Polynomial potentials and coupled quantum dots in two and three dimensions

Abstract

Keywords:.coupled quantum dots; two-and three-dimensional models; non-separable polynomial potentials; probability density bifurcations; PACS number: PACS 03.65.Ge -Solutions of wave equations: bound states 1 znojil@ujf.cas.cz AbstractPolynomial potentials V (x) = x 4 + O(x 2 ) and V (x) = x 6 + O(x 4 ) were introduced, in the Thom's purely geometric classification of bifurcations, as the benchmark models of the so called cusp catastrophes and of the so called butterfly catastrophes, respectively. Due to their asymptotically confining property, these two potentials are exceptional, viz., able to serve similar purposes even after quantization, in the presence of tunneling. In this paper the idea is generalized to apply also to quantum systems in two and three dimensions. Two related technical obstacles are addressed, both connected with the non-separability of the underlying partial differential Schrödinger equations. The first one [viz., the necessity of a non-numerical localization of the extremes (i.e., of the minima and maxima) of V (x, y, . . .)] is resolved via an ad hoc reparametrization of the coupling constants. The second one [viz., the necessity of explicit construction of the low lying bound states ψ(x, y, . . .)] is circumvented by the restriction of attention to the dynamical regime in which the individual minima of V (x, y, . . .) are well separated, with the potential being locally approximated by the harmonic oscillator wells simulating a coupled system of quantum dots (a.k.a. an artificial molecule). Subsequently it is argued that the measurable characteristics (and, in particular, the topologically protected probability-density distributions) could bifurcate in specific evolution scenarios called relocalization catastrophes.

Displaced Harmonic Oscillator V ∼ min [(x + d)2, (x − d)2] as a Benchmark Double-Well Quantum Model

“…Last but not least let us mention that among the physical systems to which our present methodical innovation could also appear, in the nearest future, to be immediately applicable are also, potentially, all of the various more-dimensional realistic multi-well models as encountered, e.g., in quantum chemistry [40] and/or in the quantum-computation-motivated mesoscopic physics of the systems of quantum dots [41]. Indeed, under the widespread interaction-analyticity constraint, just the approximate solutions of the partial differential Schrödinger equations are usually available in these systems.…”

Confluences of exceptional points and a systematic classification of quantum catastrophes

“…The latter idea found its constructive applications even in more dimensions, with at nontrivial in 89 , or at the more realistic in 90 . Nevertheless, the price had to be paid for the strictly shared locality of the benchmark potentials V ( x ).…”

Confluences of exceptional points and a systematic classification of quantum catastrophes

“…The latter idea found its constructive applications even in more dimensions, with x ∈ R d at nontrivial d = 2 in [87], or at the more realistic d = 3 in [88]. Nevertheless, the price had to be paid for the strictly shared locality of the benchmark potentials V (x).…”