Why do we need it? How does it work? An example: control of spin ensembles

Quantum Coherence via Smooth Optimal Control

Bjorn Bartels1, Florian Mintert1,Tobias Nobauer2, Johannes Majer2

1 Freiburg Institute for Advanced Studies2 Atominstitut Vienna

14 September 2012

Why do we need it? How does it work? An example: control of spin ensembles

Smooth optimal control:

1 Why do we need it?

2 How does it work?

3 An example: control of spin ensembles

Why do we need it? How does it work? An example: control of spin ensembles

1 Why do we need it?

2 How does it work?

3 An example: control of spin ensembles

Why do we need it? How does it work? An example: control of spin ensembles

How to rotate a qubit?

Why do we need it? How does it work? An example: control of spin ensembles

Apply a control pulse!

Why is it not that simple?

Why do we need it? How does it work? An example: control of spin ensembles

Nitrogen-Vacancy Centers in Diamond

Pham et al., New J. Phys. 13, 045021 (2011)

Davies, Hamer, Proc. R. Soc. London A 348, 285 (1967)

very long coherence times (up to ms): quantum memory

Why do we need it? How does it work? An example: control of spin ensembles

Quantum Computers

Kubo et al., PRL 107, 220501 (2011)

Why do we need it? How does it work? An example: control of spin ensembles

Quantum Computers

Why do we need it? How does it work? An example: control of spin ensembles

The Problem of Spin Ensembles

experiments in Vienna (Majer) and Paris (Esteve):Amsuss et al., PRL 107, 060502 (2011) + Kubo et al., PRL 107, 220501 (2011)

problems: inhomogeneous broadening of NV ensemble (→NMR),inhomogeneous field of antenna (new!)

Why do we need it? How does it work? An example: control of spin ensembles

Control of Spin Ensembles

optimal control copes with

different spin frequencies ω0

different amplitudes α of the control field

Why do we need it? How does it work? An example: control of spin ensembles

1 Why do we need it?

2 How does it work?

3 An example: control of spin ensembles

Why do we need it? How does it work? An example: control of spin ensembles

Optimal Control

objective of optimal control: maximize fidelity

max⟨∣∣〈ψf |Uf (t)(T )|ψi 〉

∣∣2⟩ω0, α︸ ︷︷ ︸

F [f (t)]

δF [f (t)]

δf (t)= 0 ?

Why do we need it? How does it work? An example: control of spin ensembles

Pulse Shaping with GRAPE/Krotov

piecewise constant functions:

ai

ai'

iterative improvement:

Fa+δa ≈ Fa + δa · ∇aFa +1

2

∑i ,j

δaiδaj∂ai∂ajFa

Glaser et al., J. Magn. Reson. 172, 296 (2005)

Krotov, Global Methods in Optimal Control Theory, Dekker (1995)

Why do we need it? How does it work? An example: control of spin ensembles

Optimization Algorithm

Why do we need it? How does it work? An example: control of spin ensembles

Typical GRAPE Pulse: High Frequency Components

Kobzar et al., J. Magn. Reson. 173, 229 (2005)

Why do we need it? How does it work? An example: control of spin ensembles

Our Approach: Smooth Control

alternative solution:

instead of

Caneva, Calarco, Montangero, PRA 84, 022326 (2011)

Romero Isart, Garcıa Ripoll, PRA 76, 052304 (2007)

⇒ use only a few frequency components:

Hc(t) = f (t)h f (t) =n∑

k=1

ak sin(kΩt)

Why do we need it? How does it work? An example: control of spin ensembles

Techniques I: What is U(t)?

dynamics periodic in time:

i∂t |ψk(t)〉 = H(t)|ψk(t)〉 H(t + 2πΩ ) = H(t)

Floquet: |ψk(t)〉 = e iεk t |φk(t)〉 with |φk(t + 2πΩ )〉 = |φk(t)〉

⇒ eigenvalue problem in Fourier space:

︸ ︷︷ ︸K

Why do we need it? How does it work? An example: control of spin ensembles

Techniques II: What is ∇aFa?

Why do we need it? How does it work? An example: control of spin ensembles

1 Why do we need it?

2 How does it work?

3 An example: control of spin ensembles

Why do we need it? How does it work? An example: control of spin ensembles

Robustness against Different Spin Frequencies

broad frequency interval: here ∆ω · T = π

Why do we need it? How does it work? An example: control of spin ensembles

Typical Control Pulse: Only 4 Frequency Components

achieved fidelities: > 99.99 % for the previous example

Why do we need it? How does it work? An example: control of spin ensembles

Robustness against Inhomogeneity in the Control Field

0 5 10 15 20 25 300.70

0.75

0.80

0.85

0.90

0.95

1.00

effective inhomogeneity H%L

fid

elit

y

0 T2 T-4

-2

0

2

4

Why do we need it? How does it work? An example: control of spin ensembles

Robustness against Inhomogeneity in the Control Field

0 5 10 15 20 25 300.70

0.75

0.80

0.85

0.90

0.95

1.00

effective inhomogeneity H%L

fid

elit

y

0 T2 T

-4

-2

0

2

4

Why do we need it? How does it work? An example: control of spin ensembles

Robustness against Inhomogeneity in the Control Field

´´ ´

0 5 10 15 20 25 300.70

0.75

0.80

0.85

0.90

0.95

1.00

effective inhomogeneity H%L

fid

elit

y

0 T2 T-4

-2

0

2

4

Why do we need it? How does it work? An example: control of spin ensembles

The more frequency components, the higher the fidelity

∆ω = 1200 ns , T = 3.14µs, Ω = 1 MHz, 20 % of inhomogeneity

(only π/2-pulse)

+

++++

+

++

+

+

+0 2 4 6 8 10 12 14

-4

-3

-2

-1

0

number of frequency components

log

10H1-

Fid

elit

yL

Why do we need it? How does it work? An example: control of spin ensembles

Generating π-pulses for the experiment

∆ω : 8 MHz Gaussian FWHM, 25 % of inhomogeneity,n = 15 frequency components, control amplitude < 1.5 MHz

〈F 〉 > 99 %

0 0.2 0.4 0.6 0.8 1

fidelity

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

100%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

112.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

125%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

137.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

150%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

162.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

175%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

187.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

200%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

212.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

212.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

225%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

237.5%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

250%

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

t ns

PHÈ

0>L

experiments by Tobias Nobauer from Atominstitut Vienna

Why do we need it? How does it work? An example: control of spin ensembles

Conclusion

Smooth optimal control

can manipulate spin ensembles with inhomogeneous controlfield

uses very simple pulses

versatile tool that can be used with any otherHamiltonian/target functional