断熱的な時間発展の結果は経路のトポロジーで決まる

量子論の諸問題と今後の発展（QMKEK6）ポスター
(発表時のポスターから多少変更があります）

Transcript

1. 1 / 10 அ೤తͳ࣌ؒൃలͷ݁Ռ͸ ܦ࿏ͷτϙϩδʔͰܾ·Δ Path topology dependence of adiabatic time evolution ాதಞ࢘ (ट౎େཧ޻) |0 |a |b ྔࢠ࿦ͷॾ໰୊ͱࠓޙͷൃలʢQMKEK6ʣϙελʔൃද
2. ͸͡Ίʹ 2 / 10 എܠ: ྔࢠܥͷஅ೤తͳ࣌ؒൃల அ೤ఆཧ (Born and Fock 1928, Kato 1950) ఆৗঢ়ଶʹ༻ҙ͞Εͨܥʹ͍ͭͯɺܥͷύϥϝʔλʔΛΏͬ͘Γม Խͤͯ͞΋ɺܥ͸ఆৗঢ়ଶʹཹ·Δɻ (஫ 1) ͜͜Ͱ͸ଋറঢ়ଶʹର͢ΔϢχλϦʔͳ࣌ؒൃలͷΈߟ͑Δɻ (஫ 2) ύϥϝʔλͷมԽʹԠͯ͡ఆৗঢ়ଶ͸࣌ʑࠁʑมԽ͢Δɻ அ೤ఆཧ͸෺ཧֶͷ༷ʑͳ৔໘Ͱར༻͞Ε͍ͯΔ: (ྫ) ݪࢠ΍෼ࢠͷѻ͍ͷجૅ (அ೤ۙࣅ)ɺྔࢠϗϩϊϛʔ (ϕϦʔҐ૬)ɺ அ೤ྔࢠܭࢉɻ
3. ͸͡Ίʹ 3 / 10 ͳʹ͕அ೤తͳ࣌ؒൃలͷ݁ՌΛܾΊΔͷ͔ʁ ࠓ೔ͷ໰୊ அ೤ܦ࿏ͷ࢝఺ͱऴ఺ɺ͓Αͼɺܥͷ࢝ঢ়ଶΛࢦఆͨ͠ͱ͢Δɻ͜ ͷͱ͖ɺऴঢ়ଶ͸Կ͕ܾΊΔͷ͔ʁ ౴ͱͦͷഎܠ (ৄࡉ͸ޙड़; AT and T. Cheon, arXiv:1512.06983 ) ▶ அ೤ܦ࿏ͷʮͱ͋ΔτϙϩδΧϧͳੑ࣭ʯͰܾ·Δɻ (୯७ͳ৔߹ɺܦ࿏ͷϗϞτϐʔಉ஋ੑ͔Βܾ·Δ) ▶ ͜ͷݱ৅ͷഎޙʹ͸ɺඃ෴ۭؒͷߏ଄͕ଘࡏ͢Δɻ (˞ ඃ෴ۭؒ ≃ ཭ࢄతͳߏ଄܈Λ࣋ͭϑΝΠόʔଋ) ▶ ͜ͷҙຯͰɺྔࢠಈྗֶʹ಺ࡏ͢ΔҐ૬زԿֶతͳߏ଄ ͕໌ Β͔ʹͳͬͨɻ(cf. ϕϦʔҐ૬͸ඍ෼زԿֶతͳߏ଄ʹରԠ)ɻ
4. ͸͡Ίʹ 4 / 10 ྫ: ֊਺ 1 ͷपظܸྗԼͷεϐϯ-1 2 ܦ࿏ͷ྆୺ (Bi, Bf) ͕ಉ͡Ͱ΋ɺஅ೤తͳ࣌ؒൃలͷ݁Ռ͕ҟͳΔ O Bi Bf Bx By ܦ࿏্Ͱͷஅ೤ঢ়ଶͷεϐϯظ଴஋ (Bloch vector)ɻݪ఺ʹ disclination (Ґ૬ܽؕͷҰछ) ͕͋Δɻ ੩࣓৔ B ԼͷεϐϯʹपظܸྗΛҹՃ: H(t) ≡ B·σ 2 +λ | ↓ ⟩⟨ ↓ | ∑ ∞ m=−∞ δ(t−m). ରԠ͢Δϑϩέԋࢉࢠ͸ U = e−iλ |↓⟩⟨↓|e−iB·σ/2. ͜͜Ͱ λ = tan−1(By/Bx) ͱ͢Δͱɺ U ͸ (Bx,By)-໘ͰҰՁɻ Ref. Combescure, JSP (1990); Milek and Seba, PRA (1990); AT and Miyamoto, PRL (2007); AT and Cheon, PLA (2015).
5. ओཁͳٞ࿦ 5 / 10 P Ͱෳ਺ͷݻ༗ۭؒͷ૊Έ׵͑Λ࿦͡Δ ͜ͷ໰୊ͷ֩৺͸ɺෳ਺ͷݻ༗ۭؒͷ૊Έ׵͕͑அ೤ܦ࿏ʹґଘ͠ಘΔ ͜ͱɻ͜ΕΛ࿦͡ΔͨΊɺॱংΛ෇͚ͯฒ΂ࣹͨӨͷྻ p = ( |0⟩⟨0|, |1⟩⟨1|, ... ) Λಋೖ͢Δɻp ͷۭؒΛ P ͱه͢ɻ |0 0|, |1 1| |a a|, |b b| |b b|, |a a|
6. ओཁͳٞ࿦ 6 / 10 ݁Ռ: அ೤ܦ࿏ͱϗϞτϐʔಉ஋ੑ ऴঢ়ଶ͸ɺϗϞτϐʔಉ஋ͳܦ࿏ͷΫϥε [C] ͷؔ਺ɻ C ∼ C′ ͷͱ͖ (C ͱ C′ ͕ϗϞτϐʔಉ஋) ࢝ঢ়ଶ͕ಉ͡ͳΒऴঢ়ଶ΋ಉ͡ɻ |0 |a C C (∼ C) C ≁ C′ ͷͱ͖ P ͷ೚ҙͷดܦ࿏͕ՄॖͳΒɺऴঢ়ଶ͸ҟ ͳΔɻ(˞ Մॖ = Ұ఺ͱϗϞτϐοΫ) ͨͩ͠ɺҰൠʹ͸ P ্ͷඇՄॖͳดܦ࿏ ͷͨΊʹऴঢ়ଶ͕ಉ͡ʹͳΓಘΔ (ޙड़)ɻ |0 |a |b C C (∼ C) ?
7. ٞ࿦ͷৄࡉ 7 / 10 ఆࣜԽ: அ೤ܦ࿏ C ͷ P ΁ͷ࣋ͪ͋͛ͱࣸ૾ ϕ[C] (p) p ͸ɺύϥϝʔλۭؒ M ্ͷܦ࿏ C ʹԊͬͨஅ೤తͳ࣌ؒൃలʹै͏ɻ p ͷي੻͸ɺC ͷ P ΁ͷ࣋ͪ͋͛ C ͱΈͳͤΔɻ cf. ྔࢠϗϩϊϛʔ (ϕϦʔҐ૬) ͷ Simon ΍ Aharanov-Anandan ͷఆࣜԽɻ Fi Ff λi λf C p φC (p) C ϕC(p) Λʮॳظ఺ p Λ༩͑ͨͱ͖ͷ C ͷऴ఺ʯͱ͢Δɻͨͩ͠ɺC ∼ C′ ͳΒ ϕC = ϕC′ ͳͷͰɺ͜ΕΛ ϕ[C] ͱه͢ɻ λ U p1 U1 p2 U2 Fλ ࣹӨ π : P → M ͸ඃ෴ࣸ૾ͷެཧ Λຬͨ͢ɻඃ෴ۭؒ͸ߏ଄܈͕཭ࢄ తͳϑΝΠόʔଋͱΈͳͤΔɻ ࠓ೔ͷ໰୊͸ʮϕ[C] ΛܾΊΔͷ͸ܦ࿏ C ͷͲͷੑ࣭͔ʁʯͱ౳Ձɻ
8. ٞ࿦ͷৄࡉ 8 / 10 ดܦ࿏Ͱͷʮ৽حͳྔࢠϗϩϊϛʔʯ΁ͷؐݩ ܦ࿏ C ͱ C′ Λดܦ࿏ α ʹ·ͱΊͯൺֱ (α ≡ C ·(C′−1)) Ϟϊυϩϛʔ࡞༻ ϕ[α] ͸ܦ࿏ α ʹΑΔݻ༗ۭؒ ͷஔ׵ (৽حͳྔࢠϗϩϊϛʔ) Λهड़͠ɺC ͱ C′ ͱͷʮͣΕʯΛ༩͑Δ (ϕ[C] = ϕ[C′] ◦ϕ[α] )ɻ ͨͱ͑͹ C ∼ C′ ͷͱ͖ α ͸Ұ఺ʹՄॖͳͷͰ ϕ[α] ͸߃౳ஔ׵Ͱ͋Γɺϕ[C] = ϕ[C′] ɻ C C α ϕ[α] ͷ׬શͳ෼ྨ (AT and T. Cheon, PLA 379, 1693 (2015)) P ͷ೚ҙͷดܦ࿏͕ՄॖͳΒɺ[α] = [β] ͱ ϕ[α] = ϕ[β] ͸ಉ஋ɻͭ ·ΓɺM ͷୈҰجຊ܈ π1(M) ͸ ϕ[α] Λ௨ͨ͡அ೤ดܦ࿏ͷ׬શ ͳ෼ྨΛ༩͑Δɻ Ұൠʹ͸ɺ அ೤ดܦ࿏ͷ׬શͳ෼ྨΛ༩͑Δͷ͸π1(M)/π∗(π1(P)).
9. ٞ࿦ͷৄࡉ 9 / 10 ྫ: ॖୀͷແ͍εϐϯ-1 2 ͱࣹӨฏ໘ P = S2 (ٿ) ॱং෇͖ͷࣹӨྻ p ͸௕͞ 1 ͷ Bloch vector a ͱ౳Ձɻ ͳͥͳΒ p = (P+,P−), P± ≡ (1±a·σ)/2 ͱ͓͚ΔͨΊɻఆৗ ঢ়ଶ P+ Ͱͷ σ ͷظ଴஋͕ a ͷ͜ͱɻ (ͳ͓ɺॖୀ͕͋Δͱ a ͸ఆٛͰ͖ͳ͍ɻ) a −a M = RP2 (ࣹӨฏ໘) ॱং෇͚ͷແ͍ࣹӨྻ b = {P+,P−} ͕அ೤ύϥϝʔλʔɻ b ͸ a ͷ director n ͱରԠ͠ɺRP2 ্ͷҰ఺ɻRP2 ͸ࣹ Ө Hilbert ۭؒͷ֦ுͷҰछɻ n ॖୀͷແ͍εϐϯ-1 2 Ͱى͖ಘΔ ϕ[α] π1(M) = {[e],[γ]} ͳͷͰɺ߃౳ࣸ૾ ϕ[e] ͱޓ׵ ϕ[γ] ͷΈ: p = (P1,P2) ʹର͠ ϕ[e] (p) = (P1,P2), ϕ[γ] (p) = (P2,P1). γ e n0 n0
10. ͓ΘΓʹ 10 / 10 จݙ [1] AT and T. Cheon, arXiv:1512.06983 (2015) (to appear in Functional Analysis and Operator Theory for Quantum Physics. A Festschrift in Honor of Pavel Exner, J. Dittrich, H. Kovaˇ r´ ık and A. Laptev eds. (Europ. Math. Soc. Publ. House, 2016)). [2] AT and T. Cheon, PLA 379, 1693 (2015). [3] H. Mehri-Dehnavi and A. Mostafazadeh, JMP 49, 082105 (2008). [4] N. Yonezawa, AT and T. Cheon, PRA 87, 062113 (2013). ˞ ϙελʔൃද࣌ʹهͨ͠ʮల๬ʯ͸ׂѪ͠·ͨ͠ɻ