Khảo sát các tính chất phi cổ điển của trạng thái hai mode kết hợp SU(1,1) thêm một và bớt một photon lẻ

✣❸■ ❍➴❈ ❍❯➌ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ P❍❆◆ ❚❍➚ ❚❹▼ ❑❍❷❖ ❙⑩❚ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ÑP ❙❯✭✶✱✶✮ ❚❍➊▼ ▼❐❚ ❱⑨ ❇❰❚ ▼❐❚ P❍❖❚❖◆ ▲➈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❱❾❚ ▲Þ ▲Þ ❚❍❯❨➌❚ ❱⑨ ❱❾❚ ▲Þ ❚❖⑩◆ ▼➣ sè ✿ ✻✵ ✹✹ ✵✶ ị ữớ ữợ ❦❤♦❛ ❤å❝ P●❙✳❚❙✳ ❚❘×❒◆● ▼■◆❍ ✣Ù❈ ❍✉➳✱ ♥➠♠ ✷✵✶✽ ✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✱ ❝→❝ sè ❧✐➺✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ tỹ ữủ ỗ t sỷ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦ý ♠ët ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ♥➔♦ ❦❤→❝✳ ❍✉➳✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ P❤❛♥ ❚❤à ❚➙♠ ✐✐ ▲❮■ ❈❷▼ ❒◆ ❍♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ♥➔②✱ tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② P rữỡ ự t t ữợ ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥✳ tổ t ỡ qỵ ổ tr t ỵ ỏ t ❙❛✉ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳❀ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❈❛♦ ❤å❝ ❦❤â❛ ✷✺ ❝ò♥❣ õ ỵ ú ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❍✉➳✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ P❤❛♥ ❚❤à ❚➙♠ ✐✐✐ ▼Ö❈ ▲Ư❈ ❚r❛♥❣ ♣❤ư ❜➻❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▲í✐ ❝↔♠ ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳ ❚r↕♥❣ t❤→✐ ❦➳t ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ữỡ é ị ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✷✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ tr↕♥❣ t❤→✐ ❦➳t ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳ ❚r↕♥❣ t❤→✐ ♥➨♥ ❝❤♦ ❤❛✐ t♦→♥ tû ❍❡r♠✐t✐❝ Aˆ ✈➔ ˆ B t❤❡♦ t❤ù tü ữủ t ỵ ❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳ ▼ët sè t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸✳✶✳ ◆➨♥ tê♥❣ ✈➔ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸✳✷✳ ◆➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸✳ ❙ü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸✳✹✳ ❚➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✹✳ ▼ët sè t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✹✳✶✳ ❚✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✷✺ ❈❤÷ì♥❣ ✷✳ ❑❍❷❖ ❙⑩❚ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ ◆➆◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ÑP ❙❯✭✶✱✶✮ ❚❍➊▼ ▼❐❚ ❱⑨ ❇❰❚ ▼❐❚ P❍❖❚❖◆ ▲➈ ✷✳✶✳ ❚r↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✶✳ ❚r↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶ ✷✳✶✳✷✳ ❚r↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ tê♥❣ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸✳ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ ❤✐➺✉ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹✳ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ ❍✐❧❧❡r② ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ❈❤÷ì♥❣ ✸✳ ❑❍❷❖ ❙⑩❚ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍❷◆ ❑➌❚ ❈❍Ò▼✱ ❙Ü ❱■ P❍❸▼ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❈❆❯❈❍❨✲ ❙❈❍❲❆❘❩✱ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ÑP ❙❯✭✶✱✶✮ ❚❍➊▼ ▼❐❚ ❱⑨ ❇❰❚ ▼❐❚ P❍❖❚❖◆ ▲➈ ✸✳✶✳ ❑❤↔♦ s→t ❝→❝ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ị♠ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✺✻ ✸✳✶✳✶✳ ❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✶✳✷✳ ❚r÷í♥❣ ❤đ♣ l = 1✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✶✳✸✳ ❚r÷í♥❣ ❤ñ♣ l = 2✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✶✳✹✳ ❚r÷í♥❣ ❤đ♣ l = 2✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✶✳✺✳ ❚r÷í♥❣ ❤đ♣ l = 3✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✶✳✻✳ ❚r÷í♥❣ ❤đ♣ l = 3✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✶✳✼✳ ❚r÷í♥❣ ❤đ♣ l = 3✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✶✳✽✳ ❚r÷í♥❣ ❤ñ♣ l = 4✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✶✳✾✳ ❚r÷í♥❣ ❤đ♣ l = 4✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✳✶✳✶✵✳❚r÷í♥❣ ❤đ♣ l = 4✱ p = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✷ ✸✳✷✳ ❑❤↔♦ s→t sü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✸✳✸✳ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ❜➡♥❣ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P✳✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ P❍Ö ▲Ö❈ ✸ ❉❆◆❍ ❙⑩❈❍ ❍➐◆❍ ❱➇ ❍➻♥❤ ✷✳✶ ❍➻♥❤ ✷✳✷ ❍➻♥❤ ✷✳✸ ❍➻♥❤ ✷✳✹ ❍➻♥❤ ✸✳✺ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ ✈➔♦ r ✈➔ q = 1, 2, ❝è ✤à♥❤ cos 2(ϕ + φ) = −1✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t số t tự tỹ tữỡ ự ợ ữớ ✤ä✱ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ D ✈➔♦ r ✈➔ q = 1, 2, ❝è ✤à♥❤ cos 2(ϕ + φ) = −1✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t số t tự tỹ tữỡ ự ợ ữớ ✤ä✱ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ H2(φ) ✈➔♦ r ✈➔ q = 1, 2, ❝è ✤à♥❤ cos 2(ϕ + φ) = −1✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t số t tự tỹ tữỡ ự ợ ữớ ✤ä✱ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ H3(φ) ✈➔♦ r ✈➔ q = 1, 2, ❝è ✤à♥❤ cos 2(ϕ + φ) = −1✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ tự tỹ tữỡ ự ợ ữớ ọ ữớ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(1, 1) ✈➔♦ r ✈➔ q = 0, 1, ❝è ✤à♥❤✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ số t tự tỹ tữỡ ự ợ ữớ ọ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ S ✹ ✸✻ ✹✵ ✹✻ ✺✺ ✻✶ ❍➻♥❤ ✸✳✻ ❍➻♥❤ ✸✳✼ ❍➻♥❤ ✸✳✽ ❍➻♥❤ ✸✳✾ ❍➻♥❤ ✸✳✶✵ ❍➻♥❤ ✸✳✶✶ ❍➻♥❤ ✸✳✶✷ ❍➻♥❤ ✸✳✶✸ ❍➻♥❤ ✸✳✶✹ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(2, 1) ✈➔♦ r ✈➔ q = 0, 1, ✳✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t tự tỹ tữỡ ự ợ ữớ ọ ữớ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(2, 2) ✈➔♦ r ✈➔ q = 1, 2, ✳✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tỹ tữỡ ự ợ ữớ ọ ữớ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(3, 1) ✈➔♦ r ✈➔ q = 0, 1, 2, ✳✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tü t÷ì♥❣ ự ợ ữớ ọ ữớ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(3, 2) ✈➔♦ r ✈➔ q = 0, 1, ✳✣÷í♥❣ ❜✐➸✉ t số t tự tỹ tữỡ ự ợ ✤÷í♥❣ ♠➔✉ ✤ä✱ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(3, 3) ✈➔♦ r ✈➔ q = 0, 1, ✳✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t số t tự tỹ tữỡ ự ợ ữớ ✤ä✱ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(4, 1) ✈➔♦ r ✈➔ q = 0, 1, ✳✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t tự tỹ tữỡ ự ợ ữớ ọ ữớ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(4, 2) ✈➔♦ r ✈➔ q = 0, 1, 2✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tỹ tữỡ ự ợ ữớ ọ ữớ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ R(4, 3) ✈➔♦ r ✈➔ q = 1, 2, 3✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tü t÷ì♥❣ ự ợ ữớ ọ ữớ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ I ✈➔♦ r ✈➔ q = 1, 2, 3✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ t số t tự tỹ tữỡ ự ợ ữớ ♠➔✉ ✤ä✱ ✤÷í♥❣ ♠➔✉ ①❛♥❤✱ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✻✷ ✻✸ ✻✹ ✻✺ ✻✻ ✻✼ ✻✾ ✼✵ ✼✸ ❍➻♥❤ ✸✳✶✺ ❍➻♥❤ ✸✳✶✻ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ RH ✈➔♦ r ✈➔ q = 1, 3, 5✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tỹ tữỡ ự ợ ữớ ọ ữớ ♠➔✉ ✤❡♥✳ ✳ ✳ ✳ ✳ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ RH ✈➔♦ r ✈➔ q = 1, 3, 5✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tü t÷ì♥❣ ù♥❣ ợ ữớ ọ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✼✺ ✼✼ ▼Ð ỵ t ❤å❝ ❝æ♥❣ ♥❣❤➺ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩✱ tr♦♥❣ ✤â t❤æ♥❣ t✐♥ ❧✐➯♥ ❧↕❝ ❧➔ ♠ët ♥❤✉ ❝➛✉ r➜t ✤÷đ❝ q✉❛♥ t➙♠ tr♦♥❣ ❝✉ë❝ sè♥❣✳ ❚r♦♥❣ ❧➽♥❤ ✈ü❝ ①û ❧➼ t❤æ♥❣ t✐♥ ✈➔ tr✉②➲♥ t❤æ♥❣✱ ❝→❝ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ✤❛♥❣ ✤÷đ❝ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➻ ❝❤ó♥❣ ❝â r➜t ♥❤✐➲✉ ❧đ✐ ➼❝❤ ♥❤÷ t➠♥❣ tè❝ ✤ë tr✉②➲♥ t✐♥✱ t➼♥❤ ❜↔♦ ♠➟t ❝❛♦ ✈➔ ❝❤è♥❣ ♥❤✐➵✉✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❝→❝ tr↕♥❣ t❤→✐ ♥➔② ❧➔ ❝ì sð ♥❣❤✐➯♥ ❝ù✉ ✈➔ →♣ ỹ ữ ỵ tt t r q✉❛♥❣ ❧÷đ♥❣ tû✱ t❤ỉ♥❣ t✐♥ ❧÷đ♥❣ tû ✈➔ ♠→② t➼♥❤ ữủ tỷ ợ t ổ tr t✐♥ q✉❛♥❣ ❤å❝✱ ❝ỉ♥❣ ♥❣❤➺ ❧❛③❡ ✈ỵ✐ ♠ư❝ ✤➼❝❤ ❧➔♠ ❝❤♦ ❝æ♥❣ ♥❣❤➺ tr✉②➲♥ t✐♥ ✈➔ ①û ❧➼ ❞ú ❧✐➺✉ ♥❣➔② ❝➔♥❣ ♥❤❛♥❤ ❝❤â♥❣✱ ❝❤➼♥❤ ①→❝ ✈➔ ❤✐➺✉ q✉↔ [13]✳ ❚❤➳ ♥❤÷♥❣ ♣❤↔✐ ❧➔♠ t❤➳ ♥➔♦ ✤➸ t➼♥ ❤✐➺✉ tr✉②➲♥ t✐♥ ❝â t➼♥❤ ❧å❝ ❧ü❛ ❝❛♦ ✈➔ ❣✐↔♠ t❤✐➸✉ ✤÷đ❝ tè✐ ✤❛ t➼♥❤ ♥❤✐➵✉✳ ❱➔♦ ♥❤ú♥❣ ♥➠♠ ✻✵ ❝õ❛ t❤➳ ❦➾ t❤ù ❳❳✱ ✈➟t ❧➼ ❤å❝ rë ❧➯♥ sü ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ tr↕♥❣ t❤→✐ ♠ỵ✐ ♠➔ ①✉➜t ♣❤→t ✤✐➸♠ ❧➔ ❤➺ t❤ù❝ ❜➜t ✤à♥❤ ❍❡✐s❡♥❜❡r❣✳ ◆â ❝❤♦ r➡♥❣ tr↕♥❣ t ổ ổ t ỗ tớ ①✉♥❣ ❧÷đ♥❣ ✈➔ tå❛ ✤ë✳ ❚r↕♥❣ t❤→✐ ✈➟t ❧➼ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❧➔ tr↕♥❣ t❤→✐ ❦➳t ❤ñ♣✱ ♥â ữủ t ỗ tứ sỹ ự r ♥➠♠ ✶✾✷✻ [16] ❦❤✐ ❦❤↔♦ s→t ❞❛♦ ✤ë♥❣ tû ✤✐➲✉ ❤á❛✱ ỉ♥❣ ❝❤♦ r➡♥❣✿ ✏❈→❝ tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ ♥❤÷ ❝→❝ ❜â sâ♥❣ ❝â t➼♥❤ ❝❤➜t ✤ë♥❣ ❧ü❝ ❤å❝ t÷ì♥❣ tü ♥❤÷ ♠ët ❤↕t ❝ê ✤✐➸♥ ❝❤✉②➸♥ ✤ë♥❣ tr♦♥❣ t❤➳ ♥➠♥❣ ❜➟❝ ❤❛✐✑✳ ❙❛✉ ✤â ❝á♥ ✤÷đ❝ ❑r❛r❞ ✈➔ ❉❛r✇✐♥ ✤÷❛ r❛ ♥➠♠ ✶✾✷✼✱ t✉② ♥❤✐➯♥ tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ ợ ữủ r [18] rs [23] ữ r tự r rr m=n ợ ❧➫✱ t❛ ✤➦t m = n = 2k + 1✱ k = 0, 1, 2, 3, ✱ ❚ø (3.33) ❝❤ó♥❣ tỉ✐ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ RH a ˆ2k+1ˆb2k+1 > a ˆ†(2k+1) a ˆ2k+1 ˆb†(2k+1)ˆb2k+1 ✭✸✳✸✾✮ ❑➯t q✉↔ t➼♥❤ t♦→♥ ❝❤♦ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ởt ợt ởt t õ ữ s ✭❦❂✵✮✳❈❤ó♥❣ tỉ✐ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ RH = a ˆ† a ˆ ˆb†ˆb − a ˆˆb ✭✸✳✹✵✮ ❚➼♥❤ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ❝❤ó♥❣ tỉ✐ ❝â ❝→❝ ❦➳t q✉↔ s❛✉ ∞ ˆb†ˆb = † a ˆa ˆ = n=0 ∞ (n+q)! n!q! [(1 − (−1)n ] ξ 2n × {n(2n + q)} (n+q)! n!q! [1 − (−1)n ] ξ 2n (2n + q + 1) n=0 ∞ (n+q)! n 2n (n + q + 1)2 + n(n n!q! [(1 − (−1) ] ξ n=0 ∞ (n+q)! [1 − (−1)n ] ξ 2n (2n + q + 1) n!q! n=0 ∞ a ˆˆb = n=0 (n+q+1)! n!q! ∞ n=0 + q) [(1 − (−1)n ] ξ 2n ξ (n + q + 2) + n (n+q)! n!q! [1 − (−1)n ] |ξ | (2n + q + 1) ✼✻ ❱➟② ∞ RH = n=0 (n+q)! n!q! [(1 − (−1)n ] tanh2n r (n + q + 1)2 + n(n + q) ∞ (n+q)! n!q! n=0 ∞ × − (n+q)! n!q! n=0 ∞     n=0 ∞ n=0 [(1 − (−1)n ] tanh2n r {n(2n + q)} (n+q)! n!q! [1 − (−1)n ] ξ 2n (2n + q + 1) (n+q+1)! n!q! ∞    [1 − (−1)n ] tanh2n r(2n + q + 1) n=0 n 2n [(1 − (−1) ] r (n+q)! n!q! n 2  ξ (n + q + 2) + n   2n [1 − (−1) ] r(2n + q + 1) ỗ t 3.16 sü ♣❤ư t❤✉ë❝ ❝õ❛ RH ✈➔♦ r ✈ỵ✐ ❝→❝ ❣✐→ tr q ỗ RH L 20 40 60 80 0.0 0.5 1.0 1.5 2.0 r ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ RH ✈➔♦ r ✈➔ q = 1, 3, 5✳ ✣÷í♥❣ ❜✐➸✉ ❞✐➵♥ ❝→❝ t❤❛♠ sè t❤❡♦ t❤ù tü tữỡ ự ợ ữớ ọ ❍➻♥❤ ✸✳✶✻✿ t❤à ❝❤♦ t❤➜② RH > ✈ỵ✐ ♠å✐ ❣✐→ trà ❝õ❛ r > 0, ✈➔ q✳ ▼➦❝ ❦❤→❝✱ ✤÷í♥❣ ❝♦♥❣ ✤✐ ①✉è♥❣ t❤➸ ❤✐➺♥ ✤ë ✤❛♥ rè✐ ❝➔♥❣ t➠♥❣✳ ❚r↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ t❤➸ ❤✐➺♥ t➼♥❤ ✤❛♥ rè✐✳ ✼✼ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t✐➳♥ ❤➔♥❤ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ tê♥❣✱ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡ ✈➔ ♥➨♥ ❦✐➸✉ ❍✐❧❧❡r②✱ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠✱ sü ✈✐ ♣❤↕♠ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ ✈➠♥ ❝â t❤➸ tâ♠ t➢t ♥❤÷ s❛✉✿ ❚❤ù ♥❤➜t ❧➔ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ♥➨♥ tê♥❣✱ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡ ✈➔ ♥➨♥ ❍✐❧❧❡r②✱ ❝❤ó♥❣ tỉ✐ ✤➣ ✤÷❛ r❛ ❝→❝ t❤❛♠ sè ♥➨♥ tê♥❣✱ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡ ✈➔ ♥➨♥ ❍✐❧❧❡r②✳ ◗✉❛ ❦❤↔♦ s→t✱ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ♥➨♥ tờ ữ ổ ố ợ trữớ ủ tê♥❣ ❤❛✐ ♠♦❞❡✱ ♠ù❝ ✤ë ♥➨♥ tê♥❣ ❝➔♥❣ t➠♥❣ ❦❤✐ ❜✐➯♥ ✤ë ❦➳t ❤ñ♣ r ✈➔ sü ❝❤➯♥❤ ❧➺❝❤ ♣❤♦t♦♥ q ❝➔♥❣ t➠♥❣✳ ❚❤ù ❤❛✐ ❧➔ ❝❤ó♥❣ tỉ✐ →♣ ❞ư♥❣ t✐➯✉ ❝❤✉➞♥ ♣❤↔♥ ❦➳t ❝❤ị♠ ❝❤♦ tr÷í♥❣ ❤đ♣ ❤❛✐ ♠♦❞❡ ✤➸ t✐➳♥ ❤➔♥❤ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫✳ ❑➳t q✉↔ ❦❤↔♦ s→t ❝❤♦ t❤➜② r➡♥❣ tr↕♥❣ t❤→✐ ♥➔② t❤➸ ❤✐➺♥ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠ ♠↕♥❤✱ ②➳✉ tị② t❤✉ë❝ ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤đ♣ r✳ ❚❤ù ❜❛ ❧➔ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✤➸ ❦❤↔♦ s→t tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ✈➔ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❜à ✈✐ ♣❤↕♠✳ ❚❤ù t÷ ❝❤ó♥❣ tỉ✐ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✶✱✶✮ t❤➯♠ ♠ët ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➫ ❜➡♥❣ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r②✳ ❑➳t q✉↔ ❦❤↔♦ s→t t❤➻ ✤è✐ ✈ỵ✐ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲ ❩✉❜❛✐r② tr↕♥❣ t❤→✐ ❝❤➾ ✤❛♥ rè✐ ❦❤✐ ❜➟❝ ❧➫ ✈➔ ❣✐→ trà ❜✐➯♥ ✤ë ❦➳t ❤đ♣ r ❧ỵ♥ ❤ì♥ ♠ët ❣✐→ trà ①→❝ ✤à♥❤✳ ✼✽ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ❑❤↔♦ s→t ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ t❤➯♠ ♣❤♦t♦♥ ❝❤➤♥✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ✶✳ ◆❣✉②➵♥ ❱➠♥ ❆♥❤ ✭✷✵✶✺✮✱ ❱➟t ❧➼✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍✉➳✳ ❚r↕♥❣ t❤→✐ ❦➳t ❤ñ♣ ♣❤✐ t✉②➳♥ ❑ ❤↕t✱ tr↕♥❣ t❤→✐ ❝→✐ q✉↕t✱ tr↕♥❣ t❤→✐ ❦➳t ❤ñ♣ ❜ë ❜❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ ❝❤ó♥❣✱ ▲✉➟♥ →♥ ❚✐➳♥ s ỵ rữỡ ✣ù❝✭✷✵✵✺✮✱ ✸✳ ▲➯ ✣➻♥❤ ◆❤➙♥ ✭✷✵✶✸✮✱ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❙❯✭✶✱✶✮✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❱➟t ❧➼✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍✉➳✳ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐ ♥➨♥ ❤❛✐ ♠♦❞❡ ❝❤➙♥ ❦❤æ♥❣ t❤➯♠ ♣❤♦t♦♥✱ ▲✉➟♥ ✈➠♥ ✹✳ ❚ỉ ❚❤à ◆❣å❝ ❚❤ó② ✭✷✵✶✶✮✱ ❚❤↕❝ s➽ ❱➟t ❧➼✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍✉➳✳ ❑❤↔♦ s→t t➼♥❤ t rố t ữủ tỷ ợ tr t❤→✐ ❤❛✐ ♠♦❞❡ ❙✉✭✶✱✶✮✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❱➟t ❧➼✱ ✣↕✐ ❤å❝ ✺✳ ▲➯ ❚❤à ❚❤õ② ✭✷✵✶✸✮✱ ❙÷ ♣❤↕♠ ❍✉➳✳ ▼ët sè ❤✐➺✉ ù♥❣ tr♦♥❣ ❤➺ ♣❤♦♥♦♥✲❡❝①✐t♦♥✲❜✐❡①❝✐t♦♥ ð ❜→♥ ❞➝♥ ❦➼❝❤ t❤➼❝❤ q✉❛♥❣✱ ▲✉➟♥ →♥ ❚✐➳♥ s➽ ❑❤♦❛ ❤å❝ ❚♦→♥ ỵ ó ó ✭✷✵✵✾✮✱ ❇➔✐ ❣✐↔♥❣ q✉❛♥❣ ❤å❝ ❧÷đ♥❣ tû ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍✉➳✳ ❑❤↔♦ s→t ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ t❤➯♠ ❤❛✐ ♣❤♦tt♦♥ t➼❝❤ ❙❯✭✶✱✶✮ ❝❤➤♥✱ ✽✳ ❚r➛♥ ❉✐➺♣ ❚✉➜♥ ✭✷✵✶✼✮✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❱➟t ❧➼✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍✉➳✳ ✼✾ ❚✐➳♥❣ ❆♥❤ ✾✳ ❆❣❛r✇❛❧✳ ●✳ ❙ ✭✶✾✽✽✮✳ ✏◆♦♥❝❧❛ss✐❝❛❧ st❛t✐st✐❝s ♦❢ ❢✐❡❧❞ ✐♥ ♣❛✐r ❝♦❤❡r✲ ❡♥t st❛t❡s✑✳ ❏✳ ❖♣t✳ ❙♦❝✳ ❆♠✳ ❇✱ ✺✱ ✶✾✹✵✳ ✶✵✳ ❆❣❛r✇✇❛❧✱ ●✳ ❙✳ ❛♥❞ ❆s♦❦❛ ❇✐s✇❛s✳ ✏◗✉❛♥t✐t❛t✐✈❡ ♠❡❛s✉r❡s ♦❢ ❡♥✲ t❛♥❣❧❡♠❡♥t ✐♥ ♣❛✐r✲ ❝♦❤❡r❡♥t st❛t❡s✳✑ ❏♦✉r♥❛❧ ♦❢ ❖♣t✐❝s ❇✿ ◗✉❛♥t✉♠ ❛♥❤ ❙❡♠✐❝❧❛ss✐❝❛❧ ❖♣t✐❝s ✼✳✶✶✭✷✵✵✺✮✿✸✺✵ ✶✶✳ ❇❡♥♥❡tt ❈✳❍ ✳✱ ❇r❛ss❛r❞ ●✳✱ ❈r❡♣❡❛✉ ❈✳✱ ❏♦③s❛ ❘✳✱ P❡r❡ ❆✳✱ ❛♥❞ ❲♦♦tt❡rs ❲✳❑✳ ✭✶✾✾✸✮✱ ✏❚❡❧❡♣♦rt✐♥❣ ❛♥ ✉♥❦♥♦✇♥ q✉❛♥t✉♠ st❛t❡ ✈✐❛ ❞✉❛❧ ❝❧❛ss✐❛❧ ❛♥❞ ❊✐♥st❡✐♥✲P♦❞♦❧s❦②✲❘♦s❡♥ ❝❤❛♥♥❡❧s✑✱ ▲❡tt✳ ✼✵✭✶✸✮✱ ♣✳✶✽✾✺✳ P❤②s✳ ❘❡✈✳ ✶✷✳ ❇❡♥♥❡tt✱ ❈❤❛r❧❡s ❍✳ ❡t ❛❧✳ ✏❈♦♥❝❡♥tr❛t✐♥❣ ♣❛rt✐❛❧ ❛♥t❛♥❣❣❧❡♠❡♥t ❜② ❧♦❝❛❧ ♦♣❡r❛t✐♥♦s✳✑ ♣❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆ ✺✸✳✹ ✭✶✾✾✻✮✿✷✵✹✻✳ ✶✸✳ ❇r❛✉♥st❡✐♥ ❙✳▲ ❛♥❞ ❑✐♠❜❧❡ ❍✳❏✳ ✭✶✾✾✽✮✱ ✏ ❚❡❧❡♣♦rt❛t✐♦♥ ♦❢ ❝♦♥t✐♥✲ ✉♦✉s q✉❛♥✲t✉♠ ✈❛r✐❛❜❧❡s✑✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✽✵✭✹✮✱ ♣✳✽✻✾✳ ✶✹✳ ❈❛♠✐❝❤❛❡❧✳ ❍✳ ❏ ❛♥❞ ❲❛❧❧s✳ ❉✳ ❋ ✭✶✾✼✻✮✳ ✏ ❆ q✉❛♥t✉♠✲♠❡❝❤❛♥✐❝❛❧ ♠❛st❡r ❡q✉❛t✐♦♥ tr❡❛t♠❡♥t ♦❢ t❤❡ ❞②♥❛♠✐❝❛❧ ❙t❛r❦ ❡❢❢❡❝t✑✱ ❇✱ ✾ ✱ ✶✶✾✾✳ ❏✳ P❤②s✳ ✶✺✳ ❈❤r✐st♦♣❤❡r✳ ❈✳ ●❡rr② ❛♥❞ P❡t❡r✳ ▲✳ ❑♥✐❣❤t ✭✷✵✵✺✮✱ ✏■♥tr♦❞✉❝t♦r② ◗✉❛♥t✉♠ ❖♣t✐❝s✑✱ ❆♠❡r✐❝❛❧ ❏♦✉r♥❛❧ ♦❢ ♣❤②s✐❝s✱ ✼✸✱ ✶✶✾✼✳ ✶✻✳ ❊♥❦ ✈❛♥ ❙✳ ❏ ✭✶✾✾✾✮✱ ✏ ❉✐s❝r❡t❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❡❧❡♣♦rt❛t✐♦♥ ♦❢ ❝♦♥✲ t✐♥✉♦✉s ✈❛r✐❛❜❧❡s✑✱ P❤②s✳ ❘❡✈✳ ❆ ✻✵✭✻✮✱ ✺✵✾✺✳ ✶✼✳ ●❛s❜r✐s ❆✳✱ ❆❣❛r✇❛❧ ●✳❙ ✭✷✵✵✼✮✱ ✏◗✉❛♥t✉♠ t❡❧❡♣♦rt❛t✐♦♥ ✇✐t❤ ♣❛✐r✲ ❝♦♥❤❡r❡♥t st❛t❡s✑✱ ■♥t✳ ❏♦✉r♥❛❧ ♦❢ ◗✉❛♥t✳ ■♥❢✳✱ ✺ ✭✶✲✷✮✱♣♣✳✶✼✲✷✷✳ ✽✵ ✶✽✳ ●❧❛✉❜❡r✳ ❘✳ ❏ ✭✶✾✻✸✮✱ ✏❈♦❤❡r❡♥t ❛♥❤ ■♥❝♦❤❡r❡♥t ❙t❛t❡s ♦❢ ❘❛❞✐❛t✐♦♥ ❋✐❡❧❞✑✱ P❤②s✳ ❘❡✈✱✾✻✭✷✮✱ ✶✸✶✱ ✷✼✻✻✳ ✶✾✳ ❍✐❧❧❡r②✳ ▼ ✭✶✾✽✾✮✱ ✏❙✉♠ ❛♥❞ ❞✐❢❢r❡♥❝❡ sq✉❡❡③✐♥❣ ♦❢ t❤❡ ❡❧❡❝tr♦♠❛❣✲ ♥❡t✐❝ ❢✐❡❧❞✑✱ P❤②s✳ ❘❡✈ ❆✱ ✹✺✱ ✸✶✹✼✲✸✶✺✺✳ ✷✵✳ ❍♦❧❧❡♥❤♦st✳ ◆✳ ◆ ✭✶✾✼✾✮✱ ✏◗✉❛♥t✉♠ ❧✐♠✐ts ♦♥ r❡s♦♥❛♥t✲♠❛ss ❣r❛✈✐t❛t✐♦♥❛❧✲ r❛❞✐❛t✐♦♥ ❞❡t❡❝t♦rs✑✱ P❤②s✳ ❘❡✈ ❉✱ ✶✾✱ ✶✻✻✾✳ ✷✶✳ ◆❣✉②❡♥ ❇❛ ❆♥ ❛♥❞ ❱♦ ❚✐♥❤ ✭✶✾✾✾✮✱ ✏●❡♥❡r❛❧ ♠✉❧t✐♠♦❞❡ s✉♠sq✉❡❡③✲ ✐♥❣✑✱ P❤②s✐❝s ▲❡tt❡rs ❆✱ ✷✻✶✱♣♣✳ ✸✹✲✸✾✳ ✷✷✳ Pr❡❧♦♠♦✈✳ ❆✳ ▼✳ ✭✶✾✼✷✮✱ ✏❈♦❤❡r❡♥t st❛t❡s ❢♦r ❛r❜✐tr❛r② ▲✐❡ ❣r♦✉♣s✑✱ ❈♦♠♠✉♥✳ ▼❛t❤✳ P❤②s✱ ✷✻✱ ✷✷✷✳ ✷✸✳ ❙❤✉❞❛rs❤❛♥✳ ❊✳ ❈✳ ● ✭✶✾✻✸✮✱ ✏❊q✉✐✈❛❧❡♥❝❡ ♦❢ ❙❡♠✐❝❧❛ss✐❝❛❧ ❛♥❞ ◗✉❛♥✲ t✉♠ ▼❡❝❤❛♥✐❝❛❧ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❙t❛t❡t✐st✐❝❛❧ ▲✐❣❤t ❇❡❛♠s✑✱ ❘❡✈✳ ▲❡tt ❉✱ ✶✵✱ ✷✼✼✳ P❤②s✳ ✷✹✳ ❙t♦❧❡r✳ ❉ ✭✶✾✼✵✮✱ ✏❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss❡s ♦❢ ▼✐♥✐♠✉♠ ❯♥❝❡rt❛✐♥t② P❛❝❦❡ts✑✱ P❤②s✳ ❘❡✈✳ ▲❡tt ❉✱ ✶✱ ✸✷✶✼✳ ✷✺✳ ❱❛✐❞♠❛♥ ▲✳ ✭✶✾✾✹✮✱ ✏❚❡❧❡♣♦rt❛t✐♦♥ ♦❢ ❝q✉❛♥t✉♠ st❛t❡s✑ ❆ ✹✾✭✷✮✱ ✶✹✼✸✳ ✽✶ P❤②s✳ ❘❡✈✳ P❍Ö ▲Ö❈ P❤ö ❧ö❝ ✶ ❈❤ù♥❣ ♠✐♥❤ e−iφ a ˆˆb (2.21) =ab ψ| e−iφ a ˆˆb |ψ ∞ = |N |2 e−2iφ (1 − |ξ|2 )1+q m=0 ab (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb+ )ˆ aˆbˆ aˆb(ˆ a† + ˆb)|n + q, n ∞ −2iφ = |N | e 1+q (1 − |ξ| ) m=0 (m + q)! m!q! [(1 − (−1) ] ξ ab ∞ m (n + q)! n!q! ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆˆbˆ aˆbˆ a† + a ˆa ˆˆbˆ aˆbˆb + ˆb† a ˆˆbˆ aˆbˆ a† + ˆb† a ˆˆbˆ aˆbˆb|n + q, n ab ∞ −2iφ = |N | e 1+q (1 − |ξ| ) m=0 × [(1 − (−1)n ] ξ n = |N | e ∞ m [(1 − (−1) ] ξ ∗m n=0 (n + q)! n!q! n(n − 1)(n + q)(n + q − 1)(2n + q − 1)δm,n−2 ∞ −2iφ (m + q)! m!q! 1+q (1 − |ξ| ) n=0 (n + q)! [(1 − (−1)n ] ξ 2n ξ n!q! × (n + q + 1)(n + q + 2)(2n + q + 3) ✭P✳✶✮ P❤ö ❧ö❝ ✷ ❈❤ù♥❣ ♠✐♥❤ (2.22) 2ˆ a†ˆb† a ˆˆb =ab ψ|2ˆ a†ˆb† a ˆˆb|ψ ∞ = 2|N |2 (1 − |ξ|2 )1+q m=0 ab (m + q)! m!q! P✳✶ ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb† )ˆ a†ˆb† a ˆˆb(ˆ a† + ˆb)|n + q, n ∞ 2 1+q = 2|N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 ab (n + q)! n!q! ˆˆbˆ a† ˆˆbˆb + ˆb† a ˆ†ˆb† a ˆˆbˆ a† + a ˆa ˆ†ˆb† a × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ†ˆb† a + ˆb† a ˆ†ˆb† a ˆˆbˆb|n + q, n ab ∞ = 2|N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n n(n + q + 1)2 + n(n − 1)(n + q) δm,n ∞ 2 1+q = 2|N | (1 − |ξ| ) n=0 (n + q)! n!q! 2 [(1 − (−1)n ] ξ 2n × n(n + q + 1)2 + n(n − 1)(n + q) ✭P✳✷✮ P❤ö ❧ö❝ ✸ ❈❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ eiφ a ˆ†ˆb† =ab ψ|ˆ a†ˆb† |ψ (2.23) ab ∞ = |N |2 eiφ (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb+ )ˆ a†ˆb† (ˆ a† + ˆb)|n + q, n ∞ = |N |2 eiφ (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! ab (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ†ˆb† a ˆ† + a ˆa ˆ†ˆb†ˆb + ˆb† a ˆ†ˆb† a ˆ† + ˆb† a ˆ†ˆb†ˆb|n + q, n ab ∞ = |N |2 eiφ (1 − |ξ|2 )1+q m=0 (m + q)! m!q! P✳✷ ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n n(n + q + 1)δm,n + (n + 1)(n + 2)(n + q + 1)(n + q + 2) × δm,n+2 ∞ iφ (n + q)! n!q! 1+q = |N | e (1 − |ξ| ) n=0 2 [(1 − (−1)n ] ξ 2n (n + q + 1) × n + ξ ∗2 (n + q + 2) ✭P✳✸✮ P❤ö ❧ö❝ ✹ ❈❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ e−iφ a ˆˆb =ab ψ|e−iφ a ˆˆb|ψ ab ∞ −iφ = |N | e (2.24) 1+q (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb+ )ˆ aˆb(ˆ a† + ˆb)|n + q, n ∞ = |N |2 e−iφ (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ab ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! (n + q)! n!q! 2 × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆˆbˆ a† + a ˆa ˆa ˆˆbˆb + ˆb† a ˆˆbˆ a† + ˆb† a ˆˆbˆb|n + q, n ab ∞ = |N |2 e−iφ (1 − |ξ|2 )1+q m=0 (m + q)! m!q! = |N |2 e−iφ (1 − |ξ|2 )1+q n=0 (n + q)! n!q! [(1 − (−1)m ] ξ ∗m n=0 × [(1 − (−1)n ] ξ n n(n + q + 1)δm,n + ∞ ∞ (n + q)! n!q! n(n − 1)(n + q)(n + q − 1)δm,n−2 2 [(1 − (−1)n ] ξ 2n (n + q + 1) × n + ξ (n + q + 2) ✭P✳✹✮ P✳✸ P❤ö ❧ö❝ ✺ ❈❤ù♥❣ ♠✐♥❤ (2.32) a ˆˆb† a ˆ†ˆb =ab ψ|ˆ aˆb† a ˆ†ˆb|ψ ∞ = |N |2 (1 − |ξ|2 )1+q m=0 ab (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb† )ˆ aˆb† a ˆ†ˆb(ˆ a† + ˆb)|n + q, n ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 ab (n + q)! n!q! a† ˆˆb† a ˆ†ˆbˆ a† + a ˆa ˆˆb† a ˆ†ˆbˆb + ˆb† a × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆˆb† a ˆ†ˆbˆ + ˆb† a ˆˆb† a ˆ†ˆbˆb|n + q, n ab ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n n(n + q + 1)(n + q + 2) + n(n − 1)(n + q + 1)δm,n ∞ 2 1+q = |N | (1 − |ξ| ) n=0 (n + q)! [(1 − (−1)n ] ξ 2n n!q! × {n(n + q + 1)(2n + q + 1)} ✭P✳✺✮ P❤ö ❧ö❝ ✻ ❈❤ù♥❣ ♠✐♥❤ (2.35) n ˆa − n ˆb = a ˆ† a ˆ − ˆb†ˆb =ab ψ|ˆ a† a ˆ − ˆb†ˆb|ψ ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! P✳✹ ab ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb† )(ˆ a† a ˆ − ˆb†ˆb)(ˆ a† + ˆb)|n + q, n ∞ 2 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 (n + q)! n!q! ab × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ† a ˆa ˆ† + a ˆa ˆ† a ˆˆb + ˆb† a ˆ† a ˆa ˆ† + ˆb† a ˆ† a ˆˆb − a ˆˆb†ˆbˆ a† − a ˆˆb†ˆbˆb − ˆb†ˆb†ˆbˆ a† − ˆb†ˆb†ˆbˆb|n + q, n ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ab ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n (n + q + 1)2 − n(n + q + 1) + n(n + q) − n(n − 1) δm,n ∞ (n + q)! [(1 − (−1)n ] ξ 2n n!q! 1+q = |N | (1 − |ξ| ) n=0 × (n + q + 1)2 − n(1 + q − nq) ✭P✳✻✮ P❤ö ❧ö❝ ✼ ❈❤ù♥❣ ♠✐♥❤ (3.26) ˆb†2ˆb2 =ab ψ|ˆb†2ˆb2 |ψ ab ∞ 2 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb† )ˆb†2ˆb2 (ˆ a† + ˆb)|n + q, n ∞ 2 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 ab (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| a ˆˆb†2ˆb2 a ˆ† + a ˆˆb†2ˆb2ˆb + ˆb†ˆb+2ˆb2 a ˆ+ P✳✺ 2 + ˆb†ˆb†2ˆb2ˆb|n + q, n ab ∞ 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n {n(n − 1)(n + q + 1) + n(n − 1)(n − 2)} δm,n ∞ = |N |2 (1 − |ξ|2 )1+q n=0 (n + q)! n!q! [(1 − (−1)n ] ξ 2n × n(n − 1)(2n + q − 1) ✭P✳✼✮ P❤ö ❧ö❝ ✽ ❈❤ù♥❣ ♠✐♥❤ (3.27) a ˆ†2 a ˆ2 =ab ψ|ˆ a†2 a ˆ2 |ψ ab ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb† )ˆ a†2 a ˆ2 (ˆ a† + ˆb)|n + q, n ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 ab (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ†2 a ˆ2 a ˆ† + a ˆa ˆ†2 a ˆ2ˆb + ˆb† a ˆ†2 a ˆ2 a ˆ† + ˆb† a ˆ†2 a ˆ2ˆb|n + q, n ab ∞ 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n (n + q)(n + q + 1)2 + n(n + q)(n + q − 1) δm,n ∞ = |N |2 (1 − |ξ|2 )1+q n=0 (n + q)! n!q! 2 [(1 − (−1)n ] ξ 2n × (n + q)(n + q + 1)2 + n(n + q)(n + q − 1) ✭P✳✽✮ P✳✻ P❤ö ❧ö❝ ✾ ❈❤ù♥❣ ♠✐♥❤ (3.28) a ˆ† a ˆˆb† b =ab ψ|ˆ a† a ˆˆb† b|ψ ∞ = |N |2 (1 − |ξ|2 )1+q m=0 ab (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb† )ˆ a† a ˆˆb† b(ˆ a† + ˆb)|n + q, n ∞ 2 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 ab (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ+ a ˆˆb† bˆ a† + a ˆa ˆ† a ˆˆb† bˆb + ˆb+ a ˆ+ a ˆˆb+ bˆ a+ + ˆb† a ˆ† a ˆˆb† bˆb|n + q, n ab ∞ 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n n(n + q + 1)2 + n(n − 1)(n + q) δm,n ∞ = |N |2 (1 − |ξ|2 )1+q n=0 (n + q)! n!q! [(1 − (−1)n ] ξ 2n × n(n + q + 1)2 + n(n − 1)(n + q) ✭P✳✾✮ P❤ö ❧ö❝ ✶✵ ❈❤ù♥❣ ♠✐♥❤ (3.37) a ˆ2ˆb2 =ab ψ|ˆ a2ˆb2 |ψ ab ∞ = |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m n=0 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb+ )ˆ a2ˆb2 (ˆ a+ + ˆb)|n + q, n P✳✼ ab ∞ 2 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ (n + q)! n!q! ∗m n=0 × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ2ˆb2 a ˆ† + a ˆa ˆ2ˆb2ˆb + ˆb† a ˆ2ˆb2 a ˆ† + ˆb† a ˆ2ˆb2ˆb|n + q, n ab ∞ 1+q = |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ (n + q)! n!q! m [(1 − (−1) ] ξ ∗m n=0 × [(1 − (−1)n ] ξ n ( (n + q)(n + q − 1)n(n − 1)(n + q + 1)2 δm,n−2 + (n + q)(n + q − 1)(n − 2)2 (n − 1)δm,n−2 ∞ = |N |2 (1 − |ξ|2 )1+q n=0 (n + q)! n!q! 2 [(1 − (−1)n ] ξ 2n ξ × {(n + q + 2)(n + q + 1)(2n + q + 3} ✭P✳✶✵✮ P❤ö ❧ö❝ ✶✶ ❈❤ù♥❣ ♠✐♥❤ (2.17) ∞ 2 1+q |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ (n + q)! n!q! ∗m n=0 × [(1 − (−1)n ] ξ n ba m, m + q| (ˆ a + ˆb+ )(ˆ a† + ˆb)|n + q, n ∞ 2 1+q ⇔ |N | (1 − |ξ| ) m=0 (m + q)! m!q! ab ∞ m [(1 − (−1) ] ξ ∗m n=0 =1 (n + q)! n!q! × [(1 − (−1)n ] ξ n ba m, m + q| a ˆa ˆ† + a ˆˆb + ˆb† a ˆ + ˆb†ˆb|n + q, n ∞ 2 1+q ⇔ |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m × [(1 − (−1)n ] ξ n ((n + q + 1)δm,n δm+q,n+q + P✳✽ √ n=0 √ ab =1 (n + q)! n!q! n + q nδm,n−1 δm+q,n+q−1 + √ n + q + n + 1δm,n+1 δm+q,n+q+1 + nδm,n δm+q,n+q ) = ∞ 2 1+q ⇔ |N | (1 − |ξ| ) m=0 (m + q)! m!q! ∞ m [(1 − (−1) ] ξ ∗m √ n=0 (n + q)! n!q! √ × [(1 − (−1)n ] ξ n ((n + q + 1)δm,n δm+q,n+q + n + q nδm,n−1 δm+q,n+q−1 √ + n + q + n + 1δm,n+1 δm+q,n+q+1 + nδm,n δm+q,n+q ) = ∞ ⇔ |N |2 (1 − |ξ|2 )1+q m=0 (m + q)! m!q! ∞ [(1 − (−1)m ] ξ ∗m × [(1 − (−1)n ] ξ n ((2n + q + 1)δm,n δm+q,n+q + √ + n + q + n + 1δm,n+1 δm+q,n+q+1 ) = ∞ 2 1+q ⇔ |N | (1 − |ξ| ) n=0 1+q √ n=0 (n + q)! n!q! √ n + q nδm,n−1 δm+q,n+q−1 (n + q)! [(1 − (−1)n ] ξ 2n (2n + q + 1) = n!q! ∞ ⇔ |N |2 = (1 − |ξ| ) n=0 (n + q)! [(1 − (−1)n ] ξ 2n (2n + q + 1) n!q! −1 ✭P✳✶✶✮ P✳✾