Optical forces applied to atomic cooling Bruno N. Santos June 22, - PowerPoint PPT Presentation

Optical forces applied to atomic cooling Bruno N. Santos June 22, 2020 IFSC-USP Instituto de Física de São Carlos 1

Summary 1. Interaction between two-level atoms and light 2. Radiation pressure and dipole gradient forces 3. Cooling of atomic gases 2

Summary 1. Interaction between two-level atoms and light 2. Radiation pressure and dipole gradient forces 3. Cooling of atomic gases 2

Summary 1. Interaction between two-level atoms and light 2. Radiation pressure and dipole gradient forces 3. Cooling of atomic gases 2

Interaction between atoms and light 3

Two-level atoms Two-level system [ ] ℏ ω 1 0 [ ˆ H atom ] = 0 ℏ ω 2 Two-level atoms Interaction between atoms and light 4 • •

Two-level atoms Two-level system [ ] ℏ ω 1 0 [ ˆ H atom ] = 0 ℏ ω 2 Two-level atoms Interaction between atoms and light 4 • •

Spherical symmetry Transition dipole moment Two-level atoms Two-level system Ideal atomic dipole µ ∗ | 1 ⟩ ⟨ 2 | d = ⃗ µ | 2 ⟩ ⟨ 1 | + ⃗ ⃗ µ ≡ ⟨ 2 | d | 1 ⟩ � �� � [ ] ℏ ω 1 0 [ ˆ H atom ] = ⟨ n | d | n ⟩ = 0 0 ℏ ω 2 � �� � Two-level atoms Interaction between atoms and light 4 • •

Spherical symmetry Transition dipole moment Two-level atoms Two-level system Ideal atomic dipole µ ∗ | 1 ⟩ ⟨ 2 | d = ⃗ µ | 2 ⟩ ⟨ 1 | + ⃗ ⃗ µ ≡ ⟨ 2 | d | 1 ⟩ � �� � [ ] ℏ ω 1 0 [ ˆ H atom ] = ⟨ n | d | n ⟩ = 0 0 ℏ ω 2 � �� � Two-level atoms Interaction between atoms and light 4 • •

Dipolar interaction Dirac picture Interaction Hamiltonian Monochromatic waves ( E 0 ) 2 e i ( k · r + ωt ) + E ∗ 2 e − i ( k · r + ωt ) 0 E = ⃗ ϵ E 0 = E 0 ( r ) → Complex amplitude Interaction Hamiltonian U † ˆ U = e − i ˆ ˆ H int = ˆ ˜ H int ˆ , ˆ H atom t/ ℏ H int = − d · E → U � �� � � �� � Interaction Hamiltonian Interaction between atoms and light 5 • •

Dipolar interaction Dirac picture Interaction Hamiltonian Monochromatic waves ( E 0 ) 2 e i ( k · r + ωt ) + E ∗ 2 e − i ( k · r + ωt ) 0 E = ⃗ ϵ E 0 = E 0 ( r ) → Complex amplitude Interaction Hamiltonian U † ˆ U = e − i ˆ ˆ H int = ˆ ˜ H int ˆ , ˆ H atom t/ ℏ H int = − d · E → U � �� � � �� � Interaction Hamiltonian Interaction between atoms and light 5 • •

Rotating wave approximation ℏ Ω ≡ ( ⃗ µ · ⃗ ϵ ) E 0 → Rabi frequency µ ∗ · ⃗ ℏ ˜ Ω ≡ ( ⃗ ϵ ) E 0 → Counter-rotating frequency ∆ ≡ ω − ω 0 → Detuning H int = ˜ ˜ H slow + ˜ H fast [ ] Ω ∗ e i ∆ t e − i k · r 0 H slow ] = − ℏ [ ˜ Ω e − i ∆ t e i k · r 2 0 [ ] ˜ Ω ∗ e i ( ω + ω 0 ) t e − i k · r H fast ] = − ℏ 0 [ ˜ ˜ Ω e − i ( ω + ω 0 ) t e i k · r 2 0 Rotating wave approximation Interaction between atoms and light 6 • •

Rotating wave approximation ℏ Ω ≡ ( ⃗ µ · ⃗ ϵ ) E 0 → Rabi frequency µ ∗ · ⃗ ℏ ˜ Ω ≡ ( ⃗ ϵ ) E 0 → Counter-rotating frequency ∆ ≡ ω − ω 0 → Detuning H int = ˜ ˜ H slow + ˜ H fast [ ] Ω ∗ e i ∆ t e − i k · r 0 H slow ] = − ℏ [ ˜ Ω e − i ∆ t e i k · r 2 0 [ ] ˜ Ω ∗ e i ( ω + ω 0 ) t e − i k · r H fast ] = − ℏ 0 [ ˜ ˜ Ω e − i ( ω + ω 0 ) t e i k · r 2 0 Rotating wave approximation Interaction between atoms and light 6 • •

Time-dependent perturbation Rotating wave approximation � �� � [ ] ∫ t ˜ 1 Ω e − i ( ω + ω 0 ) t + Ω ⟨ 2 | ˜ = − ie i k · r ∆ e − i ∆ t H int ( τ ) | 1 ⟩ dτ i ℏ ω + ω 0 0 ∆ ≪ ( ω + ω 0 ) ⇒ ˜ H fast is negligible [ ] Ω ∗ e i ∆ t e − i k · r H int ] = − ℏ 0 H int = ˜ ˜ H slow → [ ˜ Ω e − i ∆ t e i k · r 2 0 Rotating wave approximation Interaction between atoms and light 7 • •

Time-dependent perturbation Rotating wave approximation � �� � [ ] ∫ t ˜ 1 Ω e − i ( ω + ω 0 ) t + Ω ⟨ 2 | ˜ = − ie i k · r ∆ e − i ∆ t H int ( τ ) | 1 ⟩ dτ i ℏ ω + ω 0 0 ∆ ≪ ( ω + ω 0 ) ⇒ ˜ H fast is negligible [ ] Ω ∗ e i ∆ t e − i k · r H int ] = − ℏ 0 H int = ˜ ˜ H slow → [ ˜ Ω e − i ∆ t e i k · r 2 0 Rotating wave approximation Interaction between atoms and light 7 • •

Schrodinger picture Probability of fjndind Dirac picture Coherence Population inversion Density operator Definition and properties ∑ ρ = ˆ p k | ψ k ⟩ ⟨ ψ k | , ⟨ n | ˆ ρ | n ⟩ → the system at state | n ⟩ k p ≡ ⟨ 2 | ˆ ρ | 2 ⟩ − ⟨ 1 | ˆ ρ | 1 ⟩ , q ≡ ⟨ 2 | ˆ ρ | 1 ⟩ � �� � � �� � [ 1 − p ] [ ] 1 − p q ∗ q ∗ e iω 0 t 2 2 [ˆ ρ ] = , [˜ ρ ] = 1+ p 1+ p qe − iω 0 t q 2 2 � �� � � �� � Density operator Interaction between atoms and light 8 • •

Schrodinger picture Probability of fjndind Dirac picture Coherence Population inversion Density operator Definition and properties ∑ ρ = ˆ p k | ψ k ⟩ ⟨ ψ k | , ⟨ n | ˆ ρ | n ⟩ → the system at state | n ⟩ k p ≡ ⟨ 2 | ˆ ρ | 2 ⟩ − ⟨ 1 | ˆ ρ | 1 ⟩ , q ≡ ⟨ 2 | ˆ ρ | 1 ⟩ � �� � � �� � [ 1 − p ] [ ] 1 − p q ∗ q ∗ e iω 0 t 2 2 [ˆ ρ ] = , [˜ ρ ] = 1+ p 1+ p qe − iω 0 t q 2 2 � �� � � �� � Density operator Interaction between atoms and light 8 • •

Time-independent Convenient transformations � �� � [ ] [ ] [ ] 1 − p q ∗ e iωt Ω ∗ 1 0 0 int ] = − ℏ ˆ [ ˜ H ′ ρ ′ ] = 2 S = → → [˜ 1+ p e − i ∆ t qe − iωt 2 0 Ω 2∆ 2 Ω = | Ω | e iφ → q ′ ≡ qe − iωt e − iφ , p ′ ≡ p Bloch vector     q ′∗ + q ′ 2 Re ( q ′ ) i ( q ′∗ − q ′ ) ⃗  2 Im ( q ′ )    β ≡  =    p ′ p ′ Convenient transformations Interaction between atoms and light 9 • •

Time-independent Convenient transformations � �� � [ ] [ ] [ ] 1 − p q ∗ e iωt Ω ∗ 1 0 0 int ] = − ℏ ˆ [ ˜ H ′ ρ ′ ] = 2 S = → → [˜ 1+ p e − i ∆ t qe − iωt 2 0 Ω 2∆ 2 Ω = | Ω | e iφ → q ′ ≡ qe − iωt e − iφ , p ′ ≡ p Bloch vector     q ′∗ + q ′ 2 Re ( q ′ ) i ( q ′∗ − q ′ ) ⃗  2 Im ( q ′ )    β ≡  =    p ′ p ′ Convenient transformations Interaction between atoms and light 9 • •

Bloch equations Master equation Lioville superoperator Lindblat superoperator Master Equation � �� � � �� � ρ ′ ≡ i ρ ′ ≡ Γ ρ ′ , ˜ H ′ 2 ((1 + p ′ ) | 1 ⟩ ⟨ 2 | − [ | 2 ⟩ ⟨ 2 | , ˜ ρ ′ ]) L 0 ˜ ℏ [˜ int ] , L sp ˜ ρ ′ d ˜ ρ ′ dt = ( L 0 + L sp )˜ , Γ → Natural linewidth     − Γ − ∆ 0 0  → d⃗ 2 β dt = A ⃗  − Γ    A = ∆ | Ω |  , a = 0 β + a   2 0 −| Ω | − Γ − Γ � �� � Master Equation Interaction between atoms and light 10 • •

Bloch equations Master equation Lioville superoperator Lindblat superoperator Master Equation � �� � � �� � ρ ′ ≡ i ρ ′ ≡ Γ ρ ′ , ˜ H ′ 2 ((1 + p ′ ) | 1 ⟩ ⟨ 2 | − [ | 2 ⟩ ⟨ 2 | , ˜ ρ ′ ]) L 0 ˜ ℏ [˜ int ] , L sp ˜ ρ ′ d ˜ ρ ′ dt = ( L 0 + L sp )˜ , Γ → Natural linewidth     − Γ − ∆ 0 0  → d⃗ 2 β dt = A ⃗  − Γ    A = ∆ | Ω |  , a = 0 β + a   2 0 −| Ω | − Γ − Γ � �� � Master Equation Interaction between atoms and light 10 • •

Saturation parameter Saturation intensity Stationary solution d⃗ β dt ( ∞ ) = 0 = A ⃗ β ( ∞ ) + a ( ∆ ) 1 Ω − i Γ s q ( ∞ ) e iφ = e i ∆ t p ( ∞ ) = − 1 + s , 2Ω 1 + s � �� � � �� � 2 | Ω | 2 = 2Ω 2 I/I s I s ≡ = 1 + (2∆ / Γ) 2 , 4∆ 2 + Γ 2 Γ 2 I s Stationary solution Interaction between atoms and light 11 • •

Saturation parameter Saturation intensity Stationary solution d⃗ β dt ( ∞ ) = 0 = A ⃗ β ( ∞ ) + a ( ∆ ) 1 Ω − i Γ s q ( ∞ ) e iφ = e i ∆ t p ( ∞ ) = − 1 + s , 2Ω 1 + s � �� � � �� � 2 | Ω | 2 = 2Ω 2 I/I s I s ≡ = 1 + (2∆ / Γ) 2 , 4∆ 2 + Γ 2 Γ 2 I s Stationary solution Interaction between atoms and light 11 • •

Optical forces 12

Dipole gradient force Ehrenfest Theorem Radiation pressure force Deduction � �� � F = −⟨∇ ˆ ρ ∇ ˆ H int ⟩ = − Tr ˆ H int = F rp + F dp , R scatt ≡ Γ s F rp = ℏ k R scatt 2 1 + s � �� � , U dp ≡ ℏ ∆ ℏ ∆ F dp = −∇ U dp 2 ln(1 + s ) ≈ 2 s � �� � ���� far − detuned Deduction Optical forces 13 • •

Dipole gradient force Ehrenfest Theorem Radiation pressure force Deduction � �� � F = −⟨∇ ˆ ρ ∇ ˆ H int ⟩ = − Tr ˆ H int = F rp + F dp , R scatt ≡ Γ s F rp = ℏ k R scatt 2 1 + s � �� � , U dp ≡ ℏ ∆ ℏ ∆ F dp = −∇ U dp 2 ln(1 + s ) ≈ 2 s � �� � ���� far − detuned Deduction Optical forces 13 • •