Quantum Simulations with
Yb+ crystal
~5 mm
Trapped Atomic Ions
dc
rf
dc
dc
rf
dc
dc
rf
dc
dc
rf
dc
3-layer geometry:• single rf electrode• scalable to larger structures, natural for junctions
2S1/2(600 Hz/G @ 1 G)
wHF/2p = 12 642 812 118 + 311B2 Hz
| = |0,0
| = |1,0
171Yb+ hyperfine spin
2S1/2
2P1/2
369 nm
2.1 GHz
/2g p = 20 MHz
|
|
171Yb+ spin detection
(600 Hz/G @ 1 G)
wHF/2p = 12 642 812 118 + 311B2 Hz
# photons collected in 800 ms0 5 10 15 20 25
0
1
Pro
bab
ility
|z
2S1/2
2P1/2
369 nm
/2g p = 20 MHz
|
|
2.1 GHz
171Yb+ spin detection
>99% detectionefficiency
# photons collected in 500 ms0 5 10 15 20 25
0
1
Pro
bab
ility
|z |z
(600 Hz/G @ 1 G)
wHF/2p = 12 642 812 118 + 311B2 Hz
(600 Hz/G @ 1 G)
wHF/2p = 12 642 812 118 + 311B2 Hz2S1/2
2P1/2
|
|
171Yb+ spin manipulation
D = 33 THz
355 nm (10 psec @ 100 MHz)
2P3/2
/2g p = 20 MHz
National Ignition Facility: 351nm(Livermore National Laboratory)
Pavg ~ 5W at 355nm10 psec pulses, 120 MHz rep rate
0 10 20 30 pulse energy (nJ)
picosecondspin control
1
0
P(↑|↓)
See talk by Jonathan Mizrahi (Sunday)J. Mizrahi, et al., ArXiv 1307.0557 (2013)
Internal states of these ions entangled
Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)CM, et al., Phys. Rev. Lett. 74, 4714 (1995)
Q. Turchette, et al., Phys. Rev. Lett. 81, 3631 (1998)F. Schmidt-Kaler, et al., Nature 422, 408 (2003)
Trapped Ion Quantum Computer(Cirac-Zoller)
Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)CM, et al., Phys. Rev. Lett. 74, 4714 (1995)
Q. Turchette, et al., Phys. Rev. Lett. 81, 3631 (1998)F. Schmidt-Kaler, et al., Nature 422, 408 (2003)
Trapped Ion Quantum Computer(Cirac-Zoller)
Internal states of these ions entangled
Cirac-Zoller: number states of the QHO
• extreme cooling: requires a pure motional state
• not scalable: mode density problem
1
𝑘𝑥𝑟𝑚𝑠≪1𝑜𝑟 𝑛≪ℏ𝜔𝐸𝑅
=
Better: “spin-dependent displacements”
• only requires cooling to the Lamb-Dicke limit
• “virtual” coupling to phononsPossible
Mølmer & Sørensen (1999)Solano, de Matos Filho, Zagury (1999)Milburn, Schneider, James (2000)
F = F0|↑↑| - F0|↓↓|
global spin-dependent force
↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↓↑↓↑↓↑↓↓↑↓ ↑↓ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↓↑↓↑↓↑↓↓↑↓ ↑↓ ↑
|
|
ADD: Independent spin flips
B
F = F0|↑↑| - F0|↓↓|
global spin-dependent force
2S1/2
2P1/2
spin-dependent force (171Yb+)
|
|
1,-1 1,11,0
0,0
1,-11,11,0
0,0
B(x)
Magnetic field gradient
2S1/2
2P1/2
369 nm
spin-dependent force (171Yb+)
|
|
s+ s+
1,-1 1,11,0
0,0
1,-11,11,0
0,0
D
g
Position-dependent AC Stark shift
2S1/2
2P1/2
369 nm
spin-dependent force (171Yb+)
|
|
1,-1 1,11,0
0,0
1,-11,1
1,0
0,0
D
g
g
g
Red+blue sideband appliedsimultaneously
= Lamb-Dicke parameter
simultaneous sidebands
global spin-dependent oscillating force
i
ii
xi xkH )(̂
ki
tik
tik
ki
kixi
kk eaeabxk,
)()(0
)( ][ˆ †
normal mode transformation matrix: ion i, mode k
km2
1)()( 22
k
ki
i
ki bb
normal mode decomposition
Lamb-Dicke approximation:
1 rmskx
Aside: transverse Modes of an atom chain
transversemodes
frequency
. . .
transversemodes
frequency
S.-L. Zhu et al., Phys. Rev. Lett. 97, 050505 (2006)A. Serafini et al., New J. Phys. 11, 023007 (2009)
. . . . . .
axialmodes
0 1 2 3 4 5
fluorescence~ N()
Raman beatnote (MHz)
transverse x
transverse yaxial z
COM
COM
COMZigZag
ZigZag
(Dk nominally along x)
Raman spectrum of N=9 ions
Ramanbeatnotes:
wHF ± m
ki
tik
tik
ki
kixi
kk eaeabxkH,
)()(0
)( ][ˆ †
uppersidebands
frequencywHF+m
carrierlower
sidebands
wHF -m
global spin-dependent oscillating force
ki
tik
tik
ki
kixi
kk eaeabxkH,
)()(0
)( ][ˆ †
k
kkik
kii aa ])()([)(ˆ *
)sincos()(22
,
ki
k
ikiki ie
ik
†
phonon
s
k kk
k
kk
k
k
kjkijiji
bb
m
k
2
2sin
)(
)sin(
)(
)sin(
2
)()(
22
,,2
,
interaction between qubits (entangling gates etc..)
ji
jx
ixji
i
ixi iU
,
)()(,
)( )()(ˆexp)(
evolution operator
...)]](),([),([
6)](),([
2
1)(exp)(
232
0
1231
0
2
0
3
0
121
0
2
0
ttt
tHtHtHdtdtdti
tHtHdtdttdtHiU
0)sincos()(22
,
ki
k
ikiki ie
ik
How to avoid phonon creation?
(1) Pick detuning m and time t wisely “FAST MOLMER”
for all modes k
e.g.: m near single mode k only
→ ( -m wk)t = 2p m m=1,2,…
S.-L. Zhu, et al., Europhys Lett. 73 (4), 485 (2006).
Beatnote frequency
Ra
bi f
requ
enc
y
HF
x
p
“FAST MOLMER”
0)sincos()(22
,
ki
k
ikiki ie
ik
How to avoid phonon creation?
(1) Pick detuning m and time t wisely “FAST MOLMER”
for all modes k
e.g.: m near single mode k only
→ ( -m wk)t = 2p m m=1,2,…
S.-L. Zhu, et al., Europhys Lett. 73 (4), 485 (2006).
(2) “Adiabatically eliminate” phonons: | - m wk| >> hW0 “SLOW MOLMER”
1)sincos()( ,22
,
k
ikik
i
k
ikiki
iie
ik
x
p
Beatnote frequency
Ra
bi f
requ
enc
y
HF
“SLOW MOLMER”
)()(, ˆˆ j
xi
xji
jieff JH
k k
kj
kiji
ji
bb
m
kJ
22
2
, 2
)(
(2) “Adiabatically eliminate” phonons: | - m wk| >> hW0 “SLOW MOLMER”
How to avoid phonon creation?
1)sincos()( ,22
,
k
ikik
i
k
ikiki
iie
ik