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 Research and Development Activity US Army Research Office US National Security Agency National Science Foundation 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
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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)
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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)
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 )
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
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: