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.