《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

ूதθϛɿʮ୯Ұޫࢠɼྔࢠ΋ͭΕޫࢠͷൃੜͱྔࢠ‫ܭ‬ଌʯࢬদ‫ܓ‬Ұʢ౦๺େֶి‫ؾ‬௨৴‫ॴڀݚ‬ʣɹྔࢠతͳޫɼಛʹ୯Ұޫࢠɼྔࢠ΋ͭΕޫࢠͷൃੜͱͦͷར༻͸ɼޫΛ༻͍ͨྔࢠ৘ใ௨৴ٕज़ͷཁͰ͋Δ͹͔ΓͰͳ͘ɼޫͷྔࢠੑΛར༻ͨ͠৽͍͠ྔࢠ‫ܭ‬ଌɼྔࢠ෼ޫٕज़ͷૅͰ΋͋Δɽ͜ͷूதθϛͰ͸ɼޫΛ༻͍ͨ৽ͨͳྔࢠ‫ܭ‬ଌɾ෼ޫٕज़ʹ‫ڵ‬ຯΛ΋ͭ෺ੑएख‫ऀڀݚ‬Λର৅ʹɼҎԼͷ಺༰ʹ͍ͭͯͰ͖
Δ͚ͩқ͓͘͠࿩͍ͨ͠͠ɽ

1. ྔࢠޫֶͷ‫ૅج‬
2. ୯Ұޫࢠͷൃੜͱ‫؍‬ଌ
3. ྔࢠ΋ͭΕޫࢠͷൃੜͱ‫؍‬ଌ
4. ୯Ұޫࢠɼྔࢠ΋ͭΕޫࢠΛ༻͍ͨྔࢠ‫ܭ‬ଌͳ͓ɼθϛͷ಺༰ʹਂؔ͘࿈͢Δ‫ڭ‬Պॻͱͯ͠ɼҎԼΛࢀߟʹ͞Ε͍ͨɽ• ࢬদ‫ܓ‬Ұஶʮ୯Ұޫࢠͱྔࢠ΋ͭΕޫࢠɹʵྔࢠޫֶͱྔࢠޫٕज़ͷ‫ૅج‬ʵʯʢ‫ج‬ຊ๏ଇ͔ΒಡΈղ͘෺ཧֶ࠷લઢ 19ʣ‫ཱڞ‬ग़൛ʢ2018ʣISBN:978-4320035393
ҎԼͷςΩετ͸ɼຊूதθϛͷத֩ͱͳΔ߲໨ʮ2. ୯Ұޫࢠͷൃੜͱ‫؍‬ଌʯ͓Αͼʮ3. ྔࢠ΋ͭΕޫࢠͷൃੜͱ‫؍‬ଌʯʹؔ࿈͢Δࢿྉͱͯ͠ɼ্‫੶ॻه‬ͷୈ 7 ষʙୈ 8 ষΛൈਮɾ࠶ߏ੒ͨ͠΋ͷͰ͋Δɽຊ
ूதθϛͷςΩετͱͯ͠ൈਮΛ‫͢ࡌܝ‬Δ͜ͱΛշ͘‫ڐ‬୚ͯ͠Լͬͨ͞‫ཱڞ‬ग़൛ʹ‫͢ँײ‬Δɽ

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

୯Ұޫࢠͷൃੜ

1|0〉2͜ͷষͰ͸ɼྔࢠޫֶ΍ޫΛ༻͍ͨྔࢠ৘ใ௨৴|ψ〉1ٕज़ʹ͓͍ͯॏཁͳ୯Ұޫࢠ (single photon) ͷ֓೦ʹ͍ͭͯௐ΂Δɽʮ୯Ұޫࢠʯͱ͸ɼཧ૝తʹ͸ͻͱ

BSͭͷϞʔυ͋Δ͍͸͍͔ͭ͘ͷϞʔυʹ·͕ͨͬͯ

t1

ޫࢠ͕ 1 ‫͞ىྭݸ‬Ε͍ͯΔঢ়ଶΛࢦ͕͢ɼ࣮༻తʹt2

͸ɼ͋Δۭ࣌ؒʹ͓͍ͯޫࢠΛ‫ݕ‬ग़ͨ͠ࡍʹ 2 ‫ݸ‬Ҏ

᫬㛫┦㛵
ᐃჾ

্ͷޫࢠ͕‫ݕ‬ग़͞ΕΔ֬཰͕ 0 Ͱ͋Δঢ়ଶΛࢦ͢৔߹͕ଟ͍ɽ‫Ͱऀޙ‬͸ޫࢠ਺͸ 0 ·ͨ͸ 1 Ͱ͋ͬͯɼޫ
ࢠ਺͕ 1 ʹ֬ఆͨ͠ঢ়ଶͰ͸ͳ͍͕ɼ2 ‫ݸ‬Ҏ্ͷޫࢠ͕ଘࡏ͠ͳ͍͜ͱ͕֬ఆͰ͖Δ͜ͱ͔Βɼྔࢠ৘ใ௨৴ɼಛʹྔࢠ҉߸ʹ͓͍࣮ͯ༻্ॏཁͳঢ়ଶͰ

ਤ 1: ޫࢠ‫ݕ‬ग़‫ث‬Λ༻͍ͨ‫౓ڧ‬૬ؔଌఆͷ֓೦ਤɽ
g(2)

(a)

(b)

g(2)

1

͋Δɽ
0

1.1

τ

0

τ

୯Ұޫࢠͱ‫౓ڧ‬૬ؔ୯Ұޫࢠঢ়ଶΛ࿦͡Δ্Ͱॏཁͳޫࢠͷ࣌ؒతͳ୯Ұੑ͸ɼ2 ࣍ͷ‫֨ن‬Խࣗ‫ݾ‬૬ؔؔ਺ʢ‫౓ڧ‬૬ؔؔ਺ʣ

g (2) (τ ) =â† (t)â† (t + τ )â(t + τ )â(t)
â† (t)â(t)2(1)Ͱఆྔతʹද͢͜ͱ͕Ͱ͖Δɽ͜͜Ͱɼâ† (t) ͓Αͼâ(t) ͸֤ʑ࣌ࠁ t ʹ͓͚Δޫࢠͷੜ੒ɾফ໓ԋࢉࢠͰ͋Γɼ... ͸࣌ؒฏ‫ۉ‬Λද͢ɽࣜ (1) ͷ෼ࢠ͸ɼ࣌ؒࠩτ ִ͚ͩͯͯ̎ޫࢠΛ‫ݕ‬ग़͢Δ֬཰ʹൺྫ͢Δɽ୯Ұޫࢠঢ়ଶͰ͸ɼಉ࣌ʹ 2 ‫ݸ‬Ҏ্ͷޫࢠ͕‫ݕ‬ग़͞ΕΔ͜
ͱ͸ͳ͍͔Βɼg (2) (0) = 0 Ͱ͋Δɽg (2) (0) < g (2) (τ )ͱͳΔΑ͏ͳঢ়ଶΛɼޫࢠͷΞϯνόϯνϯάͱ͍͏͕ɼ୯Ұޫࢠঢ়ଶͱ͸ɼg (2) (0) = 0 ͱͳΔ࠷΋Ξϯνόϯνϯάͨ͠ঢ়ଶͰ͋Δͱ‫͑ݴ‬Δɽ࣮‫ݧ‬తʹ͸ɼ‫౓ڧ‬૬ؔؔ਺͸ਤ 1 ʹࣔͨ͠Α͏ͳ૷ஔͰଌఆ͞ΕΔɽͦͷࡍɼଌఆର৅ͱͳΔʮ୯Ұޫࢠ‫ݯ‬ʯ͕࿈ଓ (CW) ޫ‫͋Ͱݯ‬Ε͹ɼͦͷ‫౓ڧ‬૬ؔؔ਺ʹ͸ τ = 0 Λத৺ͱͯ͋͠Δ࣌ؒ෯Λ΋ͭσΟοϓਤ 2: ୯Ұޫࢠ‫ʹݯ‬ର͢Δ‫౓ڧ‬૬ؔ g (2) (τ ) ͷଌఆྫɽ(a) ࿈ଓ (CW) ޫɼ͓Αͼ (b) प‫ظ‬తύϧεޫͷ৔߹ɽͳͲɼ୯Ұͷྔࢠ‫͚͓ʹܥ‬ΔిࢠભҠʹ൐͏ൃޫΛ
ར༻͢Δํ๏Ͱ͋Δɽ͢ͳΘͪɼ͜ΕΒͷ୯Ұྔࢠ‫͚͓ʹܥ‬ΔిࢠͷϑΣϧϛΦϯੑʹ‫͖ͮج‬ɼಉ࣌͡ࠁɾঢ়ଶʹෳ਺ͷޫࢠ͕ൃੜ͠ͳ͍Α͏੍‫͞ޚ‬Εͨޫ‫ͯ͠ͱݯ‬ར༻͢Δ΋ͷͰ͋Δɽ͜ͷΑ͏ͳํ๏ʹΑΔ୯Ұޫࢠͷൃੜ͸ɼ୯Ұ‫ࢠݪ‬ͷ‫ڞ‬໐ൃޫʹ͓͍࣮ͯ‫͠ݱ‬ಘΔ͜ͱ͕ཧ࿦తʹ༧‫͞ݴ‬Ε [1]ɼఔͳͯ͘͠ Na ౳ͷ୯Ұͷ‫ࢠݪ‬ͷ‫ڞ‬໐ൃޫʹ͓͍࣮ͯ‫ݧ‬తʹ‫؍‬ଌ͞Εͨ [2, 3]ɽ͜ͷΑ͏ͳ୯Ұޫࢠঢ়ଶ͸ɼ୯Ұ‫ݪ‬ࢠҎ֎ʹ΋ɼ୯ҰΠΦϯͷ‫ڞ‬໐ൃޫ [4]ɼ୯Ұ෼ࢠͷൃޫ [5, 6]ɼ‫ݻ‬ମதͷ୯Ұෆ७෺४Ґ͔Βͷൃޫ [7]ɼ൒ಋମதͷ୯Ұྔࢠυοτ͔Βͷൃޫ [8, 9, 10]ɼ୯
Ұ༗‫ػ‬φϊ݁থ͔Βͷൃޫ [11] ͷΑ͏ʹɼ୯Ұͷྔࢠ‫͔ܥ‬ΒͷిࢠભҠʹΑΔൃޫΛ෼཭ͯ͠डޫ͢Δ͜ͱʹΑͬͯ‫؍‬ଌ͞Ε͍ͯΔɽ͜ͷΑ͏ͳ୯Ұޫࢠൃੜʹؔ͢Δղઆ࿦จͱͯ͠ɼ୯Ұ‫ࢠݪ‬Λ༻͍ͨॳ(g (2) (0) = 0) ͕‫ݱ‬ΕΔʢਤ 2(a)ʣɽ͜Εʹର͠ɼଌఆ
‫ظ‬ͷ‫ؔ͢ʹڀݚ‬Δ΋ͷ [12, 13] ΍ɼ୯Ұ෼ࢠ΍‫ݻ‬ମର৅ͱͳΔ୯Ұޫࢠ‫͕ݯ‬प‫ظ‬తύϧεޫ‫͋Ͱݯ‬Ε͹Λ༻͍ͨ‫ؔ͢ʹڀݚ‬Δ΋ͷ [14, 15] ͕ࢀߟʹͳΔͩ
‫౓ڧ‬૬ؔؔ਺΋ಉ͡‫܁‬Γฦ͠प‫ظ‬Λ΋ͭύϧεঢ়ͱΖ͏ɽͳΓɼͦͷ τ = 0 ʹ͓͚Δ஋͕ 0 ͱͳΔʢਤ 2(b)ʣɽ
͜͜Ͱ͸·ͣɼ୯Ұ‫ࢠݪ‬΍୯ҰΠΦϯͷ‫ڞ‬໐ൃޫͷ৔߹ͷΑ͏ʹɼਤ 3(a) ʹࣔ͢Α͏ͳ 2 ४Ґ‫ܥ‬Λ‫ڞ‬
1.2 ୯Ұͷྔࢠ‫ܥ‬Λ༻͍ͨ୯Ұޫࢠൃੜ ໐ྭ‫ͨ͠ى‬৔߹ʹɼྭ‫ͱޫى‬͸ҟͳΔۭؒϞʔυʹ์ग़͞ΕΔޫࢠͷ‫౓ڧ‬૬ؔؔ਺Λߟ͑Δɽ͍·ɼ࣌୯Ұޫࢠͷൃੜʹ͸͍͔ͭ͘ͷํ๏͕ఏҊ͞Εͯ
ࠁ t = 0 ʹ͓͍ͯୈ 1 ͷޫࢠ͕์ग़͞Εͨͱ͢Δɽ
͍Δ͕ɼͦͷҰ͕ͭɼ୯Ұͷ‫ࢠݪ‬ɼ෼ࢠɼଋറిࢠ

《講義ノート》(a)

物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

g(2)

(b)

(c)
1
(b)
(a)
0

τ

ਤ 3: (a) 2 ४Ґ‫ܥ‬ɼ͓Αͼ (b) 3 ४Ґ‫ܥ‬Λ༻͍ͨ୯ਤ 4: 2 ४Ґ‫ܥࢠݪ‬ͷ‫ڞ‬໐ൃޫʹ͓͚Δ‫౓ڧ‬૬ؔҰޫࢠൃੜͷ֓೦ਤ. γ1 : ᫔ࣹ؇࿨཰ɼγp : ७Ґ૬؇g (2) (τ ) ͷ‫ྫࢉܭ‬ɽ(a) ऑྭ‫͔ͭى‬७Ґ૬؇࿨͕ͳ͍
৔߹ɼ(b) ऑྭ‫͔ͭى‬७Ґ૬؇࿨͕େ͖͍৔߹ɼ(c)‫ىྭڧ‬ͷ৔߹ɽ࿨཰ɼΩ: ϥϏप೾਺ɼw: ྭ‫཰֬ى‬ɽ͜ͷͱ͖ɼ2 ४Ґ‫ܥ‬͸‫ج‬ఈঢ়ଶ |g ʹ͋Γɼྭ‫ى‬ঢ়ଶ

g(2)(τ)|e ʹ͋Δ֬཰ pe ͸ 0 Ͱ͋Δɽ͋Δ࣌ࠁʹ͓͍ͯɼୈ
2 ͷޫࢠ͕์ग़͞ΕΔ֬཰͸ɼpe ʹൺྫ͢Δɽैͬͯɼg (2) (τ ) ͸ɼt = τ (≥ 0) ʹ͓͚Δ pe ʹൺྫ͢Δ
͜ͱʹͳΔɽྭ‫ͱޫى‬͸ҟͳΔଟ਺ͷۭؒϞʔυͷͲΕ͔ʹޫࢠ͕์ग़͞ΕΔ৔߹ɼ‫ڞ‬໐ྭ‫ʹى‬ରͯ͠᫔ࣹ؇࿨߲͕ՃΘΔ͜ͱʹͳΓɼt = 0 Ͱ 0 Ͱ͋ͬͨpe ͸ɼt → ∞ Ͱ༗‫ݶ‬ͷҰఆ஋ʹ͖ۙͮɼ͜ͷͱ͖g (2) → 1 Ͱ͋Δɽैͬͯ‫౓ڧ‬૬ؔؔ਺͸ɼg (2) (0) = 0ɼg (2) (∞) → 1 ͳΔΞϯνόϯνϯάಛੑΛࣔ͢͜ͱτ (ns)
ʹͳΔɽͦͷ༷ࢠ͸ɼྭ‫౓ڧى‬ʢϥϏप೾਺ Ωʣɼ᫔ࣹ؇࿨཰ γ1 ɼ७Ґ૬؇࿨཰ γp ͷ஋ʹΑΓ࣍ͷΑ͏ਤ 5: ୯Ұ Na ‫ࢠݪ‬ͷ‫ڞ‬໐ൃޫʹ͓͚Δ‫౓ڧ‬૬ؔͳ৔߹ʹ෼͚Δ͜ͱ͕Ͱ͖Δ [1, 12, 13]ɽ(2)(a) ऑྭ‫( ى‬Ω  γ1 ) ͔ͭ७Ґ૬؇࿨͕ͳ͍ (γp = g (τ ) Λ‫ٻ‬Ί࣮ͨ‫[ ྫݧ‬2, 3]ɽ
0) ৔߹ʹͳ͓ͬͯΓɼτ = 0 Ͱ g (2) ͕૿Ճ͠ɼ1 ʹۙͮ͘͜2(2)−γ1 τ /2g (τ ) = 1 − e(2) ͱ͕Θ͔Δɽ͜ͷ͜ͱ͸ɼ̎ޫࢠ͕ಉ࣌ʹ͸์ग़͞Εͳ͍͜ͱΛ͓ࣔͯ͠Γɼ୯Ұ‫͍͓ͯʹࢠݪ‬͸̎ి(b) ऑྭ‫͔ͭى‬७Ґ૬؇࿨͕େ͖͍ (Ω  γ1  γp ,ࢠ͕ಉ࣌ʹಉ͡ΤωϧΪʔ४Ґʹྭ‫͞ى‬Εͳ͍ͱ͍1  γp τ ) ͷ৔߹͏ɼ‫ֶޫ͍ڧ‬ඇઢ‫ܗ‬ੑΛ൓өͨ͠΋ͷͰ͋Δɽ
(2)
−γ1 τ
Ұํɼ෼ࢠ΍‫ݻ‬ମෆ७෺͓Αͼ൒ಋମྔࢠυοτg (τ ) = 1 − e(3)ʹ͓͍ͯ͸ɼޫྭ‫ʹى‬Αͬͯྭ‫͞ى‬ΕΔঢ়ଶͱɼൃ(c) ‫( ىྭڧ‬Ω  γ1 , γp ) ͷ৔߹ޫΛ‫؍‬ଌ͢Δ࢝ঢ়ଶ͸ҟͳΔͷ͕ී௨Ͱ͋Δɽ͜ͷ
g (2) (τ ) = 1 − e−(γ1 +γ2 )τ /2 cos Ωτ(4) Α͏ͳ৔߹ʹద༻͞ΕΔ୯Ұͷ 3 ४Ґ‫ܥ‬Ϟσϧʹ͍ͭͯߟ͑Δʢਤ 3(b)ʣɽ‫ܥ‬͸‫ج‬ఈঢ়ଶ |g ͔Βதؒྭ͜͜Ͱɼγ2 = γ1 /2 + γp Ͱ͋Δɽ͜ΕΒͷ‫౓ڧ‬૬ؔ ‫ى‬ঢ়ଶ |i ʹྭ‫͞ى‬Εͨ‫ޙ‬௚ͪʹൃޫͷ࢝ঢ়ଶ |e ʹؔ਺ͷུ֓Λਤ 4 ʹࣔ͢ɽ(2)ʙ(4) ͷ͍ͣΕͷ৔߹ ؇࿨͠ɼͦͷ‫ޫൃޙ‬Λ൐ͬͯ‫ج‬ఈঢ়ଶʹ໭Δ΋ͷͱ΋ɼg (2) (0) = 0 ͱͳͬͯɼΞϯνόϯνϯάಛੑΛ ͢Δɽ࣌ࠁ τ = 0 ʹ͓͍ͯୈ 1 ͷޫࢠ͕์ग़͞Εͨࣔ͢୯Ұޫࢠঢ়ଶ͕࣮‫͞ݱ‬ΕΔ͜ͱ͕Θ͔Δɽ·ͨɼ ‫ޙ‬ɼ୯Ґ࣌ؒ಺ʹ |g ͔Β |i Λ‫| ͯܦ‬e ΁ྭ‫͞ى‬Ε‫ىྭڧ‬ͷ৔߹ʹ‫ݱ‬ΕΔৼಈߏ଄͸ɼ‫ڞ‬໐ྭ‫ͱޫى‬ೋ Δ֬཰Λ wɼ|e ͔Β |g ΁ͷ᫔ࣹ؇࿨཰Λ γ ɼ࣌ࠁ1४Ґ‫ͱܥ‬ͷؒͷϥϏৼಈʹ༝དྷ͢Δɽτ ʹ͓͍ͯ‫| ͕ܥ‬g, |e ʹ͋Δ֬཰Λ֤ʑpg (τ ), pe (τ )
ਤ 5 ʹɼ୯Ұ Na ‫͔ࢠݪ‬Βͷ‫ڞ‬໐ൃޫͷ‫౓ڧ‬૬ؔ(2)g (τ ) ͷଌఆྫΛࣔ͢ [2, 3]ɽτ = 0 ͷ෇ۙͰ g (2) ∼ 0

Coincidence counts, n(t )《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

CdSe/ZnS cluster

300

200

60
40
CdSe/ZnS single quantum dot

20
0

50
Time, t (ns)

100

150

ਤ 6: CdSe ୯Ұྔࢠυοτ͔Βͷൃޫͷ‫౓ڧ‬૬ؔͷ࣮‫[ ྫݧ‬8]ɽଟ਺ͷྔࢠυοτ͔ΒͳΔू߹ମʢ্ʣ
ͱ୯Ұྔࢠυοτ͔ΒͷൃޫʢԼʣͷ‫౓ڧ‬૬ؔɽਤ 7: GaAs ୯Ұྔࢠυοτͷൃޫͷ‫౓ڧ‬૬ؔͷ࣮‫ݧ‬ͱ͢Δͱɼ࣍ͷϨʔτํఔ͕ࣜ੒Γཱͭɽdpe (τ ) = wpg (τ ) − γ1 pe (τ )dτྫ [9]ɽप‫ظ‬తϨʔβʔύϧεʢ্ʣͰྭ‫͞ى‬Εͨ୯(5)Ұྔࢠυοτ͔ΒͷൃޫʢԼʣͷ‫౓ڧ‬૬ؔɽ

(a)
(6)

pe (0) = 0(7)

Ͱ͋Δ͜ͱʹ஫ҙͯ͠ (5) Λղ͘ͱɼw pe (τ ) =1 − e−(γ1 +w)τγ1 + w

0.5

0−100
X (μm)(8)ΛಘΔɽैͬͯɼ͜ͷ‫͔ܥ‬Βͷൃޫͷ‫౓ڧ‬૬ؔ͸ɼ

pe (τ )
g (2) (τ ) == 1 − e−(γ1 +w)τpe (∞)

1.0

༢୍ 㻺㼂 ୰ᚰg(2)(τ)pg (τ ) + pe (τ ) = 1(b)

Y (μm)

·ͨɼ

(9)

−50

0

50

100

τ (ns)

ਤ 8: (a) μΠϠϞϯυதͷ NV த৺ͷൃޫͷ‫ݦ‬ඍ‫ڸ‬
૾ɽ(b) ୯Ұͷ NV த৺͔Βͷൃޫͷ‫౓ڧ‬૬ؔͷଌఆྫ [18]ɽ

ͱͳΓ [8]ɼ(3) ͱಉ͡ࢦ਺ؔ਺‫ͳͱܕ‬Δ͜ͱ͕Θ͔Λࣔͯ͠͸͍Δ΋ͷͷɼg (2) (0) ͷ஋͕ 0 ·ͰԼ͕Γ

Δ1 ɽ͜ͷ৔߹΋ɼg (2) (0) = 0 ͱͳͬͯɼΞϯνόϯ

͖ͬͯ͸͓Βͣɼ୯Ұޫࢠঢ়ଶͱͯ͠͸·ͩෆ‫׬‬શνϯάಛੑΛࣔ͢୯Ұޫࢠঢ়ଶ͕࣮‫͞ݱ‬ΕΔɽ

ͳঢ়ଶͰ͋ͬͨɽಉ༷ͳ෺࣭‫ܥ‬Λ༻͍ͨ௿Թʹ͓͚൒ಋମΛ༻͍ͨ୯Ұޫࢠൃੜ͕ॳΊͯ֬ೝ͞ΕͨΔ࣮‫Ͱݧ‬͸ɼ͞Βʹ໌ྎͳʢΑΓখ͞ͳ g (2) (0) ͷ஋ͷ͸ɼίϩΠυ༹ӷ๏Ͱ࡞੡͞Εͨ CdSe ྔࢠυο Λ΋ͭʣ୯Ұޫࢠൃੜ͕‫؍‬ଌ͞Ε͍ͯΔ [17]ɽ
্ड़ͷίϩΠυྔࢠυοτͰͷ࣮‫΅΄ͱݧ‬ಉ࣌‫ظ‬τΛ༻͍࣮ͨ‫[ ݧ‬8] Ͱ͋Δɽਤ 6 ʹࣔ͢Α͏ʹɼଟ਺ͷྔࢠυοτΛ‫ؚ‬ΉΫϥελʔ͔ΒͷൃޫͰ͸Ξϯʹɼ൒ಋମද໘্ʹࣗ‫ݾ‬૊৫Խ੒௕ͨ͠୯Ұྔࢠυονόϯνϯά͸‫؍‬ଌ͞Εͳ͍͕ɼ୯Ұͷྔࢠυοττࢼྉ͔Βͷ୯Ұޫࢠൃੜ΋‫؍‬ଌ͞ΕͨɽGaAs ද͔ΒͷൃޫͰ͸໌ྎͳΞϯνόϯνϯά͕‫؍‬ଌ͞Εɼ ໘্ʹ੒௕ͨ͠ InAs ୯Ұྔࢠυοτࢼྉ͔Βͷൃ(9) ࣜͰܾ·Δͦͷ࣌ؒεέʔϧ͸਺ 10 ns Ͱ͋Δɽ ޫͷΞϯνόϯνϯάΛ‫؍‬ଌͨ͠ྫ [9] Ͱ͸ɼྭ‫ޫى‬ͳ͓ɼ௕͍࣌ؒεέʔϧͰ͸ྔࢠυοτ͔ΒͷΩϟ ͱͯ͠प‫ظ‬తͳ୹ύϧεΛൃੜ͢ΔϨʔβʔΛ༻͍ɼϦΞͷΠΦϯԽʹΑΔ఺໓‫ݱ‬৅ [16] ʹ൐͏όϯνϯ ͦͷप‫ʹظ‬ಉ‫ޫൃͨ͠ظ‬ͷ‫౓ڧ‬૬ؔΛ‫؍‬ଌ͍ͯ͠Δάʢg (2) (τ ) > g (2) (∞)ʣ͕‫؍‬ଌ͞ΕΔ͜ͱ΋஌ΒΕ ʢਤ 7ʣɽ࣮‫݁ݧ‬ՌͰ͸ಉҰ (τ = 0) ͷύϧεʹ͓͚
͍ͯΔɽਤ 6 ʹ࣮ࣔͨ͠‫Ͱݧ‬͸ɼΞϯνόϯνϯά Δ‫౓ڧ‬૬͕ؔ΄΅ 0 ͱͳΓɼ̍ͭͷྭ‫ى‬ύϧε͔Β
1 Ϩʔτํఔࣜ (5) ͕੒ཱ͢Δͷ͸ɼ|i ͔Β |e ΁ͷ଎͍؇࿨ʹΑΓҐ૬৘ใ͕ࣦΘΕΔɼ͢ͳΘͪ଎͍Ґ૬؇࿨͕͋Δ৔߹ʹ૬౰͢Δɽ͸̍‫Ͱ·ݸ‬ͷޫࢠ͔͠ൃޫ͠ͳ͍͜ͱΛ͍ࣔͯ͠Δɽ͜ͷΑ͏ͳॳ‫ظ‬ͷ࣮‫ݧ‬͸ϨʔβʔʹΑΔޫྭ‫ى‬Λ

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༻͍͍͕ͯͨɼ൒ಋମதʹຒΊࠐ·Εͨ୯Ұྔࢠυο

pumpτΛ༻͍Ε͹ɼLED ͷΑ͏ͳిྲྀ஫ೖʹΑΔ୯Ұޫ

χ(2)signalࢠൃੜ΋‫ظ‬଴͞ΕΔɽ࣮ࡍɼ͜ͷΑ͏ͳిྲྀ஫ೖʹΑΔ୯Ұޫࢠൃੜ΋ఔͳ࣮ͯ͘͠‫͞ݱ‬Εͨ [19]ɽ·

idlerͨɼྔࢠ௨৴΁ͷԠ༻ʹ͓͍ͯॏཁͳҙຯΛ΋ͭ௨৴ωp೾௕ଳͰͷ୯Ұޫࢠൃੜʹؔͯ͠͸ɼInAs/InP ྔࢠωsυοτΛ༻͍ͯ 1.3 μm ଳʹ͓͚Δ୯Ұޫࢠൃੜ [20]͕ใࠂ͞Εͨͷʹଓ͖ɼ1.5 μm ଳʹ͓͚Δ୯Ұޫࢠωiൃੜ [21, 22] ΋ใࠂ͞Ε͍ͯΔɽ͞Βʹɼ୯Ұޫࢠͷ೾௕ଳ΍ൃੜಈ࡞Թ౓Ҭͷ֦େ΋ࢼΈΒΕ͍ͯΔɽͦͷҰྫͱͯ͠ɼμΠϠϞϯυதͷ஠ૉෆ७෺ͱۭ
޸ʹัറ͞Εͨ‫ࢠిࡏہ‬த৺ʢNV த৺ʣ͕‫͛ڍ‬Βਤ 10: ࣗൃύϥϝτϦοΫԼํม‫ʹ׵‬ΑΔ૬ؔޫࢠରͷൃੜɽΕΔɽμΠϠϞϯυͷ NV த৺͸ɼࣨԹͰ΋҆ఆ͔͍ͨ΁Μ༏Ε͍ͯΔ΄͔ɼยํͷޫࢠΛ఻ྩͱͯ͠

ͭߴޮ཰ʹಈ࡞͢Δ୯Ұޫࢠ‫ͯ͠ͱݯ‬஫໨͞Ε͍࣮ͯޮతͳ୯Ұޫࢠʢ఻ྩ෇͖୯Ұޫࢠʣͱͯ͠΋༻

Δ [7, 23, 24]ɽਤ 8 ʹɼμΠϠϞϯυࢼྉதͷ NV த

͍Δ͜ͱ͕Ͱ͖ɼྔࢠޫֶɼྔࢠ৘ใ௨৴ʹ͓͚Δ

৺͔Βͷൃޫͷ‫ݦ‬ඍ‫ͱ૾ڸ‬ɼ୯Ұ NV த৺͔Βͷൃछʑͷ࣮‫͍༻ʹݧ‬ΒΕ͍ͯΔɽޫͷ‫౓ڧ‬૬ؔͷଌఆྫΛࣔ͢ [18]ɽτ = 0 ෇ۙͰ໌ྎ
ͳΞϯνόϯνϯάಛੑΛ͍ࣔͯ͠Δ͜ͱ͕Θ͔Δɽ

g (2) (0) ͕ 0 ʹୡ͍ͯ͠ͳ͍ͷ͸ɼ୯Ұ NV த৺Ҏ֎ͷޫʢଞͷ NV த৺͔Βͷൃޫ΍μΠϠϞϯυࢼྉʹΑΔഎ‫Ͳͳޫܠ‬ʣͷࠞೖ͕‫׬‬શʹ͸ഉআͰ͖ͳ͍
ͨΊͰ͋Δɽ·ͨɼτ = 0 ͷ྆ଆͰ g (2) > 1 ͱͳ͍ͬͯΔྖҬ͕͋Δͷ͸ɼલड़ͨ͠ CdSe ྔࢠυοτͰ΋ΈΒΕͨόϯνϯάʹΑΔ΋ͷͰ͋Γɼτ → ∞Ͱ͸ g (2) → 1 ͱͳΔɽ

2

ޫࢠରͷൃੜ
ҎԼͰ͸ɼ࣌ؒతɾۭؒతʹ૬ؔΛ΋ͬͯ΄΅ಉ࣌ʹൃੜ͢Δ̎‫ݸ‬ͷޫࢠɼ͢ͳΘͪޫࢠରͷൃੜํ

2.1

Χεέʔυޫ์ग़

ਤ 9 ʹࣔ͢Α͏ʹɼঢ়ଶ |a, |b, |c ͷࡾ४Ґ‫ܥ‬Λߟ͑Δɽ͜ͷΑ͏ͳࡾ४Ґ‫ܥ‬͸ɼ‫ࢠݪ‬΍෼ࢠɼ·ͨ͸‫ݻ‬ମதͷྭ‫ࢠى‬ͷΑ͏ͳ཭ࢄతͳΤωϧΪʔ४ҐΛ΋ͭ෺ཧ‫ܥ‬Λ༻͍࣮ͯ‫͖Ͱݱ‬Δɽ͜ͷ‫͍͓ͯʹܥ‬ɼ|c → |b ͷભҠʹΑΓୈ 1 ͷޫࢠʢৼಈ਺ ω1 ʣ͕,|b → |a ͷભҠʹΑΓୈ 2 ͷޫࢠʢৼಈ਺ ω2 ʣ͕ൃ
ੜ͢Δ৔߹Λߟ͑Δɽ͜ͷͱ͖ɼ‫| ͕ܥ‬b → |c ʹ࠶ྭ‫͞ى‬ΕΔ֬཰͸े෼খ͘͞ɼ|b → |a ΁͸ޫࢠ 2
Λ์ग़͢ΔҎ֎ͷ‫ܦ‬࿏͸ͳ͍ͱ͢Δͱɼޫࢠ 1 ͕ൃੜͨ͠‫ʹޙ‬ඞͣޫࢠ 2 ͕ൃੜ͢Δɽ͢ͳΘͪɼޫࢠ
1 ͱޫࢠ 2 ͷର͕ൃੜ͢Δ͜ͱʹͳΔɽ

๏ʹ͍ͭͯड़΂Δɽ͜ͷΑ͏ͳޫࢠର͸ɼ࣍ষͰड़
΂Δྔࢠ΋ͭΕΛൃੜɾอ࣋ɾ༌ૹ͢Δഔମͱͯ͠

2.2ύϥϝτϦοΫԼํม‫׵‬૬ؔޫࢠରঢ়ଶ͸ɼਤ 10 ʹࣔࣗ͢ൃύϥϝτϦοΫ

|c〉

Լํม‫׵‬ʢspontaneous parametric down-conversion:
SPDCʣΛ༻͍ͯൃੜ͢Δ͜ͱ΋Ͱ͖ΔɽSPDC ͸ɼ
2 ࣍ͷඇઢ‫ޮֶޫܗ‬ՌͰ͋ΔύϥϝτϦοΫޫֶաఔ|b〉
ͷҰछͰ͋Γɼ1 ‫ݸ‬ͷޫࢠʢϙϯϓޫࢠʣ͔Β࣌ؒతʹ‫͘ڧ‬૬ؔͨ͠ 1 ରͷޫࢠΛൃੜ͢Δ͜ͱ͕Ͱ͖Δɽ
ω2ύϥϝτϦοΫޫֶաఔͱ͸ɼ2 ࣍ͷඇઢ‫ܗ‬෼‫ʹۃ‬|a〉
Αͬͯɼޫप೾਺ͷࠞ߹·ͨ͸෼ׂ͕‫͖ى‬Δ‫ݱ‬৅Ͱ͋Δɽ͍·ɼޫʹΑͬͯ༠‫͞ى‬ΕΔ෺࣭ͷ෼‫ ۃ‬P Λਤ 9: ࡾ४Ґ‫͔ܥ‬ΒͷΧεέʔυޫ์ग़ʹΑΔޫࢠର
(1)
(2) 2
(3) 3
P
=

E
+
χ
E
+
χ
E
+
...
(10)
χ
0
ͷൃੜɽ

ω1

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ͱॻ͘2 ɽޫͱ෼‫ۃ‬ͷ૬‫ ༻࡞ޓ‬HI ͸ɼ૒‫ࣅۙࢠۃ‬ͷԼͰHI = −P · E(11)ͱॻ͚Δɽैͬͯ 2 ࣍ͷඇઢ‫ܗ‬෼‫ۃ‬SPDC ͸ 2 ࣍ͷඇઢ‫ܗ‬෼‫ ۃ‬χ(2) ʹΑͬͯੜ͡Δ೾௕ม‫׵‬աఔͰ͋Δ͔Βɼޮ཰తʹ SPDC Λੜͤ͡͞ΔͨΊʹ͸ɼେ͖ͳ χ(2) Λ΋ͭ෺࣭͕ඞཁͰ͋ΔɽͦͷͨΊͷ୅දతͳ෺࣭‫ͯ͠ͱ܈‬ɼ‫ڧ‬༠ిੑͷඇઢ‫ޫܗ‬ֶ݁থ͕ྑ͘༻͍ΒΕΔɽ·ͨɼͦͷࡍʹ͸ɼSPDC
ʹΑΔ೾௕ม‫׵‬աఔ͕݁থશମʹΘͨͬͯίώʔϨ

P (2) = 0 χ(2) E 2(12)૬੔߹৚݅ͱ͍͍ɼ݁থ௕͕ແ‫ʹݶ‬௕͍৔߹ʹ͸࣍ͱޫͱͷ૬‫༻࡞ޓ‬͸ɼHI = −P(2)ϯτʹॏͳΓ߹͏͜ͱ͕ඞཁͰ͋Δɽ͜ͷ৚݅ΛҐͷࣜͰද͞ΕΔɽ

· E = −0 χ

(2)

E

3

(13)

ͱॻ͘͜ͱ͕Ͱ͖Δɽ͍·ɼωp , ω1 , ω2 ͷ̏छͷৼಈ਺ͷϞʔυΛߟ͑ɼ͜͜Ͱɼωp ʢϙϯϓޫʣͷ‫ڧ‬

ω 1 + ω 2 = ωp

(16)

k1 + k2 = kp(17)౓͸ଞΑΓे෼‫͍ڧ‬΋ͷͱ͢Δɽ͍·ɼ(13) ͷ૬‫ޓ‬͜͜Ͱɼki (i = 1, 2, p) ͸γάφϧޫɼΞΠυϥޫɼ࡞༻ͷԼͰ ωp ʹؔ͢ΔԋࢉࢠΛ‫ؚ‬Έɼ࣌ؒฏ‫ۉ‬ͷ‫ޙ‬ϙϯϓޫͷ೾਺ϕΫτϧͰ͋Δɽ(16) ͸ɼ(15) ʹ΋ʹ΋ফ͑ͳ͍߲ʢੜ੒ɾফ໓͞ΕΔޫࢠͷΤωϧΪʔ‫ݱ‬ΕͨΑ͏ʹৼಈ਺ʢ͋Δ͍͸ޫࢠΤωϧΪʔʣͷ͕อଘ͞ΕΔ߲ʣΛर͍ग़͢ͱɼ
g â†i âp âp + h.c.
(ωi = 2ωp , i = 1, 2)(ω1 + ω2 = ωp )
g â†1 â†2 âp + h.c.

อଘଇΛද͠ɼ(17) ͸ɼ೾਺ϕΫτϧʢ͋Δ͍͸ޫࢠͷӡಈྔʣͷอଘଇΛද͢ɽ৚݅ (16) ʹΑͬͯɼൃ(14)ੜ͢ΔγάφϧޫͱΞΠυϥޫͷৼಈ਺ͷؒʹ૬ؔ

(15)

͕ੜ͡Δɽ·ͨɼ৚݅ (17) ʹΑͬͯɼͦΕΒͷ೾਺ϕΫτϧ͢ͳΘͪൃੜํ޲ͷؒʹ૬͕ؔੜ͡Δɽ͞ΛಘΔɽ͜͜Ͱɼg ͸ χ(2) ʹൺྫ͢Δఆ਺Ͱ͋Δɽ Βʹɼ2 ͭͷ৚݅Λฒཱ͢Δ͜ͱͰɼγάφϧޫ͓Α(14) ͷୈ߲̍͸ɼωp ͷϙϯϓޫΛ̎ޫࢠফ໓ͤ͞ ͼΞΠυϥޫͷৼಈ਺͕ͦΕΒͷൃੜํ޲ʹԠͯͯ͡ 2ωp ͷޫΛ̍ޫࢠੜ੒͢ΔաఔͰɼୈ̎ߴௐ೾ൃ ܾ·Δɽ࣮ࡍ্͸ɼ݁থͷ௕͕͞༗‫Ͱݶ‬ϙϯϓޫͷੜ (second harmonic generation: SHG) ʹରԠ͢Δɽ ৼಈ਺ͱ೾਺ϕΫτϧ΋༗‫ݶ‬ͷ෯Λ΋ͭͨΊɼҐ૬(15) ͷୈ߲̍͸ɼωp ͷϙϯϓޫΛ̍ޫࢠফ໓ͤͯ͞ ੔߹৚͕݅؇࿨͞Εɼγάφϧޫ͓ΑͼΞΠυϥޫω1 ͷޫͱ ω2 ͷޫΛ֤ʑ̍ޫࢠੜ੒͢ΔաఔͰɼύ ͸༗‫ݶ‬ͷৼಈ਺෯ʢεϖΫτϧ෯ʣΛ΋ͭ͜ͱʹͳϥϝτϦοΫԼํม‫( ׵‬parametric down-conversion: Δ5 ɽ·ͨɼ࣮ࡍͷ࣮‫͍͓ͯʹݧ‬Ґ૬੔߹৚݅Λຬ଍
PDC) ͱ‫ݺ‬͹ΕΔɽಛʹɼω1 =ω2 ͷ࣌Λॖୀύϥϝτ ͤ͞ΔͨΊʹ͸ɼ௨ৗɼඇઢ‫ֶ݁ޫܗ‬থͷෳ۶ંੑϦοΫԼํม‫Ϳݺͱ׵‬ɽύϥϝτϦοΫԼํม‫ ͓ʹ׵‬Λར༻͢Δɽ͜ͷͱ͖ɼγάφϧޫͱΞΠυϥޫͷ͍ͯɼೖࣹޫʢछޫʣͱͯ͠ ω1 ·ͨ͸ ω2 ͷޫ͕ଘࡏ ภޫ͕ฏߦͰ͋Δ৔߹ͱɼ‫ʹ͍ޓ‬ਨ௚Ͱ͋Δ৔߹ͷ͢Δ৔߹ʹ͸ɼͦͷޫ͸༠ಋ์ग़ʹΑͬͯ૿෯͞ΕΔ 2 छྨͷҐ૬੔߹͕ߟ͑ΒΕΔɽલऀΛ type-I Ґ૬͜ͱʹͳΓɼ͜ΕΛύϥϝτϦοΫ૿෯ (parametric ੔߹ɼ‫ऀޙ‬Λ type-II Ґ૬੔߹ͱ‫Ϳݺ‬ɽ͍·ɼϙϯϓޫΛৼ෯ αʢ࣮਺ʣͷίώʔϨϯτޫ
ampliﬁcation) ͱ‫Ϳݺ‬ɽω1 ͓Αͼ ω2 ͷೖࣹঢ়ଶ͕ਅ
ۭঢ়ଶͷ৔߹ʹ͸ɼύϥϝτϦοΫԼํม‫׵‬͸ࣗൃ ͱ͠ɼγάφϧ͓ΑͼΞΠυϥͷϞʔυʹ͸ਅۭঢ়์ग़ʹΑΔ΋ͷͱͳΓɼ͜Ε্͕ड़ͨࣗ͠ൃύϥϝ ଶΛೖྗ͢ΔͱɼSPDC ʹΑΓൃੜ͢Δޫࢠঢ়ଶ͸ɼτϦοΫԼํม‫׵‬ʢSPDCʣͰ͋Δ3 ɽSPDC ʹΑͬͯൃੜͨ͠ޫͷҰํΛγάφϧޫɼଞํΛΞΠυϥ
ޫͱ‫Ϳݺ‬4 ɽ
2 Ұൠʹ‫ײ‬ड཰͸ςϯιϧͰද͞ΕΔ͕ɼ͜͜Ͱ͸؆୯ͷͨΊ|Ψ = c0 |01 |02 + c1 |11 |12 + c2 |21 |22 + ...
(18)
cn =

(tanh θ)ncosh θ

(19)εΧϥʔྔͱͯ͠ѻ͏ɽ
3 ύϥϝτϦοΫ‫( ޫܬ‬spontaneous parametric emission) ͱͱॻ͘͜ͱ͕Ͱ͖Δ [25]ɽ͜͜Ͱɼ|ni ͸γάφϧ
΋͍͏ɽ
4 ύϥϝτϦοΫ૿෯ͷ৔߹͸ɼೖࣹޫͷ͋ΔํΛγάφϧޫɼ(i = 1) ͓ΑͼΞΠυϥ (i = 2) Ϟʔυͷޫࢠ਺ঢ়ଶΛଞํΛΞΠυϥޫͱ‫͕Ϳݺ‬ɼSPDC ͷ৔߹ʹ͸ͲͪΒΛγάφϧ
5 ‫ޙ‬ड़͢ΔΑ͏ʹɼγάφϧޫͱΞΠυϥޫͷεϖΫτϧ෯͸ɼޫͱ‫͔Ϳݺ‬͸ܾ·͍ͬͯͳ͍ɽ‫ͯ͠ͱྫ׳‬ɼৼಈ਺ͷେ͖͍ʢ೾
ͦΕΒͷؒͷ࣌ؒ૬ؔ෯ΛܾΊΔཁҼͰ͋Δɽ
௕ͷ୹͍ʣํΛγάφϧޫͱ‫Ϳݺ‬৔߹͕ଟ͍ɽ

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ද͠ɼθ ͸ gα ʹൺྫ͢Δఆ਺Ͱ͋Δɽ·ͨɼtanh2 θ =1ζ ͱஔ͘͜ͱʹΑΓɼ
|cn |2 =(tanh θ)2n
= (1 − ζ)ζ n
cosh2 θ

2’

|1〉1
BS

(20)

ΛಘΔ͕ɼζ = e−ω/(kB T ) ͱஔ͘ͱɼޫࢠ਺ঢ়ଶͷ

2

|1〉2

2

1’

֬཰෼෍ |cn | ͸೤์ࣹʢԹ౓ T , ֯ৼಈ਺ ω ʣʹ͓͚Δޫࢠ਺෼෍ʢ‫ز‬Կ෼෍ʣͱ౳͍͜͠ͱ͕Θ͔Δɽਤ 11: ̎ޫࢠ‫ׯ౓ڧ‬বͷ֓೦ਤɽ͢ͳΘͪɼSPDC ʹΑΓൃੜ͢Δޫࢠରͷ਺ͷ෼෍͸ɼ೤์ࣹͷޫࢠ਺෼෍ͱ౳͍͠ɽ͜͜Ͱɼγάφϧʢ·ͨ͸ΞΠυϥʣͷฏ‫਺ࢠޫۉ‬͸ n = ζ/(1 − ζ)Ͱ͋Δɽ·ͨɼγάφϧʢ·ͨ͸ΞΠυϥʣͷҰํͷϞʔυͷޫࢠঢ়ଶͷີ౓ߦྻ ρ1 , ρ2 ͸ɼશ‫ܥ‬ͷີ౓ߦྻ ρ = |ΨΨ| ΛଞํͷϞʔυʹ͍ͭͯ෦෼ର֯
࿨ΛͱΔ͜ͱʹΑͬͯ‫ٻ‬ΊΒΕΔɽ͢ͳΘͪ

ρ1 = Tr (2) |ΨΨ| =

|cn |2 |nn|1 ,(21)|cn |2 |nn|2

(22)

n

ρ2 = Tr (1) |ΨΨ| =nͰ͋ΔɽSPDC ͷग़ྗঢ়ଶ (18) ͸ޫࢠରʹ͍ͭͯͷॏͶ߹Θͤঢ়ଶͰ͋Δ͕ɼγάφϧ·ͨ͸ΞΠυϥ
֤ʑͷޫࢠঢ়ଶ͸ɼίώʔϨϯε͕ࣦΘΕͨ೤์ࣹ
ঢ়ଶͱͳΔͷͰ͋Δɽ͍·ɼϙϯϓޫͷ‫े͕౓ڧ‬෼ʹऑ͍৔߹ɼ͢ͳΘͪ (20) ʹ͓͍ͯ θ  1 ैͬͯ ζ  1 ͷ৔߹Λߟ͑Δɽ͜ͷͱ͖ɼ|c0 |2  |c1 |2  |c2 |2 ...ɽͱͳΔͷͰɼγάφϧ͓ΑͼΞΠυϥʹ֤ʑ2 ‫ݸ‬Ҏ্ͷޫࢠΛ‫ؚ‬Ήঢ়ଶΛແࢹͰ͖ɼ

|Ψ = c0 |01 |02 + c1 |11 |12

2.3

Hong-Ou-Mandel ͷ̎ޫࢠ‫ׯ౓ڧ‬ব

ਤ 11 ͷΑ͏ʹ 50%:50%ϏʔϜεϓϦολ (BS) ͷ̎ͭͷೖࣹϙʔτ 1, 2 ʹɼಉ͡प೾਺ ω ͓Αͼ s ภޫͷ୯Ұޫࢠঢ়ଶ |11 , |12 Λಉ࣌ʹೖࣹ͢Δɽ͜ͷͱ͖ͷೖࣹޫͷঢ়ଶ͸|Ψ = |11 |12 = â†1 â†2 |00(24)

ͱॻ͚Δɽ͜͜Ͱɼ|00 = |01 |02 ͸ਅۭঢ়ଶΛද
͢ɽBS ʹΑΔԋࢉࢠͷม‫ࣜ׵‬





1 i
â1
â1
1
=√
2
i 1
â2
â2

(25)

Λ༻͍ͯɼ(24) Λ BS ͷग़ྗଆͷঢ়ଶʹม‫͢׵‬Δͱɼ



1 †
(26a)
â1 + iâ†2 iâ†1 + â†2 |0021i(â†1 )2 + i(â†2 )2 + â†1 â†2 − â†1 â†2 |00
=
2
(26b)

|Ψ  =i= √ (|21 |02 + |01 |22 )
2

(26c)

(23)
ͱͳΔɽ͜͜Ͱɼ(26b) ͷୈ 1 ߲ͱୈ 2 ߲͸ೖࣹͨ͠ͱද͢͜ͱ͕Ͱ͖Δɽ͜ͷ͏ͪɼ|11 |12 ͕ޫࢠରঢ়

ޫࢠͷҰํ͕ BS Λಁա͠ଞํ͕௨ա͢Δ৔߹ΛදଶͰ͋Δɽਅۭঢ়ଶ |01 |02 ͸ɼޫࢠ‫ݕ‬ग़Λͨ͠ࡍ

͠ɼୈ 3 ߲͓Αͼୈ 4 ߲͸֤ʑɼೖࣹͨ͠ޫࢠ͕྆

ʹ͸ଌఆ͞Εͳ͍ͷͰɼޫࢠ͕‫ݕ‬ग़͞Εͨͱ͖ͷΈ

ํಁա͢Δ৔߹ͱ྆ํ൓ࣹ͢Δ৔߹Λද͢ɽୈ 3 ߲Λ݁Ռͱଊ͑Δ৔߹ʹ͸ແࢹͯ͠ྑ͍ɽSPDC Ͱ͸ɼ ͱୈ 4 ߲͸ූ߸͕‫͋Ͱٯ‬ΔͷͰ‫ʹ͍ޓ‬ଧͪফ͠߹͍ɼ
γάφϧ͓ΑͼΞΠυϥޫࢠ͸Ґ૬੔߹৚݅Ͱܾ·݁‫( ہ‬26c) ͷঢ়ଶʹ‫ؼ‬ண͢Δɽ͢ͳΘͪɼೖࣹͨ͠

Δൺֱత޿͍εϖΫτϧ෯Λ΋ͭɽͦͷ૬ؔ࣌ؒʢ̎

̎ޫࢠ͸̎ͭͷग़ྗϙʔτʹ෼͔Εͯग़ࣹ͢Δ͜ͱޫࢠ͕‫ݕ‬ग़͞ΕΔࡍͷ࣌ؒࠩͷ෼෍ʣ͸ɼγάφϧ͸ͳ͘ɼඞͣͲͪΒ͔ͷग़ྗϙʔτʹҰॹʹग़ࣹ͓͢ΑͼΞΠυϥͷεϖΫτϧ෯ͷͨͨΈࠐΈͷ‫਺ٯ‬ΔͷͰ͋Δɽ֤ग़ྗϙʔτ΁̎ޫࢠ͕ग़ࣹ͢Δ֬཰ఔ౓ͷ෯Λ࣋ͪɼయ‫ܕ‬తͳ৚݅ʹ͓͍ͯ͸ 100 fs ఔ͸֤ʑ1/2 Ͱ͋ΔɽͲͪΒ͔ยํͷग़ྗϙʔτ΁ͷ

౓Ͱ͋Δɽ͜ͷ࣌ؒ૬ؔ෯͸ɼҎԼͰड़΂Δ Hong,ޫࢠ਺͕ඞͣ 0 ͱͳΔ͜ͱ͔Βɼग़ྗϙʔτ 1’ ͓ΑOu, Mandel ʹΑͬͯߦΘΕͨೋޫࢠ‫ׯ‬বͷ࣮‫[ ݧ‬26]
ʹΑͬͯଌఆ͢Δ͜ͱ͕Ͱ͖Δɽ

ͼ 2’ ͷ྆ํͰಉ࣌ʹޫࢠΛ‫ݕ‬ग़͢Δ‫཰਺ܭ‬͸ɼΨ |â†1 â†2 â2 â1 |Ψ  = 0(27)

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0.5
40

20

0

–100

0

ΔL1 (μm)

100

Normarized Coincidence–1

Coincidence Rate (s )

60

1

1’’
2’

BS1

BS2
1’

2

ΔL2

2’’

0

ਤ 12: Hong-Ou-Mandel ͷ̎ޫࢠ‫ׯ౓ڧ‬বͷ࣮‫ྫݧ‬ਤ 13: ޫࢠରͱ Mach-Zehnder ‫ׯ‬ব‫ʹܭ‬ΑΔ̎ޫࢠྔࢠ‫ׯ‬বͷ࣮‫ܥݧ‬ɽ

[27]ɽ

3

×10

ͷଧͪফ͠߹͍ɼ͢ͳΘͪ‫ݟʹ͍ޓ‬෼͚͕͔ͭͳ͍
̎ޫࢠؒͷྔࢠ‫ׯ‬বʹ‫ݱͮ͘ج‬৅Ͱ͋ΓɼͦΕΛॳΊ࣮ͯ‫[ ݧ‬26] Ͱࣔͨ͠ Hong, Ou ͓Αͼ Mandel ʹͪͳΜͰ Hong-Ou-Mandel ͷ̎ޫࢠ‫ׯ౓ڧ‬ব͋Δ

Counting Rate (s–1)ͱͳΔɽ͜ͷ‫ݱ‬৅͸ɼ(26b) ʹ͓͚Δୈ 3 ߲ͱୈ 4 ߲

100
50
0
300
200
100
0

͍͸୯ʹ Hong-Ou-Mandel ‫ׯ‬বͱ‫ݺ‬͹ΕΔɽ
ਤ 12 ʹɼHong-Ou-Mandel ‫ׯ‬বͷ࣮‫[ ྫݧ‬27] Λࣔ͢ɽࣗൃύϥϝτϦοΫԼํม‫( ׵‬SPDC) ʹΑΓൃੜͨ͠ޫࢠରΛɼϏʔϜεϓϦολ (BS) ͷ྆ଆͷೖྗϙʔτʹೖࣹ͢ΔɽͦͷࡍͷɼSPDC ͔Β BS ΁
ࢸΔޫ࿏ 1 ͱ 2 ͱͷؒͷޫ࿏ࠩΛ ΔL1 ͱ͢ΔɽBSͷग़ྗϙʔτ 1’ ͓Αͼ 2’ ʹ഑ஔͨ͠ޫࢠ‫ݕ‬ग़‫ޫͰث‬ࢠΛ‫ݕ‬ग़͠ɼͦΕΒͷؒͷಉ࣌‫཰਺ܭ‬Λޫ࿏ࠩ ΔL1
ͷؔ਺ͱͯ͠ଌఆ͢ΔͱɼΔL1 = 0 ͷ෇ۙͰಉ࣌‫ܭ‬਺཰͕΄΅ 0ɼ͢ͳΘͪ (27) Ͱࣔͨ݁͠Ռͱͳͬͯ
͍Δ͜ͱ͕Θ͔Δɽ·ͨɼΔL1 = 0 ෇ۙʹ͓͚Δಉ
࣌‫཰਺ܭ‬ͷԜΈͷ෯ʢ൒஋શ෯ʣ͸ɼޫ࿏ࠩʹͯ͠
໿ 40 μmɼ࣌ؒࠩʹͯ͠໿ 130 fs Ͱ͋Δɽલड़ͨ͠
Α͏ʹɼ͜ͷ෯͸ SPDC Ͱੜ੒͞Εͨޫࢠରͷ࣌ؒ૬ؔ෯Λද͍ͯ͠Δ [26, 28]ɽ0

1

ΔL2 (μm)2

3

ਤ 14: ̎ޫࢠྔࢠ‫ׯ‬বͷ࣮‫[ ྫݧ‬27]ɽ্ஈ͸ɼBS1
ͷೖྗ 1 ΁ 1 ޫࢠɼೖྗ 2 ΁ਅۭঢ়ଶΛೖྗ͠ɼBS2ͷग़ྗϙʔτ 2” Ͱ 1 ޫࢠ‫ݕ‬ग़Λͨ͠ࡍͷ 1 ޫࢠ‫ׯ‬বɽԼஈ͸ɼBS1 ͷೖྗ 1 ͓Αͼೖྗ 2 ΁֤ʑ1 ޫࢠΛೖྗ͠ɼBS2 ͷग़ྗϙʔτ 2” Ͱ 2 ޫࢠ‫ݕ‬ग़Λͨ͠ࡍͷ 2 ޫࢠ‫ׯ‬বɽ2 ޫࢠ‫ׯ‬বͷ‫ׯ‬বࣶͷִؒ͸ 1 ޫࢠ‫ׯ‬বͷͦΕʹൺ΂ͯ 1/2 ʹͳΔɽ
ଶ (26c)ɼ͋Δ͍͸શମͷҐ૬ҼࢠΛলུͨ͠1|Ψ  = √ (|21 |02 + |01 |22 )2(28)͕ੜ੒͞ΕΔɽΔL2 Λ‫ׯ‬ব‫ܭ‬ͷ྆࿹ͷؒͷޫ࿏ࠩɼͦΕʹ൐͏Ґ૬ࠩΛ φ = 2πΔL2 /λ ͱ͢Δͱɼ(26c)Λ‫ٻ‬Ίͨͱ͖ͱಉ༷ʹͯ͠ɼBS2 Λ௨աͨ͠‫ޙ‬ͷঢ়

2.4

ޫࢠରʹΑΔྔࢠ‫ׯ‬ব –ޫࢠͷυɾϒ ଶͱͯ͠ɼ
ϩΠ೾௕ͷଌఆ–্ड़ͨ͠ Hong-Ou-Mandel ‫ׯ‬ব‫ܥ‬ʢਤ 11ʣͷ‫ʹޙ‬͞ΒʹϏʔϜεϓϦολΛ෇Ճ͢Δͱɼਤ 13 ͷΑ͏ͳ Mach-Zehnder ‫ܕ‬ͷ‫ׯ‬ব‫ߏ͕ܭ‬੒͞ΕΔɽ͜͜Ͱ|Ψ  =1 − e2iφ√(|21 |02 − |01 |22 )2 2i(1 + e2iφ )+|11 |12
2(29)

BS1, BS2 ͱ΋ 50%:50%ϏʔϜεϓϦολͰ͋Δɽ͜ ΛಘΔɽ(29) ͔Βɼग़ྗϙʔτ 1” Ͱ n1 ޫࢠɼग़ྗ
ͷ‫ׯ‬ব‫ܭ‬ͷೖࣹϙʔτ 1, 2 ʹɼಉ͡प೾਺ ω ͷ୯Ұ ϙʔτ 2” Ͱ n2 ޫࢠΛ‫ݕ‬ग़͢Δ֬཰ R(n1 , n2 ) ͸ޫࢠঢ়ଶ |11 , |12 Λಉ࣌ʹೖࣹ͢Δɽ͢ΔͱɼBS1
1
R(2, 0) = R(0, 2) = (1 − cos 2φ),(30)
Ͱͷ Hong-Ou-Mandel ‫ׯ‬বͷ݁ՌɼMach-Zehnder
41‫ׯ‬ব‫ܭ‬ͷ̎ͭͷ࿹ͷͲͪΒ͔Λ̎ޫࢠ͕‫ʹڞ‬௨Δঢ়(31)
R(1, 1) = (1 + cos 2φ)2

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ͱͳΔ͜ͱ͕Θ͔Δɽैͬͯɼ͜ͷ‫ׯ‬ব‫ܭ‬ͷग़ࣹϙʔ

Δ͜ͱ΋Ͱ͖Δɽ(23) ࣜͰද͞ΕΔঢ়ଶʹ͓͍ͯɼΞτͰͷ̎ޫࢠ‫ݕ‬ग़ʹΑΔ‫ׯ‬বࣶ͸ɼ͔͋ͨ΋྆࿹ͷΠυϥޫࢠ͕‫ݕ‬ग़͞Εͨ͜ͱΛ৚݅ͱͨ͠ͱ͖ɼ͢

ؒͷҐ૬͕ࠩ 2φ Ͱ͋Δ͔ͷΑ͏ͳ‫ׯ‬বΛࣔ͢ɽ‫ݴ׵‬ͳΘͪΞΠυϥޫࢠΛ఻ྩͱͨ͠ͱ͖ͷγάφϧޫ͢Ε͹ɼ͜ͷ‫ׯ‬ব‫ܭ‬ͷதʹੜ͡Δ̎ޫࢠঢ়ଶ (28) ͸ɼ ࢠͷঢ়ଶ͸ɼ|01 ͕আ֎͞Εɼ୯Ұޫࢠঢ়ଶ |11 ͱ‫ݹ‬యత೾௕ λ ͷ൒෼ͷ೾௕ λ/2 Λ΋ͬͯ‫ׯ‬ব͢Δɽ ͳΔɽ͜ΕΛ఻ྩ෇͖ޫࢠ (heralded photon) ͱ͍͜ͷ͜ͱ͸ɼ̎ޫࢠঢ়ଶͷʮυɾϒϩΠ (de Broglie)6͏ɽ‫ʹີݫ‬͸ (18) ͷΑ͏ʹೋޫࢠҎ্ͷঢ়ଶ΋‫·ؚ‬೾௕ʯ͕‫ݹ‬యత೾௕ͷ 1/2 ʹͳͬͨ΋ͷͱղऍ͢Δ

ΕΔͷͰ‫׬‬શͳ୯Ұޫࢠঢ়ଶͰ͸ͳ͍ɽ͢Θͳͪ఻͜ͱ΋Ͱ͖Δɽྩ෇͖ޫࢠͰ͸ɼ୯Ұྔࢠঢ়ଶ͔Βൃੜ͢Δ୯Ұޫਤ 14 ʹɼ͜ͷΑ͏ͳ 2 ޫࢠྔࢠ‫ׯ‬বͷଌఆྫΛࣔࢠͷ৔߹ͷΑ͏ʹ‫౓ڧ‬૬ؔ g (2) (0) ͷ஋͕‫ݪ‬ཧతʹ 0

͢ [27]ɽ1 ޫࢠ‫ׯ‬বʢਤ 14 ্ஈʣͷ‫ׯ‬বࣶͷִؒ͸ɼ ʹͳΔΘ͚Ͱ͸ͳ͍ɽ࣮ࡍɼ(18) ʹ͓͚Δ n = 0 ͷ༻͍ͨޫʢSPDCʣͷ‫ݹ‬యత೾௕ (860 nm) ͱಉ͡Ͱ߲Λআ֎ͨ͠ঢ়ଶʹ͍ͭͯɼγάφϧޫࢠʹର͢Δ

͋Δ͕ɼ2 ޫࢠ‫ׯ‬বʢਤ 14 Լஈʣͷ‫ׯ‬বࣶͷִؒ͸g (2) (0) Λ‫ٻ‬ΊΔͱɼͦΕΒͷ 1/2 (430 nm) ͱͳ͓ͬͯΓɼ(30) ࣜʹࣔ͢g (2) (0) =‫ׯ‬ব͕‫؍‬ଌ͞ΕΔ͜ͱ͕Θ͔Δɽ͞ΒʹਐΊͯɼn ‫ݸ‬ͷޫࢠΛͻͱ·ͱΊʹͯ͠‫ׯ‬বɾ‫ݕ‬ग़͢Ε͹ɼͦͷ‫ׯ‬ব͸ λ/n ͷʮυɾϒϩΠ೾௕ʯΛࣔ͢͜ͱ͕‫ظ‬଴͞ΕΔ [29]ɽ(28) ͷ‫ܗ‬ͷ 2 ޫࢠঢ়ଶΛ n ޫࢠʹ֦ுͨ͠ঢ়ଶ

1|Ψ  = √ (|n1 |02 + |01 |n2 )2n̂21  − n̂1 = 2ζn̂1 2(33)

ͱͳΔ͜ͱ͕Θ͔Δɽ͜͜Ͱ͸ɼ... ͸ n > 0 ʹ͍ͭͯͷ‫ظ‬଴஋Λද͢ɽैͬͯɼྭ‫͕౓ڧى‬ऑ͍ (ζ  1)ͷͱ͖ʹ͸ɼg (2) (0) ∼ 0 ͱͳͬͯ୯Ұޫࢠͷ৔߹ͱಉ͡Α͏ͳΞϯνόϯνϯάΛࣔ͢ͷͰ͋Δ7 ɽ͜Ε(32)͕ɼ఻ྩ෇͖ޫࢠ͕͠͹͠͹୯ҰޫࢠͱΈͳ͞ΕΔ
ॴҎͰ͋Δɽͳ͓ɼ఻ྩޫࢠͷ‫ݕ‬ग़Λ৚݅ͱ͠ͳ͍

͸ޫࢠ਺ʹؔ͢ΔҰछͷྔࢠ΋ͭΕঢ়ଶͰ͋Δ͕ɼͦ৔߹ʹ͸ɼγάφϧޫࢠͷঢ়ଶ (21) ͸೤์ࣹঢ়ଶͰ

ͷ‫͔ܗ‬Β NOON ঢ়ଶͱ΋‫ݺ‬͹ΕΔɽ্ड़ͨ͠Α͏

͋Δ͔Βɼྭ‫͔͔ʹ౓ڧى‬ΘΒͣ g (2) (0) = 2 ͱͳΔʹɼ2 ޫࢠͷ NOON ঢ়ଶ͸ Hong-Ou-Mandel ‫ׯ‬বόϯνϯάΛࣔ͢͜ͱʹͳΔ8 ɽ͜ͷΑ͏ͳ఻ྩ෇͖

Λ༻͍ܾͯఆ࿦తʹʢ֬཰ 1 Ͱʣੜ੒͢Δ͜ͱ͕Ͱޫࢠͷ‫౓ڧ‬૬ؔ͸ɼSPDC ʹΑͬͯൃੜͨ͠ޫࢠର

͖͕ͨɼ3 ޫࢠҎ্ͷ NOON ঢ়ଶΛܾఆ࿦తʹੜ੒Λ༻͍࣮ͯࡍʹଌఆ͞Ε͍ͯΔ [39]ɽ఻ྩ෇͖ޫࢠ

͢Δํ๏͸஌ΒΕ͍ͯͳ͍ɽͦΕͰ΋ɼޫࢠ͕‫ׯ‬ব͸ SPDC Λ༻͍ͯൺֱత؆ศʹൃੜͤ͞Δ͜ͱ͕Ͱ‫͔ܭ‬Βग़ࣹ͞Εͨ‫ʹޙ‬ଌఆ͢Δঢ়ଶΛ্ख͘બ୒͢

͖ΔͨΊɼྔࢠ҉߸ͳͲͷྔࢠ৘ใ௨৴ϓϩτίϧ

Δ͜ͱʹΑΓɼ֬཰తʹͰ͸͋Δ͕ɼ3 ޫࢠ [30]ɼ4ʹ͓͍ͯɼ҆શੑ΍ޮ཰ΛߴΊΔͨΊʹ͠͹͠͹རޫࢠ [31, 32, 33, 34]ɼ5 ޫࢠ [35, 36]ɼ͋Δ͍͸ 6 ޫ༻͞ΕΔɽࢠ [37] ঢ়ଶͷྔࢠ‫ׯ‬বΛଌఆͨ͠ྫ͕ใࠂ͞Ε͍ͯΔɽ·ͨɼ͜ͷΑ͏ͳྔࢠ‫ׯ‬বੑɼಛʹ୹ॖ͞Εͨ೾௕Λར༻͢Δ͜ͱʹΑͬͯɼޫͷճંʹΑ੍ͬͯ‫͞ݶ‬Ε͍ͯΔ‫ݦ‬ඍ‫ڸ‬΍ϦιάϥϑΟͷղ૾౓Λɼ‫ݹ‬యతճં‫ݶ‬քΛ௒͑ͯ޲্ͤ͞ಘΔٕज़ͷఏҊ [38]΋ͳ͞Ε͓ͯΓɼࠓ‫ޙ‬ͷԠ༻͕‫ظ‬଴͞Ε͍ͯΔɽ

3

ྔࢠ΋ͭΕޫࢠ
ҎԼͰ͸ɼۙ೥‫ޫ٭‬Λཋͼ͍ͯΔྔࢠ৘ใ௨৴ٕ

ज़ͷ‫ૅج‬Λͳ͢ϦιʔεͷͻͱͭͰ͋Δɼྔࢠ΋ͭΕ (entanglement) ͋Δ͍͸Τϯλϯάϧϝϯτͷੑ࣭ͱͦͷൃੜɼ‫ݕ‬ग़ํ๏ʹ͍ͭͯड़΂Δɽಛʹɼޫ

2.5ޫࢠରͱ୯Ұޫࢠʢ఻ྩ෇͖ޫࢠʣ

ࢠͷରʹྔࢠ΋ͭΕΛอ࣋ͤͨ͞ʮྔࢠ΋ͭΕޫࢠରʯ͸ɼޫͷ࣋ͭੑ࣭ͱͯ͠֎ཚʹΑΔӨ‫ڹ‬ʢσί
1.2 અͰ͸ిࢠ‫ܥ‬ͷϑΣϧϛΦϯੑΛར༻ͯ͠௚઀
ώʔϨϯεʣΛड͚ʹ͍͘͜ͱ͔Βɼۭؒతʹ཭Ε୯ҰޫࢠΛൃੜ͢Δํ๏ʹ͍ͭͯड़΂͕ͨɼ࣮༻্
ͨ৔ॴ΁ྔࢠ΋ͭΕঢ়ଶΛӡͿഔମͱͯ͠࠷దͰ͋
ʮ୯Ұʯͱ‫ͤͳݟ‬Δޫࢠঢ়ଶ͸ɼޫࢠରΛར༻ͯ͠ಘ
6 ࣭ྔΛ΋ͭ‫Ͱܥ‬ͷ௨ৗͷυϒϩΠ೾௕ͱ۠ผ͢ΔͨΊɼʮϑΥτχοΫɾυɾϒϩΠ೾௕ʯͱ‫Ϳݺ‬৔߹΋͋Δɽ

7 ྭ‫( ͍ڧ͕౓ڧى‬ζ ∼ 1) ͷͱ͖ʹ͸ɼg (2) (0) ∼ 2 ͱͳͬͯɼ
όϯνϯάΛࣔ͢ɽ
8 SPDC Ͱൃੜ͢Δޫࢠ͸ൺֱత޿͍प೾਺෯Λ΋ͭͷͰɼͦͷόϯνϯάͷ࣌ؒ෯͸ൺֱత‫͍ڱ‬ɽ

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Δɽྔࢠ΋ͭΕ͸ɼྔࢠςϨϙʔςʔγϣϯ΍ྔࢠࢠϏοτ 2ʣͷঢ়ଶΛ |b2 (b = 0, 1) ͱॻ͘ͱ͖ɼ̎

த‫Ͳͳܧ‬ͷྔࢠ৘ใ௨৴ϓϩτίϧʹඞਢͰ͋Δ΄ͭͷྔࢠϏοτͷঢ়ଶ͸ͦΕΒͷ௚ੵ͔ɼྔࢠ΋ͭΕΛར༻ͨ͠௒ߴਫ਼౓‫ܭ‬ଌΛ࣮‫͢ݱ‬Δ|a1 ⊗ |b2 ≡ |abٕज़΋ఏҊ͞ΕΔͳͲɼ༷ʑͳԠ༻͕ߟ͑ΒΕ͍ͯ
Δɽ͞Βʹɼ‫؍‬ଌ໰୊΍ඇ‫ॴہ‬ੑͳͲɼྔࢠྗֶͷࠜຊత໰୊Λ໰͍͔͚Δ‫ૅج‬త‫ʹڀݚ‬΋ྔࢠ΋ͭΕޫࢠର͕‫׆‬༂͢Δɽ

3.1

(37)

ʹ͓͚Δ‫ʹ͍ޓ‬௚ަ͢Δ̐ঢ়ଶ |00, |01, |10, |11Λ‫ج‬ఈͱͯ͠ද͢͜ͱ͕Ͱ͖ΔɽͦΕΒͷҰൠతॏͶ߹Θͤঢ়ଶ͸|Ψ = α|00 + β|01 + γ|10 + δ|11ྔࢠ΋ͭΕͷ‫ૅج‬ྔࢠ΋ͭΕͱ͸ɼෳ਺ͷཻࢠ·ͨ͸ঢ়ଶ͕ؒྔࢠ
ྗֶతͳ૬ؔΛ΋ͭ৔߹ʹ༻͍ΒΕΔ֓೦Ͱ͋Δɽྔࢠ΋ͭΕঢ়ଶ͕‫ݱ‬ΕΔ‫ͯ͠ͱܥ‬͸ɼޫࢠͷภޫ΍ిࢠͷεϐϯͷΑ͏ͳ̎४Ґ‫ͳ͢ܥ‬ΘͪྔࢠϏοτ΍ɼ(38)

Ͱ͋Δɽ͜ͷ͏ͪɼྫ͑͹ยํͷྔࢠϏοτ͕ 0 ͷͱ͖ଞํ΋ 0ɼยํ͕ 1 ͷͱ͖ଞํ΋ 1 ͱͳΔΑ͏ͳ૬ؔΛ࣋ͪɼ֤ʑͷ‫ݱ‬ΕΔ֬཰͕ 1/2 Ͱ͋ΔॏͶ߹Θͤঢ়ଶ͸

এͨ͠ EPR ͷύϥυοΫε [40] ʹ‫ݱ‬ΕΔΑ͏ͳɼҐ1|Φ+  = √ (|00 + |11)2ஔ͓Αͼӡಈྔɼ͋Δ͍͸௚ަҐ૬ৼ෯ͷΑ͏ͳ࿈ͱॻ͘͜ͱ͕Ͱ͖Δɽ͜͜Ͱɼ(35) ͓Αͼ (36) Λ༻ଓྔͷ‫Ͱ·ܥ‬ɼ༷ʑͳλΠϓ͕ߟ͑ΒΕΔɽ·ͨɼྔ͍ͯɼ(39) Λ |+ ͓Αͼ |− ͷ‫ج‬ఈͰද͢ͱ

཭ࢄతଟ४Ґ‫ܥ‬ɼ͞Βʹ͸ΞΠϯγϡλΠϯΒ͕ఏࢠ΋ͭΕΛ‫ڞ‬༗͢Δ‫ܥ‬ͷ਺΋ 2 ʹ‫ݶ‬ΒͣɼΑΓେ͖ͳ‫͚͓ʹܥ‬Δྔࢠ΋ͭΕঢ়ଶ΋ଘࡏ͢Δ͕ɼ͜͜Ͱ
͸࠷΋؆୯ͳ 2 ྔࢠϏοτ‫ܥ‬ʢ2 ‫ݸ‬ͷೋ४Ґ‫ܥ‬ʣʹ͓

1|Φ+  = √ (| + + + | − −)
2

(39)

(40)

͚Δྔࢠ΋ͭΕʹ‫ͯͬݶ‬࿩ΛਐΊΔɽ‫ʹޙ‬ड़΂ΔΑΛಘΔɽ͢ͳΘͪɼ(39) ͷঢ়ଶʹର֤ͯ͠ʑͷྔࢠ͏ʹɼྫ͑͹૬ؔޫࢠରΛߏ੒͢ΔޫࢠͷภޫΛ༻ϏοτΛ |+ ͓Αͼ |− ͷ‫ج‬ఈͰ‫؍‬ଌͨ͠ࡍʹ΋‫׬‬͍ͯ͜ͷΑ͏ͳྔࢠ΋ͭΕΛੜ੒ɾอ࣋ͤ͞Δ͜ͱ

શͳ૬͕ؔ‫؍‬ଌ͞ΕΔɽ(39) ͷΑ͏ʹɼෳ਺ͷ‫ܥ‬ͷঢ়͕Ͱ͖Δɽଶؒʹྔࢠత૬ؔΛ༗͢Δঢ়ଶΛɼྔࢠ΋ͭΕঢ়ଶ͍·ɼ͋Δ཭ࢄతೋ४Ґ‫͚͓ʹܥ‬Δ‫ج‬ఈϕΫτϧ9

Λ |0 ͓Αͼ |1 ͱ͢Δ ɽྔࢠϏοτͷঢ়ଶϕΫτϧ͸ͦΕΒͷॏͶ߹Θͤ

(entangled state) ͱ͍͏ɽྔࢠ΋ͭΕঢ়ଶͰ͸ɼྔࢠϏοτͷಛఆͷ‫ج‬ఈͰͷ‫؍‬ଌ஋͚ͩͰͳ͘ɼॏͶ߹Θͤঢ়ଶΛ‫ج‬ఈͱͨ͠‫؍‬ଌ஋ʹؔͯ͠΋૬ؔΛ༗͢Δ͜ͱ͕ಛ௃Ͱ͋Δɽ|ψ = α|0 + β|1(34)Ͱද͢͜ͱ͕Ͱ͖Δɽα ͓Αͼ β ͸ෳૉ਺Ͱɼ|α|2 +|β|2 = 1 Λຬͨ͢ɽॏͶ߹Θͤঢ়ଶͷྫͱͯ͠ɼ
1|+ = √ (|0 + |1) ,2
1|− = √ (|0 − |1)
2

(35)
(36)

10

Λ‫ ͓ͯ͛͘ڍ‬ɽ࣍ʹɼ̎ͭͷྔࢠϏοτͷঢ়ଶΛߟ͑Α͏ɽҰํʢྔࢠϏοτ 1ʣͷঢ়ଶΛ |a1 (a = 0, 1)ɼଞํʢྔ
9 ޫࢠͷภޫͷ৔߹ɼྫ͑͹ 2 ͭͷ௚ަ͢Δ௚ઢภޫঢ়ଶ |H,|V Λ‫ج‬ఈʹͱΔɽ
10 ޫࢠͷภޫͷ৔߹ɼ͜ΕΒ͸ɼ±45◦ ͷ௚ઢภޫঢ়ଶʹ͋ͨΔɽ(39) ʹ୅ද͞ΕΔΑ͏ʹɼ2 ྔࢠϏοτ‫͍͓ʹܥ‬ͯɼ‫׬‬શͳྔࢠ΋ͭΕΛ༗͠ɼ‫ʹ͍ޓ‬௚ަ͢Δ 4 ͭͷঢ়ଶ1|Φ±  = √ (|00 ± |11)2
1|Ψ±  = √ (|01 ± |10)
2

(41)
(42)

Λϕϧঢ়ଶ (Bell states) ·ͨ͸ϕϧ‫ج‬ఈ (Bell bases)11ͱ‫Ϳݺ‬ɽϕϧঢ়ଶͰද͞ΕΔΑ͏ͳঢ়ଶ͸ɼ࠷΋ߴ͍ఔ౓ͷྔࢠ΋ͭΕΛ΋ͭঢ়ଶͰɼ࠷େྔࢠ΋ͭΕঢ়ଶ (maximally entangled state) ͱ‫ݺ‬͹ΕΔɽ͜Εʹର͠ɼྔࢠϏοτΛ͋Β͔͡Ί |00 ·ͨ͸|11 ͷঢ়ଶʹ४උ͠ɼͦΕΒΛ 1/2 ͷ֬཰Ͱϥϯμ
11 2 ྔࢠϏοτ‫ܥ‬ͷ೚ҙͷঢ়ଶϕΫτϧ͸ɼ|00, |01, |10,|11 ͷ୅ΘΓʹϕϧ‫ج‬ఈͷઢ‫ͱ͘͜ॻͰ߹݁ܗ‬΋Ͱ͖Δɽ

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ϜʹૹΓग़ͨ͠৔߹͸Ͳ͏ͩΖ͏͔ʁ͜Ε͸ɼίΠঢ়ଶͰ͸‫ܥ‬ͷίώʔϨϯε͕ࣦΘΕͨ݁ՌɼྔࢠϏοϯͷදཪͷ৔߹ͷΑ͏ͳɼ‫ݹ‬యత૬ؔঢ়ଶʹ͋ͨΔɽ τΛ |+ ͓Αͼ |− ͷ‫ج‬ఈͰ‫؍‬ଌͨ͠ࡍʹ͸૬͕ؔ͜ͷͱ͖ͷશ‫ܥ‬ͷঢ়ଶ͸ɼ|00 ͱ |11 ͷࠞ߹ঢ়ଶͱ

‫؍‬ଌ͞Εͳ͍ͷͰ͋ΔɽͳΓɼ(39) ͷΑ͏ͳॏͶ߹Θͤঢ়ଶͰද͢͜ͱ͸Ͱ࣍ʹɼྔࢠ΋ͭΕঢ়ଶʹ͋Δ̎ྔࢠϏοτͷ͏͖ͪͳ͍ɽ͜ͷ৔߹ɼ֤ʑͷྔࢠϏοτΛ |0 ͱ |1 ͷͷɼยํͷྔࢠϏοτͷঢ়ଶʹ͍ͭͯߟ͑Α͏ɽҰൠ‫ج‬ఈͰ‫؍‬ଌͨ͠৔߹ʹ͸྆ऀͷؒʹ‫׬‬શͳ૬͕ؔ͋

ʹɼ͋Δ෦෼‫ܥ‬ʢ‫ ܥ‬1ʣͷີ౓ߦྻ ρ1 ͸ɼશମ‫ܥ‬ͷ

Δ͕ɼ|+ ͱ |− ͷ‫ج‬ఈͰ‫؍‬ଌ͢Δͱɼ݁Ռ͕ + ʹີ౓ߦྻʹରͯ͠ɼ‫ ܥ‬1 Ҏ֎ʢ‫ ܥ‬2ʣͷ෦෼ʹ͍ͭͯͳΔ͔ − ʹͳΔ͔͸֤ʑͷྔࢠϏοτͰશ͘ಠཱͳ

෦෼τϨʔεΛߦ͏͜ͱͰ‫ٻ‬ΊΒΕΔɽྫ͑͹ɼશ֬཰աఔͱͳΓɼͦΕΒͷ‫؍‬ଌ஋ͷؒʹ૬ؔ͸ͳ͘ମ‫͕ܥ‬ϕϧঢ়ଶ |Ψ+  Ͱ͋Δͱ͖ͷີ౓ߦྻ͸ (43)ͳͬͯ͠·͏ɽ͜Ε͸ɼ্ड़ͨ͠ྔࢠ΋ͭΕঢ়ଶͱͰද͞ΕΔ͔Βɼ͸େ͖͘ҟͳΔ఺Ͱ͋Δɽ͜ͷΑ͏ͳঢ়ଶΛ΋‫ؚ‬ΊͨҰൠతͳྔࢠঢ়ଶΛදͨ͢Ίʹ͸ɼີ౓ߦྻΛ༻͍Δɽྫ͑͹ɼ‫͕ܥ‬ঢ়ଶϕΫτϧ (39) Ͱද͞ΕΔ७ਮঢ়ଶʹ͋Ε͹ɼͦͷີ౓ߦྻ ρ ͸ρ = |Φ+ Φ+ |1= (|0000| + |0011| + |1100| + |1111|)
2⎛⎞
1 0 0 1⎜⎟
1 ⎜0 0 0 0⎟
⎟
= ⎜
(43)
⎟
2⎜
⎝0 0 0 0⎠
1 0 0 1

ρ1 = Tr(2) ρ= 0| |Φ+ Φ+ | |02 + 1| |Φ+ Φ+ | |12
1
= (|00|1 + |11|1 )
2

1 1 0
=
(46)
2 0 1
ͱͳͬͯɼྔࢠϏοτ 1 ͷঢ়ଶ͸ |0 ͱ |1 ͷ‫׬‬શࠞ߹ঢ়ଶͱͳΔ͜ͱ͕Θ͔ΔɽྔࢠϏοτ 2 ʹ͍ͭͯ΋ಉ༷Ͱ͋Δɽ͜ͷΑ͏ʹɼ2 ྔࢠϏοτ‫࠷͕ܥ‬େྔࢠ΋ͭΕঢ়ଶʹ͋Δͱ͖ɼ֤ʑͷྔࢠϏοτͷঢ়ଶ͸‫׬‬શࠞ߹ঢ়ଶͱͳΔ͜ͱ͕Θ͔Δɽ࠷େྔࢠ΋ͭΕঢ়ଶ͸ 2 ঢ়ଶؒʹίώʔϨϯεΛ༗͢Δ७ਮঢ়ଶͱද͞ΕΔɽີ౓ߦྻͷର֯ཁૉ͸ɼ‫֤͕ܥ‬ʑͷ‫ج‬Ͱ͋Δ͕ɼͦͷίώʔϨϯε͸ 2 ͭͷྔࢠϏοτͰఈϕΫτϧ ͷঢ়ଶʹ‫ݟ‬ग़͞ΕΔ֬཰Λද͠ɼඇର֯

‫ڞ‬༗͞Ε͓ͯΓɼ‫ݸ‬ʑͷྔࢠϏοτͷঢ়ଶ͸ίώʔཁૉͷେ͖͞͸ɼͦΕΒͷؒͷίώʔϨϯεΛද͠ϨϯεΛ΋ͨͳ͍‫׬‬શࠞ߹ঢ়ଶͱͳΔͷͰ͋Δɽ

͍ͯΔɽ(43) ʹ͓͍ͯ͸ɼର֯ཁૉͱඇର֯ཁૉͷઈର஋͕౳͘͠ɼ‫࠷͕ܥ‬େͷίώʔϨϯεΛอ͍ͬͯΔ͜ͱΛ͍ࣔͯ͠Δɽ͜Εʹର͠ɼ্Ͱྫࣔͨ͠
Α͏ͳ |00 ͱ |11 ͷ 1 : 1 ͷࠞ߹ঢ়ଶͷີ౓ߦྻ͸

1(|0000| + |1111|)
2⎛⎞
1 0 0 0⎜⎟
1 ⎜0 0 0 0⎟
⎟
= ⎜
⎟
2⎜
⎝0 0 0 0⎠
0 0 0 1ࠓ೔ɼզʑ͸ྔࢠ΋ͭΕΛ༗͢ΔޫࢠΛ༷ʑͳํ๏
ͦΕΒͷதͰ΋࠷΋‫ج‬ຊతͳɼภޫʹؔ͢Δྔࢠ΋ͭΕޫࢠରͷൃੜํ๏ʹ͍ͭͯड़΂ΔɽҎԼͰ͸ɼޫࢠ(44)ͷภޫঢ়ଶΛද͢‫ͯ͠ͱ߸ه‬ɼ|H, |V, |L, |R ౳Λ༻͍Δɽ͜͜Ͱɼ|H ͓Αͼ |V ͸֤ʑਫฏ͓Αͼਨ௚ͷ௚ઢภޫঢ়ଶɼ|L ͓Αͼ |R ͸֤ʑࠨճΓ͓Αͼͱͳͬͯɼඇର֯ཁૉ͕ 0 ͱͳͬͯ͠·͏ɽ͢ͳΘͪɼࠞ߹ঢ়ଶͰ͸‫ܥ‬ͷίώʔϨϯε͕ࣦΘΕ͍ͯΔɽ
͜ͷͱ͖ɼ‫ܥ‬Λ | + + ʹ‫ݟ‬ग़֬͢཰ P (++) ͸14ྔࢠ΋ͭΕޫࢠରͷൃੜͰൃੜͤ͞Δ͜ͱ͕Ͱ͖ΔΑ͏ʹͳͬͨɽ͜͜Ͱ͸ɼρ=

P (++) = + + |ρ| + + =

3.2

(45)

ͱ‫ٻ‬ΊΒΕΔɽಉ༷ʹͯ͠ɼP (++) = P (−−) =ӈճΓͷԁภޫঢ়ଶΛද͢ɽ·ͨɼޫࢠ 1 ͓Αͼޫࢠ 2
͔ΒͳΔ 2 ޫࢠͷภޫঢ়ଶΛ |H1 ⊗ |V2 ≡ |H1 |V2

≡ |HV ͳͲͱද͢ɽ

‫͔ࢠݪ‬ΒͷΧεέʔυޫ์ग़ਓྨ͕ॳΊͯྔࢠ΋ͭΕͷଘࡏΛ֬ೝ͢ΔʹࢸͬP (+−) = P (−+) = 1/4 ͱͳΔ͜ͱ͕Θ͔Δɽࠞ߹ ͨͷ͸ɼ1980 ೥୅ͷ͜ͱͰ͋ΔɽͦΕ͸ɼCa ‫͔ࢠݪ‬

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ੜ͡Δɽಛʹภޫʹؔͯ͠͸ɼ์ग़͞ΕΔ 2 ޫࢠͷ

40Ca
4p2 1S0(J=0)

551.3 nm

J=0
J=1
m=-1

|R〉1

|L〉1

ภޫ͕ฏߦͱͳΔ৔߹ʢtype-I ͷҐ૬੔߹ʣɼ͓Αͼ

2 ޫࢠͷภޫ͕‫ʹ͍ޓ‬ਨ௚ͱͳΔ৔߹ʢtype-II ͷҐ
|R〉2
|L〉2
૬੔߹ʣ͕ଘࡏ͠ɼͦΕΒΛ͏·͘༻͍Δ͜ͱʹΑͬ
422.7 nm4s S
J=0ͯภޫʹؔ͢Δྔࢠ΋ͭΕΛ࡞Γग़͢͜ͱ͕ՄೳͰ
(J=0)
͋Δɽྫ͑͹ɼtype-I ͷҐ૬੔߹Λ༻͍ͯɼ|HH ͷਤ 15: Ca ‫ࢠݪ‬ͷΧεέʔυભҠʹΑΔྔࢠ΋ͭΕ ภޫΛ΋ͭޫࢠରΛੜ੒͢Δաఔͱ |VV ͷภޫΛ΋ͭޫࢠରΛੜ੒͢ΔաఔͱΛԿΒ͔ͷํ๏Ͱίώʔޫࢠର์ग़ͷ֓೦ਤɽϨϯτʹॏͶ߹ΘͤΔͱɼ
1 ΒΧεέʔυ์ग़͞ΕΔޫࢠରΛ༻͍࣮ͯ‫͞ݱ‬Εͨ|Ψ = √ |HH + eiθ |VV
(48)
40
2
[41, 42]ɽਤ 15 ʹࣔ͢Α͏ʹɼྭ‫͞ى‬Εͨ Ca ‫ࢠݪ‬ͷ
3 ४ҐؒͷΧεέʔυભҠʢ4p2 1 S0 → 4s4p 1 P1 → ͷΑ͏ͳྔࢠ΋ͭΕঢ়ଶΛੜ੒͢Δ͜ͱ͕Ͱ͖Δɽ͜
4s2 1 S0 ʣʹؔΘΔ֯ӡಈྔมԽʢJ = 0 → 1 → 0ʣ ͜Ͱ θ ͸ɼ֤ʑͷաఔͰੜ੒͞Εͨঢ়ଶؒͷҐ૬ࠩΛ൓өͯ͠ɼ์ग़͞ΕΔ̎ޫࢠͷؒʹภޫʹؔ͢Δ Ͱ͋Δɽಉ༷ʹɼtype-II ͷҐ૬੔߹Λ༻͍ͯɼྔࢠ΋ͭΕ͕ੜ͡Δ12 ɽ͜ͷͱ͖ɼ‫์ʹ޲ํٯʹ͍ޓ‬1 |Ψ = √ |HV + eiθ |VH
(49)
ग़͞ΕΔޫࢠରΛ‫ݕ‬ग़͢ΔͱɼͦΕΒͷภޫঢ়ଶ͸2
14s4p P1

(J=1)
21

=

0

+1

+

0

1
|Ψ = √ (|LL + |RR)2
1= √ (|HH − |VV)2

Λੜ੒͢Δ͜ͱ΋Ͱ͖Δɽ͜ΕΒͷํ๏Ͱॏཁͳ͜
ͱ͸ɼ(48) ·ͨ͸ (49) ͷୈ 1 ߲ͱୈ 2 ߲ͷঢ়ଶͷؒ

(47)

ʹ͸ɼภޫҎ֎ʢྫ͑͹೾௕΍࣌ؒɼޫ࿏ͳͲʣʹ͸ҧ͍͕ͳ͍ʢ‫ݟ‬෼͚͕͔ͭͳ͍ʣΑ͏ʹ͢Δ͜ͱ͢ͳΘͪϕϧঢ়ଶͱͳΔɽAspect Β͸͜ͷޫ‫ݯ‬Λ༻Ͱ͋Δɽ͞΋ͳ͚Ε͹ɼ2 ͭͷঢ়ଶͷؒͷίώʔϨϯ

͍ͯ 2 ޫࢠͷภޫ૬ؔͷଌఆΛߦ͍ɼϕϧͷෆ౳ࣜε͕ࣦΘΕɼੜ੒͞ΕΔঢ়ଶ͸ॏͶ߹Θͤঢ়ଶͰ͸

ʢCHSH ෆ౳ࣜʣ ͕໌Β͔ʹഁΕ͍ͯΔ͜ͱΛॳΊͳ͘౷‫ܭ‬తࠞ߹ঢ়ଶͱͳͬͯ͠·͏͔ΒͰ͋ΔɽҎͯࣔͨ͠ͷͰ͋Δ [42]ɽ͜ͷํ๏Λ༻͍ͨྔࢠ΋ͭΕԼͰ͸ɼSPDC Λ༻͍ͨภޫʹؔ͢Δྔࢠ΋ͭΕͷޫࢠͷൃੜ͸ύΠΦχΞతͳ΋ͷͰ͸͕͋ͬͨɼͦ

ੜ੒ʹؔͯ͠ɼ͍͔ͭ͘ͷ۩ମతํ๏ʹ͍ͭͯ঺հ

ͷޮ཰΍೾௕Մมੑͷ໰୊Ͱࠓ೔Ͱ͸͋·ΓҰൠత

͢Δɽ

13

ʹ͸༻͍ΒΕͳ͘ͳͬͨɽ

·ͣɼSPDC ʹΑͬͯੜ੒͞ΕͨޫࢠରΛར༻ͯ͠ภޫͷྔࢠ΋ͭΕΛ࠷ॳʹ࡞Γग़࣮ͨ͠‫[ ݧ‬43, 44]Λ঺հ͢Δɽਤ 16 ͷΑ͏ʹɼType-I Ґ૬੔߹ʹΑΓɼύϥϝτϦοΫԼํม‫׵‬ࠓ೔ɼྔࢠ΋ͭΕޫࢠରͷൃੜํ๏ͱͯ͠࠷΋සൟʹ༻͍ΒΕΔํ๏͸ɼ2.2 અͰड़΂ͨࣗൃύϥϝτಉ͡೾௕Ͱਫฏ (|H) ภޫΛ΋ͭޫࢠରΛൃੜ͠ɼҰํͷޫࢠͷޫ࿏Λ 1ɼଞํͷޫ࿏Λ 2 ͱ͢Δɽͦ͠
ͯɼޫ࿏ 2 ͷޫࢠΛɼ൒೾௕൘ (HWP) Λ௨ͯ͠ਨ௚

(|V) ภޫʹ͢Δɽ͢ͳΘͪɼಉ͡೾௕ͷ |H ภޫͱ|V ภޫͷ 2 ޫࢠঢ়ଶ |H1 |V2 Λੜ੒͢Δ14 ɽͦ࣍͠ͷඇઢ‫ײܗ‬ड཰ (χ(2) ) Λ༗͢Δඇઢ‫ֶ݁ޫܗ‬থʹͯɼ֤ʑͷޫࢠΛɼϏʔϜεϓϦολ (BS) ͷ 2 ͭͷΑͬͯɼೖࣹޫࢠ 1 ‫ ͕ݸ‬2 ‫ݸ‬ͷޫࢠʹม‫͞׵‬ΕΔɽ͜ೖࣹϙʔτ͔Βޫ࿏͕ॏͳΔΑ͏ʹೖࣹ͢Δɽ͜ͷͷͱ͖ɼΤωϧΪʔ΍೾਺ϕΫτϧͷอଘʢҐ૬੔഑ஔ͸ɼ2.3 અͰड़΂ͨ Hong-Ou-Mandel ‫ׯ‬বͱಉ߹৚݅ʣ͕ཁ੥͞ΕΔ͜ͱʹΑͬͯɼൃੜͨ͠ޫࢠͷ
༷Ͱ͋Δ͕ɼޫͷঢ়ଶͱͯ͠ภޫͷࣗ༝౓΋ߟྀؒ͠ʹภޫɼΤωϧΪʔɼ࣌ؒɼ͓Αͼۭؒతͳ૬͕ؔ
ͨऔΓѻ͍͕ඞཁͰ͋ΔɽBS ΁ͷೖࣹޫͷঢ়ଶΛɼ
12 ૒‫ࢠۃ‬ભҠʹΑΔޫࢠ์ग़աఔͰ͸ɼભҠʹؔΘΔిࢠͷ‫ي‬ภޫͱޫ࿏ͷࣗ༝౓Λߟྀͨ͠ද‫͖ॻͰݱ‬௚͢ͱಓ֯ӡಈྔͷมԽͱޫࢠͷภޫͱ͕ରԠ͢Δɽ
ϦοΫԼํม‫׵‬ʢSPDCʣͰ͋Δɽ͜ͷաఔͰ͸ɼ2

13 ‫ॴہ‬త࣮ࡏੑͱ‫ݺ‬͹ΕΔԾఆΛͨ͠ͱ͖ɼ཭Εͨ஍఺ʹ͋Δ
2 ͭͷ‫ʹܥ‬ରͯ͠ɼඇՄ‫ͳ׵‬෺ཧྔͷଌఆ஋ͷؒͷ૬͕ؔຬͨ͢
΂͖ෆ౳ࣜɽJ.S. Bell ʹΑͬͯఏএ͞Εͨɽిࢠεϐϯ΍ޫࢠͷภޫͳͲͷྔࢠϏοτ‫ʹܥ‬ରͯ͠ద༻͞ΕΔϕϧͷෆ౳ࣜͷҰ‫ܗ‬͕ࣜ CHSH ෆ౳ࣜͰ͋Δɽྔࢠ࿦ʹैͬͯྔࢠ΋ͭΕ͕ଘࡏ͢Δͱ͖ɼ͜ΕΒͷෆ౳ࣜ͸ഁΕΔ͜ͱ͕஌ΒΕ͍ͯΔɽ|Ψ = |H1 |V2 = |101 |012 = â†1H â†2V |0(50)

14 ‫ݪ‬ཧతʹ͸ɼಉ͡೾௕ʢεϖΫτϧʣΛ΋ͭޫࢠରͰ͋Ε͹ɼ
type-II ͷҐ૬੔߹Λ༻͍ͯ |H ภޫͱ |V ภޫͷ 2 ޫࢠΛ௚઀ੜ੒ͯ͠΋ྑ͍ɽ

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物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

HWP

|V〉2

|V〉

1

2
|H〉1|V〉2+|V〉1|H〉2

BS
pump
SPDC

|H〉1|V〉2+|V〉1|H〉2
1

|H〉1

2

|H〉

ਤ 16: ύϥϝτϦοΫԼํม‫ͱ׵‬ϏʔϜεϓϦολ

ਤ 17: Type-II Ґ૬੔߹ύϥϝτϦοΫԼํม‫׵‬ΛΛ༻͍ͨྔࢠ΋ͭΕޫࢠରͷੜ੒ɽType-I Ґ૬੔߹༻͍ͨྔࢠ΋ͭΕޫࢠରͷੜ੒ɽType-II Ґ૬੔߹ʹʹΑΓɼಉ͡೾௕Ͱਫฏ (|H) ภޫΛ΋ͭޫࢠର͕ൃ

ΑΓɼਫฏ (|H) ͓Αͼਨ௚ (|V) ภޫΛ΋ͭޫࢠ͕

ੜ͢ΔɽͦͷҰํΛ൒೾௕൘ (HWP) Λ௨ͯ͠ਨ௚
(|V) ภޫʹ͢Δɽ֤ʑͷޫࢠΛɼϏʔϜεϓϦολ

֤ʑԁਲ਼ͷࣼ໘ʹԊͬͨํ޲ʹൃੜ͢Δɽೋͭͷԁਲ਼͕ަࠩ͢Δ 2 ํ޲ʹ์ग़͞ΕΔޫࢠର͸ɼภޫʹ(BS) ͷ 2 ͭͷೖࣹϙʔτ͔Βɼޫ࿏͕ॏͳΔΑ͏ʹ ؔ͢Δྔࢠ΋ͭΕঢ়ଶͱͳΔɽ
ೖࣹ͢Δɽ2 ޫࢠ͕ BS Λ‫ʹڞ‬ಁա͢Δ͔‫ʹڞ‬൓ࣹ͢
Δͱ͖ɼޫࢠ͸ BS ͷ 2 ͭͷग़ࣹϙʔτͷ֤ʑʹ 1 ޫ ͖ͷग़ྗঢ়ଶ͸ɼࢠͣͭग़ࣹ͞Εɼͦͷภޫঢ়ଶ͸ྔࢠ΋ͭΕঢ়ଶʹ
1|Ψ  = √ (|101 |012 + |011 |102 )ͳΔɽ
2
1= √ (|H1 |V2 + |V1 |H2 )2ͱද͢͜ͱ͕Ͱ͖Δɽ͜͜Ͱɼ|nH nV i ͸ɼޫ࿏ i(=1,12) ʹ͓͍ͯɼH ภޫͷޫࢠ͕ nH ‫ݸ‬ɼV ภޫͷޫࢠ͕(52)= √ (|HV + |VH) = |Ψ+ 
2nV ‫ݸ‬ͷ‫਺ݸ‬ঢ়ଶΛද͠ɼ|0 ≡ |001 |002 ͸ਅۭঢ়
ଶͰ͋ΔɽBS ʹΑΔԋࢉࢠͷม‫ࣜ׵‬Λ༻͍ͯɼ(50) ͱͳΔɽ͜͜Ͱɼ(51) ͷશମʹ͔͔ΔҐ૬Ҽࢠ͸আ֎͠ɼબ୒ͨ͠ঢ়ଶͷΈͰ‫֨ن‬Խͨ͠ɽ(52) ͸ɼϕΛ BS ͷग़ྗଆͷঢ়ଶʹม‫͢׵‬Δͱɼϧঢ়ଶ |Ψ+  ͢ͳΘͪྔࢠ΋ͭΕঢ়ଶͰ͋Δ16 ɽ͜ͷeiγ  ††††|Ψ  =â1 H − iâ2 H iâ1 V + â2 V |0ํ๏Ͱ͸ɼSPDC Ͱੜ੒͞Εͨঢ়ଶʹରͯ͠ɼϏʔ
2ϜεϓϦολͱࣄ‫ޙ‬બ୒Λ༻͍ͨૢ࡞Λࢪ͠ɼॴ๬
eiγ  † †=â1 H â2 V + â†1 V â†2 H
ͷ 2 ঢ়ଶʢ|HV ͱ |VHʣͷॏͶ߹Θͤঢ়ଶΛ࡞Γ
2

†
†
†
†
ग़ͨ͜͠ͱʹͳΔɽ
+iâ1 H â1 V − iâ2 H â2 V |0࣍ʹɼਤ 17 ʹࣔ͢Α͏ʹɼtype-II ͷҐ૬੔߹Λeiγ(|101 |012 + |011 |102=༻͍ͨํ๏Λ঺հ͢ΔɽҐ૬੔߹৚݅ʹΑΓɼ͜Ε2+i|111 |002 − i|001 |112 )(51) Βͷޫࢠ͸ 2 ͭͷԁਲ਼ͷࣼ໘ʹԊͬͨํ޲ʹൃੜ͢
Δ͕ɼ2 ͭͷԁਲ਼͕ަࠩ͢Δ 2 ํ޲ʹ์ग़͞ΕΔޫ
ͱͳΔ͜ͱ͕Θ͔Δɽ(51) ͷ࠷ॳͷ 2 ߲͸ޫ࿏ A ͱ ࢠରͷภޫঢ়ଶ͸ |HV ͱ |VH ͕౳͍͠ৼ෯ͰॏͶޫ࿏ B ʹ 1 ޫࢠͣͭଘࡏ͢Δঢ়ଶɼ࠷‫ޙ‬ͷ 2 ߲͸ಉ ߹Θ͞Εͨঢ়ଶ͡ޫ࿏ʹ 2 ޫࢠ͕ଘࡏ͢Δঢ়ଶͰ͋ΔɽޫࢠΛ‫؍‬ଌ
1 |ψ ≡ √ |HV + eiθ |VH
(53)
͢Δࡍʹɼޫ࿏ 1 ͱޫ࿏ 2 ͷ֤ʑʹ 1 ޫࢠͣͭ‫ݕ‬ग़
2͞Εͨ৔߹ͷΈΛબ୒͢Δ΋ͷͱ͢Ε͹15 ɼͦͷͱ͢ͳΘͪྔࢠ΋ͭΕঢ়ଶͱͳΔɽ͜͜ͰɼҐ૬ θ ͸ྫ͑͹ยํͷύεʹෳ۶ંҐ૬൘Λஔ੍͍ͯ‫͢ޚ‬Δ͜ͱ͕Ͱ͖Δɽ͜ͷํࣜͷޫ‫ݯ‬͸ɼ‫͔ࢠݪ‬ΒͷΧεέʔυ์ग़ޫʹൺ΂ܻͯҧ͍ʹߴޮ཰Ͱɼఔͳ͘͠
ͯੈքॳͷྔࢠςϨϙʔςʔγϣϯͷ࣮‫ݧ‬ʢ3.5 અࢀ15(51) ͷঢ়ଶʹ͍ͭͯɼޫ࿏ 1 ͱޫ࿏ 2 ͷ֤ʑʹ 1 ޫࢠͣͭ‫ݕ‬ग़͢Δ֬཰͸ 1/2 Ͱ͋Δɽ͜ͷΑ͏ʹɼ֬཰తʹ‫ݱ‬ΕΔࣄ৅ͷ͏ͪಛఆͷ৔߹ͷΈΛબ୒͢Δ͜ͱΛࣄ‫ޙ‬બ୒ (post-selection)ͱ͍͏ɽ

16 ޫ࿏ 1 ͱޫ࿏ 2 ͱͰӈख‫ࠨͱܥ‬ख‫͕ܥ‬൓స͢Δ࠲ඪΛ࠾༻ͨ͠৔߹ʹ͸ɼग़ྗঢ়ଶ͸ |Ψ−  ͱද͞ΕΔʢ࠲ඪ‫ܥ‬ͷҧ͍ʹ൐ͬͯද‫͕ݱ‬ม‫͞׵‬ΕΔ͚ͩͰ෺ཧత࣮ମ͕ҟͳΔΘ͚Ͱ͸ͳ͍ʣɽ

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ͱ͕Θ͔Δɽ͜͜Ͱॏཁͳ఺͸ɼ(54) ͷୈ 1 ߲ͱୈਤ 18: 2 प‫ظ‬௚ྻٙࣅҐ૬੔߹ύϥϝτϦοΫԼํม‫׵‬Λ༻͍ͨྔࢠ΋ͭΕޫࢠରͷੜ੒ [47]ɽ
রʣʹ༻͍ΒΕΔͳͲɼը‫ظ‬తͳ΋ͷͰ͋ͬͨɽͦ
ͷ‫ޙ‬ɼ2 ຕͷ type-I Ґ૬੔߹ͷ݁থΛ૊Έ߹Θͤͨ͞Βʹߴޮ཰ͳํ๏ [45]ɼ‫ׯ‬ব‫ܭ‬ͷதʹԼํม‫݁׵‬থΛೖΕΔํ๏ [46] ͳͲ΋ߟҊ͞Ε͍ͯΔɽ͞Βʹۙ೥Ͱ͸ɼٙࣅҐ૬੔߹ͱ‫ݺ‬͹ΕΔ৽͍͠

2 ߲ͱͷؒͰภޫҎ֎ʹ͸ҧ͍͕ͳ͍Α͏ʹ͢Δ͜
ͱͰ͋Δ19 ɽ͜ͷํ๏͸ɼϙϯϓޫͱൃੜ͢Δޫࢠରͷ఻ൖํ޲͕શͯ౳͍͠ಉ࣠ߏ଄ͱͳ͍ͬͯΔ͜
ͱ͕ಛ௕Ͱ͋Γɼૉࢠߏ଄Λޫಋ೾࿏ͱ͢Δ͜ͱͰɼྔࢠ΋ͭΕޫࢠରͷੜ੒ޮ཰Λඈ༂తʹߴΊΔ͜ͱ͕ՄೳͰ͋ΔɽҎ্঺հͨ͠ྫͷΑ͏ʹɼύϥϝτϦοΫԼํม‫׵‬Λ༻͍Δͱྑ࣭ͳྔࢠ΋ͭΕޫࢠରΛൺֱత؆ศ
ʹൃੜͤ͞Δ͜ͱ͕Ͱ͖ΔͨΊɼ‫Ͱࡏݱ‬͸ྔࢠ৘ใ௨৴ʹؔ࿈༷ͨ͠ʑͳ‫ݪ‬ཧ‫͍༻͘޿ʹݧ࣮ূݕ‬ΒΕ͍ͯΔɽ࠷ۙͰ͸ɼޫಋ೾࿏΍ٙࣅҐ૬੔߹Λ༻͍ͨߴޮ཰ͳඇઢ‫ֶޫܗ‬ૉࢠ΋ೖखͰ͖ɼͦͷԠ༻ൣғ͕͞Βʹ޿͕͍ͬͯΔɽٕज़Λ༻͍ͨྔࢠ΋ͭΕޫࢠରͷੜ੒ํ๏΋‫͞ڀݚ‬Ε͍ͯΔɽٙࣅҐ૬੔߹ͱ͸ɼޫֶඇઢ‫݁ܗ‬থʹ͓͚Δඇઢ‫ײܗ‬ड཰ χ(2) ʹରͯ͠प‫ظ‬తͳมௐΛࢪ͢͜ͱͰɼҐ૬੔߹৚݅Λ੍‫͢ޚ‬Δٕज़Ͱ͋Δ17 ɽͦ

3 ࣍ͷඇઢ‫ܗ‬աఔΛ༻͍ͨํ๏
SPDC ͕ 2 ࣍ͷඇઢ‫ܗ‬ੑΛ༻͍ͨ೾௕ม‫׵‬աఔͰͷҰྫͱͯ͠ɼ2 छྨͷٙࣅҐ૬੔߹ૉࢠΛ༻͍Δ

͋ͬͨͷʹର͠ɼ3 ࣍ͷඇઢ‫ܗ‬ੑΛ༻͍ͨ 4 ޫ೾ύϥํ๏Λਤ 18 ʹࣔ͢ [47]ɽ͜ͷํ๏Ͱ͸ɼ1 ͭͷඇઢϝτϦοΫաఔʹΑͬͯ 2 ‫ݸ‬ͷೖࣹޫࢠ͔Β 2 ‫ݸ‬ͷ‫ֶ݁ޫܗ‬থʢχΦϒࢎϦν΢ϜɿLiNbO3 ʣதͰ 2 छग़ྗޫࢠʹม‫͢׵‬Δաఔ΋ߟ͑ΒΕΔɽҰൠʹ 3 ࣍ྨͷҟͳΔ෼‫ۃ‬൓సप‫ظ‬ʢඇઢ‫ײܗ‬ड཰ͷมௐप‫ظ‬ʣ ͷඇઢ‫ײܗ‬ड཰ (χ(3) ) ͷ஋͸খ͘͞ɼ࣮༻తͰ͸ͳΛ΋ͭྖҬʢA ͱ Bʣ͕௚ྻʹ઀ଓ͞ΕͨٙࣅҐ૬ ͍͕ɼ͏·͘෼ࢄΛ੍‫ͨ͠ޚ‬ϑΝΠόதͰ͸ɼίΞ੔߹ૉࢠΛ༻͍Δɽ͜ͷૉࢠͰ͸ɼtype-II ͷҐ૬੔‫ܘ‬ఔ౓ͷ‫͘ڧ͕ޫʹ಺ۭ͍ؒڱ‬ด͜͡ΊΒΕΔ͜ͱ߹ʹΑΓɼ‫ʹ͍ޓ‬ਨ௚ͳภޫΛ΋ͭޫࢠର͕ੜ੒͞

ͱඇৗʹ௕͍૬‫༻࡞ޓ‬௕ͱ͕૬·ͬͯɼ3 ࣍ͷඇઢ‫ܗ‬

ΕΔ͕ɼٙࣅҐ૬੔߹৚݅ͷҧ͍ʹΑΓɼྖҬ A ͱաఔ͕ޮ཰తʹ‫͜ى‬Δɽ·ͨɼྔࢠ΋ͭΕޫࢠͷޫྖҬ B ͱͰ͸ภޫํ޲ͱৼಈ਺ͷؔ܎͕ೖΕସΘͬ ࢠΤωϧΪʔ͕ྭ‫ͱޫى‬ಉఔ౓ͱͳΔͷ΋ಛ௕Ͱ͋ͨޫࢠର͕ൃੜ͢ΔΑ͏ઃ‫͞ܭ‬Ε͍ͯΔɽ͜ͷͱ͖ɼ Δɽ͞Βʹɼଟ͘ͷϑΝΠό༻ޫֶ෦඼͕ར༻Ͱ͖
ྖҬ A ʹ͓͍ͯ͸ɼৼಈ਺ ω1 Ͱ |H ภޫͷޫࢠ͕ɼ Δ௨৴೾௕ଳͰͷྔࢠ΋ͭΕޫࢠൃੜʹ༗རͰ͋Γɼৼಈ਺ ω2 Ͱ |V ภޫͷޫࢠ͕ൃੜ͢Δɽ͜ͷޫࢠྔࢠ҉߸ͳͲͷྔࢠ৘ใ௨৴ٕज़ͱ૬ੑ͕ྑ͍ɽ͜
ରͷঢ়ଶΛɼ|Hω1 |Vω2 ͱॻ͘͜ͱʹ͢Δɽಉ༷ʹɼ ͷํ๏Λ༻͍ͯɼ௨৴೾௕ଳͰͷภޫྔࢠ΋ͭΕޫྖҬ B ʹ͓͍ͯ͸ |Vω1 |Hω2 ͷޫࢠର͕ൃੜ͢Δɽ ࢠରͷൃੜ [48] ͕ใࠂ͞Ε͍ͯΔɽ࠷ۙͰ͸ɼSi ࡉ
͜ΕΒͷྖҬ͕௚ྻʹ઀ଓ͞Ε͍ͯΔͷͰɼޫࢠର

ઢಋ೾࿏ߏ଄Λ༻͍ͨ͞Βʹߴޮ཰ͳྔࢠ΋ͭΕޫ͸ྖҬ A ͔ྖҬ B ͔ͷ͍ͣΕ͔Ͱൃੜͨ͠ঢ়ଶͷॏࢠൃੜ΋ใࠂ͞Ε͍ͯΔɽͶ߹ΘͤͱͳΓɼ
1 |Ψ = √ |Hω1 |Vω2 + eiθ |Vω1 |Hω22൒ಋମΛ༻͍ͨํ๏

(54)ͱॻ͘͜ͱ͕Ͱ͖Δɽճં֨ࢠɼϓϦζϜɼ·ͨ͸্ड़ͨ͠ํ๏͸͍ͣΕ΋ɼޫྭ‫ʹى‬Αͬͯྔࢠ΋μΠΫϩΠοΫϛϥʔ18 ʹΑͬͯޫࢠΛͦͷৼಈ਺ͭΕޫࢠରΛൃੜͤ͞Δ΋ͷͰ͋ͬͨɽ͜ΕΒͷํʢω1 ·ͨ͸ ω2 ʣʹैͬͯ 2 ͭͷޫ࿏ʹ෼͚Ε͹ɼͦͷঢ়ଶ͸ภޫʹؔ͢Δྔࢠ΋ͭΕঢ়ଶ (49) ͱͳΔ͜
17 ඇઢ‫ײܗ‬ड཰Λมௐ͢Δํ๏͸͍͔ͭ͘ఏҊ͞Ε͍ͯΔ͕ɼ‫ڧ‬༠ిମʹରͯ͠प‫ظ‬తͳ෼‫ۃ‬൓సΛࢪ͢͜ͱ͕ओྲྀͷํ๏ͱͳͬ
͍ͯΔɽ
18 प೾਺ʢ೾௕ʣʹԠͯ͡ޫΛಁա·ͨ͸൓ࣹ͢Δޫֶૉࢠɽ

19 ૉࢠͷԹ౓Λௐ੔͢Δ͜ͱͰɼൃੜ͢Δޫࢠͷৼಈ਺ʢω ͱ1
ω2 ͷ֤ʑʣΛୈ 1 ߲ͱୈ 2 ߲ͱͰҰகͤ͞Δ͜ͱ͕Ͱ͖Δɽ·ͨɼඇઢ‫݁ܗ‬থͷෳ۶ંੑͱྖҬ A ͓ΑͼྖҬ B Ͱൃੜͨ͠ޫ
ࢠରͷ݁থͷ௨ա‫཭ڑ‬ͷҧ͍ʹ༝དྷͯ͠ୈ 1 ߲ͱୈ 2 ߲ͱͷؒʹࣝผੑ͕ੜ͡Δ͕ɼͦͷิঈ΋ՄೳͰ͋Δ [47]ɽ

《講義ノート》

物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

Biexciton

J=0

(J=0)
Exciton(J=1)
Ground
state

(J=0)

=

J=1
m=-1

|L〉1
0

+1
|R〉2

D1

|R〉1

+

|L〉2

ྜྷ᫤゛ᩐ

J=0

D2

QWP

POL

ਤ 19: ྭ‫ࢠى‬෼ࢠΛ༻͍ͨྔࢠ΋ͭΕޫࢠରͷੜ੒ɽਤ 20: ภޫʹؔ͢Δྔࢠ΋ͭΕͷଌఆ૷ஔͷ໛ࣜਤɽ๏ʹର͠ɼిྲྀྭ‫ى‬΋ՄೳͱͳΔ൒ಋମΛ༻͍ͨޫൃੜͨ͠ޫࢠରͷ֤ʑʹରͯ͠ 1/4 ೾௕൘ (QWP)‫ݯ‬ͷ։ൃ΋ਐΜͰ͍Δɽͱ‫( ࢠޫݕ‬POL) Λ༻͍ͯภޫࣹӨଌఆΛߦ͍ɼೋͭ

൒ಋମʹ͓͍ͯ୯Ұޫࢠͷ์ग़Λ୲͏ͷ͸ɼిࢠͱਖ਼޸ͱ͕ΫʔϩϯྗͰ݁߹ͨ͠ྭ‫( ࢠى‬exciton) ͱͷ‫ݕ‬ग़‫( ث‬D1 , D2 ) ͕ಉ࣌ʹޫࢠΛ‫ݕ‬ग़͢Δස౓Λଌఆ͢Δɽ‫ݺ‬͹ΕΔঢ়ଶͰ͋Δ͕ɼྭ‫ ͕ࢠى‬2 ‫ݸ‬ʢ͢ͳΘͪ 2 ૊ͷిࢠɾਖ਼޸ର͕ʣ݁߹ͨ͠ྭ‫ࢠى‬෼ࢠΛ༻͍Δͱɼ ΋͠Εͳ͍ɽલड़ͨ͠‫ࢠݪ‬Χεέʔυ์ग़ͱ΄΅ಉ͡‫ݪ‬ཧΛ༻͍

·ͨɼྔࢠυοτΛ༻͍ΔҎ֎ʹɼόϧΫ݁থதͷͯภޫʹؔ͢Δྔࢠ΋ͭΕޫࢠର͕ੜ੒Ͱ͖Δ [49]ɽ ྭ‫ࢠى‬෼ࢠΛ༻͍ͨྔࢠ΋ͭΕޫࢠͷൃੜ΋ՄೳͰ
ਤ 19 ʹࣔ͢Α͏ʹɼ͜ͷաఔͰ͸ిࢠঢ়ଶͷ֯ӡಈ

͋Δ [54, 55]ɽͦͷࡍɼ2 ‫ݸ‬ͷೖࣹޫࢠΛ༻͍ͯྭ‫ى‬ྔ͕ྭ‫ࢠى‬෼ࢠ (J = 0) ͔Βதؒঢ়ଶͰ͋Δྭ‫ࢠى‬ࢠ෼ࢠΛ௚઀ 2 ޫࢠྭ‫͢ى‬Δ͜ͱʹΑͬͯɼߴ͍ޮ

(J = 1) Λ‫ͯܦ‬ऴঢ়ଶͰ͋Δ‫ج‬ఈঢ়ଶ (J = 0) ΁ͱม ཰Ͱͷޫࢠରͷൃੜ͕ՄೳͰ͋Δɽ͜ͷաఔͰੜ੒
Խ͢Δɽैͬͯɼͦͷ֯ӡಈྔมԽʢJ = 0 → 1 → 0ʣ ͞ΕΔޫࢠର΋ɼྭ‫ࢠى‬෼ࢠͷ֯ӡಈྔΛ൓өͯ͠Λ൓өͯ͠ɼಉ͡ํ޲΁์ग़͞ΕΔ̎ޫࢠͷภޫঢ় (55) ࣜͰද͞ΕΔྔࢠ΋ͭΕঢ়ଶͱͳΔɽ͜ͷաఔ͸ྭ‫ࢠى‬෼ࢠ‫ڞ‬໐ϋΠύʔύϥϝτϦοΫࢄཚͱ‫ݺ‬ଶ͸͹Εɼχ(3) Λ༻͍ͨ 2 ޫࢠ͔Β 2 ޫࢠ΁ͷม‫͍ͱ׵‬1|Ψ = √ (|LR + |RL)͏఺Ͱ͸্ड़ͨ͠ϑΝΠόதͰͷ 4 ޫ೾ύϥϝτϦο
2
1
Ϋաఔͱྨࣅ͢Δ͕ɼੜ੒‫ݪ‬ཧ͔Βࣗવʹภޫͷྔ
(55)= √ (|HH + |VV)2ࢠ΋ͭΕ͕ಘΒΕΔ͜ͱͱɼྭ‫ࢠى‬΍ྭ‫ࢠى‬෼ࢠͱ
ͱͳΔ20 ɽ͔͠΋ɼ൒ಋମྔࢠυοτதͷྭ‫ࢠى‬෼ ͍ͬͨిࢠྭ‫ى‬ঢ়ଶͱͷ‫ڞ‬໐Λར༻͢Δ͜ͱʹΑͬࢠΛ༻͍Δ͜ͱͰɼ
ʮ୯Ұͷʯྔࢠ΋ͭΕޫࢠରΛੜ ͯඇৗʹߴ͍ޮ཰Ͱม‫͜ى͕׵‬Δͷ͕ಛ௕Ͱ͋Δɽ
੒͢Δ͜ͱ΋Ͱ͖Δɽ͔͠͠ɼ௨ৗͷํ๏Ͱ࡞੡͞
ΕΔྔࢠυοτͰ͸ͦͷ‫ܗ‬ঢ়ʹҟํੑ͕ੜ͡ɼH ภ
ޫͱ V ภޫͱͰ์ग़ޫࢠͷΤωϧΪʔ͕Θ͔ͣʹҟ

3.3

ྔࢠ΋ͭΕͷ‫؍‬ଌͳͬͯ͠·͏ͨΊʹɼ|HH ͱ |VV ͱͷؒͷίώʔ͜ͷΑ͏ʹͯ͠ൃੜͨ͠ޫࢠରͷྔࢠ΋ͭΕΛ‫؍‬Ϩϯε͕ࣦΘΕɼ྆ऀͷࠞ߹ঢ়ଶ (44) ʹͳͬͯ͠·ଌɼධՁ͢ΔͨΊʹ͸ɼண໨͍ͯ͠Δ෺ཧྔΛ 2 ޫ͏໰୊఺͕஌ΒΕ͍ͯͨ [50]ɽ࠷ۙɼ‫ܗ‬ঢ়ҟํੑΛࢠʹ͍ͭͯಉ࣌ʹ‫ܭ‬ଌ͠ɼͦͷ૬ؔΛղੳ͢Δඞཁ

খͨ͘͞͠ྔࢠυοτʹԣ࣓৔ΛҹՃͯ͠ H, V ੒෼

͕͋Δɽྫ͑͹ɼภޫʹؔ͢Δྔࢠ΋ͭΕͷ‫؍‬ଌͰͷεϖΫτϧΛҰகͤ͞Δํ๏ [51] ΍ɼ෼྾ͨ͠ε͸ɼਤ 20 ͷΑ͏ʹ֤ʑͷޫࢠʹؔͯ͠ 1/4 ೾௕൘ͱϖΫτϧͷதؒͷޫࢠΤωϧΪʔྖҬͷΈΛબ୒͠‫͔ࢠޫݕ‬ΒͳΔภޫࣹӨϑΟϧλΛ༻͍ͯ೚ҙͷ௚ͯ‫؍‬ଌ͢Δํ๏ [52] ౳ʹΑͬͯɼྔࢠ΋ͭΕΛ‫؍‬ଌઢภޫ͋Δ͍͸ԁภޫঢ়ଶ΁ͷࣹӨଌఆΛߦ͍ɼภͨ͠ྫ͕ใࠂ͞Ε͍ͯΔɽͦͯ͠࠷ۙɼిྲྀྭ‫ʹى‬ޫͷ૊Έ߹Θͤ͝ͱʹɼೋͭͷ‫ݕ‬ग़‫͕ث‬ಉ࣌ʹޫࢠ

ΑΔྔࢠ΋ͭΕޫࢠൃੜͷใࠂ͕ͳ͞Εͨ [53]ɽಈ

Λ‫ݕ‬ग़͢Δස౓Λଌఆ͢Δɽ

࡞Թ౓౳ͷ఺Ͱ·ͩ໰୊͕࢒͍ͬͯΔ͕ɼ͍ۙকདྷɼ

ྫͱͯ͠ɼ্ड़ͨ͠ྭ‫ࢠى‬෼ࢠ‫ڞ‬໐ϋΠύʔύϥྔࢠ΋ͭΕޫࢠରΛൃੜ͢Δ LED ͕࣮༻Խ͞ΕΔ͔

ϝτϦοΫࢄཚʹΑͬͯੜͨ͡ޫࢠରʹ͍ͭͯภޫ

20 (47) ࣜͱԁภޫ૬ؔͷ૊Έ߹Θ͕ͤҧ͏ͷ͸ɼ(47) ࣜͰ͸‫์ʹ޲ํٯ‬ग़͞ΕΔޫࢠରΛɼ(55) ࣜͰ͸ಉํ޲ʹ์ग़͞ΕΔޫ
ࢠରΛߟ͍͑ͯΔ͔ΒͰ͋Δɽ૬ؔΛ‫؍‬ଌͨ݁͠ՌΛਤ 21 ʹࣔ͢ [55]ɽਤ 21 ͷώετάϥϜʹ͓͍ͯɼ2 ޫࢠͷ࣌ؒࠩ (τ ) ͕ 0 ͷ෇ۙ《講義ノート》

20

物性研究・電子版 Vol. 10, No. 1, 101208(2022年3月号)

RR

VV

RL

VH

LR

HV

LL

HH

|HV, |VH, |VV ͷ‫ج‬ఈΛ༻͍ͯ
⎛
1 0 0
⎜
1 ⎜0 0 0
ρ = |ΨΨ| = ⎜2⎜⎝0 0 01 0 0

10

ྜྷ᫤゛ᩐ㻃(/300s)

0
20
10
0
20
10
0
20

(56)

ͱද͞ΕΔɽີ౓ߦྻΛղੳ͢Δ͜ͱʹΑΓɼྔࢠ΋ͭΕͷఔ౓ͱͦΕΛཚ͍ͯ͠Δ‫ݪ‬Ҽͷ‫ݟ‬౰Λ͚ͭ

10
0

⎞
1
⎟
0⎟
⎟
0⎟
⎠
1

Δ͜ͱ͕Ͱ͖Δɽີ౓ߦྻ ρ Ͱද͞ΕΔঢ়ଶ͕ཧ૝
–50

0

50

–50

᫤㛣ᕣ τ (ns)050తͳྔࢠ΋ͭΕঢ়ଶ |ψ ʹͲΕ͚͍͔ͩۙΛද͢ྔͱͯ͠͸ɼ஧࣮౓ (ﬁdelity: F )ਤ 21: CuCl ݁থΛ༻͍ͨྭ‫ࢠى‬෼ࢠ‫ڞ‬໐ϋΠύʔύϥϝτϦοΫࢄཚʹΑͬͯൃੜͨ͠ޫࢠରͷภޫF = Ψ|ρ|Ψ૬ؔͷଌఆྫɽԣ࣠͸ 2 ޫࢠͷ࣌ؒࠩɼॎ࣠͸ 2 ޫ͕༻͍ΒΕΔɽF ͸ 0 ͔Β 1 ͷ஋ΛͱΓɼཧ૝తঢ়

ࢠΛ‫ݕ‬ग़ͨ͠‫͋Ͱ਺ܭ‬Δɽਤதͷ LR, HV ౳ͷ‫߸ه‬ଶͰ͸ F = 1 ͱͳΔɽ·ͨɼ‫ݹ‬యతʹ͸࠷େͷ૬ؔ

(57)

͸ɼॳΊͷจࣈ͕Ұํͷޫࢠͷภޫঢ়ଶɼ2 ൪໨ͷจ Λ΋ͭࠞ߹ঢ়ଶ (44) ͷɼྔࢠ΋ͭΕঢ়ଶ (56) ʹର͢ࣈ͕ଞํͷޫࢠͷภޫঢ়ଶͰ͋ΓɼL(R) ͕ࠨʢӈʣ Δ஧࣮౓͸ F = 0.5 Ͱ͋Γɼ͜Ε͕‫ݹ‬యతঢ়ଶʹ͓ճΓԁภޫɼH(V) ͕ਫฏʢਨ௚ʣํ޲ͷ௚ઢภޫΛ

͚Δ஧࣮౓ͷ্‫͋Ͱݶ‬Δɽද͢ɽ

2 ޫࢠͷภޫʹؔ͢Δີ౓ߦྻΛ࣮‫ݧ‬తʹ‫ٻ‬ΊΔʹ͸ɼ͞·͟·ͳภޫͷ૊Έ߹ΘͤͰภޫ૬ؔͷଌʹ‫͘ڧ‬දΕΔ৴߸͕ɼ૬ؔΛ΋ͭޫࢠରͷಉ࣌‫ݕ‬ग़ ఆΛߦ͍ɼྔࢠঢ়ଶτϞάϥϑΟͱ‫ݺ‬͹ΕΔख๏Ͱ
ͷ݁ՌͰ͋Δɽैͬͯɼ͜ͷ৴߸ͷภޫ૬ؔʹ஫໨ ີ౓ߦྻΛਪఆ͢Δ [56]ɽ͍·ɼ2 ྔࢠϏοτ‫ܥ‬ͷ‫ج‬͢Δ͜ͱͰɼൃੜͨ͠ޫࢠରͷภޫʹؔ͢Δྔࢠ૬ ఈͷ਺ʢώϧϕϧτۭؒͷ࣍‫ݩ‬ʣ͸ 2 × 2 = 4 Ͱ͋ΓɼؔΛٞ࿦͢Δ͜ͱ͕Ͱ͖ΔɽҰํɼτ = 0 ʹ‫ݱ‬ΕΔϊ ີ౓ߦྻΛਪఆ͢ΔͨΊʹ࠷௿‫ݶ‬ඞཁͳภޫͷ૊ΈΠζ͸ɼྭ‫ى‬Ϩʔβʔͷผʑͷύϧεʢύϧεִؒ 1 ߹Θͤଌఆͷ਺͸ 42 = 16 Ͱ͋Δɽ‫্ࢉܭ‬͸ɼ16 ௨
nsʣʹΑͬͯൃੜͨ͠ಠཱͳޫࢠ͕‫ۮ‬વʹಉ࣌‫ݕ‬ग़ Γͷଌఆ஋ͷ૊Έ߹Θ͔ͤΒઢ‫ܗ‬ԋࢉʹΑΓີ౓ߦ͞Εͨ݁ՌͰ͋Γɼภޫͷ૬ؔΛ΋ͨͳ͍ɽ͜ΕΒ ྻΛਪఆ͢Δ͜ͱ͸Ͱ͖Δ͕ɼ௨ৗ͸ɼ৴པͰ͖Δ
ͷώετάϥϜ͔Β໌Β͔ͳΑ͏ʹɼ‫ٯʹ͍ޓ‬ճΓ ݁ՌΛಘΔͨΊʹɼ16 ௨ΓҎ্ͷ࣮‫ݧ‬஋͔Β࠷໬ਪͷԁภޫʢLR, RLʣ͋Δ͍͸‫ʹ͍ޓ‬ฏߦͳ௚ઢภޫ ఆ๏ʹΑͬͯີ౓ߦྻΛ‫ٻ‬ΊΔ [56]ɽ
ʢHH, VVʣͷ૊Έ߹Θͤͷͱ͖ͷΈಉ࣌‫ݕ‬ग़৴߸͕ਤ 22 ͸ɼલड़ͨ͠ྭ‫ࢠى‬෼ࢠ‫ڞ‬໐ϋΠύʔύϥϝ‫ݱ͘ڧ‬Εɼޫࢠରͷঢ়ଶ͕ (55) ͔ࣜΒ༧૝͞ΕΔภ τϦοΫࢄཚʹΑͬͯੜͨ͡ޫࢠରʹ͍ͭͯɼਤ 21ޫ૬ؔΛ໌ྎʹ͍ࣔͯ͠Δ͜ͱ͕Θ͔Δɽ·ͨɼ͜ ͱಉ༷ʹͯ͠ଌఆͨ͠ 22 ௨Γͷภޫ૬ؔͷଌఆ݁Ռͷଌఆʹ༻͍ͨޫࢠରͷภޫ૬͕ؔϕϧͷෆ౳ࣜΛ ͔Βɼྔࢠঢ়ଶτϞάϥϑΟʹΑͬͯ‫ٻ‬Ίͨີ౓ߦ
ഁΔ͜ͱ΋֬ೝ͞Ε͍ͯΔ [55]ɽ
ྻΛώετάϥϜͰදͨ͠΋ͷͰ͋Δɽ͜͜Ͱɼີ
౓ߦྻͷର֯ཁૉ͸ɼਤ 21 ʹ͓͚Δ௚ઢภޫͰͷภ

3.4

ྔࢠ΋ͭΕͷධՁ

ޫ૬ؔʢHH, HV, VH, VVʣͷ஋Λ௚઀൓ө͍ͯ͠
Δɽਤ͔Βɼର֯ཁૉ |HHHH| ͓Αͼ |VVVV|ྔࢠ΋ͭΕΛఆྔతʹධՁ͢ΔͨΊʹ͸ɼ̎ޫࢠͱඇର֯ཁૉ |HHVV| ͓Αͼ |VVHH| ͷ஋͕΄ͷภޫঢ়ଶʹؔ͢Δີ౓ߦྻΛ‫ٻ‬ΊΔඞཁ͕͋Δɽ༧΅ 1/2 ͱͳΓɼ༧૝͞ΕΔঢ়ଶ (56) ʹ͍ۙ͘͝ঢ়ଶ

૝͞ΕΔྔࢠ΋ͭΕঢ়ଶ (55) ͷີ౓ߦྻ͸ɼ|HH,͕࣮‫͞ݱ‬Ε͍ͯΔ͜ͱ͕Θ͔Δɽͨͩ͠ɼ࣮‫ݧ‬త৚݅ʹΑΓɼH ภޫͱ V ภޫͷ‫ʹ౓ڧ‬Θ͔ͣͳҧ͍͕
ੜ͡ɼ|HHHH| ͱ |VVVV| ͷόϥϯε͕ѱ͍͜ͱɼ·ͨɼภޫ૬ؔΛ༗͠ͳ͍‫ۮ‬વͷಉ࣌‫਺ܭ‬ͷӨ‫Ͱڹ‬ɼ|HVHV| ͱ |VHVH| ͷ੒෼͕ए‫ݱׯ‬ΕΔ͜

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ཧղͰ͖Α͏ɽ
Re

Im

0.5

0.5

0.25
5

0.25
5
HH
HV
VH

0
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HV

VH

VV

VV

3.5
HH
HV
VH

0
HH

HV

VH

VV

VV

ਤ 22: CuCl ݁থΛ༻͍ͨྭ‫ࢠى‬෼ࢠ‫ڞ‬໐ϋΠύʔϕϧঢ়ଶଌఆͱྔࢠςϨϙʔςʔγϣϯલड़ͨ͠Α͏ʹɼ2 ྔࢠϏοτ‫ܥ‬ͷ೚ҙͷঢ়ଶϕΫτϧ͸ɼϕϧ‫ج‬ఈ (41), (42) ͷઢ‫Ͱ߹݁ܗ‬ද͢͜ͱ
͕Ͱ͖Δɽ͢ͳΘͪ (38) Λ|Ψ = α|00 + β|01 + γ|10 + δ|11= c1 |Φ+  + c2 |Φ−  + c3 |Ψ+  + c4 |Ψ− (58)√ࢠภޫঢ়ଶͷີ౓ߦྻɽਤ 21 ͷ݁ՌʹՃ͑ɼԁภ ͱॻ͖‫͑׵‬Δ͜ͱ͕Ͱ͖Δɽ͜͜Ͱɼc = (α+δ)/ 2,
1√√√
ޫʢL,Rʣ͓Αͼ௚ઢภޫʢH,V, ͓Αͼ ±45◦ ʣͷ͞ c = (α − δ)/ 2, c = (β + γ)/ 2, c = (β − γ)/ 2234·͟·ͳ૊Έ߹ΘͤʹΑΔภޫ૬ؔͷଌఆ͔Β࠷໬ Ͱ͋Δɽਪఆྔࢠঢ়ଶτϞάϥϑΟʹΑͬͯಋग़͠ɼͦͷ࣮ϕϧঢ়ଶଌఆ (Bell state measurement: BSM) ͱ෦ʢࠨʣ͓Αͼ‫ڏ‬෦ʢӈʣΛ֤ʑώετάϥϜͰද ͸ɼ2 ྔࢠϏοτ‫ܥ‬ͷঢ়ଶΛ 4 छͷϕϧ‫ج‬ఈͷͲΕ͔ʹࣔͨ͠ɽࣹӨ͢ΔଌఆͰ͋Δɽྫ͑͹ɼ(58) ͷঢ়ଶ |Ψ ͕ϕϧύϥϝτϦοΫࢄཚʹΑͬͯൃੜͨ͠ޫࢠରͷ 2 ޫ

2ͱͳͲɼཧ૝తঢ়ଶ͔Βͷҧ͍΋‫ݟ‬ΒΕΔɽ͜ͷΑ͏‫ج‬ఈ |Φ+  ʹࣹӨʢଌఆʣ͞ΕΔ֬཰͸ɼ|Φ+ |Ψ| =|c1 |2 Ͱ͋Δɽಉ༷ʹɼଞͷϕϧঢ়ଶʹࣹӨ͞ΕΔ֬
ʹ࣮ͯ͠‫ݧ‬తʹಘͨີ౓ߦྻͷɼཧ૝తঢ়ଶ (55) ͋ ཰͸֤ʑ|c |2 , |c |2 , |c |2 Ͱ͋ΓɼͦΕΒΛશͯ߹Θ2
3
4Δ͍͸ (56) ʹର͢Δ஧࣮౓ (57) ΛධՁͨ݁͠Ռ21 ɼ ͤΔͱ |c |2 + |c |2 + |c |2 + |c |2 = 1 Ͱ͋Δɽ͢ͳ
1
2
3
4
F = 0.85 ͱ͍͏஋͕ಘΒΕͨɽ্ड़ͨ͠Α͏ʹɼ‫ ݹ‬Θͪɼϕϧঢ়ଶଌఆʹ͓͍ͯ͸ɼ೚ҙͷঢ়ଶ͕ϕϧయతʹ͸ྔࢠ΋ͭΕͷ஧࣮౓͸ 0.5 Λ௒͑Δ͜ͱ͸ ঢ়ଶͷͲΕ͔ͻͱͭʹࣹӨ͞Εͯ‫؍‬ଌ͞ΕΔɽϕϧͳ͍ͷͰɼ͜ͷ݁Ռ͸ɼ‫؍‬ଌ͞Εͨޫࢠର͕‫ݹ‬య‫ ݶ‬ঢ়ଶଌఆͱ͸ଌఆର৅ͷྔࢠ΋ͭΕͷఔ౓ΛධՁ͢քΛ௒͑ͨྔࢠ΋ͭΕΛ༗͢Δ͜ͱΛ͍ࣔͯ͠Δɽ Δ΋ͷͰ͸ͳ͍ɽ͔͠͠ɼϕϧঢ়ଶଌఆͷ݁ՌΛརີ౓ߦྻ͸ྔࢠঢ়ଶͷ৘ใΛશͯ‫ؚ‬ΜͰ͍ΔྔͰ ༻͢Δ͜ͱͰɼྔࢠ৘ใ௨৴ʹ͓͚Δॏཁͳϓϩτ͸͋Δ͕ɼཁૉͷ਺͕ଟ͘ɼྔࢠ΋ͭΕͷఔ౓ΛҰٛ ίϧɼྫ͑͹ҎԼͰड़΂ΔྔࢠςϨϙʔςʔγϣϯతʹఆྔධՁ͢Δࢦඪͱͯ͠͸࢖͍ʹ͍͘ɽͦͷͨ Λ࣮‫͢ݱ‬Δ͜ͱ͕Ͱ͖ΔɽΊɼྔࢠ΋ͭΕͷఆྔධՁͷͨΊͷࢦඪͱͯ͠ɼ͞·ྔࢠςϨϙʔςʔγϣϯͱ͸ɼྔࢠ΋ͭΕΛར༻͟·ͳྔ͕ߟҊ͞Ε͍ͯΔɽྫ͑͹ɼϕϧͷෆ౳ࣜͳ ͢Δ͜ͱͰɼ͋Δର৅ͷྔࢠঢ়ଶΛผͷର৅ʹҠ͢Ͳ΋ͦͷͻͱͭͰ͋Δɽ·ͨɼebit(E), concurrence ʢసૹ͢Δʣٕज़Ͱ͋Δɽ௨ৗɼྔࢠঢ়ଶͷಛఆʹ͸ɼ(C), tangle (T = C 2 ) ౳ͷྔ΋ྑ͘༻͍ΒΕΔɽҰ 1 ͭͷ෺ཧྔ͚ͩͰ͸ͳ͘ɼྫ͑͹ҐஔͱӡಈྔͱൠʹɼE Λີ౓ߦྻ͔Β௚઀‫ٻ‬ΊΔͷ͸ࠔ೉Ͱ͋Δ ͍ͬͨɼ‫ʹ͍ޓ‬ඇՄ‫ͳ׵‬ෳ਺ͷ෺ཧྔʹؔ͢Δ৘ใ
͕ɼ2 ྔࢠϏοτ‫ܥ‬ͷ৔߹ɼC ·ͨ͸ T ͸ີ౓ߦྻ ͕ඞཁͰ͋Δɽ͔͠͠ɼҰൠʹɼ͜ΕΒͷඇՄ‫ͳ׵‬͔Βൺֱత؆୯ʹ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖ɼͦΕΒ͔Β E ෺ཧྔΛಉ࣌ʹਖ਼֬ʹ‫ٻ‬ΊΔ͜ͱ͸Ͱ͖ͳ͍ʢෆ֬
Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δ [57]ɽྫ͑͹ɼਤ 22 ͷີ౓ߦ ఆੑ‫ݪ‬ཧʣɽैͬͯɼ୯ҰޫࢠͷΑ͏ʹͻͱ͔ͭ͠ͳ
ྻ͔Β‫ٻ‬ΊΒΕΔ E ͷ஋͸ 0.65 Ͱ͋Δ22 ɽ‫ݹ‬య૬ؔ ͍ର৅ͷྔࢠঢ়ଶΛଌఆʹΑͬͯਖ਼֬ʹಛఆ͢Δ͜
ͷΈͷঢ়ଶ (44) Ͱ͸ E = 0 ͱͳΔ͜ͱʹ஫ҙ͢Ε ͱ͸Ͱ͖ͣɼͦͷଌఆ݁ՌΛ࢖ͬͯྔࢠঢ়ଶΛଞͷ͹ɼਤ 22 ͷີ౓ߦྻͰද͞ΕΔঢ়ଶ͕ɼ‫׬‬શͰ͸ͳ ର৅΁ਖ਼֬ʹసૹ͢Δ͜ͱ΋·ͨෆՄೳͰ͋Δɽ͍͠ʹͤΑߴ͍ఔ౓ͷྔࢠ΋ͭΕΛ༗͍ͯ͠Δ͜ͱ͕ ͔͠ɼྔࢠ΋ͭΕͱϕϧঢ়ଶଌఆΛར༻͢Δ͜ͱͰɼྔࢠঢ়ଶΛʢ‫ݪ‬ཧతʹ͸ʣਖ਼֬ʹసૹ͢Δૢ࡞͕Մ
21 ີ౓ߦྻ ρ ͸‫ج‬ఈ |HH, |HV, |VH, |VV Λ༻͍ͯදΘ√͞Ε͍ͯΔͷͰɼಉ͡‫ج‬ఈͰͷ ψ ͷϕΫτϧද‫( ݱ‬1, 0, 0, 1) / 2 ೳͱͳΔɽ͜ͷٕज़ΛྔࢠςϨϙʔςʔγϣϯͱ͍Ͱ ρ Λ‫ڬ‬Μͩ಺ੵΛ‫͢ࢉܭ‬Δɽ͏ [58]23 ɽ
22؆୯ʹ‫͑ݴ‬͹ɼ͜ͷঢ়ଶΛ 100 ରूΊΔͱͦΕΒ͔Β 65 ରͷ࠷େྔࢠ΋ͭΕঢ়ଶΛ࡞Δ͜ͱ͕Ͱ͖Δ͜ͱʹͳΔɽ

23 SF ͷΑ͏ͳ໊લͰ͋Δ͕ɼ෺࣭ΛҠಈͤ͞ΔΘ͚Ͱ͸ͳ͘ɼ

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2

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U

同時検出 ⇒

2

ਤ 23: ྔࢠςϨϙʔςʔγϣϯͷ֓೦ਤɽྔࢠϏοਤ 24: 2 ޫࢠ‫ׯ‬বʹΑΔϕϧঢ়ଶଌఆͷ֓೦ਤɽ+τ 2 ͓Αͼ 3 ͷؒͷྔࢠ΋ͭΕ |Ψ 23 Λར༻ͯ͠ɼ

24
ྔࢠϏοτ 1 ͷঢ়ଶ |ψ1 ΛྔࢠϏοτ 3 ʹసૹ͢Δɽ Αͬͯ ɼྔࢠϏοτ 3 ͷঢ়ଶ͸ྔࢠϏοτ 1 ͱ 2 ͱͷؒͷϕϧঢ়ଶଌఆ (BSM) ͷ|Ψ+ 12 → α|03 + β|13 = |ψ3 ,݁ՌʹԠͯ͡ɼྔࢠϏοτ 3 ʹରͯ͠ϢχλϦม‫׵‬|Ψ− 12 → α|03 − β|13 = σ1 |ψ3 ,U Λࢪ͢͜ͱʹΑΓɼྔࢠϏοτ 3 ͷঢ়ଶ͸ |ψ3 ͱͳΔɽ

|Φ+ 12 → β|03 + α|13 = σ2 |ψ3 ,(63)|Φ− 12 → −β|03 + α|13 = σ2 σ1 |ψ3
ਤ 23 ʹࣔ͢Α͏ʹɼసૹͷର৅͕ྔࢠϏοτͷ৔߹Λߟ͑Δɽసૹ͍ͨ͠ྔࢠϏοτ 1 ͷঢ়ଶ͕|ψ1 = α|01 + β|11Δɽಛʹɼϕϧঢ়ଶଌఆͷ݁Ռ͕ |Ψ+ 12 Ͱ͋ͬͨ৔

(59)

Ϗοτ 1 ͷঢ়ଶ͕ྔࢠϏοτ 3 ʹసૹ͞ΕΔ͜ͱ͕

(60)ʹ͋ͬͨͱ͠Α͏ɽ͜ͷͱ͖ɼશ‫ܥ‬ͷঢ়ଶ |Ψ123 ͸|Ψ123 = |ψ1 ⊗ |Ψ+ 23α= √ (|01 |02 |13 + |01 |12 |03 )2β+ √ (|12 |02 |13 + |11 |12 |03 )2(61)ม‫׵‬Λࢪͨ͠΋ͷͰ͋Δ͔Βɼͦͷ‫ٯ‬ม‫׵‬Λࢪ͢͜ͱʹΑͬͯɼྔࢠϏοτ 1 ͷঢ়ଶΛ࠶‫͖Ͱݱ‬Δ͜ͱ

3.6

ภޫྔࢠϏοτͷϕϧঢ়ଶଌఆͱྔࢠ
ςϨϙʔςʔγϣϯҎԼͰ͸ɼ୯ҰޫࢠͷภޫྔࢠϏοτʹର͢Δϕ1 +|Ψ123 =|Ψ 12 (α|03 + β|13 )
2+ |Ψ− 12 (α|03 − β|13 )ϧঢ়ଶଌఆͱྔࢠςϨϙʔςʔγϣϯʹ͍ͭͯड़΂ΔɽͦͷཁͰ͋Δϕϧঢ়ଶଌఆͰ͋Δ͕ɼ࢒೦ͳ͕
Βઢ‫ֶޫܗ‬ૉࢠΛ༻͍ͨૢ࡞Ͱ͸ɼ4 छͷϕϧ‫ج‬ఈ++|Φ 12 (−β|03 + α|13 )ʹ΋ɼྔࢠϏοτ 3 ͷঢ়ଶ͸ |ψ3 ʹద౰ͳϢχλϦ

͋Δ26 ɽɹ

Λ༻͍ͯϕϧ‫ج‬ఈʹॻ͖‫͑׵‬Δͱɼ

−Θ͔Δ25 ɽϕϧଌఆͷ݁Ռ͕ଞͷঢ়ଶͰ͋ͬͨ৔߹

ʹͳΔɽ͜Ε͕ྔࢠϏοτͷςϨϙʔςʔγϣϯͰ

ͱͳΔɽ͜͜ͰɼྔࢠϏοτ 1 ͱ 2 ͷঢ়ଶΛɼ(58)+ |Φ 12 (β|03 + α|13 )߹ʹ͸ɼྔࢠϏοτ 3 ͷঢ়ଶ͸ |ψ3 Ͱ͋ΓɼྔࢠϏοτ 1 ͷॳ‫ظ‬ঢ়ଶ |ψ1 ͱ౳ՁͱͳΔɽ͢ͳΘͪɼྔࢠʹ͋ΓɼྔࢠϏοτ 2 ͓Αͼ 3 ͕ϕϧঢ়ଶ1|Ψ+ 23 = √ (|02 |13 + |12 |03 )2ʹఆ·Δɽ͜͜Ͱɼσi ͸ύ΢ϦߦྻʢԋࢉࢠʣͰ͋

͢΂ͯʹܾఆ࿦తʹࣹӨͰ͖Δϕϧঢ়ଶଌఆʢ‫׬‬શ(62)ϕϧঢ়ଶଌఆʣ͸ෆՄೳͰ͋Δ͜ͱ͕஌ΒΕ͍ͯΔɽ
͔͠͠ɼͦͷ͏ͪͷ 1 ͭʢྫ͑͹ |Ψ+ ʣʹࣹӨ͢Δ

ͱͳΔɽैͬͯɼ1 ͱ 2 ͱͷؒͰϕϧঢ়ଶଌఆΛߦଌఆʢ෦෼తϕϧঢ়ଶଌఆʣ͸ɼҎԼͷΑ͏ʹ͍ͯ͠ɼͦͷ݁Ռ͕ 4 छͷϕϧ‫ج‬ఈͷԿΕͰ͔͋ͬͨʹ࣮‫͖Ͱݱ‬Δɽ

24 ͜ͷͱ͖ɼ4 छͷϕϧঢ়ଶͷ͏ͪͷಛఆͷ 1 ͕ͭଌఆ͞ΕΔ
֬཰͸ɼԿΕ΋ 1/4 Ͱ͋Δɽ
25 ϕϧঢ়ଶଌఆͷ‫ޙ‬ɼྔࢠϏοτ 1,2 ͷঢ়ଶ͸ϕϧঢ়ଶ͢ͳΘͪ࠷େྔࢠ΋ͭΕঢ়ଶͱͳΔͷͰ͋Δ͔ΒɼྔࢠϏοτ 1 ୯ಠͷঢ়ଶ͸‫׬‬શࠞ߹ঢ়ଶͱͳΔɽ͢ͳΘͪɼϕϧঢ়ଶଌఆʹΑͬͯྔྔࢠঢ়ଶΛ͋Δର৅͔Βผͷର৅΁Ҡٕ͢ज़Ͱ͋Δ͜ͱʹ஫ҙͤ ࢠϏοτ 1 ͷঢ়ଶ͸ഁյ͞ΕΔɽ
26 ͜͜Ͱ͸ɼ͋Β͔͡Ί༻ҙ͢Δྔࢠ΋ͭΕঢ়ଶͱͯ͠ |Ψ+ Αɽ·ͨɼϕϧঢ়ଶଌఆͷ݁ՌΛผͷର৅ͷͱ͜Ζ·Ͱ‫ݹ‬య௨৴
ͰૹΔඞཁ͕͋ΔͷͰɼͦͷ௨৴‫ʹ཭ڑ‬Ԡͨ࣌ؒ͡΋ඞཁͰ͋Δɽ Λ༻͍͕ͨɼଞͷϕϧঢ়ଶΛ༻͍ͯ΋ಉ༷ͷ͜ͱ͕Ͱ͖Δɽ

《講義ノート》

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BS

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検出

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ਤ 25: ޫࢠͷภޫྔࢠϏοτͷྔࢠςϨϙʔςʔγϣϯɽ
ਤ 24 ͷΑ͏ʹɼ50%:50%ແภޫϏʔϜεϓϦολ
ͷ̎ͭͷೖࣹϙʔτʹɼಉҰप೾਺ ω ͷޫࢠରΛಉ࣌ʹೖࣹ͠ɼ൓ࣹͱಁաͷޫ࿏͕ॏͳΔΑ͏ʹ‫ׯ‬বͤ͞Δɽ͜ͷ഑ஔ͸ɼ2.3 અͰड़΂ͨ Hong-Ou-Mandel‫ׯ‬ব΍ɼਤ 16 ʹࣔͨ͠ภޫྔࢠ΋ͭΕঢ়ଶͷੜ੒ํ๏ͱಉ͡Ͱ͋Δɽೖࣹޫࢠͷภޫঢ়ଶͱͯ͠ 4 छͷϕϧ‫ج‬ఈΛߟ͑ɼ(51) ͱಉ༷ʹͯ͠ग़ྗޫͷঢ়ଶΛௐ΂Δͱɼ|Φ+ , |Φ−  ͓Αͼ |Ψ−  ͷͱ͖ʹ͸ 2 ޫࢠ͸ͲͪΒ͔ͷग़ྗϙʔτ΁Ұॹʹग़ࣹ͠ɼ|Ψ+  ͷͱ͖ʹ͸ 1 ޫࢠͣͭ 2 ͭͷग़ྗϙʔτʹ෼͔Εͯग़
ࣹ͢Δ͜ͱ͕Θ͔Δɽ͢ͳΘͪɼೖࣹޫͷภޫঢ়ଶΛߏ੒͢Δ 4 ͭͷϕϧ‫ج‬ఈͷ͏ͪ 2 ޫࢠ͕ 2 ͭͷग़ྗޫ࿏ʹ෼͔Εͯग़ࣹ͢Δͷ͸ɼೖࣹޫ͕ |Ψ+  ͷ৔߹ͷΈͰ͋Δɽैͬͯɼ2 ͭͷग़ྗޫ࿏ʹஔ͍ͨ 2 ୆ͷޫࢠ‫ݕ‬ग़‫ࢠޫͰث‬Λಉ࣌‫ݕ‬ग़͢Δଌఆ͸ɼೖࣹޫࢠͷঢ়ଶ |Ψ+  ʹର͢ΔࣹӨଌఆͱͳΔ͜ͱ͕Θ͔
Δ27 ɽ͢ͳΘͪɼ͜ͷଌఆ͸ɼ4 ͭͷϕϧঢ়ଶΛશͯ
ࣝผ͢Δ‫׬‬શϕϧঢ়ଶଌఆͰ͸ͳ͍͕ɼೖྗঢ়ଶΛϕϧঢ়ଶͷ͏ͪͷ 1 ͭͷঢ়ଶ |Ψ+  ʹࣹӨ͢Δ෦෼
తϕϧঢ়ଶଌఆͱͳΔͷͰ͋Δɽ
ਤ 25 ʹɼޫࢠͷภޫྔࢠϏοτͷྔࢠςϨϙʔςʔγϣϯͷ֓೦ਤΛࣔ͢ɽϕϧঢ়ଶ |Ψ+ 23 ʹ͋Δޫࢠ 2 ͓Αͼ 3 Λ४උ͠ɼ೚ҙͷภޫঢ়ଶ (|ψ1 ) ʹ͋Δޫࢠ 1 ͱޫࢠ 2 ͱͷؒͰ্ड़ͨ͠ϕϧঢ়ଶଌఆΛߦ͏ɽϏʔϜεϓϦολ (BS) ͷ྆ํͷग़ྗͰޫࢠ
͕ಉ࣌‫ݕ‬ग़͞Εͨ৔߹ʢͦͷ֬཰͸ 1/4ʣɼϕϧঢ়
27 ࣹӨ͞ΕΔঢ়ଶ͕ |Ψ−  Ͱ͋Δͱ͢Δจ‫ݙ‬΋ଟ͘‫ݟ‬ΒΕΔʢྫ͑͹ [59], [60]ʣɽ͜Ε͸ɼޫ࿏ 1 ͱޫ࿏ 2 ͷ࠲ඪͷఆٛͷҧ͍ʹΑΔ΋ͷͱࢥΘΕΔɽຊॻͰ͸ޫ࿏ 1,2 ͱ΋ӈख‫͢ͱܥ‬Δ࠲ඪͷऔΓํΛ࠾༻͍ͯ͠Δ͕ɼޫ࿏ 1 ͱޫ࿏ 2 ͱͰӈख‫ࠨͱܥ‬ख‫ܥ‬Λ൓స͢Δ࠲ඪͷऔΓํ΋͋ΓಘΔɽͦͷࡍʹ͸൓ࣹʹ൐͏ H(p)ภޫ੒෼ͷ൓స͕ͳ͘ɼࣹӨ͞ΕΔঢ়ଶ͸ |Ψ−  ͱද͞ΕΔʢ࠲ඪ‫ܥ‬ͷҧ͍ʹ൐ͬͯද‫͕ݱ‬ม‫͞׵‬ΕΔ͚ͩͰ෺ཧత࣮ମ͕ҟͳΔ
Θ͚Ͱ͸ͳ͍ʣɽ࣮‫ݧ‬తʹ͸ɼྫ͑͹ޫ࿏ͷยํΛ‫Ͱڸ‬൓ࣹ͢Δ
͜ͱ౳ʹΑͬͯ΋༰қʹ‫͜ى‬ΓಘΔม‫͋Ͱ׵‬ΔͷͰɼ஫ҙ͢Δඞ
ཁ͕͋Δɽ

ਤ 26: ޫࢠͷภޫͷྔࢠ΋ͭΕεϫοϐϯάɽଶଌఆͷ݁Ռ͕ |Ψ+ 12 Ͱ͋Δ͜ͱʹͳΓɼͦͷ৔߹ʹ͸ޫࢠ 1 ͷঢ়ଶ (|ψ1 ) ͕ޫࢠ 3 ͷঢ়ଶ (|ψ3 ) ͱͯ͠సૹ͞Εͨ͜ͱʹͳΔɽ͜ͷछͷ࣮‫ݧ‬͸ɼΠϯεϒϧοΫେֶͷάϧʔϓ [60] ʹΑͬͯॳΊͯߦΘΕ
ͨ28 ɽ͜ͷ࣮‫Ͱݧ‬͸ɼtype-II Ґ૬੔߹ SPDCʢਤ 17ʹࣔͨ͠ํ๏ʣΛ༻͍ͯಉ࣌ʹ 2 ରͷޫࢠରΛ‫͍ޓ‬ʹ‫ൃʹ͖޲ٯ‬ੜ͠ɼͦͷ 1 ରΛϕϧঢ়ଶͱͯ͠༻͍ɼ
ଞͷ 1 ରΛ఻ྩ෇͖ޫࢠͱͯ͠༻͍ͨɽޫࢠ 1 ͷภޫঢ়ଶΛภޫࢠΛ༻͍ͯछʑͷঢ়ଶʹ४උ͠ɼϕϧঢ়ଶଌఆΛ௨ͯ͠ޫࢠ 3 ʹసૹ‫ޙ‬ɼޫࢠ 3 ͷภޫঢ়ଶ͕ޫࢠ 1 ʹରͯ͠४උͨ͠ঢ়ଶͱ΄΅ಉ͡ʹͳͬ
͍ͯΔ͜ͱɼ͢ͳΘͪྔࢠςϨϙʔςʔγϣϯ͕࣮ߦ͞Ε͍ͯΔ͜ͱΛ֬ೝͨ͠29 ɽ͞ΒʹɼྔࢠςϨϙʔςʔγϣϯΛԠ༻͢Δͱɼਤ

26 ʹࣔ͢ྔࢠ΋ͭΕεϫοϐϯάͱ‫ݺ‬͹ΕΔૢ࡞Λ࣮‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δ [61]ɽ͜ͷૢ࡞Ͱ͸ɼྔࢠ΋ͭΕঢ়ଶʹ͋Δ 2 ޫࢠʢޫࢠ 0 ͓Αͼޫࢠ 1ʣͷ͏ͪͷยํʢޫࢠ 1ʣͷঢ়ଶΛྔࢠςϨϙʔςʔγϣϯͰ
ޫࢠ 3 ʹసૹ͢Δɽ͢Δͱɼ͸͡Ί͸ޫࢠ 0 ͱޫࢠ
1 ͱͷؒʹଘࡏ͍ͯͨ͠ྔࢠ΋ͭΕ͕ɼޫࢠ 0 ͱޫ
ࢠ 3 ͱͷؒͷྔࢠ΋ͭΕͱͯ͠Ҡ͞ΕΔ͜ͱʹͳΔɽͭ·Γɼ͸͡Ί͸Կ΋૬ؔͷͳ͔ͬͨޫࢠ 0 ͱޫࢠ 3ͷؒʹྔࢠ΋ͭΕΛҠಈ͢Δ͜ͱ͕Ͱ͖Δɽ͜ͷૢ࡞Λ࣍ʑͱߦ͏͜ͱʹΑΓɼԕ͘཭Εͨޫࢠ͋Δ͍͸ྔࢠϏοτͷؒʹྔࢠ΋ͭΕΛҠಈͤͯ͞ߦ͘͜ͱ͕Ͱ͖Δɽ͜Ε͸ɼྔࢠ৘ใ௨৴ʹ͓͚Δ৘ใ఻ૹ‫཭ڑ‬ΛԆ͹ͨ͢Ίͷྔࢠத‫ثܧ‬Λ࣮‫͢ݱ‬Δࡍʹॏཁͳٕज़Ͱ͋Δͱߟ͑ΒΕ͍ͯΔɽ
28 ͦͷͨΊɼΠϯεϒϧοΫ࣮‫ͱݧ‬΋‫ݺ‬͹ΕΔɽ
29 ޫࢠ 1 ͱޫࢠ 2 ͕֤ʑ̍‫ ͭͣݸ‬BS ʹೖྗ͞Εͨͱ͖ͷϕϧঢ়ଶଌఆͷ੒ޭ֬཰͸ 1/4 Ͱ͋Δɽ༻͍ΒΕ࣮ͨ‫ݧ‬৚݅Ͱ͸ɼޫ
ࢠ 1 ͱޫࢠ 2 ͕ಉ࣌ʹೖྗ͞ΕΔ֬཰ͱɼޫࢠ 1ʢ఻ྩ෇͖ޫࢠʣ
͕ 2 ޫࢠಉ࣌ʹೖྗ͞Εͯ͠·͏֬཰͕ಉఔ౓Ͱɼ‫ ͕ऀޙ‬BS ͷग़ྗͰಉ࣌‫ݕ‬ग़͞ΕΔ֬཰͸ 1/2 ͱͳΔͨΊɼ͜ͷʮϕϧঢ়ଶଌఆʯʹ‫ͮ͘ج‬ςϨϙʔςʔγϣϯͷա൒਺͸‫ޡ‬Γʢޫࢠ 2, 3 ͕ෆࡏʣͱͳͬͯ͠·͏໰୊͕ࢦఠ͞Ε͍ͯΔɽΠϯεϒϧοΫ࣮‫ݧ‬Ͱ͸ɼޫࢠ 3 ͕ଌఆ͞Εͨ৔߹ͷΈΛ݁Ռͱͯ͠औΓѻ͍ͬͯΔɽ

《講義ノート》

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