Motion and motional qubit

Transcription

1 Quantized motion Motion and motional qubit... > > n=> > > motional qubit N ions 3 N oscillators Motional sidebands Excitation spectrum of the S / transition -level-atom harmonic trap coupled system & transitions e... g Ω Γ ω { n = S / ω n> = > > > ω ax =. MHz ω rad = 5. MHz spectroscopy: carrier and sidebands n = n = - n = Rabi frequencies Carrier: Red SB: Blue SB: Ω Ω η Ω η n n + (only one Zeeman component) Laser detuning (*) (*) Lamb-Dicke regime : ion localised to << laser wavelength :

2 Sideband cooling Sideband cooling on S / transition P 3/ Motional state preparation by sideband cooling on 79 nm transition 854 nm Cooling cascade P 3/ P / qubit 79 nm D 3/ D> S / n n = τ D = s : no spontaneous decay = Quenching on - P 3/ transition : n + > 99.9% in n=> S / S / S> Effective two-level system Qubits in a single 4 Ca + ion Qubit rotations in computational subspace internal qubit 79 nm S / > > motional qubit... > > n=> ω > > "computational subspace" D,> S,> D,> S,> COHERENT LASER MANIPULATION (Rabi oscillations) D,> S,> D,> S,> computational subspace (levels with n> ignored) Laser-driven transitions are described by unitary operators (if Ω >> Γ D, Γ Laser ) : carrier: θ = Ω t C red sideband: θ = Ω t SB iφ i ( + φ ) θ R( θ, φ) = exp i e σ + e σ iφ iφ ( ) R θ ( θ, φ) = exp i e σ + a + e σ a sideband: i i R + θ φ φ ( θ, φ) = exp i ( e σ + a + e σ a) θ = Ω SB t where S,n> D,n> : carrier transition ( = ) S,n> D,n±> : sideband transition ( = ±ω) Example: excitation on sideband with First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 474 (995).

3 D, S, D, S, carrier and sideband Rabi oscillations with Rabi frequencies Coherent state manipulation Lamb-Dicke parameter sideband carrier Each point : average of - individual measurements, preparation coherent rotation state detection Qubit rotations : phase of the wavefunction Rabi-flops on sideband D,> D,> S,> Blue sideband Ramsey Interference D,> D,> S,> S,> S,> / Blue sideband / D-state population D-state population Pulse length (µs) Pulse length (µs) ions + motion = 3 qubits With several ions, the motional qubits are shared motional qubit acts as the "bus" between the ions vibrational modes computational subspace: ions, mode Quantum gates D,D,> D,D,> laser on ion laser on ion D,S,> D,S,> S,D,> S,D,> laser on ion laser on ion S,S,> S,S,> 3

4 Quantum gate proposal(s) Cirac-Zoller two-ion controlled-not gate ε control bit bit controlled phase gate NOT ε ε ε ε target bit bit Further gate proposals: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases ion motion ion Preparation Detection S, D SWAP SWAP - S, D ε ε CNOT S> = bright D> = dark control qubit "bus" qubit target qubit Desired result, schematic Cirac-Zoller two-ion controlled-not operation S S S D D S S S S D D D ion S, D motion SWAP SWAP - ion S, D Ion CNOT Phase control bit bit target bit bit D D D S pulse sequence Ion c / / ½ / / ½ / c / control target target CNOT gate = Phase gate enclosed by /- pulses Phase gate implemented by composite pulses 4

5 Details of C-Z gate operation (Phase gate) Experimental techniques Laser pulses for coherent manipulation Addressing of ions in a string Well-focussed laser beam cw laser τ coh >> τ gate I ν t AOM ν+, Φ, Ampl to trap I Φ Φ Φ3 t (That's the difficult bit!), Φ, Ampl RF AOM AOM = acousto-optical acousto-optical modulator, modulator, based based on on Bragg Bragg diffraction diffraction "Ampl" "Ampl" = Amplitude, Amplitude, includes includes switching switching on/off on/off beam steering with electro-optical deflector addressing waist ~.5-3. mm < /4 intensity on neighbouring ion first demonstration: H.C. Nägerl et al., Phys. Rev. A 6, 45 (999) 5

6 Excitation spectrum of two ions Quantum state discrimination with ions Individual ion detection on CCD camera Two-ion histogram ( experiments) 5µm SS> DS> SS> region region SD> DS> DD> SD> DD> quantum state populations p SS,p SD,p DS,p DD Gate pulses (I) : SWAP 3-step composite SWAP operation Swap information from internal into motional qubit and back naive idea : -pulse on SB (works if initial state is not S,>) composite SWAP (from NMR) 3 computational subspace D, D, out of CS! computational subspace D, D, 4 3 S, S, Ω Rabi ~η S, S, 4 on D, S, Ω Rabi ~η A.M. Childs et al., Phys. Rev. A 63, 36 () on D, S, I. Chuang et al., Innsbruck () 6

7 Hiding qubits Detect quantum state of one ion only (needed for teleportation) Protect neighbours from addressing errors S / S / ion # ion # Cirac-Zoller quantum CNOT Gate with two trapped ions D D S / S / ion # ion # superposition state of ion # protected "Realization of the Cirac Zoller controlled-not quantum gate", F. Schmidt-Kaler et al., Nature 4, 48-4 (3). "Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate", D. Leibfried et al., Nature 4, 4 (3). Measured fidelity (truth table) Entanglement, parity and fidelity Ion Ion Ion Ion / CNOT / ϕ / ϕ SS>+ DD> SD>+ DS> "super-ramsey experiment" Parity: P SS +P DD -P DS -P SD 54% contrast Phase ϕ Oscillation with ϕ entanglement! Fidelity =.5 ( P SS + P DD + contrast) = 7(3)% 7

8 -ion gate at NIST High gate fidelity F =.97 "Geometric phase" gate uses state dependence of dipole force no addressing no motional ground state required 8