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Quentjn Ansel Bruno Bellomo Dominique Sugny Co Controlla labil ilit ity Controlla Co labil ilit ity & & Optj & tjmal al Co Control & Optj tjmal al Co Control GdR IM Gd IM GdR IM Gd IM of Spin ins Co

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SLIDE 1

Co Controlla labil ilit ity Co Controlla labil ilit ity & & Optj tjmal al Co Control & & Optj tjmal al Co Control

  • f Spin

ins Co Coupl upled ed

  • f Spin

ins Co Coupl upled ed to a a Dis issipa ipatj tjve e to a a Dis issipa ipatj tjve e Cavit vity Cavit vity

Gd GdR IM IM Gd GdR IM IM Project I- I-QUIN INS Project I- I-QUIN INS Novemb mber 2019 Novemb mber 2019

Quentjn Ansel

Bruno Bellomo Dominique Sugny

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SLIDE 2

Contents

 Context  Model systems  Controllability  Applicatjons  Conclusion

2

→ two papers in preparatjon, Available soon on ArXiv & Research gate

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SLIDE 3

Context

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Recent collaboration with the Quantronics Group (CEA- Saclay), in particular with S. Probst and P. Bertet. → Experiments involving an ensemble of spins coupled to a microwave resonator

  • S. Probst & al. Shaped pulses for transient compensation in

quantum-limited electron spin resonance spectroscopy, JMR 303, 42-47 (2019)

  • Q. Ansel & al. Optimal control of an inhomogeneous spin ensemble

coupled to a cavity, Phys. Rev. A 98, 023425 (2018)

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SLIDE 4

Context

4

Fundamental Questjons

→ Quantum measurements, quantum sensing.

→ Generatjon of entangled

states. → to what extent can we control an open quantum system? → to what extent can we generate entanglement in a system with dissipatjon?

Quantum Technologies

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SLIDE 5

Quantum Control

 What is a quantum control-

  • ptjmal problem ?

5

Initjal State Initjal State

Target State Target State

Control fjeld u(t) Evolutjon Operator

 Goal: fjnd u(t) that minimizes some constraint(s) (e.g.

control tjme) → analytjc expression (if lucky) → numerical optjmizatjon (otherwise)

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SLIDE 6

Two model systems

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Ensemble of spins ½

Damped cavity mode

Jaynes-Cummings interactjon

Control with coherent and squeezing controls on the cavity mode

Single spin ½ and max 1 quantum excitatjon.

Damped cavity mode

Jaynes-Cummings interactjon

Control of the spin energy transitjon.

→ Phys. Rev. X 7,041011 (2017) → A. Bienfait, PhD thesis → Appl. Phys.

  • Letu. 111, 202604

(2017). → Phys. Rev. A 61, 025802 (2000) → Phys. B 27, 1345053 (2013). → Sci. Rep. 8, 1 (2018)

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SLIDE 7

Two model systems

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Lorentzian distributjon of mode → efgectjve cavity mode

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Model 1

→ Use of pulse sequences Each pulse is parameterized by its amplitude and its positjon in the sequence → parameters to

  • ptjmize numerically.

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Use at most 7 pulse packages → max 28 parameters to determine

→Further details: Phys. Rev. A 98, 023425(2018), Q. Ansel, PhD thesis

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SLIDE 9

Model 1

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 Collectjve controls on the spin

ensemble.

 Spins cannot be controlled

individually

 → Make the control task very

diffjcult...

! !

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SLIDE 10

Model 2

Change of variables allows us to simplify drastjcally the system complexity.

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Lorentzian distributjon of mode → efgectjve cavity mode Max 1 quantum excitatjon.

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SLIDE 11

Two model systems

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Controllability ?

 Defjnitjon: Complete state controllability describes the ability of an

external input (the control fjeld(s)) to move the state of a system from any initjal state to any other fjnal state.

 Are the systems contr

trollable ?

12

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SLIDE 13

Model 2

 The system is described by a liner tjme-dependent ODE→

controllability can be studied using Lie algebra methods.

 Complete controllability if the group GL(2,C) can be generated  → simple calculatjon shows that the reachable set is:

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Not controllable !

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SLIDE 14

Model 2

Okay, model 2 is not controllable, but what are the spin states that can be reached?

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Numerical search of the reachable points using control fjeld

  • ptjmizatjon. Dark gray → the target state can be reached with a high

probability ! Constant controls

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SLIDE 15

Model 2

Okay, model 2 is not controllable, but what are the spin states that can be reached?

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Constant controls Numerical search of the reachable points using control fjeld

  • ptjmizatjon. Dark gray → the target state can be reached with a high

probability !

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SLIDE 16

Model 2

Okay, model 2 is not controllable, but what are the spin states that can be reached?

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Constant controls

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SLIDE 17

Model 2

Okay, model 2 is not controllable, but what are the spin states that can be reached?

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Same as before, but with another initjal state.

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SLIDE 18

Model 2

 Most of the points can be reached with simple controls (analytjc

formulas).

 Inhibitjon of the spin decay: modulated controls gives a modest

improvement than the detuning efgect.

 Effjcient controls to drive the spin on the ground state.

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SLIDE 19

Model 1

→ no relaxatjon : system is controllable. → Set of possible transformatjons: SU(N) → Squeezing fjeld: faster generatjon of the dynamical Lie algebra.

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Coherent control terms Squeezing control terms

Dimensions of the Lie algebra spanned by recursive commutators of the

  • Hamiltonians. The commutator order corresponds to the maximum

number of commutators taken into account. Calculatjons performed on a truncated Hilbert space. DimH = 18, → dim(su(18)) = N2-1 = 323.

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SLIDE 20

Applications

  • Generatjon of a symmetric state (2 spins).

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Spins Cavity

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SLIDE 21

Applications

  • Optjmal solutjon without
  • cavity damping:

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SLIDE 22

Applications

  • Generatjon of entangled states

with a measure of non-classicality.

–

  • 22
  • - Ensemble of 4 spins.
  • - Short control duratjon (tmax=π/2).
  • - Detuning distributjon:
  • Δn/g ∈ {-1,-0.5,0.5,1}
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SLIDE 23

Applications

  • Selectjvity of distjnct spins with constant or sinusoidal controls.

–

  • 23

Spin 2: p = 13,41 q Spin 1: p = 2.23 q

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Applications

  • The selectjvity process can be optjmized.
  • Cost functjon used in the calculatjons:

–

  • 24
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SLIDE 25

Applications

  • Selectjvity of distjnct spins with constant or sinusoidal controls.

–

  • 25

Spin 2: p = 13,41 q Spin 1: p = 2.23 q

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SLIDE 26

Applications

  • The selectjvity process can be optjmized.
  • Cost functjon used in the calculatjons:

–

  • 26
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Conclusion

Controllability of Spin systems coupled to a dissipatjve environement

No full controllability Simple control mechanisms (model 2) Squeezing enhance the generatjon of the dynamic Lie algebra (model 1)

Generatjon of Entangled state

Strongly limited by the relaxatjon Squeezing provides Betuer results with a measure

  • f non classicality

Selectjvity

Optjmal control Allows to reach the Physical limit

  • f parameter

selectjvity

Explore the physical limits of these control problems.

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SLIDE 28

Model 1

Use of bump pulse (coherent control) → Short pulse approximatjon → simplify the numerical calculatjons.

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→ Phys. Rev. A 98, 023425(2018) → Q. Ansel, PhD thesis

Transformatjon, To take into account The cavity dynamics Transformatjon, To take into account The cavity dynamics