Dynamical decoupling in solids
报告人：王亚
导师：杜江峰
University of Science & Technology of China
2011.8.5
Outline
 Suppressing decoherence
 Anomalous decoherence effect
 The application of DD in quantum metrology
 Future work
Decoherence suppression with DD
H  HS  HB  HSB
decoherence
Decoherence suppression
H SB  0
e
iH 1 n
e
 iH 2 2 iH 11
e
e
iHeff 1  2   n 
Hahn spin echo
 /2
SZ  A I
Z
k k

 SZ  A I
Z
k k


time
time
π
echo
Spin bath system
H k hf  S Z Ak I kZ
H hf  S Z  Ak I kZ
……
0  i j
H dd 
3(Si  eij )(S j  eij )  Si  S j 
3 
4 rij
Decoupling sequences
Dynamical decoupling(DD)
method
CPMG(PDD)
Concatenated DD(CDD)
Pulse sequence
T (2 j  1) / 2 N
even l : pl 1  pl 1
odd l : pl 1    pl 1
Optimal DD(UDD)
TSin2  j / 2N  2
locally optimized DD (LODD)
Optimization of UDD
sequence for given
noise
……
Reference
PRL 95, 180501 (2005)
PRL 98, 100504 (2007)
PRL 101, 180403 (2008)
Nature 458, 996 (2009)
PRL 103, 040501 (2009)
Impurity spin-based Quantum
computer
External field
0
1
P doped in silicon
Endohedral N@C60
[NV] center in
diamond
N
S = 1/2
……
S = 3/2
S=1
Interactions in an Electron Spin-Nuclear Bath
Coupling Solid State Computer
Magnetic Field
B
Electron Orbit
Crystal Lattice
Spin-orbital
Electron spin
Nuclear spin
Hyperfine:
Fermi Contact
Dipole-Dipole
Nuclear spin
Dipole-Dipole
Electron spin
Dipole-dipole
Typical electron spin decoherence time
in solids
Nuclear spin baths
Interaction
Decoherence
Characterized time
Crystal
Lattice(phonons)
T1
(Strongly dependent
on Temperature T)
T1
~ms (malonic acid @50K)
~ s (P:Si @6K)
T2*
T2*
~ns(P:Si or Quantum dots)
~s (NV center)
T2
T2
~s (malonic acid)
~ms (P:Si, NV center)
~s (QD)
Hyperfine
interaction
Hf & NuclearNuclear interaction
T1>>T2, spin bath is the main decoherence source
Qubit-bath model for pure dephasing
H  Sz  hz Sz  H N
Zeeman energy
Overhauser field
Nuclear spin interaction
(dipole-dipole, Zeeman
energy, etc.)
A block diagonal Hamiltonian for qubit

H    H    H

Decoherence by quantum entanglement

   I
H
  I  (t )
  I  (t )

I  I (t )  e
iH t
I
H
C


C


I


 


C   I  (t )  C   I  (t )
S (t )  C*C I  (t ) I  (t )
  I (t )    I (t )
Decoherence under Dynamical decouling


H 
H


H  
H 
t
t
DD under simulated noise
Michael J. Biercuk et al. Nature 458 996 (2009)
DD in solids: ensemble system
J.F Du et al.
Nature 461 1265 (2009)
After 7 pulses, the coherence enhancement is saturated due to the
electron-electron spin coupling.
DD in solids: single electron spin
G. Lange et. Al., Science 330, 60 (2010)
Up to 136 pulses, no limit is found
to the coherence enhancement
Pulse imperfections in DD
C.A.Ryan et al.
Phys. Rev. Lett 105, 200402 (2010)
Dynamical decoupling works in
muti-qubit case ?
Nested DD in multi-qubits case
Physical Review A 81, 012331(2010)
Physical Review A 82, 052338(2010)

 a   d 
 | a |2

 0
 ad 

 0
0
0
ad
0
0 | d |2
0
0
X1 (UDD2) X 0 (UDD1) X (UDD4)
0

0
0

0 
…
…
…
…
Phys. Rev. A 83, 022306 (2011)
Commute or anticommute
Single qubit operators, e.g.  i
n
Q2 ( XDD( N2 )) Q1 ( XDD( N1 )) QN ( XDD( N N ))
i  {x, y, z,0}
…
…
…
…
CW/Pulsed EPR sepctrometer
Anomalous Decoherence Effect in a
Quantum Bath
kcounts/s
Nitrogen Vacancy (NV) in Diamond
1μm
3A
2
Fluorescence
Optical Excitation
3E
C
C
C
N
V
B
C
C
C
Energy level
3E
1A
1
3A
2
ms=-1
ms=1
ms=0
ODMR spectrum
Rabi Oscillation
ODMR setup
Optical Parts
25
Microwave and Electronic parts
26
Quantum Description of decoherence
Y(0) = (a- - + a0 0 + a+ +
)Ä
J
Y(t ) = a- - Ä J - (t ) + a0 0 Ä J 0 (t ) + a+ e- ij ( t ) + Ä J + (t )
J a (t ) º exp(- iH ( a )t ) J
H ( a ) = a bz + H B
 Coherence is lost as which-way
information is recorded in the bath
- iH ( + )t
e
- iH ( - )t
e
- iH ( 0 )t
e
L0,± (t ) = J 0 (t ) J ± (t )
L+ ,- (t ) = J + (t ) J - (t )
3 ways for nuclear spin
of different electron state
Decoherence without DD
Observation of free induced decay of single- and multi-transition in NV
decay envelop
exp(- t 2 / T2*2 )
L0,+
L+,-
T2*=3.97
μs
T2*=1.82
μs
Anomalous Decoherence Effect in
Muti-pulse DD
Quantum Nature of ADE
H=
å
a a Ä (a bˆz + H B )
a = 0,±
in single transition (0,+) case:
H = 0 0 H B + + + (bz + H B )
= (+ + - 0 0 )
bz
b
+ ( z + HB )
2
2
in double transition (+,-) case:
H= -
- (- bz + H B ) + + + (bz + H B )
= (+ + - -
-
)bz + H B
demolished due to
bath operator
DD control
back action: cause high frequency moise
Application of dynamical decoupling in
Quantum metrology
Standard quantum limit and Heisenberg
limit
U  eiH

Generalized uncertainty relation :  h  1/ 2 

 h  [ 2 H j ]1/ 2
j
1
N
 M  m   h 
 M  m 
2
2
  1 /[  N  M  m ]
H j 
1
M 1 M N  m

2
N
 hmax   M  m 
2
  1 /[  N  M  m ]
 
Vittorio Giovannetti et al. PRL 96, 010401 (2006)
1
m
N

An example
1
1
1 0  0 1
2


1
1 0  ei 0 1

2

1
N 0 0 N
2

1
N 0  eiN 0 N

2
  1/ N

Quantum metrology with entanglement
J. M. Geremia, J. K. Stockton, H. Mabuchi, Science 304,270 (2004);
D. Leibfried , et al., Nature 438, 639 (2005);
Tomohisa Nagata, et al. Science 316, 726 (2007);
Jonathan A. Jones, et al., Science 324, 1166 (2009)
C. Gross, et al . , Nature 464, 1165-1169(2010);
……
Quantum metrology without entanglement:
Multi-round protocol
Vittorio Giovannetti et al. PRL 96, 010401 (2006)
B.L.Higgins et al.
Nature 450 393 (2007)
Geometric phase(GP):AA phase
[NV] center in
diamond
S=1
Addressing NV centre
Phase estimation with single GP
Phase estimation with multi GP
Classical:
Repeated N times
Prepare
Read out
φ
Quantum:
Repeated N times
Prepare
Read out
φ
Enhanced phase estimation with CPMG
k GP
2k GP
π
π
π
π
π/2
π
k GP
π
π
π
π
π
π
π/2
Analysis
I i  Exp ((
T 4
T
)  ( ' )) sin( wti  N )
T2
T2
I i  NExp ((
I 
 I
2
i
T 4
T
)  ( ' )) cos( wti  N )
T2
T2
/N
i
 Exp ((
 Exp ((
T 4
T
)  ( ' ))
T2
T2
 cos
i
T 4
T
)  ( ' ))
T2
T2
2
( wti  N )
Future work
 Robust gate
 Dynamically corrected gate(DCG)
PRL 102,080501(2009)…
 Combine DD with optimization control
 …
Thank you for your attention