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❙✉♣❡rr❛❞✐❛♥❝❡ ♦❢ q✉❛♥t✉♠ ✜❡❧❞s✿ ❋r♦♠ ❞r② ❢r✐❝t✐♦♥ t♦ ❜❧❛❝❦ ❤♦❧❡ r❛❞✐❛t✐♦♥ ❘♦❜❡rt ❆❧✐❝❦✐ ■♥t❡r♥❛t✐♦♥❛❧ ❈❡♥tr❡ ❢♦r ❚❤❡♦r② ♦❢ ◗✉❛♥t✉♠ ❚❡❝❤♥♦❧♦❣✐❡s ✭■❈❚◗❚✮✱ ❯♥✐✇❡rs②t❡t ●❞❛➠s❦✐✱ P♦❧❛♥❞ ❡✲♠❛✐❧✿ ✜③r❛❅✉♥✐✈✳❣❞❛✳♣❧ ❜❛s❡❞ ♦♥ t❤❡ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❡❥❛♥❞r♦ ❏❡♥❦✐♥s ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾

❙✉♣❡rr❛❞✐❛♥❝❡ ❚❤❡ ▼♦❞❡❧ ✭ ❘✳ ❆✳ ❛♥❞ ❆✳ ❏❡♥❦✐♥s✱ ❆♥♥✳ P❤②s✳ ✭◆❨✮ ✸✾✺ ✱ ✻✾ ✭✷✵✶✽✮✮ ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✶

❙✉♣❡rr❛❞✐❛♥❝❡ ❇♦s♦♥✐❝ ♦r ❢❡r♠✐♦♥✐❝ q✉❛♥t✉♠ ✜❡❧❞ ♠♦❞❡s ✭♦♣❡♥ s②st❡♠✮ ✐♥t❡r❛❝t✐♥❣ ✇✐t❤ r♦t❛t✐♥❣ ❤❡❛t ❜❛t❤ ❛t t❤❡ t❡♠♣❡r❛t✉r❡ T [ a k , a † k ′ ] ± = δ kk ′ k ✲ q✉❛♥t✉♠ ♥✉♠❜❡rs ♦❢ t❤❡ ♠♦❞❡ ◗✉❛♥t✉♠ ✜❡❧❞ ❍❛♠✐❧t♦♥✐❛♥ ❛♥❞ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ✭ z ✲❝♦♠♣♦♥❡♥t✮ ω k a † L z m ( k ) a † � � H f = ¯ h k a k , f = ¯ h k a k k k m ( k ) ✲ ♠❛❣♥❡t✐❝ q✉❛♥t✉♠ ♥✉♠❜❡r ▲✐♥❡❛r ✐♥ ✜❡❧❞s ❛♥❞ s②♠♠❡tr✐❝ ✜❡❧❞✲❜❛t❤ ✐♥t❡r❛❝t✐♦♥ � � � a k ⊗ B † k + a † H int = k ⊗ B k , k [ L z b , B k ] = − ¯ hm ( k ) B k , ❊✛❡❝t✐✈❡ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r t❤❡ r♦t❛t✐♥❣ ❜❛t❤ ✭ Ω ✲ ❛♥❣✉❧❛r ❢r❡q✉❡♥❝② ♦❢ r♦t❛t✐♦♥✮ H eff = H b − Ω L z b , b ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✷

❙✉♣❡rr❛❞✐❛♥❝❡ ▼❛r❦♦✈✐❛♥ ▼❛st❡r ❡q✉❛t✐♦♥ ❢♦r ❞❡♥s✐t② ♠❛tr✐① ♦❢ t❤❡ ✜❡❧❞ dρ ( t ) dt = − i h [ H f , ρ ( t )] + L ρ ( t ) = − i h [ H f , ρ ( t )] ¯ ¯ +1 �� � � � � � a k , ρ ( t ) a † a k ρ ( t ) , a † γ ↓ ( k ) + ] k k 2 k �� � � �� a † a † + γ ↑ ( k ) k , ρ ( t ) a k + k ρ ( t ) , a k . γ ↓ ( k ) ✲ ❛♥♥✐❤✐❧❛t✐♦♥ r❛t❡ hβ ( ωk − m ( k )Ω) ✲ ❝r❡❛t✐♦♥ r❛t❡✱ e − ¯ hβ ( ω − m Ω) ✲ ♠♦❞✐✜❡❞ ❇♦❧t③♠❛♥♥ ❢❛❝t♦r✱ γ ↑ ( k ) = γ ↓ ( k ) e − ¯ β = 1 /k B T ✱ γ ↑ ( k ) ≡ γ ↑ ( k )[ ω k ] �→ γ ↑ ( k )[ ω k + m Ω] ≥ 0 ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✸

❙✉♣❡rr❛❞✐❛♥❝❡ ❘❡❞✉❝❡❞ ❞❡s❝r✐♣t✐♦♥ ❆✈❡r❛❣❡❞ q✉❛♥t✉♠ ✜❡❧❞ ✐♥ t❡r♠s ♦❢ ❛✈❡r❛❣❡❞ q✉❛♥t✉♠ ♠♦❞❡ ❛♠♣❧✐t✉❞❡s α ≡ { α k } , α k ( t ) = Tr ( ρ ( t ) a k ) ❝♦rr❡s♣♦♥❞s t♦ ❝❧❛ss✐❝❛❧ ✜❡❧❞ ❞❡s❝r✐♣t✐♦♥ ❢♦r ❜♦s♦♥✐❝ ✜❡❧❞s✳ ✭◗✉❛s✐✮♣❛rt✐❝❧❡ ♣♦♣✉❧❛t✐♦♥ ♥✉♠❜❡rs n k ( t ) = Tr ( ρ ( t ) a † ¯ k a k ) ❛r❡ ✉s❡❞ t♦ ❝♦♠♣✉t❡ ❛✈❡r❛❣❡ ❡♥❡r❣②✱ z ✲ ❝♦♠♣♦♥❡♥t ♦❢ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠✳ ▼♦r❡ ❣❡♥❡r❛❧ r❡❞✉❝❡❞ ❞❡s❝r✐♣t✐♦♥ ✐♥✈♦❧✈❡s s✐♥❣❧❡✲♣❛rt✐❝❧❡ ❞❡♥s✐t② ♠❛tr✐❝❡s ✭❘✳❆✳ ✱ ❊♥tr♦♣② ✷✶ ✱ ✼✵✺ ✭✷✵✶✾✮✮ σ kl ( t ) = Tr ( ρ ( t ) a † l a k ) ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✹

❙✉♣❡rr❛❞✐❛♥❝❡ ❋✐❡❧❞ ❡q✉❛t✐♦♥s ❛♥❞ ❦✐♥❡t✐❝ ❡q✉❛t✐♦♥s ❚❤❡ ▼❛st❡r ❡q✉❛t✐♦♥ ❧❡❛❞s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❛✈❡r❛❣❡❞ ✜❡❧❞ dtα k ( t ) = {− iω k − 1 d 2[ γ ↓ ( k ) − ( ± ) γ ↑ ( k )] } α k ( t ) ❛♥❞ t♦ t❤❡ ❦✐♥❡t✐❝ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❛✈❡r❛❣❡ ♦❝❝✉♣❛t✐♦♥ ♥✉♠❜❡r ♦❢ ❛ s✐♥❣❧❡ ♠♦❞❡ d dt ¯ n k ( t ) = − [ γ ↓ ( k ) − ( ± ) γ ↑ ( k )]¯ n k ( t ) + γ ↑ ( k ) ✇❤❡r❡ (+) ✕ ❜♦s♦♥s ( − ) ✕ ❢❡r♠✐♦♥s✳ ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✺

❙✉♣❡rr❛❞✐❛♥❝❡ ❚❤❡ ✈❛❧✐❞✐t② ♦❢ ❝❧❛ss✐❝❛❧ ✜❡❧❞ ❞❡s❝r✐♣t✐♦♥ ❢♦r ❜♦s♦♥s ❖♥❧② ❢♦r t❤❡ ③❡r♦✲t❡♠♣❡r❛t✉r❡ ❜❛t❤ ❛t r❡st t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞s✿ ✶✮ ❚❤❡ ❝♦❤❡r❡♥t st❛t❡s ♦❢ t❤❡ ✜❡❧❞ ✲ | α � ✱ a k | α � = α k | α � ❡✈♦❧✈❡ ✐♥t♦ ❝♦❤❡r❡♥t st❛t❡s | α ( t ) � s✉❝❤ t❤❛t dtα k ( t ) = {− iω k − 1 d α k ( t ) = e {− iωk − 1 2 γ ↓ ( k ) } t α k 2 γ ↓ ( k ) } α k ( t ) , ✷✮ ❋♦r t❤❡ ✐♥✐t✐❛❧ ❝♦❤❡r❡♥t st❛t❡ ♣♦♣✉❧❛t✐♦♥s ❛r❡ ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✏❝❧❛ss✐❝❛❧ ✜❡❧❞✑ n k ( t ) = | α k ( t ) | 2 ¯ ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✻

❙✉♣❡rr❛❞✐❛♥❝❡ ❙✉♣❡rr❛❞✐❛♥❝❡ ❢♦r r♦t❛t✐♥❣ ❤❡❛t ❜❛t❤s ❇♦s♦♥✐❝ ♠♦❞❡s s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ ω k < m ( k )Ω ❛r❡ ✉♥st❛❜❧❡ ✕ ❩❡❧✬❞♦✈✐❝❤✬s r♦t❛t✐♦♥❛❧ s✉♣❡rr❛❞✐❛♥❝❡ ✐♥✏❆♠♣❧✐✜❝❛t✐♦♥ ♦❢ ❈②❧✐♥❞r✐❝❛❧ ❊❧❡❝tr♦♠❛❣♥❡t✐❝ ❲❛✈❡s ❘❡✢❡❝t❡❞ ❢r♦♠ ❛ ❘♦t❛t✐♥❣ ❇♦❞②✑✱ ❙♦✈✳ P❤②s✳ ❏❊❚P ✸✺ ✱ ✶✵✽✺ ✭✶✾✼✷✮ ✳ ❊①♣♦♥❡♥t✐❛❧ ✐♥❝r❡❛s❡ ♦❢ ♣❛rt✐❝❧❡ ♥✉♠❜❡r hβ ( m ( k )Ω − ωk ) − 1 � � � � e ¯ n k ( t ) = exp ¯ γ ↓ ( k ) t n k (0) ¯ 1 hβ ( m ( k )Ω − ωk ) − 1 � � � � � � e ¯ + exp γ ↓ ( k ) t − 1 hβ ( m ( k )Ω − ωk ) − 1 , e ¯ ❆♠♣❧✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✭❧❛s❡r ❛❝t✐♦♥✮ � 1 � hβ ( m ( k )Ω − ωk ) − 1 � � e ¯ α k ( t ) = exp 2 γ ↓ ( k ) t α k (0) ❘♦t❛t✐♦♥❛❧ ❡♥❡r❣② ♣r♦❞✉❝❡s ♣❛rt✐❝❧❡s ❛♥❞ ❤❡❛ts t❤❡ ❜❛t❤✳ ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✼

❙✉♣❡rr❛❞✐❛♥❝❡ ❇❧❛❝❦ ❤♦❧❡ r❛❞✐❛t✐♦♥ ❖✉t❡r ♠♦❞❡s ✕ t❤❡ ♦♣❡♥ s②st❡♠✱ ✕ a k , a † k ■♥♥❡r ♠♦❞❡s ✕ t❤❡ ❜❛t❤ ✕ b k ′ , b † k ′ ✱ ❛t t❤❡ ✈❛❝✉✉♠ st❛t❡ ❚✉♥♥❡❧✐♥❣ ❍❛♠✐❧t♦♥✐❛♥ � ( a k ⊗ B † k + a † H int = k ⊗ B k ) k − k ′ = time reversal of k ′ � � � f kk ′ b k ′ + g kk ′ b † B k = − k ′ k ′ ❍❛✇❦✐♥❣ ✿ str♦♥❣ ❣r❛✈✐t② ♦❢ ❇❍ ❝r❡❛t❡s ✐♥❞❡t❡r♠✐♥❛❝② ❜❡t✇❡❡♥ b k ′ ❛♥❞ b † k ′ ✳ ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✽

❙✉♣❡rr❛❞✐❛♥❝❡ ❚❤❡ ❦❡② r❡s✉❧t ♦❢ ❍❛✇❦✐♥❣ | g kk ′ | 2 | f kk ′ | 2 ≃ e − ¯ hβ H ω ( k ) , ω ( k ) = ω ( k ′ ) for ✇❤❡r❡ hc 3 1 ¯ = 6 . 2 10 − 8 K × M Sun β H = , T H = k B T H 8 πGM BH k B M BH ❚❤❡ r❡s✉❧ts ✐♠♣❧② t❤❛t✿ ❛✮ ✐♥♥❡r ♠♦❞❡s ❛t t❤❡ ✈❛❝✉✉♠ st❛t❡ ❛❝t ❛s ❛ ❤❡❛t ❜❛t❤ ❛t T H ✱ ❜✮ r♦t❛t✐♥❣ ❇❍ ✇✐❧❧ s✉♣❡rr❛❞✐❛t❡ ❜♦s♦♥s ♦❜❡②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ ω k < m ( k )Ω ✱ ❝✮ ✐♥❝✐❞❡♥t ❣r❛✈✐t❛t✐♦♥ ✇❛✈❡s ✇✐t❤ ω k < m ( k )Ω ✇✐❧❧ ❜❡ ❛♠♣❧✐✜❡❞ ❜② ❛ r♦t❛t✐♥❣ ❇❍✳ ❏✉r❡❦❢❡st✱ ❲❛rs❛✇✱ ❙❡♣t❡♠❜❡r ✶✻✲✷✵✱ ✷✵✶✾ ✾