”π‘–π‘–)

𝒒𝒒𝑝𝑝 = 𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖

Energy conservation

πœ”πœ”π‘π‘ = πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘–

Crystal length: L

οΏ½βˆ’πΏπΏ2

𝐿𝐿2𝑑𝑑𝑧𝑧 e𝑖𝑖 π‘–π‘–π‘π‘π‘§π‘§βˆ’π‘–π‘–π‘§π‘§π‘ π‘ βˆ’π‘–π‘–π‘–π‘–

𝑧𝑧 𝑧𝑧 β†’ 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§

Longitudinal momentum conservation

π‘˜π‘˜π‘π‘π‘§π‘§ = π‘˜π‘˜π‘§π‘§π‘ π‘  + π‘˜π‘˜π‘–π‘–π‘§π‘§

Page 8: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Two-photon mode function SPDCΨ𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆

(2βˆ’photon) = ∫ π‘‘π‘‘πœ”πœ”π‘ π‘  𝑑𝑑𝒒𝒒𝑠𝑠 π‘‘π‘‘πœ”πœ”π‘–π‘– 𝑑𝑑𝒒𝒒𝑖𝑖Φ(πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖) πœ”πœ”π‘ π‘ ,π’’π’’π’”π’”βŸ©|πœ”πœ”π‘–π‘– ,π’’π’’π’Šπ’Š

Ξ¦ πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖 = 𝜎𝜎𝐿𝐿 𝐸𝐸𝑝𝑝(𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖) s(πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘–) 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§

Ξ”π‘˜π‘˜π‘§π‘§(πœ”πœ”π‘ π‘ ,πœ”πœ”π‘–π‘–, 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖) = π‘˜π‘˜π‘π‘π‘§π‘§ πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– ,𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖 βˆ’ π‘˜π‘˜π‘ π‘ π‘§π‘§ πœ”πœ”π‘ π‘ ,𝒒𝒒𝑠𝑠 βˆ’ π‘˜π‘˜π‘–π‘–π‘§π‘§ πœ”πœ”π’Šπ’Š,𝒒𝒒𝑖𝑖Phase matching function:

𝐸𝐸𝑝𝑝(𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖) s(πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘–)Pump contribution:

CW Pump: s(πœ”πœ”π‘π‘) = 𝛿𝛿 πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘π‘,0 β†’ πœ”πœ”π‘–π‘– = πœ”πœ”π‘π‘,0 βˆ’ πœ”πœ”π‘ π‘  𝒒𝒒𝑠𝑠 = 𝟎𝟎,𝒒𝒒𝑖𝑖 = 𝟎𝟎

Ψ𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(2βˆ’photon) = ∫ π‘‘π‘‘πœ”πœ”π‘ π‘  𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐

𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§(πœ”πœ”π‘ π‘ ) πœ”πœ”π‘ π‘ βŸ©|πœ”πœ”π‘π‘,0 βˆ’ πœ”πœ”π‘ π‘ 

Collinear SPDC

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Phase-matchingΞ”π‘˜π‘˜π‘§π‘§ πœ”πœ”π‘ π‘  = 2πœ‹πœ‹

π‘π‘π‘›π‘›πœ‹πœ‹π‘π‘ πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– β‹… πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– βˆ’ π‘›π‘›πœ‹πœ‹π‘ π‘  πœ”πœ”π‘ π‘  πœ”πœ”π‘ π‘  βˆ’ π‘›π‘›πœ‹πœ‹π‘–π‘– πœ”πœ”π’Šπ’Š πœ”πœ”π’Šπ’Š = 𝟎𝟎Collinear phase-matching condition :

πœ”πœ”π‘–π‘– = πœ”πœ”π‘π‘,0 βˆ’ πœ”πœ”π‘ π‘ 

Ξ”π‘˜π‘˜π‘§π‘§ πœ”πœ”π‘ π‘  β‰  0For equal polarizations: π‘˜π‘˜π‘§π‘§ (πœ”πœ”π‘–π‘–) π‘˜π‘˜π‘§π‘§ (πœ”πœ”π‘ π‘ )

π‘˜π‘˜π‘§π‘§ (πœ”πœ”π’‘π’‘)

Ξ”π‘˜π‘˜π‘§π‘§

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Phase-matching

Bi-refringent phase-matching

πœ‹πœ‹π‘π‘ =πœ‹πœ‹π‘ π‘  β‰  πœ‹πœ‹π‘–π‘–

πœ‹πœ‹π‘π‘ β‰  πœ‹πœ‹π‘ π‘  = πœ‹πœ‹π‘–π‘–

Ξ”π‘˜π‘˜π‘§π‘§ πœ”πœ”π‘ π‘  = 2πœ‹πœ‹π‘π‘

π‘›π‘›πœ‹πœ‹π‘π‘ πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– β‹… πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– βˆ’ π‘›π‘›πœ‹πœ‹π‘ π‘  πœ”πœ”π‘ π‘  πœ”πœ”π‘ π‘  βˆ’ π‘›π‘›πœ‹πœ‹π‘–π‘– πœ”πœ”π’Šπ’Š πœ”πœ”π’Šπ’Š = 𝟎𝟎Collinear phase-matching condition :

πœ”πœ”π‘–π‘– = πœ”πœ”π‘π‘,0 βˆ’ πœ”πœ”π‘ π‘ 

type-II

Type-I

π‘˜π‘˜πœ‹πœ‹π‘–π‘–π‘§π‘§ (πœ”πœ”π‘–π‘–) π‘˜π‘˜πœ‹πœ‹π‘ π‘ 

𝑧𝑧 (πœ”πœ”π‘ π‘ )

π‘˜π‘˜πœ‹πœ‹π‘π‘π‘§π‘§ (πœ”πœ”π’‘π’‘)

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Phase-matching

Quasi phase-matching

Ξ”π‘˜π‘˜π‘§π‘§ πœ”πœ”π‘ π‘  = 2πœ‹πœ‹π‘π‘

π‘›π‘›πœ‹πœ‹π‘π‘ πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– β‹… πœ”πœ”π‘ π‘  + πœ”πœ”π‘–π‘– βˆ’ π‘›π‘›πœ‹πœ‹π‘ π‘  πœ”πœ”π‘ π‘  πœ”πœ”π‘ π‘  βˆ’ π‘›π‘›πœ‹πœ‹π‘–π‘– πœ”πœ”π’Šπ’Š πœ”πœ”π’Šπ’Š = 𝟎𝟎Collinear phase-matching condition :

πœ”πœ”π‘–π‘– = πœ”πœ”π‘π‘,0 βˆ’ πœ”πœ”π‘ π‘ 

πœ’πœ’ 𝟐𝟐 𝑧𝑧 β†’ πœ’πœ’ 𝟐𝟐 𝑒𝑒2πœ‹πœ‹ 𝑖𝑖 𝑧𝑧

Λ𝑝𝑝𝑝𝑝

πœ‹πœ‹π‘π‘ =πœ‹πœ‹π‘ π‘  β‰  πœ‹πœ‹π‘–π‘–

πœ‹πœ‹π‘π‘ β‰  πœ‹πœ‹π‘ π‘  = πœ‹πœ‹π‘–π‘–

type-II

Type-I

-> Collinear phase-matching condition modified:

π‘˜π‘˜πœ‹πœ‹π‘–π‘–π‘§π‘§ (πœ”πœ”π‘–π‘–) π‘˜π‘˜πœ‹πœ‹π‘ π‘ 

𝑧𝑧 (πœ”πœ”π‘ π‘ )

π‘˜π‘˜πœ‹πœ‹π‘π‘π‘§π‘§ (πœ”πœ”π’‘π’‘)

Ξ”π‘˜π‘˜π‘§π‘§ β†’ π‘˜π‘˜π‘π‘ βˆ’ π‘˜π‘˜π‘ π‘  βˆ’ π‘˜π‘˜π‘–π‘– βˆ’2πœ‹πœ‹Ξ›π‘π‘π‘π‘

2πœ‹πœ‹Ξ›π‘π‘π‘π‘

πœ‹πœ‹π‘π‘ = πœ‹πœ‹π‘ π‘  = πœ‹πœ‹π‘–π‘– Type-0

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type-0 (z-z-z): Ξ›β‰ˆ3.4Β΅m, deffβ‰ˆ11pm/V

type-I (z-y-y): Ξ›β‰ˆ2Β΅m, deffβ‰ˆ3pm/V

type-II: (y-z-y): Ξ›β‰ˆ10Β΅m, deffβ‰ˆ2pm/V

Phase-matching in ppKTP(KTiOPO4)

center wavelength spectral bandwidth

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Other typical nonlinear materials

Material Periodic poling UV absorbtion edge NL coefficient

LN Y 330 nm 18 pm/V

LT Y (3rd order) 265 nm 3 pm/V

KTP Y 350 nm 10 pm /V

LBGO Y (3rd order) 240 nm 1 pm / V

BBO N 200 nm 3 pm / V

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Generating polarization-entangled photons via SPDC: collinear type-0 sandwich scheme

Ξ¦HH πœ”πœ”π‘ π‘ 

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Kwiat et al. Phys. Rev. A 60, R773–R776 (1999) Trojek et al. Appl. Phys. Lett. 92, 21 (2008)

Generating polarization-entangled photons via SPDC: sandwich scheme

Ξ¦VV πœ”πœ”π‘ π‘  = Ξ¦HH πœ”πœ”π‘ π‘  𝑒𝑒𝑖𝑖 𝑖𝑖𝑠𝑠+𝑖𝑖𝑖𝑖 𝐿𝐿

Ξ¦HH πœ”πœ”π‘ π‘  |𝐻𝐻𝑠𝑠,π»π»π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  ⟩

|𝑉𝑉𝑠𝑠,π‘‰π‘‰π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  ⟩

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Kwiat et al. Phys. Rev. A 60, R773–R776 (1999) Trojek et al. Appl. Phys. Lett. 92, 21 (2008)

Generating polarization-entangled photons via SPDC: sandwich scheme

Ξ¦VV πœ”πœ”π‘ π‘  = Ξ¦HH πœ”πœ”π‘ π‘  𝑒𝑒𝑖𝑖 𝑖𝑖𝑠𝑠+𝑖𝑖𝑖𝑖 𝐿𝐿

Ξ¦HH πœ”πœ”π‘ π‘ 

Ξ¨ = ∫ π‘‘π‘‘πœ”πœ”π‘ π‘  Ξ¦VV πœ”πœ”π‘ π‘  [ 𝑉𝑉𝑠𝑠,π‘‰π‘‰π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  + 𝐻𝐻𝑠𝑠,π»π»π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  𝑒𝑒𝑖𝑖 𝑖𝑖𝑠𝑠+𝑖𝑖𝑖𝑖 𝐿𝐿]

|𝐻𝐻𝑠𝑠,π»π»π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  ⟩

|𝑉𝑉𝑠𝑠,π‘‰π‘‰π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  ⟩

𝐹𝐹 = |βŸ¨Ξ¨βˆ’ Ξ¨ οΏ½2

= 1 + 𝑅𝑅𝑒𝑒[∫ Ξ¦VV πœ”πœ”π‘ π‘  Ξ¦HHβˆ— πœ”πœ”π‘ π‘  ]dπœ”πœ”

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Generating polarization-entangled photons via SPDC: sandwich scheme

Ξ¨ = ∫ π‘‘π‘‘πœ”πœ”π‘ π‘  Ξ¦β€²β€²VV=HH πœ”πœ”π‘ π‘  [ 𝑉𝑉𝑠𝑠,π‘‰π‘‰π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  + 𝐻𝐻𝑠𝑠 ,π»π»π‘–π‘–βŸ©|πœ”πœ”π‘ π‘ ,πœ”πœ”π‘π‘ βˆ’ πœ”πœ”π‘ π‘  ]

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Compensation crystals

uncompensated compensated

⟩+⟩∝⟩Φisi

VVeHH is λλ

ϕλλϕ ||)(|

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polarization analysis

20-mm long PPKTP crystals

783nm

837nm

Alice

Bob

Experimental setup

Steinlechner et al. Opt. Express, 20, 9640–9649 (2012)

Fidelity 97%

Spectral brightness

>1Mcps/mW/nm

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Crystal imperfections makespectral filtering necessary

to erase which-crystal information…

…visible to the naked eye

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β€žspecial waveplateβ€œ

Linear bi-directional double-pass

Page 22: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

β€žspecial waveplateβ€œ

Folded sandwich scheme

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polarization analysis

Experimental setup

Steinlechner et al. Opt. Express 21, 11943-11951 (2013)

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Optical bandwidth 2.9 nm

Visibility 99.5 % H/V99.0 % D/A

Fidelity 99.3%

Spectral brightness

>1Mcps/mW/nm

Normalizeddetection

rate>3 Mcps/mW

Heralding efficiency 20 %

High-visibility polarization-correlations

Ο†A

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Kwiat et al, PRL 75, 4337 (1995)

Oldie but goodie…single type II emitter scheme

Kwiat et al, PRL 75, 4337 (1995)

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State-of-the-art in polarization entanglement

Page 27: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

DelinghaLijiang

β€žMICIUSβ€œ: entanglement distributionover 1200km

Page 28: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

General requirements

Total mass ≀1.5 kg

Total size ≀150 x 150 x 80 mm3

Total power consumption ≀8 W (peak)

Optical interface

Wavelength 810 +/- 5 nm

Optical bandwidth ≀0.5 nm

Output Single-mode fiber

Performance

Back-to-back pair rate >106 detected pairs

Spectral brightness >105 pairs/nm/mW

Heralding Efficiency > 50 %

Bell-state fidelity > 99 %

Environmental testing (storage)

Thermal cycling, 10 cycles -50Β°C / +75Β°C

Thermal vacuum, 4 cycles -40Β°C / +60Β°C, 10-4mbar

Humidity 95%, 24hrs

Engineering a space-proof entangled photon source (Space-EPS)

https://www.iof.fraunhofer.de/en/competences/quantum-technologies/Quantum-technology-developments.html

https://www.iof.fraunhofer.de/en/competences/quantum-technologies/Quantum-technology-developments.html
Page 29: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

VIDEO of Space EPShttps://www.iof.fraunhofer.de/de/kompetenzen/Quantentechnologie.html

https://www.iof.fraunhofer.de/de/kompetenzen/Quantentechnologie.html
Page 30: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

ESA Space-EPS engineering model

1 Million pairs / s @20mW

funded

Dimensions:β€’ < 2 kg, < 15x15x10 cmβ€’ < 10 W power cons (peak)

Performance:β€’ Detected pair rate > 50 Kcps / mW

β€’ Spectral Bandwidth < 0.3nmβ€’ Heralding efficiency > 20%β€’ Bell-state Fidelity > 99%

Page 31: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

This week on QCommβ€’ Two-photon SPDC state derivation from PDC Hamiltonian (recap)β€’ Basic nonlinear optics, properties of SPDC emission

β€’ Typical nonlinear materialsβ€’ Phase-matchingβ€’ Spectral properties

β€’ Polarization-entanglement generationβ€’ Source realizationsβ€’ Detour: Space-proof entangled photon source

β€’ Transverse-mode correlationsβ€’ Spatial Mode Entanglementβ€’ Efficient fiber-coupling

β€’ Frequency-mode correlationsβ€’ Frequency Entanglementβ€’ Multi-photon entanglement generationβ€’ Franson Interference

Page 32: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

VIDEO of Space EPShttps://www.iof.fraunhofer.de/de/kompetenzen/Quantentechnologie.html

https://www.iof.fraunhofer.de/de/kompetenzen/Quantentechnologie.html
Page 33: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Path Spatial mode Time Frequency

Qubit entanglement Qudit Entanglement

⟩+⟩⟩+⟩+ dd|...22|11|00|⟩⟩+ 11|00|

Beyond polarization qubits

We can get high-dimensional qudit entanglement from SPDC…

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SPDC biphoton mode function

Ψ𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(2βˆ’photon) = ∫ π‘‘π‘‘πœ”πœ”π‘ π‘  𝑑𝑑𝒒𝒒𝑠𝑠 π‘‘π‘‘πœ”πœ”π‘–π‘– 𝑑𝑑𝒒𝒒𝑖𝑖Φ(πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖) πœ”πœ”π‘ π‘ ,π’’π’’π’”π’”βŸ©|πœ”πœ”π‘–π‘– ,π’’π’’π’Šπ’Š

Mode-function typically entangled:

Energy/Momentum typically correlated:

Ξ¦ πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖 β‰  πœ™πœ™s πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠)πœ™πœ™π‘–π‘–( πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖

Ξ¦ πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖 β‰  πœ™πœ™πœ‡πœ‡ πœ”πœ”π‘ π‘  ,πœ”πœ”π‘–π‘–)πœ™πœ™π‘žπ‘ž(𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖

Polarization-entanglement involves at least two mode-functions

Ξ¦HsVi πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖 + Ξ¦VsHi πœ”πœ”π‘ π‘  ,𝒒𝒒𝑠𝑠 , πœ”πœ”π‘–π‘– ,𝒒𝒒𝑖𝑖

𝐹𝐹 = |βŸ¨Ξ¨βˆ’ Ξ¨ οΏ½2

= 1 + 𝑅𝑅𝑒𝑒[∫ Ξ¦V𝐻𝐻 πœ”πœ”π‘ π‘  Ξ¦Hπ‘‰π‘‰βˆ— πœ”πœ”π‘ π‘  ]dπœ”πœ”

These must overlap to achieve high state fidelity

Θ β‰ˆ |𝒒𝒒|/π‘˜π‘˜π‘§π‘§

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𝐸𝐸𝑝𝑝(𝒒𝒒𝒑𝒑) 𝒒𝒒𝒑𝒑 = 𝒒𝒒𝒔𝒔 + π’’π’’π’Šπ’Š

πœ”πœ”π‘π‘0 = πœ”πœ”π‘ π‘ 0 + πœ”πœ”π‘–π‘–0

Simplification I: Frequencies fixed (narrowband filters)

Ξ¦ πœ”πœ”π‘ π‘  = πœ”πœ”π‘ π‘ 0,𝒒𝒒𝑠𝑠 ,πœ”πœ”π‘–π‘– = πœ”πœ”π‘–π‘–0,𝒒𝒒𝑖𝑖 β†’ Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖

Spatial mode functionΘ β‰ˆ |𝒒𝒒|/π‘˜π‘˜π‘§π‘§

Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖 ∝ 𝐸𝐸𝑝𝑝(𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖)𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§(𝒒𝒒𝒔𝒔,π’’π’’π’Šπ’Š)

Page 36: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

𝐸𝐸𝑝𝑝(𝒒𝒒𝒑𝒑) 𝒒𝒒𝒑𝒑 = 𝒒𝒒𝒔𝒔 + π’’π’’π’Šπ’Š

Spatial mode functionΘ β‰ˆ |𝒒𝒒|/π‘˜π‘˜π‘§π‘§

Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖 ∝ 𝐸𝐸𝑝𝑝(𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖)𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§(𝒒𝒒𝒔𝒔,π’’π’’π’Šπ’Š)

π‘˜π‘˜π‘§π‘§ πœ”πœ”,𝒒𝒒 β‰ˆ π‘˜π‘˜0𝑧𝑧 1 βˆ’π‘˜π‘˜π‘₯π‘₯2 + π‘˜π‘˜π‘¦π‘¦2

π‘˜π‘˜0𝑧𝑧2 π‘˜π‘˜ =

𝑛𝑛 πœ”πœ” πœ”πœ”π‘π‘

πœ•πœ•πœ•πœ•π’’π’’

π‘˜π‘˜π‘§π‘§ πœ”πœ”,𝒒𝒒 = 0

Simplification II: Non-critical phase-matching

Ξ”π‘˜π‘˜π‘§π‘§(𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖) = π‘˜π‘˜π‘π‘π‘§π‘§ 𝒒𝒒𝑠𝑠 + 𝒒𝒒𝑖𝑖 βˆ’ π‘˜π‘˜π‘ π‘ π‘§π‘§ 𝒒𝒒𝑠𝑠 βˆ’ π‘˜π‘˜π‘–π‘–π‘§π‘§ 𝒒𝒒𝑖𝑖

Ξ”π‘˜π‘˜π‘§π‘§(𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖) β‰ˆ π‘˜π‘˜π‘π‘π‘§π‘§ 0 βˆ’ π‘žπ‘žπ‘ π‘ +π‘žπ‘žπ‘–π‘– 2

𝑖𝑖𝑝𝑝𝑧𝑧(0)- π‘˜π‘˜π‘–π‘–π‘§π‘§ 0 + π‘žπ‘žπ‘ π‘  2

𝑖𝑖𝑠𝑠𝑧𝑧(0)-π‘˜π‘˜π‘ π‘ π‘§π‘§ 0 + π‘žπ‘žπ‘–π‘– 2

𝑖𝑖𝑖𝑖𝑧𝑧(0)

π‘˜π‘˜π‘π‘π‘§π‘§ 0 β‰ˆ π‘˜π‘˜π‘ π‘ π‘§π‘§ 0 + π‘˜π‘˜π‘ π‘ π‘§π‘§ 0 β‰ˆ 2π‘˜π‘˜π‘ π‘ π‘§π‘§ 0

β†’ Ξ”π‘˜π‘˜π‘§π‘§(𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖) = π‘žπ‘žπ‘ π‘ +π‘žπ‘žπ‘–π‘– 2

π‘–π‘–π‘π‘π‘§π‘§βˆ’ 2 π‘žπ‘žπ‘ π‘  2

π‘–π‘–π‘π‘π‘§π‘§βˆ’ 2 π‘žπ‘žπ‘ π‘  2

𝑖𝑖𝑝𝑝𝑧𝑧= π‘žπ‘žπ‘ π‘ βˆ’π‘žπ‘žπ‘–π‘– 2

4𝑖𝑖𝑝𝑝𝑧𝑧(0)

2nd-order approx.

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Spatial poperties of the mode function

Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖 ∝ exp𝒒𝒒𝑖𝑖 + 𝒒𝒒𝒔𝒔 𝟐𝟐

4𝑀𝑀𝑝𝑝2 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐

𝐿𝐿2π‘žπ‘žπ‘ π‘  βˆ’ π‘žπ‘žπ‘–π‘– 2

2π‘˜π‘˜π‘π‘π‘§π‘§(0)Gaussian Pump:

Θ β‰ˆ |𝒒𝒒|/π‘˜π‘˜π‘§π‘§

𝒒𝒒

far field

Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖 ∝ 𝛿𝛿(𝒒𝒒𝒔𝒔 + π’’π’’π’Šπ’Š)

Thin crystal limit

Large beam waist:

𝒒𝒒𝒔𝒔 = βˆ’π’’π’’π’Šπ’Š

𝑦𝑦

π‘₯π‘₯

𝑧𝑧

-> Signal and idler emitted at anti-correlated angles

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Spatial poperties of the mode function

Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖 ∝ exp𝒒𝒒𝑖𝑖 + 𝒒𝒒𝒔𝒔 𝟐𝟐

4𝑀𝑀𝑝𝑝2 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐

𝐿𝐿2π‘žπ‘žπ‘ π‘  βˆ’ π‘žπ‘žπ‘–π‘– 2

2π‘˜π‘˜π‘π‘π‘§π‘§(0)Gaussian Pump:

Ξ¦ 𝒒𝒒𝑠𝑠,𝒒𝒒𝑖𝑖 ∝ 𝛿𝛿(𝒒𝒒𝒔𝒔 + π’’π’’π’Šπ’Š)

Thin crystal limit

Large beam waist:far field

Ξ¦ 𝒙𝒙𝑠𝑠,𝒙𝒙𝑖𝑖 ∝ 𝛿𝛿(𝒙𝒙𝒔𝒔 βˆ’ π’™π’™π’Šπ’Š)

near field

𝒒𝒒𝒔𝒔 = βˆ’π’’π’’π’Šπ’Š 𝒙𝒙𝒔𝒔 = π’™π’™π’Šπ’Š-> Signal and idler emitted at anti-correlated angles

Signal and idler β€žbornβ€œ at same position in the crystal

Θ β‰ˆ |𝒒𝒒|/π‘˜π‘˜π‘§π‘§

𝒒𝒒𝑦𝑦

π‘₯π‘₯

𝑧𝑧

Page 39: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Propagation from near to far field

Just, F., Cavanna, A., Chekhova, M. V., & Leuchs, G. (2013). Transverse entanglement of biphotons. New Journal of Physics, 15(8), 083015.

Page 40: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Application: Ghost imaging

Page 41: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Ξ¨ = ∫ 𝑑𝑑𝒒𝒒𝑠𝑠 𝑑𝑑𝒒𝒒𝑖𝑖Φ(𝒒𝒒𝑠𝑠 ,𝒒𝒒𝑖𝑖) π’’π’’π’”π’”βŸ©| π’’π’’π’Šπ’Š

So far we considered decomposition in transverse momentum modes:

other modal decomposition, e.g. a decomp into Laguerre-Gauss modes

𝑙𝑙, 𝑝𝑝 = ∫ 𝑑𝑑𝒒𝒒 𝑒𝑒𝑙𝑙𝑝𝑝𝐿𝐿𝐿𝐿(𝒒𝒒) 𝒒𝒒radial mode index ptopological winding number l

�𝐿𝐿𝑧𝑧𝑒𝑒𝑙𝑙𝑙𝑙LG 𝒒𝒒 = 𝑙𝑙 ℏ 𝑒𝑒𝑙𝑙𝑙𝑙LG 𝒒𝒒

𝑒𝑒𝑙𝑙𝑝𝑝𝐿𝐿𝐿𝐿 𝒒𝒒 =

Modal entanglement

Mair, A., Vaziri, A., Weihs, G., & Zeilinger, A. (2001). Entanglement of the orbital angular momentum states of photons. Nature, 412(6844), 313.

Page 42: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Modal entanglementMode amplitudes related to overlap integral

βˆ‘ βŸ©βˆ’βˆβŸ©Ξ¨l lAB llC ,||

0=l

2βˆ’=l2=l

OAM entanglement :

𝑙𝑙p = 0 β†’ exp 𝑖𝑖( 𝑙𝑙𝑠𝑠+𝑙𝑙𝑖𝑖 )πœ™πœ™

𝑙𝑙 𝑙𝑙 𝑙𝑙

Discrete modal decomposition:

Torres, J. P., Alexandrescu, A., & Torner, L. (2003). Quantum spiral bandwidth of entangled two-photon states. Physical Review A, 68(5), 050301.

Page 43: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Time….

Page 44: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Frequency correlations

CW Pump: s(Ξ©s+Ξ©i) = 𝛿𝛿 Ξ©s+Ξ©i

𝒒𝒒𝑠𝑠 = 𝟎𝟎,𝒒𝒒𝑖𝑖 = 𝟎𝟎

Ξ¨ = ∫ 𝑑𝑑Ω 𝛿𝛿 Ξ©s + Ξ©i 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§ Ξ© Ξ©s⟩|Ξ©i β†’ ∫ 𝑑𝑑Ω Ω⟩| βˆ’ Ξ©

Collinear SPDC πœ”πœ”π‘π‘,0 = πœ”πœ”π‘–π‘– + πœ”πœ”π‘ π‘ 

πœ”πœ”π‘–π‘– =πœ”πœ”π‘π‘,0

2+ Ξ©iπœ”πœ”π‘ π‘  =

πœ”πœ”π‘π‘,0

2+ Ξ©s

Anti-correlated joint spectral amplitude

Ξ¦ πœ”πœ”π‘ π‘  , πœ”πœ”π‘–π‘– β†’ Ξ©s

Ξ©i

Ξ¦ Ξ©s,Ξ©i =s(Ξ©s+Ξ©i) 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§(Ξ©s,Ξ©i)

Page 45: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Absence of frequency correlations

pulsed pump: s(Ξ©s+Ξ©i) = exp(βˆ’π›Όπ›Ό Ξ©s+Ξ©i 2π›Ώπ›Ώπœ‡πœ‡π‘π‘

2 )

exp(βˆ’π›Όπ›Ό Ξ©s+Ξ©i 2π›Ώπ›Ώπœ‡πœ‡π‘π‘

2 ) 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐 𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§ Ξ©s,Ξ©i β†’ πœ“πœ“ Ω𝑠𝑠 πœ“πœ“(Ω𝑖𝑖)

Un-correlated joint spectral amplitude

Un-correlated joint spectral amplitude: signal idler are in pure states HOM interference

Ξ©s

Ξ©i

𝒒𝒒𝑠𝑠 = 𝟎𝟎,𝒒𝒒𝑖𝑖 = 𝟎𝟎Collinear SPDC πœ”πœ”π‘π‘,0 = πœ”πœ”π‘–π‘– + πœ”πœ”π‘ π‘ 

πœ”πœ”π‘–π‘– =πœ”πœ”π‘π‘,0

2+ Ξ©iπœ”πœ”π‘ π‘  =

πœ”πœ”π‘π‘,0

2+ Ξ©s

Ξ¦ πœ”πœ”π‘ π‘  , πœ”πœ”π‘–π‘– β†’ Ξ¦ Ξ©s,Ξ©i =s(Ξ©s+Ξ©i) 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§(Ξ©s,Ξ©i)

Page 46: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Time-Frequency entanglement

CW Pump: s(Ξ©s+Ξ©i) = 𝛿𝛿 Ξ©s+Ξ©i

𝒒𝒒𝑠𝑠 = 𝟎𝟎,𝒒𝒒𝑖𝑖 = 𝟎𝟎

Ξ¨ = ∫ 𝑑𝑑Ω 𝛿𝛿 Ξ©s + Ξ©i 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§ Ξ© Ξ©s⟩|Ξ©i β†’ ∫ 𝑑𝑑Ω Ω⟩| βˆ’ Ξ©

Collinear SPDC πœ”πœ”π‘π‘,0 = πœ”πœ”π‘–π‘– + πœ”πœ”π‘ π‘ 

πœ”πœ”π‘–π‘– =πœ”πœ”π‘π‘,0

2+ Ξ©iπœ”πœ”π‘ π‘  =

πœ”πœ”π‘π‘,0

2+ Ξ©s

Ξ¦ πœ”πœ”π‘ π‘  , πœ”πœ”π‘–π‘– β†’ Ξ©s

Ξ©i

Ξ¦ Ξ©s,Ξ©i =s(Ξ©s+Ξ©i) 𝑠𝑠𝑖𝑖𝑛𝑛𝑐𝑐𝐿𝐿2Ξ”π‘˜π‘˜π‘§π‘§(Ξ©s,Ξ©i)

β†’ ∫ 𝑑𝑑𝑑𝑑 t⟩|𝑑𝑑FT:

Anti-correlated spectra

correlated emission time

Page 47: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

ppt δω

1∝

long pump coherence time

SPDCct δω

1∝

signal-idlercorrelation time

cp tt >>

βˆ‘ =⟩∝⟩Ψ

N

i iiAB tt1

,||Time-energy entanglement:

c

p

tt

N β‰ˆ

Time-frequency entanglement

Page 48: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

...,|,|...| 11 +⟩⟩++∝⟩Ψ ++ iiiiAB tttt

BS BS

Franson interferometer

Franson J. D. Bell inequality for position and time. Phys. Rev. Lett. 62, 2205–2208 (1989)

Page 49: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

ppt δω

1∝

long pump coherence time

SPDCct δω

1∝

signal-idlercorrelation time

cp tt >>

Polarization & time-energyEntanglement:

( )⟩⟩+βŠ—βŸ©βˆ‘ VHHVtti ii ||,|

Hyper-entanglement

β€žhyper entanglementβ€œ

Page 50: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Post-selection-free Franson interferometer

( )⟩⟩+βŠ—βŸ©βˆβŸ©Ξ¨ βˆ‘ VVHHtti iiAB ||,||

PBS PBSPOL POL

Strekalov D., Pittman T., Sergienko A., Shih Y. & Kwiat P. Postselection-free energy-time entanglement. Phys. Rev. A 54, R1 (1996).

Page 51: SPDC-based Entangled Photon Sources...PDC interaction Hamiltonian (assuming only one relevant NL tensor coeffictient) Transverse field operators (narrowband) 𝒒𝒒= (k x,k y) ,

Path Spatial mode Time Frequency

Outlook: Engineering Quantum States

How much information can photons carry?

How can we use this efficiently?

How do we benefit from coupling various degrees of freedom ?

Ongoing challenge: Engineer entanglement simultaneously in all degrees of freedom


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