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Classical analog for exotic quantum holonomy
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Atushi TANAKA
September 23, 2017
Research
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150
Classical analog for exotic quantum holonomy
(This presentation is in Japanese)
題目:新奇な量子ホロノミーの古典対応物
日本物理学会 2017年秋季大会 23pJ24-10 での講演スライド
Atushi TANAKA
September 23, 2017
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Transcript
1 / 15 ৽حͳྔࢠϗϩϊϛʔͷݹయରԠ Classical analog for exotic quantum holonomy
ాதಞ࢘ (टେཧ)ɺશथ (ߴՊେ) Atushi Tanaka (TMU) and Taksu Cheon (KUT) 2017-09-23 ຊཧֶձ 2017 ळقେձ ؠखେֶʢ্ాΩϟϯύεʣ 23pJ24-10
2 / 15 ࣍ ʮ৽حͳྔࢠϗϩϊϛʔʯͷݹయՄੵܥͰͷରԠɺͭ·Γɺ ʮτʔϥε͕அαΠΫϧʹΑͬͯผͷτʔϥεʹҠΔʯ͜ͱʹ͍ͭͯ ೋͭͷࣄྫΛհ͢Δɻ ͡Ίʹ ઢܗΩοΫυεϐϯͰͷݫີͳྫ ඇઢܗΩοΫυεϐϯͰͷྫ
·ͱΊ
͡Ίʹ 3 / 15 ৽حͳྔࢠϗϩϊϛʔ அαΠΫϧʹର͢Δ (ٖ) ݻ༗ΤωϧΪʔݻ༗ۭؒͷඇࣗ໌ͳԠɻ (1) அαΠΫϧ
C ʹԊͬͯ (ٖ) ݻ༗Τω ϧΪʔΛͨ݁͠ՌɺݩʹΒͳ͍͜ͱɻ λ 0 2π E E1 E2 E3 ਤ. λ ͕ 2π पظͷ߹ (2) ͢Δͱɺఆৗঢ়ଶΛஅดܦ࿏ C ʹԊͬͯ࣌ؒൃలͤ͞Δͱ ⟨࢝ঢ়ଶ|ऴঢ়ଶ⟩ = 0 ͱͳΔɻ Ref. AT and TC, ཧֶձࢽ 4 ݄߸ (2017)ɻ
͡Ίʹ 4 / 15 அঢ়ଶͷมԽΛݹయతʹߟ͑Δͱ. . . ԾఆɿରԠ͢Δݹయܥ͕Մੵͩͱ͢Δɻ ՄੵܥͰݹయྔࢠԽ͔݅Βݻ༗ঢ়ଶͱτʔϥε͕ରԠ͢Δɿ C
| 1 | 0 q p q p ࢝ঢ়ଶͷτʔϥεஅαΠΫϧ C ʹΑͬͯผͷτʔϥεʹҠΔͣɻ
ઢܗΩοΫυεϐϯͰͷݫີͳྫ 5 / 15 ࣍ ͡Ίʹ ઢܗΩοΫυεϐϯͰͷݫີͳྫ ඇઢܗΩοΫυεϐϯͰͷྫ ·ͱΊ
ઢܗΩοΫυεϐϯͰͷݫີͳྫ 6 / 15 ݹయઢܗΩοΫυεϐϯ εϐϯ S ʹ੩࣓ͱύϧε࣓ (पظ T)
ΛҹՃɿ H(t) = B(t)·S, ͜͜Ͱ B(t) ≡ ωe z +λn∑ n δ(t −nT). n (୯ҐϕΫτϧ) ͱ λ ύϧε࣓ͷํͱڧ͞ɻϙΞϯΧϨࣸ૾ɿ S′ = Mλ S, Mλ 3x3 ަߦྻͰ͋ΓճసߦྻͷੵʹղͰ͖Δ: Mλ = R(e z,ωT)R(n,λ) ͜͜ͰɺR(n,λ) ճస࣠ n ʹ͍ͭͯ֯ λ ͷճసɻ Mλ+2π = Mλ ͔Βɺλ Λ 0 ͔Β 2π ʹ૿Ճͤ͞Δดܦ࿏ C ΛௐΔɻ
ઢܗΩοΫυεϐϯͰͷݫີͳྫ 7 / 15 ݹయઢܗΩοΫυεϐϯɿྫ ดܦ࿏ C ʹԊͬͯ λ Λ
0 ͔Β 2π ʹஅతʹ૿ͨ͠߹ɻ ਤͷԣ࣠ํҐ֯ q ≡ arctan(Sy /Sx ). ਖ਼४ڞͳӡಈྔ p ≡ Sz ͕ॎ࣠ɻ அαΠΫϧ C ʹΑͬͯɺτʔϥεผͷτʔϥεʹҠͬͨɻ
ઢܗΩοΫυεϐϯͰͷݫີͳྫ 8 / 15 આ໌ 1: ճస࣠ͱճస֯ͷղͷᐆດ͔͞Β ઢܗΩοΫυεϐϯಛ༗ͷॳతͳઆ໌ɻ (1) ઢܗͳͷͰϙΞϯΧϨࣸ૾
Mλ = R(l λ ,∆λ ) ͱ Ͱ͖Δɻ͜͜Ͱஅతͳճస࣠ l λ ͱճస֯ ∆λ Λಋ ೖͨ͠ɻ͜ΕΒ λ ʹ͍ͭͯ࿈ଓɻ (2) తͳܭࢉ͔Β l 2π = −l 0, ∆2π = 2π −∆0. ͭ ·Γɺ͜ΕΒ λ ʹ͍ͭͯ 2π पظͰແ͍ɻ͜Ε Mλ+2π = Mλ ͱໃ६͠ͳ͍͜ͱʹҙ (ӈਤ)ɻ (3) M0 ͷݻఆ l 0 அαΠΫϧ C ͷ݁Ռͱͯ͠ −l 0 ʹҠΔɻτʔϥεͷҠಈಉ༷ɻ ∆0 2π − ∆0
ઢܗΩοΫυεϐϯͰͷݫີͳྫ 9 / 15 આ໌ 2: spin-1 2 ͰͷݫີͳྔࢠݹయରԠ εϐϯ-1
2 ಛ༗ͷઆ໌ɻ ͜ͷͱ͖ྔࢠܥͷظͷӡಈํఔࣜͱݹయܥͷӡಈํఔࣜಉ͡ܗɿ d dt S = B(t)×S (͍ΘΏΔϒϩοϗํఔࣜ). ͜ΕͱɺྔࢠܥͰஅαΠΫϧ C ͕৽حͳྔࢠϗϩϊϛʔΛى͜͢͜ͱ ͔ΒɺݹయܥͰτʔϥε͕Ҡಈ͢Δɻ ͳ͓ɺྔࢠܥͷϑϩέ࡞༻ૉ͕ C Λดܦ࿏ʹ࣋ͭͨΊʹɺ͜ͷྫͷݹయܥͷ ϋϛϧτχΞϯʹʮ͓·͚ʯ͕ඞཁ (AT and Miyamoto 2007): Hྔࢠ(t) = Hݹయ(t)+ 1 2 ¯ hλ ∑ n δ(t −nT).
ઢܗΩοΫυεϐϯͰͷݫີͳྫ 10 / 15 આ໌ 3: Ґ૬زԿֶతͳઆ໌ʢγφϦΦʣ ͜ͷܥͷτʔϥεશͯ (༿ߏ) Λߟ͑Δɻ͋Δτʔϥε͔Βग़ൃͯ͠ɺ
அαΠΫϧͰͨͲΓ͖ͭಘΔͷɺ࡞༻ม I = τʔϥε p dq ͷ͕ಉ ͡τʔϥεͷΈɻ ͪͳΈʹɺεϐϯͰ I ∝ ʢτʔϥε͕ுΔཱମ֯ʣͱͰ͖Δɻ ઢܗΩοΫυεϐϯͰɺ͜ͷΑ͏ͳτʔϥεࣗ ΛؚΊೋͭͷΈ (ॖୀ͠ಘΔ͕ྫ֎త)ɻ ͜ͷτʔϥεͷඃ෴ۭؒΛͳ͢ɻ͢Δͱɺ৽حͳ ྔࢠϗϩϊϛʔͷҐ૬زԿֶతͳఆࣜԽ (AT and TC 2015) Λར༻Ͱ͖Δɻྫ͑ҰͱϗϞτϐοΫͳஅ αΠΫϧτʔϥεΛҠಈͤ͞Δ͜ͱෆՄೳɻ
ඇઢܗΩοΫυεϐϯͰͷྫ 11 / 15 ࣍ ͡Ίʹ ઢܗΩοΫυεϐϯͰͷݫີͳྫ ඇઢܗΩοΫυεϐϯͰͷྫ ·ͱΊ
ඇઢܗΩοΫυεϐϯͰͷྫ 12 / 15 ඇઢܗΩοΫυεϐϯ ۙՄੵܥͰͷྫΛհ͢ΔɻۙՄੵܥʹҰൠతͳஅఆཧ͕ແ͍ ͜ͱʹҙɻ H(t) = B(t)·S
ͱͯ͠ɺB(t) ͕ऑ͘ Sz ʹґଘ͢Δ߹Λߟ͑Δ: H(t) = [ ωSz + 1 2 kS2 z ] +λn·S∑ n δ(t −nT). cf. Haake, K´ us and Scharf 1987. ϙΞϯΧϨࣸ૾ඇઢܗʹͳΔ͕ɺλ ͷपظੑมΘΒͳ͍ͷͰಉ͡அ αΠΫϧ C Λ͏ɻ
ඇઢܗΩοΫυεϐϯͰͷྫ 13 / 15 ඇઢܗ ΩοΫυεϐϯɿྫ (k = 0.1) C
ʹԊͬͨஅతͳ࣌ؒൃలɻ͜͜Ͱ (q,p) = (arctan(Sy /Sx ),Sz). அαΠΫϧ C ʹΑͬͯɺτʔϥεผͷτʔϥεͷۙ͘ʹҠͬͨɻ
·ͱΊ 14 / 15 ·ͱΊ ▶ ৽حͳྔࢠϗϩϊϛʔͷݹయՄੵܥͰͷରԠΛಘͨɻ͢ͳΘͪɺ அαΠΫϧ C ʹΑͬͯɺݹయܥͷτʔϥε͕ผͷτʔϥεʹҠΔ
Α͏ͳྫΛݟ͍ͩͨ͠ɻ ▶ ઢܗΩοΫυεϐϯͰͷݫີͳྫʹ͍ͭͯɺࡾͭͷղऍΛ༩͑ͨ (ճసߦྻ͔Βɺεϐϯ-1/2 ͷྔࢠݹయରԠɺҐ૬زԿֶతͳղऍ)ɻ ▶ ඇՄੵܥͷྫΛࣔͨ͠ɻ [͜ΕՄੵۙࣅͰղऍͰ͖ΔʢՄੵۙࣅʹ͍ͭͯྫ͑ɺ Hanada, Shudo and Ikeda, PRE 2015ʣ ɻ]
·ͱΊ 15 / 15 ల ۩ମྫΛ૿͢ Euler top, αΠΫϩτϩϯӡಈͷ֦ுɻ Ґ૬زԿֶతͳղऍͷඋ
ʮ༿ߏͷύϥϝʔλʔมܗʹର͢ΔϞϊυϩϛʔʯͱղऍͰ͖ Δͱ༧ɻ ࡞༻มͷΛมԽͤ͞ΔஅαΠΫϧଘࡏ͢Δ͔ʁ ৽حͳྔࢠϗϩϊϛʔ͔Βଘࡏͯ͠ ྑͦ͞͏͕ͩɺ͍·ͷͱ͜Ζଘ൱ ෆ໌ɻ C | 1 | 0 q p q p