Quantum information processing with trapped atoms Introduction Fundamentals: ion iontraps, quantum bits, bits, quantum gates Implementations: 2-qubit gates, teleportation, More recent, more advanced, Jürgen Eschner GDR-IQFA 1st colloquium, Nice, 23 March, 211 Towards quantum technology (Moore's Law) How many atoms per bit of information? W. R. Keyes, IBM J. Res. Dev. 32, 24 (1988) & C. Schuck, ICFO ENIAC (1947) ~ 22 Pentium 4 (22) 1 atom per bit 1 atom Quantum effects will play a role and open up new avenues
Quantum information with single quantum systems 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 (P. Schmidt et al., 25) (Gedanken-)experiments fundamentals of QM, decoherence, entanglement Fundamental phenomena Quantum information Technological applications Quantum Information Technology Quantum Computing & Simulation Factorization, data base search Quantum Cryptography (QKD) Quantum Random Number Generation Quantum Communication Entanglement over large distance Quantum Metrology (clocks, sensors)
Quantum processors - various approaches COLD SINGLE QUANTUM SYSTEMS WITH CONTROLLED INTERACTION Ion traps Neutral atoms in traps (opt. traps, opt. lattices, microtraps) Neutral atoms and cavity QED NMR (in liquids) Superconducting qubits (charge-, flux-qubits) Solid state concepts (spin systems, quantum dots, etc.) Optical qubits and LOQC (linear optics quantum computation) Electrons on L-He surfaces or in Penning trap Ion-doped crystals or colour centers Rydberg atoms Superconducting qubits and circuit QED and more 2 nm Single trapped and laser-cooled ions...... are cold (< 1 mk) and well-localized (< 2 nm) Application for precision measurements and frequency standards (work by NIST, PTB, MPQ, NRC, NPL, NML, Mainz, MIT,...)... are individual quantum systems Textbook experiments on fundamental quantum mechanics (work by NIST, Innsbruck, MPQ, Seattle, IBM, Hamburg,...)...... allow allow full full control, control, manipulation manipulation and and measurement measurement of of quantum quantum state state ( quantum ( quantum state state engineering ) engineering ) Application Application for for studies quantum of information entanglement processing and decoherence and and studies for quantum of entanglement information and decoherence processing (work by NIST, Innsbruck, Hamburg, Oxford, MPQ, Michigan, ICFO) (work by NIST, Hamburg, Oxford, MPQ, Ibk,...)
QIP workhorse : linear Paul trap Ions = "qubits" = q.m. coherent 2-level systems Laser-controlled Interacting through vibrational modes Individually measured Cirac & Zoller, PRL 1995 R. Blatt ~1 µm Ion traps
Classic Paul trap endcap electrode z lens fluorescence detection ring electrode x y endcap electrode cooling beam Ion storage Ion confinement requires a focusing force in 3 dimensions r r r r r r binding force F, that is F = ee = e Φ Φ ~ ~ 2 Quadrupole potential Φ 2 2 Φ = ( x + y 2z 2 r 2 ) (saddle potential) Paul trap: Φ = U + V cosωt (Intuitive picture works when ) (Penning trap: Φ = U + axial magn. field) m & 2e Equation of motion in a Paul trap: x + ( U + V cosωt) x = 2 r This is a special case of the MATHIEU EQUATION (stability diagram ) Intuitive picture: time-averaged kinetic energy of driven motion = effective potential in which the ion oscillates freely with frequency ω Full motion = secular motion at ω, with superimposed micromotion at x, ω y ωz Ω ω << Ω
Legacy : circular trap for single ions Miniature Paul trap Single atoms (Ba + ) in in trap 4.7 µm Ring Ø ~ 1 mm RF ~25 MHz, ~ 5V RF secular frequency ~ 1 MHz Linear ion trap evolution Paul mass filter Innsbruck Los Alamos München Boulder, Mainz, Aarhus Boulder, Mainz, Innsbruck,
Trapping ion strings for QIP Paul trap mechanism Trapped charged particles Linear trap ~1 mm size Ion strings Laser cooling Localization << λ Laser Coulomb repulsion Interaction by common modes of motion Stationary / single-atom qubits Examples taken from:
Quantum bits: superpositions Laser excitation switches continuously between on and off Measurement shows either on (bright) or off (dark) % population 11 5.5 (a).5 1 1.5 2 Time of excitation (µs) Qubits in trapped ions (two-level systems, TLS) Encoding of quantum information requires longlived atomic states: optical transitions (forbidden transitions, intercombination lines) S D transitions in earth alkalis: Ca +, Sr +, Ba +, Ra +, (Yb +, Hg + ) etc. Innsbruck P 1/2 D 5/2 microwave transitions (Hyperfine transitions, Zeeman transitions) earth alkalis: 9 Be +, 25 Mg +, 43 Ca +, 87 Sr +, NIST 137 Ba +, 111 Cd +, 171 Yb + P 3/2 S 1/2 TLS S 1/2 TLS
Qubits in 4 Ca + ion Ground state + long-lived excited state Zeeman substates of level -1/2 1/2 Zeeman structure of the S 1/2 D 5/2 transition Zeeman structure in non-zero magnetic field: : D 5/2 5/2 3/2 1/2-1/2-3/2-5/2 2-level-system 1/2 5/2 S 1/2-1/2 1/2 levels transitions
Optical qubit transition Superpositions of S 1/2 (m=1/2) and D 5/2 (m=5/2) forms qubit P 3/2 Manipulation by laser pulses on 729 nm transition (~ 1 ms coherence time) P 1/2 τ 1 s 729 nm D 5/2 (m=5/2)> D 3/2 qubit S 1/2 S 1/2 (m=1/2)> Qubit dynamics D> S>
Rabi oscillations Quantum bits: superpositions Laser excitation switches continuously between on and off Measurement shows either on (bright) or off (dark) % population 11 5.5 (a).5 1 1.5 2 Time of excitation (µs)
Rabi oscillations cont'd. Motion and motional qubit
Quantized motion... 2> 1> n=> 1> > motional qubit N ions 3 N oscillators Motional sidebands 2-level-atom harmonic trap coupled system & transitions e... D 5/2 g Ω Γ ω { 2 1 n = S 1/2 ω n> = > 1> 2> spectroscopy: carrier and sidebands n = n = -1 n = 1 Rabi frequencies Carrier: Red SB: Blue SB: Laser detuning (*) (*) Lamb-Dicke regime : ion localised to << laser wavelength :
Excitation spectrum of the S 1/2 D 5/2 transition ω ax = 1. MHz ω rad = 5. MHz (only one Zeeman component) Sideband cooling Motional state preparation by sideband cooling on 729 nm transition P 3/2 854 nm P 1/2 D 5/2 qubit D> D 3/2 729 nm S 1/2 S 1/2 S>
Sideband cooling on S 1/2 D 5/2 transition Cooling cascade P 3/2 D 5/2 S 1/2 n n 1 τ D = 1 s : no spontaneous decay Quenching on D 5/2 -P 3/2 transition : Effective two-level system n +1 > 99.9% in n=> Qubits in a single 4 Ca + ion internal qubit D 5/2 1> motional qubit... "computational subspace" D,> D,1> 729 nm S 1/2 > 2> 1> n=> ω 1> > S,> S,1> COHERENT LASER MANIPULATION (Rabi oscillations) S,n> D,n> : carrier transition ( = ) S,n> D,n±1> : sideband transition ( = ±ω) First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 4714 (1995).
Qubit rotations in computational subspace D,> D,1> Laser-driven transitions are described by unitary operators (if ) : carrier: S,> S,1> red sideband: computational subspace (levels with n>1 ignored) blue sideband: where Example: excitation on blue sideband with Coherent state manipulation, qubit rotations D, D,1 carrier S, S,1 carrier and sideband Rabi oscillations with Rabi frequencies sideband Lamb-Dicke parameter Each point : average of 1-2 individual measurements, preparation coherent rotation state detection
Qubit rotations : phase of the wavefunction Rabi-flops on blue sideband D,1> D,> S,> Blue sideband Ramsey Interference D,1> D,> S,> S,1> S,1> π/2 Blue sideband π/2 D-state population D-state population 1.8.6.4.2 5 1.8.6.4.2 1 15 2 25 3 Pulse length (µs) π 2π 3π 4π 5 1 15 2 25 3 Pulse length (µs) Quantum gates
Coulomb coupling : common modes of motion With several ions, the motional qubits are shared vibrational modes axial (c.o.m.) mode of 7 ions breathing mode of 7 ions Real oscillation frequencies are 2 3 khz 2 ions + motion = 3 qubits Motional qubit acts as the "bus" between the ions vibrational modes computational subspace: 2 ions, 1 mode D,D,1> D,D,> laser on ion 2 laser on ion 1 D,S,1> D,S,> S,D,1> S,D,> laser on ion 1 laser on ion 2 S,S,1> S,S,>
Quantum gate proposal(s) controlled phase gate NOT ε 1 ε 2 ε1 ε1 ε 2 1 1 1 1 1 1 1 1 11 control bit bit target bit bit Further gate proposals: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases Details of C-Z gate operation (Phase gate)
Desired result, schematic S S S D D S D D S S S D D D D S control target target Experimental techniques
Ion trap setup RF linear trap: ω.7-2mhz ω 5MHz Longitudinal confinement Radial confinement: oscillating saddle potential Double trap apparatus 21
Laser pulses for coherent manipulation etc cw laser τ coh >> τ gate I ν t AOM ν+, Φ, Ampl I Φ 2 Φ 1 Φ 3 to trap 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
Spatial addressing of ions in a string Well-focussed laser beam beam steering with electro-optical deflector addressing waist ~ 2.5-3. mm < 1/4 intensity on neighbouring ion first demonstration: H.C. Nägerl et al., Phys. Rev. A 6, 145 (1999) Spectral addressing: excitation spectrum of two ions
Discrimination of qubit states State detection by photon scattering on S 1/2 to P 1/2 transition at 397 nm P 3/2 τ 8 ns Photons observed : S 1/2 = S> No photons : D 5/2 = D> P 1/2 397 nm 866 nm τ 1 s D 5/2 D> D 3/2 qubit Detector S 1/2 S 1/2 S> Quantum jumps & "Quantum amplification" P 3/2 4 Ca + P 1/2 85 nm 854 nm 866 nm 1 sec D 5/2 397 nm 393 nm D 3/2 729 nm S 1/2 High power LED for photo ionisation (8 mw @ 385 nm, HWHM 15 nm) excites 393 nm transition Ions are pumped into D 5/2 state (life time ~ 1s) No fluorescence Detection of absorption of < 1 photon/sec
State detection: shelving P monitor S D (Shelving level can, but need not be the same as qubit state) Anzahl # of measurements der Messungen 8 7 6 5 4 3 2 1 Histogram of counts in 9 ms Poisson distribution N ± N ½ discrimination efficiency 99.85% D-Zustand D state occupied besetzt S S-Zustand state occupied besetzt 2 4 6 8 1 12 Zählrate pro 9 ms counts in 9 ms H. Dehmelt 1975 Quantum state discrimination with 2 ions Individual ion detection on CCD camera Two-ion histogram (1 experiments) 5µm SS> DS> SS> region 1 region 2 SD> DS> DD> SD> DD> quantum state populations p SS,p SD,p DS,p DD
Gate pulses (I) : SWAP Swap information from internal into motional qubit and back naive idea : π-pulse on blue SB (works if initial state is not S,1>) composite SWAP (from NMR) computational subspace D, S, π D,1 S,1 Ω Rabi ~η 1 out of CS! π 2 Ω Rabi ~η 2 computational subspace D, π S, π D,1 S,1 4π A.M. Childs et al., Phys. Rev. A 63, 1236 (21) 3-step composite SWAP operation 1 1 3 3 2 4π on D,1 S,2 π on D, S,1 I. Chuang et al., Innsbruck (22)
Hiding qubits Detect quantum state of one ion only (needed for teleportation) Protect neighbours from addressing errors D 5/2 D 5/2 π S 1/2 S 1/2 ion #1 ion #2 D D D 5/2 D 5/2 S 1/2 S 1/2 ion #1 ion #2 superposition state of ion #2 protected Cirac-Zoller quantum CNOT Gate with two trapped ions "Realization of the Cirac Zoller controlled-not quantum gate", F. Schmidt-Kaler et al., Nature 422, 48-411 (23). "Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate", D. Leibfried et al., Nature 422, 412 (23).
Quantum gate proposal phase gate 1 1 1 1 1 1 1 1 control bit bit target bit bit Cirac-Zoller two-ion controlled-not gate ε1 ε 2 Preparation Detection S> = bright D> = dark ion 1 motion ion 2 S, D S, D SWAP CNOT SWAP -1 control qubit "bus" qubit target qubit
ion 1 motion ion 2 Cirac-Zoller two-ion controlled-not operation S S Ion 1,, D D pulse sequence Ion 2 SWAP SWAP -1 blue π c π/2 blue π/2 ½ CNOT Phase blue π π/2 blue π/2 ½ blue π π/2 c π/2 π blue π π control bit bit target bit bit CNOT gate = Phase gate enclosed by π/2- pulses Phase gate implemented by composite pulses Result : full time evolution SS SS Preparation SWAP DS DD SWAP -1 Detection CNOT between motion and ion 2 SD SD DD DS every point = 1 single measurements, line = calculation (no fit)
Measured fidelity (truth table) Requirements for QC Qubits store superposition information, scalable physical system Ability to initialize the state of the qubits Long coherence times, much longer than gate operation time Universal set of quantum gates: Single bit and two bit gates Qubit-specific measurement capability ion string, but scalability? ground state cooling hard work but good progress Coherent pulses on carrier and sidebands, addressing Shelving, imaging D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (21)
Quantum teleportation "Deterministic quantum teleportation with atoms", M. Riebe et al., Nature 429, 734 (24). "Deterministic quantum teleportation of atomic qubits", M. D. Barrett et al., Nature 429, 737 (24). Teleportation idea (Bennett 1993) ALICE measurement in Bell basis recover input state unknown input state class. communication rotation BOB Bell state
Full formal procedure Very simple description (see Bouwmeester et al.) Particle 2 and 3 are prepared in an entangled state where one particle is always in the "opposite" state of the other. When the result of the Bell measurement between 1 and 2 is then 2 is in the "opposite" state of the unknown state of 1. Thus particle 3, being in the opposite state to 2, will be in the same state as 1. If the result of the Bell measurement is a different state, the appropriate rotation has to be applied to particle 3.
Teleportation of atomic qubit states Figure from Kimble, Nature N&V Teleportation protocol
Teleportation in Innsbruck (Riebe et al.) "Hide": protect ion from interaction with laser pulses (manipulation or measurement) on the other ions Quantum teleportation with atoms: result 83 % class.: 67 % no cond. op.: 5 % M. Riebe et al., Nature 429, 734 (24)
Teleportation protocol at NIST F = 78% 3 Be + ions in segmented trap, hyperfine qubits with Raman transitions, gate without addressing, instead shuttling of the ions Barrett et al., Nature 429, 737 (24) More recent progress: very long coherence time n>>1 qubit entanglement high fidelity gate
Lifetime-limited Bell states (Innsbruck) 1 Parity -1.1.2.3 Time (s) Measured decay time τ 1.5 s Limit set by spontaneous emission τ = 1.16 s Superpositions of equal energy are not affected by global phase noise (see e.g. Kilpinsky et al., Science 291, 113-115 (21)) Useful to construct "decoherence-free subspaces" Ultra-longlived Bell states (Innsbruck) D 5/2 S 1/2 Lifetime > 2 s C. Roos et al., Ibk
State of the art : The Quantum Byte Entangled W state of 8 ions @ Innsbruck Density matrix Fidelity 72% Blatt group, Nature 438, 643 (25) Entangled GHZ state of 6 ions @ NIST Boulder Wineland group, Nature 438, 639 (25) High fidelity quantum gate State populations Time (µs) 17 sequential entangling and disentangling operations individual entangling fidelity 99.3(1)% for Ψ + Bell state Blatt group, Nature Physics 4, 463 (28)
Ion trap QC architecture Scalable ion trap quantum computer Manipulation of ion strings in a scalable quantum processor with memory and processor Laser segmented ion trap coherent transport separating and combining strings This segmented trap: Schmidt-Kaler / Blatt, Innsbruck / Ulm more trap architecture: A. Steane's group, Oxford D. Winelands group, NIST C. Monroe's group, U Michigan I. Chuang's group, MIT
Quanten CCD array v 1 (t) v 2 (t) v 3 (t) v 4 (t) accumulator control qubit target qubit NIST group - Kielpinski et al, Nature 417, 79 (22) Ion chip trap at NIST NIST group
Ion chip trap at NIST NIST group Shuttling operations http://monroelab2.physics.lsa.umich.edu/research/trap/index.html#ttrap 2 ions are separated (also demonstrated earlier at NIST) single ion is shuttled around corner in a T-junction linear trap
Single neutral atom gate: Rydberg gate Dipole trap focused laser beam FORT far-off-resonance trap based on light shift (AC Stark shift)... trapping potential 2> 1> n=> Several traps with controllable distance on the < µm scale (Grangier) Lattice with single-site optical addressing (Kuhr)
Coherent Rydberg excitation pulsed two-photon excitation state measurement by recapture Grangier group Rydberg gate Dipole dipole coupling of 2 Rydberg atoms Significant coupling for > 1 µm Gate principle: Gate pulses (U Z ): Jaksch, Grangier, Saffman
Upcoming system : circuit QED Device 2 superconducting qubits coupled by microwave resonator
Control Resonant drive population control Adiabatic tuning over avoided crossing phase control Schoelkopf, Wallraff Entangling gates DiCarlo (Schoelkopf group) 29
QC with atoms : A lot has been done A lot remains to be done