Page 1
Pulse techniques for decoupling qubits from noise: experimental tests
• Bang-bang decoupling 31P nuclear spins
• Low-decoherence electron-spin qubits and global 1/f noise
• Dynamical decoupling of the qubits
– Periodic pulse sequences
– Concatenated pulse sequences
• Summary
Steve Lyon, Princeton EE
Alexei Tyryshkin, Shyam Shankar, Forrest Bradbury, Jianhua He, John Morton
Page 2
Experiments• 2-pulse Hahn echo
• Decoupling
/2
FID – T2*
Echo
Signal
Pulses
T
/2
Echo
Signal
Pulses
T
(|0 + |1)
(|0 + |1)
(|0 + |1)
(|0 + |1)
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Dynamical Decoupling• Replace single -pulse with sequence of pulses
– Refocus spins rapidly (< noise correlation time)
– “Bang-bang” – fast strong pulses (or 2 different spins)
– CP (Carr-Purcell) – periodic -pulses• x/2--X-2-X-2-…-X--echo
– CPMG (Carr-Purcell-Meiboom-Gill) – periodic -pulses• x/2--Y-2-Y-2-…-Y--echo
– Aperiodic pulse sequences – concatenated sequences• Khodjasteh, Lidar, PRL 95, 180501 (2005); PRA 75, 062310 (2007).
– x/2-(pn-1-X-pn-1-Z-pn-1-X-pn-1-Z)--X--echo with Z=XY
• Yao, Liu, Sham, PRL 98, 077602 (2007). – concatenated CPMG
– x/2-(pn-1-Y-pn-1-pn-1-Y-pn-1)--Y--echo
• Experimental pulses ~ 1s (for -pulse)
– Power ~ 1/(pulse length)2 Energy/pulse ~ power1/2
Page 4
The Qubits: 31P donors in Si
• Blue (microwave) transitions are usual ESR
• All transitions can be selectively addressed
31P donor: Electron spin (S) = ½ and Nuclear spin (I) = ½
↑e,↓n
↑e,↑n
↓e,↑n
↓e,↓nrf1
w1 w2
rf2
|3
|2
|1
|0
X-band: magnetic field = 0.35 T
w1 ~ 9.7 GHz ≠ w2 ~ 9.8 GHz
rf1 ~ 52 MHz ≠ rf2 ~ 65 MHz
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Bang-Bang control
↑e,↓n
↑e,↑n
↓e,↑n
↓e,↓n
rf1
2 rotationw1
Fast nuclear refocusing
i = a|0 + b|1 f = a|0 - b|12
0.0 0.1 0.2 0.3 0.4
0.0 0.1 0.2 0.3 0.4
0.0 0.1 0.2 0.3 0.4
(C)
(B)
Nuc
lear
Pol
ariz
atio
n
Free nuclear spin nutation(A)
One burst of 2 mw pulses
Two bursts of 2 mw pulses
Time (ms)
31P donor: S = ½ and I = ½
Nuclear refocusing pulse would be ~10 s
but electron pulse ~30 ns
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Electron spin qubits
0 2 4 6 8 10 12 14 16 18 20 2210-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
T2
T1
Temperature (K)
T1 28Si:P 9.767 GHz
T2 28Si:P 9.767 GHz
T2 Si:P 16.44 GHz
(Feher et al, 1958)
(Gordon, 28Si, 1958)
T1,
T2
(sec
)• Doping ~1015/cm3
• Isotopically purified 28Si:P
• 7K electron T1 ~ 100’s milliseconds
• 7K electron T2 ~ 60 milliseconds (extrapolating to ~single donor)
x“real” T2
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Noise in electron spin echo signals
0 2 4 6 8
0.0
0.5
1.0
1.5
Sp
in e
cho
de
cay
T (msec)
averaged
Single-pulseT2 = 2 msec
Decoherence
0 2 4 6 8
-3
-2
-1
0
1
2
Spi
n ec
ho s
igna
lsT (msec)
In-phase
Out-of-phase
Signal transferred: in-phase out-of-phase
• Must use single pulses to measure decoherence About 100x sensitivity penalty
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B-field noise
10 100 1000 10000 1000001E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01N
ois
e (
Gau
ss/H
z1/2)
Frequency (Hz)
Origin of noise unclearBackground field in lab?Domains in the iron?
Essentially 1/f
Measure noise voltage induced in coil
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0.0 0.2 0.4 0.6 0.8
****
*
*
Time (ms)
CP
*
**
* ***
Microwave Field Inhomogeneity
Sapphire
VerticalB-field
Sapphirecylinder
MetalWall
x/2--X-2-X-2-…-X--echo
Carr-Purcell (CP) sequence
Page 10
Periodic (standard) CPMG
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
*****************
* **
pulse pulse
pulse
Time (ms)
Mic
row
ave
sig
nal
x/2--Y-2-Y-2-…-Y--echoSelf correcting sequence
Page 11
Coherence after N pulses
0 4 8 12 16 20 24
0.0
0.5
1.0
T2 = 8.5ms
14
16
E
cho
Inte
nsit
y
Time (ms)
Standard CPMG
64
Page 12
Concatenated CPMG
0.0 0.5 1.0 1.5 2.0
*
*****
pulse pulse pulse
Time (ms)
Mic
row
ave
sign
al
*
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Coherence vs. concatenation level
0 2 4 6 8 10 12 14
0.0
0.5
1.0
T2 = 5.8ms
Concatenated CPMG
l = 2 (2 pulses)
l = 4 (10 pulses)
E
cho
Inte
nsit
y
Time (ms)
l = 6 (42 pulses)
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Concatenated and periodic CPMG
0 4 8 12 16 20 24 28
0.0
0.5
1.0
E
cho
Inte
nsit
y
Time (ms)
Periodic CPMG32 pulses
Concatenated CPMG42 pulses
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Fault-Tolerant Dynamical Decoupling
• x/2-(pn-1-X-pn-1-X-Y-pn-1-X-pn-1-X-Y)--X--echo
• Not obvious that it self-corrects
0.0 0.2 0.4 0.6 0.8
*
*
ConcatenatedXZXZ (p2)
Time (ms)
CPMG
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0 5 10 15 20 25 30 35 40 45 50
0.0
0.5
1.0
1.5
p5 (972 pulses)
Concatenated XZXZ pulse sequence
p3 (60 pulses)
p2 (14 pulses)
p1 (4 pulses)
p4 (242 pulses)
E
cho
Inte
nsit
y
Time (ms)
T2 = 15ms
Coherence vs. concatenation level
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Sanity check: collapse adjacent pulses• Effect of combining pairs of adjacent pulses
– Ex. Z-Z I
– nth level concatenation without combining 2*4n – 2 = 510 for n=4
– nth level concatenation with combined pulses = 306 for n=4
0 5 10 15 20 250
5
10
15 p4, with all pulses preserved p4, consecutive pulses are combined
Con
cate
nate
d X
ZX
Z E
cho
Dec
ay
Time (ms)
T2 = 11 ms
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Sanity check: white noise
0 1 2
0.1
1
XZXZ(p3) = 410 s
T2 = 330 s T
1 = 420 s
R
elax
atio
n D
ecay
Time (ms)
Si:P at 10 K
Page 19
Summary• Dynamical decoupling can work for electron spins
• Through the hyperfine interaction with the electron can generate very fast bang-bang control of nucleus
• CPMG preserves initial x/2 with fewest pulses
– But does not deal with pulse errors for y/2
– CPMG cannot protect arbitrary state• Concatenated CPMG does no better
• Can utilize concatenated XZXZ sequence out to at least 1000 pulses
– Situation with y/2 initial states is more complex
• Not clear fidelity improves monotonically with level But much better than CP
• May need to combine XZXZ with composite pulses