Quantum dynamics and quantum control of spins in diamond Viatcheslav Dobrovitski
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Transcript of Quantum dynamics and quantum control of spins in diamond Viatcheslav Dobrovitski
Quantum dynamics and quantum controlof spins in diamond
Viatcheslav Dobrovitski
Ames Laboratory US DOE, Iowa State University
Works done in collaboration withZ.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
NV center in diamond
Quantum dots
Fundamental questions
How to manipulate quantum spinsHow to model spin dynamicsWhich dynamics is typicalWhich dynamics is interestingWhich dynamics is useful
Applications
Nanoscale magnetic sensingHigh-precision magnetometryQuantum repeatersQuantum key distributionQuantum memory
Magnetic molecules
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)
ZkkZ IASH
Employ time reversal, like in spin echo
Spin echo: )exp(iHt
HHSS ZZ
)exp( iHt 1as if nothing
happened
Electron spin S Decohered by manynuclear spins Ik
Periodic DD(PDD): 1U
Central spin S is decoupled from the bath of spins Ik
τ τ ττ
Dynamical decoupling protocols
ZkkZ IASH
General approach – e.g., group-theoretic methods
Examples:
Periodic DD (CPMG, pulses along X): Period d-X-d-X (d – free evolution)
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)
ZkkZ
YkkY
XkkX ICSIBSIASH
Viola, Knill, Lloyd, PRL 1999
Performance of DD and advanced protocols
Assessing DD performance: Magnus expansion (asymptotic expansion for small delay τ, total experiment duration T )
...)]( exp[ )2()1()0( HHHTiU)1(O )(O )( 2O
Symmetrized XY PDD (XY SDD): XYXY-YXYX 2nd order protocol, error O(τ2)
Concatenated XY PDD (CDD)level l=1 (CDD1 = PDD): d-X-d-Y-d-X-d-Ylevel l=2 (CDD2): PDD-X-PDD-Y-PDD-X-PDD-Yetc.
Khodjasteh, Lidar, PRL 2005
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
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
mkkm BStCt )()(
The whole system (S+B) is isolated and is in pure quantum state
bath theand
system theof states basis - , mk BS
Numerical simulations
1. Exact solution
Very demanding: memory and time grow exponentially with NSpecial numerical techniques are needed to deal with d ~ 109
(Chebyshev polynomial expansion, Suzuki-Trotter decomposition)Still, N up to 30 can be treated
)0( )()0( )exp( )(
tUiHtTtHHHH SBBS
2. Some special cases – bath as a classical noiseRandom time-varying magnetic field acting on the spin
Spectacular recent progress in DD on single spins
Bluhm, Foletti, Neder, Rudner, Mahalu, Umansky, Yacoby: arXiv:1005.2995
de Lange, Wang, Riste, Dobrovitski, Hanson: Science 330, 60 (2010)
Pulse imperfections start playing a major roleQualitatively change the spin dynamics
Need to be carefully analyzed
Ryan, Hodges, Cory: PRL 105, 200402 (2010)
Naydenov, Dolde, Hall, Shin, Fedder, Hollenberg, Jelezko, Wrachtrup:arXiv:1008.1953
Diamond – solid-state version of vacuum:no conduction electrons, few phonons, few impurity spins, …
Simplest impurity:substitutional N
Bath spins S = 1/2Distance between spins ~ 10 nm
Nitrogen meets vacancy:NV center
Ground state spin 1Easy-plane anisotropy
Distance between centers: ~ 2 μm
Studying a single solid-state spin: NV center in diamond
ISC (m = ±1 only)
532 nm
Excited state:Spin 1
orbital doublet
Ground state:Spin 1
Orbital singlet
1A
Single NV center – optical manipulation and readout
m = 0 – always emits lightm = ±1 – not
m = +1m = –1
m = 0
m = +1m = –1
m = 0
MW
Jelezko, Gaebel, Popa et al, PRL 2004Gaebel, Jelezko, et al, Science 2006Childress, Dutt, Taylor et al, Science 2006
Initialization: m = 0 stateReadout (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
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
Hanson, Dobrovitski, Feiguin et al, Science 2008
Single central spin vs. Ensemble of similar spinsDilute dipolar-coupled baths
Spectral line – Gaussian Spectral line – Lorentzian
Rabi oscillations decay21 tSZ tSZ 1
Rabi oscillations decay
Dobrovitski, Feiguin, Awschalom et al, PRB 2008
Decoherence: Gaussian decayF ~ exp(-t2)
Decoherence: exponential decayF ~ exp(-t)
Strong variation of local environment between different NV centers
2
2
2 2exp2)(
bbbP
Prokof’ev, Stamp, PRL 1998
NV center in a spin bathNV spin
ms = 0Electron spin: pseudospin 1/214N nuclear spin: I = 1
MW
t (µs)
-0.5
0.5
0 0.2 0.4 0.6 0.8
Ramsey decay ]exp[
2*2Tt
T2* = 380 ns A = 2.3 MHz
Slow modulation:hf coupling to 14N
B
ms = +1
ms = -1
0
1
Decay of envelope:
C
C CC
C
C
N
V C
Need fast pulses
Bath spin – N atom
MW
B
m = +1/2
ms = -1/2
No flip-flops between NV and the bath
Decoherence of NV – pure dephasing
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
tB Lcos1
LB
L
xy
Rotating frame
L L
S
Spin
LOscillating field
co-rotating(resonant)
counter-rotating(negligible)
Rotating frame: static field B1/2 along X-axis
Strong driving of a single NV center
“Square” pulses: Experiment Simulation
29 MHz
109 MHz
223 MHz
Gaussian pulses:109 MHz
223 MHz
• Rotating-frame approximation invalid: counter-rotating field• Role of pulse imperfections, especially at the pulse edges
Time (ns) Time (ns)
Fuchs, Dobrovitski, Toyli, et al, Science 2009
Characterizing / tuning DD pulses for NV center
),,( )])((exp[ ZYXXX nnnnnSiU
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
Pulse error accumulation can be devastating at long timesHigh-quality pulses are required for good DD
Dobrovitski, de Lange, Riste et al, PRL 2010
• Can reliably prepare only state • Can reliably measure only SZ
“Bootstrap” problem:
“Bootstrap” protocol
Assume: errors are small, decoherence during pulse negligible
)(]2/) )((exp[ ZZYYXX iniU
Series 0: π/2X and π/2Y Find φ' and χ' (angle errors)Series 1: πX – π/2X, πY – π/2Y Find φ and χ (for π pulses)Series 2: π/2X – πY, π/2Y – πX 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
Bonuses:• Signal is proportional to error (not to its square)• Signal is zero for no errors (better sensitivity)
All errors are determined from scratch, with imperfect pulses
Bootstrap protocol: experiments
Introduce known errors: - phase of π/2Y pulse - frequency offset
Self-consistency check: QPT with corrections
Fidelity
M2
- Prepare imperfect basis states
- Apply corrections (errors are known)
- Compare with uncorrected
Ideal recovery: F = 1, M2 = 0
01 ,01 ,0 ,1 i- corrected- uncorrected 022
0
,
] [Tr
MMM
F
What to expect for DD? Bath dynamics
0 10 20 30 400.0
0.2
0.4
0.6
0.8
1.0
Time
B F2
(a)Mean field: bath as a random field B(t)
Confirmed by simulations)exp()( )0( 2 RtbtBB
)( )0( tBB
simulationO-U fitting
b – noise magnitude (spin-bath coupling) R = 1/τC – rate of fluctuations (intra-bath coupling)
t (µs)
-0.5
0.5
0 0.2 0.4 0.6 0.8
Ramsey decay
]exp[2*
2Ttfree evolution time (s)
1 10
0
0.5
Spin echo
]exp[ 32Tt
Experimental confirmation: pure dephasing by O-U noise
T2* = 380 ns
T2 = 2.8 μs
De Lange, Wang, Riste, et al, Science 2010
Dobrovitski, Feiguin, Hanson, et al, PRL 2009
CPMG
(d/2)-X-d-X-(d/2)
)](exp[Signal TW
PDD
d-X-d-X
Short times (RT << 1):
32
34)( NRbTWF
Long times (RT >> 1):
32
31)( NRbTWS
PDD-based CDD
Fast decay Slow decay
All orders: fast decay at all times, rate WF (T)
Slow decay at all times, rate WS (T)
CPMG-based CDD All orders: slow decay at all times, rate WS (T)
optimalchoice
Protocols for ideal pulses
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
11)( 22
C
S
Ornstein-Uhlenbeck noise:
Second moment is (formally) infinite – corresponds to 2BH
Cutoff of the Lorentzian: CB
UV a 1 GHz 52~~ 3
2
Protocols for realistic imperfect pulses
0 5 10 15
1.0
total time (s)
x y
simulation0.6
total time (s)0 5 10 15
1.0
x y
simulation0.6
Pulses along X: CP and CPMG
CPMG – performs like no errorsCP – strongly affected by errors
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
Sta
te fi
delit
yS
tate
fide
lity
εX = εY = -0.02, mX = 0.005, mZ = nZ = 0.05·IZ, δB = -0.5 MHz
Quantum process tomography of DD
t = 10 μs
t = 24 μs
t = 4.4 μs
-1
0
1
Ix
yz
zy
xI
-1
0
1
Ix
yz
zy
xI
-1
0
1
Ix
yz
zy
xI
-1
0
1
Ix
yz
zy
xI
-1
0
1
Ix
yz
zy
xI
-1
0
1
Ix
yz
zy
xI
Re(χ) Im(χ)
Only the elements ( I, I ) and (σZ , σZ )change with time
Pure dephasing
No preferred spin componentDD works for all states
DD on a single solid-state spin: scaling
number of pulses Np1/
e de
cay
time
(μs)
1 10 100
100
10
NV2
NV1
Normalized time (t / T2 N 2/3)0.1 1 10
0.5
1
N = 4
SE
N = 8 N = 16 N = 36 N = 72 N = 136
Sta
te fi
delit
y
33 /exp)( cohTtTS Master curve: for any number of pulses3/2
2 pcoh NTT
136 pulses, coherence time increased by a factor 26No 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
0 1 2 3 4
0
0.25
0.50
SZ
time (s)
Joint DD oncentral spin and the bath
Quantum gates with DD… and much more to come
in this field
Ultimately – sensing a single magnetic molecule
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.