Electron spin decoherence in solid-state nuclear spin baths:
Understanding, control, and applications
Ren-Bao Liu
rbliu@phy.cuhk.edu.hk
Department of Physics, The Chinese University of Hong Kong
http://www.phy.cuhk.edu.hk/rbliu
Funded by Hong Kong RGC, NSFC, CUHK Focused Investments Scheme
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Thanks to
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)
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Wen Yang
Nan Zhao
Z. Y. Wang
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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).
   t    0
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Pictorial Spin Dynamics
Schrödinger equation
z
it   H   B  S 
B
y
t S  B  S
x
The spin precesses about the magnetic field
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Hahn echo
y
x
rotation 180o about x-axis
 
 
Coherence

 a  eiX J (t  )  eiX J b 
a   eiX J t b   a   eiX J b 

echo @ 2
1
0.8
π-flip @ τ
0.6
0.4
T2*
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Works perfectly for static fluctuations.
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Dynamical Fluctuations
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Semiclassical picture of decoherence
 1

Average
B(t )e z  Sˆ 
Sˆ  exp    B  t1  B  t2  dt1dt2 
 2

Decoherence control by spin-flips (rooted in spin echo)
 1

Average
F  t  B(t )e z  Sˆ 
Sˆ  exp    B  t1  B  t2  F t1  F t2  dt1dt2 
 2



 exp   S   F  , t  d 
1
F t   
1
1
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2
3
0
4
2

 N 1  N
t
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II. Quantum theory
Local magnetic field is a Q-number (quantum field)
Bˆ (t )ez  Sˆ
or exp  iHt  S  B
Quantum fluctuation:  H , B  0
Bˆ z I  BI I but H I  EI I
eiHt I   CI  t  I
so Bz  0, the local field gets quantum fluctuation
Thermal fluctuation:    PI I I and Bˆ z I  BI I
Classical noise, static inhomogeneous broadening
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Relevant systems: Electron spin in solids for qubits
self-assembled dot
gate-defined dot
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interface fluctuation islands
donor impurity P:Si
NV center in diamond
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1 electron spin + N nuclear spins in the bath
In GaAs QD, e.g.,
e-N: MHz >> N-N: kHz
In type-IIa diamond, e.g.,
NV--13C: kHz >>
13C - 13C: 10 Hz
The nuclear spins (bath)
within a range and the
electron spin (qubit)
form a relatively close
system.
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Qubit-bath model for pure dephasing
H  B0 Sz  bz Sz  H N
Bath spin interaction (dipoledipole, Zeeman energy, etc.)
Overhauser field operator
Zeeman energy
bz   An  J n
n
Old View: Bath imposes (quantum) noise on center spin
H     H      H  with H   HN   B0  bz  2
New view: Center spin imposes interaction on bath
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Decoherence by quantum entanglement

   I
  I  (t )
  I  (t )
Bifurcated bath evolution
 which-way info known
 decoherence
I  (t )  e
iH t
I
L  S  t   I   t  I   t 
  I (t )    I (t )
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Quantum many-body theory for spin bath dynamics

H    H    H

H    An  J n   Im  Dmn  I n
n
m, n
L  I exp  iH t  exp  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).
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).
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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)
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WITHOUT fitting parameters
Nitrogen-vacancy center spin in diamond
Black: Experiment [Lukin et al
Science (06)]
Blue: CCE calculation (Nan Zhao,
unpublished)
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Recoherence by disentanglement (quantum erasure)

   I
  I  (t )

    I (t )
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  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
recoherence @ 2
 pulse @ t  
W. Yao, RBL, and L. J. Sham, Phys. Rev. Lett. 98, 077602 (07).
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)
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NV center spins in diamond: Hot qubit
 Chemical stability
 Deep level: thermal stability
 Weak Spin-orbit interaction (light C
atoms, coherence @ RT)
 Low 13C abundance
 Transparent (optical access)
 Non-toxic (medicine)
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
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Pure-dephasing model for NV center spin in nuclear spin bath
e- 13C interaction >> 13C - 13C interaction
 About 500 13C spins form a “close” bath
H  Sz2   eB  S  S  A j  I j  I j  D  I k   13 CB  I j
NV spin splitting
hyperfine
Bath spin interaction (dipoledipole + Zeeman energy)
Bath Hamiltonian conditioned on center spin state:
H

 0,1or 1
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   H   , with H    Hbath   S   A j  I j
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The stronger, the weaker
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)
Can quantum bath be approximated by a classical noise?
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Spin decoherence: The oldwife tale
doublecoherence
1
L,
  2 B
1
Singlecoherence
L0,
10   B
0
Classical noise:
Average
F  t  B(t )e z  Sˆ 
 1

L0,  exp    B  t1  B  t2  F  t1  F  t2  dt1dt2 
 2


L ,  exp 2  B  t1  B  t2  F  t1  F  t2  dt1dt2
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
L,  L40,
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Free-induction decay due to thermal (classical) noises from 13C spins
N. Zhao, Z. Y. Wang & RBL,
PRL 106, 217205 (2011).
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L,
 L40,
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FID experiment & theory
Single coherence
Multi-coherence
Time (ms)
4
L,  L0,1
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Spin decoherence: When the bath is small (therefore quantum)
Quantum bath:
 1  0
 1   B 
    B  t 
   H  

L0,  B0  t  B  t  , but L,  B  t  B1  t  . Pronged quantum evolution under control
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Anomalous decoherence in a quantum bath
Stronger “noises” 
weaker decoherence !
Stronger noises on qubit 
Stronger control over environment!
N. Zhao, Z. Y. Wang & RBL,
PRL 106, 217205 (2011)
B=0.3 Tesla
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Conditional bath evolution at high field: Nuclear spin pair-flips
 a pseudo-spin under pseudo-fields
conditioned on 
hjk   X jk , 0, 0  no hf energy cost
0
NV
Z jk 

hjk   X jk , 0,  Z jk1

X jk
Z jk  hf energy cost
X jk dipolar flip-flop rate
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h(jk)
h (jk )
h (jk )   X jk , 0,  Z jk 
hyperfine  N-N  Z jk  X jk
h (jk ) are almost anti-parallel
flip @ 3
1  1
  ,
flip @ 
1  1
Multi-transition:
Pseudo-fields for the two e-spin states are almost anti-parallel
 slower decoherence
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h(jk) and h(0)
jk are NOT (anti-)parallel
h
()
jk
 0,
h (0)
jk
flip @ 
1  1
flip @ 3
1  1
Single-transition:
Pseudo-fields for the two e-spin states are not (anti-)parallel
 faster decoherence
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Experimental verification
At this weak field,
decoherence due mainly
to single nuclear spin
precessing.
Insensitive to specific
interactions.
Observable in other
systems, e.g., singlettriplet transitions?
B=5 Gauss. Calculation w/o fitting parameters
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Atomic-scale magnetometry using NV spin coherence
2>>1+1
1 nucleus is featureless; 2 (or more) nuclei have characteristic.
N. Zhao, J. L. Hu, S. W. Ho, J. T. K. Wan, & RBL, Nature Nanotech. 6, 242 (2011).
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Decoherence by pairwise flip-flop
Hahn echo, B=0.15 T
dimer
NV
dimer only
incl. all 13C spins
Previously noted by Maze et al (PRB 2008)
Dimer: interaction strength ~ hyperfine energy cost
 large-amplitude flip-flop
Many incoherent pairs  smooth decoherence
Rare coherent pairs  coherent oscillations
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A dancing couple out of random walkers
UDD1
UDD2
UDD3
UDD4
a dimer @ 1.2nm;
B0.15 T
 j 
Uhrig DD: t j  T sin 2 

2
N

2


UDD5
Coherence time prolonged by DD,
oscillations due to the dimer are pronounced.
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Atomic-scale magnetometry of a dimer
Azimuth angle f from [1-10]
A dimer @ ~1.2nm from NV; B=.15 T, tilted from [111] by 10°
NV center spin
decoherence vs. time
& B-field direction
Contribution by
the dimmer only
crossection plot
for f  15
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Fingerprint screening
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NMR of a 13C2 molecule?
Even better if NMR of real single
molecules outside diamond could
be detected.
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13C
NV
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Noise spectrum due to weak coupling to a molecule
Weak hyperfine coupling
Transition between nuclear spin states
Noise spectrum
S     k bk2    k 
e.g., transitions in a water
molecule under zero field
H
H
O
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Many-pulse DD: Suppressing noises but one @ a certain frequency
background noise
Dynamical decoupling
suppresses noises
1
1

3
5
7
t
T
Noise @ right frequency
f  N , f 2  N 2

2
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is enhanced by a factor of N2
(N: # of pulses)
c.f. optical grating effect
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
Decoherence  exp  f 2
Toward single molecule NMR
Spin coherence of an NV center 10 nm below
5 1H216O or 12C1H4 molecules, under 100-pulse
periodic dynamical decoupling, at zero B-field
13C
NV
H
12C
H
H
H
H
H
O
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Single-molecule NMR: Cascade amplification of weak signals
single photon
detection
1015 Hz
1 eV
103 Kelvin
single electron
spin resonance
coupling to
single nuclear
spin nearby
coupling to
distant
nuclear spins
noises @ fingerprint frequencies
amplified by many-pulse
dynamical decoupling
GHz
MHz
kHz
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)
For more, visit http://www.phy.cuhk.edu.hk/rbliu
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