Open quantum systems driven by a

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Open quantum systems driven by a macroscopic current

Collaborator: M. Baro , H.-Chr. Kaiser , H. Neidhardt , J. Rehberg

Supported by: DFG: ``Kopplung von van Roosbroeck- und Schrödinger-Poisson-Systemen mit Ladungsträgeraustausch'' (Coupling between van Roosbroeck system and a Schrödinger-Poisson system including exchange of carriers)

Description: The general goal of this research is the modeling of semiconductor multi quantum well lasers on the basis of a mixed model: One part is governed by van Roosbroeck's system while a neighborhood of the active zone is described by quantum mechanics. To that end we aim at an embedding of a quantum mechanically described structure into a macroscopic flow. The problem is that usually in quantum mechanics selfadjoint Schrödinger operators (including selfadjoint boundary conditions) are regarded, but selfadjoint Schrödinger operators necessarily lead to vanishing currents within the substructure. This consequence, however, is unacceptable from a physical point of view. Having in mind a current continuity condition for both the quantum and macroscopic current, in [6]/[7] the following setup for the description of an open quantum system driven by an adjacent macroscopic flow was proposed:

Let the open quantum system be situated in a bounded spatial domain $\Omega$ of $\mathbb{R}^d$ , $d\le 3$ .There we regard the Schrödinger equation

$\begin{displaymath} - \frac{\hbar^2}{2} \nabla \cdot \left(m^{-1}\nabla\psi \right) +V\psi = f \qquad \text{in $\Omega$}\end{displaymath}$ (1)

with a mass tensor m, a complex-valued potential V, and the boundary condition

$\begin{displaymath} {\hbar} \, \nu \cdot m^{-1} \nabla\psi = {i}\,\psi\, \nu \cdot \mathfrak{v} \qquad \text{on $\partial\Omega$} ,\end{displaymath}$ (2)

where $\nu$ denotes the outer unit normal on $\partial\Omega$ .This system is partly driven by an adjacent macroscopic flow acting on the boundary $\partial\Omega$ of $\Omega$ . The macroscopic flow is assumed to be of the form $J= -\,U\,\mathfrak{v},$ where U is the density of the macroscopic transport quantity and $\mathfrak{v}$ is the corresponding velocity density. In particular, we have in mind a gradient flow $J= -\,D\,U\,\nabla\Phi$ with $U=F(\Phi)$ which is governed by a reaction-diffusion equation

$\begin{displaymath} \frac {\partial U} {\partial t} + \nabla\cdot J = R(U,J)\end{displaymath}$ (3)

in a neighborhood of $\Omega$ . With non-smooth coefficients and mixed boundary conditions such an equation allows only for little regularity of the flow it governs ([1]). That is why we make only weak assumptions on the regularity of the flow acting on the boundary of $\Omega$ .The crucial point is that the boundary condition (2) cannot be implemented by using the usual boundary integral within the weak formulation, but has to be defined by a more subtle form approach, due to the deficient regularity of the adjacent macroscopic flow. Additionally, non-vanishing imaginary parts of the Schrödinger potentials, reflecting absorptive and dispersive properties of the substrate, have to be included in this context. Naturally, in a first step, the spectral properties of the--essentially non-selfadjoint--Schrödinger operators are of interest. The abstract framework for the rigorous definition and spectral analysis of the Schrödinger-type operator (1), (2) is the form perturbation method with a symmetric principal part and a non-symmetric part which is form-subordinated to it. In [4] under minimal (and realistic ([1])) assumptions on the macroscopic flow, the following assertions about the thus defined Schrödinger operator have been proven:

1.: Its resolvent belongs to the same summability class as the resolvent of the free Hamiltonian (with Neumann boundary conditions).
2.: It generates an analytic semigroup on L².
3.: It has no real eigenvalues and is, consequently, completely non-selfadjoint.
4.: In the two- and three-dimensional case there is an Abel basis of root vectors in L²; in the one-dimensional case one obtains a Riesz basis of root vectors.
5.: If the non-selfadjoint Schrödinger operator is dissipative, then the imaginary part of the potential must be positive, and the boundary form is given by a positive measure times the imaginary unit.

In the situation of a closed quantum system the Schrödinger-Poisson system ([2], [3], [6], [7]) turned out to give an adequate description of the occurring physical quantities such as carrier densities and electrostatic potential. This was the reason for our attempt to re-establish a Schrödinger-Poisson system in the situation of an open quantum system. At first, physical notions as carrier and current densities have to be re-defined in the case of open quantum systems. The key for doing so is the observation that the open quantum system always may be embedded by dilation theory into a closed one at least if the corresponding Hamiltonian is dissipative (or anti-dissipative) ([4]). In the spatially one-dimensional case the enlarged Hilbert space and the corresponding selfadjoint operator admit an explicit description ([5]), this is why we first investigated this situation. In [5] the required dilation of the non-selfadjoint Schrödinger operator is performed, the characteristic function is determined and a (generalized) eigenfunction expansion is carried out. On the basis of this, the fundamental notions of steady state, carrier density, and current density can be defined, and the concept of a dissipative Schrödinger-Poisson system can be established. The electron/hole densities u⁺, u^- are determined by

$\begin{displaymath} \int_\omega u^{+/-}\;dx =\,tr (\rho^{\pm} \chi_\omega),\end{displaymath}$ (4)

where $\omega$ is any Borel subset of the interval $\Omega$ , and $\rho^+$ , $\rho^-$ are density matrices, related to the Schrödinger-type operators (1)-(2) for holes and electrons, respectively, which act on the Hilbert space $L^2(\Omega)$ . The dependence of the carrier density on the Schrödinger potential has been investigated concerning estimates and continuity properties. Having these things at hand, we intend to prove the existence of a solution for the dissipative Schrödinger-Poisson system.

References:

J.A. GRIEPENTROG, H.-CHR. KAISER, J. REHBERG, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions in L^p, Adv. Math. Sci. Appl., 11 (2001), pp. 87-112.
F. NIER, A stationary Schrödinger-Poisson system arising from the modelling of electronic devices, Forum Math., 2 (1990), pp. 489-510.
$\dito$ , A variational formulation of Schrödinger-Poisson systems in dimensions $d\le 3$ , Comm. Partial Differential Equations, 18 (1993), pp. 1125-1147.
H.-CHR. KAISER, H. NEIDHARDT, J. REHBERG, Macroscopic current induced boundary conditions for Schrödinger operators, WIAS Preprint no. 650, 2001, to appear in: Integral Equations Oper. Theory.
$\dito$ , On 1-dimensional dissipative Schrödinger-type operators, their dilations and eigenfunction expansions, WIAS Preprint no. 664, 2001.
H.-CHR. KAISER, J. REHBERG, About a one-dimensional stationary Schrödinger-Poisson system with Kohn-Sham potential, Z. Angew. Math. Phys., 50 (1999), pp. 423-458.
$\dito$ , About a stationary Schrödinger-Poisson system with Kohn-Sham potential in a bounded two- or three-dimensional domain, Nonlinear Anal., 41A (2000), pp. 33-72.