Protection of center-spin coherence by a dynamically polarized nuclear spin core in a quantum dot

Wenxian Zhang ( 张文献 )

复旦大学 光科学与工程系J.-L. Hu, J. Zhuang, J. Q. You, R.-B. Liu

Aug. 3rd, 2010 @ 大连理工大学

第四届全国冷原子物理和量子信息青年学者学术讨论会

PRB 82, 045314 (2010).

Outline

Introduction and experimental background

A two-region model

DNP process: formation of a polarized core

Protection effect on the center-spin coherence

Numerical simulations and discussions of

experimental results

Conclusions

Solid-state architecture of QIP

Atomic/Optical

Cavity QED

Long coherence time Not easily interacted Not easily scaled

Solid-state

Quantum dots

Easily scalableEasily interacted Not so great coherence

Marcus@Harvard

Quantum dots

A.C. Johnson’s Thesis, Harvard Univ..

Experimental conditions:

QD size ~ 100 100 10 nm3

Low temperature ~ 100 mK

Low magnetic field 100 mT

Coherence time ~ 10 ns

Spatial degree of freedom frozen

500 nm

Qubit decoherence

Qubit decoherence:Interacting with environment → qubit “forgets” its phase

2

2

e

e

bab

abai

i

2

2

0

0

b

a

No decoherence: Complete decoherence:

Classical bit Quantum bit (Qubit)

0 1 0 1 0

2 states only

0

1

10 2/2/ ii beae

Arbitrary superposition of 2 basis states

Free induction decay

10 ns

Petta et al., Science 309, 2180 (2005)

Spin decoherence in a QD

Decoherence source – hyperfine interaction:

*

1

N

e B k kk

H g B S S A I

S – electron spin-1/2 Ik – k-th nuclear spin located at rk (also assume 1/2)

* 2

6

8u

3

~ 10

k e B n nA g g

N

Merkulov et al., Phys. Rev. B 65, 205309 (2002).

Erlingsson et al., Phys. Rev. B 70, 205327 (2004).

Deng & Hu, Phys. Rev. B 73, 241303(R) (2006).

Zhang et al., Phys. Rev. B 74, 205313 (2006).

Taylor et al., Phys. Rev. B 76, 035315 (2007).

Preserve coherence via spin echoPetta et al., Science 309, 2180 (2005)

Dephasing

only

Dynamical decoupling

Zhang et al., Phys. Rev. B 75, 201302(R) (2007); 77, 125336 (2008).

FID

NRD

PDDRPD

SDD

SRPD

PCDD2

τ = 0.1

10 ns

Preserve coherence via polarization

I. Uniform nuclear polarization (> 90% )G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999).

W. A. Coish and D. Loss, Phys. Rev. B 70, 195340 (2004).

C. Deng and X. Hu, Phys. Rev. B 73, 241303(R) (2006).

1. Thermal polarizaiton in strong magnetic field – 10%

2. Spin dependent optical pumping – 60%

II. Non-uniform nuclear polarization – DNP via electron

Experiments (~1%): D. J. Reilly et al., Science 321, 817 (2008);

Phys. Rev. Lett. 104, 236802 (2010).

Theories (~1%): G. Ramon and X. Hu, Phys. Rev. B 75, 161301(R) (2007).

M. Gullans et al., Phys. Rev. Lett. 104, 226807 (2010).

W. Zhang et al., Phys. Rev. B 82, 045314 (2010).

Uniform polarization effect (I)

Deng and Hu, Phys. Rev. B 73, 241303(R) (2006); Phys. Rev. B 78, 245301 (2008).

•N = 105, envelope of correlation function G⊥

•Polarization is uniform.

•10 times extension of dephasing time if P>90%

Uniform polarization effect (II)

2 2/ 2 / 22 2 3/ 2 3

2 2 1( ) 1 cos( )

2

( ) ( ) 22 2 2

z Dt Du t e D t i eD D D

D D DErf t i Erf t i Erf i

QSA

Gaussian

Random

Gaussian

Random

p = 0.46 p = 0.76

Zhang et al., Phys. Rev. B 74, 205313 (2006).

DNP process

12

3

1 2 3

Electron spin:

0 +2 *

Nuclear spin:

0 -2 -2

Double quantum dots

Unpolarized Maximally polarized

Ak = A, uniform

Scaled with N=105

Zamboni Effect

50 times longer

Ramon and Hu, Phys. Rev. B 75, 161301(R) (2007).

Non-uniform polarization

Polarization transfer – a two-region model

H0 = 0 H0 = 6

Saturation at long time in large magnetic fields, Ikz ~ (Ak / H0)2.

A1 = 10 A2

I 1I 2

Polarization ratio (single DNP cycle)

r ≡ P1 / P2 = (A2 / A1)2 in medium-to-large magnetic fields.

Strongly coupled spins have higher polarization.

A2 / A1 = 0.1I 1 / I

2

r

Multiple DNP cycles

times.longat )tanh( s;short timeat 22 zkk

zkk

zk IAPAP

Polarized core protection effectP

olar

izat

ion

0 2 4

Two-region model

Two effects are separable:

1. T1/2 is determined by b2 = (N2)1/2 A2 instead of b = (N1

A12+N2 A2

2)1/2 ;

2. Protection effect of the polarized core: What skirt spins decoheres is not a single electron spins but a compound of an electron spin and a polarized nuclear spin core, which further makes the coherence time longer and make T1/2 increase linearly with N1.

Experimental results

Numerical methodsSmall N: Chebyshev expansion

Dobrovitski et al., PRE 67, 056702 (2003).

0 0( ) exp( ) ( )t it U t H

0

( ) exp( ) ( ) ( ) ( )kk k k

k

U t i a i J T

G G

1 1( ) 2 ( ) ( )k k kT T T G G G G

-1 G 1

Large N: P-representation Al-Hassanieh et al., PRL 97, 037204 (2006).

cos( / 2) sin( / 2) ie

0 0

( ) ({ , }, ) sinNN

j j j j j j jj j

t p t d d

0 0

Tr( ( ) ) ({ , }, ) sinNN

j j j j j j jj j

x t X p t X d d

Protect effect in a QD

N=20, Chebshev method

Large bath with P-representation

N=20 N=256

40 times extension of DNP with P = 0.7 (N=20), 0.25 (N=256) and Pk ~ Ak

2;

Linear decay at large polarizations; Small oscillations at short times;

Abrupt increase of T1/2 if P is larger than a critical value PC but much smaller than 1;

PC decreases with N increasing;

Polarized nuclear spin core is formed if P > PC;

Protection effect of the polarized core.

Protection effect: Summary

Thank You!

Polarized core protection effect

Gaussian model Two region model

Exponential increase of T1/2

1 2

2 1

b bH

b b

1 22

1 21 1

; N N

k jk j

b A b A

22

1/ 21

~ ~ PbT e

bPerturbation results

(Fermi golden rule):Gaussian Ak

DNP effect I

Sx Sz

N=20

Gaussian Ak

P = 0.68

Double quantum dot

Gaussian Ak, N = 21

DNP, Pk ~ Ak2

P = 0.7

Not related?

Experimental results