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Quantum computing … Applications in informatics and physics P. Shor, 1994: factorization of large numbers is polynomial on a quantum computer, exponential on a classical computer L. Grover, 1997: data base search N 1/2 quantum queries, N classical simulation of Schrödinger equations or any unitary evolution quantum cryptography / repeaters / quantum links improved atomic clocks understanding the fundamentals of quantum mechanics / Gedanken-Experimente Experiments with entangled matter

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The prototype

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Ion strings in a linear Paul trap H.C. Nägerl et al., Appl. Phys. B 66, 603 (1998). First observations:Raizen et al., PRA 45, 6493 (1992), Waki et al., PRL 68, 2007 (1992).

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Quantum gate proposal(s) Further gate proposals: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases control bit target bit controlled NOT

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D 5/2 729 nm |1> |0> internal qubit Qubits in a single 40 Ca + ion S 1/2 motional qubit |0> |1> 1 n=0 2 |S,n> |D,n> : carrier transition ( ) |S,n> |D,n ± 1> : sideband transition ( ) "computational subspace" |S,0> |D,0> |D,1> |S,1> COHERENT LASER MANIPULATION (Rabi oscillations) First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 4714 (1995).

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2 ions + motion = 3 qubits With several ions, the motional qubits are shared vibrational modes computational subspace: 2 ions, 1 mode |S,S,0> |D,S,0> |D,S,1> |S,S,1> |S,D,0> |S,D,1> |D,D,0> |D,D,1> laser on ion 2 laser on ion 1 laser on ion 2 laser on ion 1

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Details of C-Z CNOT gate operation (Phase gate)

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Experimental techniques conditions vs. achievements Experimental techniques conditions vs. achievements

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D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001) Qubits store superposition information,ion string, but scalability? scalable physical system Ability to initialize the state of the qubitsground state cooling Long coherence times,hard work much longer than gate operation time Universal set of quantum gates:Coherent pulses on Single bit and two bit gatescarrier and sidebands, addressing Qubit-specific measurement capabilityShelving, imaging Some requirements... See Toni's lecture today

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Innsbruck linear ion trap |1> |0> |1> |0> Two 2-level systems 5mm +HV RF GND

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P 3/2 854 nm 393 nm S 1/2 P 1/2 D 3/2 397 nm 866 nm D 5/2 729 nm Level scheme of 40 Ca + S 1/2

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Zeeman structure in non-zero magnetic field: : (+ motional degrees of freedom...) S 1/2 D 5/2 5/2 3/2 -3/2 -5/2 - /21 1 /21 1 2-level-system 1/2 5/2 Zeeman structure of the S 1/2 – D 5/2 transition

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P 3/2 S 1/2 P 1/2 D 3/2 D 5/2 729 nm Manipulation by laser pulses on 729 nm transition (~ 1 ms coherence time) |1> |0> qubit Level scheme of 40 Ca + S 1/2 Superpositions of S 1/2 (m=1/2) and D 5/2 (m=5/2) form qubits

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P 3/2 S 1/2 P 1/2 D 3/2 397 nm 866 nm D 5/2 |1> |0> qubit Level scheme of 40 Ca + S 1/2 State detection by photon scattering on S 1/2 to P 1/2 transition at 397 nm (> 99% in ~ 3 ms) Detector

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P 3/2 854 nm S 1/2 P 1/2 D 3/2 D 5/2 729 nm |1> |0> qubit Level scheme of 40 Ca + S 1/2 Motional state preparation by sideband cooling on 729 nm transition (> 99.9%)

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S 1/2 D 5/2 |n> = |0> |1> |2> coupled system & transitions 2-level-atomharmonic trap spectroscopy: carrier and sidebands Laser detuning n = 1 n = -1 n = 0 Motional sidebands Rabi frequencies Carrier: Red SB: Blue SB: k 1/2 «

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Excitation spectrum of the S 1/2 – D 5/2 transition ax = 1.0 MHz rad = 5.0 MHz (only one Zeeman component)

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Final temperature: equilibrium of heating and cooling Photon recoil: Cooling principle: cooling transitions more probable than heating transitions |p > |p + ħk laser + ħk spont > atom motion In trap: transitions between energy eigenstates |n > |n' > laserspont. Lamb-Dicke regime: only |n > |n±1 > Laser cooling of trapped atoms Resolved sidebands: ground state cooling

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Excitation spectrum of two ions

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Sideband cooling of two ions

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Laser pulses for coherent manipulation coh >> gate AOM = acousto-optical modulator, based on Bragg diffraction "Ampl" includes switching on/off AOM = acousto-optical modulator, based on Bragg diffraction "Ampl" includes switching on/off Ampl Ampl cw laser RF AOM I t I t to trap

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see also experiments at NISTRoos et al., PRL 83, 4713 (1999) S 1/2 D 5/2 0 1 1 2 Quantum state engineering

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Blue sideband /2 D-state population Rabi-flops on blue sideband Ramsey Interference 050 0 0.2 0.4 0.6 0.8 1 D-state population 100150200250300 Pulse length ( s) Qubit rotations |S,0> |D,0> |D,1> |S,1> |S,0> |D,0> |D,1> |S,1>

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Addressing of ions in a string Well-focussed laser beam beam steering with electro-optical deflector addressing waist ~ 2.5 - 3.0 mm < 1/400 intensity on neighbouring ion

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Individual ion detection on CCD camera 5µm quantum state populations p SS,p SD,p DS,p DD |SS>|DS> |DD> |SD> Two-ion histogram (1000 experiments) region 1region 2 |SS> |SD> |DS> |DD> Quantum state discrimination

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Cirac-Zoller Quantum CNOT Gate with two trapped ions

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Detection ion 1 motion ion 2 control qubit target qubit SWAP Cirac-Zoller two-ion controlled-NOT gate SWAP -1 Preparation |S> = bright |D> = dark CNOT "bus" qubit

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Result : schematic control target

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SS DS DD SD DD DS Result : full time evolution Preparation Detection every point = 100 single measurements, line = calculation (no fit)

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Details of time evolution Preparation SWAP SWAP -1 CNOT between motion and ion 2 Detection SS

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input output exp ideal >| 2 Measured fidelity (truth table) F. Schmidt-Kaler et al., Nature 422, 408 (2003)

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|SS> + e i |DD> { |SS>+|DD>, |SD>+|DS>} control target (|S>+|D>)|S> Experimental sequence: Ion 1 Ion 2 CNOT /2 /2 /2 Deterministic entanglement "Super-Ramsey experiment" Detection: Parity check... CNOT |S>|S> local /2 rotation local ( /2, ) rotation

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outputpreparationgatedetect Gate coherence Projection CNOT

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Fidelity = 0.5 ( P SS + P DD + contrast) = 71(3)% Oscillation with 2 entanglement ! Parity and fidelity Ion 1 Ion 2 CNOT /2 /2 /2 Phase Parity: P SS +P DD -P DS -P SD 54% contrast |SS>+|DD> |SD>+|DS> "super-Ramsey experiment"

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Error sourceMagnitudePopulation loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM)~ 10 % !!! Residual thermal excitation bus < 0.02 other = 6 2 % 0.4 % Laser intensity noise1 % peak to peak0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3 % Off resonant excitations for t gate 600 µs 4 % Laser detuning error~ 500 Hz (FWHM)~ 2 % TotalNovember 2002 ~ 20 % Error budget : Cirac-Zoller CNOT

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Examples of experimental problems & solutions Examples of experimental problems & solutions

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computational subspace out of CS ! naive idea : -pulse on blue SB composite SWAP (from NMR) computational subspace Gate pulses (I) : SWAP (works if initial state is not |S,1>) Swap information from internal into motional qubit and back

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1 2 3 on 1 3 I. Chuang et al., Innsbruck (2002) 3-step composite SWAP operation

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computational subspace Phase factor -1 for all except |D,0 > Phase factor -1 for |S,1 > Cirac & Zoller (1995)Composite phase gate Gate pulses (II) : Phase gate use auxiliary level M. H. Levitt, Prog. NMR Spectrosc., 1986 I. L. Chuang, Innsbruck, 2002 Phase factor conditioned on state

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1 2 3 4 2 on Composite phase gate (2 rotation)

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4 3 2 1 2 also on Action on |S,1> - |D,2> no population outside CS !

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020406080100120140160180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (µs) D 5/2 - excitation state preparation, then application of phase gate pulse sequence Single ion composite phase gate

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Time (µs) D 5/2 - excitation 050100150200250300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Phase gate state preparation, then application of CNOT gate pulse sequence 0.978 (5) - /2 /2 Single ion composite CNOT gate

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ion 1 motion ion 2 SWAP -1 SWAP Ion 1 Ion 2 pulse sequence control bit target bit Cirac-Zoller two-ion controlled-NOT operation blue 0 blue c 0 blue blue ½ 0 blue c CNOT

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SS Details of time evolution Ion 1 Ion 2 blue 0 blue c 0 blue blue ½ 0 blue c Time ( s)

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AC Stark shift & its compensation

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1/2 -5/2 1/2 -1/2 1/2 3/2

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The artist's view

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The experimentalist's view

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Why Bell states ? entangled massive particles, distinguishable resource for quantum cryptography / repeaters / quantum links improved atomic clocks understanding the fundamentals of quantum mechanics / EPR paradox, Gedanken-Experimente

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Generation of Bell states with three pulses atom 1atom 2 Carrier pulses: Blue sideband pulses result

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z x y /2 - pulse Rotation of the Bloch sphere prior to state measurement Principle of tomography (1 atom) Measurement of spin components

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SS SD DS DD SS SD DS DD SS SD DS DD SS SD DS DD F=0.91 Bell state generation & tomography

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SS SD DS DD SS SD DS DD SS SD DS DD SS SD DS DD F=0.90 Bell state generation & tomography

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SS SD DS DD SS SD DS DD SS SD DS DD SS SD DS DD F=0.88 Bell state generation & tomography

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SS SD DS DD SS SD DS DD SS SD DS DD SS SD DS DD F=0.91 Bell state generation & tomography

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Fidelity :F = 0.91 Peres-Horodecki criterion : Violation of a CHSH inequality: S (0°,90°,45°,135°) ( exp ) = 2.53(6) > 2 E( exp ) = 0.79 (4) Entanglement of formation for a pair of qubits (Wooters 98) : Entanglement characterization

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Cirac-Zoller quantum CNOT gate with two trapped ions F. Schmidt-Kaler, C. Becher, J. E., H. Häffner, C. Roos, W. Hänsel, G. Lancaster, S. Gulde, M. Riebe, T. Deuschle, I.L. Chuang, R. Blatt F. Schmidt-Kaler et al., Nature 422, 408 (2003) The works and the workers Bell States of Atoms with Ultralong Lifetimes and Their Tomographic State Analysis C. F. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) Deutsch-Jozsa quantum algorithm with a single trapped ion S. Gulde et al., Nature 421, 48-50 (2003). Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor H. Häffner et al., Phys. Rev. Lett. 90, 143602 (2003). http://heart-c704.uibk.ac.at/papers.html

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