The polarizability of a confined atomic system: an application of the Dalgarno–Lewis method

• Published in: European journal of physics 2021-07, Vol.42 (4), p.45407
• Main Authors: Antão, T V C, Peres, N M R
• Format: Article
• Language: English
• Subjects: Ciências Físicas; Ciências Naturais; Dalgarno–Lewis; polarizability; quantum mechanics; Science & Technology; Social Sciences; Stark effect
• ISSN: 0143-0807; 1361-6404
• DOI: 10.1088/1361-6404/abfd24

In this paper we give an application of the Dalgarno–Lewis method, the latter not usually taught in quantum mechanics courses. This is very unfortunate since this method allows one to bypass the sum over states appearing in the usual perturbation theory. In this context, and as an example, we study the effect of an external field, both static and frequency-dependent, on a model atom at fixed distance from a substrate. This can happen, for instance, when some organic molecule binds from one side to the substrate and from the other side to an atom or any other polarizable system. We model the polarizable atom by a short range potential, a Dirac-δ, and find that the existence of a bound state depends on the ratio of the effective 'nuclear charge' to the distance of the atom from the substrate. Using an asymptotic analysis, previously developed in the context of a single δ-function potential in an infinite medium, we determine the ionization rate and the Stark shift of our system. Using Dalgarno–Lewis theory we find an exact expression for the static and dynamic polarizabilities of our system valid to all distances. We show that the polarizability is extremely sensitive to the distance from the substrate, creating the possibility of using this quantity as a nanometric ruler. Furthermore, the line shape of the dynamic polarizability is also extremely sensitive to the distance from the substrate, thus providing another route to measure nanometric distances. The didactic value of the δ-function potential is widely accepted in teaching activities due to its simplicity, while still keeping the essential ingredients of a given problem. The authors acknowledge João Carlos Henriques and Thomas G Pedersen for many discussions and exchanges of ideas on this topic. NMRP acknowledges support from the European Commission through the project 'Graphene-Driven Revolutions in ICT and Beyond' (Ref. No. 785219), and the Portuguese Foundation for Science and Technology (FCT) in the framework of Strategic Financing UID/FIS/04650/2019. In addition, he acknowledges COMPETE2020, PORTUGAL2020, FEDER and the Portuguese FCT through Projects POCI-01-0145-FEDER-028114, POCI-01-0145-FEDER-029265, PTDC/NAN-OPT/29265/2017, and POCI-01-0145-FEDER-02888. TVCA acknowledges the FCT for a grant given in the context of the 'Quantum Matter|Materials and Concepts Summer Training Program 2020'.