Dissipação e ruído de dipolos magnéticos coletivamente acoplados a um circuito r...

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛

❉■❙❙■P❆➬➹❖ ❊ ❘❯❮❉❖ ❉❊ ❉■P❖▲❖❙ ▼❆●◆➱❚■❈❖❙

❈❖▲❊❚■❱❆▼❊◆❚❊ ❆❈❖P▲❆❉❖❙ ❆ ❯▼

❈■❘❈❯■❚❖ ❘❊❙❙❖◆❆◆❚❊

❆❧❡♥❝❛r ❏♦sé ❞❡ ❋❛r✐❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❍✉♠❜❡rt♦ ❞❡ ▼❡♥❡③❡s ❋r❛♥ç❛

❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❍✉♠❜❡rt♦ ❞❡ ▼❡♥❡③❡s ❋r❛♥ç❛ ✭■❋❯❙P✮

Pr♦❢✳ ❉r✳ ❙❛✐❞ ❘❛❤♥❛♠❛②❡ ❘❛❜❜❛♥✐ ✭■❋❯❙P✮

Pr♦❢✳ ❉r✳ ■❜❡rê ▲✉✐③ ❈❛❧❞❛s ✭■❋❯❙P✮

Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❱✐❞✐❡❧❧❛ ❇❛rr❛♥❝♦ ✭■❋●❲ ✲ ❯◆■❈❆▼P✮

Pr♦❢✳ ❉r✳ ❑❛❧❡❞ ❉❡❝❤♦✉♠ ✭■❋ ✲ ❯❋❋✮

❚❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s✳

❘❡s✉♠♦

❆❜str❛❝t

❉❡❞✐❝❛tór✐❛

❉❡❞✐❝♦ ❡st❛ t❡s❡ à ♠✐♥❤❛ ♠ã❡ ▼❛r✐❛ ■✈❡t❡ ❡ ❛♦ ♠❡✉ ❢❛❧❡❝✐❞♦ ♣❛✐ ❏♦sé ❆r❛r✐♣❡✱ q✉❡ s❡♠♣r❡ ♠❡ ❛❝♦♠♣❛♥❤❛r❛♠ ♥❛ ♠✐♥❤❛ ❥♦r♥❛❞❛✳

◆ã♦ ❡stá ♦ ❤♦♠❡♠ ❝♦♥❞❡♥❛❞♦ ❛ tr❛❜❛❧❤♦s ❢♦rç❛❞♦s ❛q✉✐ ♥❛ t❡rr❛❄ ◆ã♦ sã♦ s❡✉s ❞✐❛s ♦s ❞❡ ✉♠ ♠❡r❝❡♥ár✐♦❄ ❈♦♠♦ ♦ ❡s❝r❛✈♦ s✉s♣✐r❛ ♣❡❧❛ s♦♠❜r❛✱ ❝♦♠♦ ♦ ♠❡r❝❡♥ár✐♦ ❡s♣❡r❛ ♦ s❛❧ár✐♦✱ ❛ss✐♠ t✐✈❡ ♣♦r ❤❡r❛♥ç❛ ♠❡s❡s ❞❡ ❞❡❝❡♣çã♦✱ ❡ ❝♦✉❜❡r❛♠✲♠❡ ♥♦✐t❡s ❞❡ ♣❡s❛r✳ ◗✉❛♥❞♦ ♠❡ ❞❡✐t♦✱ ♣❡♥s♦✿ ✏◗✉❛♥❞♦ ✈✐rá ♦ ❞✐❛❄✑ ❆♦ ♠❡ ❧❡✈❛♥t❛r✿ ✏◗✉❛♥❞♦ ❝❤❡❣❛rá ❛ ♥♦✐t❡❄✑ ❊ ♣❡♥s❛♠❡♥t♦s ❧♦✉❝♦s ✐♥✈❛❞❡♠✲♠❡ ❛té ❛♦ ❝r❡♣ús❝✉❧♦✳

❏ó ✼✱✶✲✹

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ♣r✐♠❡✐r❛ ♣❡ss♦❛ ❛ q✉❡♠ ❞❡✈♦ ❛❣r❛❞❡❝❡r é ❛ ♠✐♥❤❛ q✉❡r✐❞❛ ▼❛❣❛❧✐✳ ❆ s✉❛ ❝♦♠✲ ♣❛♥❤✐❛ ❡ ♦ s❡✉ ❛❢❡t♦ sã♦ ♦s t❡s♦✉r♦s q✉❡ ❢❛③❡♠ ✈❛❧❡r ❛ ♣❡♥❛ q✉❛❧q✉❡r ❡s❢♦rç♦✳

❉❡✈♦ ❛❣r❛❞❡❝❡r ❛♦ Pr♦❢✳ ❍✉♠❜❡rt♦ ❋r❛♥ç❛ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♥✜❛♥ç❛ ♥♦ ♠❡✉ tr❛✲ ❜❛❧❤♦✱ ❛❧é♠ ❞❛ ❧✐❜❡r❞❛❞❡ ♣❛r❛ ♣❡sq✉✐s❛r ❡ ❞♦ ❡stí♠✉❧♦ ❛ ♥♦✈❛s ✐❞é✐❛s✳ ❙♦✉ ♠✉✐t♦ ❣r❛t♦ ❛♦ Pr♦❢✳ ❙❛✐❞ ❘❛❜❜❛♥✐ ♣♦r ♥♦s t❡r ✐♥tr♦❞✉③✐❞♦ ❛♦ t❡♠❛ ❞❛ r❡ss♦♥â♥❝✐❛ ♠❛❣✲ ♥ét✐❝❛ ❡ ♣❡❧❛ s✉❛ ❛❥✉❞❛ ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❛ t❡s❡✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ à Pr♦❢❛✳ ❈♦r❛❝✐ ▼❛❧t❛ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ❡♠ ❞✐✈❡rs♦s ♠♦♠❡♥t♦s✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠ã❡ ♣❡❧♦s ♠♦t✐✈♦s ó❜✈✐♦s ❡ ♣♦r ♠❡ s✉st❡♥t❛r✳ ❉❡✈♦ ❧❡♠❜r❛r ❞❛ ❛❥✉❞❛ ❞♦s ♠❡✉s ✐r♠ã♦s ❆♥❞ré ❡ ❆r❛r✐♣❡ ❡ ❞♦s ♠❡✉s ♣r✐♠♦s ❆rt✉r ❡ ❆❧❡①✱ ❛❧é♠ ❞❡ t♦❞♦s ♦s ♠❡✉s t✐♦s ❡ ♣r✐♠♦s ♣ró①✐♠♦s✳

❖s ❛♠✐❣♦s q✉❡ ❡st✐✈❡r❛♠ ♣ró①✐♠♦s ❛♦ ❧♦♥❣♦ ❞❡st❡ ♣❡rí♦❞♦ ♥♦ ■❋❯❙P t❛♠❜é♠ ❝♦♥✲ tr✐❜✉ír❛♠ ♣❛r❛ q✉❡ ❡✉ t❡r♠✐♥❛ss❡ ❡st❛ ❡t❛♣❛✱ ❡♥tr❡ ❡❧❡s ❞❡✈♦ ❝✐t❛r ▼ár❝✐♦ ❈♦r♥é❧✐♦✱ ◆❡❧s♦♥ ❨♦❦♦♠✐③♦✱ ●✉st❛✈♦ ❘♦❥❛s✱ ❆♥t♦♥✐♦ ❇❧♦✐s❡✱ ❈✐r♦✱ Pér✐❝❧❡s✱ ❆r♠❛♥❞♦✱ ❈ás✲ s✐✉s✱ ▲❡♦✱ ❘♦❞r✐❣♦ ❈✉③✐♥❛tt♦✱ P❡❞r♦ P♦♠♣é✐❛✱ ▼❛s❛②✉❦✐✱ ▼✐❧t♦♥✱ ▼❛r❝❡❧♦ P✐r❡s✱ ▼❛r❝♦ ❆♥tô♥✐♦✱ ■✈❛♥✱ ❇r✉♥♦✱ ❆♥❞ré ❙❛r❞ã♦✱ ❋❡r♥❛♥❞❛ P✐♥❤❡✐r♦✱ ❏♦ã♦ ▲✉✐s✱ ❙ér❣✐♦✱ ❘♦❞r✐❣♦ ❋r❡s♥❡❞❛✱ ▼ár✐♦✱ ❚❤✐❛❣♦✱ ❈❛r❧♦s✱ ▲❡♦♥❛r❞♦✱ ❋á❜✐♦✱ ❘♦♥✐✱ ●❡rs♦♥✱ P❛rr❛✱ ❘♦❞r✐❣♦ ❙♣♦♥❝❤✐❛❞♦✱ ❘♦♥❛❧❞♦✱ ❉✐♠❛s✱ ▼❛✉rí❝✐♦ ❚r♦t❛✱ ❋❧á✈✐♦ ✭❆❣r♦✮ ❡ ♠✉✐t♦s ♦✉t✲ r♦s ❛ q✉❡♠ ♣❡ç♦ ❞❡s❝✉❧♣❛s ♣❡❧♦ ❡sq✉❡❝✐♠❡♥t♦✳

❖s ❛♠✐❣♦s ❞♦ ♠❡✉ ❜❛✐rr♦ ❡ ❛❧❣✉♥s q✉❛s❡ ♣❛r❡♥t❡s t❛♠❜é♠ ❢♦r❛♠ ♠✉✐t♦ ✐♠♣♦r✲ t❛♥t❡s ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ â♥✐♠♦ ♥❡st❛ ❡♠♣r❡✐t❛❞❛✳ ❆❣r❛❞❡ç♦ ❛ ❆❧❞❡♥✐③ ❏ú♥✐♦r✱ ❆rt❤✉r ●✉✐♠❛rã❡s✱ ❆✉❣✉st♦ ❈❛t♦t❛✱ ❇ár❜❛r❛✱ ❈❡❝í❧✐❛ ❞❡ ❋❛r✐❛✱ ❊❧✐s♦♥ ▼❛t✐♦❧✐✱ ❊✈✲ ❡rt♦♥✱ ❋á❜✐♦✱ ❋❡r♥❛♥❞♦ ❈♦st❛✱ ❋❡r♥❛♥❞♦ ❉❡♠❡r♦✈✱ ❋✐❧✐♣❡✱ ❋❧á✈✐❛✱ ❋❧á✈✐♦✱ ❏♦②❝❡✱ ❏✉❧✐❛♥❛✱ ❑❡❦♦ P✐✈❛t♦ ❏ú♥✐♦r✱ ▲✐❛♥❛ ❑♦❤♥✱ ▲✉❛♥❛ P✐✈❛t♦✱ ▼✐❝❤❡❧✱ ❘❡♥❛t♦ P✐✈❛t♦✱ ❘✐❝❛r❞♦ ❑♦❤♥✱ ❘♦s❡❧✐ ❡ ❱✐✈✐❛♥✳

▼✉✐t♦ ♦❜r✐❣❛❞♦ às s❡❝r❡tár✐❛s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ▼❛t❡♠át✐❝❛✱ ❆♠é❧✐❛✱ ❙✐♠♦♥❡ ❡ ❊❧✐③❛❜❡t❤ ❡ ❛♦ ♣❡ss♦❛❧ ❞❡ ✐♥❢♦r♠át✐❝❛ ❏♦ã♦ ❡ ❙②❜❡❧❡✳

❊st❛ t❡s❡ t❡✈❡ ♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ ❞❛ ❋❆P❊❙P✱ ❋✉♥❞❛çã♦ ❞❡ ❆♠♣❛r♦ à P❡sq✉✐s❛ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❉✐♥â♠✐❝❛ ❜ás✐❝❛ ❞❛ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ✺

✷✳✶ ❉✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❡♠ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✶✳✶ ❚r❛t❛♠❡♥t♦ ❝❧áss✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✶✳✷ ❚r❛t❛♠❡♥t♦ q✉â♥t✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ ❉✐♥â♠✐❝❛ ❞❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❝♦♠ r❡❧❛①❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳✶ ❊q✉❛çõ❡s ❞❡ ❇❧♦❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳✷ ❉❡s❝r✐çã♦ ♠✐❝r♦s❝ó♣✐❝❛ ❞❛ r❡❧❛①❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ ❉✐♥â♠✐❝❛ ❞❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦ ✳ ✳ ✳ ✳ ✷✷

✸ ▼ét♦❞♦ ❞❡ ▲❛♥❣❡✈✐♥ ✸✵

✸✳✶ ❊q✉❛çã♦ ❡st♦❝ást✐❝❛ ❝❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✳✶ ❈✐r❝✉✐t♦ ❡❧étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✳✷ ❚❡♦r❡♠❛ ❋❧✉t✉❛çã♦✲❉✐ss✐♣❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❊q✉❛çã♦ q✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✷✳✶ ▼♦❞❡❧♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✷✳✷ ❈♦♠♣❛r❛çã♦ ❝♦♠ ♦ ❝❛s♦ ❝❧áss✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹ ❆♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦ r❡✈✐st♦ ✹✵

✹✳✶ ❙♣✐♥s ❛❝♦♣❧❛❞♦s ❛ ✉♠ ❝✐r❝✉✐t♦ ❞✐ss✐♣❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷ ❆ ❞✐♥â♠✐❝❛ ❞✐ss✐♣❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✸ ❉✐♥â♠✐❝❛ ♦s❝✐❧❛♥t❡ ❛♠♦rt❡❝✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✺ ❘✉í❞♦ ❞❡ s♣✐♥s ✻✷

✺✳✶ ❘✉í❞♦ ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✺✳✷ ❘✉í❞♦ ♣r♦❞✉③✐❞♦ ♣♦r ✉♠❛ t❡♥sã♦ ❡❧étr✐❝❛ ❡①t❡r♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾

✻ ❈♦♥❝❧✉sã♦ ❡ P❡rs♣❡❝t✐✈❛s ✼✾

❙❯▼➪❘■❖ ✈✐✐✐

✼ ❆♣ê♥❞✐❝❡s ✽✷

✼✳✶ ❆♣r♦①✐♠❛çã♦ ❞❡ ♦♥❞❛ ❣✐r❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✼✳✷ ❆♣r♦①✐♠❛çã♦ ♠❛r❦♦✈✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✼✳✸ ❊st❛❞♦s ❞❡ ❉✐❝❦❡ ❡ ♠é❞✐❛s ❞❡ ♣r♦❞✉t♦s ❞❡ ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥ ✳ ✳ ✳ ✳ ✽✻ ✼✳✹ ❙♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❘✐❝❝❛t✐ ❣❡♥❡r❛❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✼✳✺ ▼é❞✐❛s ❡st❛❝✐♦♥ár✐❛s ❞❡ ❝♦rr❡♥t❡✱ ❞❡ s♣✐♥ ❡ ❞❡ r✉í❞♦ ❞❛ s❡çã♦ ✺✳✷ ✳ ✳ ✳ ✾✶ ✼✳✻ ❆ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ΦIx(t) =hI(t0)Jx(t0+t)i ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✼✳✼ ❈á❧❝✉❧♦ ❞♦ ❡s♣❡❝tr♦ ❞❛ ❝♦rr❡♥t❡ ❞❛ s❡çã♦ ✺✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖s ❢❡♥ô♠❡♥♦s ❞❡ ❞✐ss✐♣❛çã♦ s❡♠♣r❡ ❢♦r❛♠ ❞❡ ✐♠♣♦rtâ♥❝✐❛ ❝❡♥tr❛❧ ♥♦ ❝❛♠♣♦ ❞❡ ❡st✉✲ ❞♦s ❞❛ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛✳ ❖s ♣r♦❝❡ss♦s ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❡ tr❛♥s✈❡rs❛❧ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ✉♠ ♠❛t❡r✐❛❧ sã♦ ♦s ♣r♦❝❡ss♦s ❞✐ss✐♣❛t✐✈♦s ❡ ❞❡ ♣❡r❞❛ ❞❡ ❝♦❡rê♥✲ ❝✐❛ ♠❛✐s ❝♦♥❤❡❝✐❞♦s ❡♠ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛✳ ❚❛✐s ♣r♦❝❡ss♦s ❞❡ r❡❧❛①❛çõ❡s ❧❡✈❛♠ ❛ ✉♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡ ❢♦r❛♠ ♣r✐♠❡✐r❛✲ ♠❡♥t❡ ❞❡s❝r✐t♦s q✉❛♥t✐t❛t✐✈❛♠❡♥t❡ ♣♦r ❇❧♦❝❤ ❡♠ s❡✉ ❡st✉❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❬✶❪✳ ❯♠ ♦✉tr♦ t✐♣♦ ❞❡ ❞✐ss✐♣❛çã♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✱ ❢♦✐ ❡st✉❞❛❞♦ ❡♠ ✉♠ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ♣♦r ❇❧♦❡♠❜❡r❣❡♥ ❡ P♦✉♥❞ ❬✷❪ ❡ ♠❡❞✐❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣♦r ❙③ö❦❡ ❬✸❪✱ ♠❛s só ❤á ❛❧❣✉♥s ❛♥♦s s❡ ❞❡✉ ♠❛✐♦r ❛t❡♥çã♦ ❛ ❡st❡ ❢❡♥ô♠❡♥♦✳ ❖ ❛♠♦rt❡❝✐✲ ♠❡♥t♦ r❛❞✐❛t✐✈♦ ♦❝♦rr❡ q✉❛♥❞♦ ✉♠ ❝✐r❝✉✐t♦ r❡ss♦♥❛♥t❡ ❛❝♦♣❧❛❞♦ ❛ ✉♠ ♠❛t❡r✐❛❧ ♠❛❣✲ ♥ét✐❝♦ ❡stá ❡♠ s✐♥t♦♥✐❛ ❝♦♠ s✉❛ ❢r❡qüê♥❝✐❛ ❞❡ ▲❛r♠♦r✳ ❊st❡ ♣r♦❝❡ss♦ ❞✐ss✐♣❛t✐✈♦ ♣♦ss✉✐ ✉♠ ❝❛rát❡r ❞✐st✐♥t♦ ❞❛s r❡❧❛①❛çõ❡s ♠❛❣♥ét✐❝❛s✱ s❡♥❞♦ t❛❧✈❡③ ♦ ♠❛✐s ❞❡st❛✲ ❝❛❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝♦❡r❡♥t❡ q✉❡ ♠❛♥té♠ ❝♦♥st❛♥t❡ ♦ ♠ó❞✉❧♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦✳ ❆♣❡s❛r ❞❡ t❛♥t♦s ❛♥♦s ❛♣ós ❛ s✉❛ ❞❡s❝♦❜❡rt❛✱ ❛✐♥❞❛ ♥ã♦ ❡①✐st❡ ✉♠❛ ❞❡s❝r✐çã♦ ❞❡ ♣r✐♠❡✐r♦s ♣r✐♥❝í♣✐♦s ❝♦♠♣❧❡t❛♠❡♥t❡ q✉â♥t✐❝❛ s♦❜r❡ ♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✳ ❍á ❛❧❣✉♥s ❛♥♦s ❡s♣❡❝✉❧♦✉✲s❡ s♦❜r❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ ❞❡s❝r✐çã♦ q✉â♥t✐❝❛ ❬✹✱ ✺✱ ✻✱ ✼❪✱ ❛té q✉❡ ✉♠❛ ♣r✐♠❡✐r❛ ❞❡s❝r✐çã♦ ❜❛s❡❛❞❛ ❡♠ ✉♠❛ ❡q✉❛çã♦ ♠❡str❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ ❢♦✐ ❢❡✐t❛ ❬✽❪✳ ◆♦ ❡♥t❛♥t♦ ❡st❡ ❡st✉❞♦ ♥ã♦ ❡♥❝♦♥tr♦✉ ♥❡♥❤✉♠ ♦✉tr♦ r❡s✉❧t❛❞♦ ❞✐❢❡r❡♥t❡ ❞♦s ❥á ❝♦♥❤❡❝✐❞♦s✳

❊♠❜♦r❛ ♥ã♦ s❡ t❡♥❤❛ ♣r♦❝✉r❛❞♦ ♣♦r ♥♦✈❛s ❞❡s❝r✐çõ❡s q✉â♥t✐❝❛s ❞♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✱ s✉r❣✐r❛♠ ❡st✉❞♦s ♥♦s q✉❛✐s ❡♥❝♦♥tr❛r❛♠ ♥♦✈❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❝❧ás✲ s✐❝❛s ❞♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✳ ◆❡st❡s tr❛❜❛❧❤♦s ✉s♦✉✲s❡ ♦ ♠ét♦❞♦ ❞❡ ♣r♦❥❡çã♦ ❡st❡r❡♦❣rá✜❝❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♣❛r❛ s❡ ❡st✉❞❛r ❛ r❡s♣♦st❛ ❞♦ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦ ❛ ❞✐❢❡r❡♥t❡s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ♣✉❧s♦s ♠❛❣♥ét✐❝♦s ❛♣❧✐✲

❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✷

❝❛❞♦s ❬✾✱ ✶✵✱ ✶✶✱ ✶✷❪✳

◆❡st❛ t❡s❡ ❞❡ ❞♦✉t♦r❛♠❡♥t♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ✉♠❛ ❞❡s❝r✐çã♦ q✉â♥t✐❝❛ ❞♦ ❛♠♦rt❡❝✲ ✐♠❡♥t♦ r❛❞✐❛t✐✈♦✱ ♣❛rt✐♥❞♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞♦ s♣✐♥s ❞♦ ♠❛t❡r✐❛❧ ♠❛❣✲ ♥ét✐❝♦ ❛❝♦♣❧❛❞♦s ❛ ✉♠ ❝✐r❝✉✐t♦ r❡ss♦♥❛♥t❡ ❡ ❞✐ss✐♣❛t✐✈♦✳ ❯s❛♠♦s ♦ ♠ét♦❞♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ▲❛♥❣❡✈✐♥ q✉â♥t✐❝❛s ❡ r❡❝♦rr❡♠♦s ❛♦ ♠ét♦❞♦ ❞❡ ♣r♦❥❡çã♦ ❡st❡r❡♦❣rá✜❝❛ ♣❛r❛ ♦❜t❡r ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ♠❛❣♥❡t✐③❛çã♦✳ ▼♦str❛♠♦s q✉❡✱ ❛❧é♠ ❞❡ ♥♦ss♦ ♠♦❞❡❧♦ ❞❡s❝r❡✈❡r ♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✱ t❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s ✉♠❛ ❞✐♥â♠✐❝❛ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛ ❞❡♣❡♥❞❡♥❞♦ ❞❛s ❝♦♥❞✐çõ❡s ❞♦ ❝✐r❝✉✐t♦ r❡ss♦♥❛♥t❡✳

❆❝♦♠♣❛♥❤❛♥❞♦ ♦s ❢❡♥ô♠❡♥♦s ❞✐ss✐♣❛t✐✈♦s✱ s❡♠♣r❡ s❡ ♣♦❞❡ ❡s♣❡r❛r ❛ ♣r❡s❡♥ç❛ ❞❡ ✢✉t✉❛çõ❡s ❛❧❡❛tór✐❛s ❞❡❝♦rr❡♥t❡ ❞❛ ✐♥t❡r❛çã♦ ❞♦ s✐st❡♠❛ ❞❡ ✐♥t❡r❡ss❡ ❝♦♠ ♦ ❛♠❜✐❡♥t❡ ❛ s✉❛ ✈♦❧t❛✳ ◆♦ ❝❛s♦ ❡st❛❝✐♦♥ár✐♦ ❡ ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ❞✐ss✐♣❛çã♦ ❡ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ❛♠❜✐❡♥t❡ é ❡st❛❜❡❧❡❝✐❞❛ ♣❡❧♦ t❡♦r❡♠❛ ✢✉t✉❛çã♦✲❞✐ss✐♣❛çã♦✳ ❈♦♠ r❡❧❛çã♦ ❛ ✉♠ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦ ❡ ✉♠ ❝✐r❝✉✐t♦ ❛❝♦♣❧❛❞♦s ♥ã♦ é ❞✐❢❡r❡♥t❡✳ ❈♦♠♦ ❢♦✐ ♣r❡✈✐st♦ ❡ ♠❡❞✐❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣♦r ❙❧❡❛t♦r✱ ❍❛❤♥✱ ❍✐❧❜❡rt ❡ ❈❧❛r❦❡ ❬✶✸✱ ✶✹❪✱ ✉♠ r✉í❞♦ ♣r♦✈❡♥✐❡♥t❡ ❞♦s s♣✐♥s ❞♦ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦ ❛❧t❡r❛ ❛ ❝♦rr❡♥t❡ ✢✉t✉❛♥t❡ ❞♦ ❝✐r❝✉✐t♦ r❡ss♦♥❛♥t❡✳ ❆ ♦r✐❣❡♠ ❞♦ r✉í❞♦ ♥♦ ❝✐r❝✉✐t♦ ♣r♦✈é♠ t❛♥t♦ ❞♦ s❡✉ ❛❝♦♣❧❛♠❡♥t♦ ❝♦♠ ♦ ❛♠❜✐❡♥t❡ ♣♦r ♠❡✐♦ ❞❡ s✉❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛✱ ❝♦♠♦ ❞♦ r✉í❞♦ ❣❡r❛❞♦ ♥♦s s♣✐♥s ❞♦ ♠❛t❡r✐❛❧ ❞❡✈✐❞♦ ❛ s✉❛ ✐♥t❡r❛çã♦ ❝♦♠ ❛ r❡❞❡ ♠❛t❡r✐❛❧ q✉❡ ♦ ❝♦♥st✐t✉✐ ❡ ❣❡r❛ ♦s ♣r♦❝❡ss♦s ❞❡ r❡❧❛①❛çã♦✳ ❘❡❝❡♥t❡♠❡♥t❡ ♦ ❢❡♥ô♠❡♥♦ ❞❡ r✉í❞♦ ❞❡ s♣✐♥s ❢♦✐ ✉s❛❞♦ ♣❛r❛ ♦❜t❡r ✐♠❛❣❡♥s ❡♠ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ♥✉❝❧❡❛r ❬✶✺❪✱ ♠♦str❛♥❞♦ ❣r❛♥❞❡s ♣❡rs♣❡❝t✐✈❛s ♥❡st❛ té❝♥✐❝❛ ❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ♥♦ t❡♠❛✳

❆ ♣❛rt✐r ❞♦ ♠❡s♠♦ ♠ét♦❞♦ q✉❡ ❞❡s❡♥✈♦❧✈❡♠♦s ♣❛r❛ ❡st✉❞❛r ❛ ❡✈♦❧✉çã♦ t❡♠✲ ♣♦r❛❧ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✱ ❡st✉❞❛♠♦s t❛♠❜é♠ ♦ r✉í❞♦ ❞❡ s♣✐♥s✳ ▼♦str❛♠♦s q✉❡ ♦ r✉í❞♦ ❞❡ s♣✐♥s ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❡stá ❛ss♦❝✐❛❞♦ à r❡❧❛①✲ ❛çã♦ tr❛♥s✈❡rs❛❧ ❞❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛✱ ♦ q✉❡ r❡♣r♦❞✉③ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❙❧❡❛t♦r✱ ❍❛❤♥✱ ❍✐❧❜❡rt ❡ ❈❧❛r❦❡ ❬✶✸✱ ✶✹❪✳ ❈♦♥✜r♠❛❞❛ ❛ ❡✜❝á❝✐❛ ❞♦ ♥♦ss♦ ♠ét♦❞♦ ♣❛r❛ ❡st✉❞❛r ♦ r✉í❞♦ ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦✱ ✐♥✈❡st✐❣❛♠♦s ♦ ❝❛s♦ ❡♠ q✉❡ ♦ s✐st❡♠❛ é ❝♦♥st❛♥t❡♠❡♥t❡ ❡①❝✐t❛❞♦ ♣♦r ✉♠❛ t❡♥sã♦ ♦s❝✐❧❛♥t❡ ❡①t❡r♥❛✳ ❉❡❞✉③✐♠♦s q✉❡✱ ❛❧é♠ ❞❡ ✉♠❛ r❡s♣♦st❛ ❧✐♥❡❛r ❡ ❝♦❡r❡♥t❡ ❛♦ s✐♥❛❧ ❛♣❧✐❝❛❞♦✱ ✉♠❛ ❝♦♥tr✐❜✉✐çã♦ ✐♥❝♦❡r❡♥t❡ é ♥♦t❛❞❛ ♥♦ r✉í❞♦ ❞♦ ❝✐r❝✉✐t♦✳ ❊st❛ ❝♦♥tr✐❜✉✐çã♦ ♣r♦❞✉③ ❛❧t❡r❛çõ❡s ♥❛s ✢✉t✉❛çõ❡s ❞❡ ❝♦rr❡♥t❡ ❞♦ ❝✐r❝✉✐t♦ ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛ s✉♣r❡ssã♦ ❞♦ s❡✉ r✉í❞♦ ❞❡ ◆②q✉✐st ❡♠ t♦r♥♦ ❞❡ três ❢r❡qüê♥❝✐❛s ❝❡♥tr❛❞❛s ♥❛ ❢r❡qüê♥❝✐❛ ❞❡ ▲❛r♠♦r ❞❛ ❛♠♦str❛✳ ❯♠ ❢❡♥ô♠❡♥♦ ❛ss❡♠❡❧❤❛❞♦ ❛ ❡st❡ q✉❡ ❡♥❝♦♥tr❛♠♦s é ✈✐st♦ ♥❛ Ó♣t✐❝❛ ◗✉â♥t✐❝❛ ❡ é ❝♦♥❤❡❝✐❞♦ ♣♦r ✢✉♦r❡s❝ê♥❝✐❛ r❡ss♦♥❛♥t❡ ❬✶✻✱ ✶✼❪✳

❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✸

❝✐❛ é ✐❣✉❛❧ à ❢r❡qüê♥❝✐❛ ❞❡ tr❛♥s✐çã♦ ❞❡ ❡st❛❞♦ ❛tô♠✐❝❛✳ ❖ ❡s♣❡❝tr♦ ❞❛ ❧✉③ ❡♠✐t✐❞❛ ♣♦ss✉✐ ✉♠❛ ♣❛rt❡ ✐♥❝♦❡r❡♥t❡ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r três ♣✐❝♦s ❝❡♥tr❛❞♦s ♥❛ ❢r❡qüê♥❝✐❛ ❞♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳ ❆ ❡①✐stê♥❝✐❛ ❞♦s três ♣✐❝♦s ❞❛ ❧✉③ ❡♠✐t✐❞❛ ❞❡✈❡✲s❡ ❛ ✉♠ ❞❡s❞♦❜r❛✲ ♠❡♥t♦ ❞♦s ❡st❛❞♦s ❛tô♠✐❝♦s ♣r♦✈♦❝❛❞♦ ♣❡❧♦ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❡❢❡✐t♦ ❡st❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ s❡♣❛r❛çã♦ ❙t❛r❦ ❞✐♥â♠✐❝❛ ❬✶✽❪✳ ◆♦ ❝❛s♦ ❛♣r❡s❡♥t❛❞♦ ♥❡st❛ t❡s❡✱ ♣♦r s✉❛ ✈❡③✱ ❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ é ❛❝♦♣❧❛❞❛ ❛ ✉♠ ❝✐r❝✉✐t♦ r❡ss♦♥❛♥t❡ s✉❜♠❡t✐❞♦ ❛ ✉♠❛ t❡♥sã♦ ❡①t❡r♥❛ ❛♣❧✐❝❛❞❛✳ ❊st❛ t❡♥sã♦ ❡①t❡r♥❛ ❧❡✈❛ ♦ ❝✐r❝✉✐t♦ ❛ ♣r♦❞✉③✐r ✉♠ ❝❛♠♣♦ ♠❛❣✲ ♥ét✐❝♦ ♦s❝✐❧❛♥t❡ s♦❜r❡ ❛ ❛♠♦str❛✱ q✉❡ s❡❣✉❡ ❛ ✉♠❛ ✐♥t❡r❛çã♦ ❞❡ ❩❡❡♠❛♥✳ ❖s s♣✐♥s ❞❛ ❛♠♦str❛ s✉❜♠❡t✐❞♦s ❛ ❡st❛ ✐♥t❡r❛çã♦ s♦❢r❡♠ ♦ ❞❡s❞♦❜r❛♠❡♥t♦ ❞♦s s❡✉s ❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ ❛❜s♦rçã♦ ❞♦ r✉í❞♦ ❞♦ ❝✐r❝✉✐t♦ ♣❡❧❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛✱ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ ❡s♣❡❝tr♦ ❞❡ três ♣✐❝♦s✳

❈♦♠♦ é ❛♣♦♥t❛❞♦ ♣♦r ❍❛♥❤ ❬✶✾❪✱ ♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦ ❡ ♦ r✉í❞♦ ❞❡ s♣✐♥s ❡s✲ tã♦ ❝♦♥❡❝t❛❞♦s✱ ❛ss✐♠ ❝♦♠♦ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦ ❝❛♠♣♦ ❞❛ r❡ss♦♥â♥✲ ❝✐❛ ♠❛❣♥ét✐❝❛ ❡ ❛ Ó♣t✐❝❛ ◗✉â♥t✐❝❛✱ ❡♠❜♦r❛ ❡st❛ ú❧t✐♠❛ r❡❧❛çã♦ ♥ã♦ s❡❥❛ ♣❡r❢❡✐t❛✳ P♦r ❡st❡ ♣♦♥t♦ ❞❡ ✈✐st❛✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦ é ✉♠ ❢❡♥ô♠❡♥♦ ❞❡ ❡♠✐ssã♦ ❡s♣♦♥tâ♥❡❛ ❝♦❧❡t✐✈❛ ❡ ♦ r✉í❞♦ ❞❡ s♣✐♥ é ❛♥á❧♦❣♦ ❛ ❡♠✐ssã♦ ❡s♣♦♥tâ♥❡❛ ❞❡ át♦♠♦s ❡♠ ✉♠❛ ❝❛✈✐❞❛❞❡ ❡❧❡tr♦♠❛❣♥ét✐❝❛✳

❊st❛ t❡s❡ ❞❡ ❞♦✉t♦r❛♠❡♥t♦ ❡stá ♦r❣❛♥✐③❛❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ ◆♦ ❝❛♣í✲ t✉❧♦ ✷ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛✳ ■♥✐✲ ❝✐❛❧♠❡♥t❡ ♠♦str❛♠♦s ❛❧❣✉♥s ❢❡♥ô♠❡♥♦s ❜ás✐❝♦s ❞❡ ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s s♦❜ ❛ ❛çã♦ ❞❡ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❝♦♥st❛♥t❡s ❡ ♦s❝✐❧❛♥t❡s✳ ◆❛ s❡çã♦ s❡❣✉✐♥t❡ ❛♣r❡s❡♥t❛♠♦s ❛ ❞✐♥â♠✐❝❛ ❜ás✐❝❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ♦s ❡❢❡✐t♦s ❞❡ r❡✲ ❧❛①❛çã♦✳ ❚❡r♠✐♥❛♠♦s ♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♥❞♦ ❛ ❞✐♥â♠✐❝❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♥♦ ❝❛s♦ ❞♦ ❛♠♦rt❡❝✐♠❡♥t♦ r❛❞✐❛t✐✈♦✳

◆♦ ❝❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s ♦ ♠ét♦❞♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ▲❛♥❣❡✈✐♥ ♣❛r❛ ❞❡s❝r✲ ❡✈❡r s✐st❡♠❛s ❢ís✐❝♦s ❛❜❡rt♦s✱ ✐st♦ é✱ ❡♠ ❝♦♥t❛t♦ ❝♦♠ ✉♠ r❡s❡r✈❛tór✐♦ tér♠✐❝♦✳ ❉❡✲ s❝r❡✈❡♠♦s ♦ ♠ét♦❞♦ ♥❛ s✉❛ ✈❡rsã♦ ❝❧áss✐❝❛ ❡ q✉â♥t✐❝❛✳ ❉✐s❝✉t✐♠♦s ❞❡♣♦✐s s♦❜r❡ ❛s s✐♠✐❧❛r✐❞❛❞❡s ❡ ❞✐❢❡r❡♥ç❛s ❞♦s ❞♦✐s ❢♦r♠❛❧✐s♠♦s✳

❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✹

❈❛♣ít✉❧♦ ✷

❉✐♥â♠✐❝❛ ❜ás✐❝❛ ❞❛ r❡ss♦♥â♥❝✐❛

♠❛❣♥ét✐❝❛

■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❛s♣❡❝t♦s ❜ás✐❝♦s ❞❛ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♣♦ss❛♠♦s ✐♥tr♦❞✉③✐r ❛ ♥♦t❛çã♦ ✉s❛❞❛ ♥❡st❛ t❡s❡ ❡ ❝♦♥t❡①t✉❛❧✐③❛r ♦s ❢❡♥ô♠❡♥♦s r❡❧❛❝✐♦♥❛❞♦s ❛♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ♦r✐❣✐♥❛✐s ❛♣r❡s❡♥t❛❞♦s ♥♦s ❝❛♣ít✉✲ ❧♦s ✹ ❡ ✺✳ ❆q✉✐ ✈❛♠♦s ❝❤❛♠❛r ❞❡ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ✉♠❛ s✉❜stâ♥❝✐❛ ❝♦♠♣♦st❛ ♣♦r ♣❛rtí❝✉❧❛s ❝♦♠ ♠♦♠❡♥t♦s ❞❡ ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ✐❞ê♥t✐❝♦s✳ ❆s ♣❛rtí❝✉❧❛s ♠❛❣♥ét✐❝❛s ❞❡ ✐♥t❡r❡ss❡ ❡stã♦ ❞✐str✐❜✉í❞❛s ❞❡ ❢♦r♠❛ ❛❧❡❛tór✐❛ ❡ ❤♦♠♦❣ê♥❡❛✱ ✐st♦ é✱ ❛ ❞❡♥s✐❞❛❞❡ ♠é❞✐❛ ❞❡ ♣❛rtí❝✉❧❛s é ❛ ♠❡s♠❛ ❡♠ t♦❞❛ ❛ s✉❜stâ♥❝✐❛✳ ❆❧é♠ ❞✐ss♦✱ ❛s ♣❛rtí❝✉❧❛s ✐♥t❡r❛❣❡♠ ♠✉✐t♦ ❢r❛❝❛♠❡♥t❡ ✉♠❛s ❝♦♠ ❛s ♦✉tr❛s✱ ❛ss✐♠ ♥ã♦ ❧❡✈❛♠♦s ❡♠ ❝♦♥t❛ ✐♥✲ t❡r❛çõ❡s ❞✐r❡t❛s ❞❛s ♣❛rtí❝✉❧❛s ❡♥tr❡ s✐✳ ◆♦ ❡♥t❛♥t♦ ❛s ✐♥t❡r❛çõ❡s ❞❡ ❝❛❞❛ ♣❛rtí❝✉❧❛ ❝♦♠ ♦ ❛♠❜✐❡♥t❡ ❛ s✉❛ ✈♦❧t❛✱ ♦ q✉❡ ✐♥❝❧✉✐ ♦ ♣ró♣r✐♦ ♠❛t❡r✐❛❧ q✉❡ ❝♦♠♣õ❡ ❛ ❛♠♦str❛✱ s❡rã♦ ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ❛ ✐♥t❡r❛çã♦ ❝♦♠ ✉♠ ❜❛♥❤♦ tér♠✐❝♦ ❞❡ ✐♥✜♥✐t♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ r❡❞❡ ♠❛t❡r✐❛❧✳ ❉✉❛s ❡①❝❡❧❡♥t❡s ❡①♣♦s✐çõ❡s s♦❜r❡ ♦s ❢❡♥ô♠❡♥♦s ❞❡ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ sã♦ ❡♥❝♦♥tr❛❞❛s ♥♦s ❧✐✈r♦s ❞❡ ❙❧✐❝❤t❡r ❬✷✵❪ ❡ ❆❜r❛❣❛♠ ❬✷✶❪✳

✷✳✶ ❉✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❡♠ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s s✐♠✲

♣❧❡s

❖ ♣r✐♠❡✐r♦ ♣r♦❜❧❡♠❛ ❛❝❡r❝❛ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s ♠❛❣♥ét✐❝❛s ❞❡ ✉♠❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ é ♦ ❝❛s♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❧✐✈r❡s ❡♠ ❝❛♠♣♦s ♠❛❣♥ét✐✲ ❝♦s ❝♦♥st❛♥t❡s ❡ ♦s❝✐❧❛♥t❡s✳ ❱❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❡st✉❞❛r ❡st❡ s✐st❡♠❛ ❝❧❛ss✐❝❛♠❡♥t❡ ❡ ❞❡♣♦✐s q✉❛♥t✐❝❛♠❡♥t❡✳

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✻

✷✳✶✳✶ ❚r❛t❛♠❡♥t♦ ❝❧áss✐❝♦

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❊❧❡tr♦♠❛❣♥❡t✐s♠♦ ❈❧áss✐❝♦ ❬✷✷❪✱ ♦ t♦rq✉❡ _T~ _{s♦❢r✐❞♦ ♣♦r ✉♠❛} ♣❛rtí❝✉❧❛ ❝♦♠ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ~µs♦❜ ❛ ❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦

~ H é

~

T = d ~L

dt =~µ×H,~ ✭✷✳✶✮

t❛❧ q✉❡ ❛ ♣❛rtí❝✉❧❛ t❡♠ ✉♠ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r _L~ _{♣❛r❛❧❡❧♦ ❛♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦} ♠❛❣♥ét✐❝♦✱~µ✱ ❞❡ ♠♦❞♦ q✉❡ é ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ❛ r❡❧❛çã♦

~µ=γ ~_L, ✭✷✳✷✮

♥❛ q✉❛❧ γ é ❛ r❛③ã♦ ❣✐r♦♠❛❣♥ét✐❝❛ ❞❛ ♣❛rtí❝✉❧❛✳ ❆ r❛③ã♦ ❣✐r♦♠❛❣♥ét✐❝❛ ❞❡♣❡♥❞❡ ❞❛ ❡str✉t✉r❛ ✐♥t❡r♥❛ ❞❛ ♣❛rtí❝✉❧❛ ❡♠ q✉❡stã♦✳ ◆♦ ❝❛s♦ ❞❛ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ♥✉❝❧❡❛r✱γ ❞❡♣❡♥❞❡ ❞❛ ❡str✉t✉r❛ ❞❡ s♣✐♥s ✐♥t❡r♥❛ ❞♦ ♥ú❝❧❡♦ ❛tô♠✐❝♦ ❡♠ ❡st✉❞♦✳ ❏á ♥♦ ❝❛s♦ ❞❛ r❡ss♦♥â♥❝✐❛ ♣❛r❛♠❛❣♥ét✐❝❛ ❞♦ ❡❧étr♦♥✱ γ ❞❡♣❡♥❞❡ ❞♦s ♠♦♠❡♥t♦s ❛♥❣✉❧❛r❡s ♦r❜✐t❛✐s ❡ ❞❡ s♣✐♥ ❡ s❡✉s ❛❝♦♣❧❛♠❡♥t♦s✳ ◆❡st❛ t❡s❡ ♥ã♦ ♥♦s ♣r❡♦❝✉♣❛r❡♠♦s ❝♦♠ t❛✐s ♣r♦❜❧❡♠❛s✳

Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ _H~ _{é ✉♠ ❝❛♠♣♦ ❝♦♥✲} st❛♥t❡ ❛♣❧✐❝❛❞♦✱

~

H(t) =H0z.ˆ ✭✷✳✸✮

❉❛❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ①✲②✲③✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❝♦♠♦

~µ=µ0(sin(θ) cos(φ)ˆx+ sin(θ) sin(φ)ˆy+ cos(θ)ˆz), ✭✷✳✹✮

t❛❧ q✉❡θ é ♦ â♥❣✉❧♦ ❞❡~µ❝♦♠ r❡❧❛çã♦ ❛♦ ❡✐①♦z ❡φ é ♦ â♥❣✉❧♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ❡✐①♦x ❞❛ ♣r♦❥❡çã♦ ❞❡~µ♥♦ ♣❧❛♥♦ x✕y✳ ❈♦♠ ❛s ❡①♣r❡ssõ❡s ✭✷✳✸✮ ❡ ✭✷✳✹✮ ❡ ❛s ❡q✉❛çõ❡s ✭✷✳✶✮ ❡ ✭✷✳✷✮ ♣♦❞❡♠♦s ♠♦♥t❛r ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❝✉❥❛s s♦❧✉çõ❡s sã♦

µx(t) = µ0sin(θ0) cos(φ0−ω0t), ✭✷✳✺✮

µy(t) = µ0sin(θ0) sin(φ0−ω0t) ✭✷✳✻✮ ❡

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✼

t❛✐s q✉❡µ0 é ♦ ♠ó❞✉❧♦ ❞♦ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❡❧❡♠❡♥t❛r ❞❛ ♣❛rtí❝✉❧❛✱ θ0 ❡ φ0 sã♦ ♦s â♥❣✉❧♦s ❡♠ ✭✷✳✹✮ ♥♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧t = 0 ❡

ω0 =γH0 ✭✷✳✽✮

é ❛ ❢r❡qüê♥❝✐❛ ❛♥❣✉❧❛r ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♣r❡❝❡ssã♦ r❡❛❧✐③❛❞♦ ♣❡❧❛ ♣❛rtí❝✉❧❛ ❡♠ t♦r♥♦ ❞❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛♣❧✐❝❛❞♦✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢r❡qüê♥❝✐❛ ❞❡ ▲❛r♠♦r✳

❆ss✐♠ ❞❡s❝r❡✈❡♠♦s ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♣r❡❝❡ssã♦ ❞❡ ✉♠❛ ú♥✐❝❛ ♣❛rtí❝✉❧❛ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❝♦♥st❛♥t❡✳ ◆♦ ❡♥t❛♥t♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ✉♠❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❝♦♠♣♦st❛ ♣♦r ✉♠ ♥ú♠❡r♦ ♠❛❝r♦s❝ó♣✐❝♦ ✭❞❛ ♦r❞❡♠ ❞♦ ♥ú♠❡r♦ ❞❡ ❆✈♦❣❛❞r♦✮ ❞❡ ♣❛rtí❝✉❧❛s✳ P❛r❛ ❡♥t❡♥❞❡r♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛❝r♦s❝ó♣✐❝♦✱ ❞❡✈❡♠♦s s♦♠❛r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ t♦❞❛s ❛s ♣❛rtí❝✉❧❛s✱ s❡❣✉✐♥❞♦ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❡st❛tíst✐❝❛✳ ◆❡st❡ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ❛ ❛♠♦str❛ ✐♠❡rs❛ ♥♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡stá ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦✳ ❈♦♠♦ ♣❡❧♦ ❊❧❡tr♦♠❛❣♥❡t✐s♠♦ ❈❧áss✐❝♦ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞♦ s✐st❡♠❛ é

H_{0 =−~µ·H}~ _{=−H0µ0cos(θ),} ✭✷✳✾✮ ♦ ♠♦✈✐♠❡♥t♦ ♠é❞✐♦ ❞❛s ♣❛rtí❝✉❧❛s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛✱ ❞❡✈❡ s❡r ❝❛❧❝✉❧❛❞♦ ♣♦r ❬✷✸❪

h~µithermal =

R

dΩµ~µeH0µ0cos(θ)/kBT

R

dΩµeH0µ0cos(θ)/kBT

. ✭✷✳✶✵✮

❖ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ♠é❞✐♦ r❡s✉❧t❛♥t❡ ❞❛ ❡①♣r❡ssã♦ ✭✷✳✶✵✮ ♣♦ss✉✐ ❛♣❡♥❛s ❛ ❝♦♠♣♦✲ ♥❡♥t❡ z ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❖ ♠♦t✐✈♦ ❞✐st♦ é q✉❡ ❝❛❞❛ ♣❛rtí❝✉❧❛ ❞❡s❝r❡✈❡ ✉♠ ♠♦✈✐✲ ♠❡♥t♦ ❞❛❞♦ ♣❡❧❛s ❡①♣r❡ssõ❡s ✭✷✳✺✮✱ ✭✷✳✻✮ ❡ ✭✷✳✼✮✱ ❝✉❥♦s â♥❣✉❧♦s ✐♥✐❝✐❛✐s sã♦ ❛❧❡❛tór✐♦s✳ ❈♦♥t✉❞♦✱ ❡♥q✉❛♥t♦ φ0 t❡♠ ✉♠❛ ❡st❛tíst✐❝❛ ❝♦♥st❛♥t❡ ♣❛r❛ t♦❞♦s ♦s â♥❣✉❧♦s ❞❡ 0 ❛

2π✱ θ0 t❡♠ ✉♠❛ ❡st❛tíst✐❝❛ ❞❛❞❛ ♣❡❧♦ ❢❛t♦r ❞❡ ❇♦❧t③♠❛♥♥ e−H0/kBT✳ ▲♦❣♦ ❛s ❝♦♠✲ ♣♦♥❡♥t❡s tr❛♥s✈❡rs❛✐s ❞♦s ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❝❛♥❝❡❧❛♠✲s❡ ❡ ❛♣❡♥❛s ❛s ❝♦♠♣♦♥❡♥t❡s ♥❛ ❞✐r❡çã♦z ♣♦ss✉❡♠ ✉♠❛ s♦♠❛ ♥ã♦ ♥✉❧❛✳

❆❧é♠ ❞❛ ♠é❞✐❛ ❡st❛tíst✐❝❛✱ ♣❛r❛ ♦❜t❡r♠♦s ❛ ♠❛❣♥❡t✐③❛çã♦ ♠❛❝r♦s❝ó♣✐❝❛ ❛ ♣❛r✲ t✐r ❞♦ ♠♦✈✐♠❡♥t♦ ❞❛s ♣❛rtí❝✉❧❛s ♠✐❝r♦s❝ó♣✐❝❛s✱ ❞❡✈❡♠♦s ❢❛③❡r ✉♠❛ ♠é❞✐❛ ❞❛ ❞✐s✲ tr✐❜✉✐çã♦ ❡s♣❛❝✐❛❧ ❞❛s ♣❛rtí❝✉❧❛s✳ P❛r❛ ✐st♦ ❞❡✈❡♠♦s ❧❡♠❜r❛r ❞❡ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ é ❛ ❞❡♥s✐❞❛❞❡ ❡s♣❛❝✐❛❧ ❞♦ ♠♦♠❡♥t♦ ❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❬✷✷❪✳ ❙❡ ❡①✐st❡♠ ✐♥♦♠♦✲ ❣❡♥❡✐❞❛❞❡s ❞❛ ❛♠♦str❛✱ ❡♥tã♦✱ ♣❛r❛ ✉♠ ❞❛❞♦ ✈♦❧✉♠❡ ∆V~r ❡♠ t♦r♥♦ ❞❡ ✉♠❛ ❞❛❞❛

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✽

r❡❧❛çã♦ ❛ t♦❞❛ ❛♠♦str❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ❝♦♠♦

~

M(~r) = N~rh~µi~r ∆V~r

. ✭✷✳✶✶✮

P❛r❛ ❡✈✐t❛r ❝♦♠♣❧✐❝❛çõ❡s ❞❡s♥❡❝❡ssár✐❛s✱ ♥❡st❛ t❡s❡ s❡♠♣r❡ ❝♦♥s✐❞❡r❛♠♦s ❛♠♦str❛s ♠❛❣♥ét✐❝❛s ❡s♣❛❝✐❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛s✱ ❛ss✐♠ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥ã♦ é ✉♠❛ ❢✉♥çã♦ ❞❛ ♣♦s✐çã♦✱ ❞❡ ♠♦❞♦ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

~ M = N

Vsh

~µithermal ✭✷✳✶✷✮

♥❛ q✉❛❧N é ♦ ♥ú♠❡r♦ ❞❡ ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❡ Vs é ♦ ✈♦❧✉♠❡ ❞❛ ❛♠♦str❛✳

❋❡✐t❛ ❛s ♠é❞✐❛s ❞❡s❝r✐t❛s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❛ ❛♠♦str❛ ❞❡✈✐❞♦ ❛♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦H0zˆ❡ ♦❜t❡♠♦s

~

M =M0z,ˆ ✭✷✳✶✸✮

❝♦♠ ❛ ♠❛❣♥❡t✐③❛çã♦ ❡stát✐❝❛ ❞❛❞❛ ♣♦r

M0 = N µ0

Vs

coth

µ0H0 kBT

− kBT

µ0H0

. ✭✷✳✶✹✮

P❛r❛ ❛❧t❛s t❡♠♣❡r❛t✉r❛s ❛ ❡①♣r❡ssã♦ ✭✷✳✶✹✮ s❡ r❡❞✉③ ❛

M0 = N µ2

0H0

3VskBT

. ✭✷✳✶✺✮

❆ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ M0 ♣❡❧♦ ✐♥✈❡rs♦ ❞❛ t❡♠♣❡r❛t✉r❛✱ 1/T✱ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❧❡✐ ❞❡ ❈✉r✐❡✳

❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞✐r❡t❛♠❡♥t❡ ❛♣❧✐❝❛♥❞♦✲ s❡ ❛ r❡❧❛çã♦ ✭✷✳✶✶✮ ♥❛ ❡q✉❛çã♦ ✭✷✳✶✮✳ ❖ r❡s✉❧t❛❞♦ é s✐♠♣❧❡s♠❡♥t❡

d ~M

dt =γ ~M ×H.~ ✭✷✳✶✻✮

❆ ❡q✉❛çã♦ ✭✷✳✶✻✮ ♣♦❞❡ r❡s✉❧t❛r ♥♦ ❝❛s♦ ❡stát✐❝♦ ❛♥t❡r✐♦r ❛♣❡♥❛s t♦♠❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧_M~_{(0) = M0z✳ P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ ♦ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦ ♣♦ss✉✐ ✉♠❛ ❝♦♠♣♦✲ˆ} ♥❡♥t❡ ♦s❝✐❧❛♥t❡ ❢r❛❝❛✱

~

H(t) =H0zˆ+H1(t)ˆx=H0zˆ+H1cos(νt)ˆx, ✭✷✳✶✼✮

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✾

❝❡ssã♦ ❡♠ t♦r♥♦ ❞❛ ❞✐r❡çã♦ z ❡ ✉♠ ♠♦✈✐♠❡♥t♦ ❞❡ ♥✉t❛çã♦ q✉❡ ❞❡♣❡♥❞❡ ❞❛ s✐♥t♦♥✐❛ ❡♥tr❡ω0 ❡ ν✳ P❛r❛ s❡ ❡①♣❧✐❝❛r ❡st❡ ❢❡♥ô♠❡♥♦✱ ✈❛♠♦s ✉s❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦ ❡ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡✳ ❖ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦ é ♦ r❡❢❡r❡♥❝✐❛❧ ✐♥❡r✲ ❝✐❛❧ ❞♦ ❛♣❛r❛t♦ ❡①♣❡r✐♠❡♥t❛❧✱ ❞❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❡t❝❀ ♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ é ✉♠ r❡❢❡r❡♥❝✐❛❧ q✉❡ ❣✐r❛ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ z ❝♦♠ ✉♠❛ ❢r❡qüê♥❝✐❛ ❛♥❣✉❧❛r

~ω =₋ωrz,ˆ ✭✷✳✶✽✮

❡♠ r❡❧❛çã♦ ❛♦ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦✳ ❆ ♠❛❣♥❡t✐③❛çã♦ ❞❛ ❛♠♦str❛ ❞❡♣❡♥❞❡ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♥❛ q✉❛❧ é ♠❡❞✐❞❛✳ ■st♦ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ❡❢❡t✉❛♥❞♦✲s❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞♦ ✈❡t♦r ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❞❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♣❛r❛ ♦ ♦✉tr♦✳ ❆ss✐♠ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦

d ~M dt |lab =

"

d ~M

dt −M~ ×~ω

#

rot

, ✭✷✳✶✾✮

♥❛ q✉❛❧ ♦ s✐♥❛❧|lab✐♥❞✐❝❛ ❛ ✈❛r✐á✈❡❧ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦ ❡|rot✐♥❞✐❝❛ ❛ ✈❛r✐á✈❡❧ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡✳ ▼❛✐s ❛❞✐❛♥t❡ ✈❛♠♦s ✐♥❞✐❝❛r ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♦✉ ♦s ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦ ❝♦♠ ♦s í♥❞✐❝❡s ♠✐♥ús❝✉❧♦sx✱ y ❡z ❡ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ ❝♦♠ ♦s í♥❞✐❝❡s ♠❛✐ús❝✉❧♦s X✱Y ❡ Z✳

P♦r ❝♦♥✈❡♥✐ê♥❝✐❛ ✈❛♠♦s ❢❛③❡r ωr =ν ❡ ✉s❛r ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♦♥❞❛ ❣✐r❛♥t❡ ✭♦✉

♥❛ s✐❣❧❛ ✐♥❣❧❡s❛ ❘❲❆✱ ❘♦t❛t✐♥❣ ❲❛✈❡ ❆♣r♦①✐♠❛t✐♦♥✮✱ ✐st♦ é✱ ❛♣r♦①✐♠❛r ♦ ❝❛♠♣♦ ❧✐♥❡❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦ ♣♦r ✉♠ ❝❛♠♣♦ ❝✐r❝✉❧❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦✱ ♣♦✐s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❇❧♦❝❤ ❡ ❙✐❡❣❡rt ♦ ❝❛♠♣♦ ♦s❝✐❧❛♥t❡ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❝❛♠♣♦ ❣✐r❛♥t❡ ❝♦♠ ❛ ❛♠♣❧✐t✉❞❡ ❞✐✈✐❞✐❞❛ ♣♦r ❞♦✐s ❬✷✹❪✳ P❛r❛ ✉♠ ❞❡t❛❧❤❛♠❡♥t♦ ❞❡st❛ ❛♣r♦①✐♠❛çã♦✱ ✈❡❥❛ ♦ ❛♣ê♥❞✐❝❡ ✼✳✶✳ ▲♦❣♦ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡

d ~M

dt =M~ ×

γH1

2 xˆ+ (ω0−ν)ˆz

. ✭✷✳✷✵✮

◆♦t❛✲s❡ q✉❡ ♥❡st❡ r❡❢❡r❡♥❝✐❛❧ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡❢❡t✐✈♦ ❛❣✐♥❞♦ s♦❜r❡ ❛ ❛♠♦str❛✱

~ Hef =

H1

2 xˆ+ 1

γ(ω0−ν)ˆz, ✭✷✳✷✶✮

é ❝♦♥st❛♥t❡✳ ❆ss✐♠✱ ❧❡♠❜r❛♥❞♦ ❞♦ ❝á❧❝✉❧♦ ❞❛ ♣r❡❝❡ssã♦ ❞♦ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❡♠ ✉♠ ❝❛♠♣♦ ❡stát✐❝♦✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ ❛❞q✉✐r❡ ✉♠❛ ♣r❡❝❡ssã♦ ❡♠ t♦r♥♦ ❞❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦_H~_{ef ❝♦♠ ❢r❡qüê♥❝✐❛ ❛♥❣✉❧❛r}

ωef =

r

(ω0 −ν)2+ ω2

1

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✵

♥❛ q✉❛❧ω1 =γH1✳ ◆♦ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦ ♦ q✉❡ s❡ ♦❜s❡r✈❛ é ✉♠❛ ♠❛❣♥❡t✐③❛✲ çã♦ ❞❡ ♠ó❞✉❧♦M0 ❡ ❝♦♠ s✉❛ ❞✐r❡çã♦ ❞❡s❝r❡✈❡♥❞♦ ✉♠ ♠♦✈✐♠❡♥t♦ ❝♦♠♣♦st♦ ♣♦r ✉♠❛ ♣r❡❝❡ssã♦ ❞❡ ❢r❡qüê♥❝✐❛ν ❡ ♣♦r ✉♠❛ ♥✉t❛çã♦ ❞❡ ❢r❡qüê♥❝✐❛ ωef✳ ❆ ❡st❡ ♠♦✈✐♠❡♥t♦ ❞❡ ♥✉t❛çã♦ é ❞❛❞♦ ♦ ♥♦♠❡ ❞❡ ♦s❝✐❧❛çã♦ ❞❡ ❘❛❜✐ ❬✷✺❪✳

❆ ❛♠♣❧✐t✉❞❡ ❞❛ ♦s❝✐❧❛çã♦ ❞❡ ❘❛❜✐ ❞❡♣❡♥❞❡ ❞❛ s✐♥t♦♥✐❛ ❡♥tr❡ ♦ ❝❛♠♣♦ ♦s❝✐❧❛♥t❡ ❡①t❡r♥♦ ❡ ❛ ❢r❡qüê♥❝✐❛ ❞❡ ▲❛r♠♦r✳ ◆♦ ❝❛s♦ ♣❡r❢❡✐t❛♠❡♥t❡ r❡ss♦♥❛♥t❡✱ ω0 = ν✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ t❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♠♣♦♥❡♥t❡s

MX = 0, ✭✷✳✷✸✮

MY =M0sin(ω1t/2) ✭✷✳✷✹✮

❡

MZ =M0cos(ω1t/2). ✭✷✳✷✺✮

◆♦ r❡❢❡r❡♥❝✐❛❧ ❞♦ ❧❛❜♦r❛tór✐♦ ❛ ♠❛❣♥❡t✐③❛çã♦ ✜❝❛

Mx =M0sin(ω1t/2) sin(ω0t), ✭✷✳✷✻✮ My =M0sin(ω1t/2) cos(ω0t) ✭✷✳✷✼✮ ❡

Mz =MZ. ✭✷✳✷✽✮

P♦❞❡♠♦s ♥♦t❛r q✉❡ s❡ ♦ ❝❛♠♣♦ ♦s❝✐❧❛♥t❡ é ❛♣❧✐❝❛❞♦ ❛♣❡♥❛s ❡♠ ❝❡rt♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ∆t✱ ❛ ♠❛❣♥❡t✐③❛çã♦ é r♦❞❛❞❛ ♣♦r ✉♠ â♥❣✉❧♦ ∆θ = ω1∆t/2✳ ❯♠❛ ✈❡③ ❞❛❞♦ ❡st❡ ♣✉❧s♦✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ♣❛ss❛ ❛ ❣✐r❛r ❡♠ ♣r❡❝❡ssã♦ ❝♦♠ ❢r❡qüê♥❝✐❛ω0 ❡ ✉♠ â♥❣✉❧♦ ∆θ ❡♠ r❡❧❛çã♦ ❛ s✉❛ ❞✐r❡çã♦ ✐♥✐❝✐❛❧✱ ♥❡st❡ ❝❛s♦ ♦ ❡✐①♦ z✳ ❈♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡st❡s ♣✉❧s♦s ❞♦ ❝❛♠♣♦H1 ✭❡♠ ♠❛❣♥❡t✐s♠♦ ♥✉❝❧❡❛r sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣✉❧s♦ ❞❡ r❛❞✐♦✲ ❢r❡qüê♥❝✐❛ ♦✉ ❘❋✮ ♣♦❞❡♠♦s tr❛♥s❢♦r♠❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❡stát✐❝❛ ❞❡ ✉♠❛ ❛♠♦str❛ ❡♠ ✉♠❛ ♠❛❣♥❡t✐③❛çã♦ ❣✐r❛♥t❡✳ ❆♣ós ❛ ❛♠♦str❛ s❡r ❡①❝✐t❛❞❛✱ ❡❧❛ ♣♦❞❡ ❛♣r❡s❡♥t❛r ✉♠❛ ♠❛❣♥❡t✐③❛çã♦ ❣✐r❛♥t❡ ♣♦r ❛❧❣✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✱ q✉❡ ♣r♦❣r❡ss✐✈❛♠❡♥t❡ r❡t♦r♥❛ ❛♦ ✈❛❧♦r ❡stát✐❝♦ ❡♠ r❛③ã♦ ❞❡ ❡❢❡✐t♦s ❞❡ r❡❧❛①❛çã♦✱ q✉❡ ❡st✉❞❛r❡♠♦s ♠❛✐s ❛ ❢r❡♥t❡✳

✷✳✶✳✷ ❚r❛t❛♠❡♥t♦ q✉â♥t✐❝♦

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✶

❞❡ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ é

~µ=γ~_S,~ ✭✷✳✷✾✮

♥❛ q✉❛❧~ é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦ ❞✐✈✐❞✐❞❛ ♣♦r _2π ❡ _S~ é ♦ ✈❡t♦r ♦♣❡r❛❞♦r ❞❡ s♣✐♥✱

q✉❡ ❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♦r❞❡♥❛❞❛s é

~

S =Sxxˆ+Syyˆ+Szz.ˆ ✭✷✳✸✵✮

❖s ♦♣❡r❛❞♦r❡s ❞❡ s♣✐♥Sx✱Sy ❡Sz ❞❡✈❡♠ r❡s♣❡✐t❛r ❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ ❞❡ s♣✐♥ ❬✷✻❪

[Sj, Sk] =iǫjklSl , j;k;l∈ {x, y, z}, ✭✷✳✸✶✮

♥❛ q✉❛❧ ǫjkl é 1 ♣❛r❛ ♣❡r♠✉t❛çõ❡s ❝í❝❧✐❝❛s ❞❡ j, k, l = x, y, z✱ −1 ♣❛r❛ ♦✉tr❛s ♣❡r✲

♠✉t❛çõ❡s ❡ 0 ♣❛r❛ ♦s ♦✉tr♦s ❝❛s♦s✳ ❚♦♠❛♥❞♦ _|S~_|2 _{= S}2

x +Sy2 +Sz2 ❡ ❛ r❡❧❛çã♦ ❞❡

❝♦♠✉t❛çã♦ ✭✷✳✸✶✮✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡

[|S~|2_{, S}

k] = 0, k ∈ {x, y, z}. ✭✷✳✸✷✮

❉❛❞♦ ✉♠ ❞✐♣♦❧♦ ♠❛❣♥ét✐❝♦ ❝♦♠♦ ❞❡s❝r✐t♦ ❛❝✐♠❛✱ ❛ s✉❛ ❤❛♠✐❧t♦♥✐❛♥❛✱ q✉❛♥❞♦ s✉❜♠❡t✐❞♦ ❛ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ q✉❛❧q✉❡r_{H✱ ❞❡✈❡ s❡r ✉♠❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ❩❡❡✲}~ ♠❛♥✱

H ₌₋~_{γ ~S·H.}~ ✭✷✳✸✸✮

▲♦❣♦ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞♦ s✐st❡♠❛ é

i~∂

∂t|ψ;ti=H |ψ;ti, ✭✷✳✸✹✮

♥❛ q✉❛❧|ψ;t_ié ♦ ✈❡t♦r ❞❡ ❡st❛❞♦ ❞♦ ❞✐♣♦❧♦✱ ❝♦♠ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ❞❛❞❛ ♣♦r_|ψ;t= 0_i✳

◆♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ é ❛♣❡♥❛s ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡stát✐❝♦_H~ _=H0z✱ˆ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ✭✷✳✸✸✮ ✜❝❛

H_{0 =−}~_γH0Sz. ✭✷✳✸✺✮

❈♦♠♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ✈❡t♦r ❞❡ ❡st❛❞♦ |ψ;ti ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ♦rt♦♥♦r♠❛❧ ❡

❝♦♠♣❧❡t♦ ❞❡ ✈❡t♦r❡s ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❞♦ s✐st❡♠❛✱ ✉♠❛ ❜♦❛ ❡s❝♦❧❤❛ s❡r✐❛ ♦s ❛✉t♦❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛✳ ❯♠❛ ✈❡③ q✉❡ ♦s ♦♣❡r❛❞♦r❡s Sz ❡ |S~|2 ❝♦♠✉t❛♠✱ ❡♥tã♦ ♦s

❛✉t♦❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛ ❞❡✈❡♠ s❡r ❛✉t♦❡st❛❞♦s s✐♠✉❧tâ♥❡♦s ❞❡Sz ❡|S~|2✱ s❡❣✉✐♥❞♦ ❛s

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✷

Sz|s, mi=m|s, mi ✭✷✳✸✻✮

❡

|S~_|2_|s, m_i=s(s+ 1)_|s, m_i, ✭✷✳✸✼✮ ❝♦♠ −s_≤m_≤s✳

❖ ✈❡t♦r ❞❡ ❡st❛❞♦ |ψ;t_i é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡s❝r❡✈❡r t♦❞❛ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠ ❞✐♣♦❧♦ ✐s♦❧❛❞♦✳ ◆♦ ❡♥t❛♥t♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ ❛❣❧♦♠❡r❛❞♦ ❞❡ ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❡♠ ❝♦♥t❛t♦ ❝♦♠ ♦ ❛♠❜✐❡♥t❡ ❛ s✉❛ ✈♦❧t❛✳ ▲♦❣♦ ❞❡✈❡♠♦s r❡❝♦rr❡r ❛ ✉♠ r❡❝✉rs♦ ♠❛✐s ❣❡r❛❧✱ q✉❡ ❞❡ ❝♦♥t❛ ❞❛s ❝♦♥tr✐❜✉✐çõ❡s ❡st❛tís✲ t✐❝❛s ❞❡ ✉♠ ❡♥s❡♠❜❧❡ ❞❡ ♣❛rtí❝✉❧❛s ♦✉ ❞❛ ♠é❞✐❛ ❞❛ ✐♥t❡r❛çã♦ ❝♦♠ ♦✉tr♦s s✐st❡♠❛s✱ ❝♦♥❤❡❝✐❞♦ ♣♦r ♠ét♦❞♦ ❞❡ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡✳ ❯♠❛ ♣❛rtí❝✉❧❛ ❝♦♠ ✉♠ ❞❛❞♦ ✈❡t♦r ❞❡ ❡st❛❞♦ t❡♠ ✉♠❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❞❛❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❡①t❡r♥♦ ❞♦ s❡✉ ✈❡t♦r ❞❡ ❡st❛❞♦ ♣♦r ❡❧❡ ♠❡s♠♦✱ ρ = _|ψ_ihψ_|✳ ❊st❛ s✐t✉❛çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡st❛❞♦ ♣✉r♦✳ P♦❞❡✲s❡ ✐♠❛❣✐♥❛r t❛♠❜é♠ q✉❡ ♦ ❡st❛❞♦ ❢ís✐❝♦ ❞❡ ✉♠ s✐st❡♠❛ é ❛ s♦♠❛ ❡st❛tíst✐❝❛ ❞❡ ✈ár✐♦s ❡st❛❞♦s ♣✉r♦s ❞✐st✐♥t♦s✳ ❆ss✐♠ ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❞♦ s✐st❡♠❛ é

ρ=X

k

Pk|ψkihψk|, ✭✷✳✸✽✮

♥❛ q✉❛❧Pk é ♦ ♣❡s♦ ❡st❛tíst✐❝♦ ❞♦ ✈❡t♦r ❞❡ ❡st❛❞♦ |ψki✱ t❛❧ q✉❡ ❞❡✈❡ ✈❛❧❡r ❛ ❝♦♥❞✐çã♦

P

kPk = 1✳ ❯♠❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❞❡st❡ t✐♣♦ é ❝❤❛♠❛❞❛ ❞❡ ❡st❛❞♦ ♠✐st♦✳

❆ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ é

dρ(t)

dt = i

~[ρ,H ], ✭✷✳✸✾✮

❝✉❥❛ s♦❧✉çã♦ ❣❡r❛❧ é

ρ(t) =e−(i/~₎Ht_ρ(0)e(i/~₎Ht _=U(t)ρ(0)U†_{(t), ✭✷✳✹✵✮}

♦♥❞❡ ✐♥tr♦❞✉③✐♠♦s ♦ ♦♣❡r❛❞♦r ✉♥✐tár✐♦ U(t) = e−(i/~₎H_t

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✸

hO(t)i = X

m

hm|Oρ(t)|mi= Tr[Oρ(t)] =

= Tr[OU(t)ρ(0)U†(t)] = Tr[U†(t)OU(t)ρ(0)] =

= Tr[_O(t)ρ(0)]. ✭✷✳✹✶✮

❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❛ ♠❛❣♥❡t✐③❛çã♦ ♠❛❝r♦s❝ó♣✐❝❛ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣❡❧♦ ✈❡t♦r ❞❡ s♣✐♥ ♠é❞✐♦✱

~

M(t) = N

Vs

~_γhS~_(t)i. ✭✷✳✹✷✮

❘❡♣❡t✐♥❞♦ ♦ q✉❡ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡ ♣❡❧♦ ❢♦r♠❛❧✐s♠♦ ❝❧áss✐❝♦✱ ✈❛♠♦s ❝❛❧❝✉✲ ❧❛r ❛ ♠❛❣♥❡t✐③❛çã♦ ❡stát✐❝❛ ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦✳ P♦❞❡♠♦s ♠♦str❛r q✉❡ ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❞♦ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ é

ρ(th) = e

−H_0/kBT

Tr [e−H0/kBT]. ✭✷✳✹✸✮

▲♦❣♦ é ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡ ❛ ú♥✐❝❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ♥ã♦ ♥✉❧❛ é ❬✷✸❪

Mz =M0 = N Vs

Tr

~_γSze(~γH0/kBT)Sz Tr [e(~_γH0/kBT)Sz

] =

N~_γ

Vs

Ps

m=−sme(

~_γH0/kBT)m

Ps

m=−se(

~_γH0/kBT)m , ✭✷✳✹✹✮ ♥❛ q✉❛❧ ❛ ♠❛❣♥❡t✐③❛çã♦ ❡stát✐❝❛ é

M0 = N Vs

~_γ

s+ 1 2

coth

s+ 1 2

_~

γH0 kBT

− 1

2coth

_~

γH0

2kBT

. ✭✷✳✹✺✮

P♦❞❡♠♦s ♥♦t❛r q✉❡ ♣❛r❛ ❛❧t❛s t❡♠♣❡r❛t✉r❛s ❛ ❡①♣r❡ssã♦ ✭✷✳✹✺✮ t❡♥❞❡ t❛♠❜é♠ ♣❛r❛ ❛ ❧❡✐ ❞❡ ❈✉r✐❡✱

M0 =

N(~_γ)2_{s(s+ 1)}

3VskBT

H0. ✭✷✳✹✻✮

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✹

é ❬✷✼❪

M0 = N Vs

µ0

coth

2µ0 γ~ tanh

_~

γH0

2kBT

− ~γ

2µ0

coth

_~

γH0

2kBT

. ✭✷✳✹✼✮

❖ tr❛❜❛❧❤♦ ❞❡ ❇♦②❡r ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❇❛rr❛♥❝♦✱ ❇r✉♥✐♥✐ ❡ ❍✉♠❜❡rt♦ ❋r❛♥ç❛ ❡ ❢♦✐ ♠♦str❛❞♦ ♣♦r ❡❧❡s q✉❡ ❛ ❡①♣r❡ssã♦ ✭✷✳✹✼✮ t❛♠❜é♠ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❬✷✽❪✳ ❊st❡s ❡st✉❞♦s ❢♦r❛♠ r❡❛❧✐③❛❞♦s ♥♦ ❝❛♠♣♦ ❞❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❊st♦❝ást✐❝❛ ✭♦✉ ♥❛ s✐❣❧❛ ✐♥❣❧❡s❛ ❙❊❉✱ ❙t♦❝❤❛st✐❝ ❊❧❡❝tr♦❞②♥❛♠✐❝s✮✳ ❊st❡ ❝❛♠♣♦ ❞❡ ♣❡sq✉✐s❛ ❛♣❧✐❝❛ ❛ t❡♦r✐❛ ❞❡ ♣r♦❝❡ss♦s ❡st♦❝ást✐❝♦s à ❊❧❡tr♦❞✐♥â♠✐❝❛ ❈❧áss✐❝❛✱ ♣♦s✲ t✉❧❛♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ r❛❞✐❛çã♦ ❞❡ ♣♦♥t♦✲③❡r♦ ❝❧áss✐❝❛✳ ❈♦♠ ❡st❡s ❡❧❡♠❡♥✲ t♦s ♠♦str❛✲s❡ q✉❡ ✉♠❛ ❣❛♠❛ ♠✉✐t♦ ❣r❛♥❞❡ ❞❡ ❢❡♥ô♠❡♥♦s ❝♦♥s✐❞❡r❛❞♦s ♣✉r❛♠❡♥t❡ q✉â♥t✐❝♦ ♣♦❞❡♠ s❡r ❡♥t❡♥❞✐❞♦s✱ ❞❡ ❢❛t♦✱ ❝♦♠♦ ❢❡♥ô♠❡♥♦s ❝❧áss✐❝♦s✳ ❆t✉❛❧♠❡♥t❡ ❛ ❊❧❡tr♦❞✐♥â♠✐❝❛ ❊st♦❝ást✐❝❛ ❡stá ❡♠ ❢r❛♥❝♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ ❯♠❛ ♣❡rs♣❡❝t✐✈❛ ❣❡r❛❧ ❞❡st❛ t❡♦r✐❛ é ❞❛❞❛ ♣❡❧♦ ❧✐✈r♦ ❞❡ P❡♥ã ❡ ❈❡tt♦ ❬✷✾❪✳

❱❛♠♦s ❛❣♦r❛ ❝❛❧❝✉❧❛r ❛s ♦s❝✐❧❛çõ❡s ❞❡ ❘❛❜✐ q✉❛♥t✐❝❛♠❡♥t❡ ❬✷✺❪✳ ❆ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ s♣✐♥ s♦❜ ❛ ❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡stát✐❝♦ ❡ ♦✉tr♦ ♦s❝✐❧❛♥t❡ é

H_s=−~_γH0Sz−~_{γH1cos(νt)Sx.} ✭✷✳✹✽✮

❆ss✐♠ ❝♦♠♦ ✜③❡♠♦s ❝❧❛ss✐❝❛♠❡♥t❡✱ é ♥❡❝❡ssár✐♦ ❡❢❡t✉❛r♠♦s ♦s ❝á❧❝✉❧♦s ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ ❞❡ ❢r❡qüê♥❝✐❛ ν ❡ ❡✐①♦ z✳ P❛r❛ t❛❧ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞❡ r♦t❛çã♦✱

Ur =e−iνtSz. ✭✷✳✹✾✮

❆ss✐♠ ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ♥♦ ♥♦✈♦ r❡❢❡r❡♥❝✐❛❧ é

ρr =UrρUr−1. ✭✷✳✺✵✮

❆ r❡s♣❡❝t✐✈❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ é

dρr(t)

dt = i

~[ρr,Hef], ✭✷✳✺✶✮

♥❛ q✉❛❧ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❡❢❡t✐✈❛ é

H_{ef =−}~_{(ω0−ν)Sz−}~ω1

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✺

❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♦♥❞❛ ❣✐r❛♥t❡✳ ❆ss✉♠✐♥❞♦ q✉❡ ♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ❞❛ ❛♠♦str❛ é ♦ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦✱ ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ✐♥✐❝✐❛❧ ❞❡✈❡ s❡r

ρr(0) =ρ(th). ✭✷✳✺✸✮

❆♣❧✐❝❛♥❞♦ ❛ s♦❧✉çã♦ ✭✷✳✹✵✮ à ❡q✉❛çã♦ ✭✷✳✺✶✮✱ ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ é

ρr(t) =

1

Tr [e−H₀/kBT]e

iω1tSx/2_e−H0/kBT_e−iω1tSx/2_. ✭✷✳✺✹✮

P♦rt❛♥t♦ ❛ ♠❛❣♥❡t✐③❛çã♦ ♠❛❝r♦s❝ó♣✐❝❛ ♥♦ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ ❞❡✈❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r

~ M_|rot =

N Vs

~_γTrh_Sρ~ _r(t)i ₌ N

Vs

~_γ

Trhe−iω1tSx/2_Se~ iω1tSx/2_e−H0/kBT

i

Tr [e−H_0/kBT

] , ✭✷✳✺✺✮

q✉❡✱ ♥♦ ❝❛s♦ r❡ss♦♥❛♥t❡ ❞❡ω0 =ν✱ r❡s✉❧t❛ ♥❛s ❡①♣r❡ssõ❡s ✭✷✳✷✸✮✱ ✭✷✳✷✹✮ ❡ ✭✷✳✷✺✮✱ ❥á ♦❜t✐❞❛s ❝❧❛ss✐❝❛♠❡♥t❡✳

✷✳✷ ❉✐♥â♠✐❝❛ ❞❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❝♦♠ r❡❧❛①❛çã♦

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✻

❇❧♦❝❤ ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❡✈❛♥t❡s✳ ◆❛ s✉❜s❡çã♦ s❡❣✉✐♥t❡ ✈❡r❡♠♦s ✉♠❛ ❜r❡✈❡ ❡①✲ ♣♦s✐çã♦ ❞❛ ❡①♣❧✐❝❛çã♦ ♠✐❝r♦s❝ó♣✐❝❛ ❞❛ r❡❧❛①❛çã♦✳

✷✳✷✳✶ ❊q✉❛çõ❡s ❞❡ ❇❧♦❝❤

❙❡❣✉♥❞♦ ❇❧♦❝❤✱ ♣❛r❛ ✐♥❝❧✉✐r♠♦s ♦ ❡❢❡✐t♦ ❞❡ r❡❧❛①❛çã♦ ♥❛ ❞✐♥â♠✐❝❛ ❞❛ ❛♠♦str❛ ♠❛❣✲ ♥ét✐❝❛✱ ❞❡✈❡♠♦s ❞✐st✐♥❣✉✐r ❞♦✐s t✐♣♦s ❞❡ r❡❧❛①❛çã♦ ❛❣✐♥❞♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❯♠ ❞❡st❡s t✐♣♦s é ❛ r❡❧❛①❛çã♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ♠♦✈✐♠❡♥t♦ ❣✐r❛tór✐♦ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ ❞♦ ❝❛♠♣♦ ❝♦♥st❛♥t❡ H0✳ ❈♦♠♦ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ q✉❛♥❞♦ ♦s ❞✐♣♦❧♦s ♠❛❣♥ét✐❝♦s ❡stã♦ t♦❞♦s ❡♠ ♠♦✈✐♠❡♥t♦ ❞❡ ♣r❡❝❡ssã♦ ❝♦♠ ❢❛s❡s ✐♥❝♦❡r❡♥t❡s✱ ❛ ♠❛❣♥❡t✐③❛çã♦ tr❛♥s✈❡rs❛❧ r❡s✉❧t❛♥t❡ é ♥✉❧❛✳ P♦r ♦✉tr♦ ❧❛❞♦ é ♣♦ssí✈❡❧ ♣r❡♣❛r❛r ❛ ❛♠♦str❛ ♠❛❣♥ét✐❝❛ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦s ❞✐♣♦❧♦s ❡st❡❥❛♠ ❡♠ ❢❛s❡✱ r❡s✉❧t❛♥❞♦ ❡♠ ✉♠❛ ♠❛❣♥❡t✐③❛çã♦ ❣✐r❛♥t❡ ♥ã♦ ♥✉❧❛✱ ❝♦♠♦ ✈✐♠♦s q✉❛♥❞♦ ✉♠ ❝❛♠♣♦ ♦s❝✐❧❛♥t❡ é ❛♣❧✐❝❛❞♦✳ ❖ ❡❢❡✐t♦ ❞❡ r❡❧❛①❛çã♦ ♥❡st❡ ❝❛s♦ ❧❡✈❛r✐❛ ♦s ❞✐♣♦❧♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❡♠ ❢❛s❡ ❛ ♣❡r❞❡r❡♠ ❛ ❝♦❡rê♥❝✐❛ ❡♥tr❡ s✐✱ ❡♠ r❛③ã♦ ❞♦ ❡❢❡✐t♦ ❞❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ❞✐♣♦❧♦s ❡ ❞❡ ♣❡rt✉r❜❛çõ❡s ❞♦s ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❧♦❝❛✐s ✐♥♦♠♦❣ê♥❡♦s ✳ ❊st❡ ♣r♦❝❡ss♦ ♣r♦❞✉③ ✉♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❛s ❝♦♠♣♦♥❡♥t❡s Mx ❡ My✳ ❖♠✐t✐♥❞♦ ♦s t❡r♠♦s r♦t❛❝✐♦♥❛✐s✱ ✐st♦ é

r❡♣r❡s❡♥t❛❞♦ ♣♦r

dMx;y

dt =−

1

T2

Mx;y, ✭✷✳✺✻✮

♥❛ q✉❛❧ ♦ t❡♠♣♦ ❞❡ ❞❡❝❛✐♠❡♥t♦T2 é ❝❤❛♠❛❞♦ ❞❡ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ tr❛♥s✈❡rs❛❧ ♦✉ ❞❡ r❡❧❛①❛çã♦ s♣✐♥✲s♣✐♥✳

❖ ♦✉tr♦ t✐♣♦ ❞❡ r❡❧❛①❛çã♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛ ♣❡rt✉r❜❛çã♦ tér♠✐❝❛ ❞❛ r❡❞❡ ♠❛✲ t❡r✐❛❧✳ ◆❡st❡ ❝❛s♦ ♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ r❡❞❡ ❧❡✈❛r✐❛♠ ♦s ❞✐♣♦❧♦s ❛ ❡♥tr❛r❡♠ ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ❝♦♠ ♦ ❛♠❜✐❡♥t❡ r❡t✐r❛♥❞♦ ✐rr❡✈❡rs✐✈❡❧♠❡♥t❡ ❛ s✉❛ ❡♥❡r❣✐❛✱ ♦✉ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ q✉â♥t✐❝♦✱ ❡st✐♠✉❧❛♥❞♦ ❛ tr❛♥s✐çã♦ ❞♦s ❡st❛❞♦s ❡①❝✐t❛❞♦s ❞♦s ❞✐♣♦❧♦s ♣❛r❛ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❛té ❛❧❝❛♥ç❛r❡♠ ✉♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ❡♥tr❡ ❛s ♣♦♣✉❧❛çõ❡s✳ ❆ss✐♠ ❛ ❝♦♠♣♦♥❡♥t❡ Mz✱ ✈✐♥❝✉❧❛❞❛ à ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛✱ ❡✈♦❧✉✐ ❞♦

✈❛❧♦r ✐♥✐❝✐❛❧ ❡①❝✐t❛❞♦ ♣❛r❛ ♦ ✈❛❧♦r ❞❡ ❡q✉✐❧í❜r✐♦ t❡r♠♦❞✐♥â♠✐❝♦M0✱ s❡❣✉♥❞♦ ✉♠❛ ❧❡✐ ❡①♣♦♥❡♥❝✐❛❧✳ ▲♦❣♦

dMz

dt =−

1

T1

(Mz(t)−M0), ✭✷✳✺✼✮

♥❛ q✉❛❧T1 é ❝❤❛♠❛❞♦ ❞❡ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦ ❧♦♥❣✐t✉❞✐♥❛❧ ♦✉ r❡❧❛①❛çã♦ s♣✐♥✲r❡❞❡✳ ❈♦♠❜✐♥❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✷✳✺✻✮ ❡ ✭✷✳✺✼✮ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✷✳✶✻✮ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞❡ ❇❧♦❝❤✱

dMx

dt =γ(My(t)Hz(t)−Mz(t)Hy(t))−

1

T2

❈❆P❮❚❯▲❖ ✷✳ ❉■◆➶▼■❈❆ ❇➪❙■❈❆ ❉❆ ❘❊❙❙❖◆➶◆❈■❆ ▼❆●◆➱❚■❈❆ ✶✼

dMy

dt =γ(Mz(t)Hx(t)−Mx(t)Hz(t))−

1

T2

My(t) ✭✷✳✺✾✮

❡

dMz

dt =γ(Mx(t)Hy(t)−My(t)Hx(t))−

1

T1

(Mz(t)−M0). ✭✷✳✻✵✮ ❚♦♠❛♥❞♦ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ ❝♦♠♦

~

H =H0zˆ+H1cos(νt)ˆx ✭✷✳✻✶✮

❝♦♠ H0 ≫H1✱ ❛s ❡q✉❛çõ❡s ❞❡ ❇❧♦❝❤ ✜❝❛♠

dMx

dt =γH0My(t)−

1

T2

Mx(t) ✭✷✳✻✷✮

dMy

dt =−γH0Mx(t) +γHx(t)Mz(t)−

1

T2

My(t) ✭✷✳✻✸✮

dMz

dt =−γMy(t)Hx(t)−

1

T1

(Mz(t)−M0) ✭✷✳✻✹✮

❊❢❡t✉❛♥❞♦ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♦♥❞❛ ❣✐r❛♥t❡ ✭❘❲❆✮ ❡ ♣❛ss❛♥❞♦ ♣❛r❛ ✉♠ r❡❢❡r❡♥❝✐❛❧ ❣✐r❛♥t❡ ❝♦♠ ❡✐①♦z ❡ ❢r❡qüê♥❝✐❛ ❛♥❣✉❧❛r ✐❣✉❛❧ ❛ ❢r❡qüê♥❝✐❛ ❞♦ ❝❛♠♣♦ ♦s❝✐❧❛♥t❡✱ν✱ ❛s ❡q✉❛çõ❡s ❞❡ ❇❧♦❝❤ ✜❝❛♠

dMX

dt = ∆ωMY(t)−

1

T2

MX(t), ✭✷✳✻✺✮

dMY

dt =−∆ωMX(t) + γH1

2 MZ(t)− 1

T2

MY(t) ✭✷✳✻✻✮

❡

dMZ

dt =− γH1

4 MY(t)− 1

T1

(MZ(t)−M0), ✭✷✳✻✼✮ ♥❛s q✉❛✐s∆ω =ω0−ν✳

◆❛ s✐t✉❛çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡st❛❝✐♦♥ár✐♦✱ ✐st♦ é✱ q✉❛♥❞♦ ❢❛③❡♠♦st_{→ ∞}✱ ❛s ❡q✉❛çõ❡s ✭✷✳✻✺✮✱ ✭✷✳✻✻✮ ❡ ✭✷✳✻✼✮ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s t♦♠❛♥❞♦ _M˙_{X = ˙MY = ˙MZ = 0✳ ❖❜t❡✲}

♠♦s✱ ❡♥tã♦✱

M_X(eq) = γM0T2 2

∆ωT2H1

1 + ∆ω2_T2

2 + (ω1/2)2T12T22

, ✭✷✳✻✽✮

M_Y(eq) = γM0T2 2

H1

1 + ∆ω2_T2

2 + (ω1/2)2T12T22