electron spin decoherence in solid-state nuclear spin baths: understanding, control, and...
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Electron spin decoherence in solid-state nuclear spin baths: Understanding, control, and applications
Ren-Bao Liu
Department of Physics, The Chinese University of Hong Kong
http://www.phy.cuhk.edu.hk/rbliu
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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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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)
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