ren-bao liu [email protected] department of physics, the chinese university of hong kong

Electron spin decoherence in solid-state nuclear spin baths: Understanding, control, and applications

Ren-Bao Liu

[email protected]

Department of Physics, The Chinese University of Hong Kong

http://www.phy.cuhk.edu.hk/rbliu

ASI 08/12/2011 1www.phy.cuhk.edu.hk/

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Funded by Hong Kong RGC, NSFC, CUHK Focused Investments Scheme

Wen Yang (postdoc, now @UCSD) Nan Zhao (Postdoc)Jian-Liang Hu (PhD student)Zhen-Yu Wang (PhD student)Sai-Wah Ho (MPhil student)Jones Z. K. Wan (Postdoc)

Jiangfeng Du, Xing Rong, Ya Wang, Jiahui Yang, Pu Huang, Xi Kong, Pengfei Wang, Fazhan Shi (experimentalists @ USTC)

Lu J. Sham (UCSD)Wang Yao (UCSD, now @ HKU)

Thanks to

Wen Yang Nan Zhao

Z. Y. Wang


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ASI 08/12/2011www.phy.cuhk.edu.hk/

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Outline

Introduction - Semiclassical theory: Gone can be back

Introduction - Quantum theory: Passive can be active

A difference between the two: Strong can be weak

An application of decoherence: Bad can be good


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I. Spin decoherence & control:Semiclassical theory

R. Kubo, J. Phys. Soc. Jpn. 9, 935 (1954).P. W. Anderson, J. Phys. Soc. Jpn. 9, 316 (1954).


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Coherence of the slow and the swift

It works when the snails’ speeds are kept constant (but random).

0t


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Pictorial Spin Dynamics

t S B S

ti H B S

y

z

x

B

The spin precesses about the magnetic field

Schrödinger equation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

t

Cohere

nce

π-flip @ τ

echo @ 2

*2T

Hahn echo

Works perfectly for static fluctuations. Dynamical Fluctuations

JiX ta e b JiXa e b ( )J JiX t iXa e e b

rotation 180o about x-axis

xy


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Decoherence control by spin-flips (rooted in spin echo)

Average1 2 1 2 1 2

1ˆ ˆ( ) exp2zF t B t S B t B t F t F t dt dt

e S

Semiclassical picture of decoherence

Average1 2 1 2

1ˆ ˆ( ) exp2zB t S B t B t dt dt

e S

2

0exp ,S F dt

F t 1

1 N t1 3 4 1N 2


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II. Quantum theory

or exp iHt S B ˆˆ( ) zB t e S

Quantum fluctu a 0tion: ,H B

ˆ

theso 0, local field gets quantum fluctuati

t

o

u

n

bz I

iHt

I

I

z

B I B I

e I C t

H

B

I

I E I

Local magnetic field is a Q-number (quantum field)


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ˆ Thermal fluctuation: and I z IP I I B I B I Classical noise, static inhomogeneous broadening

Relevant systems: Electron spin in solids for qubits

self-assembled dot interface fluctuation islands

gate-defined dot donor impurity P:Si NV center in diamond


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1 electron spin + N nuclear spins in the bath

The nuclear spins (bath) within a range and the electron spin (qubit) form a relatively close system.

In type-IIa diamond, e.g., NV--13C: kHz >>13C - 13C: 10 Hz


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In GaAs QD, e.g., e-N: MHz >> N-N: kHz

Qubit-bath model for pure dephasing

0 z z z NH B S b S H

Zeeman energy

Overhauser field operator

Bath spin interaction (dipole-dipole, Zeeman energy, etc.)

New view: Center spin imposes interaction on bath

0 with 2N zH H H H H B b

z n nn

b A J


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Old View: Bath imposes (quantum) noise on center spin

Decoherence by quantum entanglement

( )I t ( )I t

I Bifurcated bath evolution which-way info known decoherence

( ) ( )I t I t

( ) iH tI t e I

L S t I t I t


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Quantum many-body theory for spin bath dynamics

Cluster-correlation expansion (a generalization of textbook cluster expansion to finite systems, good for nano-science):W. Yang & RBL, Phys. Rev. B 78, 085315 (2008).

H H H

,

mnn n m nn m n

H D A J I I��

exp exp ?L I iH t iH t I

Step stones:0. Semiclassical spectral diffusion theory, Anderson, Kubo (1956)1. Cluster expansion, Witzel & Das Sarma (2005)2. Pair-correlation: Yao, RBL & Sham (2006).


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Experiments vs. theory

Phosphorus donor spins in silicon

Black: Experiment [Lyon et al, PRB (2003)]Red: CCE calculation (Nan Zhao, unpublished)

Nitrogen-vacancy center spin in diamond

WITHOUT fitting parameters

Black: Experiment [Lukin et al Science (06)]Blue: CCE calculation (Nan Zhao, unpublished)


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Recoherence by disentanglement (quantum erasure)

( )I t ( )I t

I

( )I t

Bifurcated bath evolution which-way info known less coherence left

qubit flip bath pathways exchange directions pathway intercross which-way info erased recoherence


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Resurrecting from ashes: When disentangled

W. Yao, RBL, and L. J. Sham, Phys. Rev. Lett. 98, 077602 (07).

pulse @ t

recoherence @ 2


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Observable if thermal fluctuation suppressed:Duncan Steel, Amir Yacoby, …?


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Dynamical disentanglement and dynamical decoupling

Talks in this ASI by Lu Sham, Goetz Uhrig, Jiangfeng Du, Jiangbin Gong, S. Das Sarma, Amir Yacoby, Joerg Wrachtrup

Reviews, e.g., W. Yang, Z. Y. Wang and R. B. Liu, Front. Phys. 6, 2 (2011).Z. Y. Wang and R. B. Liu, Chapter 15 in Quantum Error Correction, eds. D. Lidar et al (Cambridge U Press, in press)

NV center spins in diamond: Hot qubit


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Quantum coherence time is long @ RT in this US$10.8M worth type-IIa diamond, good for for solid-state quantum computing & magnetometry, http://news.yahoo.com

Chemical stability Deep level: thermal stability Weak Spin-orbit interaction (light C

atoms, coherence @ RT) Low 13C abundance Transparent (optical access) Non-toxic (medicine)

http://news.yahoo.com/

Pure-dephasing model for NV center spin in nuclear spin bath

13

2

Cjz kj je jAH S D S I BB I IS I��

NV spin splitting hyperfine Bath spin interaction (dipole-dipole + Zeeman energy)

Bath Hamiltonian conditioned on center spin state:

bath

0, 1or 1

, with j jH H H H A

S I��

e- 13C interaction >> 13C - 13C interaction About 500 13C spins form a “close” bath


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Anomalous decoherence effect in a quantum bath

Theory: N. Zhao, Z. Y. Wang & RBL, PRL 106, 217205 (2011).Experiments: P. Huang et al. Nature Comm. 2, 570 (2011)


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The stronger, the weaker

Can quantum bath be approximated by a classical noise?

0

1

1

0,L

,L

10 B

2 B

Single-coherence

double-coherence


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Average

0, 1 2 1 2 1 2

, 1 2 1 2 1 2

ˆ( )

1exp

2

exp 2

Classical noise:

zF t B t

L B t B t F t F t dt dt

L B t B t F t F t dt dt

e S

4, 0,L L

Spin decoherence: The oldwife tale

Free-induction decay due to thermal (classical) noises from 13C spins

40,, LL N. Zhao, Z. Y. Wang & RBL,

PRL 106, 217205 (2011).


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FID experiment & theory


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4, 0,1L L

Time (s)

Single coherence Multi-coherence

0, 0 , 1, but . Pronged quantum evolution

1 0

under contr

1

ol

Quantum bath: H

L B t B

B

t L B t

B t

B t


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Spin decoherence: When the bath is small (therefore quantum)

Anomalous decoherence in a quantum bath

Stronger “noises” weaker decoherence !

N. Zhao, Z. Y. Wang & RBL,PRL 106, 217205 (2011)


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Stronger noises on qubit Stronger control over environment!

B=0.3 Tesla

Conditional bath evolution at high field: Nuclear spin pair-flips

NV

jkX jkZ

dipolar flip-flop ratejkX

hf energy costjkZ

0

1

a pseudo-spin under pseudo-fields

conditioned on

no hf energy cost

,0,0

,0,

jk jk

jk jk jk

X

X Z

h

h


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( )jk h ( )

jkh

flip @

1 1

flip @ 3

1 1

,

( )

( )

,0,

hyperfine N-N

are almost anti-parallel

jk jk jk

jk jk

jk

X Z

Z X

h

h

Multi-transition: Pseudo-fields for the two e-spin states are almost anti-parallel slower decoherence


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(0)jkh

( )jkh

flip @

1 1

0,

flip @ 3

1 1

Single-transition: Pseudo-fields for the two e-spin states are not (anti-)parallel faster decoherence

( ) (0)and are NOT (anti-)paralleljk jkh h


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Experimental verification


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B=5 Gauss. Calculation w/o fitting parameters

At this weak field, decoherence due mainly to single nuclear spin precessing.

Insensitive to specific interactions.

Observable in other systems, e.g., singlet-triplet transitions?

Atomic-scale magnetometry using NV spin coherence


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N. Zhao, J. L. Hu, S. W. Ho, J. T. K. Wan, & RBL, Nature Nanotech. 6, 242 (2011).

2>>1+1

1 nucleus is featureless; 2 (or more) nuclei have characteristic.

Decoherence by pairwise flip-flop

B=0.15 THahn echo,

incl. all 13C spins

Rare coherent pairs coherent oscillations

dimerNV

Many incoherent pairs smooth decoherence

Dimer: interaction strength ~ hyperfine energy cost large-amplitude flip-flop

dimer only


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Previously noted by Maze et al (PRB 2008)

A dancing couple out of random walkers

UDD1

UDD2

UDD3

UDD4

UDD5Coherence time prolonged by DD, oscillations due to the dimer are pronounced.

22sin 2

N

jTt j

Uhrig DD:

a dimer @ 1.2nm; B0.15 T


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Atomic-scale magnetometry of a dimer


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Azi

mut

h an

gle

fro

m [

1-10

]

Contribution by the dimmer only

crossection plotfor 15

A dimer @ ~1.2nm from NV; B=.15 T, tilted from [111] by 10°

NV center spin decoherence vs. time & B-field direction

Fingerprint screening


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NMR of a 13C2 molecule?

Even better if NMR of real single molecules outside diamond could be detected. NV

13C

Noise spectrum due to weak coupling to a molecule

Weak hyperfine couplingTransition between nuclear spin states Noise spectrum

2k kk

S b

e.g., transitions in a water molecule under zero field

O

HH


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1

1T

t 5 73

is enhanced by a factor of N2 (N: # of pulses)

Noise @ right frequency

Many-pulse DD: Suppressing noises but one @ a certain frequency


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Dynamical decoupling suppresses noises

c.f. optical grating effect

2 2

2

,

Decoherence exp 2

N N

background noise

Toward single molecule NMR

NV13C

Spin coherence of an NV center 10 nm below 5 1H2

16O or 12C1H4 molecules, under 100-pulse periodic dynamical decoupling, at zero B-field

OHH

H

H

HH

12C


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Single-molecule NMR: Cascade amplification of weak signals

1015 Hz1 eV103 Kelvin

single photon detection

coupling to distant

nuclear spins

GHz

single electron spin resonance

MHz

kHz

coupling to single nuclear

spin nearby

noises @ fingerprint frequencies amplified by many-pulse

dynamical decoupling

fingerprint oscillation of nuclear spin clusters

Features:

Full information about nuclear spin interaction (c..f. liquid-state NMR: dipolar intra-molecule interaction averaged to zero by rapid rotation of molecules under B field)

High-resolution of resonances (c.f., solid-state NMR: inter-molecule interaction causes large broadening)


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Summary

quantum theory and hence control schemes;

Anomalous effect in quantum bath: Stronger “noises” may cause slower decoherence;

Atomic-scale magnetometry of single nuclear spin clusters at distance;

Single-molecule NMR by many-pulse DD

Perspective: Single center spins as media for detecting physics and manipulating information in a quantum bath (e.g., nuclear spins)


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For more, visit http://www.phy.cuhk.edu.hk/rbliu

ren-bao liu [email protected] department of physics, the chinese university of hong kong

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