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Quantum optimal control using the adjoint method

100%
Borzì A.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
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2012
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tom 1
93-111
EN
Control of quantum systems is central in a variety of present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures, including the emerging fields of quantum computation and quantum communication. In this paper, a review of recent developments in the field of optimal control of quantum systems is given with a focus on adjoint methods and their numerical implementation. In addition, the issues of exact controllability and optimal control are discussed for finite- and infinitedimensional quantum systems. Some insight is provided considering ’two-level’ models. This work is completed with an outlook to future developments.
2
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Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

100%
Rajchel K.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2019
|
tom 18
157-165
EN
The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.
3
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Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential

88%
Franco J.
Open Mathematics
|
2012
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tom 10
|
nr 3
927-941
EN
We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V λ(x) = λx −2 as a representation of $\widetilde{SL(2,\mathbb{R})}$ . The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of $\widetilde{SL(2,\mathbb{R})}$ ⋉ H 3, where H 3 is the three-dimensional Heisenberg group.
4
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Hybrid fluid-quantum coupling for the simulation of the transport of partially quantized particles in a DG-MOSFET

75%
Jourdana C., Vauchelet N.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
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2015
|
tom 4
|
nr 1
EN
This paper is devoted to numerical simulations of electronic transport in nanoscale semiconductor devices forwhich charged carriers are extremely confined in one direction. In such devices, like DG-MOSFETs, the subband decomposition method is used to reduce the dimensionality of the problem. In the transversal direction electrons are confined and described by a statistical mixture of eigenstates of the Schrödinger operator. In the longitudinal direction, the device is decomposed into a quantum zone (where quantum effects are expected to be large) and a classical zone (where they are negligible). In the largely doped source and drain regions of a DG-MOSFET, the transport is expected to be highly collisional; then a classical transport equation in diffusive regime coupled with the subband decomposition method is used for the modeling, as proposed in N. Ben Abdallah et al. (2006, Proc. Edind. Math. Soc. [7]). In the quantum region, the purely ballistic model presented in Polizzi et al. (2005, J. Comp. Phys. [25]) is used. This work is devoted to the hybrid coupling between these two regions through connection conditions at the interfaces. These conditions have been obtained in order to verify the continuity of the current. A numerical simulation for a DG-MOSFET, with comparison with the classical and quantum model, is provided to illustrate our approach.
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