Transcript Slide 1

Pulse techniques for decoupling qubits
from noise: experimental tests
Steve Lyon, Princeton EE
Alexei Tyryshkin, Shyam Shankar,
Forrest Bradbury, Jianhua He, John Morton
• 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
Experiments
• 2-pulse Hahn echo
/2


Pulses

(|0 + |1)
FID – T2*
Signal
Echo
T
(|0 + |1)
• Decoupling
/2


Pulses
(|0 + |1)
Signal
(|0 + |1)
T
Echo
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 ~ 1s (for -pulse)
– Power ~ 1/(pulse length)2  Energy/pulse ~ power1/2
The Qubits:
31P
31P
donors in Si
donor: Electron spin (S) = ½ and Nuclear spin (I) = ½
↓e,↓n
X-band: magnetic field = 0.35 T
w1 ~ 9.7 GHz ≠ w2 ~ 9.8 GHz
↓e,↑n
w1
rf1
↑e,↓n
|2
w2
↑e,↑n
rf1 ~ 52 MHz ≠ rf2 ~ 65 MHz
|3
|1
rf2
|0
• Blue (microwave) transitions are usual ESR
• All transitions can be selectively addressed
Bang-Bang control
Fast nuclear refocusing
donor: S = ½ and I = ½
↓e,↓n
|3
↓e,↑n
|2
w1
Free nuclear spin nutation
0.0
2 rotation
|1
↑e,↑n
rf1
↑e,↓n
i = a|0 + b|1
(A)
2
|0
f = a|0 - b|1
Nuclear refocusing pulse
would be ~10 s
but electron pulse ~30 ns
Nuclear Polarization
31P
0.1
0.1
0.4
0.2
0.3
0.4
Two bursts of 2 mw pulses
(C)
0.0
0.3
One burst of 2 mw pulses
(B)
0.0
0.2
0.1
0.2
Time (ms)
0.3
0.4
Electron spin qubits
Doping ~1015/cm3
Isotopically purified 28Si:P
7K  electron T1 ~ 100’s milliseconds
7K  electron T2 ~ 60 milliseconds (extrapolating to ~single donor)
103
T1, T2 (sec)
•
•
•
•
(Feher et al, 1958)
102
T1
101
T2
T2
T1
100
“real” T2
10-1
28Si:P
9.767 GHz
28Si:P 9.767 GHz
Si:P 16.44 GHz
x
10-2
(Gordon, 28Si, 1958)
10-3
T2
10-4
10-5
10-6
10-7
0
2
4
6
8
10
12
14
Temperature (K)
16
18
20
22
Noise in electron spin echo signals
2
Decoherence
Spin echo signals
Spin echo decay
1.5
1.0
Single-pulse
T2 = 2 msec
0.5
1
In-phase
0
-1
Out-of-phase
-2
-3
0
averaged
0.0
0
2
4
T (msec)
2
4
6
8
T (msec)
6
8
Signal transferred: in-phase  out-of-phase
• Must use single pulses to measure decoherence
 About 100x sensitivity penalty
B-field noise
0.01
1/2
Noise (Gauss/Hz )
1E-3
Measure noise voltage
induced in coil
1E-4
1E-5
1E-6
1E-7
1E-8
10
100
1000
Frequency (Hz)
Origin of noise unclear
Background field in lab?
Domains in the iron?
 Essentially 1/f
10000
100000
Microwave Field Inhomogeneity
Vertical
B-field
Sapphire
cylinder
Metal
Wall
Sapphire
*
*
*
*
Carr-Purcell (CP) sequenceCP
x/2--X-2-X-2-…-X--echo
*
*
*
*
*
*
*
*
*
0.0
0.2
0.4
Time (ms)
0.6
0.8
Periodic (standard) CPMG
x/2--Y-2-Y-2-…-Y--echo
Self correcting sequence
2 pulse
Microwave signal
 pulse
*
*
*
*
*
 pulse
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Time (ms)
Coherence after N pulses
Standard CPMG
Echo Intensity
1.0
T2 = 8.5ms
1
0.5
4
16
64
0.0
0
4
8
12
Time (ms)
16
20
24
Concatenated CPMG
2 pulse
 pulse
 pulse
Microwave signal
*
*
0.0
*
0.5
*
*
1.0
Time (ms)
*
1.5
*
2.0
Coherence vs. concatenation level
Concatenated CPMG
l = 2 (2 pulses)
l = 4 (10 pulses)
l = 6 (42 pulses)
Echo Intensity
1.0
0.5
T2 = 5.8ms
0.0
0
2
4
6
8
10
Time (ms)
12
14
Concatenated and periodic CPMG
Echo Intensity
1.0
Concatenated CPMG
42 pulses
0.5
Periodic CPMG
32 pulses
0.0
0
4
8
12
16
Time (ms)
20
24
28
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
*
Concatenated
XZXZ (p2)
*
CPMG
0.0
0.2
0.4
Time (ms)
0.6
0.8
Coherence vs. concatenation level
Concatenated XZXZ pulse sequence
p1 (4 pulses)
1.5
Echo Intensity
p2 (14 pulses)
p3 (60 pulses)
1.0
p4 (242 pulses)
p5 (972 pulses)
0.5
T2 = 15ms
0.0
0
5
10
15
20
25
30
Time (ms)
35
40
45
50
Sanity check: collapse adjacent pulses
• Effect of combining pairs of adjacent pulses
Concatenated XZXZ Echo Decay
– 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
p4, with all pulses preserved
p4, consecutive pulses are combined
15
10
T2 = 11 ms
5
0
0
5
10
15
Time (ms)
20
25
Sanity check: white noise
Si:P at 10 K
Relaxation Decay
1
T2 = 330 s
T1 = 420 s
0.1
XZXZ(p3) = 410 s
0
1
Time (ms)
2
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