Pulse techniques for decoupling qubits from noise: experimental tests Bang-bang decoupling 31 P nuclear spins Low-decoherence electron-spin qubits and.

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

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)

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

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

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

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

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

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

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

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

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

Concatenated CPMG

0.0 0.5 1.0 1.5 2.0

*

*****

pulse pulse pulse

Time (ms)

Mic

row

ave

sign

al

*

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)

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

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

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

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

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

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