Habilitation à diriger des recherches

Transcription

1 Habilitation à diriger des recherches Ferruccio Renzoni To cite this version: Ferruccio Renzoni. Habilitation à diriger des recherches. Atomic Physics. Université Pierre et Marie Curie - Paris VI, <tel > HAL Id: tel Submitted on 4 Sep 2003 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 P P q r s t t s ❷❶❸7 ❹ ❺ ❻ ❼ ❽ 6 ❾ 4 ➀❿ ➁ ➁ ❾ 4 ➂ ➃ ➄ ➅ ➆ ➇ 9➉➈ 7 ➊ ➈ ➋ ➌➅ ➍➎ 5 ➏➑➐➒ ➆ ❹ 9➉➈ ➓ ➈ ➌ 5 ➏➑➐➒ ➆ ❹ 9➉➈ ➏ ➈ ➆ 9➉➈ ➈ 9➉➈ 7 ➊ ➈ ❺ 8

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4 P P qr P s t ❶ ❷ 5❸ t t t t t 10 ❶ ❹❺ ❶ 1 t3❻ ❼❹❺ ❶ 6 ❽ t P 6 6 ❾ 6 5 P 1 ➀❿ 2 ❽ 6 2 t t ❼ 1 r ❾r 6 ➁ 9 ❽➂ 1❷ 5❸ ❽ ➁ 2 ❾ ➃ 2 4 ➄ ➅6 8 ➆ 2 ➇ ➈ ➉ ➊ ➋➃ ➄ ➊ ❿ ➌ ➍ ➄ ➎➇➏ ➐ ➑5➒ ➓ ➓ ➐ ➑ ➎ ➃ ➓ r➐ ➑ ➂ P ❻ r t ➆ ❽ ➉ ➁ 2 ➀10 P10 ➈ P ❽ P P t P t ➀ P ❿ 10 ➁ ➇3 P ❼ ❼ ➃ t 6 1 ➁➐ ➒ ➁ ➓ 1 ➃ P P ➊ P ❺ ❾ ➆P ❼ 4 s 1 1 ➃ ➃ ➃ ❾ 1 ➄ ➈ ❼ P ➀❿ ➃❾ 3 t ➁

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6 ❻ 6 ❻ ➂ 8 ➅ ➅ q ❹ 5 ➂ 6 ➅ 8 ➅ q P ➍ P q ❿ q q ❻ q 8 ➀ ➀ P 1 1 q P ➋ ❻ ❻ ➄ ➌➆ ➍➎ ➂➆ ➌ ➐ 8 ➍➎ q q P ➃ q q ➀ ➃ q q ➈ ➈ q q q q ➆ t ➀ s ➀ ❾ q q ➆ ➁ q q 3 ➃ ➊ q q q q ➀ ➃ ➈ ➅ 8 4 ➅ ➊ ➍ ➐ ➇ 10 8 ➅ ➆ 4 5 ➐ ➇ ❹ 6 ➅ ➐ 10 ➇ ➒ ➐ ➎ ➐ ➓ ➒ ➃ ➁➒ ➓ ➎ ➓ ➒5➑ ➑5➐ ➑ ➒ ➑5➐ ➎ ➐ ➒ ➓ ➐ ➓ ➐ ➐ ➀➐ ➒ ➐ P ➀➑5➒ ➓ ➐ ❾ ❿

7 ➐ ➇ ➎ ➎ ➐ ➓ ➑ ➒ ➐ ➎ ➐ ➓ ➎ ➓ ➒ ➐ ➐ ➎ ➒ ➓ ➑ ➓ ➓ ➒ ➑ ❾ ➐ ➏ ➇ ➓ ➓ ➐ ➉ ➑5➒ ➎ ➁➒ ➓ ➓ ➐ ➐ ➀➐ ➓ ➐ ➒ ➓ ➎ ➁➒ ➓ ➒ ➎ ➒5➑ ➑5➐ ➄ ➃ ➑5➐ ➎ ➐ ➓ ➓ ➐ ➁ ➐ ➐ ➓ ➇ ➎ ➒ ➓ ➎ ➒ ➁➒ ➓ ➒ ➐ ➑ ➓ ➒ ➁ ➃ ➁ ➐ ➑ ➓ r➐ ➎ ➎ ➐ ➁ ➁➒ ➓ ➓ r➐ ➎ t➑ ➒ ➓ ➒ ➓ ➒ ➓ ➐ ➓ ➁➒ ➓ ➑ r➒5 ➑ ➁ ➎ ➐ ➑5➒ ➓ ➓ ➐ ➐ ➐ ➇ ➒5 ➑ ➁ ➎ ➐ ➑5➒ ➓ ➓ ➐ ➓ ➐ ➎ ➁➒ ➁ ➒ ➐ ➓ ❽➑ ➎ ➐ 5 ➁➒ ➓➅ ➓ ➑ ➑ ➒ ➁ ➃ ➁ ➐ ➌➁ ➐ ➓ ➇ P ➎ ➎ ➒ ➁➒ ➓ ➒ ➐ ➑ ➓4➒ ➁ ➃ ➁ ➐ ➑ ➐ ❻ ➇ ➐ ➑5 ➐ ➎ ➓ ➓ r➒ ➁ ➓ ➁ ➑ ➓ ➑ ➑ ➐ ➒ ➁ ➃ ➁ ➐ ➑ ➁ ➐ ➋ ➇ ➑5➐ ➒ ➓ ➎ ➒ ➒ ➎ ➐ ➐ ➑ ➁➒ ➓ ➓➄ ➑ ➐ ➎ ➁ t➒ ➁ ➃ ➁ ➐ ➁

8 ❿ ➁ 2 t ➀ ❼ P 6 6 ➇ ➇ P t ➁6 5 t 6 ➇ P ➇ ➃ ➇ ❾r 6 ➁ 9 ➃ ➃ P➂ ➇ P 6 6 P P 1 ❿ ➅ ❶ ❷ 5❸ ➀ 10 ❶1 ❹❺ ❶ 6 ➀ 6 ❽ ❾ P ❻6 2 t P 6 1 ➀ P 6 t ❼ ➀ ❾r 6 t ❽9 ➀ ➂ P ❽ ➁6 ➀ P ❽ 1❷ ➋➃ ❸ P 6 ❾ t P ➈ P 6 ➋ ➁ 8 ➃ ➁6 ➁6 ❻ ➄ P P ❾r ➁ ➂ 6 ➁ 9 t ➂ t P 1 2 t P ➁ r ➈ ❾ P➆P ❼ P 1 ➀ ➈ ➇ ➇ ❼ 1 P P 6 ➃ t ➈ ➀ P ➁ 2 ➀ 1 5➈ ➀ ➈ 6 6 P 6 ➈ ❾ ➄ ➍ P ➀ ➈ 8 ➁6 ➀ 1 ➄ ➉ ❻ ➂ ➃ 1 ➃ ➀

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10 P ➀ ❷ ❿ 8❺ ➀ ❽ ❿ 8 ❸ 1 ❻ P ❻ ➈ 6 5 ❷ ❸P ❿ 1 P 6 ➁6 6 t ❾ ➀ 6 P 6 ➀ ❾r ➀ 1 t 2 ❻ t 6 4 ➉ P 6 ➇ 1 ω L1 g1 e g2 ω L2 Fluorescence (unités arbitraires) δ 1 =0 δ 1 =Γ/ δ 2 /Γ q 101 P ➈ 6 t 5 ❺ ➀ ➃ P 6 ❼ P P 6 P ➉ P ➃ 4 ❾

11 ❻ ➈ 6 6 ❷ q ❸ ➄ 6 ❾ 6 ❾r P 6 ➄❷ ❸ ❽ P ➀ ➃ P 2 6 ➁ r P ➃ ❷ ➐ ➏ ➇➓ ➒ ➎ ❸ ❿ ➀ P 6 t ➃ 6 ❷ ➇q ➃ 5❸ 1 6 ➀ P 6 ➀ 1 P 1 ➈ 6 5 ➃ ❿ ❾ t t t❿ 8 ➀ 2 ➀ t 6 ❿ 8 ➃ P P ➀ P 1 ➀ P P 6 P ➊ t P 6 ➁ 2 ➃ ➃ ❽➂ t ➄ ➇ P 6 ➁ 1 ❾ P P P ➁6 2 P 6 ➋ t t 4 6 ➃❾ ➁ 2 P 6 ❻ 6 ❼ t t P ➀10 ❶ ❿ ❿ ➌ 6 8 ➄ 1 2 ➃ P P P ❸P6 P P ➇ ➇ P ➇ 1 P 1 ➀ 2 ➄ q ➈ 6 ➄ ❾r P ➃ t ➀ ➀ ➈ 6 2 t ➀ ➀ ❾ 6 ❾➁ ➀ P ❾ ➊ ➊ ➁6 ➃ ➅ ➅ P ➊❷ ❻ q ❸ ➍ ➃ ➉ 2 t ➃ ❽➂ 2 P ➈ 6 ➃ ➉ t P 2 76

12 6 1 ➃ 6 ❷ q ❸ qr 6 ➀ 6 ❷P ❸ ❷ q ❸ ➃ 1 P ❻ ➀ P 6 1 ➀ ➇ t❷ ❸ j ❷ q ❸ ω j,j 1 ω j+1,j j 1 j N ω 1 (a) ω 21 Ν1 q ➉❷ ❸ ❿ 2 ➍ ➉ 2 2 ❾ 1 (b) 3 6 7❷ ❼❸ P P ➁ ➂ ❷ q ❸ ➈ P ➈ 6 P P ➇❿ P 6 t t P ➋ ❺ 1 P ➀ ➈ 6 6 ❾ ❷ ➉q ❷ ❼❸ ❸t ❾r ➈ ➁ 1 ➀ ➀ 6 ❷ q ❸ ➃ ❶ P ➁ ❾r ➀ P t ➊

13 6 P ➍ ➈ ➀ 1 P 2➈ r t ➀ ❿ 8 P 6 ➃ t6 ❻ ➅ ➅ ➈ 6 P 4 2 P ➄ ➉❿ 8 2 ❻ P 2 ➊6 ❻ ➃ ❷ q ❸ ➈❻ 1 ➈ 6 6 ➉ ❺ ❾r q ➄qr ❻ ➍ ❿ ➍ ➄ 4 ➊ P P 4 4 t 6 P ➍ ➃ 4❷ ❸ ➂ ❶ 4❷ ❼❸ ➂ ❶ P P P6 1 P 5 ❾ ➃ ➄ ➈ 6 6 ➀ ➃ ➄ ➈ P t ➉ 1 P 1 P ❻6 ➀ t t 1 ❻❿ 8 ➁ ➈ 6 6 ➀ P t ❾ P 1 ➀ ➀ P 6 ➇ ➃ 1 t ❿ ➁ 2 P ➇ P ➃ ➀ ❾r ➇ 5 P❿ P 5

14 6 P ❼ 6 ❾ 9 P ➀ ❾ ➀ ➀ ❾ 5 1 P 1 ➅ ➍ ❻ 1 P ❷ q ❸ t ➀ ❷ ➁ 2 ➃ 1 6 ➃ ❼❸ 2 ➉ ➈ 6 6 ❷ 1q ❸ 6 6 ❻ ❽ P ➁ t6 ➁ r ❷ q ❸ ➇❿ ➀ 6 ➁ 2 ➃ P 1 t t ➄ 5 6 P ➁ 2 ➃ ➍❿ ➃ 6 6 ➄ ❻ ➀ ❻ P 1 6 ➄ ➉ ➈ 6 ❻ ➄ ❻ ❻ ➄❶ ➃ ➄ ❻ ➀ ➄ t P P ➈ 6 6 ➍ P ❾ ➁ P P 6 P ➇ 1 P 6 t P ➇ P ➀ ➁ ➃ P 1 P P P P P ➉6 1 ➁ 2 P ➀ P ➀ ❾ ➀ P 1 t 6 t6 ➁ 2 ➃ ➀ ➄ ➃ 2 P P P ❿ 8 ➀ ➈ ➉ ❻➂ P 1 5 ❷ q ❸ ❷ q ❸

15 ➉ ❻ ❻ ➁6 P P ❾r ➃ ❾ ➅➂ ➉ ❽ P ❿ 8 ➅ ➃ 2 6 t P ❷ ➁6 P ❸ ➇q Fe=2 Fe=1 Fe=2 Fe=1 F g=2 F g =1 F g=2 Fe=1 F e =2 Fe=1 F e =2 Fe=1 F g=2 F g=1 F g=2 F g=1 q t P 2 ➄ 1 ➈ ➄ 7 ❾r ➃ ➃ ❾r P ➃ ❾r ➀ P P P P 5➈ 6 ➈ ➀ ➃ ➃ ➀ r➈ ❻ q ❿ ➄ P P ➉ ➇ t ➀ ➀ 1 ❾ ➄ ❾ ➃ 1 ❺ ❾r ❿ 1 ➇ 1 ❾ P 5 6 ➁ 2 ➃ t t P ➁ r ➀ P ➍ t 1 ❻ ➉ 1 ➀ P

16 Iint (arb. units) Fg=2->Fe= B(G) Iint (arb. units) Fg=1->Fe= B(G) Iint (arb. units) Fg=2->Fe= B(G) Iint (arb. units) Fg=1->Fe= B(G) ➅q qr ➃ ❺ P ➉ ➃ ➈ 1 t t P t P 5 ➃ ➈ 6 P P ➀ 4 t 5 ➉ ❻ ➆ 4 t 1 ➃ ➀ ➄ ➀ ❾ ➃ ➁ 6 ❻ ➃ P ❻ ➃ 6 qr ➃ t t t ➄

17 ➁ 2 P ➀ ➁ 2 ➃ 1 P P ❷ ❸ ❷ 6 ➇q ❸ ❷ q ❸ F g =1 > F e =1 F g =2 > F e =1 F g =2 > F e = I int (a. u.) 0.5 I int (a. u.) 0.5 I int (a. u.) B (G) B (G) B (G) 1q ❷ ❸ t P 5 ❿ ➀ ➀ 6 t P ➃ 4 1 ➃ 1 ❿ 4 ❻ 1 6 P P ❻ t ➄ ➀ ➁ r ➁ ➃ ➀ ➃ ➁6 P P 4 P ➇qr ❼ ❼ P 1 5 P ➉ ❻ 4 2 ➀ ➐ ➎ ❼➐ 5 P 2 ➅ 6 ➀ ➁ t P t ➀ ➄ P ➍ ❻➆ ❺ ➃

18 6 ❸ ➉ ❻ ❾ ➀ t ➇ P t ❷ t ❸ ➃ ❷ ❸ 2 P ➃ t 6 ❾r ➇q B (mg) t f (µs) B (mg) Γ t (s 1 ) q ➇ P ➇ 2 10 ❾ 1 ➃ ➀ t ➃ 4❷ 5❸ ➅6 ➃ ➀ ➃ ➃ ➃ 1 6 ➃ ➃ t ❾➁ 6 ➀ P ➀ ➃ t ➃ 6 ❾ 1 6 t 1 6 ➀ 2 6 ➃ P ➄ P t ➁ P 6 ➄6 t ➃ 49 ➁6 ➃ ➇ P ➋ ❶ P ❿ P P 1 ➁ 2 ➃ P ❷ q ❷ ❸ ❷ ❸ ❷ q ❸ ❿ P ❷ ❸ 6 ❷ ❸ ➀ q 6 5 ❷ ❸ ❻ ➇ 6 ❿ ➃ ➄ 7 7

19 6 ➄ ➅ 2 ❾ 6 ❾ ➅ ➀ 1 P 6 P ➀ P ➃ ➀ ❷ ❸ 6 ❷ ❸ ➄ P t 2 ➀ t t ❷ ❸ ➃ ❿ t 1.0 I(B,t) (unités arbitraires) magnetic field (G) q ➇qr ➃ ❷ ❸ ➃ ❷ 5❸ ➅6 ➃ 6 ➃ 5 P P P ❾r P P 4 P ➀ t❿ 10 1 ❶ 6 1 ❼ P ➀❿ P➌ ❾ ➊6 8 t ➉ 4 ❻ 1 t P 3 ➁6 ➁ 2 P ➄ P ❾ 6 1 t ➀ 9 ➀ P 1 1 ➁6 1 ❾ ➄ ❼ ❿ ➄➌ ❺ 5 ➁ 2 ❾

20 t ➁ r ➁ ➀ ❾ ➄➌ 6 ➁6 ➁ ➀ 2 ➀ ➁ P ➃ ❿ ❿ ❾r8 ➁ P P 6 ➇ ❾ P 9 P 1 r➈ ➂ 4 P P P P ➁6 6 t t ➁6 t ➈ P t ➀ t 1 ❷ ❸ ➀ ❾ ➀ ➇ P 6 t ❷ ➇q ❸ F e = F g =1 M F = q ➃ 5 ➁6 P ➊ ➃ ➍ 4 ➄ 4 ➃ 4 ➁6 ➉ 2 ➁6 ➁ 2 ❻ 5 ➁6 ❻6 1 ❷ ❼❸ ➁6 6 1 ➁6 1 ➉ ➃ ❷ ❸ 1 P P 6 ➉ ❿ ❽ ❽ ➃ ➁6 ➉6 ➄ ➁6 ➁6 t t t ➀ ❾r ➃ ❻➌ ❾ 6 P P 1 P ➉ ➁ r ➁ 4 4 P ➁ ❾ P ❾r ➃ ➀ ❽ t ➇ P ➇ ❾ ➇ 6 ❷5 ❸ ➃ 2 P 6 4 ❽ ➃ ❾ ➌ ➀ P ❾ ➌ ❽➂ ➀ ➀ ❾r ➃ 6 ➁ r ➃ P P

21 ❿ 6 t 6 t ❿ ❻ P ❽ ➄ 4 ➁6 2 4 ➅ ➅ 2 ➅ ➅ 2➈ ➄ ➍ P ➁6 4 2 ➁6 t ❽ t ❾ ➁ ➇ ➁6 6 ➄ ❽ ➃ ❾ ➍ ❻ ❾ 1 ➁ ➁ 6 ➅ ➅ P ➁6 6 ➀ P P ➌ ➀ P 6 P P 6 t P 2 P ➌ ➁ 6 P t 1 ➃ 1 1 ❺❷ q ❸ 5 1 P ➁ mw/cm x Π e st B (G) q 4 P ➃ ❽ t 6 ➁6 ➀ P ➀ ➃ 6 ➀ 1 ➁ 2 ➀ 1 ➂ ➇ ➃ 6 1 ➃ ➄ t ❷ ❸ ❿ 2 1 ➄❷ ❸ t 1 P ➀6 P ➁6 ➁ ➃ q

22 6 ➁ ➇ 6 ➁ ➃ mw/cm x Π e st B (G) q ➃ P 1 ➁ ➂ P ❷ ❸ P P PP P P P P ➇ t ➇ ❾r 6 ❽9 ➂ t ❽ 16 2 t ➄ t ❿ P 8 q ➀ P ➁ 1 ➇ 2 ➇ ➈ 6 t6 P 2 ➉ P 6 ➁ ❿ 1 P ➁ 2 ❾r ❷ r➒ ➐ ➑ ❸ ➁ r 6 ➁6 ❺❷ ➒ ➐ ❸t ➁ r ➀6 ❷ q ❸ ➃ ➃ 5 P ➈ 6 P 6 ➉ P ➇ t 5 ➋ ❽ ❻ 6 ➀ ➈ 6 P6 ➁ ➃ ❷ t 1 P ❽ ➄ ❻ P 5❸ 1 6 ➀

23 6 e Ω P Ω S Ω S Ω P, Ω S Ω P f i q 10 ➈ 6 ➃ ➀ 1 t6 time ➀ ➃ t q ➀ ➈ 6 t ➃ ➅ 5 ➅ ➊ 8 q ➀ P P t ❾ P P P 8 q ➀ ❾ ➀ P ➀ P 8 q 5 ➃ 1 t 8 q ❾ P 2 ➈ 6 6 ➁ e Ω S Ω S+ Ω P Ω P+ f f+ i i+ q ➈ 6 6 t ➃ ➀ P 1 P 1 ➈ 6

24 6 6 6 ➃ ❷ q ❸ ➄ ❾ ❻ 6 ❻ ➅ P ➃ ➀ ➀ ❾ ➍ 1❷ ❾r ➃ ❸ 4 8 q ➀ ➁ ➊ ❾ ➁ P 1 ➃ 1 ➇ 1 ➃ ➀ ➃ 1 6 ➁ ➉ 6 P P P ❿ 4 4 ➅ P 7 8 q ❺ ❾ 1 P ➀ ➀ P ➄ 7 ➄ 4 ➄ P 8 q 6 t ➀ 2 t 8 q ➀ ➅ P 1 P ➁ 4 Ω 1 Ω 2 Ω 3 Ω Ω Ω Ω 1, Ω 2, Ω t tq 10 ➀ ➈ 6 6 ➄ ➀ ➄ ➀ 8 q ➀ ➀ ➃ ➇ 2 P ➀ ➇ ➃ ➇ 8 q ➉ ➇ ➈ 6 ➍❷ q ❸ ❾r ➃ ❻ ➁ ➇ t ➃ ➃ ➀ ➀ ❷ 1 ❸ ➀ 4 t ❾ ❺❷ ❸➀ P ➁6 1 ❾ ➅❷ ❸ ➀ ➀ ➀ ❷ ❸ ❷ ❸ ❷ ❸ ❷ ❸ ❷ ❸ ❼ 5 P P

25 ➅ ➃ t t ➁ ❷ ➇ ➇ ❸ 2 6 ❷ ❸ ❷ q ❸ ❷ ❸ ➁ 1 9 ❷ ❸ ❷ q ❸ 6 ❷ ❸ P ➉ ➁ ❷ ❸P ❽ ❽ P P P 8 q ➁6 8 q t 1 ➅ t 8 q ➀ 1 ➀ P ❾ ➃ ➃ 5 6 P ➃ ➀ 1 ❾ P ❾ ➂ ➅ ❾ ❺ ❾r ❺ ❷5 ❸ ➅ ❷P ❸ ❷ q ❸ ➅ ❷ ❸ ❷ q ❸ ➄ ❺ ➃ ➄ ➉ ➁6 ❻ 8 q ➀ 6 ➁ ❷ q ❸ ➀ ➁ 2 t ➁6 8 q ➀ P ➁ r 1 ❼ 1 ➁ ❻ ➁6 8 q ➀ ❻ P ➁ t ➃ t ➀❿ ➄ ➃❾ ➁ 2 6 ➁ 2 ➇ ➁6 P ➄ 8 q ➃ 6 ➁6 t P ❻ 8 q ➀ ➃ ➀ ❿ P ❻ 8 q 6 ❷ ❸ ❷ q ❸

26 ➀ 1❷ q q ❸ ➅ ❷ ❸ ❷ q ❸ ❷ t 1 ❷ ❸ ❷ ❸ ❷ q ❸ ❸ t 5 ❿ ➁ 2 P P P P6 P6 P 6 ➃ ❾ ❿ P ➉6 ➄ 2 ➈ ➀❷ 8 q ➀ q ❸ ➇ ❽ t ➇ P 2 1 t ➀ P ➌ ➀ t 6 ❷ ➃➈ ❾ ➈ ❾ qr8 ❸➀ 1 1 ❾ 6 ➀ ➁ ❷ 7 ❸ ➃ ➈ ➃ ➃ P 2 ❾ P P 6 t 7 ➊ P ❾ 6 ➁ 2 ➃ 6 ➁ r ❾ 1 ❻ P ➉6 ❾ ➀ P ➋➃ ➀ 6 ❾r t ➄ ➈ 6 ➈ ➉ ➄ ➋➃ t t ❼ t P 6 ➃ ➈ 6 ❾ ➋ ➍ ❻ ➈ 6 ➃ ➁ ❾ ➀6 ➁ 9 ➂ ❽ P ➃ ➁6 ➀ ➁ P 6 ➋ 5 ➀8 ➆P ➀ ➈ 8 P ➀ 1 9 ❾ P 1

27 2 P ➀ ➆P ❼ ➁ 9 3 ❿ ➈ 6 ➈ 5 P ➋➃ P 6 ➁ ➉6 3 ➊ 1 ➆ P ➇q Γ ε 0 ε 0 ε G ε G T c Γ µ ε 2 ε 0 µ Current [pa] Γ= 0.1 Γ 0 Γ= 0.5 Γ 0 Γ= 1.0 Γ 0 Γ= 5.0 Γ 0 Γ= 10 Γ 0 Γ 0 = 10 9 s -1 ε = 10 µev Ω R = 1.0Γ 0 T c = 1µeV ε 1 ω 1 ω Raman Detuning δ R [µev] q 101 ➀ ➀6 ❻ ➄ P ➋ ➁ ❿ P ➋ 1 P t ❿ ➀ 1 t ➀ 6 ➋➃ ➀ ❿ 1 P 2 ➀6 ➀ ➉ 6 5 ➈ 6 1 ➋ 1 ➉ P 6 ➁ ➄ 5 ❾ ➁ 5 ❺ 6 t ➋➃ t ➃ 1 P 6 t P ➃ ➁6 1 1 ➁ 2 1 ➄ ➉ P r ❷ ➋ ➇ ❸ ➇ 5 ❾ 6 t 5 1 ➃❾ P ➁ 2 P P 6 t6 t3 P ➇ ❼ P ➋ 2 P ➋➃ 1 ➋➃ ❿ t ➂ t ➅ ➈ 6 1 P 6 6 ❺

28 6 ➊6 ➅ t ➃ ➈ ❷ q 5❸ P ➃ ❷ ❸ P ❷5 ➀ ➋➃ ❾ t ❸ ❷ ❸ ➃ t ➁ r t 1 ➁ P ➀ ➈ 1 r 6 ➃❾r ❷ 5❸ ❷ ❸ ➀ 7 ❷ ❸ 1 t ➁ 1 6 ➁ r 1 ➃ ❻❷ ❸ t ❷ ➃ t 5❸ P 1 P ❷ ➃❸➇ P ➃ ➁ r ❾r ➃ 6 4 ➇ P ❼6 t 1 1 ➇ ❾ 6 ➃❾ P 6 3 ❻ P 7❷ q ➃ 5❸ 4 6 ➃ ➃❾ ➍ ➁ ➉6

29

30 ❿ ➁6 t 1 t ❾ t P ➉ t 1 ➆P t8 ➃ ❾ ❺ ➊ ❽ ❾r P ➈ 1 P ➀ 6 t ➈ ➀ ➀ P 1 ➁6 ➀ 5 ➁ 1 ➄ 1 ➀ ➀ 1 ➀ ➀ 1 ❾ t ❺6 ➉ ➃ ➄ 1 t s P ➃ 2 5 P P ➈ 6 4 P ➇ ➃ P P P ➁6 1 ➈ qr 7 ➈ 4 t ➁ t s ➀ ➃ P ➀ t ➈ ❽ P 1 ➀ ➈ t 6 ➀ s ➃ t 6 P t P 5 ➅ ➅ ➅ 6 1 P 6 1 ➃ P 1 ➃ t ❻ ➎ ➐ r➐ ➁ ❼

31 ➁6 ➁ 2 ➇ 1 1 ➃ t ➃ ➀ P ➀ 2 ➃ ➀ 1 P ➄ ➀ ➄ ❽ ➄ ❿ ➄ ➀ ➈ 4 ➁6 4 4 P ➈ 6 ❻ 6 ➃ ➈ 6 ❾r P P 6 ➀ ➀ P P ➁ ➄❿ 4 ➁6 ➉ t 4 ➈ 6 P ➃ ➇ t ➉ ❾ ➉ ➈ ➃ 1 ➈ ➈ 6 P P ➈ ➄ P ➃ 4 t t t ❻ 1 ➈ ❿ P ➃ 6 5 ➀ 1 ➇ P ❿ 1 P P ➃ ❷ q q ❷ ❸ ❸ ❷ ❷ ❸ ❸ ❷ ❷ ❸ ❷ q q ❸ ❸ ❷ q q ❸ 6 ➀ ➀ 1 t ➀ ❷ ❸ ❷ ❸ ❷ ❸ ❷ ❸ 5 t❷ ❸ 6 4 ➃ 6 5 ➇ P ➁ 2 ➇ P P ❷ ❸ ❷ ❸ ❷ q q ❸ ❷ q q ❸ ❷ q q ❸ 1 ❺ 1 t ➃ 2 P ❿ ❾ 2 ➃ 7 P 4 P 7

32 ❾ ➊➌ ➍ ➁ r ❷ q q ❷ ❼❸ ❸ P ➀ ❷ q q ❸ 6 t P ➀ t ❿ ❾ ➀ ➁ 2 ➇ ❷ q q q q ❸ P 6 ➃ 1 ❷ ➇q q ❸ ❷ q q ❸ ❷ q q ❸ ❷ q q ❸ (a) E 1 σ σ + σ lin lin y g,+1/2 x 0 λ/8 λ/4 3λ/8 λ/2 z E 2 E 0 (b) U 0 g,-1/2 Pq q ❷ ❸ ❿ t 2 ❷ ❼❸ 4 ❾ ➍➌ t t ➈ t ➁ ➉ P ❾ t t ➀ ➀ tq q 1❿ P

33 P ➃ ➃ ➇ 1 ❻ ➁ ❻ P ❽ ➃ t ➀ t ❿ 2 ❼ t P 1 1 ➁ 1 ➁6 ➇ ➁ ➀ ➀ ➃ t ❾ ❷ ❸ ➁6 ➀ ➁ 6 ➃ P ➁ ❾ ➀ ❽ ❾ ❽ ➀ 2 ➀ ➀ ➉ P ❿ 6 6 ❽ ❽ ➃ s ➀ 2 ➁ 2 ➅ ➃ ➅ ➅ ➅ ➄ ➀ 1 ➉ ❻ ➁ qr t 1 ➉ ➃ 6 ➉ 1 ➁6 5 ➇ ➈ ❽ ❾ 1 ➃ ❿ 2 6 t ➀ ➀ r➈ ➎➇➏ ➐ ➑5➐ ➎ ➃ ➓ E U + m g=+1/2 U σ+ σ σ+ σ m g = 1/2 ➀q q ➈ P 6 ➀ ➇ ❾ 1 P 1 P 6 ➈ 6 ➁ P ➀ ➇ t

34 6 Pq q ➇❿ P 1 ➃ t ❾ ➁ ➇ ➃ ➇ ❼ 1 1 ❻ ➍ ❻ ➇ q q t P 1 ❷ q q ❸ ❷ q q ❸ ❷ q q ❸ P ➀ 1 t P ➀ t ➀ t ➀ ➉ ➇ ❷ ➇ ❸ 2θ x O 2θ y y z x q q ➉10 ➍ ➊ ❿ ❷ 5❸t ❷ ➃ ❸ ➀ P ❷ P P P ➈ ❻ ❾ 4 ➁ ➀ ➀ ➃ ➀ P 6 ➃ ➅ ➄q q ❾r ➃

35 6 t 2 ➃ 6 ➃ 1 ➇ ➉ ➉9 ➁6 ➅ ➃ ➈ t ❼ ➁ P ➃ ➀ ➀ ➈ 6 ❽ ➃ ➃ t ➃ 1 ➀ ❾r ➃ ➁ g;+1/2> e> < x i 2 > (mm 2 ) < ξ 2 > < z 2 > g;-1/2> t (ms) q q ➇101 ➃ ➇ ➃ ❾r ➃ 7 ➉ ➁ ➀ ➃ ➅ P ➅ P 5➈ ➅ ➍ 6 ❻ ➁ 2 1❿ ❿ ➉ ➃ 5 P 6 ➀ ➀ 6 ❽ ➀ ➀ P 1 ❺ ➇ P P P 5➈ t ❷ q q ➇ ❸1 ➁ 2 76 ➉ t P P 5➈ P ➀ ❷ ❸ ❷ q q ❸ P P s ➀ P P9 ➃ ➇q q ❿ ➀ ➀ s 1 ❾ ➃ ➈ ➈

36 6 2πMD ξ /h, 2πMD z /h 2πMD ξ /h, 2πMD z /h 2πMD ξ /h, 2πMD z /h = 24 MHz D ξ D z = 34 MHz = 44 MHz I L (mw/cm 2 ) = 30 MHz = 40 MHz = 50 MHz I L (mw/cm 2 ) q q ❿ s ➉ ➄ P t ➁ PP P P πMD ξ /h, 2πMD z /h 2πMD ξ /h, 2πMD z /h 2πMD ξ /h, 2πMD z /h 6 ➁6 t 1 ❾r 4 ➀ ❾ P P 6 ➄ ➈ P t ➁6 ➁ 2 ➃ ➄ ➈ ❷ ➃❾ 5❸ ➁ t ➈ ➀ ❿ P ❺ P ➄ ❻ 5➈ ➊ ➅ ➁ 2 ➃ ➃ ❻ 5

37 6 ➉ ➁ 2 4 ➁6 ❷ ➀ ❸ t 6 P ❷ r➈ P ❸ ➀ t t 1 ➀ ➁6 P 6 1 ➀ 2 ➁6 ➀ t 1 t ➃ t t ➁6 ➀ ➀ ➇qr ➀ P 2 ➃ ❻ 1 t 2 ❾ ➃ P P ➁ ➍ ❷ ❸P 1 t ❷ ❸ ❷ q q ❸ ➁6 P ➈ P P t ❻ t ➀ ➁ ➀ 9 ➀ r➈ q q t ➄ ➀ t ➈ x (T T 0 )/T [ 5:5] [ 20:20] (khz) δ/(2π) (MHz) q q 8 P ➇ ➈ ➁ 2 ➉6 ➈ P t ➈ ➅ t ➈ 1 t ➀ ➀ t P

38 s P ➃ ➀ ➀ t ➈ 4 ➅ ➅ ➃ P t 1 ➐ ➓ ➑ ➏ ➐ ➊ ➄ ➉ ➄ ➃ ❷ P t ❸ P ❻ 5➈ 4 s t ➃ t 1 ➇ t ➀ ➈ ➀ P 1 ❷ ❸ ❷ ❸ ❷ q q ❸ t t 1 ➈ 4 t ➉ ➉ ➇ ➀ t ❻ t ➈ ➀ ➀ 6 ➀ ➄ t 1 ❻q q ➅❷ 5❸ t ❷ t 4q q ❸ t ➃ ❷ q q ❸➀ 1 4 q q ❻❷ 5❸ ➉ ➃ 6 ❻ 1 t 5➈ 2 t t ➃ P ➃ P P P γ R (khz) = 24 MHz = 30 MHz = 34 MHz = 40 MHz = 44 MHz = 50 MHz γ D (MHz) I L (mw/cm 2 ) I L (mw/cm 2 ) q q 10 ➀ ➀ P ➃❾ t ➈ ➀ ➇8 ➀ P s ➁ ❼ q q ➈ 1 t ➁ 2 t 6 ➀ ➇ ➀ ➈ 3 ❿ ➀ P 4 ➈ 1 P ❷ ❸ ❷ ❸ ❷ q q ❸

39 6 P ❷ ❸ ➀ ➈ 2 ➀ t P P P P6 P ❺ ❽ 6 ➍ ❾ ➀ 1 ➁ r ➀ ❺6 ➄ ❷➁ ➄q q ❸ 1 ➄ ➃ ➄ ❾ 46 ➀ ❷ 1q q ❸ 3➉ t t 2 1 ➈ ➆ ➀ ➁6 ❼ 6 t6 6 ➃ P ➄ ❺➆P ➍ ❺ ➄ ➃ ➁6 t 3 ❿ ➀ 1 ➀ ❽ ➀ ➀ ❿ 1 ❾r ➃ ➀ ➃ ❷ q q ❸ t 1➆ 1 1 ➁ 2 6 ❽ ➃ ➁ ➇ 2 ❾r 6 t 1 P❿ 6 ➃ 5➈ ❷ q q ❸ 6 5 ➀ ➇ ➄ ❺ ➊ ➄ ➃ P ➀ 2 ➈ ➇ P ➎ ➐ ➐ ➁ 2 P ➀ ➃ t ➈ 1 1 ➈ P ➉ P ➀ ➃ s ➃ ➄ ❻ 1 ➁ P ➃

40 exc energy g+ g λ x x 1q q 8 1 ➆ 4 ❷ ❸ 1 t 1 P 2 ➆ ➀ ➀ 1q q ❷ 5❸ ❾ P 4 ➁ 4 ❽ ❿ ➉ ➁6 ❺❿ 2 ➃ ➅ ➍ ❾ ➃ ❷ ❸ 6 2 ❷ ❼❸ qr ➀ P P ➃ ➀ ➃ 6 ➃ ➄ ➄ ❾ 2 ➃ ➆P ❷ q q ❸ 6 5 ➃ P 5 ➃ ❾ ➉6 1 P P P s ➍ s

41 θ 2θ 4 D x, D z πMD x /h 2πMD z /h sonde 1 y 500 O z x δ/ω x q q 10 ❿ 4 ➉ t 5 ❺ ❺ ❷ 5❸ ➃ ➄ 7 ➀ P ➊ s ➄ ➃ ➍ P 1 ➃ 1 t P ➇ ❿ ➁ 2 ➆ P P P P P rq s r r t r r r q t P 65 s 8 9 P 10 ❶ ❷ 65 ❹ 7 P ❶ ❸ ❹ q❺ P ❻q ❼

42 3 4 3 M 1 2 2ϕ 2θ 2θ 4 v c,x, v c,z (mm/s) v c,x v c,z M 2 1 y O z x δ (khz) P ❼ r P ❸ P ❸ ❸ ❸ 2 2 r 3 r r r 6 6 P P r r ❸5 r r P ❼ s 3 ❷ r ❼2 3 ❹ t r P q ❼ t P r P r r P P r P P t P P 4 8 P 18 P 4 P 2 ❹ P 6 6 r

43 q 8 ❸ ❸ 6 r t P 3 r r P P 6 2 P P s r P t 65 ❺ r q s P r r ❷ q2 2 t 3 M 1 2 2ϕ 2θ 2θ 4 M 2 1 y z O x P P ❸ 65 r r 3 r P r ❻ 3 4 t 3 s 6 P q P 3 P 2 t P q 3

44 v cm,x (mm/s) (a) v φ / v δ/ω x , expt., expt., MC, MC 1 (b) sinϕ 4 P q P P ❷ ❷ 8 8 ❸ ❸ ❸ ❸ ❸ 4 8 ❸ ❸ P t 3 3 r s P ❺ t s ❺ P q P q q t 6 t r 6 r q❺ P ❻q P ❻q q ❷ P q t 3 t 2 ❸ ❸ r r r r r r t r r r P t r P r 3 r P q P q q

45 t 2 q P ❻q P ❻q P ❻q P ❻q ❺ 6 ❺ P R E E A S R S P s s ❶ 2 P ❻q 1 P ❻q

46 t 9 ❷ P 2 ❼ ❸ 6 3 r s t P ❸ ❼ r 3 ❸ ❸ t t q s t P P P P t 2 P r r ❺ ❺ P P q r t P 6 ❷ 2 t s P 3 2 P t s ❺ 2 P r q s ❸ P 3 ❸ 2 P P P P ❼ P 6 P t P 3 r 6 6 ❺ s 4 P ❼ P ❹ ❹ P ❻q 1 r t t P t P ❺ s P ❺ P 2 P t P

47 ❺ t 2 P P t ξ (mm/s) Γ 0 /ω r P P ❷ 6 3 ❷ 8 P t P P 2 s ❺ 2 ❷ r 6 P 2 ❺ ❺ ❺ ❷ ❺ r 2 t ❺ r r 2 r

48 r ❷ 2 P ❼ q P ❸ t 8 6 ❷ P 3 r tq 3 6 ❸ t 3 ❸ 6 3 t 3 2 ❺ ε y out ω + α(t) φm in Ω RF out clock in 2ω, φ out atoms z x clock in Ω RF out y v (mm/s) z (mm) 0.6 φ=π/2 0.0 φ=3π/ t (ms) AOM 1 AOM 2 L L 1 2 ε x 8 0 π/2 π 3π/2 2π φ P r P P P ❷ 2 3 s r P r P 2 P 8 ❺ ❷ 9 s ❷ P P P q r 4 3 q 6 2 P

49 t r 2 P P 2 q q q P P q P r 6 ❶ q r 2 3 ❺ P r ❼ ❼ 2 ❼ ❼ 2 q 3 t r t 6 6 s P 6 P ❸ P t ❷ ❷ r t 3 ❷ 3 r q ❸ ❺ ❸ 1 ❷ 3 ❷ r 9 q ❷ r t r P t

50 t P P r 2 2 t 9 t t P r 2 t P r 2 6 ❷ P P P P r 2 P 2 r r r r P P 6 6 ❷ ❷ t ❼ r 2 t 3 3 P r P P ❺ P 3 P P r 6 ❼ P 3 r t ❸ 6 ❸ s 2 r ❷ t 6 2 ❶ 3 4 t 6 ❸ r 3 3 ❸

51 ❼ q P 2 t 3 P r P ❸ 6 t 3 3 ❷ ❷ ❸ 3 P ❸ 6 P 4 4 t t ❷ ❷ P ❸ P P P ❺ s P r q r P P r r r P ❺ P r 8 r 2 t

52 P q r s r t q P❶ ❷ ❹❸ ❺ 6 ❻ ❽❼ 1 ❾ ➀❿ ➁ ➂ 4 5 ➃ ➄➅ ❸ ❸ ❸ P ➇➆ ➅ ❺ ➉➈➊ ➋➊ ➌➍ ➎ ➏ 5 5 ➐ ➒➑ ➏ 4 ➉➓ ➁❿ ➏ ❾ ➐ ➈ 8 ➃ P❶ ❷ 9 ➋ ➑ ➐ 4 ❽ ❹ ➐ ➂ ➈ ➅ ➐ ➄ ❷ ❸ P❶ ❼ ➏ 4 ➉ ❺ ➏ ➌ ➏ ❷ ❽ ➐ ➈➊ ❺ 4 ➐ ❸ P ➋ ➐ ❻ ➅ ➅ ➋ ❽ ➈➊ ➅ ❺ ➂ 7 ➄➅ ➆ ➆ P ➉ ➐ ➈❻ ➏ ❺ ➋ 4 ➎ ➏ ➐ t ➐ ➈ 7 ❸➏➆ ➆ ➆ P ➇➌ ➋➊ ➐ ➅ ➂ ❼ ➍4 ➇ ➈ ➐ ➎ ➊❿ ➐ ➅ ➆ ❸ P ➈ ➅ ➐ ➐ ❺ ➍ ❿ ❻ ➐ ➈ ❸ P❶ ➈ ❺ 6 ➁❿ ➐ ➅➈❻ ➅ ➆ P

53 ➑ 9 ➉ ➈ ➂ 10 ❺ 6 6 ➉ ➐ ➅➈❻ ➅ 8 ➆ 9 P ❸ ➏ ❼ ❼ ➍4 ❹➌ ➈ ➐ ➎ ➐ ❶ ➎ ➐ 7 8 ❸ P❶ ❸ ➆ ➌ ➅ 6 ➑ 4 ➏ ➎ ❽ ❽➌ ➍4 ❿ r ❾ ➐ ❺ ➑ 8 ❸ P ➉9 ❺ q ❷ ➊ ➐ ➈ ➂ P❶ ➐ ➏ ❼❽ ➅ 5 ➏ ❻➑ ➎ 5 ➐ ➏ ➅ q❻ ➊ 5 ➃ ➄ ❷ ❸ P ➅ P ➈❻ ➌ ➎ ➎ ❺ P ➏ ➉ 6 ➏ 5 5 ➐ ➑ 4 ➁➓ ➁❿ 10 ➏ 8 ❸ P ➈❻ ➅ 5 ➏ ➏ ➏ ❼❽ ❺ ➐ ➈ ➆ P ➉ 10 ❾ ➐ ➈ ➂ ➈ ➆ P❶ ➉ ❹➌ ➌ ➉ ➋❽ 4 ➇ ➐ ❻ q ➐ ➐ 7 ➆ 9 ❸ 910P❶ ❸ ➈➊ ❼ ❹ ➐ ➏ ➋ 6 ❾ ➐ ➈ 8 7 ➆ ❸ ➆ 9 P ➆ ➈ ❾ ➏ ➅ ➐ 5 ➃ P ➉ 9 ➉➋ ➁➈ ➅q 6 q ➏ ➐ ➈ 8 ❸ ❸ 10P ➅ ❺ ➐ ➑ ❷ ❸ P❶ ❷

54 9 ❸ ➇➌ ➏4 ❺ ➍ 4 5 ➐ 4 7 ➅➄➅ 9 ➆ P❶ ❷ ❾ ➂ ➈ ➐ ➅➈❻ ➅ ➄ ➆ ❸ P❶ 9 ❼ 5 ➂ ➈ ❷ 6 ➅ ❺ 4 ➐ ➅ ➂ ➆ ➆ 910P ❸ ➅ ❻ ➐ ➑ ❺ ➅ ❼ ❷ 5 ➏ ➐ ❷ ➉ ➐ ➍ ➄ ➄➅ ➆ P ❸ ➏ ❺ ➏ ❷❼ ➏ 6 ❷ ➎ ➁ ➅ ➋➊ ➂ ➏ ❼ ❾ ➆ ➊P❶ ❷ ❸ ➈ ➏4 ➋ 6➋ ➋ t ➑ ➐ P❶ ➆ ❸ ❸ ➊➑ ❽ ➈ ➂ ❿ ➐ ➈❻ ➅ ➆ ➆ P ❸ ➆ ➊➑ ➈ ➈➊ ➎ ➐ ➈ 8 ➂ ❸ 9 ❸ 10P ➉ ❸ 9 ➍➈ ➑ ➐ ➈❻ 8 ➂ ➆ 9 ❸ P ➉ ❸ ➈➊ ➎ ❹ ➏ ➈ ➏4 ➋ ❺ 10 ❿ q ➐ ➈❻ ➅ ➃ P ❸ ➈➊ 5 6 ➂ ➐ ❷ ➐ ➈ ➍ ❼ ❼➊ ❶ t ➏ ❺ 4 ➐ ➃ ➄ ➆ P ➉ ❸ ➈➊ 5 6 ➐ ➏ ➐ ❷ ➈ ➍ ❼ ❼➊ ❶ q 10 ➐ ➈ ➆ ❸ P ❸ ❸ 5 4 ➎ 4 ➋ ➎ ➎ ➏ ❶ ➐ ➈❻ ➅ ➄ ➄➅ 10P❶ ➆ ➅ ➍❼ 4 5 ➍ ➈ ➂ 4 5 ➑ ➐ ➏ 5 4 ❼ ❼ ➅ q ➀ ➐ ➅➈ ➄ ➆ 10P❶ ❷

55 9 ➆ ➆ ➐ ➏ ➈➊ 5 6 ➂ ➐ ❷ ➈ ➍ ❼ ❼➊ ❶ ➐ ➈ ❸ ❸ ❸ P ➆ ➐ ❷ ➈➊ 5 ➐ ➈ ➍ ❼ ❼➊ ❶ ❽ ❺ 4 ➐ ➅ ➂ ➆ ❸ P ➆ ❸ ➐ ❷ ➐ ➈➊ 5 ➈ ➍ ❼ ❼➊ ❶ ➐ ➅➈❻ ❸ 6P ➏ ➆ ➆ ➈➊ 5 ➏ ➐ ❷ ➑ ➍ ➈ ❾ ❼ ❼ ➅ ❾ ➏ ❺ 4 ➐ ➅ 4 ❷ ➆ 9 ➐ ➐ ➏ ➈ ❼ ❼ ➅ ➐ ➈ ➆ 6 ➏ 5 5 ➐ ➑ 4 ➁➓ ➁❿ 10 ➏ 8 ❸ P ➆ 5 4 ➅ ❼➊4 ➅ ➈➊ ❷ ➐ ➂ ➌ ➎ ➎ ❼ ❼ ➐ ➈❻ ➅ ➅➄ ➄➅ ➆ P❶ ❷ ➆ 5 4 ❼❽ ❼➊ ❶ q ➅ ➐ 7 8 P ➆ ➎ 6 3 ❹➌ ❺ 5 ➎ 4 7 ➃ ❸ ➆ ➆ P❶ 9 ➅❼ ➏ 6 6 ➐ ➄ ➂ ❸ 4 P❶ ➏ ➅ ➏ ➐ q ➈ 9 ➈➊ ➅ 4 ➌ 4 t ➐ ➃ ➂ ➆ 9 ❸ P❶ ➉ 9 5 ➑ 4 ❻7 ➂ 6P❶ 9 ❸ ➏ ➋❽ ➈ ➈❻ ➐ ➅➈❻ ➅ 8 P❶ 9 ➆ ➋ 3 4 ➎➏ ❺ 4 1 ❷ ❸ ❸ 10P❶ ❸ 9 9 ❾ ➎ 6 ❼❽ ➍ 4 ➐ ➎ ➈➊ ❾ ➏ ❾ ➇➌ ❺ 5 ➎ ❼ ➎ 1 ➐ ➄ ➆ P❶ ❸

56 9 9 9 ❼ ❻4 6 ➊ ➐ ➅ ➂ ➃ ➄ ➂ ➆ 10P ❸ 9 ❾ ➈ ➐ ➄ 9➏ 6P❶ ❸ 9 ➁ ➐ ❽❿ ➍ ➏ ➐ 7 ➃ P 9 ❹ ➋ 4 ➐ ➅➈❻ ➅ ❺ ➄➅ ➈ ❸ P 5 ➐ ❷ ➏ ➐ ➈ ❺ 8 ➆ P ❾ 1 ➀ ➐ ➈ ❺ 8 ➆ 10P ➈➊ ➏ ➈➊ ➐ ➇➌ ❼➊ ➐ ➈ ➂➃ ➆ ➆ ❸ P ❸ ❺ ➋ 6 ➎ ➅➑ 5 4 ➐ ➈❻ ➅ ➄ 910P❶ 9 ➆ 4 ➐ ➓ ➏ ➈ ➐ 8 ➉ 10P 9 s ➐ ➈❻ ➅ ➄ ➃ ➆ P ❸ ➓ ➏ 4 ➎➁ q 6 ➐ ➂ ➉9 9 P❶ ➆ ➂ 4 ❺ ➂ ➐ ➏ P 6 ❷ ➅ ➏ ➐ ➍ 4 ➐ ➎ 6 ❶ 4 ➎ ➍ t ➐ ➈❻ ➅ ➂ ❸ 9 P ➅ ➅ ➐ ➐ ➎ ➂ ➍ 4 ➐ ➎➏ 4 ➎ ➁ ➐ ➈❻ ❺ 8 8 ➆ ➆ P ➎ 6 ➈❻ ➌ ➎ 6 10 ❺ ➌ 4 q ➍ ➐ ➅➈ ➄ 10P ➈➊ ❺ ➐ ➐ ➈ 10 ➁❿ ➀ ➐ ➅➈❻ ➅ 6P ➏ ❼ 6 ➑ ➐ 4 10 ➐ ➅➈❻ ➂ 8 ➄➅ 9 P❶ ➉ ➏ 6t

57

58 ➌ P ❷ ❷ ❻ ➂ 4 ❷ ➐ 4 ❸ ❷ 4 ➓ 4 ❶ P ➐ 4 ➏ ❺ ➐ 4 P t ❼➊ 4 ➐ ➏ ➓ 4 3 ➂ 6 ➑ ➐ P 4➁➓ 5 ➐ 4 ❾ ➏ 4 ❽ 4 4 ❻ 4 ➅ ➓ 4 ❾ ➁ 4 4 P ➐ 4 ❾ ➂ 4 P ➓ 4 ➇ ➒ ➏ ❷ 4 ❻ 4 P ➓ ➓ 4 ❽ q ❻ ➓ 4 4 P ➊ s ➏ ➁ ➒ ➂ ➋➊ ❺ ❻ ➏ 4 P 4 ❷ ➑❻ ➈ ➓ 4 4 ➏ ➈ ➑ ➑ 4 t 4 ❻➑ 4 t 4 ➍4 ❹ 109 ➐ 4 ❻ 4 ➑ 4 ➑ ➐ 4 P ➆ ➐ 4 4 ➏ ➑ ❺ 4 P ➆ ➐ 4 ➏4 ➊ ➌➍ ➑ ❺ 6 4 ❺ ❼ t P ❸ ➐ 4 4

59 9 5 ➏ ➒ ➏ ❷ ❼ ➅ ❼➊ ❶ ➏ ➂ 4 ➊ ➂ ➂ ➐ ➈ 5 6 P ➅4 ❻ ➏ ➈❻ ❷ ➐10 ❷ ➏4 4 5 ➏ ➒ ❷ ❼ ❼➊ ❶ 4 ➏ ❺ 6 ➏ ➎ ➑ P ➏ ➂ ➏ 5 ➏ ➒ ❷ ❼➊ ❼➊ ❶ 4 ➂ ➈ ➉ P ➏ ➒ ➅ ❷ ❼➊ ❼➊ ❶ 4 ➐ ❼➊ ➍P 4 5 ➏ ➒ ❻ ❷ ❼➊ ❼ ➅ ➐ ➐ ➐ ❷ ❽P➀ 4 ➏ 4 ➊ ➐ 4 ➊ 4 6 ❷ ➏4 4 ➈ ➅ 4 ➅q ❿ ➊ q ❿

60 ❾ ➐ ➈ ❻ ➏ ❺ ➋ 4 ➎ ➐ t ➐ ➈ 7 ❸ ➆ ➆ ➆ P❶ ❷ ➈ ➐ ➐ ❺ ➅ 10❿ ➐ ➊➈ ❸ P❶ ❸ ➈ ➐ 4 ❺ ❾ ➐ ➅➄ ❸ 9 P❶ ➆ ➈ ➈ ➅ ➐ 4 ❾ ➃ ➂ 6P❶ ❷ 9 ➈❻ ❻ ➐ 4 ❺ ❾➋➊ 4 ➎ ➐ ➐ ➂ 7 9 6P ➁➈❻ ➏ ❺ ➁❿ ➐ ➈❻ ➆ P ➈ 10 ➏ 5 ❿ ➀ ➁1 6 ➏ ➏ ➑ ➐ ➃ ➂ 9 10P ➍➈❻ ➍ ➂ ➏ ❺ ➉ ➐ ➈❻ 8 ➆ 9 P ➈ ➏ ❺ 10 ❺ 4 ➐ 8 P❶ 4 ➈❻ ➐ ❶ ➐ ❺ 6 ➍ 4 4 ➃ ❸ 6P❶ ➈ ❺ ➅ ❿ 5 ❾➃ ➄ ❸ ➆ 910P ➉ ➑ ❾ 10 ➈❻ ❽ ❿ ➐ ➈ ➍ ➆ ➆ 10P ➏ ❸ ➈❻ ➌ ➎➏ ➎ ❺ P ➏ ➉ ➆ ➈ ➐ 5 ➃ P ➏ ➉

61 ➉9 ➈ ➅ 5 ❼ ❺ ➐ ➈ ➆ P ➉ ➑ ❾ ➈ ❻ ➈ ➎ ➐ ➈ 8 ➂ ❸ 9 ❸ 10P ➉ ➇ ➈ 5 6 ➐ ❷ ➐ ➅ ➈❻ ➏ ❼ ➅❼➊ ❶ t ➏ ❺ 4 ➍ ➐ ➃ ➄➅ ➆ P ➈ ➑ ➊ ➉ ➐ ➈ ➍ 8 ➂ ➆ 9 ❸ P ➉ ➋ ➍➈ q q ➐ ➈❻ 8 ❸ ❸ 10P ➏ ➏ ➐ ➇ ➈ 5 6 ➐ ❷ ➈❻ ➏ ❼ ➅❼➊ ❶ 10 ➐ ➈ ➍ ❸ ❸ ❸ P ❺ 5 ➐ ➍ ➈ ❺ q ❷ ➁❿ 5 ➂ P ❺ 5 ➐ ➈❻ ❺ 6 ➁❿ 5 ➉ 910P ❸ ❻ ➐ ❷ ➈➊ 5 ➐ ➅ ➈❻ ➏ ❼ ➅❼➊ ❶ ❺ 4 ➐ ➆ ❸ P ➆ ❺ 5 ➐ ➈ ❺ ➁❿ ❽ ➏ 10P ➏ 9 ❻ ➐ ❷ ➐ ➅ ➈➊ 5 ➈❻ ➏ ❼ ➅❼➊ ❶ ➐ ➈ ❸ 6P ➏ ➇ ➈ 5 6 ➐ ➏ ➐ ❷ ➈❻ ➏ ❼ ➅❼➊ ❶ q ➐ ➊➈ ➂ ➆ ❸ P ➏ ❸ ➇ ➈ 5 ➐ ➑ ➅ ➈ ➍ ❼ ❼ ➅ ➏ ❺ 4 ➐ ❸ 6P ➏ ❸ ❻ ➐ ❷ ➂ ➐ ➂ ➈ ❼ ❼ ➊ ➐ ➈❻ ➅ ➆ P ❸

62 ➏ ➐ ➐ ❷ ➈➊ 5 ➈❻ ➏ ❼ ➅❼➊ ❶ ➏

63

64 P qsr t t

65 PHYSICAL REVIEW A VOLUME 55, NUMBER 5 MAY 1997 Coherent population trapping with losses observed on the Hanle effect of the D 1 sodium line F. Renzoni, 1 W. Maichen, 1 L. Windholz 1, and E. Arimondo 2 1 Institut für Experimentalphysik, Technische Universität Graz, A-8010 Graz, Austria 2 Unità Istituto Nazionale di Fisica della Materia, Dipartimento di Fisica dell Università, I Pisa, Italia Received 21 October 1996 We consider the coherent-population-trapping phenomenon in a thermal sodium atomic beam. We compare the different coherent-population-trapping schemes that can be established on the D 1 line using the Zeeman sublevels of a given ground hyperfine state. The coherent-population-trapping preparation is examined by means of a Hanle-effect configuration. The efficiency of the coherent-population-trapping phenomenon has been examined in presence of optical pumping into hyperfine levels external to those of the excited transition. We show that both the contrast and the width of the coherent population trapping resonance strongly decrease when the optical pumping rate is increased. In the experiment, the loss rate due to optical pumping has been controlled by means of a laser repump of variable intensity. S PACS number s : Pj, Rk I. INTRODUCTION Recently, the coherent-population-trapping CPT phenomenon has received much interest, especially in connection with its applications, for example, the laser cooling below the recoil limit or lasing without inversion see 1 for a review. At the beginning CPT theoretical studies were restricted to the three-level system 2,3, but recently, they have been extended to different atom-light interaction schemes 4,5. Despite the large amount of data available on the CPT phenomenon exploited in several atom-light interaction schemes, the features related to different level schemes and established on a given atomic species have never been directly compared. However, the increasing interest in CPT applications requires realistic calculations, taking in account processes such as the loss towards the external states, the Doppler broadening of the absorbing transition, and the collisions, all of them present in the experiment and determining the strength of the CPT resonance. The main goal of the present investigation is to study the dependence of the CPT phenomenon on the atom-laser interaction parameters. In this paper we focus our attention on the level schemes that can be established on the sodium D 1 line using as ground states the Zeeman sublevels of the same hyperfine component. We investigate theoretically and experimentally the CPT features associated with the different hyperfine optical transitions. As shown in Fig. 1, the hyperfine transitions composing the D 1 line are F g 1 F e 1, F g 2 F e 1, F g 2 F e 2, and F g 1 F e 2. Within all those optical transitions, only a limited number of Zeeman sublevels contributes to the preparation of the CPT coherent superposition of states, and precisely those connected by heavy lines in Fig. 1. For instance, the F g 1 F e 1 transition contains a system that when excited by, light produces the coherent superposition of ground states not interacting with the laser radiation. The F g 2 F e 2 transition contains an M-shaped system that again when excited by, light produces a coherent superposition of three ground states not interacting with the laser radiation. For the transition F g 2 F e 1 both the coherent dark superpositions listed above for the F g 1 F e 1 and F g 2 F e 2 transitions are present. Finally the transition F g 1 F e 2 is not relevant for CPT in the, light configuration, because it does not contain a coherent superposition noncoupled to the laser field. The experiment is based on the laser excitation of a sodium atomic beam. To produce CPT in the sodium atoms and to investigate its production we have used a Hanle effect configuration. The sodium atoms are excited by monochromatic linearly polarized laser light resonant with an hyperfine optical transition; the degeneracy of the ground-state FIG. 1. Different CPT schemes that can be established on the hyperfine components of the sodium D 1 line through excitation by, laser lights /97/55 5 / /$ The American Physical Society

66 55 COHERENT POPULATION TRAPPING WITH LOSSES OBSERVED FIG. 2. Theoretical results for the time-dependent intensity I versus time t and integrated fluorescence intensity I int at t f 45 for the F g 1 F e 1 hyperfine transition in correspondence with different values of the loss parameter Fe 1 F g 1;F g 2. In a and b Fe 1 F g 1;F g 2 0, i.e., a hypothetical closed transition; in c and d Fe 1 F g 1,F g 2 5 as for the real hyperfine transition of the D 1 line. The time t and t f are measured in units of 1/ Fe 1 F g 1. The laser intensity is 26 mw/cm 2. Zeeman sublevels is removed by the introduction of an external magnetic field that is parallel to the laser propagation direction. For zero magnetic field the atomic system is optically pumped into a coherent superposition of ground states noninteracting with the laser field, i.e., the dark or noncoupled state. For a fixed laser frequency, scanning the external magnetic field around the zero value, the atomic fluorescence emitted at right angles with respect to the directions of the laser field propagation and of the magnetic field exhibits a minimum at zero magnetic field with a line-shape dip that is typical of the Hanle-CPT phenomenon. We have measured the contrast and the linewidth of the resonance in the Hanle-CPT line shape versus the intensity of applied laser field. These measurements are compared to analytical and numerical analyses. For a closed optical transition the CPT process is quite straightforwardly understood and described in the frame of the optical Bloch equations. With an atomic sample in an initial uniform Zeeman distribution, the atomic preparation into the coherent superposition increases with a time constant determined by the optical pumping rate into the noncoupled state. For an open optical transition, i.e., in the presence of atomic levels that are external to the hyperfine transition resonant with the laser light, the atomic preparation into the coherent superposition is modified by the optical pumping into those levels. The atomic time evolution is governed by the competition between optical pumping into the noncoupled state and the optical pumping into the external hyperfine states. We have investigated, both theoretically and experimentally, how the contrast and linewidth features of the CPT resonance depend on the optical pumping rate towards the external levels. In effect for the different hyperfine optical transitions of the sodium D 1 line, owing to their different optical pumping rates, the CPT resonances have different linewidths and contrasts. The optical pumping rate into external levels may be compensated by the application of a repumping laser of variable intensity. Thus, we have studied the CPT features of a given scheme of hyperfine levels as a function of the external optical pumping loss rate by applying a repumping laser of variable intensity. Some of our results, as the narrow linewidth realized on open transitions, can be applied to the magnetometry based on coherent population trapping recently introduced by Wynands et al. 6.

67 3712 F. RENZONI, W. MAICHEN, L. WINDHOLZ, AND E. ARIMONDO 55 Our investigation should be compared to previous CPT studies. The more recent and detailed investigation of the CPT resonance has been performed by Ling et al. 7. Previous accurate studies of the Hanle effect in the ground state 8 12 have examined the line-shape dependence on the laser intensity. Moreover, optical pumping and narrow lineshapes have been studied in a variety of nonlinear spectroscopy techniques 13. None of the previous studies analyzed the role of the different hyperfine transitions as presented here. Ling et al. 7 have examined the role of the Doppler broadening on CPT. Their results show that in our experiment the Doppler broadening associated to the residual divergence of the sodium atomic beam has a negligible influence on the measured contrast and linewidth. Thus, we have not included the Doppler broadening in our analysis. In the comparison between our data and the theoretical analysis we have discovered that our measured contrast could not reach the theoretical value because the magnetic-field compensation was not accurate as required. On the contrary, in the experiment by Picqué 12 the very good magnetic-field compensation allowed the author to reach the contrast predicted by the theory. Thus, we have used the data of Ref. 12 for completing the comparison with our theoretical analysis. In the present work Sec. II contains a theoretical analysis of the CPT process based on the analytical and numerical solution of the optical Bloch equations. Section III describes the experimental setup and the experimental results. Section IV contains the comparison between the theoretical analysis and the experimental results. In Sec. V conclusions are presented. II. OPTICAL BLOCH EQUATION We consider a sodium atom interacting with a linearly polarized monochromatic laser light resonant with one of the hyperfine transition of the D 1 line and propagating in the direction Oz E z,t E 2 x e i kz t c.c. 2E 4 e i kz t c.c., 1 with i the unit vector of the i polarization. We indicate by F g F e the transition pumped by the laser, F g and F e being the quantum numbers of the total angular momentum of the hyperfine levels of the 2 S 1/2 and 2 P 1/2 levels, respectively. The quantum number of the total angular momentum of the other hyperfine level of the 2 S 1/2 level will be denoted by F g. A magnetic field B is applied in the direction Oz. For the z axis as quantization axis, the optical Bloch equations OBE for the system F g F e under examination have the following form ( e j JIF e j, g j JIF g j ): FIG. 3. Theoretical results for the integrated fluorescence intensity I int at t f 45/ Fe 1 F g 1 for the real open transitions of the sodium D 1 line, calculated for the laser intensity of 26 mw/cm 2. a corresponds to the transition F g 1 F e 1, b to F g 2 F e 1, and c to F g 2 F e 2.

68 55 COHERENT POPULATION TRAPPING WITH LOSSES OBSERVED ei e j i ei e j Fe F g 1 Fe F g ;F g ei e j i g k ei g k V gk e j V ei g k gk e j, 2a ei g j Fe F g 1 Fe F g ;F g i ei g j 2 ei g j i ei e k V ek g j ek V ei g k gk g gk j, 2b gi g j i gi g j gi g j i e k gi e k V ek g j V gi e k ek g j d dt g i g j 2c SE. Fe F g Fe F g ;F g 2F g 1 Fe F g 2F g F g 2 1 F e F g 2 1 F e The term Fe F g ;F g describes the loss due to spontaneous decay to the ground level F g external to the transition pumped by the laser. Note that the term Fe F g (1 Fe F g ;F g ) in Eqs. 2 is equal to so that the description through evidences the role of the spontaneous decay towards external levels. For the hyperfine components of the D 1 transition the values of are Fe 1 F g 1;F g 2 5, 4 5a The quantities i, j, with, (e,g) represent the frequency separation between the levels i and j, including the Zeeman splittings of the ground and excited levels due to the applied magnetic field B, Fg 2 F e 2;F g 1 1, Fg 2 F e 1;F g b 5c E i E j i, j. 3 is the total spontaneous emission rate for any excited level, Fe F g denotes the spontaneous decay rate on the F e F g transition and Fe F g ;F g the ratio between the spontaneous decays on the F e F g and F e F g transitions. This ratio is given by 14 In the dipole approximation the atom-laser interaction has matrix elements V ei,g j e i d x g j E. 6 2 The spontaneous-emission repopulation terms are 15,16 FIG. 4. Experimental setup for the investigation of the Hanle-CPT resonance on a sodium atomic beam for the different hyperfine transitions on the D 1 line.

69 3714 F. RENZONI, W. MAICHEN, L. WINDHOLZ, AND E. ARIMONDO 55 p k q d dt g k g k SE 2F e 1 Fe F g F g 1 F e q F k e 1 F g 1 k p q e q e q p. 7 q,q F e, F e, p 1,1 In order to examine the influence of the external levels on the Hanle-CPT resonance around zero magnetic field, we have solved numerically the time-dependent OBE with the initial condition gi,g j (t 0) 1 8 i j, gi,e j (t 0) ei,e j (t 0) 0. We have calculated the time-dependent fluorescence intensity emitted from the atomic system I B,t ee. m F F e,f e In our experiment on an atomic beam, we detect a signal proportional to the integrated fluorescence intensity emitted by an atom interacting with the laser light during a time t f, I int B 0 t fi B,t dt. In the case of a closed atomic system, if the timeintegrated detected signal corresponds to a long interaction times t f, the transient initial regime produces a negligible contribution to the overall intensity. The contrast between the maximum and the minimum of the emitted fluorescence intensity, defined as in Ref. 1,17, approaches 100% when all the atoms are pumped into the noncoupled state. In the case of an open system the excited-state occupation at the steady state is equal to zero: all population is lost because of the presence of the external state. In this case the transient regime produces the most important contribution to the integrated emitted intensity, which exhibits a Hanle-CPT resonance with contrast strongly depending on the atomic transition. Figure 2 shows results for I versus B at different interaction times and I int versus B at t f 45/ Fe 1 F g 1 for the F g 1 F e 1 hyperfine transition of the D 1 line, at different values of Fe F g ;F g. The case 0 of a and b corresponds to an ideal close transition, the case Fe 1 F g 1;F g 2 5 of c and d corresponds to the real F g 1 F e 1 transition of the D 1 line. In a and b for the case of a closed atomic system, where Fe F g ;F g 0, the time-dependent fluorescence intensity exhibits a sharp and well pronounced Hanle-CPT resonance around zero magnetic field. The contrast of I int increases with the interaction time and results around 100% at larger interaction times. In c and d a large loss towards external states produces a reduced intensity I, and a smaller contrast in the integrated intensity I int. The limiting value for the contrast observed on I is 100% independently of the loss rate and is reached at interaction times shorter than in the case of a closed system, because only those atoms already in the noncoupled state contribute to the contrast, the remaining ones having been pumped into the external states. The integrated fluorescence I int is obtained summing up the contributions at different times, and those at earlier times have a larger 8 9 weight on the sum. Also for an open system the contrast increases with the interaction time, reaching a value smaller than 100% at larger interaction times. Figure 3 shows the results for the integrated fluorescence intensity I int in the case of the real open transitions of sodium, for a choice of experimental parameters corresponding to the conditions of the experimental investigation. The different contrast of the Hanle-CPT resonance for the different hyperfine transitions is quite evident. A large contrast is obtained for the F g 2 F e 1 transition with the smallest value of losses towards external states. The results of Figs. 2 and 3 show that the loss towards the external states modifies strongly also the linewidth. In Fig. 2 the resonance linewidth for the case of Fe 1 F g 1;F g 2 5 is much narrower than the linewidth for the case 0. In Fig. 3 the real hyperfine transitions are directly compared and once again open transitions with a larger loss towards external states have a more narrow resonance. The comparison between different hyperfine transitions is not straightforward because the matrix element V e,g of the atom-laser interaction between excited and ground states depends on the hyperfine levels, and furthermore, in the system F g 2 F e 1, two noncoupled states are present. However, the CPT analysis for a closed symmetrical system performed in the basis of the coupled and noncoupled states is very useful for the interpretation. In that analysis 1,17, in absence of ground-state relaxation, at the steady state the linewidth of the CPT resonance is determined by the loss rate of the coupled state, with V e,g /( 2 exc ), exc being the excited-state lifetime. In Fig. 2, the main difference between the closed, in a and b, and open, in c and d, systems is the excited-state lifetime, with exc 1 1 in the first case, and exc 1 1 (1 1 1;2 ) in the second case. An increase in that lifetime produces a smaller and a more narrow CPT resosnce, as observed in the figure for both I and I int. It should be noted that in an open system the coupled-noncoupled approach applied to states without population because of the optical pumping towards external states is not really meaningful. The narrow linewidths obtained in Fig. 3 have a different explanation, because for the real D 1 line hyperfine transitions the excited states have the same lifetime, whichever Zeeman or hyperfine component. The narrow resonances are obtained in the curve of I int versus B and the contribution of I at different interaction times should be considered. They arise because of the simultaneous action of the pumping into the noncoupled nonabsorbing state and of the optical pumping towards external states. The optical pumping towards external states is less efficient whenever the optical pumping in the noncoupled states is more efficient: in the noncoupled state the atomic wave function has no contribution of excited state, whence the atom does not decay towards the external states. Thus, the narrow CPT resonance, being produced by the contribution of only those atoms

70 55 COHERENT POPULATION TRAPPING WITH LOSSES OBSERVED remaining in the noncoupled state and nondecaying towards the external states, is a consequence of an atomic selection. III. EXPERIMENTAL SETUP The experimental setup is shown in Fig. 4. We used a single-mode CW dye laser; the light polarization was linear and the propagation direction orthogonal to the thermal sodium atomic beam. An external magnetic field was applied parallel to the laser beam propagation direction. Magneticfield Helmholtz coils, applied around the atom-laser interaction region, were driven by a programmable power supply in order to produce a linear scan of the magnetic field. The earth magnetic field was compensated to better than 0.05 G by additional pairs of Helmholtz coils. By tuning the laser to different hyperfine optical transitions of the sodium D 1 line, we excited different atomic configurations. The laser frequency was tuned to the center of the homogeneous hyperfine absorption profile. The fluorescence signal emitted by the sodium atoms at right angles with respect to the directions of the laser field propagation and of the magnetic field was detected through a photomultiplier and recorded by means of a lock-in amplifier and a standard data acquisition system. In our experiment the interaction time was of the order of few s, to be compared to the 3P 1/2 excited-state lifetime of 16 ns. FIG. 5. Experimental results for the fluorescence intensity I int as a function of the magnetic field B on the different hyperfine transitions with no repumping laser. The interaction time is 4.5 s. Laser intensity I L 26 mw/cm 2 in all the data sets.

71 3716 F. RENZONI, W. MAICHEN, L. WINDHOLZ, AND E. ARIMONDO 55 FIG. 6. Experimental results for the full width at half maximum linewidth of the Hanle-CPT resonance for the different hyperfine transitions as a function of the laser intensity. Black triangles correspond to laser excitation on the transition F g 2 F e 2, while open squares to F g 1 F e 1 and closed circles to F g 2 F e 1. Typical error bars are marked. IV. EXPERIMENTAL RESULTS AND COMPARISON WITH THEORY Examples of the time-integrated fluorescence signals recorded on the different hyperfine transitions are shown in Fig. 5. Three of the four transitions on the D 1 line exhibit the Hanle-CPT typical dip. The transition F g 1 F e 2 does not exhibit any dip because there are no noncoupled states into which the atom could be pumped. The dip contrasts for the other three transitions, F g 1 F e 1, F g 2 F e 1, F g 2 F e 2, are very different. The dip for the transition F g 2 F e 1 is very pronounced, and the corresponding contrast is large. The contrasts of the dips associated with the transitions F g 1 F e 1 and F g 2 F e 2 are smaller. The observed behaviors correspond to those predicted by the analysis and shown in Fig. 2. The hyperfine transitions with a larger loss factor towards external states present a narrower Hanle-CPT resonance. However, it should be noted that different hyperfine transitions cannot be directly compared, as already discussed. The full width at half maximum linewidth and the contrast of the Hanle-CPT resonance for the various hyperfine transitions have been studied as a function of the laser intensity. The experimental results are shown in Figs. 6 and 7 a. The experimental results of Figs. 6 and 7 a for the linewidth and contrast of the Hanle-CPT resonance have been compared to the theoretical numerical analysis. The theory predicts an increase of the linewidth and contrast with the applied laser intensity, as observed in the experiment. However, the maximum contrast achieved in our experimental observations is lower than that predicted by the theory. For instance, for the F g 2 F e 1 hyperfine transition, the maximum theoretical contrast is 67%, while the measured value of the maximum contrast on that transition is around 50%. In order to examine more carefully the comparison between theory and experiments, our theoretical analysis has been applied also to the Hanle-CPT measurements published by Picqué 12 in an experimental configuration very similar to the present one, on the F g 2 F e 1 hyperfine transition of the sodium D 1 line as a function of the applied laser field FIG. 7. In a experimental results for the contrast of the Hanle- CPT resonance for the different hyperfine transitions as a function of the laser intensity. Black triangles correspond to the transition F g 2 F e 2, open squares to F g 1 F e 1, and closed circles to F g 2 F e 1. Typical error bars are marked. In b experimental results for the F g 2 F e 1 transition from Ref. 12 for the contrast of the Hanle-CPT resonance as a function of the laser intensity. Laser intensity derived from 12 ; error bar of the contrast estimated on the figures of that reference. intensity. The contrast of the Hanle-CPT resonances, as derived from the five curves published in 12, and the results of a theoretical analysis for the interaction times of that reference are reported in Fig. 7 b. In this case the agreement between our theory and the experimental results is very good, confirming the validity of the theoretical approach 18. The experiment of 12 reported a compensation of the stray magnetic fields within 10 mg over the whole atomlaser interaction volume. Thus, our failure to reproduce the theoretical contrast is produced by an imperfect compensation of the magnetic field in our apparatus. More precisely, owing to the presence of some magnetic-field gradient present in the atom-laser interaction region, the magnetic field could not be compensated over the whole interaction region through our Helmholtz coils. For the D 1 excitation on sodium atoms none of the hyperfine optical transitions is closed. However, experimentally, a closedlike situation can be realized through the application of a repumping laser, which compensates the losses term ei e i Fe F g ;F g Fe F g ei e i for the population decay. We examined, for a given transition, the dependence of contrast and linewidth on the population loss rate, by apply-

72 55 COHERENT POPULATION TRAPPING WITH LOSSES OBSERVED ing a repumping laser to partially compensate the losses towards external states. Varying the intensity of the repumping laser it was possible to study the features of the CPT resonance as a function of the rate of population losses. Obviously the repumping transition should be chosen in order to not produce an additional noncoupled coherent superposition. In the case of the sodium D 1 line the most favorable transition for this study is the F g 2 F e 2 transition because the transition F g 1 F e 2, without noncoupled state, as verified by the record in Fig. 5 d, can be used for repumping. The CPT results in presence of a repumping laser are reported in Fig. 8: it is clearly visible that the contrast and the width of the CPT resonance increase for increasing power of the repumping laser, i.e., for a decreasing rate of population losses. These experimental results can be compared to the theoretical ones of Fig. 2. The presence of a repumping laser, compensating for the losses towards external states, is equivalent to a longer lifetime of the excited state. Thus starting from the configuration of an open system as in Fig. 2 b in absence of the repumping laser, its application produces an effective closed system as in Fig. 2 a. It should be noted that the repumping laser does not compensate for the decay rate of the optical coherences. However, after preparation of the atoms in the noncoupled states, the optical coherences are zero and their decay is not relevant for the atomic preparation. V. CONCLUSIONS The CPT phenomenon on the sodium D 1 line has been investigated by means of an Hanle effect configuration comparing different level schemes ( and M and studying the influence of the losses towards external states. Different atom-laser interaction schemes involving as ground states the sublevels of the same hyperfine component have been considered. On the sodium D 1 line the relevant transition are F g 1 F e 1, F g 2 F e 1, F g 2 F e 2. As the original contribution of the present investigation we have investigated CPT in atomic configurations that are not closed for spontaneous-emission decay. We have demonstrated that CPT may be realized on those open transitions with an efficiency decreasing with the amount of spontaneous emission towards external states. Because in the D 1 line the different level configurations correspond to different rates of population losses, the features, i.e., width and contrast, of the Hanle-CPT resonance depend on the explored transition. A careful study of the dependence of the width and contrast of the Hanle-CPT dip on the interaction parameters has been performed. A numerical analysis of the density-matrix equations confirms most experimental observations, except that the maximum contrast is lower than the expected one because of the imperfect compensation of the magnetic field in the experiment. However, our numerical analysis is in agreement with the contrast measured in a previous experiment 12, where a very good magnetic-field compensation was applied. The experimental data show that the contrast of the Hanle-CPT dip is less than 100%. It depends strongly on the population loss rate and it can be controlled by means of a repumping laser that compensates the rate of population losses towards the external levels. The contrast of the Hanle- CPT resonance is only weakly affected by the small Doppler FIG. 8. Experimental results for the integrated fluorescence intensity, I int, as a function of the applied magnetic field, with one laser tuned to the transition F g 2 F e 2 to establish the M-scheme; another laser tuned to the transition F g 1 F e 2 to repump the atoms. The intensity of the laser used to establish the M scheme was I L 250 mw/cm 2. The intensity of the repumping laser was in a I R 0, b I R 20 mw/cm 2, c I R 200 mw/cm 2.

73 3718 F. RENZONI, W. MAICHEN, L. WINDHOLZ, AND E. ARIMONDO 55 broadening associated to the atomic beam. Note that in the regime of very large saturation, the power broadening of the spectral lines does not allow the interpretation of the experiment as a pure F g F e transition with losses on F g. Let us consider, for example, the case of F g 1 F e 1 ( scheme. At large laser saturation the absorption lines F g 1 F e 1 and F g 1 F e 2 are excited simultaneously. Thus, we do not have a pure system but a W system, where the states are not completely dark for the laser absorption. ACKNOWLEDGMENT This work has been partially supported by the Austrian Science Foundation Grant No. S E. Arimondo, in Progress in Optics, edited by E. Wolf Elsevier, Amsterdam, 1996, Vol. 35 p E. Arimondo and G. Orriols, Lett. Nuovo Cimento 17, H.R. Gray, R.W. Whitley, and C.R. Stroud, Jr., Opt. Lett. 3, V.S. Smirnov, A.M. Tumaikin, and V.I. Yudin, Zh. Éksp. Teor. Fiz. 96, Sov. Phys. JETP 69, F. Papoff, F. Mauri, and E. Arimondo, J. Opt. Soc. Am. B 9, R. Wynands, S. Brandt, A. Nagel, and D. Meschede, in International Quantum Electronics Conference, 1996 OSA Technical Digest Series Optical Society of America, Washington, D.C., 1996, paper TuL62; in European Quantum Electronics Conference, 1996 OSA CLEO/Europe Technical Digest Series Optical Society of America, Washington, D.C., 1996, paper QThJ1. 7 H.Y. Ling, Y.-Q. Li, and M. Xiao, Phys. Rev. A 53, J.C. Lehmann and C. Cohen-Tannoudji, C. R. Acad. Sci. Paris 258, ; J. Dupont-Roc, S. Haroche, and C. Cohen-Tannoudji, Phys. Lett. 28A, ; C. Cohen- Tannoudji, J. Dupont-Roc, S. Haroche, and F. Laloë, Rev. Phys. Appl. 5, ; C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, edited by R. Balian, S. Haroche, and S. Liberman North-Holland, Amsterdam, 1977, Vol. 1, p W. Rasmussen, R. Schieder, and H. Walther, Opt. Commun. 12, R. Schieder and H. Walther, Z. Phys. 270, H. Brand, W. Lange, J. Luther, B. Nottbeck, and R. Schröder, Opt. Commun. 13, J.L. Picqué, J. Phys. B 11, L For a recent review, see Frontiers in Laser Spectroscopy, edited by M. Inguscio and T. W. Hänsch North-Holland, Amsterdam, 1994, and references therein. 14 I.I. Sobel man, in Atomic Spectra and Radiative Transitions, Springer Series in Chemical Physics Vol. 1 Springer, Berlin, W. Happer, Rev. Mod. Phys. 44, A. Omont, Prog. Quantum Electron. 5, E. Arimondo, Phys. Rev. A 54, The presence in the experiment of a distribution of the interaction times owing to the velocity distribution in the thermal beam does not affect the results. In effect the interaction times were so long that the large majority of the atoms reached the steady state of empty population.

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75 VOLUME 85, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 6 NOVEMBER 2000 Current Switch by Coherent Trapping of Electrons in Quantum Dots T. Brandes 1, * and F. Renzoni 2 1 Universität Hamburg, 1. Institut für Theoretische Physik, Jungiusstrasse 9, D Hamburg, Germany 2 Universität Hamburg, Institut für Laserphysik, Jungiusstrasse 9, D Hamburg, Germany (Received 26 April 2000) We propose a new transport mechanism through tunnel-coupled quantum dots based on the coherent population trapping effect. Coupling to an excited level by the coherent radiation of two microwaves can lead to an extremely narrow current antiresonance. The effect can be used to determine interdot dephasing rates and is a mechanism for a very sensitive, optically controlled current switch. PACS numbers: Hk, Gk, Pz The analogy between real and artificial atoms (quantum dots) suggests the transfer of concepts from atomic physics to ultrasmall semiconductor structures. If methods such as optical coherent control are combined with the tunability of quantum dots, basic quantum mechanical effects such as preparation in a superposition of states and quantum interference can be realized and controlled in artificial microscopic devices. The interaction with light has been used to create coherent superpositions of states in single [1] and double quantum dots [2]. Furthermore, external radiation fields lead to nonlinear electron transport effects such as photoassisted tunneling and photosidebands [3,4]. In this Letter, we propose a new transport mechanism through tunnel-coupled quantum dots based on the coherent population trapping effect, a well-known effect in atomic laser spectroscopy [5]. We predict that the interaction with coherent light of two frequencies can be used to pump a current through a double dot. As a function of the relative detuning of the two frequencies the current shows an extremely narrow antiresonance, i.e., an optically controlled abrupt transition from a conducting to a nonconducting state. We furthermore show that the vanishing of the current antiresonance due to dephasing of the coupled ground states coherence (which can be controlled by tuning the tunnel coupling) can be used to obtain quantitative estimates for inelastic dephasing rates in coupled dots. The effect appears in double quantum dots where electron transport involves tunneling through two bonding and antibonding ground states j1 and j2 and one additional excited state j0 ; see Fig. 1. Leads coupled to both dots have chemical potentials such that electrons can tunnel into the ground states but leave the dot only through the excited state. The system is driven by two light (microwave) sources with frequencies v 1 and v 2 that are detuned off the two excitation energies by hd 1 : hv 1 and hd 2 : hv 2. Relaxation from the excited level by acoustic phonon emission traps the dot in a coherent superposition of the bonding and the antibonding state, if d R : d 2 2 d 1 is tuned to zero. In this case, the excited level becomes completely depopulated. In the case of real atoms, the resulting trapping of the electron in a radiatively decoupled coherent superposition leads to dark resonances in the fluorescence emission. In the double dot case discussed here, the dark resonance effect appears as a suddenly vanishing electron current for d R 0. We suggest that for low enough microwave intensity, the effect can serve as a very sensitive, optically controlled current switch. Atomic dark states have been found to be extraordinary stable against a number of perturbations [6]. In the quantum dot case, due to the Pauli blocking of the leads, a trapped electron cannot tunnel out of the ground state coherent superposition. Furthermore, this superposition is protected from incoming electrons due to Coulomb blockade (no second electron can tunnel in). These two mechanisms guarantee the robustness of the effect, which is limited only by dephasing from inelastic processes. The latter are due to spontaneous emission of phonons in double dots [7,8] and can be controlled by tuning system parameters with gate voltages. In our model, we consider a double quantum dot in the strong Coulomb blockade regime that is determined by transitions between states of fixed particle number N and N 1 1. The two tunnel-coupled N 1 1-particle ground states jg and jg 0 (see Fig. 1, inset) have energy difference : G0 2 G and hybridize into states j1 and Γ ε 0 ε G ε G T c ω 1 ε 0 Γ µ ε 2 ε 1 FIG. 1. Level scheme for two coupled quantum dots in the Coulomb blockade regime. Two tunnel-coupled ground states jg and jg 0 (small inset) form states j1 and j2 from which an electron is pumped to the excited statej0 by two light sources of frequency v 1 and v 2. Relaxation by acoustic phonon emission is indicated by dashed arrows. ε 0 ω 2 µ (19) 4148(4)$ The American Physical Society