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

INTERACTING IONS: BLACKBOARD

Trapped Atom-Light Interaction 2-level atom trapped in harmonic potential |e> ⊗ " ! ω k ! |g> ⎛ ⎞ H ext = ! ν a † a + 1 H int = 1 2 ! ωσ z ⎜ ⎟ 2 ⎝ ⎠ Interaction with near resonant lin.pol. travelling wave; lowest order in multipole expansion Rabi frequency Ω R ≡ d eg ⋅ F 0 ! H L = ! Ω R σ x cos( kz − ω L t + φ )

Trapped Atom-Light Interaction 2-level atom trapped in harmonic potential |e> ⊗ " ! ω k ! |g> ⎛ ⎞ H ext = ! ν a † a + 1 H int = 1 2 ! ωσ z ⎜ ⎟ ⎝ 2 ⎠ Interaction with near resonant lin.pol. travelling wave; lowest order in multipole expansion H L = ! Ω R σ x cos( kz − ω L t + φ ) Rabi frequency Ω R ≡ d eg ⋅ F 0 ! = 1 ! Ω R ( σ + + σ − )( e i ( kz − ω L t + φ ) + e − i ( kz − ω L t + φ ) ) 2 ! ( a † + a ) = Δ z ( a † + a ) With position operator ˆ z = 2 m ν and Lamb-Dicke parameter η ≡ Δ z k = 2 π Δ z = ( ! k ) 2 / ! ν λ 2 m ⇒ H L = 1 ! Ω R ( σ + + σ − )( e i η ( a + + a ) − ω L t + φ ⎡ ⎤ ⎥ + H . c .) ⎢ ⎣ ⎦ 2

Trapped Atom-Light Interaction i − i  Unitary transformation  H 0 t H L e  H 0 t H L = e with H o = H ext + H int =  ν ( a † a + 1 ) + 1  ω σ z 2 2 H L = 1 ⎡ ⎤ ⎡ ⎤ ⎥ + H . c . i η a † ( t ) + a ( t ) ⎡ ⎤ ⇒  i ( ω − ω L ) t + φ ⎢  Ω R e ⎦ σ + e ⎣ ⎣ ⎦ ⎢ ⎥ 2 ⎣ ⎦ where a † ( t ) = a † e i ν t and a ( t ) = ae − i ν t Expansion in η : H L = 1 ⎦ σ + 1 + i η ( a + e i ν t + ae − i ν t ) + .... ⎡ ⎤ ⎡ ⎤  i ( ω − ω L ) t + φ ⎡ ⎤  Ω R e ⎦ + H . c ⎣ ⎣ ⎢ ⎥ ⎣ ⎦ 2 Lowest order in η : H L = 1 ⎦ σ + + i η e i ( ω − ω L + ν ) t σ + a + + e ⎡ ⎤ ⎡ ⎤ ⎡ ⎤  i ( ω − ω L ) t + φ i ( ω − ω L − ν ) t σ + a  Ω R e ⎦ + H . c ⎣ ⎢ ⎥ ⎣ ⎣ ⎦ 2 ω L = ω − ν , φ = 0, "red sideband" ω L = ω , "Carrier" H L = 1 ( ) H L = 1 ⇒  ⎡ ⎤  Ω R σ + e i φ + σ − e − i φ  Ω R η σ + a + σ − a + ⇒  ⎣ ⎦ 2 2

Single Qubit Gate |1> ω L = ω ( ) H L = 1  Ω R σ + e i φ + σ − e − i φ ⇒  |0> 2 ⎛ ⎞ Time evolution operator (interaction picture) U ( t ) = exp − i  H L t ⎜ ⎟ ⎝ ⎠  ϑ ϑ ⎛ ⎞ − cos isin ϑ 2 2 ⎜ ⎟ With φ = 0: ϑ = − σ = U( ) exp( i ) where ϑ ≡ Ω t x ⎜ ϑ ϑ ⎟ 2 − isin cos ⎝ ⎠ 2 2

Trapped Atom-Light Interaction ω L = ω , φ = 0, "Carrier" H L = 1 ⇒   Ω R σ x 2 ω L = ω + ν , ( φ =0) "blue sideband" ω L = ω − ν , ( φ =0) "red sideband" H L = 1 H L = 1 " Ω R η σ + a † + σ − a ⇒ ! ⎡ ⎤ ⇒ ! ⎡ ⎤ " Ω R η σ + a + σ − a + ⎣ ⎦ ⎣ ⎦ 2 2 ω − ν ω ω + ν ω L

Two-Qubit Gate Electromagnetic radiation used to · couple internal and external degrees of freedom η ≡ ! k = Δ z 2 π 2p 0 λ |1> |0>

Coupling internal and motional states Semi-classical illustration. QM calculation p p 0 |1>  k  k p 0 ⊗ zz 0 |0> ( ) ⎡ ⎤ H I ∝ σ + exp i η a + a + ⎦ + H.c. ⎣