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Dynamical decoupling with imperfect pulses Viatcheslav Dobrovitski Ames Laboratory US DOE, Iowa State University

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Dynamical decoupling with imperfect pulses

Viatcheslav Dobrovitski

Ames Laboratory US DOE, Iowa State University

Works done in collaboration with

Z.H. Wang, B. N. Harmon (Ames Lab),

G. de Lange, D. Riste, R. Hanson (TU Delft),

G. D. Fuchs, D. D. Awschalom (UCSB),

L. Santos (Yeshiva U.), K. Khodjasteh, L. Viola (Dartmouth College)

S. Lyon, A. Tyryshkin (Princeton)

W. Zhang (Fudan), N. Konstantinidis (Fribourg)

1. Introduction – what are we doing and why.

2. Quantum dots – some lessons and caveats.

3. P donors in Si – how pulse errors qualitatively change

the spin dynamics.

4. Dynamical decoupling of a single spin – decoupling

protocols for a NV center in diamond.

P donor in silicon

Localized electron S=1/2

NV center in diamond

Localized electron spin S=1

Quantum dots

Localized electron S=1/2

Fundamental questions:

How to reliably manipulate quantum spins

How to accurately model dynamics of driven spins

Which dynamics is typical

Which dynamics is interesting

Which dynamics is useful

Magnetometry with nanoscale resolution

STM

ODMR nanoprobe:

quantum dot, NV center, …

Quantum computation

NV centers in a waveguide

Array of quantum dots

Quantum repeater

2-qubit quantum computer

NV center with

an electron and a nuclear spin (15N or 13C)

Influence of environment: nuclear spins,

phonons, conduction electrons, …

Decoherence:

phase is forgotten

Dynamical decoupling: applying a sequence of pulses to

negate the effect of environment

Spectacular recent progress in DD on single spins

Bluhm, Foletti, Neder, Rudner, Mahalu, Umansky, Yacoby, 2010:

16-pulse CPMG sequence on quantum dot

arXiv:1005.2995

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

DD on a single solid-state spin (NV center in diamond)

136 pulses, ideal scaling with Np

Coherence time increased by a factor of 26

arXiv:1008.2119

Pulse imperfections start playing a major role

Qualitatively change the spin dynamics

Need to be carefully analyzed

Studying dirt can be useful

Antoni van Leeuwenhoek

Delft, 17th century

Studied dirt – discovered germs

Ames Lab + TU Delft, 21st century

Studied dirt, achieved DD on a single solid-state spin

Dynamical decoupling protocols

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 – see further)

Performance of DD and advanced protocols

Assessing DD performance: Magnus expansion

(asymptotic expansion for small period duration T )

Symmetrized XY PDD (XY SDD): XYXY-YXYX

2nd order protocol, error O(T2)

Concatenated XY PDD (CDD)

level l=1 (CDD1 = PDD): d-X-d-Y-d-X-d-Y

level l=2 (CDD2): PDD-X-PDD-Y-PDD-X-PDD-Y

etc.

- 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

1. Exact solution

The whole system (S+B) is isolated and is in pure quantum state

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

for a single-electron quantum dot

Single electron spin in a quantum dot

Single electron QD

Hyperfine spin coupling

Fermi contact interaction

electron spin

(delocalized)

nuclear spins

(Ga, As nuclei)

Hahn echo : from T2* ~ 10 ns to T2 ~ 1 μs

Universal DD: protect all three components of the spin

control Hamiltonian

PDD

SDD

CDD2

Is Magnus expansion sufficient ?

Periodic DD (PDD) d-X-d-Y-d-X-d-Y

Symmetrized DD (SDD) XYXY-YXYX

Concatenated, level 2 (CDD2) PDD-X-PDD-Y-PDD-X-PDD-Y

Magnus expansion

is valid

only for ≤ 10 ps

Preserving unknown state of the spin

Decoherence:

Worst-case scenario: minimum fidelity

- 8 different protocols
- Large τ(up to 5 ns)
- Long times
- Imperfections considered
- Finite-width pulses
- Intra-bath interactions

DD works very well – but ME is not valid

Long times: fidelity saturation

SX (t)

SZ (t)

XY PDD

τ= 0.01

τ= 0.1

τ= 0.01

τ= 0.1

τ= 1

τ= 1

– commutes with Sz

Sz is a “quasi-conserved” quantity

Quantum tomography is a must to confirm decoupled qubit

pulse errors and fidelity saturation

DD for P donors in silicon, fidelity for different states

(S. Lyon and A. Tyryshkin)

XZ PDD

SY

quasi-conserved

Initial state along Y

Initial state along X

Initial state along Y

XY PDD

SZ

quasi-conserved

Initial state along X

1. Ensemble experiments: ESR on a large number of P spins

2. 29Si – depleted sample: f = 800 ppm (naturally, f=4.67%)

3. Inhomogeneous broadening: cw ESR linewidth 50 mG

4. However, T2 = 6 ms – plenty of room for DD

Dephasing by almost static bath – decoupling should be perfect

Model: pulse field inhomogeneity

Bpulse (x)

- Rotation angle is not exactly π
- everywhere
- Rotation axis is not exactly X
- (or Y) everywhere

Sample

x

Freezing in Si:P, qualitative picture

Consider some spin

PDD, after 1/2-cycle:

(composition of rotations = rotation)

After N cycles:

Each spin rotates around its own axis, by

its own angle

But all axes are close to Y (for PDD XZ)

Total spin component along Y – conserved,

other components average to zero

Simplified analytics (leading order in pulse errors)

XZ PDD

XY PDD

All rotation axes close to Y

Rotation angle – 1st order inεX , εY

SY – frozen, SX and SZ decay fast

All rotation axes close to Z

Rotation angle – 2nd order inεX , εY

SZ – frozen, SX and SY decay slow

In agreement with experiment

Quantitative treatment: numerics vs. experiment

XZ PDD

XY PDD

SZ

SY

SX

SY

SX

SZ

Hollow squares – experiment, dots – theory

Rotation angle errors (εX , εY) – distribution width 0.3 (~15º)

Rotation axis errors (nZ, mZ) – distribution width 0.12 (~7º)

Concatenation: single-cycle fidelity

XZ CDDs

XY CDDs

Nothing to show

All fidelities are 1 (within 2%)

SY

SZ

SX

Analytical result:

CDDs of all levels have the same error,

in spite of exponentially increasing number of cycles

Periodic DD (PDD) d-X-d-Y-d-X-d-Y

Symmetrized DD XYXY-YXYX

(called XY-8 in the original paper)

Hollow circles – PDD XY

Solid circles – SDD (XY-8)

SX

SZ

SY

- Less freezing
- Overall better fidelity

Aperiodic sequences: Uhrig’s DD

Optimization of the inter-pulse intervals: UDD

Np = 20:

SX

SX

SY

SZ

SY

SZ

All errors

nZ errors only

UDD is not robust wrt pulse errors

Very susceptible to the rotation angle errors

Aperiodic sequences: Quadratic DD

3rd order QDD: U4(Y)-X-U4(Y)-X-U4(Y)-X-U4(Y)-X

U4(Y) = Uhrig’s DD with 4 pulses

Np = 20

εX only

All errors

SX

SY

εY only

SZ

- Pulse errors are important
- 2. Pulse errors can accumulate pretty fast
- 3. Concatenated design is very good: errors stay the same
- in spite of exponentially growing number of pulses
- 4. Fidelity of different initial states must be measured.
- 5. Freezing is a sign of low fidelity
- 6. UDD and QDD require very precise pulses

Nitrogen-vacancy centers

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

Nitrogen meets vacancy:

NV center

Bath spins S = 1/2

Distance between spins ~ 10 nm

Ground state spin 1

Easy-plane anisotropy

Distance between centers: ~ 2 μm

NV center – solid-state version of trapped atom

3E

ISC (m = ±1 only)

1A

532 nm

3A

m = 0 – always emits light

m = ±1 – not

Initialization: m = 0 state

Readout (PL level): population of m = 0

Ground state triplet:

Individual NV centers can be

initialized and read out:

access to a single spin dynamics

m = ±1

2.87 GHz

m = 0

- 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

C

C

N

V

C

C

C

C

0.5

-0.5

0

0.2

0.4

0.6

0.8

t(µs)

NV center in a spin bath

NV spin

Bath spin – N atom

ms=+1

m=+1/2

ms=-1

ms=-1/2

MW

MW

ms=0

Electron spin: pseudospin 1/2

14N nuclear spin: I = 1

B

B

No flip-flops between NV and the bath

Decoherence of NV – pure dephasing

Ramsey decay

Decay of envelope:

T2* = 380 ns

A = 2.3 MHz

Need fast pulses

Slow modulation:

hf coupling to 14N

Strong driving of a single NV center

Pulses 3-5 ns long → Driving field in the range close to GHz

Standard NMR / ESR, weak driving

Rotating frame

Spin

Oscillating field

x

co-rotating

(resonant)

y

counter-rotating

(negligible)

Rotating frame: static field B1/2 along X-axis

Strong driving of a single NV center

Experiment

Simulation

“Square” pulses:

29 MHz

109 MHz

223 MHz

Time (ns)

Time (ns)

Gaussian pulses:

109 MHz

223 MHz

- Rotating-frame approximation invalid: counter-rotating field
- Role of pulse imperfections, especially at the pulse edges

Characterizing / tuning DD pulses for NV center

Pulse errors - important: see Si:P DD

- unavoidable: counter-rotating field, pulse edges

- all errors (nX, nY, nZ, εX)

We want to determine and/or reduce the pulse errors

- 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

Can not be directly applied to strong driving

– linear relation between “in” and “out”

1. Prepare full set of basis states

2. Apply process L[ρ] to each of them

3. Measure in the same basis: determine χ

Our situation:

- Can reliably prepare only state
- Can reliably measure only SZ

Quantum process tomography

Describes most of experimental situations – QM is linear !

a’s and b’s are linearly related – matrix χ– complete description of L

“Bootstrap” problem

Assume: errors are small, decoherence during pulse negligible

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

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)

- uncorrected

Bootstrap protocol: experiments

Introduce known errors:

- phase of π/2Y pulse

- frequency offset

Self-consistency check: QPT with corrections

- Prepare imperfect basis states

- Apply corrections

(errors are known)

- Compare with uncorrected

Ideal recovery: F = 1, M2 = 0

M2

Fidelity

0.5

Spin echo

0

-0.5

1

10

0

0.2

0.4

0.6

0.8

free evolution time (ms)

t(µs)

What to expect for DD? Bath dynamics

Mean field: bath as a random field B(t)

simulation

O-U fitting

Gaussian, stationary, Markovian noise:

Ornstein-Uhlenbeck process

b – noise magnitude(spin-bath coupling)

R = 1/τC – rate of fluctuations (intra-bath coupling)

Agrees with experiments: pure dephasing by O-U noise

Ramsey decay

T2* = 380 ns

T2 = 2.8 μs

Long times (RT >> 1):

PDD

d-X-d-X

Fast decay

Slow decay

PDD-based CDD

All orders: fast decay at all times, rate WF (T)

optimal

choice

CPMG

(d/2)-X-d-X-(d/2)

Slow decay at all times, rate WS (T)

CPMG-based CDD

All orders: slow decay at all times, rate WS (T)

x

y

0.6

simulation

0

5

10

15

total time (ms)

1.0

x

y

simulation

0.6

0

5

10

15

total time (ms)

Protocols for realistic imperfect pulses

εX = εY = -0.02, mX = 0.005, mZ= nZ= 0.05·IZ, δB = -0.5 MHz

Pulses along X: CP and CPMG

CPMG – performs like no errors

CP – strongly affected by errors

State fidelity

Pulses along X and Y: XY4

(d/2)-X-d-Y-d-X-d-Y-(d/2)

(like XY PDD but CPMG timing)

State fidelity

Very good agreement

CPMG

CPMG

1

UDD

UDD

1/e decay

time (μs)

State fidelity

exp.

Np = 6

sim.

5

0.5

5

10

15

5

0

10

15

Np

Total time (ms)

Aperiodic sequences: UDD and QDD

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

Robustness to errors:

UDD vs XY4

QDD6 vs XY4

UDD,SX

QDD,SX

UDD,SY

QDD,SY

Np= 48

XY4,SX

XY4, SY

Np= 48

XY4,SX

XY4, SY

Small times:

QDD: F = 0.992

XY4: F = 0.947

QDD,SX

QDD,SY

XY4,SX

XY4, SY

XY4:

QDD:

Sensitive to different kind of errors

Solution: symmetrization

XY8

No 1st-order errors.

Initial F = 0.9999

but decays slowly as XY4

XY8,SX

XY8, SY

Master curve: for any number of pulses

100

1

NV2

SE

N = 4

State fidelity

1/e decay time (μs)

N = 8

10

N = 16

N = 36

NV1

0.5

N = 72

N = 136

0.1

1

10

1

10

100

Normalized time (t / T2 N 2/3)

number of pulses Np

DD on a single solid-state spin: scaling

136 pulses, coherence time increased by a factor 26

No limit is yet visible

Tcoh = 90 μs at room temperature

0

Re(χ)

Im(χ)

-1

1

0

-1

1

0

-1

1

0

-1

1

1

0

0

-1

-1

Quantum process tomography of DD

t = 4.4 μs

Pure dephasing

t = 10 μs

Only the elements

(I, I) and (σZ , σZ )

change with time

t = 24 μs

- Standard analytics (Magnus expansion) is often insufficient
- Numerical simulations are useful and often needed for
- realistic assessment of DD protocols
- In-out fidelity for a single state is not enough (freezing happens)
- Tomography is needed, at least partial
- Pulse errors are more than a little nuisance: can seriously plague
- advanced DD sequences
- Pulse errors need to be seriously addressed, theoretically and
- experimentally
- All taken into account, DD on a single solid-state spin achieved

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