Quantum Information Experiments with Ion Crystals in
Penning Traps
John Bollinger
NIST-Boulder
Ion storage group
+V
Michael Biercuk, Hermann Uys, Joseph
Britton, Wayne Itano, and Nobuysau Shiga
Long term collaborators: David Wineland,
Jim Bergquist, Dietrich Leibfried, Till
Rosenband, Joseph Tan, Pei Huang, Brana
Jelenkovic, Travis Mitchell, Brad King, Jason
Kriesel, Marie Jensen, Taro Hasegawa
Dan Dubin – UCSD(theory)
+V
0.5 mm
B
Outline:
- Penning traps
Monday: Quint, Wada
Tuesday: Charlton
Friday: Schweikhard
- High magnetic field qubit
124 GHz
2s 2S1/2
- Qubit coherence and dynamical decoupling techniques
- Current effort: quantum simulation
H
x
i
Bx
J z2
i
- Decoherence due to elastic Rayleigh scattering
2
Rayleigh
2
adJ
R
J
auJ
d
J
u
Penning trap
NIST Penning
trap
B=4.5 T
For r , z
trap dimensions ,
1
m
2
trap ( r , z )
z (t )
zo sin( 2
12 cm
r (t ) rc sin 2 (
2
z
z2
t
z
)
c
m
)t
z
rm sin( 2
9Be+
c~
4 cm
7.6 MHz
z ~ 800 kHz,
m ~ 40 kHz
r2
2
m
t
c
m
)
Penning trap: many particle confinement
axial confinement ↔
conservation of energy
radial confinement ↔
conservation of angular momentum
P
mv
j
qB
2c
j
q
A (rj ) rj
c
2
rj for A (r )
j
Br
and B large
2
O’Neil, Dubin, UCSD
axial symmetry important – trap and B field
aligned to 0.01o
radial confinement due to rotation –
ion plasma rotates v = r r due to ExB fields
in rotating frame, r r ˆ Bzˆ Lorentz force is
directed radially inward
precisely control r (and the radial binding force)
with a rotating electric field (rotating wall)
Planar ion arrays in Penning traps
Density no
nB
B
m
c
Rotation frequency
c
r
m
cyclotron
frequency
0.5 mm
Features of Penning traps for quantum information/simulation experiments:
- static trapping fields enable large traps to be used; ions are far from electrode
surfaces ➯ low heating rates
- ion crystals form naturally from minimization of the Coulomb potential energy
- for a single plane, the minimum energy lattice is triangular ➯ good for magnetically
frustrated simulations
- ion crystals rotate (~50 kHz) but rotation precisely controlled ➯ individual particle
detection still possible
High magnetic field qubit
9Be+
, B~4.5 T,
/2 ~124.1 GHz
(mI, +3/2)
~ 80 GHz
2p 2P3/2
(mI, +1/2)
F = 0,1,2,3
(mI, -1/2)
(mI, -3/2)
2p
2P
(mI, +1/2)
F = 1,2
1/2
(mI, -1/2)
cooling
2s 2S1/2
F=2
optical dipole force beams
(+3/2, +1/2)=|
F=1
high B qubit:
~ 40 GHz
repump
↑›
124 GHz
(+3/2,-1/2)=| ↓
(+1/2,-1/2)
(-1/2,-1/2)
(-3/2,-1/2)
(mI , mJ)
›
Rabi flopping on 124 GHz electron spin flip
60 cm
Fluorescence →
Rabi time
Pulse duration (ms) →
random benchmarking experiment:
Biercuk, et al., Quantium Information and
Control 9, 920 (2009).
–pulse fidelity > 99.9%
Ramsey (T2) coherence on 124 GHz electron spin flip
0
1
2
3
Ramsey Precession time (ms)
s)
B
~ 10 9 , T2 ~ 2 ms
B
coherence can be extended to 10’s of ms with spin echo (dynamical decoupling)
Biercuk, Uys, VanDevender, Shiga, Itano, Bollinger, Nature 458, 996 (2009)
4
Dynamical decoupling sequences
• Carr-Purcell-Meiboom-Gill (CPMG) sequence – evenly spaced -pulses
/2
T
2T
refocus
/2
refocus
refocus
refocus
usually only need refocusing at sequence end ⇒
-pulses need not be evenly spaced
UDD construction
• Uhrig sequence (UDD) - Uhrig, PRL 98, 100504 (2007)
/2
/2
refocus
n -pulse UDD sequence cancels lowest n-1 derivatives of the noise
B(t) = B(0) + B’(0)t +(1/2)B’’(0)t2 + (1/6)B’’’(0)t3 + …..
• Theoretical determination of coherence:
Uhrig
Cywinski, Das Sarma
Martinis
F( )
S( )
coherence e
noise filtration function of sequence
noise power spectrum
( )
,
( )
2 S( )
2
0
F(
)d
Qubit coherence extended and accurately predicted by dynamical decoupling
Biercuk, et al., Nature 458, 996 (2009)
Uys, et al., PRL 103, 040501 (2009)
CPMG
UDD
- Fits use analytical filter function and
measured noise spectrum
- CPMG, UDD perform similarly for 1/f2
ambient noise
- UDD performs better for noise spectra
with sharp cutoffs
- Ohmic noise with sharp cutoff injected
improved performance through feedback optimization
- vary inter-pulse delays for fixed precession time (Nelder-Mead simplex method)
- results shown for injected Ohmic noise
- improved performance with feedback
optimization at each total precession time
- no prior knowledge of S( ) required
Magnetic field noise reduced through vibration reduction
bore tube coil emf (dBV/Hz)
floor
isolation pads, bore covered
0 Hz
Normalized fluorescence
after spin echo sequence
isolation pads, bore open
frequency (Hz)→
200 Hz
3 ms → 15 ms increase in spin
echo coherence time
(single arm time)
Present effort: quantum simulation (see Thursday tutorial by Schaetz)
- Porras, Cirac, PRL 92 (2004); Deng, Porras, Cirac, PRA 72 (2005)
- Schaetz group, Nature Physics 4 (2008); Monroe group, Nature 465 (2010)
uniform Ising model
H
x
i
Bx
dipolar anti-ferromagnetic Ising interaction
J z2
H
i
x
i
Bx
i
z
i
J
i j
z
j
d o3
 
ri rj
3
- Bx derived from 124 GHz microwaves
-
, J derived from state-dependent optical
dipole forces
B
L+
~3/4º
off-resonant
laser beams
L
J z2
z
uniform Ising interaction,
squeezing
z
dipolar anti-ferromagentic
Ising interation
0
Optical dipole force excitation of ion planar arrays (N>100)
triggered Doppler velocimetry
internal state dependence of
optical dipole drive
Biercuk et al., Nature Nanotechnology 5 (2010)
optical dipole
force
124 GHz
detection
/2
/2
optical dipole
force
detection
Decoherence ∝
normalized fluorescence
dipole force frequency
decoherence due to spin/axial COM mode entanglement
z
Quantum simulation with planar ion arrays
uniform Ising model
H
x
i
Bx
dipolar anti-ferromagnetic Ising interaction
J z2
H
i
i
Bx = 5 kHz; > 50 kHz with new wave source
first configuration:
= 0.72o, 10 mW/beam
waist ~ 500 m x 50 m
~(2 ∙20 GHz, N=100
presently implementing:
→ 5o, 10 mW/beam
waist ~ 500 m x 50 m
future configuration?
→ 35o, 10 mW/beam
waist ~ 500 m x 300 m
sub-Doppler cooling required
x
i
Bx
~ 2 ∙36 Hz
J ~ 2 ∙7 Hz
/ ~1
i j
z
j
Spontaneous emission limits coherent interaction
effects; squeezing and entanglement still possible
at short times t ~ 0.02( /2), but …..
~ 2 ∙1.7 kHz
J ~ 2 ∙320 Hz
~0.07
~ 2 ∙2.1 kHz
J ~ 2 ∙380 Hz
~0.01
with high power fiber laser based system
10 mW/beam → 100 mW/beam
z
i
J
d o3
 3
ri rj
→ 100∙
J → 100∙J
Decoherence due to elastic Rayleigh scattering
Uys, et al., arXiv:1007.2661
J, MJ
~ 80 GHz
2 2P3 / 2
2 2P1/ 2
3 / 2,
3 / 2,
3 / 2,
3 / 2,
1 / 2,
1 / 2,
3/ 2
1/ 2
1/ 2
3/ 2
1/ 2
1/ 2
qubit superposition state described by density matrix
uu
ud
du
dd
o
o
2
2 S1/ 2
313 nm
non-resonant light (
u
1/ 2
d
1/ 2
124 GHz
Be+ energy levels, B=4.5 T
o)
scattering causes decoherence,
d ud
dt
2
???
ud
Raman scattering vs Rayleigh scattering
u
d ; d
u
Raman scattering:
final qubit state entangled with the polarization or frequency of the scattered photon
⇒ decoherence after single scattering event
J, M J
~ 80 GHz
2 2P3 / 2
2
2 P1/ 2
3 / 2,
3 / 2,
3 / 2,
3 / 2,
1 / 2,
1 / 2,
3/ 2
1/ 2
1/ 2
3/ 2
1/ 2
1/ 2
Raman scattering rate given by Kramers-Heisenberg formula
2
 *

j d ˆ (i j ) J ,
i J,
id ˆ i
2
J
J
a
,
a
i j
ij
R
i j
i; J ,
J
d ud
dt
Raman
2
ud
,
Raman
ud
i
du
o
o
2
2 S1/ 2
313 nm
u
1/ 2
d
1/ 2
124 GHz
Ozeri et al. , Phys. Rev. Lett. 95, 030403 (2005) –
precise test of the above prescription for calculating
decoherence due to Raman scattering
Raman scattering vs Rayleigh scattering
elastic Rayleigh scattering:
u
u; d
d
Rayleigh
???
literature indicates that elastic Rayleigh scattering should not produce decoherence
when the elastic scatter rates are equal
J, M J
~ 80 GHz
2 2P3 / 2
2 2P1/ 2
3 / 2,
3 / 2,
3 / 2,
3 / 2,
1 / 2,
1 / 2,
3/ 2
1/ 2
1/ 2
3/ 2
1/ 2
1/ 2
- “when of equal rate from both qubit levels, off-resonance
Rayleigh scattering of photons did not affect the coherence of a
hyperfine superposition”, PRA 75, 042329 (2007)
- “In our system, Rayleigh scattering occurs at the same rate for
the two clock states, does not reveal the atomic state, and so
does not harm the coherence”, PRL 104, 073602 (2010)
o
o
2 2S1/ 2
313 nm
estimate based on difference in elastic scatter rates,
from Ozeri et al., PRA 75, 042329 (2007)
u
1/ 2
d
1/ 2
124 GHz
Rayleigh, diff
( uu
~
( uu
2
)
dd
dd ) 2
Master equation treatment of decoherence due to light scattering
- consistent treatment of decoherence due to both Raman and Rayleigh scattering
- credit to: Hermann Uys
ud
t
J, M J
~ 80 GHz
2
2 P3 / 2
2 2P1/ 2
3 / 2,
3 / 2,
3 / 2,
3 / 2,
1 / 2,
1 / 2,
3/ 2
1/ 2
1/ 2
3/ 2
1/ 2
1/ 2
1
(
2
uu
Raman
du
1
(
2
dd
Rayleigh
)
du
du
Rayleigh
dd
ud
)
ud
uu
2
2
Rayleigh
Raman
adJ
R
J
auJ
d
u
J
decoherence due to difference in the
elastic scattering amplitudes !
o
o
2
2 S1/ 2
313 nm
- large decoherence expected if scattering amplitudes have
opposite sign
u
1/ 2
- sign of scattering amplitude determined by the detuning
d
1/ 2
- physically the state of the qubit can be determined from the
phase of the scattered photon
124 GHz
Decoherence measured from decrease in the Bloch vector
Normalized Counts
0.5
0.4
0.3
- off-resonant laser beam
polarization adjusted to null
light shift
measured slope
0.2
0.1
off-resonant beam blocked
0.0
2
4
6
Delay Time (s)
8
10x10
-3
total laser
on time
Good agreement between theory and experiment
1
2
Raman
Rayleigh
2
2
adJ
R
J
1
2
Raman
1
2
auJ
d
u
J
theory based
on uu dd
Raman
- Laser electric field calibrated from measured light shift and Raman rates
Contribution of
Rayleigh
to low field trapped ion experiments
Be+ low field levels
Hyperfine Coherence in the Presence of Spontaneous
Photon Scattering, Ozeri , et al., PRL 95 (2005)
• no decoherence due to Rayleigh scattering
• conditions: 1. clock states
2. >> hf
•
For clock states and
>>
hf
Raman
~ 1/ 4 → gate errors due to
spontaneous emission reduced with
large detunings
:
2
2
Rayleigh
a
R
J
J
d
J
u u
a
d
~
1
2
2
hf
~
1
4
J
For qubits that are not clock states:
Rayleigh
two-ion gate strength:
~
2
1
~
2
V2
2
~
No apparent advantage of large
clock states
1
2
if qubits are not
Summary:
- ion crystals in Penning traps look like a good platform for simulating
quantum magnetic models with an intractable number of ions
- qubit coherence limited by 20 -200 Hz magnetic field fluctuations;
3 ms → 15 ms coherence time through vibration reduction
- current effort: Ising model simulation using state dependent
optical dipole forces to push the ions in the axial direction
~5o