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 [ ˆ Hatom] = [ ℏω1 ℏω2 ]

Two-level atoms

• Interaction between atoms and light
• 4

Two-level atoms

Two-level system [ ˆ Hatom] = [ ℏω1 ℏω2 ]

Two-level atoms

• Interaction between atoms and light
• 4

Two-level atoms

Two-level system [ ˆ Hatom] = [ ℏω1 ℏω2 ] Ideal atomic dipole d = ⃗ µ |2⟩ ⟨1| + ⃗ µ∗ |1⟩ ⟨2| ⃗ µ ≡ ⟨2|d|1⟩

• Transition dipole moment

⟨n|d|n⟩ = 0

• Spherical symmetry

Two-level atoms

• Interaction between atoms and light
• 4

Two-level atoms

Two-level system [ ˆ Hatom] = [ ℏω1 ℏω2 ] Ideal atomic dipole d = ⃗ µ |2⟩ ⟨1| + ⃗ µ∗ |1⟩ ⟨2| ⃗ µ ≡ ⟨2|d|1⟩

• Transition dipole moment

⟨n|d|n⟩ = 0

• Spherical symmetry

Two-level atoms

• Interaction between atoms and light
• 4

Interaction Hamiltonian

Monochromatic waves E = (E0 2 ei(k·r+ωt) + E∗ 2 e−i(k·r+ωt) ) ⃗ ϵ E0 = E0(r) → Complex amplitude Interaction Hamiltonian ˆ Hint = −d · E

• Dipolar interaction

→ ˜ Hint = ˆ U † ˆ Hint ˆ U

• Dirac picture

, ˆ U = e−i ˆ

Hatomt/ℏ Interaction Hamiltonian

• Interaction between atoms and light
• 5

Interaction Hamiltonian

Monochromatic waves E = (E0 2 ei(k·r+ωt) + E∗ 2 e−i(k·r+ωt) ) ⃗ ϵ E0 = E0(r) → Complex amplitude Interaction Hamiltonian ˆ Hint = −d · E

• Dipolar interaction

→ ˜ Hint = ˆ U † ˆ Hint ˆ U

• Dirac picture

, ˆ U = e−i ˆ

Hatomt/ℏ Interaction Hamiltonian

• Interaction between atoms and light
• 5

Rotating wave approximation

ℏΩ ≡ (⃗ µ · ⃗ ϵ)E0 → Rabi frequency ℏ˜ Ω ≡ (⃗ µ∗ · ⃗ ϵ)E0 → Counter-rotating frequency ∆ ≡ ω − ω0 → Detuning ˜ Hint = ˜ Hslow + ˜ Hfast [ ˜ Hslow] = −ℏ 2 [ Ω∗ei∆te−ik·r Ωe−i∆teik·r ] [ ˜ Hfast] = −ℏ 2 [ ˜ Ω∗ei(ω+ω0)te−ik·r ˜ Ωe−i(ω+ω0)teik·r ]

Rotating wave approximation

• Interaction between atoms and light
• 6

Rotating wave approximation

ℏΩ ≡ (⃗ µ · ⃗ ϵ)E0 → Rabi frequency ℏ˜ Ω ≡ (⃗ µ∗ · ⃗ ϵ)E0 → Counter-rotating frequency ∆ ≡ ω − ω0 → Detuning ˜ Hint = ˜ Hslow + ˜ Hfast [ ˜ Hslow] = −ℏ 2 [ Ω∗ei∆te−ik·r Ωe−i∆teik·r ] [ ˜ Hfast] = −ℏ 2 [ ˜ Ω∗ei(ω+ω0)te−ik·r ˜ Ωe−i(ω+ω0)teik·r ]

Rotating wave approximation

• Interaction between atoms and light
• 6

Rotating wave approximation

Time-dependent perturbation

• 1

iℏ ∫ t ⟨2| ˜ Hint(τ)|1⟩dτ = −ieik·r [ ˜ Ω ω + ω0 e−i(ω+ω0)t + Ω ∆e−i∆t ] ∆ ≪ (ω + ω0) ⇒ ˜ Hfast is negligible ˜ Hint = ˜ Hslow → [ ˜ Hint] = −ℏ 2 [ Ω∗ei∆te−ik·r Ωe−i∆teik·r ]

Rotating wave approximation

• Interaction between atoms and light
• 7

Rotating wave approximation

Time-dependent perturbation

• 1

iℏ ∫ t ⟨2| ˜ Hint(τ)|1⟩dτ = −ieik·r [ ˜ Ω ω + ω0 e−i(ω+ω0)t + Ω ∆e−i∆t ] ∆ ≪ (ω + ω0) ⇒ ˜ Hfast is negligible ˜ Hint = ˜ Hslow → [ ˜ Hint] = −ℏ 2 [ Ω∗ei∆te−ik·r Ωe−i∆teik·r ]

Rotating wave approximation

• Interaction between atoms and light
• 7

Density operator

Definition and properties ˆ ρ = ∑

k

pk |ψk⟩ ⟨ψk| , ⟨n|ˆ ρ|n⟩ →

Probability of fjndind the system at state |n⟩

p ≡ ⟨2|ˆ ρ|2⟩ − ⟨1|ˆ ρ|1⟩

• Population inversion

, q ≡ ⟨2|ˆ ρ|1⟩

• Coherence

[ˆ ρ] = [ 1−p

2

q∗ q

1+p 2

]

• Schrodinger picture

, [˜ ρ] = [

1−p 2

q∗eiω0t qe−iω0t

1+p 2

]

• Dirac picture

Density operator

• Interaction between atoms and light
• 8

Density operator

Definition and properties ˆ ρ = ∑

k

pk |ψk⟩ ⟨ψk| , ⟨n|ˆ ρ|n⟩ →

Probability of fjndind the system at state |n⟩

p ≡ ⟨2|ˆ ρ|2⟩ − ⟨1|ˆ ρ|1⟩

• Population inversion

, q ≡ ⟨2|ˆ ρ|1⟩

• Coherence

[ˆ ρ] = [ 1−p

2

q∗ q

1+p 2

]

• Schrodinger picture

, [˜ ρ] = [

1−p 2

q∗eiω0t qe−iω0t

1+p 2

]

• Dirac picture

Density operator

• Interaction between atoms and light
• 8

Convenient transformations

ˆ S = [ 1 e−i∆t ] →

Time-independent

• [ ˜

H′

int] = −ℏ

2 [ Ω∗ Ω 2∆ ] → [˜ ρ′] = [

1−p 2

q∗eiωt qe−iωt

1+p 2

] Ω = |Ω|eiφ → q′ ≡ qe−iωte−iφ , p′ ≡ p Bloch vector ⃗ β ≡    2Re(q′) 2Im(q′) p′    =    q′∗ + q′ i(q′∗ − q′) p′   

Convenient transformations

• Interaction between atoms and light
• 9

Convenient transformations

ˆ S = [ 1 e−i∆t ] →

Time-independent

• [ ˜

H′

int] = −ℏ

2 [ Ω∗ Ω 2∆ ] → [˜ ρ′] = [

1−p 2

q∗eiωt qe−iωt

1+p 2

] Ω = |Ω|eiφ → q′ ≡ qe−iωte−iφ , p′ ≡ p Bloch vector ⃗ β ≡    2Re(q′) 2Im(q′) p′    =    q′∗ + q′ i(q′∗ − q′) p′   

Convenient transformations

• Interaction between atoms and light
• 9

Master Equation

Lioville superoperator

• L0˜

ρ′ ≡ i ℏ[˜ ρ′, ˜ H′

int] , Lindblat superoperator

• Lsp˜

ρ′ ≡ Γ 2 ((1 + p′) |1⟩ ⟨2| − [|2⟩ ⟨2| , ˜ ρ′]) d˜ ρ′ dt = (L0 + Lsp)˜ ρ′

Master equation

, Γ → Natural linewidth A =    − Γ

2

−∆ ∆ − Γ

2

|Ω| −|Ω| −Γ    , a =    −Γ    → d⃗ β dt = A⃗ β + a

• Bloch equations

Master Equation

• Interaction between atoms and light
• 10

Master Equation

Lioville superoperator

• L0˜

ρ′ ≡ i ℏ[˜ ρ′, ˜ H′

int] , Lindblat superoperator

• Lsp˜

ρ′ ≡ Γ 2 ((1 + p′) |1⟩ ⟨2| − [|2⟩ ⟨2| , ˜ ρ′]) d˜ ρ′ dt = (L0 + Lsp)˜ ρ′

Master equation

, Γ → Natural linewidth A =    − Γ

2

−∆ ∆ − Γ

2

|Ω| −|Ω| −Γ    , a =    −Γ    → d⃗ β dt = A⃗ β + a

• Bloch equations

Master Equation

• Interaction between atoms and light
• 10

Stationary solution

d⃗ β dt (∞) = 0 = A⃗ β(∞) + a p(∞) = − 1 1 + s , q(∞)eiφ = ei∆t (∆ Ω − i Γ 2Ω ) s 1 + s

Saturation parameter

• s ≡

2|Ω|2 4∆2 + Γ2 = I/Is 1 + (2∆/Γ)2 ,

Saturation intensity

• I

Is = 2Ω2 Γ2

Stationary solution

• Interaction between atoms and light
• 11

Stationary solution

d⃗ β dt (∞) = 0 = A⃗ β(∞) + a p(∞) = − 1 1 + s , q(∞)eiφ = ei∆t (∆ Ω − i Γ 2Ω ) s 1 + s

Saturation parameter

• s ≡

2|Ω|2 4∆2 + Γ2 = I/Is 1 + (2∆/Γ)2 ,

Saturation intensity

• I

Is = 2Ω2 Γ2

Stationary solution

• Interaction between atoms and light
• 11

Optical forces

12

Deduction

Ehrenfest Theorem

• F = −⟨∇ ˆ

Hint⟩ = −Trˆ ρ∇ ˆ Hint = Frp + Fdp Frp = ℏkRscatt

• Radiation pressure force

, Rscatt ≡ Γ 2 s 1 + s Fdp = −∇Udp

• Dipole gradient force

, Udp ≡ ℏ∆ 2 ln(1 + s) ≈ ℏ∆ 2 s

• far−detuned

Deduction

• Optical forces
• 13

Deduction

Ehrenfest Theorem

• F = −⟨∇ ˆ

Hint⟩ = −Trˆ ρ∇ ˆ Hint = Frp + Fdp Frp = ℏkRscatt

• Radiation pressure force

, Rscatt ≡ Γ 2 s 1 + s Fdp = −∇Udp

• Dipole gradient force

, Udp ≡ ℏ∆ 2 ln(1 + s) ≈ ℏ∆ 2 s

• far−detuned

Deduction

• Optical forces
• 13

Deduction

Ehrenfest Theorem

• F = −⟨∇ ˆ

Hint⟩ = −Trˆ ρ∇ ˆ Hint = Frp + Fdp Frp = ℏkRscatt

• Radiation pressure force

, Rscatt ≡ Γ 2 s 1 + s Fdp = −∇Udp

• Dipole gradient force

, Udp ≡ ℏ∆ 2 ln(1 + s) ≈ ℏ∆ 2 s

• far−detuned

Deduction

• Optical forces
• 13

Interpretation

Scattering force

Saturation

• lim

s→∞ Rscatt = Γ

2 lim

∆→∞ Rscatt = 0 , ∂Rscatt

∂∆

• ∆=0

= 0

• Resonant

Interpretation

• Optical forces
• 14

Interpretation

Scattering force

Saturation

• lim

s→∞ Rscatt = Γ

2 lim

∆→∞ Rscatt = 0 , ∂Rscatt

∂∆

• ∆=0

= 0

• Resonant

Interpretation

• Optical forces
• 14

Interpretation

Scattering force

Saturation

• lim

s→∞ Rscatt = Γ

2 lim

∆→∞ Rscatt = 0 , ∂Rscatt

∂∆

• ∆=0

= 0

• Resonant

Dipole force I(r) ∝ |Ω(r)|2 lim

∆→0 Udp

• Non-resonant

= 0 = lim

∆→∞ Udp

• Saturation

Interpretation

• Optical forces
• 14

Cooling atomic gases

15

Overview

Overview

• Cooling atomic gases
• 16

Overview

Overview

• Cooling atomic gases
• 17

Overview

Overview

• Cooling atomic gases
• 18

Overview

Overview

• Cooling atomic gases
• 19

Exemplyfing with Dy

Dysprosium atom

• Magnetic dipole

moment (10µB)

• Long-range dipolar

interaction

• 164Dy (28.3%),

m = 163.93u Relevant transitions J = 8 → J′ = 9

• Transition 1

Transition 2 λ0 626 nm 421 nm Γ 854 kHz 202 MHz Is 72 µW/cm2 56 mW/cm2

Exemplyfing with Dy

• Cooling atomic gases
• 20

Exemplyfing with Dy

Dysprosium atom

• Magnetic dipole

moment (10µB)

• Long-range dipolar

interaction

• 164Dy (28.3%),

m = 163.93u Relevant transitions J = 8 → J′ = 9

• Transition 1

Transition 2 λ0 626 nm 421 nm Γ 854 kHz 202 MHz Is 72 µW/cm2 56 mW/cm2

Exemplyfing with Dy

• Cooling atomic gases
• 20

Zeeman Slower

Frp = ℏkRscatt → Scattering force

∆ = δ −

Doppler efgect

• k · v

−

Zeeman efgect

ωzee(z) δ = ω − ω0

• Laser detuning

, ℏωzee = µ′B(z)

• Zeeman shift

, µ′ ≡ µB(g′

Jm′ J − gJmJ) Zeeman Slower

• Cooling atomic gases
• 21

Zeeman Slower

Frp = ℏkRscatt → Scattering force

∆ = δ −

Doppler efgect

• k · v

−

Zeeman efgect

ωzee(z) δ = ω − ω0

• Laser detuning

, ℏωzee = µ′B(z)

• Zeeman shift

, µ′ ≡ µB(g′

Jm′ J − gJmJ) Zeeman Slower

• Cooling atomic gases
• 21

Zeeman Slower

Frp = ℏkRscatt → Scattering force

∆ = δ −

Doppler efgect

• k · v

−

Zeeman efgect

ωzee(z) δ = ω − ω0

• Laser detuning

, ℏωzee = µ′B(z)

• Zeeman shift

, µ′ ≡ µB(g′

Jm′ J − gJmJ) Zeeman Slower

• Cooling atomic gases
• 21

Zeeman Slower

Ideal case amax = ℏkΓ 2m , l0 = v0 2amax B(z) = B0 √ 1 − z l0 + Bbias B0 = hv0 λ0µ′ , Bbias = ℏδ µ′ mJ = 8 (gJ = 1.24) m′

J = 9 (g′ J = 1.22)

v0 = 645 m/s

Zeeman Slower

• Cooling atomic gases
• 22

Zeeman Slower

Ideal case amax = ℏkΓ 2m , l0 = v0 2amax B(z) = B0 √ 1 − z l0 + Bbias B0 = hv0 λ0µ′ , Bbias = ℏδ µ′ mJ = 8 (gJ = 1.24) m′

J = 9 (g′ J = 1.22)

v0 = 645 m/s

Zeeman Slower

• Cooling atomic gases
• 22

Zeeman Slower

Ideal case amax = ℏkΓ 2m , l0 = v0 2amax B(z) = B0 √ 1 − z l0 + Bbias B0 = hv0 λ0µ′ , Bbias = ℏδ µ′ mJ = 8 (gJ = 1.24) m′

J = 9 (g′ J = 1.22)

v0 = 645 m/s

Transition 2 (large Γ421)

δa = 0 , δb = −18Γ

Zeeman Slower

• Cooling atomic gases
• 22

Optical molasses technique

1D Model

Scattering force

• Frp = ℏkRscatt , ∆ = δ − kv

kv ≪ δ and I/Is small Frp(v) ≃ Frp(0) + v ∂Frp ∂v

• F = Frp(v) − Frp(−v) ≃ −αv
• Damping force

α = 4ℏk2 I Is −2δ/Γ [1 + I/Is + (2δ/Γ)2]2

Optical molasses technique

• Cooling atomic gases
• 23

Optical molasses technique

1D Model

Scattering force

• Frp = ℏkRscatt , ∆ = δ − kv

kv ≪ δ and I/Is small Frp(v) ≃ Frp(0) + v ∂Frp ∂v

• F = Frp(v) − Frp(−v) ≃ −αv
• Damping force

α = 4ℏk2 I Is −2δ/Γ [1 + I/Is + (2δ/Γ)2]2

Optical molasses technique

• Cooling atomic gases
• 23

Optical molasses technique

1D Model

Scattering force

• Frp = ℏkRscatt , ∆ = δ − kv

kv ≪ δ and I/Is small Frp(v) ≃ Frp(0) + v ∂Frp ∂v

• F = Frp(v) − Frp(−v) ≃ −αv
• Damping force

α = 4ℏk2 I Is −2δ/Γ [1 + I/Is + (2δ/Γ)2]2

Transition 1 (small Γ626)

I = 0.15Is , δ = −Γ/2 TD = 3 µK

Optical molasses technique

• Cooling atomic gases
• 23

Optical molasses technique

Energy decay dE dt = −αv2 = − E τdamp E(t) = E(0)e−t/τdamp τdamp = m 2α ∼ µs Doppler limit Frp → Average δFrp → Fluctuations TD = ℏΓ 2kB

• Doopler limit

Optical molasses technique

• Cooling atomic gases
• 24

Optical molasses technique

Energy decay dE dt = −αv2 = − E τdamp E(t) = E(0)e−t/τdamp τdamp = m 2α ∼ µs Doppler limit Frp → Average δFrp → Fluctuations TD = ℏΓ 2kB

• Doopler limit

Optical molasses technique

• Cooling atomic gases
• 24

Magneto-Optical Trap

Cooling (Optical Molasses) + Trapping (Magnetic Field + Polarization)

Magneto-Optical Trap

• Cooling atomic gases
• 25

Magneto-Optical Trap

Cooling (Optical Molasses) + Trapping (Magnetic Field + Polarization)

B = B0(xex + yey − 2zez)

• Magnetic fjeld near to origin

1D Model B = B(z)k , B(z) = B0z v = vez , J = 0 → J′ = 1 ∆ = δ − δzeez − kv ℏδzee = µ′ ∂B ∂z = µB(g′

Jm′ J − gJmJ)B0 Magneto-Optical Trap

• Cooling atomic gases
• 25

Magneto-Optical Trap

Cooling (Optical Molasses) + Trapping (Magnetic Field + Polarization)

B = B0(xex + yey − 2zez)

• Magnetic fjeld near to origin

1D Model B = B(z)k , B(z) = B0z v = vez , J = 0 → J′ = 1 ∆ = δ − δzeez − kv ℏδzee = µ′ ∂B ∂z = µB(g′

Jm′ J − gJmJ)B0 Magneto-Optical Trap

• Cooling atomic gases
• 25

Magneto-Optical Trap

F = F(∆σ+) − F(∆σ−) F = −αv − βz

• Taylor expansion

β ≡ αδzee k , ωMOT ≡ √ β m d2z dt2 + ( 1 2τdamp ) dz dt + (ωMOT )2z = 0 → Damping oscillator

Magneto-Optical Trap

• Cooling atomic gases
• 26

Magneto-Optical Trap

F = F(∆σ+) − F(∆σ−) F = −αv − βz

• Taylor expansion

β ≡ αδzee k , ωMOT ≡ √ β m d2z dt2 + ( 1 2τdamp ) dz dt + (ωMOT )2z = 0 → Damping oscillator

Magneto-Optical Trap

• Cooling atomic gases
• 26

Magneto-Optical Trap

F = F(∆σ+) − F(∆σ−) F = −αv − βz

• Taylor expansion

β ≡ αδzee k , ωMOT ≡ √ β m d2z dt2 + ( 1 2τdamp ) dz dt + (ωMOT )2z = 0 → Damping oscillator

Magneto-Optical Trap

• Cooling atomic gases
• 26

Optical-Dipole Trap

Gaussian beams I(r, z) = 2P πw2 exp ( −2r2 w2 ) w(z) = w0 √ 1 + ( z zR )2 zR = πw2 λ → Rayleight length Trapping potential |δ| ≫ Γ , |δ| ≫ |kv| → ∆ ≃ δ Udp(r) ≃ ℏΓ2 8δIs I(r, z) U0 ≡ Udp(0, 0) ≃ ℏΓ2 8δIs 2P πw2

• Trap depth

δ < 0 → Red-detuned traps δ > 0 → Blue-detuned traps

Optical-Dipole Trap

• Cooling atomic gases
• 27

Optical-Dipole Trap

Gaussian beams I(r, z) = 2P πw2 exp ( −2r2 w2 ) w(z) = w0 √ 1 + ( z zR )2 zR = πw2 λ → Rayleight length Trapping potential |δ| ≫ Γ , |δ| ≫ |kv| → ∆ ≃ δ Udp(r) ≃ ℏΓ2 8δIs I(r, z) U0 ≡ Udp(0, 0) ≃ ℏΓ2 8δIs 2P πw2

• Trap depth

δ < 0 → Red-detuned traps δ > 0 → Blue-detuned traps

Optical-Dipole Trap

• Cooling atomic gases
• 27

Optical-Dipole Trap

Transition 1

(small Γ626) P = 100 mW w0 = 41 µm δmax = 1854Γ

Taylor expansion in (r, z)

• Udp(r, z) = −U0 + 1

2m(ω2

rr2 + ω2 zz2) , Strong

• ωr ≡

√ 4U0 mω2 ,

Weak

• ωz ≡

√ 2U0 mz2

R Optical-Dipole Trap

• Cooling atomic gases
• 28

Optical-Dipole Trap

Transition 1

(small Γ626) P = 100 mW w0 = 41 µm δmax = 1854Γ

Taylor expansion in (r, z)

• Udp(r, z) = −U0 + 1

2m(ω2

rr2 + ω2 zz2) , Strong

• ωr ≡

√ 4U0 mω2 ,

Weak

• ωz ≡

√ 2U0 mz2

R Optical-Dipole Trap

• Cooling atomic gases
• 28

Thank you for your attention!

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