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 Coupl upled ed of Spin ins Co Coupl upled ed Project I- I-QUIN INS Project I- I-QUIN INS to a a Dis issipa ipatj tjve e to a a Dis issipa ipatj tjve e Novemb mber 2019 Novemb mber 2019 Cavit vity Cavit vity
Contents 2 Context Model systems Controllability Applicatjons Conclusion → two papers in preparatjon, Available soon on ArXiv & Research gate
3 Context 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)
4 Context → Quantum measurements, quantum sensing. → Generatjon of entangled states. Quantum Technologies Fundamental Questjons → to what extent can we control an open quantum system? → to what extent can we generate entanglement in a system with dissipatjon?
Quantum Control 5 What is a quantum control- Control fjeld u ( t ) optjmal problem ? Target Initjal Target Initjal State State State State Evolutjon Operator Goal: fjnd u ( t ) that minimizes some constraint(s) (e.g. control tjme) → analytjc expression (if lucky) → numerical optjmizatjon (otherwise)
Two model 6 systems Ensemble of spins ½ Single spin ½ and max 1 quantum excitatjon. Damped cavity mode Damped cavity mode Jaynes-Cummings interactjon Jaynes-Cummings interactjon Control with coherent and Control of the spin energy transitjon. squeezing controls on the cavity mode → Phys. Rev. A → Phys. Rev. X 61, 025802 (2000) 7,041011 (2017) → A. Bienfait, → Phys. B 27, 1345053 PhD thesis (2013). → Appl. Phys. Letu. 111, 202604 → Sci. Rep. 8, 1 (2018) (2017).
Two model 7 systems Lorentzian distributjon of mode → efgectjve cavity mode
Model 1 8 → Use of pulse sequences Each pulse is parameterized by its amplitude and its positjon in the sequence → parameters to optjmize numerically. Use at most 7 pulse packages → max 28 parameters to determine →Further details: Phys. Rev. A 98, 023425(2018), Q. Ansel, PhD thesis
Model 1 9 ! ! Collectjve controls on the spin ensemble. Spins cannot be controlled individually → Make the control task very diffjcult...
Model 2 10 Change of variables allows us to simplify drastjcally the system complexity. Lorentzian distributjon of mode → efgectjve cavity mode Max 1 quantum excitatjon.
Two model 11 systems
12 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 ?
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: Not controllable !
14 Model 2 Okay, model 2 is not controllable, but what are the spin states that can be reached? Constant controls Numerical search of the reachable points using control fjeld optjmizatjon. Dark gray → the target state can be reached with a high probability !
15 Model 2 Okay, model 2 is not controllable, but what are the spin states that can be reached? Constant controls Numerical search of the reachable points using control fjeld optjmizatjon. Dark gray → the target state can be reached with a high probability !
16 Model 2 Okay, model 2 is not controllable, but what are the spin states that can be reached? Constant controls
17 Model 2 Okay, model 2 is not controllable, but what are the spin states that can be reached? Same as before, but with another initjal state.
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.
Model 1 19 → no relaxatjon : system is controllable. → Set of possible transformatjons: SU(N) → Squeezing fjeld: faster generatjon of the dynamical Lie algebra. Coherent Squeezing control terms 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)) = N 2- 1 = 323.
20 Applications ● Generatjon of a symmetric state (2 spins). Spins 0 0 Cavity
21 Applications ● Optjmal solutjon without ● cavity damping: 0 0
22 Applications ● Generatjon of entangled states ● - Ensemble of 4 spins. with a measure of non-classicality. ● - Short control duratjon ( t max =π/2 ). – ● - Detuning distributjon: ● ● Δ n /g ∈ {-1,-0.5,0.5,1} ●
23 Applications ● Selectjvity of distjnct spins with constant or sinusoidal controls. – ● ● Spin 2: p = 13,41 q Spin 1: p = 2.23 q
24 Applications ● The selectjvity process can be optjmized. ● Cost functjon used in the calculatjons: – ● ●
25 Applications ● Selectjvity of distjnct spins with constant or sinusoidal controls. – ● ● Spin 2: p = 13,41 q Spin 1: p = 2.23 q
26 Applications ● The selectjvity process can be optjmized. ● Cost functjon used in the calculatjons: – ● ●
27 Conclusion Explore the physical limits of these Selectjvity control problems. Controllability of Optjmal control Spin systems coupled Allows to reach the to a dissipatjve Generatjon of Physical limit of parameter environement Entangled state selectjvity No full controllability Strongly limited by the relaxatjon Simple control Squeezing provides mechanisms (model 2) Betuer results with a measure Squeezing enhance the of non classicality generatjon of the dynamic Lie algebra (model 1)
Model 1 28 Use of bump pulse (coherent control) → Short pulse approximatjon → simplify the numerical Transformatjon, Transformatjon, To take into account To take into account calculatjons. The cavity dynamics The cavity dynamics → Phys. Rev. A 98, 023425(2018) → Q. Ansel, PhD thesis
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