Quantum Computing with Trapped Atomic Ions

APS March Meeting - Montréal: March 21, 2004Brian King

Dept. Physics and Astronomy, McMaster University

http://physserv.mcmaster.ca/~kingb/King_B_h.html

Innsbruck Oxford

Ann Arbor

Boulder

Garching

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:

1. qubits• 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.!)

* the Devil is in the details...

Why atomic qubits?

• unparallelled persistence of quantum superposition

• control over quantum states - internal and external

• atomic clocks - accuracy, precision

• 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...

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

• “data bus:” (2) common-mode motion

• transitory• tdecoh tgate

• 10 2 10 3

E|0|1

|1|0

2. map qubit i state to motion with lasers

How it works...

A quantum logic gate between 2 different ions:1. prepare qubits using single-qubit gates

3. 2-qubit gate between motion and ion j4. put information from motion “back into” ion i

i j

laser laserlaser laserlaser

Dynamical RF trapping:

• want to confine charged atoms E fields!• Eherenfest/Gauss can’t use static fields

+V

+V

0

0

F

0

0

+V

+V

F

+V

+V

0

0

F

0

0

+V

+V

F

+V

+V

0

0

F

0

0

+V

+V

F

+V

+V

0

0

F

use oscillating fields!

• in 3-D:

• 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:ra

dial

axial

V0,U0

• axial confinement - static!

(z) = (mz2/2q) (z2/2)

• z2=2qU0/m ~ 1 (geom.)

-1.0

-0.5

0.0

0.5

1.0

po

siti

on

1086420 time

"secular" motion

"micromotion"

• micromotion small, at different freq.

• radial confinement -dynamic!

(r) = (m/2q) (r2 z

2/2) (r2)• r

2 = q2V02/(2mRF4r4) ~ 1 (geom.)

• r < RF

V0,U0

Innsbruck MPQ/GarchingOxford

Ion Traps - initial micromachining:

2”

1 cm 0.2 mm

• DC: U0 ≈ 10 V• RF: V0 ≈ 750 V ≈ 230 MHz

HO ≈ 10 MHz

• pressure < 2×1011 torr• single ion lifetime: > 10

h.(cryogenic up to 100

days...)

centre of mass (COM)x

“stretch”x

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

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...)

1. motion only sensitive to noise spectrum near mot

2. heating scales strongly with trap size ~ 10 4

• fluctuating patch potentials?• RF-assisted tunnelling?

3. heating seems related to atom source shield trap!

• 21st century: NIST < 1 /(4 ms)IBM: 1/(10 ms)Innsbruck: 1/(190 ms)

Q.A. Turchette et al. Phys. Rev. A 62, 053807, 2000.

• plus sympathetic cooling (multi-species...)

Internal-State Qubits:

• long-lived electronic states:

Ca+, Sr+, Ba+,Hg+

S1/2

D3/2

D5/2

P1/2

P3/2

397 nm

866 nm,1092 nm

422 nm194 nm

729 nm674 nm282 nm

= 1 s = 345 ms = 90 ms

199Hg+: Qmeas = 1.6·1014

@ 282 nm

Ene

rgy

Internal-State Qubits:

• ground-state hyperfine levels:

= 19 MHz= 8 ns

0

1

Be+ (313 nm),Mg+ (280 nm),Cd+ (215 nm)

9Be+: Qmeas = 3.4·1011

@ 303 MHz173Yb+: Qmeas = 1.5·1013

P1/2

P3/2

Be+

> 10,000 yr

313 nm

1.25 GHzS1/2

Ene

rgy

State Detection:

1

det.0

• cycling transition - excited state decays back to |0

State preparation:

• electronic:

•optical qubit - kT free!

•hyperfine qubit: optical pumping

• vibrational: Doppler & sideband laser cooling

optical: laser• single-photon• requires L ¿ mot

01

2P1/2

• strong E-gradients (optical)• motional coupling

• RF frequency diff. coupling• controllable strength• RF phase stability

2-photon stimulated Raman transitions

Single-qubit logic gate:

1086420

543210t (sec)

Avg

# c

oun

ts3020100

|0

3020100

|0 + |1

3020100

|1

Classically: · E0 m im Jm(kz0) eimzt e-iLt

sidebands!

L – 00

z

Quantum: HI ½E0 (S+ + S-) ei(kz0 (a + a†)- Lt)

= (S+ + S-) ei( (a + a†)- Lt)

• can change motion!(k z0nvib ~ [z0 / ]nvib(... and resonance...)

Coupling qubit levels:

• oscillating field induces dipole moment

• HI · E0 ei(kz - Lt)

+

• can change electronic level(resonance?)

• if ion vibrates, interaction strength modulated

• HI · E0 ei(kz0 cos(zt)- Lt)

CZ Realized:

• motion-dependent spin transitions (conditional logic)

|1m|0m

|1m|0m ||e

|1m|0m

|aux

( phase shift)

( phase shift)

( phase shift)

( phase shift)

c t c’t’| | | | | | | | | | | | | | | |

Controlled-Phase Gate (‘95):

-30 -20 -10 0 10 20 300.0

0.5

1.0

/2-pulse detuning (kHz)

initially |0m|0 initially |1m|0

Pr[|0]

phase

( phase shift)

/2

Initial State Final StateP(m=1) P() P(m=1) P()

0.02 0.01 0.09 0.160.03 0.99 0.04 0.880.92 0.05 0.77 0.880.94 0.98 0.88 0.19

/2 C-Phase /2 Controlled-NOT:

||e

CZ Realized - a two-ion logic gate!

• two 40Ca+ ions - CZ scheme

• but no |aux needed...

F. Schmidt-Kaler, et al., Nature 422, 408 (2003)

1 0 000 1 000 0 010 0 10

theoretical: measured: F ~ 70%

CZ Realized - a two-ion logic gate!

• doesn’t use |aux - uses clever NMR trick!

|1m|0m

|1m|0m ||e

( phase shift)

||e

|2m

( phase shift)

( phase shift)

coupling strength ~n> !•2 for n>=1 but 2 for n>=2

use (,x) (/2,y) (,x) (/2,y)

Scaling up:

• problem:• as Nions :

• ion string gets heavier gates get slower!• more motional modes greater “noise”

fibre

R. DeVoe, PRA 58, 910 (98)J.I. Cirac, et al. PRL 78, 3221 (97)

1. optical multiplexing:

laser(stim. Raman)

to other cavity/qubits

cavity mode(spont. Raman)

Solutions (1) - optical:

• MPQ, Garching (Ca+): 4 2S1/24 2P1/2G.R. Guthöhrlein, et al., Nature 414 (01)

res. /10

• U. Innsbruck (Ca+): 4 2S1/23 2D5/2A.B. Mundt, et al., quant-ph/0202112

• sweep PZT Doppler shift• Pex. > 0.5 coherent

Excitation Laser Det. (MHz)-0.2 0.2

Exc

itat

ion

Pro

b.

blueshift

redshift

• positioning:node/antinode• res. /100

• differential coupling to motional sidebands

Scaling up:

segmented electrodes

accumulator

memory register

• problem:• as Nions :

• ion string gets heavier gates get slower!• more motional modes greater “noise”

2. “quantum CCD:”

“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:

no transport: 96.8 ± 0.3% contrastline triggered: 96.6 ± 0.5% contrast!

• 60 Hz fields...

360 m

400 m

(1) Ramsey interferometer:

“spin echo”96% contrast

95% sep. eff. (5000 shots)

n=200 quanta (2.9 MHz) for 10 ms sep. time(separation electrode too wide!)

(2) separating ions:

Solutions (2) - physical multiplexing:

• “gold foil” traps:

• silicon traps:

alumina silicon

• easily micro-machined, smooth

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 algorithmGulde 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)

http://physserv.mcmaster.ca/~kingb/King_B_h.htmlNIST Boulder, MPQ, IBM Almaden, U. Innsbruck, Oxford, U. Michigan, McMaster

References:

1. Cirac & Zoller: “New Frontiers in Quantum Information With Atoms and Ions,” Physics Today 57, #3, 38 (March '04).

2. Steane: Appl. Phys. B 64 , 623 ('97).

3. Ghosh: Ion Traps, (Clarendon Press, '97), ISBN: 0198539959.

4. Leibfried et al.: “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281 ('03).

5. Wineland, et al.: “Quantum information processing with trapped ions,” Phil. Trans. Royal Soc. London A 361, 1349, ('03).

6. Wineland, et al., “Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions”, J. Research NIST 103, 259 ('98).

7. 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?

t

T/2:create

superposition

T/2, phase :try to undo

superposition!

tR:phase evolves(Schrodinger)

*

• interferometer

-30 -20 -10 0 10 20 300.0

0.5

1.0N

t

2 is better than one!...

• spin-dependent motional Berry’s phase

D. Leibfried, et al., Nature 422, 412 (2003)

• 2 lasers with L create “standing wave”• dipole force

P1/2

S1/2

• 2 lasers with L z create “walking standing wave” which can resonantly drive ion motion

2 is better than one!...

• resonant oscillating force = displacement operator in phase space

x

p

D()

D()•|| set by strength of force•phase set by phase between motion and lasers

D() D() = ei Im(*)D()

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

0

1

P1/2

differentcouplingstrengths

S1/2

“walking standing wave” has different strengths for ,

z

pz

| ei |

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

ei/2 ei/2

= ei (ei/2 ) ( ei/2) •controlled-Phase + single-qubit rotations (F ~ 97%)

and some 2’s are “better” than others

• 2-qubit gates utilize the motion• > cough, cough, mumble…<

• higher motional gives faster gates

shining laser on only one ion!• Motional gates (Mølmer-Sørensen, Milburn, etc.) can

be done illuminating all ions!

- keep high fast motional gates

- with expt. gate, can have different illuminations

• single-qubit operations can be done with weak trap• the “accordion quantum computer!”

…in the lab…

Coupling qubit levels:

• laser-ion interaction: messy details:

• in interaction picture:

• rotating-wave approximation:

• expand exponential: