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),
G. de Lange, D. Riste, R. Hanson (TU Delft),
G. D. Fuchs, D. Toyli, D. D. Awschalom (UCSB)
Quantum spins in the solid state settings
Quantum dots
NV center in diamond
Fundamental questions
How to manipulate quantum spins
How to model spin dynamics
Which dynamics is typical
Which dynamics is interesting
Which dynamics is useful
Magnetic molecules
Applications
Nanoscale magnetic sensing
High-precision magnetometry
Quantum repeaters
Quantum key distribution
Quantum memory
General problem: decoherence
Decoherence:
nuclear spins,
phonons, conduction
electrons, …
Quantum control of spin state in presence of decoherence
Spin control – important topic (>10,000 items on Amazon.com)
Preserving coherence: dynamical decoupling (DD)
Employ time reversal, like in spin echo
Electron spin S
Spin echo:
Decohered by many
nuclear spins Ik
H  SZ  A I
Z
k k
exp(iHt)
exp(iHt)
SZ   SZ
as if nothing
happened
H  H
Periodic DD
(PDD):
τ
τ
τ
1
τ
U 1
Central spin S is decoupled from the bath of spins Ik
Dynamical decoupling protocols
General approach – e.g., group-theoretic methods
Viola, Knill, Lloyd, PRL 1999
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)
Performance of DD and advanced protocols
Assessing DD performance: Magnus expansion
(asymptotic expansion for small delay τ, total experiment duration T )
U  exp[i T ( H (0)  H (1)  H ( 2)  ...)]
O( )
O (1)
Symmetrized XY PDD (XY SDD):
O( 2 )
XYXY-YXYX
2nd order protocol, error O(τ2)
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
Khodjasteh, Lidar, PRL 2005
Why we need something else?
Traditional NMR and ESR:
• Only one spin component is preserved – others are often lost
• Only macroscopic systems
• Our focus: preserve complete quantum spin state for a single spin
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 simulations
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
Spectacular recent progress in DD on single spins
de Lange, Wang, Riste, Dobrovitski, Hanson: Science 330, 60 (2010)
Ryan, Hodges, Cory: PRL 105, 200402 (2010)
Bluhm, Foletti, Neder, Rudner, Mahalu, Umansky, Yacoby:
arXiv:1005.2995
Naydenov, Dolde, Hall, Shin, Fedder, Hollenberg, Jelezko, Wrachtrup:
arXiv:1008.1953
Pulse imperfections start playing a major role
Qualitatively change the spin dynamics
Need to be carefully analyzed
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
Single NV center – optical manipulation and readout
Excited state:
Spin 1
orbital doublet
m = +1
m = –1
m=0
Jelezko, Gaebel, Popa et al, PRL 2004
Gaebel, Jelezko, et al, Science 2006
Childress, Dutt, Taylor et al, Science 2006
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
Theoretical picture: NV center and the bath of N atoms
Most important baths:
• Single nitrogens (electron spins)
• 13C nuclear spins
Long-range dipolar coupling
Hanson, Dobrovitski, Feiguin et al, Science 2008
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
Single central spin vs. Ensemble of similar spins
Dilute dipolar-coupled baths
Spectral line – Gaussian
Decoherence: Gaussian decay
F ~ exp(-t2)
Rabi oscillations decay
SZ  1  t 2
Spectral line – Lorentzian
Decoherence: exponential decay
F ~ exp(-t)
Rabi oscillations decay
SZ 1   t
Prokof’ev, Stamp, PRL 1998
Strong variation of local environment between different NV centers

P(b)  2
b
 2 
exp  2 

 2b 
2
Dobrovitski, Feiguin, Awschalom et al, PRB 2008
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 of 0.1-1 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
Fuchs, Dobrovitski, Toyli, et al, Science 2009
Characterizing / tuning DD pulses for NV center


U X  exp[i(   X )(Sn)]
n  (nX , nY , nZ )
Pulse error accumulation can be devastating at long times
High-quality pulses are required for good DD
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
“Bootstrap” problem: • Can reliably prepare only state 
• Can reliably measure only SZ
Dobrovitski, de Lange, Riste et al, PRL 2010
“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)
Confirmed by simulations
1.0
simulation
O-U fitting
0.8
B(0) B(t )  b exp(Rt)
2
0.6
b – noise magnitude (spin-bath coupling)
R = 1/τC – rate of fluctuations (intra-bath coupling)
0.4
0.2
0.0
Dobrovitski, Feiguin, Hanson, et al, PRL 2009
0
10
20
30
40
Time
Experimental confirmation: pure dephasing by O-U noise
0.5
0.5
Ramsey decay
T2 = 2.8 μs

0
0.2
0.4
t (µs)
]
* 2
2
exp[ t T
-0.5
0.6
Spin echo
T2* = 380 ns
0.8
exp[t T2  ]
3
0
1
10
free evolution time (ms)
De Lange, Wang, Riste, et al, Science 2010
Protocols for ideal pulses
Signal  exp[W (T )]
Short times (RT << 1):
PDD
d-X-d-X
4 2
WF (T )  b NR 3
3
Fast decay
PDD-based CDD
CPMG
(d/2)-X-d-X-(d/2)
CPMG-based CDD
Long times (RT >> 1):
1 2
WS (T )  b 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)
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
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 (ms)
15
1.0
x
y
simulation
0.6
0
5
10
total time (ms)
15
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
t = 10 μs
Iˆ ˆ
x
ˆ y ˆ
z
z
Only the elements
( I, I ) and (σZ , σZ )
change with time
ˆ z
ˆ ˆ y
Iˆ x
No preferred spin component
DD works for all states
t = 24 μs
Iˆ ˆ
x
ˆ y ˆ
Pure dephasing
ˆ yˆ z
ˆ

Iˆ x
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
What I will not show (for the lack of time)
Single-spin
magnetometry
with DD
Ultimately – sensing a single magnetic molecule
0.50
SZ
Joint DD on
central spin
and the bath
0.25
0
0
1
2
3
4
time (ms)
Quantum gates with DD
… and much more to come
in this field
Summary
• Dynamical decoupling – important for applications and for
fundamental reasons
• DD on a single spin – challenging but possible
• Accumulation of pulse errors – careful design of DD protocols
• (Careful theoretical analysis) + (advanced experiments)
=
First implementation of DD on a single solid-state spin.
• Further advances: DD for control and study of the bath, DD with
quantum gates, DD for improved magnetometry, etc.