Trapped Atom-Light Interaction 2-level atom trapped in harmonic - - PowerPoint PPT Presentation

INTERACTING IONS: BLACKBOARD

Trapped Atom-Light Interaction

H

L = !Ω Rσ x cos(kz − ω Lt + φ)

Rabi frequency ΩR ≡ deg ⋅F0 !

2-level atom trapped in harmonic potential

Hint = 1

2 !ωσ z

|e> |g>

!ω

⊗

Hext = !ν a†a + 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Interaction with near resonant lin.pol. travelling wave; lowest order in multipole expansion

! " k

Trapped Atom-Light Interaction

HL = !ΩRσ x cos(kz − ωLt + φ) = 1 2 !ΩR(σ + + σ −)(e i (kz −ωLt +φ) + e

−i (kz −ωLt +φ))

⇒ HL = 1 2 !ΩR(σ + + σ −)(e i η(a+ +a )−ωLt +φ

⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + H.c.)

Rabi frequency ΩR ≡ deg ⋅F0 !

With position operator ˆ z = ! 2mν (a† + a) = Δz(a† + a) and Lamb-Dicke parameter η ≡ Δz k = 2π Δz λ = (!k )2 2m / !ν

2-level atom trapped in harmonic potential

Hint = 1

2 !ωσ z

|e> |g>

!ω

⊗

Hext = !ν a†a + 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

Interaction with near resonant lin.pol. travelling wave; lowest order in multipole expansion

! " k

Trapped Atom-Light Interaction

Expansion in η :  HL = 1 2 ΩR e

i (ω −ωL )t +φ ⎡ ⎣ ⎤ ⎦σ + 1 + iη(a +e iνt + ae −iνt ) + ....

⎡ ⎣ ⎤ ⎦ + H.c ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Lowest order in η :  HL = 1 2 ΩR e

i (ω −ωL )t +φ ⎡ ⎣ ⎤ ⎦σ + + iη e i (ω −ωL +ν )tσ +a + + e i (ω −ωL −ν )tσ +a

⎡ ⎣ ⎤ ⎦ + H.c ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

with Ho = Hext + Hint = ν (a†a + 1 2 ) + 1 2 ω σ z ⇒  HL = 1 2 ΩR e

i (ω −ωL )t +φ ⎡ ⎣ ⎤ ⎦σ +e iη a†(t )+a(t ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + H.c.

⎡ ⎣ ⎢ ⎤ ⎦ ⎥ where a†(t ) = a†e iνt and a(t ) = ae −iνt

Unitary transformation ωL = ω, "Carrier" ⇒  HL = 1 2 ΩR σ +e iφ + σ −e −iφ

( )

ωL = ω − ν, φ = 0, "red sideband" ⇒  HL = 1 2 ΩRη σ +a + σ −a + ⎡ ⎣ ⎤ ⎦  HL = e

i H0tHLe − i H0t

Single Qubit Gate

|0> |1>

ωL = ω ⇒  HL = 1 2 ΩR σ +e iφ + σ −e −iφ

( )

Time evolution operator (interaction picture) U(t ) = exp − i   HLt ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

x

cos isin 2 2 U( ) exp( i ) 2 isin cos 2 2 ϑ ϑ ⎛ ⎞ − ϑ ⎜ ⎟ ϑ = − σ = ϑ ϑ ⎜ ⎟ − ⎝ ⎠

where

t Ω ≡ ϑ = With 0: φ

Trapped Atom-Light Interaction

ωL = ω, φ = 0, "Carrier" ⇒  HL = 1 2 ΩRσ x ωL = ω − ν, (φ=0) "red sideband" ⇒ ! HL = 1 2 "ΩRη σ +a + σ −a + ⎡ ⎣ ⎤ ⎦ ωL = ω + ν, (φ=0) "blue sideband" ⇒ ! HL = 1 2 "ΩRη σ +a† + σ −a ⎡ ⎣ ⎤ ⎦

L

ω

ω − ω ν + ω ν

Two-Qubit Gate

· couple internal and external degrees of freedom Electromagnetic radiation used to

|0> |1>

η ≡ !k 2p0 = Δz λ

2π

Coupling internal and motional states

Semi-classical illustration. QM calculation

zz p p

HI ∝σ + exp iη a + a+

( )

⎡ ⎣ ⎤ ⎦ + H.c.

|0> |1>

⊗

k p0

k