Stark resonances for a double   quantum well: crossing scenarios, exceptional points and geometric phases

Abstract

The complex energy resonances of a double δ potential well in a constant (Stark) field are studied. Varying the two system parameters (well distance and field strength) we investigate the behaviour of the resonance energies and wavefunctions both analytically and numerically. Different crossing scenarios for the real and imaginary parts of two resonance energies are observed and compared with a simple two-state model. In addition, a point in parameter space where both the real and imaginary parts of the two energies degenerate, an exceptional point, is found. Varying the system parameters around this exceptional point, the behaviour of energies and wavefunctions is discussed and the corresponding geometric phases, or Berry phases, for this non-Hermitian system are considered.

Decay of a negative molecular ion in a constant electric field

“…As a result, the expression for ⑀ ± in approxi mation (20) assumes the form (26) The solution to system (14) for coefficients f 1 and f 2 , which determine the degree of localization of the electron near the first and second centers (f 1 = ±f 2 in zero field), gives (27) …”

“…Expressions (26)- (28) are inapplicable in orthogo nal geometry (F ⋅ R = 0) when A + = A -≡ A(⑀) and con dition (20) does not hold. In this case, the iterative inclusion of the exchange interaction involves the solution of system (16), which can be reduced to the equations in this case, the right hand side is treated as a pertur bation.…”

“…Exactly solvable analytic models are known only for the case of short range atomic centers, which corresponds, for example, to the interaction of a negative molecular ion with an electric field. Detailed analysis of the simplest 1D model (electron in the field of two 1D zero range (δ) potentials in a constant electric field) was carried out in [20][21][22]; in recent publication [23], this model was used for a qualitative description of photodetach ment of an electron from the negative molecular ion in parallel geometry (F || R). More realistic results should be expected in simulating atomic centers by 3D δ potentials whose action is equivalent to imposition of appropriate boundary conditions on the wavefunc tion near the centers [24].…”

Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

“…Existing approaches for finding matrices with multiple eigenvalues [7,9,11,12,16,18,19,23,26,33,40,41] assume arbitrary perturbations of a matrix and, hence, they do not work for multiparameter matrix families. We also mention the topological method for the localization of double eigenvalues in twoparameter matrix families [22].…”

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

“…In modern problems of quantum physics, crystal optics, physical chemistry, acoustics and mechanics it is important to know how the diabolic point bifurcates under arbitrary complex perturbations forming topological singularities of eigenvalue surfaces like a double coffee filter with two exceptional points or a diabolic circle of exceptional points, see e.g. [Mondragon and Hernandez (1993)], [Mondragon and Hernandez (1996)], [Shuvalov and Scott (2000)], [Keck et al (2003)], [Berry and Dennis (2003)], [Korsch and Mossman (2003)], [Berry (2004)]. …”