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.

Resonance states have been used to solve a wide range of problems in the fields of nuclear physics [

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., [

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 [

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

Due to the non-square-integrable character of wave functions

We consider the Hamiltonian

An example contour in the complex plane that serves as a “complexified” spatial axis in a model of an open quantum system in which a non-Hermitian Hamiltonian (

The differential expression

In this work we aim to avoid any reliance on formal operator properties. Instead, we show by explicit calculation of the underlying contour integrals that the following orthogonality relation holds for outgoing Stark resonances:

We have recently presented a proof of principle for application of metastable electronic states to calculate nonlinear response in a time-dependent field of an optical pulse [

If we represent a particular quantum state

For the purposes of this work, (

The coupling terms can be related to the dipole moment matrix elements with the help of the following argument utilizing the parametric dependence of the Hamiltonian on

Moreover, for the normalized resonances the coupling terms are antisymmetric in indices

In this section we outline the properties of resonant wave functions for two different systems, and we find associated resonant eigenvalue equations. We also calculate explicit normalization factors for all Stark resonances using the orthogonality relation (

In this work we assume that the potential

To obtain the remaining portion(s) of energy eigenstates, one has to “fill in” the wave function in the central region of

The asymptotic form of the energy eigenstates as shown in (

We present our final results, including those on resonant state normalization, in the form that is independent of how one chooses to parameterize the eigenfunctions. Whenever we show intermediate results, it is for the wave functions written as follows. For the Dirac-delta model, we take the unnormalized ansatz for the outgoing resonance in the form

For the Dirac-delta potential

Longer calculations are required to obtain the analogous equation [

It is helpful to visualize the resonance energy “landscapes” illustrated in Figure

Outgoing resonance eigenvalue equation landscapes for Dirac-delta ((a)

To establish formulas for normalization and to verify the orthogonality relation (

To calculate the contour integral(s), the corresponding primitive functions are evaluated at points of discontinuities of the potential and at both ends of the contour, and this is where the choice of the contour is important. The resonant wave functions decay exponentially for

Thus, direct integration along the contour, followed by simplifications making use of the eigenvalue equation and the Wronskian for Airy functions, yields the following normalization factor for the Stark resonance in the Dirac-delta model:

To verify the mutual orthogonality with respect to (

To conclude this subsection, we note that our direct verification of the orthogonality relation (

Next we calculate the generalized dipole matrix elements, both diagonal and off-diagonal. Expressed with the help of unnormalized resonance eigenfunctions

For the diagonal matrix element in the Dirac-delta potential model, we obtain the following expression in terms of Airy functions:

For the off-diagonal dipole matrix elements, we use (VS 3.54). Utilizing

Let us proceed with the dipole moment calculations for the square-well potential. The matrix elements can be found by again making use of (VS 3.51) and (VS 3.54). Thanks to the continuity properties of the wave function, many terms that arise in the course of this calculation cancel, and the resulting diagonal terms are

Now we calculate the off-diagonal elements. When using (VS 3.54), taking into account that continuity of the wave function allows us to ignore terms that do not involve

We now turn our attention to the terms

To calculate the coupling terms, we first differentiate the wave function with respect to the field

For the model with the square-well potential we have a combination of integrals of similar types; namely,

The above integrals for both models contain terms of two kinds. The first group can be evaluated making use of known, previously published Airy integrals. These have the form

For the Dirac-delta model, the individual integrals on the RHS of (

The first two lines can be simplified using the Wronskian and normalization factors, while the third can be rewritten with the help of the eigenvalue equation. Combining these shows that the first three lines sum up to zero, leaving only the last term which we write in terms of derivatives of the wave functions at the origin:

Coupling terms (

To conclude this section, we have shown that the coupling terms can be directly calculated using a new Airy integral technique detailed in the appendix. While our explicit calculations do not justify the formal steps taken to obtain (

Comparing results (

This is an intriguing result, because an identical formula can be derived for the discrete-energy eigenstates of a self-adjoint Hamiltonian by evaluating its double commutator with the position operator. However, here we have Stark resonances represented by complex-valued functions living on the contour

We have used numerical simulations (not shown here) to verify that the relation between the Stark resonance pseudonorm and the generalized expectation value of the “atomic” potential gradient could also be valid for three-dimensional systems. It is tempting to speculate that the off-diagonal dipole element relation (

We have derived analytic expressions for a number of quantities that characterize the Stark resonance states in two exactly solvable systems. The first model studied in this work is the one-dimensional particle in a Dirac-delta potential with additional homogeneous field, and the second has the square-well potential.

We have studied these systems as open, non-Hermitian models, and identified a natural choice for the pairing connecting the states in the domain of the Hamiltonian with the states in the domain of its adjoint operator. With respect to this pairing, Stark resonances form an orthogonal system, and many of their properties can be evaluated analytically.

Despite the fact that both models have been studied for years, explicit expressions for their (pseudo-) norms, dipole moment expectation values, and their relations connecting the resonant state wave functions at different field values are new. Our results thus further the understanding of the mathematical properties that underline the Stark effect.

Moreover, we have shown that certain results naturally extend to a wide class of one-dimensional models and we have also identified relations that appear to be candidates for properties generally applicable to three-dimensional Stark systems. In particular, we have found that the generalized dipole moment matrix elements between the nonphysical resonant states can be related to the expectation values of the atomic potential gradient in a way that is completely analogous to relations that hold for real-valued discrete-energy eigenfunctions in Hermitian systems. We speculate that these relations apply generally in three dimensions and could be used in numerical calculations to assess the fidelity of the resonant eigenfunctions.

An important by-product of this study is a new integration technique applicable to combinations of Airy functions that represent Stark resonances in one-dimensional models with piecewise constant potentials.

Results presented in this work have also an immediate practical impact on modeling of light-matter interactions in strong time-dependent optical fields in the framework of the Metastable Electronic State Approach.

The Airy integral technique outlined in this appendix is used to evaluate integrals that contain linear combinations of Airy functions

The two unknown integrals that we are required to solve are

Multiply integrand

Differentiate with respect to

Repeat (

Subtract the two equations.

The procedure relies on the fact that wave functions and their derivatives are continuous at jumps of

To solve the integral of

To solve this integral, we use the same procedure: multiply integrand by

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the United States Air Force Office for Scientific Research under Grants nos. FA9550-13-1-0228 and FA9550-10-1-0561.