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

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...

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.!)

unparallelled persistence of quantum superposition 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...

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

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 laser laser laser laser i j

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

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

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! radial axial F(r) = (m/2q) (wr2 - wz2/2) (r2) wr2 = q2V02/(2mWRFb4r4) b ~ 1 (geom.) wr < WRF Innsbruck MPQ/Garching Oxford micromotion small, at different freq.

Ion Traps - initial micromachining: 2” 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...) 1 cm 0.2 mm

“normal modes” - the string moves as one... 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 “stretch” 2 wx centre of mass (COM) wx

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...)

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

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

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

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

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...) Classically: m · E0 Sm im Jm(kz0) eimwzt e-iwLt sidebands! wL – w0 wz

CZ Realized: motion-dependent spin transitions (conditional logic) c t c’t’ | 0ñ| 0ñ | 0ñ| 0ñ | 0ñ| 1ñ | 0ñ| 1ñ | 1ñ| 0ñ | 1ñ| 0ñ | 1ñ| 1ñ -| 1ñ| 1ñ Controlled-Phase Gate (‘95): p/2-pulse detuning (kHz) · initially |0ñm|0ñ · initially |1ñm|0ñ Pr[|0ñ] phase 2p (p phase shift) p/2 |1ñm 2p (p phase shift) |ñº |1ñe |0ñm 2p (p phase shift) Initial State Final 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 p/2 · C-Phase · p/2 º Controlled-NOT: |1ñm |¯ñº |0ñe 2p (p phase shift) |1ñm |auxñ |0ñm 2p (p phase shift) |0ñm

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% 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0

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

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

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 Excitation Laser Det. (MHz) -0.2 0.2 Excitation Prob. red shift blue shift sweep PZT Þ Doppler shift Pex. > 0.5 Þ coherent positioning: node/antinode res. » l/100 differential coupling to motional sidebands

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

Solutions (2) - physical multiplexing: M. Rowe, et al., Quantum Information and Computation 1, x (‘01). transporting ions between traps: (1) Ramsey interferometer: 360 mm 400 mm 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)

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

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

References: Cirac & Zoller: “New Frontiers in Quantum Information With Atoms and Ions,” Physics Today 57, #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 A 361, 1349, ('03). Wineland, et al., “Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions”, J. Research NIST 103, 259 ('98). Monroe, et al.: “Experimental Primer on the Trapped Ion Quantum Computer,” Forschr. Physik 46, 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”

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

spin-dependent motional Berry’s phase 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 P1/2 S1/2 D 2 lasers with dwL  wz create “walking standing wave” which can resonantly drive ion motion

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 D(b) x p D(a) D(a) D(b) = ei Im(ab*)D(a+b) geometric Berry’s phase!

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 , 0 1 P1/2 different coupling strengths S1/2 z pz |  eij |

 motional “Berry’s phase”  phase shift 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%)

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!”

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