Imperial College London Cooling Using the Stark Shift Gate M.B. Plenio (Imperial) A. Retzker (Imperial) Maria Laach 7/3/007 Department of Physics and Institute for Mathematical Sciences Imperial College London www.imperial.ac.uk/quantuminformation
Cooling Using the Stark Shift Gate Introduction to Ion trap Quantum Computing Stark Shift Gate Cooling
Cold ion crystals Aarhus, Denmark: 40 Ca + (red) and 4 Mg + (blue) xford, England: 40 Ca + Innsbruck, Austria: 40 Ca + Boulder, USA: Hg + (mercury)
Ion Trap Quantum Processor Laser pulses manipulate individual ions row of qubits in a linear Paul trap forms a quantum register Effective ion-ion interaction induced by laser pulses that excite the ion`s motion A CCD camera reads out the ion`s quantum state Courtesy of R. Blatt
Penning Trap
Scaling on a chip Wineland - Nist
Orders of Magnitude Physical size of the ground state: harmonic trap ν =(π)μηz m = 40u x Size of the wave packet << wavelength of visible light / = nm mν ν Energy scale of interest: ν ν = kt B T = 50μK k B Seperation between ions: d 5μm
Detection of Ions e g Lifetime of excited state: Maximum photon scattering rate: Efficiency : η 0 3 Rate of detected photons: r τ 0ns = τ 50MHz R= ηr 50kHz 50 photons per ms Detection within ms feasible.
40 40 Ca Ca + : Important energy levels P / t = 7 ns t = s D 5/ S / φ = α + β Qbit quadropole transition
Center-of-mass and breathing mode excitation center-of-mass mode stretch mode Courtesy of R. Blatt
Laser Ion Interactions Hamiltonian: H = H + H + H ( Internal ) ( External ) ( Interaction) ω ( ) ( Internal ) 0 H = e e g g H = ν a a ( External ) i i i i mode frequencies ( ) H = Ω g e + e g cos( kx ωt+ φ) ( Interaction) ˆ Rabi frequency Laser frequency
Laser Ion Interactions Ω { ( i t i t )} i t i Hint = σ exp iη e ν a e ν a e δ + φ + + + hc.. Lamb-Dicke parameter η = kx = k 0 mν relates size of ground state to wave length of light δ = ω ω 0 Detuning of laser with respect to atomic transition In ion trap experiments, usually η
Lamb Dicke regime Taylor expansion of the exponentiel up to first order: Ω { ( ν ν )} δ φ int σ η i t i t = + i t + i H + + i e a+ e a e + hc.. Carrier resonance: δ = 0 Ω Hint = e + e { + φ φ σ i i σ } + gn, en, Red sideband: δ = ν Ω Hint = iη σ ae a e { + i φ + iφ σ } + gn, en, Blue sideband: δ =+ ν Ω Hint = iη σ a e ae { + + i φ iφ σ } + gn, en, +
Vacuum Entanglement in in an Ion Trap Introduction to Ion trap Quantum Computing Fast Cooling
The Stark Shift Gate e g ω atom + Ω + + 0 υ trap 0 υ trap D. Jonathan M.B. Plenio P.L. Knight PRA(00)
The Stark Shift Gate In the Frame Rotating with H D. Jonathan M.B. Plenio P.L. Knight PRA(00) x ( iνt + + iνt) H = Ωσ + ηωσ ae + a e i H = a a ss + y i( Ω ν )t i( )t e σ Ω ν a e σ + + a = iηω i( Ω+ ν )t i( Ω ν )t e σ + + a e σ + a For: υ Ω = in RWA ην σ σ + n + n
Γ Regular Side Band Cooling e 0 Γ e Γ e Coupling to a dissipative Level Cooling Laser: Ω ν n n Final Population and Final Rate: n Γ α = +, W < ηγ ν 4 Winleand et. Al. PRL 40 - Two Level Side band cooling Monroe et. Al. PRL 75 - Raman Side band cooling Vuletic et. Al. PRL 8 - Side band cooling for Atoms
Γ e Stark Shift Cooling e e Γ Γ 0 + + + Cooling Laser: n + n Coupling to a dissipative Level
Stark Shift Cooling - Pulsed e e e 0
The Hamiltonian e Δ Γ Ω Ω Γ + H = ω + ω + ω e e + νa a 3 ( i( ω + Δ) t i( ω + Δ) t ee ee H.c) Ω + + ( i( k ) ) cx ωct e H.c Ω + Ω c
The Master Equation H t ω 0 0 + = 0 ω 0 + νaa ω 0 0 3 0 Γ ρ = L ρ = [H, ρ] + L ρ i int iω3t iω3t i(k x ωt ) ( 3 ) ( 3 ) c ( ) H = Ω P e + Ω P e + Ω P e EIT Lasers Cooling Laser Relaxation part rel ( ) ρ = Γ + Γ P ρp 3 3 33 33 d Ω (q 3 ) iq3 x iq3 x + Γ3 φ(q 3 )e P3ρP3e 4π d Ω (q 3 ) iq3 x iq3 x + Γ 3 φ(q 3 )e P3ρP3e 4π Γ3 + Γ 3 + + + ( P ρp P ρp P ρp P ρp ) 33 33 33 Pij = i j
Cirac & Zoller The Master Equation Γ ρ = L ρ = [H, ρ] + L ρ t i Γ Steady State Solution: [H, ρ] + L ρ= i Expansion in the Lamb Dicke Parameter (Adiabatic elimination): 0 Lindberg, Stenholm & Javanainen L = L + ηl + η L +... 0 ρ = ρ + ηρ + η ρ + 0... No coupling Rabi flipping 0 η :L( ρ ) = 0 0 Second order rotation + dissipation 0 η :L( ρ ) + L( ρ ) = 0 0 Dark State Mixture of number states η :L( ρ ) + L( ρ ) + L( ρ ) = 0 0 0 0
The Solution The expansion is valid Γν ν Δν η, η, η under the conditions: Ω Ω Ω Detailed balance: d P(n) = η A ( (n )P(n ) np(n) ) A (np(n ) (n )P(n)) + + + + + dt { } n+ (n+)a+ n na+ n- (n+)ana- p( n ) = ( q )q q = A A + n A( ν ) = ΓΩ Ω c c c c ( ) ( ) + + ( )( ) Γ ν Ω Ω ν Ω Δ νω A = A( υ ) +
Final Temperature and Rate n = A A + A + ( ) W = A A η + The Optimal Point: W ηω This point is achieved for the validity conditions: The rate at this point: Γν ν Δν η, η, η Ω Ω Ω 8 η Ω c Gate period: π η Ω c
Numerical Results - Rate 0.8 0.6 0.4 n Solution of the Master Equation Cooling of One Phonon 0. 50 00 50 00 50 t W 0.035 0.03 0.05 0.0 0.05 0.46 0.48 0.5 0.54 c The rate as a function of the Rabi Frequency, Comparison to Numerical Results
The Rate at the Optimal Point W ηω W 0.03 0.05 0.0 0.05 0.0 0.005 0.0 0.04 0.06 0.08 Η T 750 500 50 000 750 500 50 0.0 0.04 0.06 0.08 0. Η The analytical result versus the numerics. As can be seen from this figure the rate is proportional to the Lamb Dicke parameter and the fit between the rate equation results and the numerical results improves with the decrease of the Lamb Dicke parameter. The squares are Tc, Tc is the time that takes to reduce the population from to 0.0. The green line corresponds to a period of a Rabi frequency and the blue line to the analytical cooling rate
The Final Population n 0.04 0.0 0.0 0.008 0.006 0.004 η = 0 η = 0 The final population as a function of the Rabi frequency. The result of the rate equation in comparison with the solution of the Master equation η = / 0, Γ = 0, Ω = / 0, ν =, Δ = 0 0.00 0.45 0.5 0.55 0.6 c n 0.00 0.005 0.00 0.0005 0.0 0.04 0.06 0.08 0. Η The final population as a function of the Lamb Dicke Parameter. Solution of the Master equation Γ = 6, Ω = / 0, Ω = /, ν =, Δ = 0 c
The Final Population The final population at the optimal point: Γν ν Δν η, η, η Ω Ω Ω n 4 Ω = 3 ν ν νγ + Δ+ ν ν + Δ+ ν Ω n η Recoil Energy
The final population and the optimal point -Numerics n 0.005 0.004 0.003 0.00 0.00 0.0 0.04 0.06 0.08 0. Η Numerics at the point: Ω = Γνη, Δ = 0 Versus: η n 0.005 0.00 0.005 0.00 0.0005 0.0 0.04 0.06 0.08 0. Η Numerics at the point: Ω = Γνη, Δ = Γ Versus: η 4
center-ofmass mode stretch mode 0.5 Cooling of a Chain mode:,n: 0 0 5000 0000 mode:,n: 0.5 P ( 0 ) P ( ) 0.5 mode:,n: 0 0 5000 0000 mode:,n: 0.5 0 0 5000 0000 mode:3,n: 0.5 0 0 5000 0000 0 0 5000 0000 mode:3,n: 0.5 0 0 5000 0000 Monte Carlo Simulation of cooling three modes simultaneously. The Rabi Frequency is set to the third mode
Summary The Cooling Time Gate Time The Final Temperature is the Recoil below Energy 3 Cooling of few modes or even the whole chain is possible