Quantum dynamics and quantum control
of spins in diamond
Viatcheslav Dobrovitski
Ames Laboratory US DOE, Iowa State University
Works done in collaboration with
Z.H. Wang (Ames Lab – now USC),
T. der Sar, G. de Lange, T. Taminiau, R. Hanson (TU Delft),
G. D. Fuchs, D. Toyli, D. D. Awschalom (UCSB),
D. Lidar (USC)
Individual quantum spins in solid state
Quantum dots
NV center in diamond
Donors in silicon
Quantum spin coherence: valuable resource
 Quantum information processing
 Single-spin coherent spintronics and photonics
 High-precision metrology and magnetic sensing at nanoscale
Grand challenge – controlling single quantum spins in solids
Fundamental problems:
1. Understand dynamics of individual
quantum spins
2. Control individual quantum spins
3. Preserve coherence of quantum spins
4. Generate and preserve entanglement
between quantum spins
Spins in diamond – excellent testbed for quantum studies
• Long coherence time
• Individually addressable
• Controllable optically and magnetically
Jelezko et al, PRL 2004; Gaebel et al, Nat.Phys. 2006; Childress et al, Science 2006
Dynamical decoupling protocols
Traditional analysis and classification: Magnus expansion
Uper  exp[i T ( H (0)  H (1)  H ( 2)  ...)]
O (1)
Simplest – Periodic DD :
Symmetrized protocol:
O (T )
O(T 2 )
Period τ -X- τ -X
τ-X-τ-X-X- τ -X- τ = τ -X- τ - τ -X- τ
CPMG sequence
2nd order protocol, error O(τ2)
Concatenated protocols (CDD)
level l=1 (CDD1 = PDD): τ -X- τ -X
level l=2 (CDD2):
etc.
PDD-X-PDD-X
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
(the UV cutoff is too large) but DD works
• Behavior at long times – unclear
• Accumulation of pulse errors and imperfections – unknown
Assessing the quality of coherence protection
1. Exact numerical modeling
H  H S  H B  H SB
(t )  exp(iHt)(0)
Up to 32 spins (Hilbert space d = 4×109) on 128 processors
Parallel code, 80 % efficiency
2. Approximate – but very accurate – numerics: coherent spin states
3. Analytical mean-field techniques
Outline
1. Quantum control and dynamical decoupling of NV center:
protecting coherence
Spectacular recent progress: DD on a single NV spin
de Lange, Wang, Riste, Dobrovitski, Hanson: Science 2010
Ryan, Hodges, Cory: PRL 2010
Naydenov, Dolde, Hall, Fedder, Hollenberg, Jelezko, Wrachtrup: PRB 2010
2. Decoherence-protected quantum gates
3. Decoherence-protected quantum algorithm:
first 2-qubit computation with invidivual solid-state spins
NV center in diamond
Simplest impurity:
substitutional N (P1 center)
Environment (spin bath)
S = 1/2
Long-range dipolar coupling
Nitrogen meets vacancy:
NV center
Central spin
S = 1, I = 1
HF coupling onsite
Dipolar coupling to the bath
Single NV spin can be initialized, manipulated and read out
Single NV center – optical manipulation and readout
Excited state:
Spin 1
orbital doublet
m = +1
m = –1
m=0
ISC
(m = ±1 only)
1A
532 nm
m = +1
m = –1
m = 0 – always emits light
m = ±1 – not
MW
Ground state:
Spin 1
Orbital singlet
m=0
Initialization: m = 0 state
Readout (PL): population of m = 0
Decoherence: NV center in a spin bath
C
N
V
C
C
Bath spin – N atom
NV spin
C
ms = –1
C
m = +1/2
C
ms = +1 
0
ms = 0
1
 0
0
B
ms = -1/2
B
NV electron spin: pseudospin S = 1/2 (qubit)
No flip-flops between NV and the bath: energy mismatch
H  0 SZ  SZ  Ak S  H B  0 SZ  Bˆ (t )SZ
Z
k
k
ˆ (t ) – field created by the bath spins
B
Time dependence governed by HB
Mean field picture: bath as a random field
Gaussian, stationary, Markovian noise
B(0) B(t )  b2 exp( t  C )
b – noise magnitude (spin-bath coupling)
τC – correlation time (intra-bath coupling)
Direct many-spin modeling: confirms mean field
simulation
O-U fitting
1.0
0.8
B
F2
B(0) B(t )
0.6
(a)
0.4
0.2
0.0
0
10
20
30
40
Time
Dobrovitski et al, PRL 2009
Hanson et al, Science 2008

0.5
*2
2
exp  t T
2

Free decoherence
Decay due to field inhomogeneity
from run to run
b  2 T2*
T2* = 380 ns
-0.5
0
0.2
0.4
0.6
t (µs)
Modulation: HF coupling to 14N of NV
0.8
Spin echo: probing the bath dynamics
exp(iHt)
exp(iHt)  1
0.5
SZ   SZ
H  H
0
τC = 25 μs
T2 = 2.8 μs

exp  t 3 T23

1
10
free evolution time (ms)
Quantum control and Dynamical decoupling:
Extending coherence time
of a single NV center
Choice of the DD protocol: theory
Signal(T )  exp[i]  exp[W (T )]
Long times (T >> τC):
Short times (T << τC):
PDD
 
4
2
WF (T )  (b C ) N  
3
C 
τ -X- τ -X
Fast decay
CPMG
τ-X- 2τ -X-τ
3
 
1
WF (T )  (b C ) 2 N  
3
C 
3
Slow decay
Slow decay at all times, rate WS (T)
optimal
choice
Concatenated PDD
Fast decay at all times, makes things worse
Concatenated CPMG
Slow decay at all times, no improvement
and many other protocols have been analyzed…
Qualitative features
• Coherence time can be extended well beyond τC as long as
the inter-pulse interval is small enough: τ/τC << 1
• Magnus expansion (also similar cumulant expansions) predict:
W(T) ~ O(N τ4) for PDD but we have W(T) ~ O(N τ3)
Symmetrization or concatenation give no improvement
Source of disagreement: Magnus expansion is inapplicable
S ( ) 
Ornstein-Uhlenbeck noise:
1
 2 C2  1
Second moment is (formally) infinite – corresponds to H B2  
Cutoff of the Lorentzian:
UV ~
m B2
a
3
~ 2  5 GHz  1  C
Pulses only along X:
τ-X-2τ-X- τ
X component – preserved well
Y component – not so well
State fidelity
DD “as usual”
1.0
x
y
0.6
simulation
0
What is wrong?
Control pulses are not perfect
5
10
total time (ms)
15
Fast rotation of a single NV center
Experiment
Example pulse shape:
Simulation
29 MHz
109 MHz
223 MHz
Time (ns)
Time (ns)
• Rotating-frame approximation invalid: counter-rotating field
• Pulse imperfections important
Fuchs et al, Science 2009
1. Bootstrap protocol - characterize all pulse errors from scratch
Dobrovitski et al, PRL 2010
2. Understand well the accumulation of the pulse errors
Wang et al, arXiv:1011.6417; Khodjasteh et al PRA 2011
Pulses only along X:
τ -X-2 τ -X- τ
X component – preserved well
Y component – not so well
State fidelity
Protecting all initial states
1.0
x
y
0.6
simulation
0
5
10
total time (ms)
15
Pulses along X and Y:
τ -X-2 τ -Y-2 τ -X-2 τ -Y- τ
State fidelity
Solution: two-axis control
1.0
x
y
simulation
0.6
Both components are preserved
Coherence extended far beyond echo time
0
5
10
total time (ms)
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 (ms)
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 (ms)
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 (ms)
40
XY4, SX
XY4, SY
Extending coherence time with DD
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
Tcoh = 90 μs at room temperature, and no limit in sight
De Lange, Wang, Riste, et al, Science 2010
Using DD for other good deeds
Single-spin
magnetometry
with DD
de Lange, Riste, Dobrovitski et al, PRL 2011
Taylor, Cappellaro, Childress et al. Nat Phys 2008
Naydenov, Dolde, Hall et al. PRB 2011
Detailed probe of the
mesoscopic spin bath
SZ
0.50
0.25
0
0
1
2
3
4
time (ms)
de Lange, van der Sar, Blok et al, arXiv 2011
Combining DD and quantum operation
Gates with resonant decoupling
Coupling NVs to each other – hybrid systems
Hybrid systems: different types of qubits for different functions
NV centers – qubits
Nanomechanical oscillators – data bus
Rabl et al, Nat Phys 2010
NV centers – qubits
Spin chain (other spins) – data bus
Cappellaro et al PRL 2010; Yao et al. PNAS 2011
Electron spins – processors
Nuclear spins – memory
Many works since Kane 1998, maybe before
“Standard” quantum operation
Unprotected
quantum
gate
Bath
Protected storage:
decoupling
Bath
Contradiction:
DD efficiently preserves the qubit state
but
quantum computation must change it
Gates with integrated decouplind
Unprotected
quantum
gate
Protected storage:
decoupling
Bath
Bath
DD
gate
Tg
DD
Protected
gate
Bath
Gate with resonant decoupling (GARD) for hybrid systems
Different qubits have different coherence and control timescales
One qubits decoheres before another starts to move
Nuclear 14N spin: memory, Electronic NV spin: processing
(quantum memory, quantum repeater, magnetic sensing, etc.)
Childress, Taylor, Sorensen et al. PRL 2006
Taylor, Marcus, Lukin PRL 2003
Hint  A SZ I Z
C
C
N
V
C
C
Jiang, Hodges, Maze et al. Science 2009
Neumann, Beck, Steiner et al. Science 2011
C
But control of nuclear spin takes much longer than T2*
C
Poor choice:
either decouple the electron – no gates possible
or gating without DD – no gates possible
A way out: use internal resonance in the system
How the GARD works - 1
Rotating frame (ωN << A )
Rotating frame:
H  A S Z I Z  N I X
0, 
1, 
A
A = 2π ∙ 2.16 MHz
ωN = 2π ∙ 18 kHz
1, 
0, 
ωN
100 times smaller
Electron: 1
Nuclear rotation around Z
Electron: 0
Nuclear rotation around X
How the GARD works - 2
Main problem: electron switches very frequently between 0 and 1
and slow nuclear spin should keep track of this
Contradiction with the very idea of DD?
Motion of the nuclear spin: conditional single-spin rotation

0-X-1-1-Y-0
H0  exp[i (n0 )]
XY4 unit:

τ -X- τ - τ -Y- τ
H

exp[

i

(
n
1-X-0-0-Y-1
1
1 )]
 
2 smth 1
Axes n0 and n1 are both close to z (A >> ω1): n0 n1  1  ( N / A)
smth 2
small
Resonance: smth 2 also becomes small when   (2n  1) A

n0n1  1
How the GARD works - 3
IN:
1
1
0
OUT:
1, 
1, X
RZ(π)
IN:
0
RX(2α)
RZ(π)
0
1
OUT:
0, 
0, X
RX(α)
XY-4
unit:

RZ(2π)
2  2 A
RX(α)

Experimental implementation of GARD
Resonances are very narrow, ~ (ωN /A)2
Timing jitter < 1 ns over 100 μs time span
Error by 10 ns – fidelity drops by 10%
Nuclear spin rotation conditioned on the electron:   (2n  1)  A
Unconditional nuclear spin rotation:   2n  A
All nuclear gates are produced only by changing τ
Experimental implementation: proof of concept
CNOT gate
Protected C-Rot gate
electron
Total gate time (us)
100μs200
TG0 = 60
>>
1.0
*
T300
2
400
500
nucleus
0.5
P(Nucleus = Up)P(Nucleus = Up)
Total gate time (us)
IZ
IZ
0.00
0.5
1.0
0.5
100
200
300
400
500
1
-0.5
0.0
1.0
1.0
0.5
0.5
0
mostly, T1 decay
-0.5
0.0
1.00
0.5
4
8
12
16
20
24
Total # of XY2
Fidelity 97%
How good is GARD: protected CNOT gate
Controllable decoherence: inject a noise into the system
Decoherence time: T2 = 50 μs; Gate time TG = 120 μs
 in   0  i 1   
Overlap    out  
Fidelity 
   out  
  0   1
GARD implementation of Grover’s algorithm
2 qubits – Grover’s algorithm converges in one iteration
Total time: 330 μs, T2 time only 250 μs
First quantum computation on two individual solid-state spins
GARD implementation of Grover’s algorithm
1
2
3
4
5
6
Fidelity: 95% for 1, 
For other states: 0.93, 0.92, 0.91
High fidelity beyond coherence time
Conclusions
1. Diamond-based QIP becomes truly competitive
2. Coherence time can be extended, 25-fold demonstrated
3. DD can be efficiently combined with gates
4. GARD algorithms demonstrated, 50% longer than T2
Fidelity above 90%
First 2-qubit computation on individual solid state spins