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 – 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 information processing Single-spin coherent spintronics and photonics High-precision metrology and magnetic sensing at nanoscale

Quantum spin coherence: valuable resource

NV center in diamond

Quantum dots

Donors in silicon

Fundamental problems:

1. Understand dynamics of individual quantum spins

2. Control individual quantum spins

3. Preserve coherence of quantum spins

4. Generate and preserve entanglementbetween quantum spins

Grand challenge – controlling single quantum spins in solids

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

Symmetrized protocol: τ-X-τ-X-X- τ -X- τ = τ -X- τ - τ -X- τ 2nd order protocol, error O(τ2)

Concatenated protocols (CDD)level l=1 (CDD1 = PDD): τ -X- τ -Xlevel l=2 (CDD2): PDD-X-PDD-Xetc.

Simplest – Periodic DD : Period τ -X- τ -X

CPMG sequence

...)]( exp[ )2()1()0(per HHHTiU

)1(O )(TO )( 2TO

Traditional analysis and classification: Magnus expansion

2. Approximate – but very accurate – numerics: coherent spin states

Assessing the quality of coherence protection

)0()exp()( iHttHHHH SBBS

Up to 32 spins (Hilbert space d = 4×109) on 128 processors Parallel code, 80 % efficiency

3. Analytical mean-field techniques

1. Exact numerical modeling

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

Outline

de Lange, Wang, Riste, Dobrovitski, Hanson: Science 2010

Ryan, Hodges, Cory: PRL 2010

Naydenov, Dolde, Hall, Fedder, Hollenberg, Jelezko, Wrachtrup: PRB 2010

Spectacular recent progress: DD on a single NV spin

1. Quantum control and dynamical decoupling of NV center:protecting coherence

2. Decoherence-protected quantum gates

3. Decoherence-protected quantum algorithm:

first 2-qubit computation with invidivual solid-state spins

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 onsiteDipolar coupling to the bath

NV center in diamond

Single NV spin can be initialized, manipulated and read out

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

Initialization: m = 0 stateReadout (PL): population of m = 0

Decoherence: NV center in a spin bath

NV electron spin: pseudospin S = 1/2 (qubit)

NV spin

ms = 0

B

ms = –1

ms = +1

0

1

Bath spin – N atom

B

m = +1/2

ms = -1/2

No flip-flops between NV and the bath: energy mismatch

ZZBk

ZkkZZ StBSHSASSH )(ˆ

00

00

– field created by the bath spins

Time dependence governed by HB

)(ˆ tB

C

C CC

C

C

N

V C

Mean field picture: bath as a random field

Gaussian, stationary, Markovian noise

)exp()( )0( 2CtbtBB

b – noise magnitude (spin-bath coupling)τC – correlation time (intra-bath coupling)

Direct many-spin modeling: confirms mean field

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

Time

B F2

(a)

)( )0( tBB

simulationO-U fitting

Dobrovitski et al, PRL 2009 Hanson et al, Science 2008

t (µs)

-0.5

0.5

0 0.2 0.4 0.6 0.8

2*2

2exp Tt

free evolution time (s)1 10

0

0.5

32

3exp Tt

T2 = 2.8 μs

Free decoherence

T2* = 380 ns

)exp(iHt

HHSS ZZ

1)exp( iHt

Spin echo: probing the bath dynamics

Decay due to field inhomogeneity from run to run

*22 Tb

τC = 25 μs

Modulation: HF coupling to 14N of NV

Quantum control and Dynamical decoupling:

Extending coherence time

of a single NV center

CPMG

τ-X- 2τ -X-τ

)](exp[]exp[)(Signal TWiT

PDD

τ -X- τ -X

Short times (T << τC):3

2 )(34)(

CCF NbTW

Long times (T >> τC):

Fast decay Slow decay

Slow decay at all times, rate WS (T)optimalchoice

Choice of the DD protocol: theory

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…

32 )(

31)(

CCF NbTW

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

DD “as usual”

0 5 10 15

1.0

total time (s)

x y

simulation0.6

Pulses only along X:

τ-X-2τ-X- τ

X component – preserved wellY component – not so well

Sta

te fi

delit

y

What is wrong?

Control pulses are not perfect

Fast rotation of a single NV center

Example pulse shape: Experiment Simulation

29 MHz

109 MHz

223 MHz

• Rotating-frame approximation invalid: counter-rotating field• Pulse imperfections important

Time (ns) Time (ns)

Fuchs et al, Science 2009

1. Bootstrap protocol - characterize all pulse errors from scratchDobrovitski et al, PRL 2010

2. Understand well the accumulation of the pulse errorsWang et al, arXiv:1011.6417; Khodjasteh et al PRA 2011

Protecting all initial states

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 only along X:

τ -X-2 τ -X- τX component – preserved well

Y component – not so well

Pulses along X and Y:

τ -X-2 τ -Y-2 τ -X-2 τ -Y- τ

Both components are preservedCoherence extended far beyond echo time

Sta

te fi

delit

y

Sta

te fi

delit

y

Solution: two-axis control

0 10 20 30 400.0

0.5

1.0

Total time (s)

B C

Aperiodic sequences: UDD and QDD

Are expected to be sub-optimal: no hard cut-off in the bath spectrum

Sta

te fi

delit

y

0 5 10 15

0.5

1

Total time (s)

UDD CPMG

Np = 6sim.

1/e

deca

ytim

e (μ

s)

Np

5 10 15

5

20

exp.

CPMG UDD

Robustness to errors:

QDD, SXQDD, SYXY4, SX

XY4, SY

QDD6 vs XY4

Np= 48

0 10 20 30 40-0.5

0.0

0.5

1.0

Total time (s)

B C B CUDD, SX

UDD, SYXY4, SX

XY4, SY

UDD vs XY4

Np= 48

Extending coherence time with DD

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 26Tcoh = 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

0 1 2 3 4

0

0.25

0.50

SZ

time (s)

Detailed probe of the mesoscopic spin bath

de Lange, Riste, Dobrovitski et al, PRL 2011Taylor, Cappellaro, Childress et al. Nat Phys 2008Naydenov, Dolde, Hall et al. PRB 2011

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 – qubitsNanomechanical oscillators – data bus

Rabl et al, Nat Phys 2010

NV centers – qubitsSpin chain (other spins) – data bus

Cappellaro et al PRL 2010; Yao et al. PNAS 2011

Electron spins – processorsNuclear spins – memory

Many works since Kane 1998, maybe before

Bath

Unprotected quantum

gate

Bath

Protected storage:decoupling

“Standard” quantum operation

Contradiction:

DD efficiently preserves the qubit statebut

quantum computation must change it

Bath

Unprotected quantum

gate

Bath

Protected storage:decoupling

Gates with integrated decouplind

Bath

Protectedgate

gate DDDD

Tg

Nuclear 14N spin: memory, Electronic NV spin: processing(quantum memory, quantum repeater, magnetic sensing, etc.)

But control of nuclear spin takes much longer than T2*

Poor choice: either decouple the electron – no gates possible

or gating without DD – no gates possible

ZZ ISAH int

Gate with resonant decoupling (GARD) for hybrid systems

Childress, Taylor, Sorensen et al. PRL 2006Taylor, Marcus, Lukin PRL 2003

Jiang, Hodges, Maze et al. Science 2009Neumann, Beck, Steiner et al. Science 2011

C

C CC

C

C

N

V C

A way out: use internal resonance in the system

Different qubits have different coherence and control timescalesOne qubits decoheres before another starts to move

How the GARD works - 1

Rotating frame (ωN << A )

A ωN

,0 ,0

Nuclear rotation around XNuclear rotation around Z

0 :Electron1 :Electron

,1

,1

XNZZ IISAH

Rotating frame:

A = 2π ∙ 2.16 MHzωN = 2π ∙ 18 kHz

100 times smaller

How the GARD works - 2

Main problem: electron switches very frequently between 0 and 1and slow nuclear spin should keep track of this

Contradiction with the very idea of DD?

0-X-1-1-Y-0 )](exp[ 00 niH

1-X-0-0-Y-1 )](exp[ 11 niH

XY4 unit:τ -X- τ - τ -Y- τ

Motion of the nuclear spin: conditional single-spin rotation

Axes n0 and n1 are both close to z (A >> ω1): 2 1 )/(1 2

10 smthsmthAnn N

small

Resonance: smth 2 also becomes small when An )12(

110 nn

How the GARD works - 3

RZ(π) RX(2α) RZ(π)

RX(α) RZ(2π) RX(α)

IN:,1

,0IN:

1

1

10

00

OUT:X,1

X,0OUT:

XY-4 unit:

A 22

Experimental implementation of GARD

Nuclear spin rotation conditioned on the electron: An )12(

Unconditional nuclear spin rotation: An 2

All nuclear gates are produced only by changing τ

Resonances are very narrow, ~ (ωN /A)2 Timing jitter < 1 ns over 100 μs time span

Error by 10 ns – fidelity drops by 10%

Protected C-Rot gate

TG = 60 μs >> T2*

Experimental implementation: proof of concept

IZ

electron

nucleus

0

10.5

0.5-0.5

-0.5

IZ

Fidelity 97%

mostly, T1 decay

CNOT gate

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

out

out

Fidelity

Overlap

10 iin 10

GARD implementation of Grover’s algorithm2 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

1 2 3

4 5 6

Fidelity: 95% for

GARD implementation of Grover’s algorithm

,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