Evolution of electric-field-induced quasibound states and resonances in one-dimensional open quantum systems

Evolution of electric-field-induced quasibound states and resonances in one-dimensional open quantum systems

O. Olendski1
11Department of Applied Physics and Astronomy, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates; E-mail: oolendski@sharjah.ac.ae

A comparative analysis of three different time-independent approaches to studying open quantum structures in a uniform electric field was performed using the example of a one-dimensional attractive or repulsive -potential and the surface that supports the Robin boundary condition. The three considered methods exploit different properties of the scattering matrix as a function of energy : its poles, real values, and zeros of the second derivative of its phase. The essential feature of the method of zeroing the resolvent, which produces complex energies, is the unlimited growth of the wave function at infinity, which is, however, eliminated by the time-dependent interpretation. The real energies at which the unitary scattering matrix becomes real correspond to the largest possible distortion, , or its absence at which in either case leads to the formation of quasibound states. Depending on their response to the increasing electric intensity, two types of field-induced positive energy quasibound levels are identified: electron- and hole-like states. Their evolution and interaction in the enlarging field lead ultimately to the coalescence of pairs of opposite states, with concomitant divergence of the associated dipole moments in what is construed as an electric breakdown of the structure. The characteristic features of the coalescence fields and energies are calculated and the behavior of the levels in their vicinity is analyzed. Similarities between the different approaches and their peculiarities are highlighted; in particular, for the zero-field bound state in the limit of the vanishing , all three methods produce the same results, with their outcomes deviating from each other according to growing electric intensity. The significance of the zero-field spatial symmetry for the formation, number, and evolution of the electron- and hole-like states, and the interaction between them, is underlined by comparing outcomes for the symmetric geometry and asymmetric Robin wall.

1 Introduction

The most fundamental quantity to analyze in the study of the elastic motion of non-relativistic quantum particles in a potential that undergoes sufficiently fast decays at infinity is the scattering matrix , which describes the distortion of the field-free wave function by the force exerted on the corpuscle by [1, 2, 3, 4]. is generally considered to be a dimensionless complex function of the energy of the particle and, as such, can have poles in the plane, which determine the resonances characterized by (usually) complex energy. If the potential allows the existence of bound states, the scattering matrix also has poles at their negative energies. Another important property of the matrix is its unitarity, , at real energies that do not coincide with the energies of the bound states; this physically expresses the conservation of the number of particles in the elastic collisions. It is natural to expect that for the complex unitary function its real values can have a special meaning. Additionally, due to the unitarity, the scattering matrix can be expressed in the form


with the real phase being a function of the energy, . It is known [1, 4] that the region where the fastest change of occurs is most significant, and the Wigner delay time [5] is arrived at in this manner. is defined as a derivative of the phase with respect to energy


and is an essential characteristic of the scattering process [6, 7], with the extreme energies at which it achieves its maxima being the most important. Thus, three sets of energy have been identified, all of which are significant for the scattering matrix

  • (in general) complex energies, at which ,

  • real energies, at which ,

  • real energies, at which and .

Each of these sets defines its own specific type of behavior of at and around the corresponding energy, and even within each set the physical processes that are mathematically described by the scattering matrix may be quantitatively different. The overwhelming majority of mathematicians prefer to analyze only the first case where, on the basis of the time-independent Schrödinger equation, the complex energies are located and calculated for each particular potential without requiring details of the associated waveforms. These functions, as the first stage of the physical consideration reveals, exhibit unrestricted growth at infinity, known as the ’exponential catastrophe’ [8]. However, even more careful interpretation allows it to be eliminated by proper reasoning involving the temporal evolution of the wave function [3, 8, 9]. The real part of the complex energy is customarily associated with the location of the resonance on the axis, whereas its imaginary component describes its half width or lifetime. For each specific potential, physicists also analyze the last two energy sets, but quite often this is performed separately for each case and no parallels are drawn between them and the first (complex energy) counterpart.

In the present study, a comparative analysis was performed of the outcomes of the three above-mentioned approaches, as applied to the behavior of electrons in the uniform electric field superimposed on (a) a one-dimensional (1D) attractive or repulsive -potential, which is the extreme limit of the finite width and depth quantum well (QW) or finite height quantum barrier, and (b) the Robin wall, i.e., a surface that in the general 3D geometry supports the boundary condition (BC) for the wave function of the form[10]


where is a unit inward vector and the extrapolation length , which is considered to be real, is called the Robin or de Gennes [11] distance. The model that represents an extremely localized finite strength interaction in the form of the -potential is a very appealing one due to its relative simplicity and ease with which the necessary calculations can be carried out. It also reflects the essential properties of more complicated structures [12]. The model is characterized by only one parameter, whose continuous variation from positive to negative values describes its repulsive or attractive strength; in particular, it possesses one localized level in the latter configuration. On the other hand, the quantum system, which at zero voltage is able to support bound orbitals, no longer has any discrete stationary levels when placed into the time and space unvarying electric field but instead only a continuum of states with their energies covering the whole axis, . As a result, the corresponding scattering matrix has poles only at the complex energies. A great deal of attention has been devoted to the analysis of the finite-width QW [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and its counterpart [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] subjected to a dc electric field with the outcome that one approach is sometimes completely at odds to the predictions of another scheme; in particular, the peculiarities of the transformation of the resonances into the true bound states at were scrutinized and the formation of new field-induced complex energy quasibound states and resonances was predicted.

Building on previous studies, below a detailed analysis was conducted of the resonances and quasibound states of the -potential calculated from the three above-specified requirements. It can be seen that the complex energies derived as solutions of the stationary Schrödinger equation with a non-zero field inevitably imply the exponential growth of the associated wave functions that, however, can be correctly construed with the help of the time-dependent picture. The development, based on the assumption that the energies have to remain real, leads to the conclusion that the infinitely small applied voltage at maximal scattering, , generates an infinite number of quasibound states in the positive energy continuum, the first set of which bears the features of electrons while the second exhibits the properties of a positively-charged particle that in solid state physics corresponds to hole excitations; for example, corpuscles residing in these two types of levels move in opposite directions as the field grows, and their evolution with increasing voltage ultimately forces them to coalesce with each other in what can be considered as the electric breakdown of the structure. The highest breakdown field is predicted to occur for mergers involving the level that developed from the zero-field bound state. Amalgamation of the levels is accompanied by the divergence of the associated dipole moments. This phenomenon, previously predicted only for the ground state [50, 60], is calculated and analyzed in subsection 2.2 for all quasibound levels. It should be noted that the corresponding eigenvalue equation has, at intensities greater than the breakdown voltage, a pair of complex conjugate solutions that can be a mathematical indication of the formation in the electric field of a composite electron-hole-like structure. Corresponding maxima of the Wigner delay time were also computed as a function of the applied voltage. It can be seen that at the vanishing electric intensities, the predictions of the three methods coincide for the field-free bound state; however, even in this regime each approach has its own peculiarities. The difference between the results that coincided at grows with the field. Contrary to the model of the -potential that is symmetric at , the motion of the particle in the presence of the Robin wall takes place only on the half line. This lack of spatial symmetry has a drastic effect on the emergence and evolution of the quasibound states when the voltage is applied; in particular, for this system the field induces only hole-like quasibound levels and the lowest of them merges with the state that evolved from the field-free orbital that exists for the negative de Gennes distance, while the higher lying states survive any electric intensity.

2 -potential

The starting point of our analysis is the 1D stationary Schrödinger equation


where the Hamiltonian is given by


for the wave function of the particle with mass and charge (with being an absolute value of the electronic charge) moving along an infinite straight line in a uniform electric field with a potential being of the -like form


Here, is a real coefficient , which has a dimension of length, being either positive or negative. Due to its presence, the matching conditions at are:


with the prime denoting a derivative of the function with respect to its argument. Eq. (7b) demonstrates that there is a jump in the derivative of the wave function at the origin that is inversely proportional to the distance . In the absence of an electric field, , the attractive potential, , in addition to the continuous spectrum at (which is also characteristic for ) binds the particle at negative energy


while its normalized to unity,


wave function exponentially decreases away from the origin:


An applied electric field changes the charge distribution in the system. The quantitative measure of this influence is provided by the polarization, or dipole moment, , defined as [62, 63, 64]


where the angular brackets denote a quantum mechanical expectation value:


with the integration carried out over all available space, which in the case of the -potential, reduces the polarization to


As will be shown below, it is possible to calculate this quantity even in the case of open structures, such as the ones considered in the present study.

It is convenient to switch to dimensionless scaling from the outset so that all distances are measured in units of , energies – in units of , time – in units of , polarization – in units of , velocity – in units of , electric fields – in units of , and current density – in units of . Then, Eq. (4) using the potential from Eq. (6) takes a universal form:


while the matching condition from Eq. (7b) is transformed to


where the upper (lower) sign refers to the repulsive (attractive) potential. The same convention will be used, as necessary, throughout the whole section.

Due to its generic definition, the scattering matrix describes the results of wave reflection from the structure when the total function includes both the incoming [first term on the right-hand side of Eq. (16)] and reflected (second item) components:



(obviously, the superscript at refers to the sign of its imaginary part and not to the -potential), and and are Airy functions [65, 66]. To the left of the well, the fading at the negative infinity solution is:


where the explicitly-included subscript , counts the corresponding resonances (see below) and is a normalization constant. Matching according to Eqs. (7a) and (15) leads to the scattering matrix:


Note that for real energies, this complex function, which also depends on the parameter , is unitary.

2.1 Poles of the Scattering Matrix: Gamow-Siegert States

Zeroing the denominator of the right-hand side of Eq. (18) produces a universal equation for calculating the complex energies :


Note that this equation can be derived in an alternative way; because the applied field is created by the potential that unrestrictedly decreases with , it follows that at the non-zero voltage even for the attractive well there are, strictly speaking, no true bound states since the electron localized near the origin at lowers its potential energy by tunneling away from the attractive center when the electric intensity is not zero. It is then contended [20, 30, 31, 35, 58, 59, 29] that the solution of the Schrödinger equation should represent the outgoing waves at infinity and, since this requirement infers the non-zero imaginary component of the wave function , the energy also becomes complex. This results in a transformation of the true bound level into the resonance state with a finite lifetime


where the positive is a half width of the corresponding resonance:


with being real. Accordingly, the function for positive is written as


where is a normalization constant, and its match with the waveform from Eq. (17) leads to Eq. (19).

From the properties of the Airy functions [65, 66] it is easy to derive the evolution of the field-free bound level at small electric intensities:


It can be seen that the real part of the energy depends quadratically on the voltage, which is a result of the symmetry of the field-free structure with respect to the inversion . An exponentially small increase of the half width


which is typical for a wide range of potentials that decay at infinity [59], physically means quite a small probability of tunneling away from the well and a particularly long lifetime of the state, as follows from Eq. (20).

Figure 1: (a) Real (solid line, left axis) and negative double imaginary (dashed curve, right axis) components of the complex energy as a function of the field calculated from Eq. (19). (b) Wave function (normalized to its value at ) in terms of the distance and electric field .

Panel (a) of Fig. 1 shows the zeroth level energy dependence on the field. A quadratic decrease of the real part and an exponentially small increase of the half width at the small intensities derived above are clearly seen in the plot. The real part of the energy reaches a minimum of at , after which it demonstrates a permanent growth; in particular, it crosses its zero-field value of at and enters the positive part of the spectrum at . The half width after an almost flat profile at small exhibits a nearly linear dependence on the applied field, which at higher voltages approaches . This growth implies a drastic decrease of the lifetime by field-enhanced tunneling.

The evolution of the associated wave function is depicted in panel (b) of Fig. 1. Its most striking feature is an exponential growth with positive distance at the non-zero fields. Indeed, the claim that the function from Eq. (22) describes the outgoing wave [20, 30] is correct only for the real energies, but for the complex it inevitably leads to an exponential increase of the function at large , which can be easily shown by employing the asymptotic behavior of the Airy functions [65, 66]. Complex eigenvalues were first introduced by Thomson [67] in his analysis of the modes supported by an electric sphere and were later implemented in Gamow’s explanation of alpha decay [68]. The most serious criticism of these Gamow, or Siegert [69], states denies their physical legitimacy due to the unrestricted growth of the wave function, with the vivid manifestation of this ’exponential catastrophe’ presented in Fig. 1. However, this divergence is elegantly eliminated by considering the temporal evolution of the structure; as a matter of fact, if the time decay of the state is considered and even the corresponding lifetime from Eq. (20) is introduced, it is logical to investigate the associated time-dependent function . For the large negative argument in the waveform from Eq. (22), the total function reads:


where also a smallness of the half width has been assumed. The first exponent in Eq. (25) is a plane wave that describes free motion in the linear potential and is just the classical time needed for the electron to travel the distance between the quasi classical turning point and the coordinate :

This is particularly clear when this expression is rewritten in the alternative equivalent form:

where is a field-dependent classical speed at :


The above equations in this paragraph, similar to the analysis of alpha decay [8, 3, 9], can be construed as follows. In the derivation of Eq. (25) it was tacitly assumed that it is valid for all times . However, in reality the decay does not start in the infinitely remote past since the corresponding state has to be created first by, say, the adiabatic varying of the field or any other means. Accordingly, it is natural to choose as the origin the moment when the emitted particle emerges at the turning point after tunneling through the triangular barrier, which the electron with the negative energy located inside the -well ”sees” to its right. This also means that at this point, the prehistory of the formation of the scattering level at is of no concern. Then, at any positive time the particle travels with an average speed to reach the observation point at moment from Eq. (26b). Consequently, it does make sense to talk about measuring the probability density at detector position at times only:


The corresponding current density in the direction


which is calculated from the general expression [2]


apart from the familiar velocity-dependent term [first item in Eq. (29)], contains an additional contribution that is proportional to the half width . Such a viewpoint eliminates the ’exponential catastrophe’, yielding instead the anticipated exponential decay law [8, 3, 9] with its lifetime taken from Eq. (20). However, neglecting all times smaller than and, in particular, , completely ignores the processes of under-barrier tunneling at and this semi classical reasoning is therefore not a strictly quantum one. To correctly account for the build-up of the levels at earlier times , a solution of Eq. (19) with the positive imaginary component, which is a complex conjugate of its counterpart from Eq. (21), is required. Negative half widths specify the system in the growing state, which is called antiresonance (see Sec. 2.2). They are also appropriate to describe a particle traveling back in time towards the past. A separation of the complex energy eigenfunctions into those corresponding to the physical states at the earlier times (with ) and those associated with the configuration for the later times, , is a peculiar property of the Gamow vectors with complex energies [70]. From the point of view of mathematical formalism, the Gamow-Siegert states do not represent vectors from the Hilbert space of the quantum structure under consideration, being instead eigenvectors of the rigged Hilbert space (see Refs. [70, 71, 72] and references therein).

In addition to the resonance that, at vanishing electric intensities, transforms into the field-free bound state, Eq. (19) has other solutions for either attractive [45, 57] or repulsive potentials. The easiest way to show their existence is to substitute into the equation an Airy functions relation [65, 66]

and to consider the resulting transcendental formula


in the limit of the low voltages. After some algebra involving the properties of the Airy functions, two sets of solutions are achieved, which at are:


Here, (all negative) coefficients , , are solutions of equation [65, 66]. Note the opposite signs of the real parts of the energies and and different powers of the field dependence of their imaginary components. We will address more of the properties of these states while comparing them to the results obtained via other methods.

2.2 Real-energy Quasibound States:

Having seen the properties of the complex Gamow-Siegert states, let us now return to the scattering matrix from Eq. (18). To remain rigorously within the time-independent quantum treatment without divergences, only the real energies will be used in this and the following subsections, unless otherwise stipulated. For our geometry, the matrix characterizes the influence of the -potential on the incident particle; namely, it is well known [2] that without it, , the corresponding waveform is proportional to the Airy function , which corresponds to in Eq. (16):


Then, in the superposition of both forces, the scattered wave is the difference between the total function from Eq. (16) and its unperturbed counterpart, Eq. (33):


This equation shows that is a purely outgoing wave. The squared modulus of its amplitude with its maximum normalized to unity is called the scattering probability:


Note that the extremely localized potential, as expected, does not scatter the wave of the arbitrary strength electric field when its position coincides with one of the nodes of the unperturbed orbital :


In this way, the significance of the special value of the matrix mentioned in the Introduction is established. Another important property is the identical vanishing of the current density of the total function at any energy, as it easily follows from its substitution into Eq. (30). Thus, mathematical models of the Gamow-Siegert states and that based on the real energy analysis describe different physical situations: while the former one calculates the temporal leakage from the well of the level that was prepared at the earlier times, the subject of the latter method is the stationary configuration that emerged as a result of the interference between the incident and reflected waves, with the resulting net current being an exact zero. As a consequence of this, real values of the energy guarantee that the waveform is finite everywhere.

Distortion of the electron motion by the -potential is maximal for energies , , at which the scattering matrix changes to positive unity:


and the associated total function to the right degenerates to the Airy function :


For the attractive potential, Eq. (37) was derived previously with the help of the Green functions [47, 51] without any detailed analysis. The same configuration was discussed by C. A. Moyer [50, 60], who concentrated on finding the energy and associated polarization of its lowest level only, which at the vanishing electric intensities tends to the zero-field bound state:

In addition, for either the -well or barrier, Eq. (37) has two infinite sets of positive solutions that will be denoted below by the superscripts or corresponding to their behavior at low voltages; namely, for the weak fields, , one finds:

. Here, negative is the th zero of the Airy function : [65, 66]. It is important to stress that Eq. (37), in addition to the real solutions, has complex roots too. This follows from the fact that at low electric intensities the set is essentially determined by with . This equation is also satisfied by the complex numbers , in addition to the real coefficients [65, 66]. However, these states are disregarded due to the convention of avoiding the ’exponential catastrophe’. Therefore, under the assumption of keeping the energies real we have found that quantization results in a countably infinite number of solutions. To place them correctly within the nomenclature of the other solutions of the Schrödinger equation, it should be noted that historically, the terms ”quasibound (or quasi-stationary) state” and ”resonance” were used interchangeably to describe the complex-energy Gamow-Siegert level. The standard procedure for categorizing the poles of the -matrix implements their location in the complex -plane, where with real and [4, 73]. Bound states, , lie on the imaginary semi axis in the upper -halfplane, , , which leads to fading function at large distances, , as expected. A complete mathematical set of solutions should also include those dependencies with the purely imaginary negative wave vector, , . Accordingly, the waveforms of such anti-bound states [4, 73] diverge at infinity: . Despite this unlimited growth, these levels can have a physical meaning too [74, 75]. All other poles of the scattering matrix lie in the lower -halfplane, , with the positive real part of the wave vector corresponding to the above-discussed Gamow quasi-stationary states (resonances) while those with are called antiresonances and describe an ingoing wave [4]. Obviously, solutions from Eqs. (39) do not fall into one of these categories, since in the first instance they are not poles of the -matrix and, second, all their energies (except the lowest one of the attractive potential) are positive. Moreover, the corresponding functions at large positive neither exponentially diverge nor fade, presenting instead oscillatory damped modes, as it follows from Eq. (38) and asymptote of the Airy function. Nevertheless, in the recent analysis of the lowest level evolution in the field [60] this orbital was called the quasibound state and was defined broadly as the level having a connectedness to the true bound state through the variation of some physical parameter. In our extension to all the solutions (including those with ), we found it relevant to call them real energy quasibound (REQB) states. These are bound states, since for each of them the wave function has a finite absolute value everywhere including the point while the prefix ’quasi’ in their definition means that due to the slow decrease at large positive of the function from Eq. (38), they cannot be normalized according to Eq. (9) [60]. Contrary to these states, the divergent-at-infinity Gamow-Siegert solutions will be called resonances [4, 60]. It is important to underline that the REQB states with the discrete energies , , and are embedded into the continuum of the delocalized levels, with its energies ranging from the negative to positive infinity; as such, they mathematically represent only a measure zero part of all continuum energy eigenstates of the given Hamiltonian, while physically they describe the largest possible, , disturbance by the -potential of the motion in the uniform electric field. To end this part of the discussion, it should be mentioned that, in addition to the levels discussed above, open systems can also support under very special conditions true bound states in the continuum, i.e., waves that remain localized (with square integrable functions) even though they coexist with a continuous spectrum of radiating oscillations that can carry energy away [76].

The first term on the right-hand side of Eq. (39b) states that the set of solutions reflects the creation by the applied voltage and the -potential of the triangular QW [64, 63, 77], with the wave function taken from Eq. (17) and the subsequent terms representing an admixture due to its coupling to the right half space. Since the leading term in this formula is independent of the sign of the term, the formation of the triangular well takes place at either a positive or negative electrostatic potential while the interaction between the left and right semi-infinite areas carries its sign. In the same way, the first expression on the right-hand side of Eq. (39c) describes the formation in the region terminated by the impenetrable wall of the standing wave from Eq. (38) with the linear in the field and higher order factors there describing the correction due to coupling to the left-hand territory. The first part of the mathematical inequality chain


physically means that the solutions are located to the left of the -potential, with the energies being larger than those of the corresponding states with their distribution spreading at . The sequence from Eq. (40) also states that the and levels alternate on the energy axis.

The product of the two Airy functions on the left-hand side of Eq. (37) is a bounded function of the energy: it decreases to zero as for large negative , reaches its global maximum at , and for positive energies it presents sinusoidal oscillations with its amplitude modulated again by the same factor . Hence, for quite large electric intensities, this equation does not have any solutions. The disappearance of the levels at increasing voltage occurs as a coalescence of the two adjacent states at the electric fields that are different for the well and the barrier:


and the energies at the merger are:




and non-positive is the th solution, , of equation , or in the expanded form


Since [65, 66], the breakdown field of the two lowest levels is




where is the -function [65]. For quite large it is elementary to derive an asymptote


that leads to the approximate formula for :


Table 1 lists the exact values of and their approximations by Eq. (47) together with the exact and approximate coalescence fields and energies. It shows that the estimates from Eqs. (47) and  (48) provide reasonably good accuracy, even for small .

Exact Eq. (47) Exact Eq. (48a) Exact Eq. (47) Exact Eq. (48b)
0 0 0 2.58106 0 -1.76475 -1.77068 0.39591 0.42441 0.95150
1 -2.80824 -2.81078 0.20773 0.21221 0.98501 -3.68166 -3.68317 0.14006 0.14147 0.99293
2 -4.46080 -4.46184 0.10549 0.10610 0.99593 -5.1767 -5.1775 0.084566 0.084883 0.99736
3 -5.84606 -5.84667 0.070551 0.070736 0.99815 -6.47897 -6.47947 0.060514 0.060630 0.99864
4 -7.08232 -7.08273 0.052973 0.053052 0.99895 -7.66095 -7.66130 0.047102 0.047157 0.99917
5 -8.21847 -8.21878 0.042401 0.042441 0.99933 -8.75768 -8.75795 0.038553 0.038583 0.99944
6 -9.28076 -9.28100 0.035344 0.035368 0.99953 -9.78949 -9.78971 0.032629 0.032647 0.99960
7 -10.28532 -10.28552 0.030300 0.030315 0.99966 -10.76947 -10.76965 0.028282 0.028294 0.99970
8 -11.24297 -11.24313 0.026516 0.026526 0.99974 -11.70670 -11.70685 0.024957 0.024965 0.99977
9 -12.16141 -12.16155 0.023572 0.023579 0.99979 -12.60778 -12.60791 0.022332 0.022338 0.99981
10 -13.04638 -13.04650 0.021216 0.021221 0.99983 -13.47773 -13.47784 0.020206 0.020210 0.99985
20 -20.70998 -20.71003 0.010610 0.010610 0.99996 -21.05373 -21.05377 0.010351 0.010352 0.99996
50 -38.14819 -38.14820 0.0042441 0.0042441 0.99999 -38.40209 -38.40210 0.0042021 0.0042021 0.99999
100 -60.55649 -60.55650 0.0021221 0.0021221 1 -60.75818 -60.75819 0.0021115 0.0021115 1
Table 1: Exact and approximate solutions of Eq. (44) together with the dissociation fields and energies for the -potential
Figure 2: Energies of the quasibound states of the attractive -potential (solid lines) as a function of the electric field . The upper edge of each petal corresponds to the level. Dotted curves depict real components of the complex solutions of Eq. (19). The thin vertical dash-dot-dotted line shows the location of the fundamental dissociation field . Solid and dashed lines at denote real and positive imaginary parts, respectively, of the first complex solution of Eq. (37). The inset depicts an enlarged view of several positive energy states, where levels of the repulsive barrier from Eq. (37) (dash-dotted curves) and  (19) (dash-dot-dotted lines) are also shown. Contrary to the attractive well, in the latter geometry the lower rim of each petal is formed by the level.

Fig. 2 shows the energies of REQB states calculated from Eq. (37) together with the real parts of the complex energies being solutions of Eq. (19). At the weak fields, the energy decreases quadratically with electric intensity, as derived in Eq. (39a). This can be interpreted as an increase in the binding of the electron by the small voltages. The ground state energy reaches a minimum of at . A comparison with the corresponding Gamow-Siegert data provided above shows a conspicuous difference at these fields, while for small the energies calculated by either method are the same. The deviation of the energies at is clearly seen in the figure. A subsequent increase of the voltage leads to the growth of the energy until it approaches zero at , where it amalgamates with the lowest level. All positive energies grow from their zero value as at the weak fields, with the steepness being higher for larger . This energy increase for the electric potential that seemingly has to force them downward is explained for the levels by the formation of the triangular QW, as discussed above. The lowest level that does not have its complex counterpart passes at through the broad maximum of , after which it decreases towards zero. To determine the energy behavior close to this merger, it is convenient to represent Eq. (37) in the parametric form [50]


where the coefficient , which is equal to , varies from zero to positive infinity (for the lower level) or to (for the upper state). Close to the coalescence, this parameter is small, , and the Taylor expansion simplifies Eqs. (49) to


Eliminating from these equations, one gets after some simple algebra


For the higher lying amalgamations, this expression is generalized as


which is also valid for the repulsive potential. The coalescence of the two states physically results in ionization of the structure by the growing field when the -potential can no longer bind the charged particle. Higher lying states dissociate at weaker electric fields, as they are less bound by the potential. As Fig. 2 demonstrates, for larger quantum numbers the energies of the levels deviate less from their complex-Airy-functions counterparts, while the petals formed by the and energies become narrower. For comparison, the curves of the repulsive potential are also shown in the inset. In this case, the energies can take positive values only with all basic features described for the QW being observed too. One major difference lies in the fact that for the barrier, the lower edge of each petal is formed by the level. Additionally, it should be noted that to the right of the field at which the levels with real energies merge, Eq. (37) has two complex conjugate solutions. Real and positive imaginary components of the lowest levels are also shown in the figure. The real part grows with the field from its value at the breakdown while the magnitude of the imaginary part increases more rapidly from its zero value. The physical interpretation of these mathematically correct solutions of Eq. (37) will be discussed below.

A clear manifestation of the electric breakdown of the QW is revealed by the analysis of the polarization that, in the coordinate representation, can be written as


Note that the primitives in Eq. (53) can be readily calculated analytically [66] but applying the limits of integration to the second terms in the numerator and denominator leads to their divergence. Scattering theory has developed special regularization procedures for treating such integrals [4] that go back to early efforts in the 1960s [78, 79]. However, to avoid handling of the divergent integrals and their subsequent division in our particular case, it is much easier to use another method for finding the dipole moment that employs the Hellmann-Feynman theorem, which on application to the Hamiltonian from Eq. (5) is


This immediately leads to the following reworking of Eq. (11) [63, 50]:


The field-free -potential is symmetric with respect to the change , which results in . Applying the rule of differentiation of the implicit functions to Eq. (37) leads to


The implementation of Eq. (55) to the ground state weak-field limit from Eq. (39a) shows that in this regime, the dipole moment increases linearly with applied voltage


meaning that the electron moves in the direction of the external force acting upon it. However, close to the fundamental critical field , the negative divergence of the ground state polarization is gained from Eqs. (55) and (51)


To understand this reversal of polarization, the structure of the associated wave functions needs to be considered. Starting from the levels, the behavior in the very weak fields to the left of the potential is determined by the Airy functions entering the first integrands in Eq. (53), which together with the asymptotes from Eq. (39b) yields:


Applying the properties of the Airy functions [65, 66], it is found that this waveform has extrema that are located at


[the negative is the th zero of the derivative of the Ai Airy function, ], and the value of the functions at these points is:


The wave function to the right possesses an infinite number of fading oscillations, whose amplitudes at low voltages are times smaller than their counterpart(s) at :


Hence, at extremely weak fields, the function is located far to the left, which is in accordance with the corresponding negatively diverging polarization derived from Eqs. (55) and (39b):


Note that with the increase of the small voltage the particle, as follows from Eq. (60), moves to the right: this results in the growth of the dipole moment from Eq. (63) and agrees with the electron behavior in the electric field. This shift in the positive direction is clearly seen in panel (a) of Fig. 3. On the contrary, the level under the same assumption of the small electric intensities resides mainly to the right of the -potential with its wave function


exhibiting an infinite number of peaks and dips with the following values


[negative coefficients form an infinite set of roots of the derivative of the Bi Airy function, ] located at


Observe that the period of swinging and the distance between extrema increase for decreasing electric intensity. Oscillations to the left, if they exist, are characterized by the amplitudes that are times smaller, after which the waveform exponentially decays with tending to negative infinity:


The most important properties to deduce from Eq. (66) are: i) at extremely weak fields the particle is located far to the right and ii) with the increase of the (still small) voltage it moves to the left, which is the opposite direction compared to the state, and which has just been associated with the electron. This can only be possible if the charge of particle dwelling at the level is opposite that of its counterpart. In this way, the natural conclusion is that the levels correspond to the hole states carrying positive charge. This explains the growth of their energies at small intensities, Eq. (39c); namely, as the particle moves into the area of the higher potential, its energy increases correspondingly. The general definition of the polarization, Eq. (55), has to be amended to take into account the different charges of the electrons and holes:


The last formula together with Eq. (39c) results in positively diverging dipole moments at the weak fields:


conforming to our earlier conclusion regarding the hole locations in this electric regime. Fig. 3(b) exemplifies the wave function shift in the left-hand direction for the lowest state at small . Note that in this regime, the and waveforms do not exhibit a mutual mirror symmetry with respect to since, as discussed above, the former (latter) is characterized at () by a finite (infinite) number of extrema and with the distance from the origin growing fades as .

Figure 3: The waveforms of the lowest [panel (a)] and [panel (b)] levels at several electric fields , normalized to the maximum of their absolute values. In the upper plot, the dash-dot-dotted line represents , the dotted curve represents , the dashed curve represents , the dash-dotted curve represents , which corresponds to the maximum value of the associated polarization (see Fig. 5), and the solid line describes the wave function at the coalescence field . For panel (b) the thin solid line represents , dotted and dashed curves denote the same voltages as in the upper part, the dash-dotted line denotes , the dash-dot-dotted line denotes , and the thick solid line corresponds to the fundamental field . Note different and ranges for each of the panels.
Figure 4: Evolution of the wave function of the lowest quasibound state (normalized to its value at ) with the electric field (in units of ).

It has thus been shown that, while the particle localization at is completely undetermined for the flat geometry with a short-range potential, the vanishingly weak electric intensity creates electron- and hole-like REQB states in the positive energy continuum that are split in opposite directions. It is important to stress that we use the word ”hole” just to underline that the corresponding level behaves like the particle with the positive charge. However, this interesting analogy has its limitations since in the considered system there is no positive charge whatever while in semiconductors the hole carries it because its host atom has a missing electron. From this point of view, it is better to use for our configuration the terms ”hole-like state” or quasi-hole, what is tacitly assumed below.

Spatial electron-hole separation can be defined as the distance between their nearest largest extrema, which for the low voltages reduces to


This length decreases for larger quantum numbers:


These equations, together with Fig. 3, manifest that at weak electric intensities the electron and hole states are well separated and, accordingly, barely affect each other. The growing field decreases the partition and, as a result, their mutual distortion increases with the corresponding changing shape of the wave functions. The same remains true for the interaction of the ground state with its closest counterpart. Evolution with the field of the ground level wave function is shown in Fig. 4. At very low voltages, it exhibits fading trigonometric oscillations with the largest amplitude of after only; accordingly, the associated polarization , which, together with its counterparts for several higher lying quasibound states, is shown in Fig. 5, is determined by the redistribution of the charges near , which results in the linear dependence from Eq. (57). At the same time, the nearest