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Mark Acton (grad)Kathy-Anne Brickman (grad)Louis Deslauriers (grad)Patricia Lee (grad)Martin Madsen (grad)David Moehring (grad)Steve Olmschenk (grad)Daniel Stick (grad)
http://iontrap.physics.lsa.umich.edu/
US Advanced Researchand Development Activity
US Army Research Office
US National Security Agency
National ScienceFoundation
FOCUS FOCUS Center
Boris Blinov (postdoc)Paul Haljan (postdoc)Winfried Hensinger (postdoc)Chitra Rangan (postdoc/theory – to U. Windsor)
Luming Duan (Prof., UM)Jim Rabchuk (Visiting Prof., West. Illinois Univ.)
David Hucul (undergrad)Rudy Kohn (undergrad)Mark Yeo (undergrad)
NSF
Trapped Atomic Ions IQuantum computing and motional quantum gates
Christopher MonroeFOCUS Center & Department of PhysicsUniversity of Michigan
“When we get to the very, very small world – say circuits of seven atoms - we have a lot of new things that would happen that represent completely new opportunities for design. Atoms on a small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics…”
“There's Plenty of Room at the Bottom”(1959 APS annual meeting)
Richard Feynman
A quantum computer hosts quantum bits that can store superpositions of 0 and 1
classical bit: 0 or 1 quantum bit: |0 + |1
Benioff (1980)Feynman (1982)
examples of “qubits”:
N
S
N
Sh
V
H
atomsparticlespins
photons
GOOD NEWS…quantum parallel processing on 2N inputs
Example: N=3 qubits
= a 0 |000 + a 1 |001 + a 2 |010 + a 3 |011 a 4 |100 + a 5 |101 + a 6 |110 + a 7 |111
f(x)
…BAD NEWS…Measurement gives random result
e.g., |101 f(x)
depends on all inputs
quantumlogic gates
|0 |0 |0 |0|0 |1 |0 |1|1 |0 |1 |1|1 |1 |1 |0
e.g., (|0 + |1)|0 |0|0 + |1|1 quantumXOR gate:
superposition entanglement
|0 |0 + |1|1 |1 |0
quantumNOT gate:
…GOOD NEWS!quantum interference
Key resource: Quantum Entanglement
• not just a “choice of basis” e.g. vs. |0,0
must be able to access subsystems individually (see Bell )
= ( + )( + ) Contrast = 2|| = 0.5
( i)( i) = or
( i)( i)
…
Contrast = 2|| = 0.5
= + Contrast = 2|| = 1
• not hard to qualify (entanglement thresholds)
ideal:
1 = | + |
2 = | + | + | + | + | + |
• very hard to quantify (esp. mixed states)
Classical Information: S(AB) S(A) + S(B)
Quantum Information: S(AB) < S(A) + S(B) possible!
Information Entropy
Quantum computer hardware requirements
1. Must make states like
|000…0 + |111…1
2. Must measure state with high efficiency
•strong coupling to environment
• strong coupling between qubits• weak coupling to environment
xx+
see E. Schrödinger (1935)
N qubits controlledcoupling
… to >99% accuracy*
* provided things have been done right
Quantum Information and Atomic Physics
N
i
jiN
jiij
ii tgtH
1
)()(
1,
)( ˆˆ)(ˆ)(2
1
0.3 mm
199Hg+
J. Bergquist, NIST
AarhusBoulder (NIST)Munich (MPQ)HamburgInnsbruck
Los AlamosMcMasterMichiganOxfordTeddington (NPL)
Ion Trap QC Groups:
Trapped Atomic Ions
J. Bergquist (NIST)
|
|
qubit storedinside eachtrapped ion
2 Cd+ ions
S
P
D
|
|
Ca+, Sr+, Ba+, Yb+
optical(1015 Hz)
1 sec
Energy
Atomic Ion Internal Energy Levels (think: HYDROGEN)
S
P
||
Be+, Mg+, Hg+, Cd+, Zn+
microwave(1010 Hz)
hyperfine qubit levels
State |
N
S
N
S
Hyperfine Structure: States of relative electron/nuclear spin
State |
S
N
N
S
111Cd+ atomic structure
1,11,01,-1
0,0
=215nm
2S1/2
2P3/22,22,1
14.53 GHz
|
|
magnetic insensitivequbit (to 2nd order) (400 Hz/G2)·B·B
(1400000 Hz/G)·B
1,11,01,-1
0,0
=215nm
2S1/2
2P3/22,22,1
14.53 GHz
|
|
/2 = 50 MHz
“bright”
# photons collected in 100 s0 5 10 15 20 25
0
1
Pro
babili
ty111Cd+ qubit measurement
1,11,01,-1
0,0
=215nm
2S1/2
2P3/22,22,1
14.53 GHz
|
|
/2 = 50 MHz
99.7% detectionefficiency
“dark”
0 5 10 15 20 25
Pro
babili
ty
# photons collected in 100 s
0
1
111Cd+ qubit measurement
1,11,01,-1
0,0
=215nm
2S1/2
2P3/22,22,1
14.53 GHz
|
|
111Cd+ qubit manipulation: microwaves
microwaves
coupling rate: g
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1,11,01,-1
0,0
1,11,01,-1
0,0
Microwave Rabi FloppingP
rob(
10|0
0)P
rob(
11|0
0)
prepare00
wavesmeasure
fluorescence(bright or dark)
: :
sweep g 10100kHz
(s)
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200 250 300 350 400
prepare00
wavesmeasure
fluorescence(bright or dark)
: :increment
“Single shot” Rabi Flopping
Pro
b(10
|00)
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Time (ms)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1,11,01,-1
0,0
1,11,01,-1
0,0
Microwave Ramsey InteferometryP
rob(|)
Pro
b(|)
prepare00
waves measurefluorescence
: :
sweep
/2 /2
1,11,01,-1
0,0
=215nm
2S1/2
2P3/22,22,1
14.53 GHz
|
|
/2 = 50 MHz
111Cd+ qubit manipulation: optical Raman transitions
/2 0.1-1 THz
coherent coupling rate (good):gR = g1g2/
direct coupling to P (bad): Rdec = g1g2/
want small (but <FS!)
0.3 mm J. Bergquist, NIST
Thanks: R. Blatt, Univ. Innsbruck
40Ca+
logical |0m
logical |1m
Another Qubit: The quantized motion of a single mode of oscillation
harmonic motion of a collective single mode described byquantum states |nm = |0m, |1m, |2m,..., where E = ħ(n+½)PHONONS: FORMALLY EQUIVALENT TO PHOTONS
motional “data-bus” quantum bit spans |nm = |0m and |1m
•••
01
2
Coupling (internal) qubits to (external) bus qubit
radiation tuned to 0
| | | | |
| | | | |
|
|
|1m
|0m
|1m
|0m
•••
01
2
•••
01
2S1/2
P3/2
|
|
excitation on 1st lower (“red”) motional sideband (n=0)
~ few MHz
•••
01
2
•••
01
2S1/2
P3/2
|
|
excitation on 1st lower (“red”) motional sideband (n=0)
Mapping: (| + |) |0m | (|0m + |1m)
•••
01
2
•••
01
2S1/2
P3/2
|
|
•••
01
2
•••
01
2S1/2
P3/2
|
|
Mapping: (| + |) |0m | (|0m + |1m)
•••
01
2
•••
01
2S1/2
P3/2
|
|
•••
01
2
•••
01
2S1/2
P3/2
|
|
Spin-motion coupling: some math
)21( aa
)ˆ(ˆˆ2
1
2
ˆˆ 22
2
0 xExmm
pH z
interaction frame; “rotating wave approximation”
)ˆˆ( ˆˆ tixiktixik eegH
= L 0 = detuningk = 2 = wavenumber
)(ˆ 0titi eaaexx
mx
20
x0
))(ˆˆ(2
ˆˆ00
tixiktixik LL eeE
frequency ofapplied radiation
tieaaeikxtieaaeikx titititi
eegH
)()( 00 ˆˆ
stationary terms arise in H at particular values of
“Lamb-Dicke Limit”110 nkx
,n|H0|,n = ħg)ˆˆ(0 gH = 0
“CARRIER”
110 nkx
,n1|H0|,n = ħg n)ˆˆ)(( 01
aakxgH = +
“1ST BLUE SIDEBAND”
110 nkx
,n1|H+1|,n = ħg 1n)ˆˆ)(( 01 aakxgH
= “1ST RED SIDEBAND”
110 nkx
DopplerCooling
Raman spectrum of single 111Cd+ ion (start in |)
|
|
n=0
0.0
0.5
1.0
P
“Red”Sideband
|,n |,n+1
“Blue”sideband
|,n |,n-1
-3.6 +3.6 (MHz)
nnblue
nnred
ngtPI
ngtPI
)(sin
)1(sin
2
2
sideband strengths:
n
n n
nP
1
thermaloccupationdistribution
Thermometry: 1
n
n
I
I
red
blue
n 6
n
|
|
Raman Sideband Laser-Cooling
.
n1
. n1
|
|
n1
stimulated Raman ~-pulse on blue sideband
spontaneous Ramanrecycling
.
.
n=-1 n recoil/trap << 1
DopplerCooling
Raman spectrum of single 111Cd+ ion (3.6 MHz trap)
|
|
n=0
L. Deslauriers et al., Phys. Rev. A 70, 043408 (2004)
Doppler+ Raman Cooling
|
|
n=0
P
0.5
0.0
1.0
n < 0.05
0.0
0.5
1.0
P
“Red”Sideband
|,n |,n+1
“Blue”sideband
|,n |,n-1
n 6
-3.6 +3.6
-3.6 +3.6
(MHz)
(MHz)
x0 ~3 nm
Heating of asingle Cd+ ion from n0
Delay Time (msec)
0 10 20 30 40 500.0
0.5
1.0
1.5
n
Trap Frequency (MHz)
Heating rate dn/dt(quanta/msec)
1 2 3 4 5 60.01
0.1
1
10
Quadrupole Trap (160 m to nearest electrode)
Linear Trap (100 m to nearest electrode)
Heating Ratedn/dt
(quanta/msec)
Decoherence of Trapped Ion Motion
40Ca+
199Hg+
111Cd+
Heating history in 3-6 MHz traps
9Be+
Distance to nearest trap electrode [mm]
0.04 0.1 0.2 0.3 0.610-3
10-2
10-1
100
101
102
137Ba+
heating rate (quanta/msec)
137Ba+ IBM-Almaden (2002)
40Ca+ Innsbruck (1999)
199Hg+ NIST (1989)9Be+ NIST (1995-)
111Cd+ Michigan (2003)
Q. Turchette, et. al., Phys. Rev. A 61, 063418-8 (2000)L. Deslauriers et al., Phys. Rev. A 70, 043408 (2004)
Trap dimension [mm]
0.04 0.1 0.2 0.3 0.610-2
10-1
100
101
102
SE() 10-12 (V/m)2/Hz
40Ca+
199Hg+111Cd+
137Ba+9Be+
1/d4 guide-to-eye
Electric Field Noise History in 3-6 MHz traps
~ 1/d 4
Heating due tofluctuating patch potentials (?)
)(4
2
ES
m
q
d
est. thermal noise
Quantum Gate Schemes for Trapped Ions
1. Cirac-Zoller2. Mølmer-Sørensen3. Fast Impulsive Gates
Universal Quantum Logic Gateswith Trapped Ions
Step 1 Laser cool collective motion to rest
Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)
n=0
Universal Quantum Logic Gateswith Trapped Ions
laser
j k
Step 2 Map jth qubit to collective motion
Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)
Universal Quantum Logic Gateswith Trapped Ions
laser
j k
Step 3 Flip kth qubit depending upon motion
Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)
|
|10
10
10
10
/2, 2, /2
sign flip ||n=1 only !
2/2
Universal Quantum Logic Gateswith Trapped Ions
laser
j k
Step 4 Remap collective motion to jth qubit (reverse of Step 1)
Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)
Net result: [|j + |j] |k |j |k + |j|k
n=0
• CNOT between motion and spin (1 ion): F=85%C.M., et. al., Phys. Rev. Lett. 75, 4714 (1995)
• CNOT between spins of 2 ions: F=71%F. Schmidt-Kaler, et. al., Nature 422, 408-411 (2003).
Demonstrations of Cirac-Zoller CNOT Gate
= m + m
During the gate (at some point), the state of an ion qubit and motional bus state is:
Decoherence Kills the Cat
Direct coupling between | and | with bichromatic excitation ?
uniformillumination
| + ei|2
|
|
g 2
+ = Rabi Freq =
= 0
g 2
Bichromatic coupling to sidebands
uniformillumination
|, |
|
|
n1n
n+1
n
n
Mølmer/SørensenMilburn/Schneider/James
(1999)
kx0gn+1) 2
kx0gn) 2
+ = Rabi Freq =
=kx0g) 2
as long as kx0n+1<< 1: “Lamb-Dicke regime”)
independentof motion !
1n n
1n n
2ˆ xH
i
i
e
eMS
MS
i
i
Mølmer/Sørensen 2-ion entangling quantum gate – a “super” /2-pulse
Big improvement –• no focussing required• no n=0 cooling required• less sensitive to heating
|
|
|
|
n1n
n+1
n n
n1n
n+1
Can scalable to arbitrary N!
|··· |··· + |···
2
e.g., 6 ions
|3,-3 = |
|3,3 = |
|3,-1 = | + ···
|3,1 = | + ···
| J,Jz
Coupling: H = Jx2
flips all pairs of spins
Entangling rate N-1/2
Four-qubit quantum logic gate
Sackett, et al., Nature 404, 256 (2000)
| | + ei|
x
p
N=1 ion: Force = F0|| (spin-dependent force)
Same idea in a different basis
ei
( enclosed area)
laser
N=2 ions ei ei
e.g., force on stretch mode only
= /2: -phase gateNIST (2003): 97% Fidelity
2ˆ zH
Strong Field Impulsive Gates
2S1/2
2P1/2
|
|
+
0,0
1,11,01,-1
0,01,1
1,0
1,-1
e.g. 111Cd+
14.5 GHz
strong coupling: Rabi>> and Rabi~ 1
(a) off-resonant laser pulse; differential AC Stark shift provides qubit-state-dependent impulse
| | = |’’= linear shift = nonlinear shift = 2Udd/ħ
++
++
“dipole engineering”: Udd = 12/r3 = (e)2/r3
r
| e+i-i/2 | = |’’| e-i-i/2 | = |’’| | = ei|’’
quantum phase gate
(t)
t
sub s
Cirac & Zoller (2000)
Poyatos, Cirac, Blatt & Zoller, PRA 54, 1532 (1996)Garcia-Ripoll, Zoller, & Cirac, PRL 91, 157901 (2003)
p = 2ħk
| ||e |e
U = ||e2iaa †
| |
|e
| |
|e
-pulseup
-pulsedown
two sequential -pulses
spin-dependent impulse
(b) resonant ultrafast kicks
The trajectory of a normal motional mode of two ions in phase space under the influence of four photon kicks. Gray curve: free evolution. Black curve: four impulses kick the trajectory in phase space, with an ultimate return to the free trajectory after ~1.08 revolutions.
2S1/2
2P1/2
| |
+
0,0
1,11,01,-1
=226.5 nm10 psec no
kick
2P3/2
1/(15 fsec) = FS splittinge 3nsec|e
Fast version of z phase gate
does not require Lamb-Dicke regime!
e.g. 111Cd+
require FS << pulse << e
Summary
Trapped Ions satisfy all “DiVincenzo requirements” for quantum computing:
1. identifiable qubits2. efficient initialization3. efficient measurement4. universal gates5. small decoherence
SO WHAT’S THE PROBLEM?!
ENIAC(1946)
Next: Ion Traps and how to scale them!