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Gregynog QIP meeting
QIP Experiments with ions, atoms and molecules
Christopher Foot, University of [email protected]
Objectives of these lectures:
Describe QIP experiments that use atomic and molecular physics (not including NMR)
• General requirements for a quantum gate between two qubits – spin-dependent interaction
• Review experimental techniques and some current experiments with ion in traps
• Neutral atoms in optical lattices: simulation of condensed matter physics and QIP applications
• Recent ideas for QIP with polar molecules = hybrid of atoms and ions
Quantum gate - controlled operation
• CNOT gate:
• CROT gate (or controlled-Z gate)
Exercise: Show how to construct a CNOT gate from a controlled-Z gate
and 2 Hadamard gates. (Ex 4.17 in Nielson & Chuang)
CROT gate (or controlled-Z gate) based on state-dependent (spin-dependent) interaction
or
Interaction
0 01 1
where
Qubit A Qubit B
Comment on quantum gates
Equivalent to
Difficult to implement since requires very precise controlof experimental timing.
`Pushing gate’– laser beam exerts state-dependent force on ions
Harmonic trapping potential
RepulsiveCoulomb force
1
0
E
Phase factor
Phase difference
Cf. `wobble gate’ discussed later
Repulsive
Attractive
Summary of Lecture 1: Ions
1. Requirements for QIP2. Ion trap principles3. Read out and Quantum jumps4. Manipulation of single qubit by Raman transitions5. Laser cooling to the lowest vibrational level6. Current experimental capability in Oxford and
elsewhere7. Survey of ideas being actively explored in
experimental groups8. How to make a computer,
Requirements for quantum computing
• Excellent quality qubits
Q = Tcoherence / Tgate = 106 now realised for ions
• move qubits around quickly and without error
• High precision gate between neighbours (error = 10-4 )• Single-qubit gates • Measurement of qubits (read out)
D. Deutsch, Proc R Soc Lond A 400, 97 (1985) & 425, 73 (1989).A.Steane, Phys. Rev. A 68, 042322 (2003) & Quant. Inf. Comp. 2, 297 (2002).
*
Basic ion trap methods
N.B.Most of the slides in this lecture come from the Ion trapping group in Oxford (part of the IRC)
Paul Trap
r.f. quadrupole+ ‘end caps’++
trapped ions1000 volts,10 MHz
end view
Vx
tim e‘=’
Axial confinement by electrostatic (quadratic) potential.Radial confinement by oscillating quadrupole potential.
Electrostatic trapping not possible, see Foot, Atomic Physics, OUP 2005
The trap
Axial motional freq. of order MHz
Ion-electrode distance = 0.1 to 1 mm
7 mm
Alkali-like ions
Choice of ion:Want simple energy level structure when singly-ionisedGroup II or other metals:X = Be, Ca, Sr, Ba, Yb, Cd, Hg
Ground configuration of X+ ion has electron spin s = ½.
Hyperfine structure arising from interaction of nuclear spin (nuclear magnetic moment) withmagnetic field created by the unpaired electron.
The hyperfine structure of Ca-43, the ion used in Oxford experiments, is inverted
S 1/2
F=3
F=4 m F=0
3.2 G H z
m F=+4
B field-independent
1st order Zeeman sensitive
• Use transition with no first-order Zeeman effect, which is therefore insensitive to magnetic field fluctuations,cf. atomic clocks.• Qubit coherence time of order seconds (see later)
Readout by fluorescence on cycling transition
2S1/2
F=1, MF=1
F=2, MF=21.25 GHz
2P3/2
2P1/2
F=3, MF=3
E.g. Be+Selection rules cycling transition
PM tube or CCD camera
Electron shelving or “quantum jumps”
0 1 2 3 4 5 6 7 8 90
100
200
300
400
500
600
700(d)
C
ount
s pe
r 8
ms
Time (s)
0 1 2 3 4 5 6 7 8 90
100
200
300
400(c)
Observed fluorescence signals from one (a,b), two (c) and three ions (d). (a): random telegraph (850 nm laser is left permanently on)(b-d): controlled shelving (850 nm laser pulsed on when desired)
Cou
nts
per
8 m
s
Time (s)
0 1 2 3 4 5 6 7 8 90
50
100
150
200
250
300(b)
Cou
nts
per
8 m
sTime (s)
0 1 2 3 4 5 6 7 8 90
100
200
300
400
500 (a)
Time (s)
Cou
nts
per
20 m
s
Time needed to measure a qubit at 99.9% fidelity
• Collection & photon detection efficiency = 0.02• excited state lifetime = 5 ns• required photon count for P(error) < 0.001 is 10
photons• time to count 10 photons = 2 x 10 / = 5 s
• Current experiments allow 100 s to 1 ms
Poissonian distribution of photon count
Scattering rate
The trap
Excellent signal-to-noise in detection of individual ions
7 mm
Comment: There is well-developed and efficient scheme for reading out the state of ions. This is not yet achieved for neutral atoms or molecules.
Single bit rotations
Microwaves:U on all qubits at once
Stimulated Raman transition:U on a chosen individual qubit
1-bit gate time' 1 micro-second
Raman transition
2S1/2
F=1, MF=1
F=2, MF=21.25 GHzhyperfine structure
313 nm
2P3/2
2P1/2
AOM: shift 1.25 GHZ
Very high-precisionand stable phase
Raman transition: effective 2-level system
1
2
100 GHz
500 MHz500 MHz
10 MHz
photon scattering = 10-4
= 1.25 MHz
(unwanted) photon scattering rate
2S1/2
F=1, MF=1
F=2, MF=2
2P3/2
2P1/2
Qubit decoherence per gate time
Photons scattered per gate time
PRL 95, 030403 (2005)
Trapped atom: quantum simple harmonic motion
z
z
12
0
z
12
0
0
0
Laser cooling of trapped ions: “Sideband cooling”
3
2
0
L 0 z
12
0
1
3
<< 1
Resultant thermal distribution:
Laser cooling a trapped ion very different to cooling of free atoms
Measure temperature of trapped ion
10
12
2
Pg
L
Compare excitation probability of first red sideband and first blue sideband
0Cf Fig. 12.10 in Foot (2005)
Laser cooling of trapped ions: “Sideband cooling”
3
2
0
L 0 z
12
0
1
3
Experimentalresults:
2005 Oxford 0.02
<< 1
Current experimental capability in Oxford
Dr David LucasProf. Andrew SteaneProf. Derek Stacey…………….N.B.Most of the slides in this lecture come from the Ion trapping group in Oxford (part of the IRC)
Main results
VERY LONG COHERENCE
DETERMINISTIC ENTANGLEMENTOF ION SPIN QUBITS
Very long (1s) qubit coherence time
S 1/2
F=3
F=4 m F=0
3.2 G H z
m F=+4
field-independent,prepare with ~15% efficiency
1st order Zeeman sensitive,prepare with 100% efficiency
• Readout by shelving with 95% efficiency
• Rabi flopping and Ramsey experiments using 3.2 GHz microwaves
Ca-43 hyperfine qubit
Rabi flopping
microwave pulse length (ms)
data fit
P(F
=4,
MF=
0)
Coherence (T2 ) time control experiment (small delay) Ramsey experiment (long delay)
Detuning (Hz) from 3,225,611,696
Fringe visibility vs. delay
Ramsey fringes
= 0.8(2) s
Ramsey gap (ms)
Long-lived memory qubit
p/2 p/2 p/2 p/2854
393
854
393
PMT
PMT0.1ms 300ms
Dop
ple r
cool
prep
are
shor
tR
amse
yex
pt
shel
ve
dete
ct A
Observe many (~270) Rabi flops on the m =0 ->0 field-independent transition, lasting >30ms.
F
A single spin-echo pulse can be used to protect the memory qubit from the residual field-sensitivity. We detect no decoherence in a 1 second experiment (Ramsey fringe , right), implying an effective coherence time of
.
contrast >99%=10s–100sT2
SE
Spin
-dow
n (F
=4) P
opul
atio
n (a
.u.)
Spin-echo
p/2
p/2
( )
p
1s
T2
SE > 10s
Exercise
Show why the pulse sequence /2 - - /2 is more
robust than a Ramsey experiment (/2 - /2 sequence)
Also called Square root of NOT gate
Cf Exercise 7.3 in Foot, Atomic Physics
NOT gate:
Entanglement
Spin-dependent oscillating force
Dipole force in standing wave
B
50 m
Raman beams
F
• Laser standing wave produces an oscillating force on a pair of ions
• Robust and “fast”• To be described in detail later
+
( Explain how state-dependence of the force arises in nextlecture. Force/potential depends on the polarization of the light. )
*
Leibfried or “wobble” gate
D. Leibfried et. al. Nature 422, 412 (2003)
Equivalent to controlled-not
Really nice gate
1. Only the area, not the shape of the loop matters
2. Non-zero ion temperature?– just displace the starting point,– same loop no problem
3. Shift of laser standing wave phase?– rotate the loop about the initial point– same loop no problem
Deterministic entanglement by phase gate
spin qubit (40Ca ground state)
stretch mode freq. 866 kHzion separation 9 m (= 22
<n> = 0.35 , < 0.1
gate time 77 s(s = 67) fidelity 75 (5) %
cool
| ↓ ↓
2, )2
measure
2
analysis pulseprepare
gate
Wobble gate results
July 2005 data:
use twin loop,
fidelity 82(2)%
limited largely by photon scattering( = 30 GHz)and laser intensity noise
D. Lucas, M. McDonnell, S. Webster, J. Home, B. Keitch, D. Stacey, A. Ramos, A. Steane
Tomography
Deduced density matrix
hence entanglement of formation E = 0.52
Two-qubit gates
• Use laser-driven oscillatory motion of ions:• Current experiments (2 to 8 ions):
– fidelity » 90% (97% reported)– gate time » 10 to 100 s
• Future:– fidelity 99.99% (10-4 ) with good (bright and
stable) lasers
– time 100 ns to 1 s
Requirements for quantum computing
• Excellent quality qubits
Q = Tcoherence / Tgate = 106 now realised for ions
• move qubits around quickly and without error
• High precision gate between neighbours (error = 10-4 )• Single-qubit gates • Measurement of qubits (read out)D. Deutsch, Proc R Soc Lond A 400, 97 (1985) & 425, 73 (1989).
A.Steane, Phys. Rev. A 68, 042322 (2003) & Quant. Inf. Comp. 2, 297 (2002).
Moving information around the machine
m
Logical information encoded in large groups of ions.QEC uses a lot of parallel ops. (Animated version, Chuang website.)
Kielpinski, Monroe, Wineland, Nature 417,709 (2002)
Moving ions around
-Already mentioned by Prof Knight
7-zone trap, Oxford/Liverpool collaboration
ion-electrode distance = 0.7 mmtrap-trap separation = 0.8 mmtest open design conceptBuilt by University of Liverpool, S. Taylor
Moving quantum information around
• Array of ion traps with ions transported between traps
• Possibly use large numbers of ions in same trap
• Possibly use ion—photon coupling
Ion entangled with photon (2004)
March 2004Blinov,Moehring,DuanandMonroe
Scalable computer by cluster methods
Duan et al, Quant. Inf. Comp. 4, 165 (2004)
multi-qubit controlled entanglement
4,5,6- ion “cat” state |0000 + |1111F > 0.76, W< 0.51 |00000 + |11111 F > 0.60, W< 0.2 |000000 + |111111F > 0.51, W< 0.02NIST, Boulder, USA
3 to 8- ion “W” state
|001 + |010 + |100 F = 0.82, W= 0.53 |0001 + |0010+|0100 + |1000 F = 0.60, W= 0.46 ...
|00000001+|00000010+F = 0.72, W= 0.029
Innsbruck, Austria
Requirements for quantum computing
• Excellent quality qubits
Q = Tcoherence / Tgate = 106 now realised for ions
• move qubits around quickly and without error
• High precision gate between neighbours (error = 10-4 )• Single-qubit gates • Measurement of qubits (read out)
Conclusion:Ion trapping fulfils these requirements.No obvious `roadblock’ in the way of QIP with ions,`just’ requires development of technology
How to make a computer
quant-ph 2004
