lecture 1: ion qubits, trapping, and qubit initialization
TRANSCRIPT
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The trapped-ion qubit tool box
x = 1000x2 + 3x = 5R(µ)R(µ)R(theta)
Roee Ozeri
Weizmann Institute of Science
Rehovot, 76100, Israel [email protected]
Contemporary Physics, 52, 531-550 (2011)
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Physical Implementation of a
quantum computer
1. Well defined qubits.
2. Initialization to a pure state
3. Universal set of quantum gates.
5. Long coherence times (compared with gate & meas. time).
4. Qubit specific measurement.
David Divincenzo’s criteria:
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Universal Gate set
- A finite set of unitary gates that spans any such U.
U
- The Deutsch-Toffoli gate
- D. Deutsch, Proc. R. Soc. London, Ser. A, 425, 73, (1989).
- For N qubits, a general unitary transformation U acts on a 2N- dimension Hilbert space.
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Universal Gate set
- A finite set of unitary gates that spans any such U.
Single qubit gate
- Barenco et. al. Phys. Rev. A, 52, 3457, (1995).
- For N-qubits, and unitary transformation U on a 2N-dimension Hilbert space.
U
2-qubit C-not gate
- Rotations can be approximated to e by concatenating k gates, from a finite
set {Vi}, where k < polylog(1/e).
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Physical Implementation of a
quantum computer
How well???
David Divincenzo’s criteria:
Well enough to allow for a large scale computation:
Fault tolerance
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ECC
Noisy operations
Fault-tolerant Quantum
Computation
One level quantum error-correction codes
concatenation; threshold theorem
k-levels of fault-tolerant encoding
if
• Fault tolerance threshold.
• Heavy resource requirements when
• Depends on code, noise model, arch. constraints etc.
• Current estimates for
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Tutorial overview
1. The ion-qubit: different ion-qubit choices, Ion traps.
2. Qubit initialization.
4. Universal set of quantum gates:
single qubit rotations; two-ion entanglement gates
5. Memory coherence times
3. Qubit measurement.
How well???
Benchmarked to current threshold estimates
Disclaimer: non exhaustive; focuses on laser-driven gates
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Different ion-qubit choises
- One electron in the valence shell; “Alkali like” 2S1/2 ground state.
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Electronic levels Structure
n 2P1/2
n 2P3/2
n 2S1/2
n 2P Fine structure
n-1 2D3/2
n-1 2D5/2
n-1 2D Be+
Mg+
Zn+
Cd+
w/o D
Ca+
Sr+
Ba+
Yb+
Hg+
with D
313 nm
280 nm
206 nm
226 nm
397 nm
422 nm
493 nm
369 nm
194 nm
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2S1/2 Zeeman qubit
n 2S1/2
(Isotopes w/o nuclear spin)
Turn on small B field
2.8 MHz/G
m = -1/2
m = 1/2
24Mg+
64Zn+
114Cd+
40Ca+ 88Sr+ 138Ba+ 174Yb+ 202Hg+
e.g. Advantages
- RF separation.
- Tunable.
- Infinite T1.
Disadvantages
- Energy depends
linearly on B.
- Transition photon caries no
momentum
-Momentum transfer with off-
resonance lasers: photon
scattering.
- Detection.
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2S1/2 Hyperfine qubit
2S1/2
1 - 40 GHz
F = I – 1/2
Hyperfine structure (order depends on the sign of Ahf)
Turn on small B field
F = I + 1/2
m = 0
m = 0
9Be+
25Mg+
67Zn+
111Cd+
43Ca+ 87Sr+ 137Ba+ 171Yb+ 199Hg+
e.g.
Advantages
- MW energy separation.
- B-field independent qubit.
- Infinite T1.
- State selective fluorescence Detection.
Disadvantages
- Few GHz energy separation.
-Transition photon caries no momentum
- Off resonance photon scattering.
- initialization to clock transition can be more
tricky
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Optical qubit
2S1/2
2D5/2
Ca+
Sr+
Ba+
Yb+
Hg+
with D
729 nm
674 nm
1760 nm
411 nm
282 nm
Innsbruck/Weizmann
Disadvantages
- Finite (~ 1 sec) lifetime.
- Coherence time is limited by laser linewidth.
Advantages
- Single optical photon (momentum).
- B-field independent qubit (if nuclear spin ≠ 0).
- Hardly any spontaneous decay during gates.
- State selective fluorescence Detection: excellent
discrimination.
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Trapping
- Trap ions: a minimum/maximum to f, the electric potential.
- Impossible in all directions; Laplace’s equation:
02
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Trapping
RF electrode
High dc potential
control electrode
Low dc voltage
control electrode
Positive ion
Linear RF Paul trap
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Dynamic trapping
(pondermotive forces)
Large and slow small and fast
Newton’s E.O.M:
Oscillating electric field:
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Dynamic trapping
Field expansion:
To 0th order:
Next order:
Average over one period:
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+
+
Dynamic trapping
d
2~
dm
eV
rf
rf
trap
Pseudo-potential:
For electric quadruple:
Harmonic frequency:
Sign of e
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•Drive frequency ~ 20-30 MHz
•RF amplitude ~ 200-300 V
•Secular frequency
–Radial ~ 2-3 MHz
– Axial ~ 1 MHz
eFXM
0)2cos(22
2
iiii Xqa
d
Xd
2
0 )cos( i
i
rfii XtV
(Mathieu eq‘: )
Solve E.O.M :
2~
dm
eV
rf
RFtrap
Trapping
Linear RF Paul trap
Potential :
Stable solution:
d – the distance between the ion and the electrodes
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Weizmann trap.
0.6 mm
- Laser machined Alumina
0 . 3
m m
n
n z
r
= 1 – 2 MHz (axial)
= 2 – 3 MHz (radial)
200 Vpp AC
21 MHz
10 V DC
1.2 mm
+ + 0.2 mm
+
+
~ 2-5 mm Nitzan Akerman
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Scale up?
One (out of many) problem:
isolating one mode of motion for gates
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interconnected multi-trap structure subtraps completely decoupled
routing of ions by controlling
electrode voltages
Subtrap for different purpose: Gates, readout, etc.
D. J. Wineland, et al.,
J. Res. Nat. Inst. Stand. Technol. 103, 259 (1998);
D. Kielpinski, C. Monroe, and D. J. Wineland,
Nature 417, 709 (2002).
Other proposals: DeVoe, Phys. Rev. A 58, 910 (1998) .
Cirac & Zoller, Nature 404, 579 ( 2000) .
The ion vision: Multiplexed trap array
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Multi-zone ion trap
control
control rf
rf
view along axis:
rf filter board •Gold on alumina
construction
•RF quadrupole realized in two
layers
•Six trapping zones
•Both loading and experimental
zones
•One narrow separation zone
•Closest electrode ~140 mm from ion
segmented
linear trapping
region
NIST
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Ion transport
100 mm
6-zone alumina/gold trap (Murray Barrett, John Jost)
200 mm separation zone
•Ions can be moved between traps.
–Electrode potentials varied with time
•Ions can be separated efficiently in sep. zone
–Small electrode’s potential raised
•Motion (relatively) fast –Shuttling (adiabatic):
several 10 ms
–Separating: few 100 ms
NIST
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Surface-electrode traps
RF electrodes
Control electrodes
1 mm
Microfabricated
filter resistor
Filter capacitor
Control lead
Field lines:
Elbows and tee-
junctions possible:
J. Chiaverini et al. Quant. Inform. Comp. 5, 419 (2005).
Trapping center
NIST
S. Seidelin et al. Phys. Rev. Lett. 96, 253003 (2006).
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Micro-fabricated traps
Allcock; Oxford
Allcock et al. App. Phys. Lett. 102, 044103 (2012).
Shappert et al. arXiv quant-ph\1304. 6636 (2013).
Shappert; Georgia Tech
Amini; NIST
Amini et al. New J. Phys. 12, 033031 (2010).
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Trapped ions on the tabletop
• Vacuum ~ 10-11 Torr
• Room temperature (or a bit above)
• RF created with coax/helical resonator
• Atoms created by oven.
• Ions created by photo ionization.
• Approx. F1 imaging optics to
EMCCD/PMT (res = 0.8 um).
EMCCD
PMT
Flip Mount
f/1 lens
Microwaves
cooling ,detection and excitation beams
B-field
RF resonator
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Harmonic oscillator levels
Vibrational mode
quantum number
1
0
1
0
- Doppler cooling.
- Resolved side band cooling.
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Sideband Spectroscopy
5 2S1/2
4 2D5/2
674
nm
• Scan the laser frequency
across the S →D transition
axial radial
carrier
Blue sidebands Red sidebands
Dn < 80 Hz
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n = 0 n = 1 n = 2 n = 3
S
D
Red sideband ~4 MHz
1033 nm
674 nm
S
D
P
Resolved-sideband Cooling
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• Uncertainty in ion position = ground state extent nmm ho
62
• T ≈ 2 mK
Sideband cooling to the ground state
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Anomalous heating
Hite et al. Phys. Rev. Lett. 109, 103001 (2012) Deslauriers et al. Phys. Rev. Lett. 97, 103007 (2006)
• Fluctuating charges
• f-1.5 noise
• Thermally activated
• Due to monolayer of C on eletrodes: gone after Ar+ ion cleaning
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Qubit Initialization
-Optical pumping into a dark state.
2S1/2
2P1/2
s
Zeeman qubit
e ~ 10-6 – 10-3
Error sources:
- Polarization purity.
-CPT possible into any superposition..
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Qubit Initialization Zeeman qubit
Process tomography of optical pumping Limited to 10-3 due to stress-
induced birefringence in vacum
chamber optical viewports
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2P1/2
2P3/2
2S1/2
2D5/2
2D3/2
g = 0.4 Hz
-1/2 +3/2
Qubit Initialization Zeeman qubit: the D level option
• Limited by off-resonance coupling of ↓ to D • Power broadening and in-coherent noise
• e 10-4
Keselman, Glickman, Akerman, Kotler and Ozeri, New J. of Phys. 13, 073027, (2011)
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Qubit Initialization
- Optical pumping into a dark state.
2S1/2
2P1/2
s
Hyperfine qubit
F=1
F=0
p
C. Langer, Ph.D Thesis, University of Colorado, (2006).
Estimated:
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Qubit Initialization
- state initialization: same as previous.
2S1/2
2D5/2
Optical qubit
- state initialization:
- Rapid adiabatic passage.
2P3/2
e ~ 10-2
- p-pulse.
(Wunderlich et. al. Journal of Modern Optics 54, 1541 (2007))
e ~ 10-2