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    Journals >Photonics Research >Volume 7 >Issue 11 >Page A45 > Article
    • Photonics Research
    • Vol. 7, Issue 11, A45 (2019)
    Generation of optical Fock and W states with single-atom-based bright quantum scissors
    Ziv Aqua1, M. S. Kim2, and Barak Dayan1、*
    Author Affiliations
  • 1AMOS and Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel
  • 2QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK
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    DOI: 10.1364/PRJ.7.000A45 Cite this Article Set citation alerts
    Ziv Aqua, M. S. Kim, Barak Dayan. Generation of optical Fock and W states with single-atom-based bright quantum scissors[J]. Photonics Research, 2019, 7(11): A45 Copy Citation Text show less

    Abstract

    We introduce a multi-step protocol for optical quantum state engineering that performs as “bright quantum scissors,” namely truncates an arbitrary input quantum state to have at least a certain number of photons. The protocol exploits single-photon pulses and is based on the effect of single-photon Raman interaction, which is implemented with a single three-level Λ system (e.g., a single atom) Purcell-enhanced by a single-sided cavity. A single step of the protocol realizes the inverse of the bosonic annihilation operator. Multiple iterations of the protocol can be used to deterministically generate a chain of single photons in a W state. Alternatively, upon appropriate heralding, the protocol can be used to generate Fock-state optical pulses. This protocol could serve as a useful and versatile building block for the generation of advanced optical quantum states that are vital for quantum communication, distributed quantum information processing, and all-optical quantum computing.
    H=−iℏg∫dωκ/πκ−iω(|e⟩⟨gh|a^ω+|e⟩⟨gv|b^ω)e−i(ω+δ)t+h.c.,(1)

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    A^(t)≡12π∫dωa^ωe−iωt,(2)

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    B^(t)≡12π∫dωb^ωe−iωt,(3)

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    |Nh⟩=1N![∫dtf(t)A^†(t)]N|0⟩,(4)

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    |Nh,1vkth⟩≡(N+1)!(k−1)!(N−k+1)!×∫−∞∞dtB^†(t)f(t)[∫−∞tdt1A^†(t1)f(t1)]k−1⁢[∫t∞dt2A^†(t2)f(t2)]N−k+1|0⟩.(5)

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    |1h,0v,gh⟩→−|0h,1v,gv⟩.(6)

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    |Nh,1vkth,gv⟩→|Nh,1v(k+1)th,gv⟩,(7a)

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    |Nh,1v(N+1)th,gv⟩→−|N+1h,0v,gh⟩.(7b)

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    |Nh,1v⟩=1N+1(|Nh,1v1st⟩+⋯+|Nh,1v(N+1)th⟩).(8)

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    |1h,1v,gv⟩=12(|1h,1v2nd⟩+|1h,1v1st⟩)⊗|gv⟩→atom12(−|2h,0v,gh⟩+|1h,1v2nd,gv⟩)→h112(|2h,0v,gv⟩⊗|v1⟩+|1h,1v2nd,gv⟩⊗|h1⟩)→atom12(|2h,0v,gv⟩⊗|v1⟩−|2h,0v,gh⟩⊗|h1⟩)→h212|2h,0v,gv⟩⊗(|v1,h2⟩+|h1,v2⟩)→…⏟2≤k+1 iterations→|2h,0v,gv⟩⊗12(|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩).(9)

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    |Nh,1v,gv⟩→iterationsk+1{for  k≥N:|N+1h,0v,gv⟩⊗1N+1(|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩+⋯+|h1,…,hN,vN+1,hN+2,…,hk+1⟩)for  k≤N−1:|N+1h,0v,gv⟩⊗1N+1(|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩+⋯+|h1,…,hk,vk+1⟩)+1N+1(|Nh,1v(k+2)th,gv⟩+⋯+|Nh,1v(N+1)th,gv⟩)⏟(N−k) terms⊗|h1,…,hk+1⟩.(10)

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    |ψinitial⟩=|ϕh,1v⟩=∑N=0∞CN|Nh,1v⟩.(11)

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    |ψfinal⟩=|v1,h2,h3,…,hk+1⟩⊗(∑N=0∞CNN+1|N+1h⟩)|0v,gv⟩+|h1,v2,h3,…,hk+1⟩⊗(∑N=1∞CNN+1|N+1h⟩)|0v,gv⟩+⋯+|h1,…,hk,vk+1⟩⊗(∑N=k∞CNN+1|N+1h⟩)|0v,gv⟩+|h1,…,hk+1⟩⊗∑N=k+1∞CNN+1(|Nh,1v(k+2)th⟩+⋯+|Nh,1v(N+1)th⟩)|gv⟩.(12)

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    a^a^−1=I;a^−1a^=I−|0⟩⟨0|.(13)

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    a^−1=∑n=0∞1n+1|n+1⟩⟨n|.(14)

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    a^†=∑n=0∞n+1|n+1⟩⟨n|,(15)

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    S^+=∑n=0∞|n+1⟩⟨n|,(16)

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    ⟨v1|ψfinal⟩=(∑N=0∞CNN+1|N+1h⟩)|0v,gv⟩→⋆∑N=0∞CNN+1|N+1h⟩=a^−1∑N=0∞CN|Nh⟩=a^−1|ϕh⟩,(17)

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    η1=∑N=0∞|CN|2N+1.(18)

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    ⟨vk|ψfinal⟩=|h1,h2,…,hk+1⟩⏟trace  out⊗(∑N=k−1∞CNN+1|N+1h⟩)⊗|0v,gv⟩⏟trace out→N∑N=k∞CN−1N|Nh⟩≡|k+⟩,(19)

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    η2=∑N=k∞|CN−1|2N.(20)

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    P(N≥k+1)=∑N=k+1∞|CN|2≪1.(21)

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    ηBQS=1−∑N=k+1∞(N−k)|CN|2N+1.(22)

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    a^N∑N=k∞CNN+1|N+1h⟩=N∑N=k∞CN|Nh⟩,(23)

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    O^=I^−∑n=0k−1|n⟩⟨n|(24)

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    |χ⟩=|vk⟩|hk+1⟩|ψ1⟩+|hk⟩|vk+1⟩|ψ2⟩.(25)

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    |χ⟩=|V1H2⟩|ψ1⟩+|H1V2⟩|ψ2⟩.(26)

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    ±12⟨ψ(−)|=±12(⟨V1H2|−⟨H1V2|).(27)

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    ±12⟨ψ(−)|χ⟩=±12(|ψ1⟩−|ψ2⟩).(28)

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    12(⟨vk,hk+1|−⟨hk,vk+1|)|ψfinal⟩=12|h1,…,hk−1⟩⊗|0v,gv⟩⏟trace out⊗(∑N=k−1∞CNN+1|N+1h⟩−∑N=k∞CNN+1|N+1h⟩)=12Ck−1k|kh⟩.(29)

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    η3=|Ck−1|22k.(30)

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    η4=12k!(k−1e)k−1.(31)

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    |Wn⟩=1n(|VHH…H⟩+|HVH…H⟩+⋯+|HH…HV⟩).(32)

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    |ψfinal⟩=∑N=0k−1CNN+1|N+1h,0v,gv⟩⊗(|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩+⋯+|h1,…,hN,vN+1,hN+2,…,hk+1⟩)+∑N=k∞CNN+1|N+1h,0v,gv⟩⊗(|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩+⋯+|h1,…,hk,vk+1⟩)+∑N=k+1∞CNN+1(|Nh,1v(k+2)th,gv⟩+⋯+|Nh,1v(N+1)th,gv⟩)⊗|h1,…,hk+1⟩.(33)

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    ⟨Mh,0v|ψfinal⟩=CM−1M|gv⟩⊗{for  3≤M≤k:|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩+⋯+|h1,…,hM−1,vM,hM+1,…,hk+1⟩for  M≥k+1:|v1,h2,h3,…,hk+1⟩+|h1,v2,h3,…,hk+1⟩+⋯+|h1,…,hk,vk+1⟩.(34)

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    ⟨Mh,0v|ψfinal⟩={|WM⟩⊗|hM+1,…,hk+1⟩for  3≤M≤k|Wk+1⟩for  M≥k+1.(35)

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    Ziv Aqua, M. S. Kim, Barak Dayan. Generation of optical Fock and W states with single-atom-based bright quantum scissors[J]. Photonics Research, 2019, 7(11): A45
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