Dynamical decoupling with imperfect pulses
Viatcheslav Dobrovitski
Ames Laboratory US DOE, Iowa State University
Works done in collaboration with
Z.H. Wang, B. N. Harmon (Ames Lab),
G. de Lange, D. Riste, R. Hanson (TU Delft),
G. D. Fuchs, D. D. Awschalom (UCSB),
L. Santos (Yeshiva U.), K. Khodjasteh, L. Viola (Dartmouth College)
S. Lyon, A. Tyryshkin (Princeton)
W. Zhang (Fudan), N. Konstantinidis (Fribourg)
Outline
1. Introduction – what are we doing and why.
2. Quantum dots – some lessons and caveats.
3. P donors in Si – how pulse errors qualitatively change
the spin dynamics.
4. Dynamical decoupling of a single spin – decoupling
protocols for a NV center in diamond.
Quantum spins in solid state
NV center in diamond
Localized electron spin S=1
P donor in silicon
Localized electron S=1/2
Quantum dots
Localized electron S=1/2
Fundamental questions:
How to reliably manipulate quantum spins
How to accurately model dynamics of driven spins
Which dynamics is typical
Which dynamics is interesting
Which dynamics is useful
Possible applications
Magnetometry with nanoscale resolution
STM
ODMR nanoprobe:
quantum dot, NV center, …
Quantum computation
Array of quantum dots
NV centers in a waveguide
Quantum repeater
2-qubit quantum computer
NV center with
an electron and a nuclear spin (15N or 13C)
General problem: decoherence
  a   be i 
 a2
  
i
ab
e

ab e  i 

2
b 
Influence of environment: nuclear spins,
phonons, conduction electrons, …
 a2
  
0
0

2
b 
Decoherence:
phase is forgotten
Dynamical decoupling: applying a sequence of pulses to
negate the effect of environment
Spectacular recent progress in DD on single spins
Bluhm, Foletti, Neder, Rudner, Mahalu, Umansky, Yacoby, 2010:
16-pulse CPMG sequence on quantum dot
arXiv:1005.2995
de Lange, Wang, Riste, Dobrovitski, Hanson, 2010:
DD on a single solid-state spin (NV center in diamond)
136 pulses, ideal scaling with Np
Coherence time increased by a factor of 26
arXiv:1008.2119
Pulse imperfections start playing a major role
Qualitatively change the spin dynamics
Need to be carefully analyzed
Talking about dirt
Studying dirt can be useful
Antoni van Leeuwenhoek
Delft, 17th century
Studied dirt – discovered germs
Ames Lab + TU Delft, 21st century
Studied dirt, achieved DD on a single solid-state spin
Dynamical decoupling protocols
General approach – e.g., group-theoretic methods
Examples:
H  SZ  Ak I kZ
Periodic DD (CPMG, pulses along X):
Period d-X-d-X
(d – free evolution)
H  SX  Ak IkX  SY  Bk IkY  SZ Ck IkZ
Universal DD (2-axis, e.g. X and Y):
Period d-X-d-Y-d-X-d-Y
Can also choose XZ PDD, or YZ PDD – ideally, all the same
(in reality, different – see further)
Performance of DD and advanced protocols
Assessing DD performance: Magnus expansion
(asymptotic expansion for small period duration T )
Uper  exp[i T ( H (0)  H (1)  H ( 2)  ...)]
O (1)
Symmetrized XY PDD (XY SDD):
O (T )
O(T 2 )
XYXY-YXYX
2nd order protocol, error O(T2)
Concatenated XY PDD (CDD)
level l=1 (CDD1 = PDD):
d-X-d-Y-d-X-d-Y
level l=2 (CDD2):
etc.
PDD-X-PDD-Y-PDD-X-PDD-Y
Why we need something else?
Deficiencies of Magnus expansion:
• Norm of H(0), H(1),… – grows with the size of the bath
• Validity conditions are often not satisfied in reality
(but DD works)
• Behavior at long times – unclear
• Role of experimental errors and imperfections – unknown
• Possible accumulation of errors and imperfections with time
Numerical simulations:
realistic treatment and independent validity check
Numerical approaches
1. Exact solution
The whole system (S+B) is isolated and is in pure quantum state
H  H S  H B  H SB
(t )  T exp(iHt) (0)  U (t ) (0)
(t )  Ckm (t ) Sk Bm
Sk , Bm - basis statesof thesystem
and thebath
Very demanding: memory and time grow exponentially with N
Special numerical techniques are needed to deal with d ~ 109
(Chebyshev polynomial expansion, Suzuki-Trotter decomposition)
Still, N up to 30 can be treated
2. Some special cases – bath as a classical noise
Random time-varying magnetic field acting on the spin
Dynamical decoupling
for a single-electron quantum dot
Single electron spin in a quantum dot
Hyperfine spin coupling
Fermi contact interaction
Single electron QD

H   Ak S I k
electron spin
(delocalized)
N
k 1
nuclear spins
(Ga, As nuclei)
Hahn echo : from T2* ~ 10 ns to T2 ~ 1 μs
Universal DD: protect all three components of the spin

H  H C (t )   Ak S I k  H B
N
k 1
control Hamiltonian
Is Magnus expansion sufficient ?
Periodic DD (PDD)
d-X-d-Y-d-X-d-Y
Symmetrized DD (SDD)
XYXY-YXYX
Concatenated, level 2 (CDD2)
PDD-X-PDD-Y-PDD-X-PDD-Y
PDD
Magnus expansion
is valid
only for  ≤ 10 ps
SDD
CDD2
ME valid
Preserving unknown state of the spin
t
Decoherence:  0 

S (t )
Fmin (t )  min  0  S (t )  0
Worst-case scenario: minimum fidelity
• 8 different protocols
1.0
0.9
SRPD
Minimum fidelity
• Imperfections considered
CDD2
SHD
• Large τ (up to 5 ns)
• Long times
0
0.8
SDD
RPD
PDD
NRD
0.7
• Finite-width pulses
FID
0.6
FID
PDD
SDD
CDD2
• Intra-bath interactions
0.5
0.1
1
10
100
Time
DD works very well – but ME is not valid
NRD
RPD
SRPD
SHD
1000
Long times: fidelity saturation

H 0   Ak S I k
N
k 1
SX (t)
1.0
XY PDD
0.8
0.6
τ = 0.01
0.4
100
τ = 0.01
0.8
τ = 0.1
0.6
200
300
τ=1
0.2
τ=1
0
B
B
B
1.0
0.4
τ = 0.1
0.2
0.0
SZ (t)
400
0.0
0
Time
50
100
150
Time
  0 : U p / 2  X U ( ) Y U ( )  exp(i S z ) – commutes with Sz
Sz is a “quasi-conserved” quantity
Quantum tomography is a must to confirm decoupled qubit
DD for P donors in silicon:
pulse errors and fidelity saturation
DD for P donors in silicon, fidelity for different states
(S. Lyon and A. Tyryshkin)
XZ PDD
SY
quasi-conserved
Fidelity
1.0
Initial state along Y
0.5
Initial state along X
0.0
0
5
10
15
Number of Repeats
XY PDD
SZ
quasi-conserved
Fidelity
1.0
Initial state along Y
0.5
Initial state along X
0.0
0
5
10
15
20
25
Number of Repeats
30
P donors in Si: key features
1. Ensemble experiments: ESR on a large number of P spins
2.
29Si
– depleted sample: f = 800 ppm
(naturally, f=4.67%)
3. Inhomogeneous broadening: cw ESR linewidth 50 mG
4. However, T2 = 6 ms – plenty of room for DD
Dephasing by almost static bath – decoupling should be perfect
Model: pulse field inhomogeneity
Bpulse (x)
Sample
x
• Rotation angle is not exactly π
everywhere
• Rotation axis is not exactly X
(or Y) everywhere
Freezing in Si:P, qualitative picture
Consider some spin

U X  exp[i (   X )(Sn )]

UY  exp[i (   Y )(Sm )]
PDD, after 1/2-cycle: U1/ 2
Ud  exp[i B SZ ]

 exp[i(   ) ( a )]
(composition of rotations = rotation)
After N cycles: U N  exp[i 2 N ( a)]
Each spin rotates around its own axis, by
its own angle
But all axes are close to Y (for PDD XZ)
Total spin component along Y – conserved,
other components average to zero
Simplified analytics (leading order in pulse errors)
XZ PDD
XY PDD
  O ( X ,  Y )
  O ( X2 ,  Y2 )
aX  0
aX  0
aY  1
aY  0
aZ  0
a Z  1
All rotation axes close to Y
Rotation angle – 1st order in εX , εY
All rotation axes close to Z
Rotation angle – 2nd order in εX , εY
SY – frozen, SX and SZ decay fast
SZ – frozen, SX and SY decay slow
In agreement with experiment
Quantitative treatment: numerics vs. experiment
XZ PDD
XY PDD
1.0
1.0
Fidelity
0.8
SY
0.8
0.6
0.6
0.4
0.4
0.2
SX
SZ
0.0
5
10
NC (the number of cycles)
15
SZ
SX
SY
0.2
0.0
0
B
C
D
B
C
0
5
10
15
20
25
30
NC (the number of cycles)
Hollow squares – experiment, dots – theory
Rotation angle errors (εX , εY) – distribution width 0.3 (~15º)
Rotation axis errors (nZ, mZ) – distribution width 0.12 (~7º)
35
Concatenation: single-cycle fidelity
XZ CDDs
XY CDDs
1.0
B
Nothing
to show
C
D
All fidelities
are 1 (within 2%)
Fidelity
0.8
0.6
SY
SZ
SX
0.4
0.2
0.0
0
1
2
3
4
Concatenation level
Analytical result:
CDDs of all levels have the same error,
in spite of exponentially increasing number of cycles
Symmetrization: XY-8 sequence
Periodic DD (PDD)
d-X-d-Y-d-X-d-Y
Symmetrized DD
XYXY-YXYX
(called XY-8 in the original paper)
1.0
Hollow circles – PDD XY
Fidelity
0.8
Solid circles
0.6
SX
SZ
0.4
0.2
0.0
SY
0
5
10
15
20
25
NC (number of cycles)
30
35
dXdYdXdY-YdXdYdXd, (cycle
PDD (<Sx>)
–<Sy>
SDD (XY-8)
B
C
D
• Less freezing
• Overall better fidelity
Aperiodic sequences: Uhrig’s DD
Optimization of the inter-pulse intervals: UDD
j
t j  T sin
2( N p  1)
2
Np = 20:
1.0
SX
0.5
Fidelity
Fidelity
1.0
Y Axis Title
B
SZ
SY
0.0
X Axis Title
SX
10000 (pulse)
SY
0.5 number (2<Sy>)
Np= (2<Sz>)
SZ
0.0
0
100
Total time (s)
All errors
200
0
50
100
150
Total time (s)
nZ errors only
UDD is not robust wrt pulse errors
Very susceptible to the rotation angle errors
200
Aperiodic sequences: Quadratic DD
3rd order QDD: U4(Y)-X-U4(Y)-X-U4(Y)-X-U4(Y)-X
Np = 20
U4(Y) = Uhrig’s DD with 4 pulses
Fidelity
1.0
0.5
All errors
0.0
0
50
100
B
C
D
0.5
εX only
0.0
150
200
0
Total time (s)
50
100
150
200
Total time (s)
1.0
SX
SY
SZ
Fidelity
Fidelity
1.0
0.5
εY only
B
C
D
0.0
0
50
100
150
Total time (s)
200
Lessons learned so far:
1. Pulse errors are important
2. Pulse errors can accumulate pretty fast
3. Concatenated design is very good: errors stay the same
in spite of exponentially growing number of pulses
4. Fidelity of different initial states must be measured.
5. Freezing is a sign of low fidelity
6. UDD and QDD require very precise pulses
DD for spins in diamond
Nitrogen-vacancy centers
Studying a single solid-state spin: NV center in diamond
Diamond – solid-state version of vacuum:
no conduction electrons, few phonons, few impurity spins, …
Simplest impurity:
substitutional N
Bath spins S = 1/2
Distance between spins ~ 10 nm
Nitrogen meets vacancy:
NV center
Ground state spin 1
Easy-plane anisotropy
Distance between centers: ~ 2 μm
NV center – solid-state version of trapped atom
3E
ISC (m = ±1 only)
1A
532 nm
3A
m = 0 – always emits light
m = ±1 – not
Initialization: m = 0 state
Readout (PL level): population of m = 0
Ground state triplet:
m = ±1
2.87 GHz
m=0
Individual NV centers can be
initialized and read out:
access to a single spin dynamics
NV center and bath spins
Most important baths:
• Single nitrogens (electron spins)
• 13C nuclear spins
Long-range dipolar coupling
DD on a single NV center
• Absence of inhomogeneous broadening
• Pulses can be fine-tuned: small errors achievable
• Very strong driving is possible
(MW driving field can be concentrated in small volume)
• NV bonus: adjustable baths – good testbed for DD and
quantum control protocols
NV center in a spin bath
C
C
N
V
C
C
Bath spin – N atom
NV spin
ms = +1
ms = -1 
C
C
MW
Electron spin: pseudospin 1/2
14N nuclear spin: I = 1
ms = 0 
m = +1/2
1
MW
ms = -1/2
0
B
B
No flip-flops between NV and the bath
Decoherence of NV – pure dephasing
0.5
Ramsey decay
Decay of envelope:

exp[ t T
-0.5
0
0.2
0.4 0.6
t (µs)
0.8
]
* 2
2
Slow modulation:
hf coupling to 14N
T2* = 380 ns
A = 2.3 MHz
Need fast pulses
Strong driving of a single NV center
Pulses 3-5 ns long → Driving field in the range close to GHz
Standard NMR / ESR, weak driving

BL
B1 cosLt
L
Rotating frame
y
L
Spin
x
Oscillating field
L

S
L
co-rotating
(resonant)
counter-rotating
(negligible)
Rotating frame: static field B1/2 along X-axis
Strong driving of a single NV center
Experiment
“Square” pulses:
Simulation
29 MHz
109 MHz
223 MHz
Time (ns)
Gaussian pulses:
Time (ns)
109 MHz
223 MHz
• Rotating-frame approximation invalid: counter-rotating field
• Role of pulse imperfections, especially at the pulse edges
Characterizing / tuning DD pulses for NV center


U X  exp[i(   X )(Sn)]
n  (nX , nY , nZ )
Pulse errors - important: see Si:P DD
- unavoidable: counter-rotating field, pulse edges
- all errors (nX, nY, nZ, εX)
We want to determine and/or reduce the pulse errors
Known NMR tuning sequences:
• Long sequences (10-100 pulses) – our T2* is too short
• Some errors are negligible – for us, all errors are important
• Assume smoothly changing driving field – our pulses are too short
Can not be directly applied to strong driving
Quantum process tomography
out  L[ in ] – linear relation between “in” and “out”
Describes most of experimental situations – QM is linear !
in  a0  a1 X  a2 Y  a3 Z
out  b0  b1 X  b2 Y  b3 Z
a’s and b’s are linearly related – matrix χ – complete description of L
1. Prepare full set of basis states  ,  , X     , Y    i 
2. Apply process L[ρ] to each of them
3. Measure in the same basis: determine χ
Our situation: • Can reliably prepare only state 
• Can reliably measure only SZ
“Bootstrap” problem
“Bootstrap” protocol
Assume: errors are small, decoherence during pulse negligible

U X  exp[i(   )( n) / 2]    i( X   Y Y   Z Z )
Series 0:
Series 1:
Series 2:
π/2X and π/2Y
πX – π/2X, πY – π/2Y
π/2X – πY, π/2Y – πX
Find φ' and χ' (angle errors)
Find φ and χ (for π pulses)
Find εZ and vZ (axis errors, π pulses)
Series 3:
π/2X – π/2Y,
π/2Y – π/2X
π/2X – πX – π/2Y, π/2Y – πX – π/2X
π/2X – πY – π/2Y, π/2Y – πY – π/2X
Gives 5 independent equations for 5 independent parameters
All errors are determined from scratch, with imperfect pulses
Bonuses:
• Signal is proportional to error (not to its square)
• Signal is zero for no errors (better sensitivity)
Bootstrap protocol: experiments
Introduce known errors:
- phase of π/2Y pulse
- frequency offset
Self-consistency check: QPT with corrections
- Prepare imperfect basis states
1 , 0 , 1  0 , 1 i 0
- corrected
- uncorrected
- Apply corrections
(errors are known)
- Compare with uncorrected
Ideal recovery: F = 1, M2 = 0
F  T r[  0  ]
M 2  M 2 , M    0
M2
Fidelity
What to expect for DD? Bath dynamics
B(0) B(t )
Mean field: bath as a random field B(t)
1.0
Gaussian, stationary, Markovian noise:
Ornstein-Uhlenbeck process
0.8
B(0) B(t )  b2 exp(Rt)
0.4
simulation
O-U fitting
0.6
b – noise magnitude (spin-bath coupling)
R = 1/τC – rate of fluctuations (intra-bath coupling)
0.2
0.0
0
10
20
30
Time
Agrees with experiments: pure dephasing by O-U noise
0.5
0.5
Ramsey decay
Spin echo
*
T2 = 380 ns

exp[ t T
-0.5
0
0.2
0.4
t (µs)
]
* 2
2
0.6
0.8
T2 = 2.8 μs
exp[t T2  ]
3
0
1
10
free evolution time (s)
40
Protocols for ideal pulses
…
B(t)
X
Pulses
 (t )
τ
X
τ
X
τ
X
X
τ
…
+1
…
–1
T=Nτ
T
Total accumulated phase:
    (t ) B (t )dt
0
X
τ
X
τ


T
T s
0
0
Signal(T )  exp i  exp  ib W (T ) , W (T )  ds e  Rs  (t ) (t  s)dt


2
Short times (RT << 1):
PDD
d-X-d-X
4
WF (T )  NR 3
3
Fast decay
PDD-based CDD
CPMG
(d/2)-X-d-X-(d/2)
CPMG-based CDD
Long times (RT >> 1):
1
WS (T )  NR 3
3
Slow decay
All orders: fast decay at all times, rate WF (T)
Slow decay at all times, rate WS (T)
optimal
choice
All orders: slow decay at all times, rate WS (T)
Protocols for realistic imperfect pulses
Pulses along X: CP and CPMG
CPMG – performs like no errors
CP – strongly affected by errors
State fidelity
εX = εY = -0.02, mX = 0.005, mZ = nZ = 0.05·IZ, δB = -0.5 MHz
1.0
x
y
0.6
simulation
Pulses along X and Y: XY4
(d/2)-X-d-Y-d-X-d-Y-(d/2)
(like XY PDD but CPMG timing)
Very good agreement
State fidelity
0
5
10
total time (s)
15
1.0
x
y
simulation
0.6
0
5
10
total time (s)
15
Aperiodic sequences: UDD and QDD
Are expected to be sub-optimal: no hard cut-off in the bath spectrum
20
State fidelity
1
0.5
UDD
Np = 6
0
CPMG
1/e decay
time (μs)
CPMG
UDD
exp.
sim.
5
5
10 15
Total time (s)
5
10
Np
15
Robustness to errors:
QDD6 vs XY4
1.0
B
C
QDD, SX
QDD, SY
0.5
Np= 48
0.0
0
10
20
30
Total time (s)
40
XY4, SX
XY4, SY
1.0
UDD vs XY4
B
C
B
CX
0.5
UDD, S
UDD, SY
0.0
Np= 48
-0.5
0
10
20
30
Total time (s)
40
XY4, SX
XY4, SY
Visibility issue
Small times:
1.0
QDD, SX
QDD, SY
XY4, SX
XY4, SY
0.5
0.0
0
10
20
30
40
Total time (s)
QDD: F = 0.992
XY4: F = 0.947
B
C
XY4: U  1  i  24(mX  nY ) Z
QDD: U  1  i3 X  X  3 Y Y 
Sensitive to different kind of errors
1.0
Solution: symmetrization
XY8, SX B
XY8, SY C
XY8
B
C
B
C
No 1st-order errors.
Initial F = 0.9999
0.5
0
but decays slowly as XY4
10
20
Total time (s)
30
DD on a single solid-state spin: scaling
Master curve: for any number of pulses
3

S (T )  exp t 3 / Tcoh
Tcoh  T2 N p2 / 3
100
1/e decay time (μs)
State fidelity
1
SE
N=4
N=8
N = 16
N = 36
0.5
N = 72
NV2
10
NV1
N = 136
0.1
1
Normalized time (t / T2 N 2/3)
10
1
10
100
number of pulses Np
136 pulses, coherence time increased by a factor 26
No limit is yet visible
Tcoh = 90 μs at room temperature
Quantum process tomography of DD
Re(χ)
Im(χ)
1
1
0
0
-1
Iˆ ˆ
x
ˆ y ˆ
z
Iˆ
ˆ z
ˆ ˆ y
-1
x
1
1
0
0
-1
Iˆ ˆ
x
ˆ y ˆ
z
ˆ ˆ z
ˆ
Iˆ x
-1
y
1
1
0
0
-1
Iˆ ˆ
x
ˆ y ˆ
z
ˆ ˆ z
ˆ  y
Iˆ x
-1
t = 4.4 μs
Iˆ ˆ
x
ˆ y ˆ
z
ˆ yˆ z
ˆ

Iˆ x
Pure dephasing
t = 10 μs
Iˆ ˆ
x
ˆ y ˆ
z
ˆ z
Only the elements
( I, I ) and (σZ , σZ )
change with time
ˆ ˆ y
Iˆ x
t = 24 μs
Iˆ ˆ
x
ˆ y ˆ
z
ˆ yˆ z
ˆ

Iˆ x
Summary
• Standard analytics (Magnus expansion) is often insufficient
• Numerical simulations are useful and often needed for
realistic assessment of DD protocols
• In-out fidelity for a single state is not enough (freezing happens)
Tomography is needed, at least partial
• Pulse errors are more than a little nuisance: can seriously plague
advanced DD sequences
• Pulse errors need to be seriously addressed, theoretically and
experimentally
• All taken into account, DD on a single solid-state spin achieved