Advances in Mathematical Physics

Volume 2015, Article ID 125832, 11 pages

http://dx.doi.org/10.1155/2015/125832

## Properties of Stark Resonant States in Exactly Solvable Systems

College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA

Received 15 September 2015; Revised 23 November 2015; Accepted 29 November 2015

Academic Editor: Emmanuel Lorin

Copyright © 2015 Jeffrey M. Brown and Miroslav Kolesik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Properties of Stark resonant states are studied in two exactly solvable systems. These resonances are shown to form a biorthogonal system with respect to a pairing defined by a contour integral that selects states with outgoing wave boundary conditions. Analytic expressions are derived for the pseudonorm, dipole moment, and coupling matrix elements which relate systems with different strengths of the external field. All results are based on explicit calculations made possible by a newly designed integration method for combinations of Airy functions representing resonant eigenstates. Generalizations for one-dimensional systems with short-range potentials are presented, and relations are identified which are likely to hold in systems with three spatial dimensions.

#### 1. Introduction

Resonance states have been used to solve a wide range of problems in the fields of nuclear physics [1, 2], quantum chemistry [3], nonlinear optics [4–6], and semiconductor physics [7, 8]. Despite their widely recognized utility, relatively little is known about their general properties since they do not live in the familiar Hilbert space associated with Hermitian quantum mechanics [9]. The properties and issues that are less well-understood than in Hermitian quantum mechanics include inner products, normalization and completeness [10–17], complex expectation values [18, 19], and their physical interpretation [20]. Despite the mathematical difficulties related to their applications, resonance states do contain valuable physical information and it is important to investigate systems that could provide some guidance.

Here we add to the present understanding of resonance systems by analytically calculating a number of useful quantities for two exactly solvable quantum systems: the 1D Dirac-delta potential and 1D square-well models in the presence of a homogeneous field. Despite the latter model being a textbook example, and the former being studied and used in applications for decades (e.g., [21–24]), their resonances have so far been studied mainly with numerical tools [8, 25]. The quantities of interest, explicitly evaluated for the first time in this work, are the normalization factors, eigenvalue equations, dipole matrix elements, and off-diagonal transition elements that characterize the dependence of resonant basis states on the external field.

We also generalize our results to more complex models with piecewise constant potentials, and for general one-dimensional systems with finite-range potentials. We identify relations between the generalized dipole moments and the gradient of the atomic potentials, which resemble similar properties in systems with self-adjoint Hamiltonians.

Last but not least, all of our new results are based on direct evaluation of integral expressions, for which we have developed a new integration technique that is applicable to functions representing Stark resonances in one dimension with a piecewise constant potential.

Beyond developing a deeper understanding of exactly solvable systems, the additional motivation for this work is in the use of resonance states as a basis for time-dependent Schrödinger evolution, with applications in modeling electron ionization and nonlinear polarization due to a time varying optical pulse field [5]. Detailed study of exactly solvable systems with Stark resonant states brings multiple benefits. First, having explicit expressions for complex-valued observables and the ability to study their field and time dependence gives intuition of how one maps these complex values and open-system dynamics back to the real expectation values and the norm-preserving evolution found in Hermitian quantum mechanics of a closed system. Such a connection is crucial for applications in nonlinear optics (e.g., [5, 26, 27]). Secondly, the ability to compare different resonance systems may indicate which properties or relations are universally valid or common to all resonance systems. For example, we have witnessed in numerical simulations a field-dependent relation connecting the expectation values of the gradient of the atomic potential to the resonant state pseudonorm for one- and three-dimensional systems.

#### 2. Non-Hermitian Hamiltonians

In this section, we give some background of the class of Hamiltonians that we want to investigate. We begin with a 1D Hamiltonian that is parameterized by the strength of the external field :where the function represents the atomic potential. To study the Stark resonances, one usually assumes outgoing wave boundary conditions at and seeks solutions of where is the eigenvalue. With outgoing boundary conditions, the system is open as the particle can escape toward and the operator is non-Hermitian. Therefore, energy , along with many other observables, is complex-valued. Without Hermiticity, we lose many of the guarantees of Hermitian quantum mechanics, such as conservation of the number of particles, real-valued observables, and square integrable wave functions. There is not yet a full consensus on how to handle and interpret many of these quantities, including normalization and inner products.

Due to the non-square-integrable character of wave functions , the standard inner product and normalization prescriptions do not apply, since the integrals normally used to calculate them are divergent. Some regularization method must be used, and a number of approaches can be found in the literature [10, 13, 28]. However, it is important to appreciate that there may not be as much choice as it may seem in how the Stark resonant states should be normalized. For example, if a resonant state expansion of Green’s operator exists, the eigenstates appear in it with a definite “norm” [29]. In what follows we utilize biorthogonality of the Stark resonant system and obtain the eigenstates with such preferred normalization factors.

We consider the Hamiltonian to act on functions living on a complex contour , where the contour follows the real axis in the vicinity of the atom and then deviates from the real axis far from the origin. To select the space of outgoing wave functions, the contour departs into the upper complex plane as . The shape of this contour is inconsequential, except its property that it approaches infinity in the sector of the complex plain in which all outgoing waves, and in particular the resonance states, decay exponentially. One possible example utilizing a piecewise linear path is shown in Figure 1. At the far end of the contour, outgoing wave functions that behave as (with positive ) decay exponentially. Thus, the introduction of the contour is in the spirit of the external complex scaling.