1. 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

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. 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...

5. 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

6. 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

7. 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:

8. 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

9. 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.

10. 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

11. “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

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 ( )
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, , 2000.
21st century: NIST < 1 /(4 ms)
IBM: 1/(10 ms)
Innsbruck: 1/(190 ms)
plus sympathetic cooling (multi-species...)

13. 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

14. 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+

15. 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

16. 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

17. 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

18. 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()
p/2 · C-Phase · p/2
º Controlled-NOT:
|1ñm
|¯ñº |0ñe 2p
(p phase shift)
|1ñm
|auxñ
|0ñm 2p
(p phase shift)
|0ñm

19. 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. 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

21. 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)

22. 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/
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

23. 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)

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

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

27. 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:
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).
D. Kielpinski, “Entanglement and Decoherence in a Trapped-Ion Quantum Register”
B.E. King, “Quantum State Engineering and Information Processing withTrapped Ions”

28. 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

29. 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

30. 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!

31. 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 |

32.  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%)

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: