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Quantum Computing with Trapped Atomic Ions

Explore the latest developments in building quantum computers using atomic qubits, ion trapping, logic gates, and coupling qubits for future applications. Understand the potential of atomic ions in quantum state engineering.

amery-herman
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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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