Quantum Coherence via Smooth Optimal Control orn Bartels 1 , Florian - - PowerPoint PPT Presentation

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Quantum Coherence via Smooth Optimal Control orn Bartels 1 , Florian - - PowerPoint PPT Presentation

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Why do we need it? How does it work? An example: control of spin ensembles Quantum Coherence via Smooth Optimal Control orn Bartels 1 , Florian Mintert 1 , Bj obauer 2 , Johannes Majer 2 Tobias N 1 Freiburg Institute for Advanced Studies 2

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

Quantum Coherence via Smooth Optimal Control

Bj¨

  • rn Bartels1, Florian Mintert1,

Tobias N¨

  • bauer2, Johannes Majer2

1 Freiburg Institute for Advanced Studies 2 Atominstitut Vienna

14 September 2012

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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

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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

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

How to rotate a qubit?

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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?

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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

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

Quantum Computers

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

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

Quantum Computers

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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 (Est` eve):

Ams¨ uss 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!)

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

Control of Spin Ensembles

  • ptimal control copes with

different spin frequencies ω0 different amplitudes α of the control field

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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

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

Optimal Control

  • bjective of optimal control: maximize fidelity

max

  • ψf |Uf (t)(T)|ψi
  • 2

ω0, α

  • F[f (t)]

δF[f (t)] δf (t) = 0 ?

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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)

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

Optimization Algorithm

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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)

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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, Garc´ ıa Ripoll, PRA 76, 052304 (2007)

⇒ use only a few frequency components: Hc(t) = f (t)h f (t) =

n

  • k=1

ak sin(kΩt)

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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 + 2π

Ω ) = H(t)

Floquet: |ψk(t) = eiǫkt|φk(t) with |φk(t + 2π

Ω ) = |φk(t)

⇒ eigenvalue problem in Fourier space:

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

Techniques II: What is ∇aFa?

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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

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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

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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

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

Robustness against Inhomogeneity in the Control Field

5 10 15 20 25 30 0.70 0.75 0.80 0.85 0.90 0.95 1.00 effective inhomogeneity fidelity

T2 T 4 2 2 4

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

Robustness against Inhomogeneity in the Control Field

5 10 15 20 25 30 0.70 0.75 0.80 0.85 0.90 0.95 1.00 effective inhomogeneity fidelity

T2 T 4 2 2 4

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

Robustness against Inhomogeneity in the Control Field

  • 5

10 15 20 25 30 0.70 0.75 0.80 0.85 0.90 0.95 1.00 effective inhomogeneity fidelity

T2 T 4 2 2 4

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

The more frequency components, the higher the fidelity

∆ω =

1 200 ns, T = 3.14 µs, Ω = 1 MHz, 20 % of inhomogeneity

(only π/2-pulse)

  • 2

4 6 8 10 12 14 4 3 2 1

number of frequency components log101Fidelity

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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 %

0.2 0.4 0.6 0.8 1

fidelity

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

First experimental data

100

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

112.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

125

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

137.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

150

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

162.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

175

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

187.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

200

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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SLIDE 36

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

First experimental data

212.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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SLIDE 37

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

First experimental data

212.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

225

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

237.5

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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Why do we need it? How does it work? An example: control of spin ensembles

First experimental data

250

100 200 300 400 500 600 0.25 0.5 0.75 1 t ns P0

experiments by Tobias N¨

  • bauer from Atominstitut Vienna
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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 control field uses very simple pulses versatile tool that can be used with any other Hamiltonian/target functional