Brian King. SQuInT summer school June, Dept. Physics and Astronomy, McMaster University

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1 Ion Traps for Quantum Computing Ann Arbor Garching Innsbruck Boulder SQuInT summer school June, 2003 Brian King Dept. Physics and Astronomy, McMaster University Oxford

M c Master Physics and Astronomy: 16, 000 students -

condensed matter, nuclear physics, AMO/quantum physics

2 M c Master Physics and Astronomy: 16, 000 students - medical, nursing schools Hamilton, Southwestern Ontario (CANADA!) astrophysics, soft condensed matter, biophysics, hard condensed matter, nuclear physics, AMO/quantum physics Southwestern ONTARIO World-Class Vinyards and Wineries! Hamilton Niagara Falls Buffalo Toronto Lake Ontario Rochester New York State

3 Outline: physical requirements ion traps atomic physics identifying, initializing, reading qubits rogue s gallery of gates the future (? ) 1. ion trap as quantum micro-laboratory 2. ion trap as small-scale QC proof of principle demonstrations error correction for Q Communication 3. ion trap as QC technology test-bed 4. ion trap as scalable QC technology

4 Building Quantum Need: 1. qubits Computers: two-level quantum systems superpositions isolated from outside world confined, characterizable, scalable 2. preparation prepare computer in standard start state 3. read-out 4. logic gates controllable interactions with outside world! single- and two-qubits gate sufficient (not nec.!)

5 Proposed Tecnologies: strong, switchable, controllable qubit interactions no other interactions! photons nuclear spins inside liquid-state molecules (NMR)? Josephson junctions nuclear spins of impurities in Si crystal? electron dots electrons floating above liquid He etc... trapped atomic ions

6 Ion traps for quantum computing: store quantum information inside atoms need way to hold atoms in place and protect them ion traps:? Oxford Boulder G. Werth, Progress in Atomic Spectroscopy, H.J. Beyer, H. Kleinpoppen,eds Innsbruck MPQ/Garching

7 Ion Traps: want electric field pointing inwards everywhere positive charges trapped! problem: Gauss Law V 0,Ω U 0 +V 0 +V 0 F radial +V 0 +V 0 2-D: dynamic trapping axial 3-D: axial - static radial - dynamic

8 Ion Traps: V 0,Ω U 0 axial confinement - static! F(r) = (mw z2 /2q) (z 2 /2) w z2 =2aqU 0 /m a ~ 1 (geom.) radial confinement -dynamic! radial axial F(r) = (m/2q) (w r 2 - w z2 /2) (r 2 ) w r 2 = q 2 V 02 /(2mW RF b 4 r 4 ) b ~ 1 (geom.) w r < W RF " m ic ro m o t io n " Innsbruck " s e c u la r" m o t io n Oxford micromotion small, at different freq. 0 t im e

9 Ion Motion in Trap: single ion: like a mass on a spring multiple, cold ions: normal modes - the string moves as one... N ions: N modes per direction centre stretch of mass (COM) 2 ω xx

10 Ion Traps - initial micromachining: 2 DC: U 0 10 V RF: V V 1 cm 0.2 mm F = k z : harmonic oscillator Ω 230 MHz ω HO 10 MHz single ion lifetime: > 10 h. (up to 100 days...)

11 Putting it all together

12 Trapped-Ion QC (Cirac, Zoller('95)) a collection (string) of trapped atomic ions: qubits: (1) internal atomic levels E 1æ 0æ data bus: (2) common-mode motion quantum memory t decoh >> t gate T 2 > 10 min. clocks 1æ 0æ transitory t decoh > t gate

13 Internal (electronic) Qubits: long-lived electronic states: Energy P 3/2 P 1/2 397 nm 422 nm 194 nm S 1/2 866 nm,1092 nm 729 nm 674 nm 282 nm D 5/2 D 3/2 Ca +, Sr +, Ba +, Hg + τ = 1 s τ = 345 ms τ = 90 ms 199 Hg + : Q meas = nm why long-lived? conservation of angular momentum! - regular (dipole) photon can carry one 1 unit of L

14 Internal (electronic) Qubits: ground-state hyperfine levels: Energy S 1/2 P 3/2 P 1/2 γ/2π = 19 MHz τ = 8 ns 313 nm Be + 1 τ > 10,000 yr GHz Be + (313 nm), Mg + (280 nm), Cd + (215 nm) 9 Be + : Q meas = MHz 173 Yb + : Q meas = hyperfine????!! (!!@#%%!!) energy of interaction between electron current and nuclear magnetic moment

15 State preparation: atoms come in thermal equilibrium distribution of levels... must prepare in definite quantum state electronic: optical qubit: kt free! hyperfine - optical pumping polarized light carries angular momentum pumps atom into sub-level of highest (quantized) L unstable excited electronic state lasers 1 Γ 0 : state with most L

16 State preparation: vibrational: laser cooling light carries momentum photon kicks can slow atom re-emission is symmetric State Detection: cycling transition Γ p = hk v ion 1 0 det.

17 Single-qubit logic gates: analog of classical NOT gate - expanded! equivalent to preparation of arb. qubit state apply laser/microwaves! Energy E = hν absorption stimulated emission spontaneous emission (long-lived ex. state...) classical: random hopping (either/or) quantum: flow of wave-function E 0 E 1

18 Single-qubit logic gate: P 1/2 1 Avg # counts t (µsec) optical: 2-photon laser stimulated Raman single-photon transitions strong E-gradients (optical) motional coupling RF frequency diff. coupling controllable strength RF phase stability easier to focus laser than RF!

19 Nobel Sidebar - Ramsey s expt.: superpositions - how do we characterize phase? T/2: create superposition ~ Hadamard t R : phase evolves (Schrodinger) T/2, phase φ: try to undo superposition! t interferometer 0.0 ω t * f

20 Coupling qubit levels: oscillating field induces dipole moment + H I µ E 0 e i(kz - ω Lt) can change electronic level (resonance?) if ion vibrates, interaction strength modulated H I µ E 0 cos(kz 0 cos(ω z t)- ω L t) Quantum: Classically: Hµ I E-½µE 0 (S + + S - )[e i(kz 0 (a + a )- ω L t) 0 Σ m i m J m (kz 0 ) e imω z t e -iω L t + H.C.] sidebands! = hω (S + + S - ) [e iη (a + a ) e i - ωlt + H.C. ] ω z can change motion! (k z 0 n vib ~ [z 0 / λ ] n vib ) (... and resonance...) 0 ω L ω 0

21 CZ Realized: motion-dependent spin transitions (conditional logic) 2π 2π (π phase shift) (π phase shift) 1 m 0 m aux 1 m 0 m 1 e 2π (π phase shift) 2π (π phase shift) π/2 1 m 0 m Controlled-Phase Gate ( 95): 0 e c t c t initially 0 m 0 initially 1 m 0 Initial 1.0 State Final State P(m=1) 0.5 P( ) P(m=1) P( ) Pr[ 0 ] π/2 C-Phase π/2 Controlled-NOT: π/2-pulse phase 0.04 detuning (khz)

22 CZ Realized - a two-ion logic gate! F. Schmidt-Kaler, et al., Nature 422, 408 (2003) two 40 Ca + ions - CZ scheme theoretical: measured: F ~ 70%

23 2 is better than one!...twice! D. Leibfried, et al., Nature 422, 412 (2003) spin-dependent motional Berry s phase oscillating field induces dipole moment + dipole moment interacts with laser field gradient oscillating force 2 lasers whose frequency differs by ω z create walking standing wave which can resonantly drive ion motion drive stretch mode: need different force on each ion to drive can only excite if ions in different electronic levels!

24 2 is better than one!...twice! IF ions in different electronic states, move quantum motional state in closed loop in phase space motional Berry s phase phase shift Ψ Ψ Ψ e iπ/2 Ψ Ψ e iπ/2 Ψ Ψ Ψ = e iπ (e iπ/2 ) ( e iπ/2 ) Ψ p z z flæ fi e ij flæ = controlled-phase + single-qubit rotations (F ~ 97%)

25 and some 2 s are better than others in the lab 2-qubit gates utilize the motion > cough, cough, mumble < higher motional ν gives faster gates shining laser on only one ion! Motional gates (Mølmer-Sørensen, Milburn, etc.) can be done illuminating all ions! - keep ν high fast motional gates - with expt. gate, can have different illuminations single-qubit operations can be done with weak trap the accordion quantum computer!

26 problem: Scaling up: as N ions : ion string gets heavier gates get slower! more motional modes greater noise 1. optical multiplexing: fibre to other cavity/qubits cavity mode (spont. Raman) laser (stim. Raman) R. DeVoe, PRA 58, 910 (98) J.I. Cirac, et al. PRL 78, 3221 (97)

, quant-ph/0202112 Excitation Prob. red shift blue shift -0.2 0.2 Excitation Laser Det.

27 Solutions (1) - optical: MPQ, Garching (Ca + ): 4 2 S 1/2 4 2 P 1/2 G.R. Guthöhrlein, et al., Nature 414 (01) res. λ/10 U. Innsbruck (Ca + ): 4 2 S 1/2 3 2 D 5/2 A.B. Mundt, et al., quant-ph/ Excitation Prob. red shift blue shift Excitation Laser Det. (MHz) sweep PZT Doppler shift P ex. > 0.5 coherent positioning: node/antinode res. λ/100 differential coupling to motional sidebands

28 problem: Scaling up: as N ions : ion string gets heavier gates get slower! more motional modes greater noise 2. quantum CCD: segmented electrodes accumulator quantum CCD Wineland, et al. J. Res. NIST 103, 259 (98) D. Kielpinski, et al. Nature 417, 709 (02) memory register

Solutions (2) - physical multiplexing: Boulder, data to be

interferometer: 400 µm 360 µm (2) separating ions: no

29 Solutions (2) - physical multiplexing: Boulder, data to be published transporting ions between traps: (1) Ramsey interferometer: 400 µm 360 µm (2) separating ions: no transport: 96.8 ± 0.3% contrast line triggered: 96.6 ± 0.5% contrast! 60 Hz fields... spin echo 96% contrast n=200 quanta (2.9 MHz) for 10 ms sep. time (separation electrode too wide!) 95% sep. eff. (5000 shots)

30 Solutions (2) - physical alumina silicon multiplexing: gold foil traps: silicon traps: easily micro-machined, smooth

31 Ion Trap QC: Wither thou?......ion-trap QC progress: single-qubit logic gates ( 40 s) (>98% fidelity) single-ion 2-qubit logic gate ( 95) (80% fidelity) 2-ion 2-qubit logic gates 2 (80% / 97% fidelity) state preparation (fidelity > 98%) spin qubit t / t gate > 1000* motional data bus/qubit heating NIST< 1 /(4 ms), t / t gate ~ 100 1/(10 ms) - IBM, 1/(190 ms) - Innsbruck NIST Boulder, MPQ Garching, IBM Almaden, U. Innsbruck, Oxford, U. Michigan, McMaster U

32 Quantum Computing: Wither thou?... the present reality... the dream...

33 Quantum Computing: Wither thou?... but, oh! the road...

Ion trap quantum processor

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