Nội dung chính của bài viết nghiên cứu nghiên cứu các tính chất phi cổ điển bậc cao của trạng thái đơn mode chồng chất nén kết hợp tăng cường. Khi khảo sát tính chất nén Hillery bậc cao, nhận thấy trạng thái này thể hiện tính nén Hillery bậc chẵn càng mạnh khi tham số nén càng lớn.
◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❇❾❈ ❈❆❖ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ✣❒◆ ▼❖❉❊ ❈❍➬◆● ❈❍❻❚ ◆➆◆ ❑➌❚ ❍ÑP ❚❿◆● ❈×❮◆● ◆●❯❨➍◆ ❚❍➚ ❚❍❯1 ✱ ▲➊ ❚❍➚ ❍➬◆● ❚❍❆◆❍2 ì 3, rữớ ữ P ỵ ✲ ❍â❛ ✲ ❙✐♥❤✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉↔♥❣ ◆❛♠ t ỵ rữớ ữ ❍✉➳ ✯❊♠❛✐❧✿ tr✉♦♥❣♠✐♥❤❞✉❝❅❞❤s♣❤✉❡✳❡❞✉✳✈♥ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ ❝❛♦ ❝õ❛ tr t ỡ ỗ t t ủ t ❝÷í♥❣✳ ❑❤✐ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦✱ ❝❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② tr↕♥❣ t❤→✐ ♥➔② t❤➸ ❤✐➺♥ t➼♥❤ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❤➤♥ ❝➔♥❣ ♠↕♥❤ ❦❤✐ t❤❛♠ sè ♥➨♥ r ❝➔♥❣ ❧ỵ♥✱ t✉② ♥❤✐➯♥ ♠ù❝ ✤ë ♥➨♥ t❤➸ ❤✐➺♥ ②➳✉ ❦❤✐ ❜➟❝ ❝➔♥❣ ❝❛♦✳ ❍ì♥ ♥ú❛✱ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦✱ ❦➳t q✉↔ ❦❤↔♦ s→t ❝❤♦ t❤➜② tr↕♥❣ t ỡ ỗ t t ủ t ữớ t❤➸ ❤✐➺♥ ❝↔ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦✱ ❤❛✐ t➼♥❤ ❝❤➜t ♥➔② t❤➸ ❤✐➺♥ ❝➔♥❣ ②➳✉ ❦❤✐ r ✈➔ ❝➔♥❣ ❧ỵ♥✳ ❚ø ❦❤â❛✿ ❚♦→♥ tû ♥➨♥ t➠♥❣ ❝÷í♥❣✱ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦✱ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦✱ ❤✐➺✉ ù♥❣ ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦✳ ❚â♠ t➢t✿ ✶ ●■❰■ ❚❍■➏❯ ▼ët tr♦♥❣ ♥❤ú♥❣ ♥❣✉②➯♥ t➢❝ ❝ì ❜↔♥ ♥❤➜t tr♦♥❣ ❝ì ❤å❝ ❧÷đ♥❣ tỷ sỹ ỗ t ữủ tỷ ởt ổ ❝ư q✉❛♥ trå♥❣ ✤➸ ✤÷❛ r❛ ✈➔ t❤❛♦ t→❝ ❝→❝ tr↕♥❣ t❤→✐ ❧÷đ♥❣ tû✳ ❚r♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ♥❤✐➲✉ tr↕♥❣ t❤→✐ ❧÷đ♥❣ tû ♣❤✐ ❝ê ✤✐➸♥ ❦❤→❝ ♥❤❛✉ ✤÷đ❝ ợ t ỗ t tr t t ❤đ♣ ✈ỵ✐ ❝→❝ ♣❤❛ ❦❤→❝ ♥❤❛✉ ❬✶❪✱ ❬✷❪✳ ❘♦② ✭✶✾✾✽✮ ✤➣ ①➙② ❞ü♥❣ ❝→❝ tr↕♥❣ t❤→✐ ❦➳t ❤ñ♣ ❝❤➤♥ ✈➔ ❧➫✳ ❚➜t ❝↔ ❝→❝ tr↕♥❣ t❤→✐ ♥➔② t❤➸ ❤✐➺♥ ❝→❝ t ú ỵ ữ tố sPss ù♥❣ ♥➨♥ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠✳ ▼➦t ❦❤→❝✱ t➼♥❤ ❝❤➜t ♥➨♥ ❝õ❛ tr↕♥❣ t❤→✐ ❧÷đ♥❣ tû ✤➣ ✤÷đ❝ t❤↔♦ ❧✉➟♥ ✤➸ ❧➔♠ ❣✐↔♠ sü ♣❤➙♥ t→♥ ð ♠ët tr♦♥❣ ❤❛✐ t❤➔♥❤ ♣❤➛♥ ✈✉æ♥❣ ❣â❝ ❝õ❛ tå❛ ✤ë ✈➔ ①✉♥❣ ❧÷đ♥❣✳ ❈→❝❤ ✤ì♥ ❣✐↔♥ ✤➸ t↕♦ r❛ tr↕♥❣ t❤→✐ ♥➨♥ t➠♥❣ ❝÷í♥❣ ❧➔ t→❝ ❞ư♥❣ t♦→♥ tû ♥➨♥ t➠♥❣ ❝÷í♥❣ ❧➯♥ ♠ët tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ t❤ỉ♥❣ t❤÷í♥❣ ❬✸❪✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ♠ỵ✐ ♥➔② ✤➣ ♠ð r❛ ♥❤ú♥❣ ù♥❣ ❞ư♥❣ ♠ỵ✐ tr♦♥❣ ❦ÿ t❤✉➟t✳ ⑩♣ ❞ö♥❣ ❝→❝ tr↕♥❣ t❤→✐ ♥➔② ✈➔♦ t❤ü❝ ♥❣❤✐➺♠ ❝❤♦ ♣❤➨♣ ❝❤ó♥❣ t❛ t↕♦ r❛ ❝→❝ t❤✐➳t ❜à q✉❛♥❣ ❤å❝✱ ❝→❝ t❤✐➳t ❜à ✤✐➺♥ tû ✈ỵ✐ ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦ ✤➸ ✤→♣ ù♥❣ sü ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳ ■❙❙◆ ✶✽✺✾✲✶✻✶✷✱ ❙è ✸✭✺✾✮✴✷✵✷✶✿ tr✳✻✼✲✼✼ ◆❣➔② ♥❤➟♥ ❜➔✐✿ ✷✼✴✶✶✴✷✵✷✵❀ ❍♦➔♥ t❤➔♥❤ ♣❤↔♥ ❜✐➺♥✿ ✶✺✴✶✷✴✷✵✷✵❀ ◆❣➔② ♥❤➟♥ ✤➠♥❣✿ ✶✻✴✶✷✴✷✵✷✵ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ✻✽ ✈➔ ❝s✳ ♣❤→t tr✐➸♥ ❝õ❛ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t ♥❣➔② ♥❛②✳ ❱➔♦ ♥➠♠ ✷✵✶✾✱ tr↕♥❣ t❤→✐ ❣å✐ ❧➔ tr↕♥❣ t ỡ ỗ t t ủ t ữớ ✤➣ ✤÷đ❝ ✤➲ ①✉➜t ♥❤÷ s❛✉ ❬✷❪✿ |ψ V,θ = N V (λ, r) exp − |α|2 ∞ αn + eiθ (−α)n √ |n , n! n=0 ✭✶✮ ❧➔ ❤➺ sè ❝❤✉➞♥ ❤â❛✳ ❚♦→♥ tû ♥➨♥ t➠♥❣ ❝÷í♥❣ N = + e−2|α| ❝♦sθ iλ † †2 V (λ, r) = e− [a cosh r+(2a a+1) sinh r+a cosh r] ợ a a ữủt ❧➔ t♦→♥ tû s✐♥❤ ✈➔ t♦→♥ tr♦♥❣ ✤â −1/2 tû ❤õ② ❤↕t tr÷í♥❣ ❜♦s♦♥✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ t❤➜♣ ❝õ❛ tr↕♥❣ t❤→✐ ♥➔② ✤➣ ✤÷đ❝ t❤➸ ❤✐➺♥ ❝❤✐ t✐➳t ❬✷❪✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ ❝❛♦ ❝õ❛ ❝❤ó♥❣ ✈➝♥ ❝❤÷❛ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉✳ ❱➻ ✈➟②✱ tr♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tæ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ tr t ỡ ỗ t t ❤đ♣ t➠♥❣ ❝÷í♥❣ t❤ỉ♥❣ q✉❛ t➼♥❤ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦✱ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦✳ ✷ ❚➑◆❍ ❈❍❻❚ ◆➆◆ ❍■▲▲❊❘❨ ❇❾❈ ❈❆❖ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ✣❒◆ ▼❖❉❊ ❈❍➬◆● ❈❍❻❚ ◆➆◆ ❑➌❚ ❍ÑP ❚❿◆● ❈×❮◆● ❍✐➺✉ ù♥❣ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦✱ ✤➛✉ t✐➯♥ ữủ ợ t r s õ ữủ ❝→❝ t→❝ ❣✐↔ ❦❤→❝ ♣❤→t tr✐➸♥ t❤➯♠ ❬✺❪✱ ❬✻❪✱ ❬✼❪✳ ▼ët tr↕♥❣ t❤→✐ ✤÷đ❝ ❣å✐ ❧➔ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ ♥➳✉ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✮ F , ∆Q (ϕ) < = ! (τ ) ( − τ )!τ ! tr♦♥❣ ✤â ❬✺❪ F a , a = =1 ợ ( ) = ữ s ( − 1) ( − τ + 1)✳ a† a S = a† −τ a −τ ✭✸✮ , ✣➸ t❤✉➟♥ t✐➺♥✱ ❤➺ sè ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ ✤➣ ✤÷đ❝ ✤÷❛ r❛ + e−2i ϕ a2 ! (t) t=1 ( −t)!t! e−i −2 (a† ) −t ϕ a a −t ✭✹✮ ✣✐➲✉ ❦✐➺♥ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ ❝õ❛ ♠ët tr↕♥❣ t❤→✐ ♥➔♦ ✤â ❧➔ ❤➺ sè ♥➨♥ S ♣❤↔✐ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ −1 ≤ S < ✈➔ tr t ỵ tữ S = t tr tr tr t ỡ ỗ ❝❤➜t ♥➨♥ ❦➳t ❤đ♣ t➠♥❣ ❝÷í♥❣✱ t❛ ❝â a† a V,θ = 2N |α|2 |µ|2 + |ν|2 B + T1 A +|ν|2 A , ✭✺✮ ◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳ a†2 a2 V,θ ✻✾ |µ|4 + |ν|4 |α|4 A + |µ|2 T1 2|α|2 B + A + T2 A = 2N + |µν|2 A + 8|α|2 B + 4|α|4 A + |ν|2 T2 5A + 2|α|2 B +|ν|4 2A + 4|α|2 B a†3 a3 V,θ , ✭✻✮ |µ|6 + |ν|6 |α|6 B + 3|µ|4 T1 |α|4 A + |α|2 B = 2N + 9|µ|4 |ν|2 |α|6 B + 3|α|4 A + |α|2 B + 9|µν|2 T1 |α|4 A + 4|α|2 B + 2A + 3|ν|2 T2 |α|2 B + 4A + 9|µ|2 |ν|4 [|α| B + 6|α|4 A + 7|α|2 B +A] +3|ν|4 T1 |α|4 A + 7|α|2 B + 9A + |ν|6 9|α|4 A + 18|α|2 B + 6A a†4 a4 V,θ ✭✼✮ , |µ|8 + |ν|8 |α|8 A + 2|µ|6 T1 2|α|6 B + 3|α|4 A + T4 A = 2N + 3|µ|4 T2 A + 4|α|2 B + 2|α|4 A + 2|µ|2 T3 2|α|2 B + 3A + 4|à|6 ||2 ì 4||8 A + 16||6 B + 9|α|4 A + 6|µ|4 |ν|2 T1 4|α|6 B + 22|α|4 A +22|α|2 B + 3A + 4|µν|2 T2 4|α|4 A + 24|α|2 B + 21A + 2|ν|2 T3 × 11A + 2|α|2 B + 9|µν|4 A + 24|α|2 B + 60|α|4 A + 32|α|6 B +4|α|8 A + 6|µ|2 |ν|4 T1 4|α|6 B + 38|α|4 A + 86|α|2 B + 39A + 3|ν|4 T2 41A + 20|α|2 B + 2|α|4 A + 4|µ|2 |ν|6 4|α|8 A + 48|α|6 B +153|α|4 A + 132|α|2 B + 18A + 2|ν|6 T1 84A + 96|α|2 B + 27|α|4 A +2|α|6 B + |ν|8 24A + 96|α|2 B + 72|α|4 A + 16|α|6 B a a2 a3 V,θ = 2iN e−2|α| sin θ (µα − να∗ ) , = 2N µ2 α2 A + µν 2|α|2 B + A + ν α∗2 A , V,θ = (2i) N e−2|α| sin θ µ3 α3 + |α|2 − a4 V,θ = 2N µν α∗ − µ2 να − ν α∗3 , µ4 α4 + ν α∗4 A + µ3 να2 + µν α∗2 +3µ2 ν A + 4|α|2 B + 2|α|4 A , ✭✽✮ ✭✾✮ V,θ , ✭✶✵✮ ✭✶✶✮ 2|α|2 B + 3A ✭✶✷✮ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ✼✵ a6 V,θ = 2N µ6 α6 + ν α∗6 A + µ5 να4 + µν α∗4 + 15 µ4 ν α2 + µ2 ν α∗2 V,θ = 2N 6|α|2 B + 15A |α|4 A + 4|α|2 B + 3A +5µ3 ν 4|α|6 B + 18|α|4 A + 18|α|2 B + 3A a8 ✈➔ ❝s✳ ✭✶✸✮ , µ8 α8 + ν α∗8 A + µ7 να6 + µν α∗6 4|α|2 B + 14A 210A + 168|α|2 B + 28|α|4 A + µ6 ν α4 + µ2 ν α∗4 + µ5 ν α2 + µ3 ν α∗2 210A + 420|α|2 B + 210|α|4 A + 28|α|6 B +5µ4 ν 21A + 168|α|2 B + 252|α|4 A + 112|α|6 B + 14|α|8 A , ✭✶✹✮ ✈ỵ✐ A = + e−2|α| ❝♦sθ , B = − e−2|α| ❝♦sθ ✱ ✈➔ Tn = (µ∗να∗2)n + (µν ∗α2)n ✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ tỉ✐ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ ❜➟❝ ❝❛♦ ố tr t ỡ ỗ t ♥➨♥ ❦➳t ❤đ♣ t➠♥❣ ❝÷í♥❣✿ ❛✮ ◆➨♥ ❍✐❧❧❡r② ❜➟❝ ❤❛✐ 2 S2 = a†2 a2 V,θ + e−4iϕ a4 V,θ −2 e−2iϕ a2 V,θ F2 ✭✶✺✮ ❙û ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✻✮✱ ✭✶✵✮✱ ✭✶✷✮ ❝❤♦ ❝→❝ sè ❤↕♥❣ a†2a2 V,θ , e−2iϕ a2 V,θ , e−4iϕ a4 ✳ ❚❤❛② = ✈➔♦ ❝æ♥❣ t❤ù❝ ✭✸✮ ✈➔ t❤ü❝ ❤✐➺♥ ❝→❝ ♣❤➨♣ ❜✐➳♥ t t ữủ F2 = 4N 2B||2 |à|2 + |ν|2 + 2AT1 + A 2|ν|2 + ❍➻♥❤ ✶✿ ❍➺ sè ♥➨♥ ❜➟❝ ❤❛✐ ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ r✳ ✭✶✻✮ ◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳ ✼✶ ❍➻♥❤ ✭✶✮ ♠æ t↔ ❤➺ sè ♥➨♥ ❜➟❝ tr t ỡ ỗ t t ủ t ữớ tở ợ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ t❤❛♠ sè ♥➨♥ r = 0, 0.2, 0.5 ỗ t t ự ợ t số r ❝➔♥❣ t➠♥❣ t❤➻ t❤❛♠ sè S2 ❝➔♥❣ ➙♠✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❦❤✐ t❤❛♠ sè ♥➨♥ r ❝➔♥❣ t➠♥❣ t❤➻ t➼♥❤ ❝❤➜t ♥➨♥ t❤➸ ❤✐➺♥ ❝➔♥❣ ♠↕♥❤✱ t➼♥❤ t t tữỡ tỹ ợ t t ❜➟❝ ♠ët ❬✷❪✳ ❜✮ ◆➨♥ ❍✐❧❧❡r② ❜➟❝ ❜❛ S3 = a†3 a3 + V,θ e−6iϕ a6 V,θ −2 e−3iϕ a3 V,θ F3 ✭✶✼✮ ❙û ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✼✮✱ ✭✶✶✮✱ ✭✶✸✮ ❝❤♦ ❝→❝ sè ❤↕♥❣ a†3a3 V,θ , e−6iθ a6 V,θ ✳ ❚❤❛② = ✈➔♦ ❝æ♥❣ t❤ù❝ ✭✸✮ ✈➔ t❤ü❝ ❤✐➺♥ ❝→❝ ♣❤➨♣ ❜✐➳♥ e−3iϕ a3 , ✤ê✐ t❛ t❤✉ ✤÷đ❝ F3 = 6N 3A|α|4 |µ|4 + |ν|4 + 6B|α|2 |µ|2 + |ν|2 + 2|ν|4 + 3T1 A |µ|2 + 5|ν|2 + + 2|α|2 |µ|2 + |ν|2 B + 3AT2 + 3|µν|2 [A + 8B|α|2 +4A|α|4 +2A 3|ν|4 + 3|ν|2 + ❍➻♥❤ ✷✿ ✭✶✽✮ ❍➺ sè ♥➨♥ ❜➟❝ ❜❛ ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ r✳ ❍➻♥❤ ✭✷✮ ♠æ t↔ ❤➺ sè tr t ỡ ỗ t t ủ t ữớ tở ợ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ t❤❛♠ sè ♥➨♥ r = 0, 1, 1.5 ỗ t t ự ợ t❤❛♠ sè r ❝➔♥❣ t➠♥❣ t❤➻ t❤❛♠ sè S3 ❝➔♥❣ ❣➛♥ ❣✐→ trà 0✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❦❤✐ t❤❛♠ sè ♥➨♥ r ❝➔♥❣ t➠♥❣ t❤➻ t➼♥❤ ❝❤➜t ♥➨♥ t ữủ ợ t t ♠ët ✈➔ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❤❛✐ ❦❤✐ t❤❛♠ sè ♥➨♥ r ❝➔♥❣ t➠♥❣ t❤➻ t➼♥❤ ❝❤➜t ♥➨♥ t❤➸ ❤✐➺♥ ❝➔♥❣ ♠↕♥❤✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ✼✷ ✈➔ ❝s✳ ❝✮ ◆➨♥ ❍✐❧❧❡r② ❜➟❝ ❜è♥ S4 = a†4 a4 V,θ + e−8iϕ a8 V,θ F4 −2 e−4iϕ a4 V,θ ✭✶✾✮ ❙û ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✽✮✱ ✭✶✷✮✱ ✭✶✹✮ ❝❤♦ ❝→❝ sè ❤↕♥❣ a†4a4 V,θ , e−4iϕ a4 , e−8iϕ a8 V,θ ✳ ❚❤❛② = ✈➔♦ ❝æ♥❣ t❤ù❝ ✭✸✮ ✈➔ t❤ü❝ ❤✐➺♥ ❝→❝ ♣❤➨♣ ❜✐➳♥ t t ữủ F4 = 16N 2B||6 |à|2 + |ν|2 + 6T1 |µ|4 A|α|4 + |α|2 B + 6T2 |µ|2 B|α|2 + A + 2AT3 + 18|µ|4 |ν|2 B|α|6 + 3A|α|4 + B|α|2 + 18|µν|2 T1 A|α|4 + 4B|α|2 + 2A + 6T2 |ν|2 B|α|2 + 4A + 18|µ|2 |ν|4 B|α|6 + 6A|α|4 + 7B|α|2 + A + 3A + 6T1 |ν|4 × A|α|4 + 7B|α|2 + 9A + 2|ν|6 9A|α|4 + 18B|α|2 + 6A + 9A |µ|4 + |ν|4 |α|4 + 9|µ|2 T1 2B|α|2 +A] + 9T2 A + 9|µν|2 A + 8B|α|2 + 4A|α|4 + 12|ν|2 A + 9|ν|2 T2 × 5A + 2B|α|2 + 9|ν|4 2A + 4B|α|2 +12|α|2 B |µ|2 + |ν|2 + 12AT1 ❍➻♥❤ ✸✿ ✭✷✵✮ ❍➺ sè ♥➨♥ ❜➟❝ ❜è♥ ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ r✳ ❍➻♥❤ ✭✸✮ ♠æ t↔ ❤➺ sè ♥➨♥ ❜➟❝ ❜è♥ tr t ỡ ỗ t t ủ t ữớ tở ợ tr ❦❤→❝ ♥❤❛✉ ❝õ❛ t❤❛♠ sè ♥➨♥ r = 0, 0.2, 0.5 ỗ t t t số r ❝➔♥❣ t➠♥❣ t❤➻ t➼♥❤ ❝❤➜t ♥➨♥ t❤➸ ❤✐➺♥ ❝➔♥❣ ♠↕♥❤✱ t t t tữỡ tỹ ợ t t ♥➨♥ ❜➟❝ ♠ët ✈➔ t➼♥❤ ❝❤➜t ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❤❛✐✳ ◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳ ✼✸ ❍➺ sè ♥➨♥ ❜➟❝ ❤❛✐ ✭✤÷í♥❣ ✤ùt ♥➨t✮ ✈➔ ❜➟❝ ❜è♥ ✭✤÷í♥❣ ❧✐➲♥ ♥➨t✮ ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ t❤❛♠ sè ♥➨♥ r = 0.4✳ ❍➻♥❤ ✹✿ ❳➨t ✈➲ ♠ù❝ ✤ë ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ t❤➻ rã r➔♥❣ tr t ỡ ỗ t t ủ t ❝÷í♥❣ ✤➲✉ ①✉➜t ❤✐➺♥ ❤✐➺✉ ù♥❣ ♥➨♥ ❜➟❝ ❤❛✐ ✈➔ ❜➟❝ ❜è♥ ✈ỵ✐ ❣✐→ trà ❝→❝ t❤❛♠ sè α, λ, , θ ✤➣ ❝❤♦✱ ♥❤÷♥❣ ❤✐➺✉ ù♥❣ ♥➨♥ ❜➟❝ ❤❛✐ ♠↕♥❤ ❤ì♥ ❤✐➺✉ ù♥❣ ♥➨♥ ❜➟❝ ❜è♥✳ ❍➻♥❤ ✭✹✮ ❝❤♦ t❛ t❤➜② r➡♥❣✱ sü t➠♥❣ ❞➛♥ ❝õ❛ ❜➟❝ ❝❤➤♥ t❤➻ t❤❛♠ sè Sc ❝➔♥❣ ❧ó❝ ❝➔♥❣ ❣➛♥ ✈ỵ✐ ❣✐→ trà 0✱ ♥❣❤➽❛ ❧➔ t➼♥❤ ❝❤➜t ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ ❝➔♥❣ ②➳✉ ❦❤✐ ❜➟❝ c ❝➔♥❣ ❧ỵ♥✳ ✸ ❚➑◆❍ ❚❍➮◆● ❑➊ ❙❯❇✲P❖■❙❙❖◆ ❇❾❈ ❈❆❖ ❱⑨ ❚➑◆❍ P❍❷◆ ❑➌❚ ❈❍Ò▼ ❇❾❈ ❈❆❖ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ✣❒◆ ▼❖❉❊ ❈❍➬◆● ❈❍❻❚ ◆➆◆ ❑➌❚ ❍ÑP ì tố sPss ữủ ❣✐ỵ✐ t❤✐➺✉ tr♦♥❣ ❬✽❪✳ ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ n( ) = n (n − 1) (n − + 1) = a† a , ✈ỵ✐ n = a† a t❤❛♠ sè P ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ ❬✾❪✿ a† a V,θ P = − 1, ✭✷✶✮ ( a† a ) V, ợ số ữỡ t số P < ❝❤➾ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦ss✐♦♥ ❜➟❝ ❝❛♦✳ ❚❤❡♦ ✤â✱ ●❧❛✉❜❡r ✤à♥❤ ♥❣❤➽❛ ❤➔♠ t÷ì♥❣ q✉❛♥ ❜➟❝ ❝❛♦ ♥❤÷ s❛✉ ❬✶✵❪✿ g( ) = a† a a† a ✭✷✷✮ ❚➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ✈➔ ♠ù❝ ✤ë ♣❤↔♥ t ũ ợ tữỡ q ọ ❤ì♥ s♦ ✈ỵ✐ ✶✳ ❙♦ s→♥❤ ❜✐➸✉ t❤ù❝ g( ) ợ P t t ú tữỡ tỹ ♠➦t t♦→♥ ❤å❝✳ ❈↔ g( ) ✈➔ P ✤➲✉ ❧➔ ❤➔♠ ♣❤ö t❤✉ë❝ ✈➔♦ ❝→❝ t❤❛♠ sè , α, θ, λ, r✳ ❚ø ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠✱ ❝❤ó♥❣ t❛ ♥❤➟♥ t❤➜② r➡♥❣ ♥➳✉ ♠ët tr↕♥❣ t❤→✐ t❤➸ ❤✐➺♥ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ t❤➻ ❝ơ♥❣ s➩ t❤➸ ❤✐➺♥ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠✳ ❚ø ✤➙② ❝â t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ ❝↔ ❤❛✐ t➼♥❤ ❝❤➜t ♥➔② ❝â ♠è✐ ❧✐➯♥ q✉❛♥ ✈ỵ✐ ♥❤❛✉✱ ♥❣❤➽❛ ❧➔ ♥➳✉ tr t ỡ ỗ t t ủ t ❝÷í♥❣ ✤➣ t❤➸ ❤✐➺♥ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ t❤➻ ❝ơ♥❣ s➩ t❤➸ ❤✐➺♥ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠ ❜➟❝ ❝❛♦✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ✼✹ ✈➔ ❝s✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ tæ✐ s➩ ❦❤↔♦ s→t t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐ ỡ ỗ t t ủ t ữớ ợ tr ọ ữủ ữ r ữợ ❛✮ ❱ỵ✐ = ❚❤❛② = ✈➔♦ ❜✐➸✉ t❤ù❝ ✭✷✶✮ ✈➔ sû ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✺✮✱ ✭✻✮ t❛ ✤÷đ❝ t❤❛♠ sè P2 ❝õ❛ tr t ỡ ỗ t t ủ t ữớ | V, õ P2 = 2N |à|4 + |ν|4 |α|4 A + |µ|2 T1 2|α|2 B + A + |ν|2 T1 5A + 2|α|2 B +T2 A + |µν|2 A + 8|α|2 B + 4|α|4 A + ||4 2A + 4||2 B ì 2Nm ||2 |à|2 + |ν|2 B + T1 A + |ν|2 A −2 − ✭✷✸✮ ❚❤❛♠ sè P2 ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ r = 0, 0.8, 1.2 ✈ỵ✐ λ = 0.2 ✈➔ α = eiπ/3 ỗ t t r t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ♠ët ❝õ❛ tr↕♥❣ t❤→✐ ✤ì♥ ỗ t t ủ t ữớ r ợ ữ ự t t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ♠ët ❝õ❛ tr↕♥❣ t❤→✐ ♥➔② ❝➔♥❣ ♠↕♥❤ ❦❤✐ r ❝➔♥❣ ❧ỵ♥✳ ❜✮ ❱ỵ✐ = ❚❤❛② = ✈➔♦ ❜✐➸✉ t❤ù❝ ✭✷✶✮ ✈➔ sû ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✺✮✱ ✭✼✮ t❛ ✤÷đ❝ t❤❛♠ sè P3 ❝õ❛ tr↕♥❣ t❤→✐ ✤ì♥ ỗ t t ủ t ữớ | V, ❝â ❞↕♥❣ P3 = 2N |µ|6 + |ν|6 |α|6 B + 3|µ|4 T1 |α|4 A + |α|2 B + 3|µ|2 T2 |α|2 B + A + T3 A + 9|µ|4 |ν|2 |α|6 B + 3|α|4 A + |α|2 B + 9|µν|2 T1 |α|4 A + 4|α|2 B + 2A + 3|ν|2 T2 |α|2 B + 4A + 9|µ|2 |ν|4 |α|6 B + 6|α|4 A + 7|α|2 B + A +3|ν|4 T1 |α|4 A + 7|α|2 B + 9A + |ν|6 9|α|4 A + 18|α|2 B + 6A × 2Nm |α|2 |µ|2 + |ν|2 B + T1 A + |ν|2 A −3 − ✭✷✹✮ ◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳ ✼✺ ❚❤❛♠ sè P3 ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ r = 0, 0.8, 1.2 ✈ỵ✐ λ = 0.2 = ei/4 ỗ t ❤➻♥❤ ✭✻✮ ❝❤♦ t❤➜② r➡♥❣ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ tr t ỡ ỗ t t ❤đ♣ t➠♥❣ ❝÷í♥❣ ②➳✉ ❞➛♥ ❦❤✐ r t➠♥❣ ❞➛♥✳ ◆❤÷ ✈➟②✱ ♠ù❝ ✤ë t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❤❛✐ ❝õ❛ tr↕♥❣ t❤→✐ ♥➔② ❝➔♥❣ ②➳✉ ❦❤✐ r ❝➔♥❣ ❧ỵ♥✳ ❝✮ ❱ỵ✐ = ❚❤❛② = ✈➔♦ ❜✐➸✉ t❤ù❝ ✭✷✶✮ ✈➔ sû ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t➼♥❤ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✺✮✱ ✭✽✮ t❛ ✤÷đ❝ t❤❛♠ sè P4 tr t ỡ ỗ t t ❤đ♣ t➠♥❣ ❝÷í♥❣ |ψ V,θ ❝â ❞↕♥❣ P4 = 2N |µ|8 + |ν|8 |α|8 A + 2|µ|6 T1 2|α|6 B + 3|α|4 A + A + 4|α|2 B + 2||4 A ì 3|à|4 T2 + 2|à|2 T3 2||2 B + 3A + T4 A + 4|µ|6 |ν|2 4|α|8 A + 16|α|6 B + 9|α|4 A + 6|µ|4 |ν|2 T1 4|α|6 B +22|α|4 A + 22|α|2 B + 3A + 4||4 A + 24||2 B + 21A ì 4|à|2 T2 + 9|µν|4 A + 24|α|2 B + 60|α|4 A + 32|α|6 B +4|α|8 A + 2|ν|2 T3 × 11A + 2|α|2 B + 6|µ|2 |ν|4 T1 4|α|6 B + 38|α|4 A + 86||2 B + 39A + 4|à|2 ||6 ì 4|α|8 A + 48|α|6 B +153|α|4 A + 132|α|2 B + 18A + 2|ν|6 T1 84A + 96|α|2 B +27|α|4 A + 2|α|6 B +|ν|8 24A + 96|α|2 B + 72||4 A + 16||6 B ì 2Nm ||2 |à|2 + |ν|2 B + T1 A + |ν|2 A −4 − ✭✷✺✮ ✼✻ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ✈➔ ❝s✳ ❍➻♥❤ ✼✿ ❚❤❛♠ sè P4 ❧➔ ❤➔♠ ❝õ❛ θ ✈ỵ✐ ❝→❝ ❣✐→ trà ❦❤→❝ ♥❤❛✉ ❝õ❛ r = 0, 0.4, 0.8 ợ = 0.2 = ei/4 ỗ t❤à ❤➻♥❤ ✭✼✮ ❝❤♦ t❤➜② r➡♥❣ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ tr t ỡ ỗ t ❦➳t ❤đ♣ t➠♥❣ ❝÷í♥❣ ②➳✉ ❞➛♥ ❦❤✐ r t➠♥❣ ❞➛♥✳ ◆❤÷ ✈➟②✱ ♠ù❝ ✤ë t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❜❛ ❝õ❛ tr↕♥❣ t❤→✐ ♥➔② ❝➔♥❣ ②➳✉ ❦❤✐ r ❝➔♥❣ ❧ỵ♥✳ ❈↔ ❤❛✐ t➼♥❤ ❝❤➜t ♥➔② ❤♦➔♥ t♦➔♥ tữỡ tỹ ợ ứ õ t t r➡♥❣ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐ ♥➔② ❝➔♥❣ ②➳✉ ❦❤✐ ✈➔ r ❝➔♥❣ ❧ỵ♥✳ ✹ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ ❦❤↔♦ s→t ❝→❝ ❤✐➺✉ ù♥❣ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦✱ t➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t ỡ ỗ t t ủ t ữớ q tr st t t ỗ t❤à t❤ỉ♥❣ q✉❛ ❝→❝ t❤❛♠ sè✱ ❝❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② ♠ù❝ ✤ë ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❤➤♥ tr♦♥❣ tr↕♥❣ t❤→✐ ✤ì♥ ỗ t t ủ t ữớ t ❝➔♥❣ ♠↕♥❤ ❦❤✐ r ❝➔♥❣ ❧ỵ♥✱ t✉② ♥❤✐➯♥ t➼♥❤ ❝❤➜t ♥➔② t❤➸ ❤✐➺♥ ❝➔♥❣ ②➳✉ ❦❤✐ ❜➟❝ ❝➔♥❣ ❝❛♦✳ ❍ì♥ ♥ú❛✱ ♠ù❝ ✤ë t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ trð ♥➯♥ ②➳✉ ❤ì♥ ❦❤✐ t➠♥❣ ❜➟❝ ✈➔ t➠♥❣ t❤❛♠ sè r✳ ❚➼♥❤ t❤è♥❣ ❦➯ s✉❜✲P♦✐ss♦♥ ❜➟❝ ❝❛♦ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠ ❜➟❝ ❝❛♦ ❣➛♥ ♥❤÷ ❣✐è♥❣ ♥❤❛✉ ✈➲ ❜✐➸✉ t❤ù❝ t♦→♥ ❤å❝✱ ❝→❝ ❤➻♥❤ ✈➩ ❝ơ♥❣ t÷ì♥❣ tü ♥❤❛✉✳ ◆❤÷ ✈➟②✱ ♠ù❝ ✤ë t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐ ♥➔② s➩ ❝➔♥❣ ②➳✉ ❦❤✐ , r ❝➔♥❣ ❧ỵ♥✳ ❉♦ ✤â✱ ❝❤ó♥❣ tỉ✐ ❝â t❤➸ t tr t ỡ ỗ t ♥➨♥ ❦➳t ❤đ♣ t➠♥❣ ❝÷í♥❣ t❤➸ ❤✐➺♥ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ ❝❛♦✳ Ð t➼♥❤ ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦✱ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝➔♥❣ ✤÷đ❝ t➠♥❣ ữớ ỡ s ợ t t t❤❛② ✤ê✐ t❤❛♠ sè t➠♥❣ ❝÷í♥❣ ♠ët ❝→❝❤ ♣❤ị ❤đ♣✳ ▲❮■ ❈❷▼ ❒◆ ◆❣❤✐➯♥ ❝ù✉ ♥➔② ✤÷đ❝ t➔✐ trđ ❜ð✐ ◗✉ÿ P❤→t tr✐➸♥ ❦❤♦❛ ❤å❝ ✈➔ ❝æ♥❣ ♥❣❤➺ ◗✉è❝ ❣✐❛ ✭◆❆❋❖❙✲ ❚❊❉✮ tr♦♥❣ ✤➲ t➔✐ ♠➣ sè ✶✵✸✳✵✶✲✷✵✶✽✳✸✻✶✳ ◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳ ✼✼ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❙❝❤❧❡✐❝❤ ❲✳✱ P❡r♥✐❣♦ ▼✳✱ ❑✐❡♥ ❋✳ ▲✳ ✭✶✾✾✶✮✱ ✧◆♦♥❝❧❛ss✐❝❛❧ st❛t❡ ❢r♦♠ t✇♦ ♣s❡✉❞♦❝❧❛s✲ s✐❝❛❧ st❛t❡s✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✹✹✱ ✷✶✼✷✳ ❬✷❪ ❏✐❛♥✲♠✐♥❣ ❉✳✱ ●❛♥❣ ❘✳✱ ❍❛✐✲❥✉♥ ❨✳✱ ❲❡♥✲❤❛✐ ❩✳ ✭✷✵✶✾✮✱ ✧❙q✉❡❡③✐♥❣✲❡♥❤❛♥❝❡❞ ❙✉♣❡r✲ ♣♦s✐t✐♦♥ ♦❢ ❈♦❤❡r❡♥t ❙t❛t❡s ❛♥❞ ❚❤❡✐r ◆♦♥❝❧❛ss✐❝❛❧ Pr♦♣❡rt✐❡s✧✱ ❖♣t✐❝s✳ ❬✸❪ ❍♦♥❣ ❈✳ ❑✳ ❛♥❞ ▼❛♥❞❡❧ ▲✳ ✭✶✾✽✺✮✱ ✧❍✐❣❤❡r✲♦r❞❡r ❙q✉❡❡③✐♥❣ ♦❢ ❛q✉❛♥t✉♠ ❢✐❡❧❞✧✱ P❤②s✲ ✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✺✹✭✹✮✱ ♣♣✳ ✸✷✸✲✸✷✺✳✶✷✳ ❬✹❪ ❍✐❧❧❡r② ▼✳ ✭✶✾✻✸✮✱ ✧❈♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❛♥❞ ♥♦♥❝❧❛ss✐❝❛❧ st❛t❡s ✐♥ ♥♦♥❧✐♥❡❛r ♦♣t✐❝❛❧ s②st❡♠s✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✸✶✱ ♣♣✳✸✽✲✸✹✷✳ ❬✺❪ ❚r✉♦♥❣ ▼✳ ❉✳ ❛♥❞ ◆❣✉②❡♥ ❇✳ ❆✳ ✭✷✵✵✹✮✱ ✧❍✐❧❧❡r②✲❚②♣❡ ❙q✉❡❡③✐♥❣ ✐♥ ❋❛♥ ❙t❛t❡s✧✱ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❑♦r❡❛♥ P❤②s✐❝❛❧ ❙♦❝✐❡t②✱ ✹✹✱ ♣♣✳ ✶✹✷✶✲✶✹✷✻✳✶✻✳ ❬✻❪ ❉♦❞♦♥♦✈ ❱✳ ❱✳✱ ▼❛♥✬❦♦ ❱✳ ■✳ ✭✷✵✵✸✮✱ ✧❚❤❡♦r② ♦❢ ◆♦♥❝❧❛ss✐❝❛❧ ❙t❛t❡s ♦❢ ▲✐❣❤t✧✱ ❚❛②❧♦r ❛♥❞ ❋r❛♥❝✐s✱ ▲♦♥❞♦♥✱ ♣♣✳✷✶✾✲✷✹✵✳ ❬✼❪ ❙✉❞❛rs❤❛♥ ❊✳ ❈✳ ●✳ ✭✶✾✻✸✮✱ ✧❊q✉✐✈❛❧❡♥❝❡ ♦❢ s❡♠✐❝❧❛ss✐❝❛❧ ❛♥❤ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝❛❧ ❞❡s❝r✐♣t✐♦♥s ♦❢ st❛t✐st✐❝❛❧ ❧✐❣❤t ❜❡❛♠s✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✶✵✱ ♣♣✳✷✼✼✲✷✼✾✳ ❬✽❪ ❉❛♥✐❡❧ ❊✳✱ ❘❡❡t❛ ❱✳✱ ❛♥❞ ❙✉r❡♥❞r❛ ❙✳ ✭✷✵✵✶✮✱ ✧❍✐❣❤❡r✲♦r❞❡r s✉❜✲P♦✐ss♦♥✐❛♥ ♣❤♦t♦♥ st❛t✐st✐❝s ✐♥ t❡r♠s ♦❢ ❢❛❝t♦r✐❛❧ ♠♦♠❡♥ts✧✱ ❖♣t✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❆♠❡r✐❝❛✱ ✶✾✭✻✮✱ ♣♣✳✶✹✼✶ ❬✾❪ ❚r✉♦♥❣ ▼✳ ❉✳✱ ❏❛❡✇♦♦ ◆✳ ✭✷✵✵✽✮✱ ✧❍✐❣❤❡r✲♦r❞❡r ♣r♦♣❡rt✐❡s ♦❢ ♣❤♦t♦♥❛❞❞❡❞ ❝♦❤❡r❡♥t st❛t❡s✧✱ ❖♣t✐❝s ❈♦♠♠✉♥✐❝❛t✐♦♥s✱ ✷✽✶✱ ♣♣✳✷✽✹✷✲✷✽✹✽✳ ❬✶✵❪ ●❧❛✉❜❡r ❘✳ ❏✳ ✭✶✾✻✸✮✱ ✧❈♦❤❡r❡♥t ❛♥❞ ■♥❝♦❤❡r❡♥t ❙t❛t❡s ♦❢ t❤❡ ❘❛❞✐❛t✐♦♥ ❋✐❡❧❞✧✱ P❤②s✲ ✐❝❛❧ ❘❡✈✐❡✇ ❇✱ ✶✸✶✭✻✮✱ ♣♣✳✷✼✻✻✲✷✼✽✽✳ ❙❚❯❉❨■◆● ❚❍❊ ❍■●❍❊❘✲❖❘❉❊❘ ◆❖◆❈▲❆❙❙■❈❆▲ P❘❖P❊❘❚■❊❙ ❖❋ ❚❍❊ ❙◗❯❊❊❩■◆●✲❊◆❍❆◆❈❊❉ ❙❯P❊❘P❖❙■❚■❖◆ ❖❋ ❈❖❍❊❘❊◆❚ ❙❚❆❚❊ ❚✐t❧❡✿ ❆❜str❛❝t✿ ■♥ t❤❡ ♣❛♣❡r✱ ✇❡ st✉❞② t❤❡ ❤✐❣❤❡r✲♦r❞❡r ♥♦♥❝❧❛ss✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ sq✉❡❡③✐♥❣✲ ❡♥❤❛♥❝❡❞ s✉♣❡r♣♦s✐t✐♦♥ ♦❢ ❝♦❤❡r❡♥t st❛t❡ ❛s ❤✐❣❤❡r✲♦r❞❡r ❍✐❧❧❡r② sq✉❡❡③✐♥❣✱ ❤✐❣❤❡r✲♦r❞❡r s✉❜✲P♦✐ss♦♥✐❛♥ ❞✐str✐❜✉t✐♦♥s✱ ❤✐❣❤❡r✲♦r❞❡r ❛♥t✐❜✉♥❝❤✐♥❣ ❡❢❢❡❝t✳ ❚❤❡ r❡s✉❧ts s❤♦✇ t❤❛t t❤✐s st❛t❡ ❡①❤✐❜✐ts ❡✈❡♥ ♦r❞❡r ❍✐❧❧❡r② sq✉❡❡③✐♥❣✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❍✐❧❧❡r② sq✉❡❡③✐♥❣ ❛❧✇❛②s ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ ✐♥❝r❡❛s❡ ♦❢ t❤❡ sq✉❡❡③✐♥❣ ♣❛r❛♠❡t❡r r ❜✉t t❤✐s ♣r♦♣❡rt② ❜❡❝♦♠❡s ✇❡❛❦❡r ✇❤❡♥ ✐♥❝r❡❛s✐♥❣ t❤❡ ♦r❞❡r✳ ■♥ ❛❞❞✐t✐♦♥✱ ✐♥ t❤❡ sq✉❡❡③✐♥❣✲❡♥❤❛♥❝❡❞ s✉♣❡r♣♦s✐t✐♦♥ ♦❢ ❝♦❤❡r❡♥t st❛t❡✱ t❤❡ r❡s✉❧t ❛❧s♦ s❤♦✇s t❤❛t t❤❡ ❤✐❣❤❡r✲♦r❞❡r s✉❜✲P♦✐ss♦♥✐❛♥ ❞✐str✐❜✉t✐♦♥s ❛s ✇❡❧❧ ❛s t❤❡ ❤✐❣❤❡r✲♦r❞❡r ❛♥t✐❜✉♥❝❤✐♥❣ ❡❢❢❡❝t ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ ✇❡❛❦❡r ✇❤❡♥ ✐♥❝r❡❛s✐♥❣ t❤❡ ♦r❞❡r ❛♥❞ sq✉❡❡③✐♥❣✲❡♥❤❛♥❝❡❞ ♣❛r❛♠❡t❡r✳ ❑❡②✇♦r❞s✿ ❙q✉❡❡③✐♥❣✲❡♥❤❛♥❝❡❞ ♦♣❡r❛t♦r✱ ❤✐❣❤❡r✲♦r❞❡r ❍✐❧❧❡r② sq✉❡❡③✐♥❣✱ ❤✐❣❤❡r✲♦r❞❡r s✉❜✲P♦✐ss♦♥✐❛♥ ❞✐str✐❜✉t✐♦♥s✱ ❤✐❣❤❡r✲♦r❞❡r ❛♥t✐❜✉♥❝❤✐♥❣ ❡❢❢❡❝t✳