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Quantum Computing with Trapped Atomic Ions

Oxford. Ann Arbor. Innsbruck. Quantum Computing with Trapped Atomic Ions. APS March Meeting - Montr éal: March 21 , 2004 Brian King Dept. Physics and Astronomy, McMaster University http://physserv.mcmaster.ca/~kingb/King_B_h.html. Garching. Boulder. Outline:.

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Quantum Computing with Trapped Atomic Ions

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  1. Oxford Ann Arbor Innsbruck Quantum Computing with Trapped Atomic Ions APS March Meeting - Montréal: March 21, 2004 Brian King Dept. Physics and Astronomy, McMaster University http://physserv.mcmaster.ca/~kingb/King_B_h.html Garching Boulder

  2. Outline: • building “quantum computers” • overview of ion trap quantum information processor • ion trapping • initialization and detection • single-qubit gates (internal) • coupling internal and external qubits • directions for the future...

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

  4. Why atomic qubits? • unparallelled persistence of quantum superposition • atomic clocks - accuracy, precision control over quantum states - internal and external • BEC, Fermi degeneracy (controllable), Mott insulator transition, quantum squeezing, quantum state engineering... atomic ions - demonstration of building blocks for scalable* quantum “computer” architecture * the Devil is in the details... * the Devil is in the details...

  5. |1 E |0 “data bus:” (2) common-mode motion |0 • transitory • tdecoh  tgate • 10  2  10  3 |1 Trapped-Ion QC (Cirac, Zoller('95)) • a collection (string) of trapped atomic ions: • qubits: (1) internal atomic levels • quantum memory • tdecohÀ tgate • T2 > 10 min. • clocks • accuracy, stability > 1/1015

  6. laser laser laser laser laser How it works... A quantum logic gate between 2 different ions: • prepare qubits using single-qubit gates • map qubit i state to motion with lasers • 2-qubit gate between motion and ion j • put information from motion “back into” ion i i j

  7. in 3-D: 0 0 0 +V +V +V +V +V +V +V 0 0 0 0 e.g. z: F F F assume: F F F F 0 0 0 +V +V +V 0 0 0 0 +V +V +V +V Dynamical RF trapping: • want to confine charged atoms  E fields! • Eherenfest/Gauss  can’t use static fields  use oscillating fields!

  8. average over 1 RF period: Quantum Motion: Dynamical RF trapping: full solution: Mathieu equation (same results...) same results: • quantum harmonic oscillator • wavepackets “breathe” at T

  9. radial axial Oxford Innsbruck MPQ/Garching Linear Ion Traps for QC: axial confinement - static! U0 U0 V0,W V0,W F(z) = (mwz2/2q) (z2/2) • wz2=2aqU0/m a ~ 1 (geom.) radial confinement -dynamic! F(r) = (m/2q) (wr2 - wz2/2) (r2) • wr2 = q2V02/(2mWRFb4r4)b ~ 1 (geom.) • wr < WRF micromotion small, at different freq.

  10. 2” 0.2 mm 1 cm Ion Traps - initial micromachining: • DC: U0 ≈ 10 V • RF: V0 ≈ 750 V •  ≈ 230 MHz wHO ≈ 10 MHz • pressure < 2×1011 torr • single ion lifetime: > 10 h. (cryogenic  up to 100 days...)

  11. multiple, cold ions: • “normal modes” - the string moves as one... N ions: N modes per direction centre of mass (COM) wx “stretch” 2 wx Ion Motion in Trap: • single ion: • like a mass on a spring

  12. Dirty little secrets - motional heating: • after cooling to the ground state of motion, the ion heats back up! • timescale for motional manipulation ~ 10 s • |0  |1 in ~ 100 s (1998...) • motion only sensitive to noise spectrum near mot • fluctuating patch potentials? • RF-assisted tunnelling? • heating scales strongly with trap size ~ 10  4 • heating seems related to atom source  shield trap! Q.A. Turchette et al. Phys. Rev. A 62, 053807, 2000. • 21st century: NIST < 1 /(4 ms) IBM: 1/(10 ms) Innsbruck: 1/(190 ms) plus sympathetic cooling (multi-species...)

  13. P3/2 P1/2 866 nm,1092 nm 397 nm D5/2 Energy t = 1 s 422 nm t = 345 ms 194 nm D3/2 t = 90 ms 729 nm 674 nm 282 nm S1/2 Internal-State Qubits: • long-lived electronic states: Ca+, Sr+, Ba+, Hg+ 199Hg+: Qmeas = 1.6·1014 @ 282 nm

  14. Energy t > 10,000 yr Internal-State Qubits: ground-state hyperfine levels: Be+ (313 nm), Mg+(280 nm), Cd+(215 nm) P3/2 g/2p = 19 MHz t = 8 ns P1/2 313 nm 1 9Be+: Qmeas = 3.4·1011 @ 303 MHz 173Yb+: Qmeas = 1.5·1013 1.25 GHz S1/2 0 Be+

  15. State Detection: cycling transition - excited state decays back to |0 G 1 det. 0 State preparation: • electronic: • optical qubit - kT free! • hyperfine qubit: optical pumping • vibrational: Doppler & sideband laser cooling

  16. 2P1/2 optical: laser • single-photon • requires L¿wmot 2-photon stimulated Raman transitions D • strong E-gradients (optical) • motional coupling • RF frequency diff. coupling • controllable strength • RF phase stability |0 + |1 |0 |1 0 10 20 30 0 10 20 30 0 10 20 30 10 8 Avg # counts 6 4 2 0 1 0 1 2 3 4 5 0 t (sec) Single-qubit logic gate:

  17. + Classically:m· E0SmimJm(kz0) eimwzt e-iwLt sidebands! wz wL – w0 0 Coupling qubit levels: • oscillating field induces dipole moment • HI m· E0 ei(kz - wLt) • can change electronic level (resonance?) • if ion vibrates, interaction strength modulated • HI  m· E0 ei(kz0 cos(wzt)- wLt) Quantum: HI  ½mE0 (S+ + S-) ei(kz0 (a + a†)- wLt) = W(S+ + S-) ei(h (a + a†)- wLt) • can change motion! (k z0nvib ~ [z0 / l]nvib) (... and resonance...)

  18. Controlled-Phase Gate (‘95): p/2·C-Phase·p/2 º Controlled-NOT: p/2 2p (p phase shift) 2p (p phase shift) 2p (p phase shift) 2p (p phase shift) 2p (p phase shift) Initial StateFinal State P(m=1) P() P(m=1) P() 0.02 0.01 0.09 0.16 0.03 0.99 0.04 0.88 0.92 0.05 0.77 0.88 0.94 0.98 0.88 0.19 c t c’t’ | 0ñ| 0ñ| 0ñ| 0ñ | 0ñ| 1ñ| 0ñ| 1ñ | 1ñ| 0ñ| 1ñ| 0ñ | 1ñ| 1ñ-| 1ñ| 1ñ · initially |0ñm|0ñ · initially |1ñm|0ñ Pr[|0ñ] p/2-pulse detuning (kHz) phase CZ Realized: • motion-dependent spin transitions (conditional logic) |1ñm |ñº|1ñe |0ñm |1ñm |¯ñº|0ñe |1ñm |auxñ |0ñm |0ñm

  19. 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 CZ Realized - a two-ion logic gate! F. Schmidt-Kaler, et al., Nature 422, 408 (2003) • two 40Ca+ ions - CZ scheme • but no |aux needed... theoretical: measured:F ~ 70%

  20. coupling strength ~n> ! • 2p for n>=1 but  2p for n>=2 2p (p phase shift) 2p (p phase shift) 2p? (p phase shift) CZ Realized - a two-ion logic gate! • doesn’t use |aux - uses clever NMR trick! |2ñm |1ñm |ñº|1ñe |0ñm use (p,x) (p/2,y) (p,x) (p/2,y) |1ñm |¯ñº|0ñe |0ñm

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

  22. Excitation Prob. -0.2 0.2 Excitation Laser Det. (MHz) Solutions (1) - optical: • MPQ, Garching (Ca+): 4 2S1/2«4 2P1/2 G.R. Guthöhrlein, et al., Nature 414 (01) res. » l/10 • U. Innsbruck (Ca+): 4 2S1/2«3 2D5/2 A.B. Mundt, et al., quant-ph/0202112 red shift blue shift • sweep PZT Þ Doppler shift • Pex. > 0.5 Þcoherent • positioning: node/antinode • res. » l/100 differential coupling to motional sidebands

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

  24. 400 mm 360 mm Solutions (2) - physical multiplexing: M. Rowe, et al., Quantum Information and Computation 1, x (‘01). • transporting ions between traps: (1) Ramsey interferometer: no transport: 96.8 ± 0.3% contrast line triggered: 96.6 ± 0.5% contrast! • 60 Hz fields... “spin echo” 96% contrast (2) separating ions: Dn=200 quanta (2.9 MHz) for 10 ms sep. time (separation electrode too wide!) 95% sep. eff. (5000 shots)

  25. alumina silicon Solutions (2) - physical multiplexing: • “gold foil” traps: silicon traps: easily micro-machined, smooth

  26. Ion Trap QC: Wither thou?... • single-qubit logic gates (´40’s) (>98% fidelity) • single-ion 2-qubit logic gate (´95) (80% fidelity) C. Monroe et al. Phys. Rev. Lett. 75, 4714 (‘95). • 2-ion 2-qubit logic gates  2(80% / 97% fidelity) Gulde et al. Nature 422, 408 (‘03). Leibfried et al. Nature 422, 412 (‘03). • Deutsch-Jozsa algorithm Gulde et al. Nature 421, 48 (‘03). • state preparation (fidelity > 98%) • spin qubit: t / tgate > 1000* • motional data bus/qubit • heating < 1/(4, 10, 190 ms) (NIST, IBM, Innsbruck) NIST Boulder, MPQ, IBM Almaden, U. Innsbruck, Oxford, U. Michigan, McMaster http://physserv.mcmaster.ca/~kingb/King_B_h.html

  27. References: • Cirac & Zoller: “New Frontiers in Quantum Information With Atoms and Ions,” Physics Today57, #3, 38 (March '04). • Steane: Appl. Phys.B 64 , 623 ('97). • Ghosh: Ion Traps, (Clarendon Press, '97), ISBN: 0198539959. • Leibfried et al.: “Quantum dynamics of single trapped ions,” Rev. Mod. Phys.75, 281 ('03). • Wineland, et al.: “Quantum information processing with trapped ions,” Phil. Trans. Royal Soc. London A361, 1349, ('03). • Wineland, et al., “Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions”, J. Research NIST103, 259 ('98). • Monroe, et al.: “Experimental Primer on the Trapped Ion Quantum Computer,” Forschr. Physik46, 363 ('98). http://jilawww.colorado.edu/pubs/recent_theses/ D. Kielpinski, “Entanglement and Decoherence in a Trapped-Ion Quantum Register” B.E. King, “Quantum State Engineering and Information Processing withTrapped Ions”

  28. interferometer * f N tR: phase evolves (Schrodinger) T/2, phase f: try to undo superposition! T/2: create superposition w t Nobel Sidebar - Ramsey’s expt.: superpositions - how do we characterize phase? t

  29. 2 lasers with dwL  wz create “walking standing wave” which can resonantly drive ion motion P1/2 D S1/2 2 is better than one!... D. Leibfried, et al., Nature 422, 412 (2003) • spin-dependent motional Berry’s phase • 2 lasers with dwL  0 create “standing wave” • dipole force

  30. D(b) p D(a) D(a) D(b) = ei Im(ab*)D(a+b) x geometric Berry’s phase! 2 is better than one!... • resonant oscillating force = displacement operator in phase space |a| set by strength of force phase set by phase between motion and lasers

  31. pz P1/2 different coupling strengths z |  eij | 1 S1/2 0 2 is better than one!... • “stretch mode:” • need different force on each ion to drive • can only excite if ions in different electronic levels! • move ions in closed loop in phase space “walking standing wave” has different strengths for ,

  32. 2 is better than one!... • IF ions in different electronic states, move quantum motional state in closed loop in phase space  motional “Berry’s phase”  phase shift Y  Y   Y   eip/2 Y   Y   eip/2  Y   Y   Y  = e-ip(eip/2 ) ( eip/2) Y  • controlled-Phase + single-qubit rotations (F ~ 97%)

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

  34. Coupling qubit levels: • laser-ion interaction: messy details: in interaction picture: rotating-wave approximation: expand exponential:

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