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 slides courtesy of Hartmut Haeffner, Innsbruck Group with some notes by Andreas Wallraff, ETH Zurich
Trapping Individual Ions
Experimental setup Fluorescence detection by CCD camera photomultiplier atomic beam Photomultiplier CCD camera oven Laser beams for: Vacuum pump photoionization cooling quantum state manipulation fluorescence excitation
Ions as Quantum Bits Ions with optical transition to metastable level: 40 Ca +, 88 Sr +, 172 Yb + P τ 1 s 1/2 1/2 metastable D 5/2 D 5/2 e> τ =1s Qubit levels: S detection Doppler, D 1/2 5/2 optical cooling Sideband transition 40 cooling Ca + Qubit transition: Quadrupole transition S 1/2 S 1/2 g> S 1/2 D 5/2 1/2 stable
Detection of Ion Quantum State Quantum jumps: spectroscopy with quantized fluorescence P short lifetime monitor S D long lifetime metastable level weak transition absorption and emission cause fluorescence steps (digital quantum jump signal) lots of photons S Quantum jump technique Electron shelving technique no photons time in excited state (average is lifetime) D Observation of quantum jumps: Nagourney et al., PRL 56,2797 (1986), Sauter et al., PRL 57,1696 (1986), Bergquist et al., PRL 57,1699 (1986) Electron shelving for quantum state detection P 1/2 D 5/2 One ion : Fluorescence histogram 8 D 7 5/2 state 6 5 4 3 2 1 Quantum state τ =1s Fluorescence manipulation detection S 1/2 40 Ca + S 1/2 S 1/2 state 0 0 20 40 60 80 100 120 counts per 2 ms 1. Initialization in a pure quantum state 2. Quantum state manipulation on S 1/2 D 5/2 transition 3. Quantum state measurement by fluorescence detection 50 experiments / s Repeat experiments 100-200 times
Electron shelving for quantum state detection P 1/2 D 5/2 1. Initialization in a pure quantum state Fluorescence detection 2. Quantum state manipulation on S 1/2 D 5/2 transition S 1/2 3. Quantum state measurement by fluorescence detection Two ions: 5µm Spatially resolved detection with CCD camera: 50 experiments / s Repeat experiments 100-200 times Mechanical Motion of Ions in their Trapping Potential: Mechanical Quantum harmonical oscillator Extension of the ground state: harmonic trap Size of the wave packet << wavelength of visible light Energy scale of interest: ions need to be very cold to be in their vibrational ground state
An Ion Coupled to a Harmonic Oscillator e 2-level-atom harmonic trap... joint energy levels e,0 e,1 e,2 g tensor product g,0 g,1 g,2 dressed states diagram Laser ion interactions Approximations: Ion: Electronic structure of the ion approximated by two-level system (laser is (near-) resonant and couples only two levels) Trap: Only a single harmonic oscillator taken into account External degree of freedom: ion motion 2-level-atom harmonic trap joint energy levels... ion transition frequency 400 THz trap frequency 1 MHz
A closer look at the excitation spectrum (3 ions) (red / lower) motional sidebands carrier transition (blue / upper) motional sidebands Laser detuning Δ at 729 nm (MHz) many different vibrational modes of ions in the trap red and blue side bands can be observed because vibrational motion of ions is not cooled (in this example) Stretch mode excitation no differential force differential force differential force no differential force
Cooling of the vibrational modes e, n 1 e,n Sideband absorption spectra 99.9 % ground state population g, n 1 red side band transition g,n 0.8 0.6 after Doppler cooling n z =1.7 0.8 0.6 P D 0.4 P D 0.4 after sideband cooling 0.2 0-4.54-4.52-4.5-4.48 Detuning δω (MHz) red sideband 0.2 0 4.48 4.5 4.52 4.54 Detuning δω (MHz) blue sideband But also controlled excitation of the vibrational modes Coherent excitation on the sideband Blue sideband pulses: coupled system D, n 1 D,n D, n+1...... S, n 1 S,n S, n+1 Entanglement between internal and motional state! D state population
Single qubit operations Laser slightly detuned from carrier resonance or: Arbitrary qubit rotations: Concatenation of two pulses with rotation axis in equatorial plane z-rotations (z-rotations by off-resonant laser beam creating ac-stark shifts) x,y-rotations Gate time : 1-10 μs Coherence time : 2-3 ms D state population limited by magnetic field fluctuations laser frequency fluctuations (laser linewidth δν<100 Hz) Addressing the qubits electro-optic deflector coherent manipulation of qubits Paul trap Excitation 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-10 -8-6 -4-2 0 2 4 6 8 10 Deflector Voltage (V) dichroic beamsplitter Fluorescence detection CCD inter ion distance: ~ 4 µm addressing waist: ~ 2 µm < 0.1% intensity on neighbouring ions
Generation of Bell states generation of entanglement between two ions Pulse sequence: Generation of Bell states Pulse sequence: Ion 1: π/2, blue sideband creates entangled state between qubit 1 and oscillator
Generation of Bell states Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier excites qubit 2 Generation of Bell states Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier Ion 2: π, blue sideband takes qubit 2 (with one oscillator excitation) back to ground state and removes excitation from oscillator SD0> is non-resonant and remains unaffected
Bell state analysis Coherent superposition or incoherent mixture? What is the relative phase of the superposition? Fluorescence detection with CCD camera: tomography of qubit states (= full measurement of x,y,z components of both qubits and its correlations) Ψ + Measurement of the density matrix: SS SDDS DD SS SD DD DS Obtaining a single qubit density matrix (a naïve persons point of view) A measurement yields the z-component of the Bloch vector => Diagonal of the density matrix Rotation around the x- or the y-axis prior to the measurement yields the phase information of the qubit. as discussed before! => coherences of the density matrix
Bell state reconstruction F=0.91 SS SDDS SS SDDS DD DD DS SD SS DD DD DS SD SS Controlled Phase Gate CNOT implementation of a CNOT for universal ion trap quantum computing Both, the phase gate as well the CNOT gate can be converted into each other with single qubit operations. Together with the three single qubit gates, we can implement any unitary operation! controlled phase gate
Quantum gate proposals with trapped ions...allows the realization of a universal quantum computer! Some other gate proposals by: Cirac& Zoller Mølmer & Sørensen, Milburn Jonathan, Plenio & Knight Geometric phases Leibfried & Wineland Cirac - Zoller two-ion phase gate ε1 ε 2 D,0 D,1 S,0 S,1 ion 1 motion ion 2 S, D 0 0 S, D SWAP
Cirac - Zoller two-ion phase gate ε1 ε 2 Phase gate using the motion and the target bit. ion 1 motion ion 2 S, D 0 0 S, D SWAP Z Cirac - Zoller two-ion phase gate ε1 ε 2 D,0 D,1 S,0 S,1 ion 1 motion ion 2 S, D S, D SWAP SWAP 0 0 Z
Cirac - Zoller two-ion phase gate ε1 ε 2 Phase gate using the motion and the target bit. ion 1 motion ion 2 S, D 0 0 S, D Z How do you do this with just a two-level system??
Phase gate Composite 2π-rotation: A phase gate with 4 pulses (2π rotation) ( ) ( ) ( ) ( ) R( θφ, ) = R ππ, 2 R π 2,0 R ππ, 2 R π 2,0 + + + + 1 1 1 1 1 2π on S,0 D,1 3 2 4
Continuous tomography of the phase gate A single ion composite phase gate: Experiment state preparation S,0, then application of phase gate pulse sequence 1 π 2 π π 2 π φϕ = 0 φ = π 2 φϕ = 0 φ ϕ = π 2 0.9 0.8 D 5/2 - excitation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 Time (μs) Population of S,1> - D,2> remains unaffected all transition rates are a factor of sqrt(2) faster at the same Laser power ( ) ( ) ( ) ( ) R( θφ, ) = R π 2, π 2 R π,0 R π 2, π 2 R π,0 + + + + 1 1 1 1 2 4 3 1
Testing the phase of the phase gate 0,S> 1 0.9 π 2 π Phase gate 2 97.8 (5) % 0.8 D 5/2 - excitation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 Time (μs) ion 1 motion ion 2 Cirac - Zoller two-ion controlled-not operation S, D S, D SWAP SWAP 0 0 control qubit target qubit pulse sequence: ion 1 ion 2
Cirac Zoller CNOT gate operation SS -SS DS -DD SD -SD DD -DS Measured truth table of Cirac-Zoller CNOT operation input output
Superposition as input to CNOT gate prepare gate output detect Error budget for Cirac-Zoller CNOT Error source Laser frequency noise (Phase coherence) Residual thermal excitation Laser intensity noise Addressing error (can be corrected for partially) Off resonant excitations Laser detuning error Total Magnitude ~ 100 Hz (FWHM) <n> bus < 0.02 <n> spec = 6 1 % peak to peak 5 % in Rabi frequency (at neighbouring ion) for t gate = 600 µs ~ 500 Hz (FWHM) November 2002 Population loss ~ 10 %!!! 2 % 0.4 % 0.1 % 3 % 4 % ~ 2 % ~ 20 %
Scaling ion trap quantum computers Easy to have thousands of ions in a trap and to manipulate them individually but it is hard to control their interaction! Scaling of the Cirac-Zoller approach? Problems : Coupling strength between internal and motional states of a N-ion string decreases as -> Gate operation speed slows down (momentum transfer from photon to ion string becomes more difficult) More vibrational modes increase risk of spurious excitation of unwanted modes Distance between neighbouring ions decreases -> addressing more difficult -> Use flexible trap potentials to split long ion string into smaller segments and perform operations on these smaller strings
Scaling ion trap quantum computers Easy to have thousands of ions in a trap and to manipulate them individually but it is hard to control their interaction! The solutions: solution: Kielpinski, Monroe, Wineland Cirac, Zoller, Kimble, Mabuchi Tian, Blatt, Zoller Idea #1: move the ions around D. Wineland, Boulder, USA Kieplinski et al, Nature 417, 709 (2002) accumulator control qubit target qubit
Segmented ion traps as scalable trap architecture (ideas pioneered by D. Wineland, NIST, and C. Monroe, Univ. Michigan) Segmented trap electrode allow to transport ions and to split ion strings State of the art: Transport of ions 1 mm within 50 μs no motional heating Splitting of two-ion crystal t separation 200 μs small heating Δn 1 Architecture for a large-scale ion-trap quantum computer, D. Kielpinski et al, Nature 417, 709 (2002) Transport of quantum states, M. Rowe et al, quant-ph/0205084 D. Wineland (NIST) : Alumina / gold trap
Coherent transport of quantum information C. Monroe (Michigan) : GaAs-GaAlAs trap