Optimally Sharp Energy Filtering of Quantum Particles via Homogeneous Planar Inclusions

Abstract

Some of the most influential players from academia and industry have recently expressed concrete interest for quantum engineering applications, especially for new concepts in controlling and processing the quantum signals traveling into condensed matter. An important operation when manipulating particle beams behaving as matter waves concerns filtering with respect to their own energy; such an objective can be well-served by a single planar inclusion of specific size and texture embedded into suitable background. A large number of inclusion/host combinations from realistic materials are tried and the optimally sharp resonance regimes, which correspond to performance limits for such a simplistic structure, are carefully identified. These results may inspire efforts towards the generalization of the adopted approach and the translation of sophisticated inverse design techniques, already successfully implemented for nanophotonic setups, into quantum arena.

Introduction

Quantum interactions of particle beams with crystalline matter are present in numerous and diverse effects with significant quantum engineering potentialities. One can indicatively refer to matter waves tunneling through arbitrary potential distributions that support complex interference schemes¹ or controllable transitions between vibrational states, crucial in quantum sensing². Importantly, proper coupling between impinging particles and materials has been utilized for emulation of exotic electronic properties in heterojunctions³ and led to the systematic formulation of cavity quantum electrodynamics illustrating fundamental aspects of measurement theory⁴.

All these inherently fascinating phenomena being vital in disruptive quantum applications, have recently attracted huge funding interest; indeed, along with 5G communications and Artificial Intelligence, is one of the technologies that presidents of both, countries and companies, love to cite⁵. In particular, impressive State initiatives have been taken⁶, echoing major investments from Google, IBM, Intel, Microsoft and other industry giants, that are expected to ignite extensive quantum engineering research efforts in the near future. Already, National Science Foundation (NSF) is efficiently supporting research teams in quest of suitable hybrid materials used in quantum signal processing and computing⁷ while US Army Research Office (ARO) is currently funding studies on growing of matter for noise suppression towards high fidelity qubit operations⁸. Finally, several Multidisciplinary University Research Initiatives (MURIs) are executed with main objective to reformulate the range of light-matter interactions for quantum optics and photonic configurations⁹.

There are various alternative media employed in setups hosting the aforementioned effects; more specifically, one can use from isolated elements (germanium, silicon, carbon etc) and semiconductors (arsenides, antimonides, tellurides etc) to arbitrary alloys, mixtures and heterojunctions¹⁰. Not only reliable measurements and computational simulation models for the effective parameters¹¹ describing the transport of quantum particles¹² are available, but also several fabrication processes can be implemented for constructing, measuring and testing the corresponding prototypes. In particular, simple chemical precipitation methods have been adopted towards the synthesis of sulfide nanocomposites in graphene oxide building solar cell devices¹³ while exfoliation approaches are followed in growing telluride multilayers supporting preferential scattering observable through transport measurements¹⁴.

In addition, the coupling between arsenide quantum dots and an external fiber-mirror-based microcavity can be investigated by using mostly self-assembly techniques¹⁵ while quantum interference between single photons heralded from two independent micro-ring resonators has been measured fully-integrated onto a monolithic silicon photonic chip¹⁶. Diamond-based quantum layouts like nanowires into polycrystalline diamond fabricated via top-down methods allow for large collection efficiency of emitted photons¹⁷; moreover, unprecedented coherence of atoms combined with the scalability of a solid-state platform has been achieved by creating suitable defects into diamond¹⁸. It should be noted that diamond nanophotonic structures are additionally used for efficient light collection in hybrid integration with other material subsystems¹⁹ but, mainly, as solid-state quantum sensors exploiting nitrogen defects²⁰.

One of the most typical objectives in modeling a quantum setup is the efficient energy filtering of the incident matter waves, namely the fabrication of a planar structure letting the impinging particles pass only if they possess a specific amount of energy, otherwise it blocks them²¹. Over thirty years ago, solutions based on sequential tunneling exploiting the, well-known from Photonics, Fabry-Perot mechanism²² led to tunable wavelength-selective detectors²³. Moreover, systematic efforts towards filter design based on matter waves interference into potential superlattices by manipulating the emerged resonances have been recorded²⁴, with the obtained results being experimentally tested²⁵. Since then, interference-based energy filters have been used to perform very accurate scanning tunneling spectroscopic measurements²⁶ and to execute electron counting with high resolution and sensitivity²⁷. Most significantly, similar potential distributions with increased selectivity have played the role of modules in interferometric experiments complementing the field of electron holography²⁸ and controlling the quantum transport phenomena in nanoscale systems via time -dependent fields²⁹. It should be finally remarked that energy filtering plays a crucial role in nanostructured thermoelectric generators³⁰, where the output power can be further boosted with help from Fabry-Perot cavities³¹ or in coolers with increased coefficient of performance³². Similarly, response selectivity with respect to the impinging energy of the beams is important in the operation of magnetic tunnel junctions with band pass utilities³³ and quantum Hall setups³⁴ considering the electron-nuclear spin flip-flops in the parametric vicinity of interest.

In this work, our aim is to provide realistic quantum designs with sharp response that can work as energy filters for the impinging matter waves. The structure is the simplest possible: a planar homogeneous slab embedded into a suitable environment; however, our search for optimal sizes and material combinations is exhaustive. In particular, we try and test every single quantum medium from a long directory in the role of host or inclusion and we conclude to the best designs that are transparent to a specific level of incoming particles energy, while being opaque to all the others energies belonging to an extensive band around it. Several alternative optimized designs are presented whose selectivity scores constitute performance limits of the considered simplistic setup provided the list of available media; similar conclusions have been drawn for different type of inclusions serving alternative purposes³⁵. Fabry-Perot resonances of different orders, in proportion to what is the level of selected energy and the used materials, are activated at each filter layout giving different inclusion thicknesses and oscillation frequencies. The selectivity robustness of the designs with respect to fabrication defects of the slab size or engineering flaws influencing the effective parameters of the media is also found substantial even though shifts at the centrally filtered energies are noted.

This study reports numerous highly selective setups that deploy actual quantum matter and thus provides the interested experimentalists with additional degrees of freedom in fabrication of sharp energy filters for quantum particles and their respective matter waves. This primitive library of optimal setups performing such a ubiquitous operation at each selected energy level can be also useful in integrated systems design with state-of-the-art quantum engineering applications of a huge range spanning from attosecond time resolution³⁶ and all-optical particle acceleration³⁷ to optimal field detection³⁸ and quantum signal processing^39,40.

Results

Proposed setup

We consider the physical configuration depicted in Fig. 1, where the employed Cartesian coordinate system (x, y, z) is also defined, comprising a planar inclusion of thickness d into a specific background, scattering a normally incident electron beam (e-beam) of energy E. The periodic crystals into the two regions excite Bloch waves following certain band structures affecting the particles moving within⁴¹; such an interaction is approximated by assuming that the motion of electrons is in free space with a different mass. In this way, the effective mass characterizing the travel into background is m₀ and the corresponding quantity describing the particle trip into the cavity is denoted by m.

Figure 1

(a) The setup of the presented energy filter comprising a homogeneous planar inclusion with effective mass m, macroscopic potential energy V and thickness d into a specific background of effective parameters (m₀, V₀). The structure is excited by a normally incident electron beam of kinetic energy E. The considered structures correspond to quantum well configurations as shown in the schematic, namely to V < V₀ . (b) Indicative variation of the transmissivity of the filter τ(E) for 0 < E < 2E₀, where E₀ is the selecting energy. The residual transmissivity τ₂(E₀) at E = 2E₀ and the half-power energy bandwidth B(E₀) are also defined. (c) Combinations of relative effective masses m/m_e and macroscopic potential energies V (in eV) of the materials populating our list used in the followed optimization.

The macroscopic potential energy in each medium is defined as the minimal energy needed to extract an electron from the material into vacuum. It is taken spatially invariant into each of the two regions, namely the transition from the background (potential V₀) to the filter (potential V) is assumed abrupt; note also that only the difference in the energy levels counts and thus we can use as reference the energy into the background (V₀ ← 0). In this sense, we may take V ← V − V₀ ≡ −ΔV and E ← E − V₀, namely E will, from now on, denote the difference of the incoming particle energy from the host potential level. It should be also remarked that when considering the interface between two materials (like the ones normal to z axis), the potential-energy difference is not in general given by (V − V₀). In particular, there is a charge redistribution across the boundaries, paving the way to bipolar charge development and causing an additional potential drop; such band offsets are neglected in the followed approach.

If the incident matter wave is described by the wave function Ψ_inc = exp(ik₀z), the response of the slab from the other side is expressed by a similar matter wave with Ψ = Texp(ik₀z), where ${k}_{0}=\sqrt{2{m}_{0}E}/\hslash$ is the wave vector norm into the background material and $\hslash$ is the reduced Planck constant. It is straightforward to find that the transmissivity τ = |T|² is given by:

$$\tau =\frac{\mathrm{4(}{k}_{0}mk{m}_{0}{)}^{2}}{{({k}_{0}^{2}{m}^{2}+{k}^{2}{m}_{0}^{2})}^{2}-{({k}_{0}^{2}{m}^{2}-{k}^{2}{m}_{0}^{2})}^{2}{\cos }^{2}(kd)},$$

(1)

where $k=\sqrt{2m(E+\Delta V)}/\hslash$. Let us confine ourselves to quantum well configurations (V₀ > V ⇒ ΔV > 0 ⇒ k > 0), as indicated in the schematic of Fig. 1, since we aim at observing interfering particles of arbitrarily small energy E higher than the potential level of background. By inspection of (1), it is clear that the transmissivity takes its maximal (unitary, τ = 1) value when cos(kd) = 1 ⇒ kd = nπ for n ∈ ℕ^*, under the obvious constraint of k > 0. Accordingly, the quantity in (1) is minimized for cos(kd) = 0 ⇒ kd = nπ − π/2 (for positive integers $n\in {{\mathbb{N}}}^{\ast }$) with minimal values τ = [2k₀mkm₀/(k₀²m² + k²m₀²)]².

If one demands full transmissivity (τ = 1) at a single incoming particle energy level E = E₀, defined by the application, it becomes feasible for specific sizes of the homogeneous planar inclusions: $d=n\pi \hslash /\sqrt{2m({E}_{0}+\Delta V)}$ for $n\in {{\mathbb{N}}}^{\ast }$. Our aim is to propose designs working as effective particle energy filters for a pre-determined energy across an E-range around the operational point E = E₀. In particular, if we take this band to start from vanishing energies (E = 0), it is natural to assume a maximum level E = 2E₀, symmetric with respect to E = E₀. Needless to say that the followed approach can be also used for alternative functional ranges or multiple filtering energy levels. A typical graph τ(E) for 0 < E < 2E₀ is illustratively depicted in Fig. 1, where the transmissivity gets trivially nullified for E = 0, it exhibits a single resonant peak at E = E₀ with maximal value τ = 1 and decreases for E₀ < E < 2E₀; the half-of-maximum response τ = 1/2 is produced for two energies differing by B, which defines the resonance bandwidth. The merit of our filter is judged based on how suppressed is the response at the right extremum of the operational range E = 2E₀ and how rapidly decays far from E = E₀. With reference to Fig. 1, we are in search of designs possessing tiny residual transmissivities τ₂ = τ(2E₀) and as narrow half-power bands B(E₀)/E₀ as possible, at a given E₀ each time.

Optimal designs

This quest concerns an exhaustive search of all possible material combinations picked from a long list of quantum materials filling either the planar cavity⁴² or the background environment¹⁰, for a given level of selecting energy E₀ that can vary within an interval 0.1 eV < E₀ < 2 eV; once again, the range of E₀ is just indicative and can be modified at will. The effective parameters of the used media are shown in Fig. 1(c), namely the combinations of effective masses m normalized by the inertial mass of electron m_e and their macroscopic potential energies V (in eV) are depicted on a plane. In our quest for optimal pairing of quantum media, we consider isolated semiconducting elements from group IV of the periodic table and various compounds between two elements of different groups (pairs of III and V or pairs of II and VI). Note that there is a proportional relation between the two represented quantities (m/m_e, V) with the II-VI compounds possessing the middle values of both features, while the rest semiconductors cover much larger ranges. In this way, a coherent and extensive parametric area on (m/m_e, V) map is occupied solely by regarding realistic and simple quantum texture, which can definitely serve as constituent in our optimization with respect to the effective quantities m and V. One major novelty of the present work is that these parameters are not taken as free continuous variables but correspond to actual quantum media and thus the proposed layouts, carrying certain beneficial characteristics, can be directly fabricated. As far as the evaluation of (m, V) is concerned, the potential energies V are derived through identifying the electronic energy-band structure for each material, where the quasi-cubic band model is employed¹⁰; on the other hand, the effective masses m are computed via estimating slopes of effective Hamiltonian for conduction and valence bands assisted from a theory of invariants⁴³.

It should be stressed that the variation of τ(E) within a specified 0 < E < 2E₀ is obviously more rapid when the thickness d becomes larger, namely the integer n gets higher as long as the condition τ(E₀) = 1 is imposed⁴⁴. For this reason, we have to select the integer n as big as possible but with care of retaining increasing transmission for 0 < E < E₀ and decreasing for E₀ < E < 2E₀, otherwise queues from neighboring peaks will intrude into the considered energy interval 0 < E < 2E₀. In Methods Section, it is shown that the optimal order of the occurred Fabry-Perot resonance is given by:

$$n=\lfloor \frac{\sqrt{{E}_{0}+\Delta V}\mathrm{/2}}{\sqrt{2{E}_{0}+\Delta V}-\sqrt{{E}_{0}+\Delta V}}\rfloor .$$

(2)

It should be also stressed that if we demand only selectivity around an energy level E = E₀, without caring about what is happening across a much broader range (like 0 < E < 2E₀), the obvious optimal resonance order is n → +∞. Indeed, for a giant n, which implies a huge thickness d, the response will be extremely sharp but with many other peaks at neighboring energies.

Some of the results from this trial-and-error process that correspond to a maximum d provided that a single resonance is “entrapped” into our parametric box, are shown in Table 1, where E₀ = 0.1 eV; the rows of the Table refer to different background media while its columns indicate the employed quantum material of the planar inclusion. Only those cases that lead to sufficiently low τ₂ are presented, meaning that numerous additional quantum texture combinations have been tested and rejected; note also that the best scores (in terms of τ₂) are reached when the contrast between the potential energies of the two materials is substantial (${V}_{0}\gg V$). By inspection of Table 1, one observes extremely suppressed τ₂ for most of the considered scenarios while the half-power bands B are tiny fractions of E₀, designating very sharp filtering designs. There is a clear correlation between low τ₂ and shrunk B/E₀; importantly, the best filters are thicker, demanding slabs of several dozen of nanometers. Diamond, with its huge potential energy (V₀ $\cong$ 15.3 eV), makes deep quantum well configurations and, when loaded by suitably-sized cavities, it creates ultra-performing quantum energy selectors. Indeed, diamond with certain nitrogen-based inclusions of significant stability in their electronic level structure has been additionally found to possess remarkable properties for quantum sensing applications^18,20.

Table 1 Optimal combinations of planar inclusions media (columns) and background materials (rows) that block all the impinging quantum particles except those of energy E₀ = 0.1 eV. The transmissivity τ₂(E₀) at E = 2E₀, the half power energy band B(E₀)/E₀ and the optimal thickness d of the slab are shown in each box.

In Table 2, we show the results of our optimization when the central filtering energy is tenfold higher: E₀ = 1 eV. Obviously, the size of the structures is shrunk compared to the corresponding ones of Tables 1 since the propagating wavenumber in the background medium k₀ is higher. Furthermore, the performance of the optimal designs is certainly lower as both the relative half-power bandwidth B/E₀ and the residual transmissivity τ₂ at E = 2E₀ are much more substantial. The empty box indicates poor performance, namely τ₂ climbs above 20% preventing the planar slab to work as an efficient energy quantum filter.

Table 2 Same as in Table 1 but for increased selecting energy E₀ = 1 eV. Empty box indicates poor performance.

In Fig. 2, we select specific material combinations and represent the two basic figures of merit (τ₂, B/E₀) for all the considered amplitudes of filtering energies E₀; in other words, the content of each box in Tables 1 and 2 correspond to a single point in the graphs of Fig. 2. In particular, Fig. 2(a) shows the variation of residual transmission τ₂ with respect to operational energy E₀ for four characteristic pairs of quantum media. There is a clear increasing trend of τ₂ with E₀ as also indicated by comparing Tables 1 and 2; it is thus demonstrated the much higher selectivity scores by such a simple setup when the energy of the propagating matter waves is low. One also directly notices the jumps of the curves at specific energy levels E₀ which correspond to change of resonance orders n so that the appearance of a second peak within the considered energy range 0 < E < 2E₀ is avoided. Note that the represented quantity exhibits stability between two successive discontinuities, while the jump turns larger when E₀ gets more significant.

Figure 2

Optimal scores. (a) The residual transmissivity τ₂(E₀) at E = 2E₀ and (b) The normalized half power energy bandwidth B(E₀)/E₀ of the resonances as functions of selecting energy E₀, for several inclusion/background combinations optimized at every single level E₀.

In Fig. 2(b), we show the relative half-power band B/E₀ as function of E₀ for the same texture combinations considered at Fig. 2(a). The selectivity of the filter certainly worsens for increasing E₀ while the jumps to different resonance order n occur at the same E₀-levels as in Fig. 2(a). Due to the waveform τ(E), and unlike with what is happening to τ₂ at Fig. 2(a), the quality factor B/E₀ of the filtering peak increases between two consecutive discontinuities. By inspection of Fig. 2(a,b), one can again understand the excellent job done by diamond as a host; the corresponding designs when loaded with dielectrics (InSb or InN) of specific thicknesses d work almost flawlessly as energy filters, especially at small magnitudes E₀.

Sharp energy filtering

It is meaningful to test the energy response for some of the best filter designs picked from Tables 1, 2 and Fig. 2. Therefore, in Fig. 3(a), we show the variation of τ(E) for several layouts with filtering energies of level E₀ = 0.25 eV. We notice very rapid drops away from the central energy E = E₀, given the fact that the vertical axis is logarithmic. One can find even better performing designs that are not examined in these set of results since our aim is to show the variety of material combinations. In fact, we have also considered a structure of moderate performance (SiC in diamond) not included in the Tables 1 and 2 (which, of course, are referring to different central energies) in order to show that a diamond host is not always enough for a super-selective output. Note also that the transmissivity vanishes for E = 0 ⇒ k₀ = 0, as becomes obvious from (1).

Figure 3

Transmissivity τ(E) as a function of the e-beam energy level E for (a) several optimal setups with E₀ = 0.25 eV and (b) several optimal cavities hosted into diamond with E₀ = 1 eV (included in Table 2).

In Fig. 3(b), we test three of the best designs from Table 2 (with E₀ = 1 eV) recommending suitably grown slabs into diamond background and again the extremely sharp response of our reported setups is demonstrated. Our finding regarding the positive correlation of residual transmissivity τ₂ and the half-power energy band B/E₀ originating from graphs of Fig. 2 and Tables 1 and 2 is also verified; indeed, the more selective are the bell-shaped curves τ(E), the lower is the response at the right extremum E = 2E₀ of the regarded energy interval 0 < E < 2E₀. Finally, the positive influence of the contrast, either in terms of potential energies ΔV or regarding the effective masses m/m₀, between the two utilized media on the filtering operation can be identified (InSb has a smaller potential V and a tinier effective mass m than the ones of InN).

Apart from the energy profile of the device response, it is also important to observe the variation of the wave function Ψ(z) across the propagation axis of the considered matter wave. In Fig. 4(a), we pick a specific optimal design (InSb planar inclusion into diamond with E₀ = 0.5 eV) and evaluate the spatial distribution of squared magnitude |Ψ(z)|² (which is proportional to the probability of finding the particle at position z) for three different impinging energies. When the filter is fed by an e-beam of energy exactly equal to the operational one (E = E₀), the particles penetrate the cavity (whose boundaries are notated by dashed lines) with probability one. On the contrary, when the energy of the incoming matter waves is perturbed by only a small fraction of E₀ (5%), the transmissivity drops substantially (to almost 10%) accompanied by sizeable reflections for z < 0. However, in all the three cases, the represented quantity oscillates many times within the planar slab (0 < z < d) which is the outcome of suitable matter waves interference in order for the desired behavior at E = E₀ to be achieved. The principles behind this effect are identical to the ones giving photonic²² and quantum²³ Fabry-Perot resonators. In Fig. 4(b), we examine the same material combination (InSb in diamond) as in Fig. 4(a) but optimized for different filtered energy E₀ = 1 eV, namely has the size indicated by Table 2. As expected, the thickness is smaller since k₀ is higher while the performance is deteriorated. Indeed, the transmissivities for E = E₀(1 ± 5%) reach the quarter of its maximal value (25%); they are also almost equal each other (as happens in Fig. 4(a) too) revealing the locally quasi-symmetric nature of resonance around E = E₀.

Figure 4

Spatial distribution of squared magnitude of the wave function |Ψ(z)|² as a function of position z when the particle has energy exactly equal to the selected one (E = E₀) and when E = E₀(1 ± 5%) for InSb optimal planar inclusions into diamond with: (a) E₀ = 0.5 eV, (b) E₀ = 1 eV (included in Table 2). Black dashed lines denote the vertical boundaries of the filter.

It should be remarked that not only the magnitude of the wave function Ψ(z) has a physical meaning but also its real and imaginary parts are quantities employed in quantum signal processing^39,45 and computing^46,47. Therefore, in Fig. 5, we depict the signal Re[Ψ(z)] for two characteristic optimal designs as a function of position z. In Fig. 5(a), we consider a selected energy E₀ = 0.1 eV (Table 1) and the structures are fed by an e-beam possessing random mixture of energies, where the optimal E₀ participates with unitary magnitude. One can clearly observe that the chaotic input signal pattern for z < 0 is transformed into a harmonic output in both the considered designs propagating into the region right to the dashed boundaries of respective color (z > d in each case). Such a feature demonstrates the ability of the structure to let the matter waves of E = E₀ pass, while blocking all the others with 0 < E < 2E₀. The output signals are not identical; there are always some bi-products (unequal in the two scenarios) of small amplitudes corresponding to non-optimal incident energies.

Figure 5

The quantum wave function signal Re[Ψ(z)] as a function of position z for a random mixture of various energies in two designs optimized for: (a) E₀ = 0.1 eV and (b) E₀ = 1 eV (included in Table 2), where the excitation includes a unitary matter wave operated at the optimal level of energy. Dashed lines denote the vertical boundaries of the filters (painted black for the common one at z = 0 and in the corresponding color for z = d).

In Fig. 5(b), we consider the same material combinations as in Fig. 5(a) but the structure is optimized for E₀ = 1 eV (designs included in Table 2). Once again, the filtering operation exhibits a decreased efficiency compared to the cases working at smaller E₀; however, the output of the devices still reminds us of a harmonic tone exp(ik₀z) with specific spatial frequency k₀ (different in the two graphs). Of course, the amplitude of the transmissive matter wave exhibits an envelop fluctuation being bigger in the less successful design that uses BN as host. Special mention should be made to the lower number of complete oscillations into the slab when the impinging energy E₀ is more substantial, despite the fact that the operational wavelengths 2π/k₀ of the incoming particles are smaller; the physical thickness d of the filter shrinks more. Note finally that the change of Re[Ψ(z)] signal into the planar filter resembles damped oscillations, indicating the asymmetry between the two oppositely propagating waves within the inclusion.

Fabrication and engineering defects

As has been pointed out, there are several reliable fabrication methods of constructing quantum setups as the considered one in Fig. 1(a), namely creating in a specified host a planar cavity of certain thickness. The alternative approaches involve chemical characterization techniques combined with surface functionalization¹³ top-down lithographic fabrication¹⁷, plasma-sintering nano-structuring processes, or even self-assembled inclusions coupled with cavities¹⁵. Most of the aforementioned methodologies¹⁹ can give fine nanometer-sized⁴⁸ structures; however, it is important to examine how differently the proposed filters are behaving under imperfect growing of sample thickness d. We also assume that the traveling of the matter wave into the background can be perfectly modeled, namely the parameters (m₀, V₀) and the energy E are exactly selected; accordingly, it would be interesting to investigate scenarios of wrongly estimated effective mass m or falsely engineered macroscopic potential V for the slab material.

In Fig. 6, we regard the most successful design of Table 2, namely the InSb planar inclusion into diamond (for E₀ = 1 eV). We consider various thicknesses d′, effective masses m′ and potentials V′ around the optimal value d and the measured or simulated parameters (m, V) from standard textbooks¹⁰; wherever none or more than one τ = 1 peaks appear within the permissible interval 0 < E < 2E₀, the design is labeled as infeasible and white color is used in the corresponding maps. Once such imperfections occur, the transmissivity maximum is usually observed at a different energy E = E′≠E₀ and thus a quantity we selected to represent is the relative difference between the two peaks, denoted as (E′ − E₀)/E₀ = ΔE/E₀.

Figure 6

Performance sensitivity of the optimal planar inclusion of InSb into diamond for E₀ = 1 eV (Table 2). (a) The relative difference in peak energy ΔE/E₀ and (b) the half-power bandwidth B in eV as function of thickness misselection d′/d and effective mass misestimation m′/m. (c) The relative difference in peak energy ΔE/E₀ and (d) the half-power bandwidth B in eV as function of thickness misselection d′/d and macroscopic potential energy engineering error V′/V. White regions corresponds to infeasible parametric combinations where our demand for a single peak of unitary transmissivity within the range 0 < E < 2E₀, gets violated. White crosses denote optimal operational points.

In Fig. 6(a), we show this indicator ΔE/E₀ as function of misselected thickness d′/d and incorrectly estimated effective mass m′/m. We observe several forbidden parametric regions followed by zones of feasible designs that form a rather periodic pattern with respect to both d′/d and m′/m as an outcome of the undamped oscillatory nature of the phenomenon. Indeed, we have lossless Fabry-Perot interference resulting to periodic waveforms for the device response as d′ is being swept; since infinite resonances (each one of a different order n), namely infinite slab sizes, give the desired τ(E₀) = 1, the same happens for all the other values of transmissivity τ, which makes a periodic variation with respect to d′. Between two successive infeasible sections, large shifts of the peak transmission energy are recorded covering the entire working range 0 < E′ < 2E₀; in particular, the energy E′ gets smaller than E₀ for thinner slabs and greater effective masses. Note also that the boundaries between acceptable and non-acceptable parametric sets are not smooth due to the discrete nature of the resonances, similarly to Fig. 2; importantly, these boundaries are parallel to the smooth isocontours (into the colored region) demanding the size of the filters to decrease combined with a proper (almost linear) increase of the effective mass, to keep the represented quantity constant.

In Fig. 6(b) we represent across the same parametric plane (d′/d, m′/m) the half-power band B of the peak, expressed in dB. We notice that the infeasible regions are slightly more extended since we reject the designs whose range E′ − B/2 < E < E′ + B/2 does not fully belong in the permissible energy span; not just demanding 0 < E′ < 2E₀ as in Fig. 6(a). One directly notices the small values of B since, in the worst (maximum B) case, the energy range is as short as 0.1 eV; such a feature demonstrates that the selectivity of our designs remains high under fabrication or engineering defects even though the peak occurs at another e-beam energy E′≠E₀. Mostly surprisingly, the band B, within which the transmissivity falls at half, can be even better (smaller) than in the optimal design (for E′ = E₀) yielding a sharper filtering. Such a finding does not question the followed optimization since it happens around very small energies $E^{\prime} \ll {E}_{0}$ where our best scores are much more substantial (like in Table 1).

In Fig. 6(c,d), we examine the effect of improper estimation for the macroscopic potential V of the planar inclusion, which is found mild given the fact that the isocontour lines are almost vertical. Such a feature is attributed to the huge potential of diamond V₀ compared to that of InSb; whatever relative change it experiences, we always obtain ${V}_{0}\gg V$. In Fig. 6(c), where the difference ΔE/E₀ is depicted, the trade-off between thicker designs and higher peak energies E′ is again noticed. Similarly, in Fig. 6(d), the half-power band B is larger when the peak energy E′ (at which τ = 1) increases, while keeping small values below 0.1 eV.

In Fig. 7, we repeat the evaluations of the quantities of Fig. 6 but for a less efficient design from Table 1, namely CdTe into diamond; cadmium telluride is an isotropic noncentrosymmetric material with interesting properties exploited in photonic applications too⁴⁹. The variations are similar with those of Fig. 6; the only difference is the wilder dependence of the represented quantities from the potential energy into the inclusion since V for CdTe is much higher than this into InSb. Finally, the values for the half-power energy band B are higher, a property related to the diminished performance of the considered layout as a filter.

Figure 7

Similar calculations as in Fig. 7 but for the optimal planar inclusion of CdTe into diamond for E₀ = 1 eV (Table 2). (a) The relative difference in peak energy ΔE/E₀ and (b) the half-power bandwidth B in eV as function of thickness misselection d′/d and macroscopic potential energy engineering error V′/V.

Discussion and Conclusions

Selecting matter waves with respect to the energy they carry is a significant operation in quantum signal processing, namely when e-beams are channeled, guided and funneled at different modules of an integrated quantum engineering setup. In this work, a simple configuration of a single planar inclusion into a dense background is used to filter an impinging particle beam via a maximally sharp transfer function. The optimal resonance order is determined by choosing the most abrupt variation without letting secondary maximum appear within a considered wide energy range. This process is meticulously repeated for a long list of quantum media combinations by using their effective properties. We show the results of our optimization both in tables and graphs for various energies and the positive correlation of the selectivity performance and the slab thinness with the operational level of energy has been identified. To this end, the functionality of certain designs is demonstrated by feeding them with particles of a variety of energies and observe them to practically block all the others except for the filtered ones. Finally, the device response is tested when the size and the texture of the inclusions are not properly estimated; it has been found that the energy at which unitary transmissivity is exhibited gets shifted but the designs remains selective.

This work reports the best scores of a simple, easy-to-fabricate setup given the set of available media; in this sense, presents limits in the selectivity of the configuration that cannot be surpassed unless a smarter structure (multilayers) or more complicated quantum texture (alloys, hybrid heterojunctions) is employed. An interesting next step would be to generalize our approach to find the best structure that mimics a given response at various distinct energies or even across a band of them. In this way, the purpose of inverse design will be served in quantum layouts. Thus, inspired approaches applied to nanophotonics involving sophisticated gradient optimizations⁵⁰ or deep learning methods that have attracted the interest of both agenda-setting academic groups⁵¹ and industry research teams⁵², will be recasted and applied into quantum systems.

Methods

The physical configuration of the device is depicted in Fig. 1(a), where a quantum particle traveling along an axis z with equal probability across each point of the formed normal xy plane, meets normally a homogeneous planar inclusion. The wave function Ψ(r) describing the probabilistic motion of the quantum particle into an arbitrary inhomogeneous medium with effective mass m(r) and potential V(r), respects the time-independent Schrödinger equation⁵³:

$${\rm{\nabla }}[\frac{1}{m(r)}{\rm{\nabla }}]\Psi ({\bf{r}})+\frac{E-V({\bf{r}})}{{\hslash }^{2}/2}\Psi ({\bf{r}})=0,$$

(3)

where r is the related position vector. Our approach is semi-classical since the background medium and the considered inclusion is treated classically; only the impinging beam is assumed to exhibit quantum behavior. In addition, the effective quantum characteristics in each region are taken to vary abruptly without linear or other transition as one crosses the constant-z boundaries. The matter wave defining the behavior of the aforementioned particles in the absence of the slab inhomogeneity, possesses the form: Ψ_inc(z) = exp(ik₀z); however, in the presence of that thin cavity which is supposed to work as a quantum filter, the wave function for z < 0 changes by the reflecting-wave term Ψ_ref(z) = R exp(−ik₀z). In a similar manner, the entire matter wave behind the slab (for z > d) is given by: Ψ_tran(z) = T exp(ik₀z). As far as the wave function into the planar slab (0 < z < d) is concerned, it is written as: Ψ_slab(z) = C exp(ikz) + D exp(−ikz). The followed technique is the well-known Wentzel-Kramers-Brillouin (WKB) approximation, where the wavefunctions are assumed to possess exponential forms with either amplitude or phase taken to be slowly changing relative to the de Broglie wavelength⁵⁴.

The boundary conditions across the interface between two regions with effective masses {m₁, m₂} and wave functions {Ψ₁, Ψ₂} demand continuity of the probability amplitudes and the probability currents (u is the unitary vector normal to the boundary):

$${\Psi }_{1}={\Psi }_{2},\,\frac{1}{{m}_{1}}{\bf{\text{u}}}\cdot {\rm{\nabla }}{\Psi }_{1}=\frac{1}{{m}_{2}}{\bf{\text{u}}}\cdot {\rm{\nabla }}{\Psi }_{2}.$$

(4)

After imposing these requirements at z = 0, d, the unknown complex quantities {C, D, R, T} are determined; in particular, the reflection and the transmission coefficients are given by:

$$R=\frac{({k}_{0}^{2}{m}^{2}-{k}^{2}{m}_{0}^{2})\,\sin \,(kd)}{({k}_{0}^{2}{m}^{2}+{k}^{2}{m}_{0}^{2})\,\sin \,(kd)+2ik{k}_{0}m{m}_{0}\,\cos \,(kd)},\,T=-\frac{4\,\exp \,[id(k-{k}_{0})]k{k}_{0}m{m}_{0}}{\exp \,\mathrm{(2}ikd)({k}_{0}m-k{m}_{0}{)}^{2}-{({k}_{0}m+k{m}_{0})}^{2}}.$$

(5)

By assuming that effective masses {m₀, m} and macroscopic potentials {V₀, V} are selected so that k₀, k > 0, one can evaluate the reflectivity ρ = |R|² and the transmissivity τ = |T|², which is shown in (1).

If one requests unitary transmissivity at a specific energy level E = E₀, the thickness of the slab is permitted to take only discrete values $d=n\pi \hslash /\sqrt{2m({E}_{0}+\Delta V)}$ for $n\in {{\mathbb{N}}}^{\ast }$. By replacing the explicit forms for the wavenumbers ${k}_{0}=\sqrt{2{m}_{0}E}/\hslash$ and $k=\sqrt{2m(E+\Delta V)}/\hslash$, we obtain the following expressions:

$$\rho (E)=\frac{{[(1-m/{m}_{0})E+\Delta V]}^{2}{\sin }^{2}\,(n\pi \sqrt{\frac{E+\Delta V}{{E}_{0}+\Delta V}})}{4(m/{m}_{0})E(E+\Delta V)+{[(1-m/{m}_{0})E+\Delta V]}^{2}{\sin }^{2}(n\pi \sqrt{\frac{E+\Delta V}{{E}_{0}+\Delta V}})},$$

(6)

$$\tau (E)=\frac{4(m/{m}_{0})E(E+\Delta V)}{{[(1+m/{m}_{0})E+\Delta V]}^{2}-{[(1-m/{m}_{0})E+\Delta V]}^{2}{\cos }^{2}(n\pi \sqrt{\frac{E+\Delta V}{{E}_{0}+\Delta V}})},$$

(7)

where n is a positive integer.

The transmissivity (7), is maximized by taking unitary value (τ = 1) for $E+\Delta V=({E}_{0}+\Delta V){(\frac{\nu }{n})}^{2}$ with ν ∈ ℕ^*; similarly, it gets minimized for $E+\Delta V=({E}_{0}+\Delta V){(\frac{\nu }{n}-\frac{1}{2n})}^{2}$ with minimum values τ = [2k₀mkm₀/(k₀²m² + k²m₀²)]². In this way, one can directly evaluate the optimal order n of Fabry-Perot resonance so that the energy response of the filter gets as sharp as possible; indeed, the first minimum of τ(E) when E > E₀ occurs for ν = n + 1 and should be exhibited outside of the considered energy band 0 < E < 2E₀. Namely:

$$({E}_{0}+\Delta V){(\frac{n+1}{n}-\frac{1}{2n})}^{2}-\Delta V > 2{E}_{0}\Rightarrow n < \frac{\sqrt{{E}_{0}+\Delta V}/2}{\sqrt{2{E}_{0}+\Delta V}-\sqrt{{E}_{0}+\Delta V}},$$

(8)

which is identical to (2). The residual transmission τ₂ at the right extremum of the interval 0 < E < 2E₀ can be also evaluated from (7) at E = 2E₀ with use of (8).