CMOS-Compatible Electrostatic SWAP Gate in Silicon Quantum Dots: Tight-Binding Model and Beyond

Abstract

We present a generalized electrostatic SWAP gate realized in a chain of two double quantum dots operated in the single-electron regime. Using a minimalist tight-binding model, we derive analytical results and corroborate them with numerical simulations. We exploit the charge anticorrelation arising from Coulomb repulsion and quantify the resulting entanglement generation. We contrast classical and quantum descriptions and show how device geometry and coupling strengths govern entanglement dynamics and gate performance. The results are relevant to cryogenic, CMOS-compatible quantum technologies and suggest a route toward large-scale semiconductor implementations of quantum logic. Finally, we outline a systematic procedure for translating classical electrostatic logic gates into single-electron quantum gates.

1. Current Status of Quantum Technologies

The purpose of this section is to establish why electrostatically defined silicon quantum dots as given by [1,2,3,4,5,6] that are a relevant platform for the SWAP-gate architecture studied in the manuscript. The analysis narrows the broader quantum-hardware landscape to devices that can plausibly support dense integration, cryogenic electrical control, and lithographically defined coupling. Silicon is emphasized because 300 mm processing, gate-defined confinement, and cryogenic CMOS are compatible with a position-based architecture in which the principal controls are voltages that tune local energies and tunnel barriers. Figure 1, Figure 2 and Figure 3 therefore connect system-level scalability to the double-quantum-dot network analyzed in the present model.

Several cited experiments provide more specific physical benchmarks for this reduction. Gustavsson et al. [7] resolved individual tunneling events in real time with a quantum point contact and measured the full counting statistics of transport through a semiconductor quantum dot, including the second and third moments of the current distribution. Wang et al. [8] measured the interdot relaxation time $T_1$ of a single electron in a Si/SiGe double quantum dot as a function of detuning and tunnel coupling, finding a tuning range of about four orders of magnitude and a maximum $T_1$ of 45 microseconds. Stehlik et al. [9] used repeated passages through an avoided crossing to demonstrate Landau–Zener–Stuckelberg interferometry in a single-electron GaAs charge qubit, including coherent destruction of tunneling and multiphoton excitation.

The single-electron circuit context is further clarified by Durrani [10], who reviewed Coulomb-blockade devices in silicon, including room-temperature single-electron transistors, few-electron memories, and binary-decision-diagram logic, and by Korotkov [11], who analyzed single-electron-transistor operation, ultradense digital circuit families, and the limits introduced by discrete spectra and finite tunnel barriers. For CMOS-compatible readout, Gonzalez-Zalba et al. [12] observed coherent charge oscillations in a silicon nanowire field-effect transistor using a gate-coupled radio-frequency resonator and resolved Landau–Zener–Stuckelberg–Majorana interference, with a reported charge-coherence time of approximately 100 ps. Schaal et al. [13] subsequently demonstrated conditional dispersive readout of a CMOS single-electron memory cell in FD-SOI, integrating the control transistor and quantum dot on the same chip.

These experiments motivate the present modeling choice but remain distinct from the authors’ contribution. The manuscript uses a compact position-basis Hamiltonian (as in contrast to mostly used in literature eigenenergy based Hamiltonian) to relate detuning, tunneling, Coulomb interaction, and geometry to SWAP-related dynamics. Device-level numerical tools provide a complementary route to realistic structures: Beaudoin et al. [14] demonstrated robust nonlinear Poisson–Schrödinger QTCAD simulations of an industrial FD-SOI quantum-dot device at 100 mK using adaptive meshing, whereas Kriekouki et al. [15] used 3D quantum TCAD to analyze the single-electron regime, corner-dot formation, and Coulomb-blockade behavior in a 28 nm FD-SOI device against experimental data. These works support the use of electrostatic device simulation as a bridge between the analytical Hamiltonian and foundry-scale geometry.

The remainder of the article therefore treats the tight-binding model as a controlled low-energy description rather than a universal microscopic theory. Detuning, tunnel coupling, Coulomb interaction, and device geometry are carried forward as the dominant effective variables, while later sections explicitly examine the conditions under which occupation of additional states requires a larger Hilbert space or a more microscopic treatment.

More than twenty hardware platforms are being explored across quantum computing, communication, and sensing. From a scalability point of view, two families currently show a credible path toward wafer-scale integration: (i) silicon spin/charge qubits compatible with CMOS process flows [16,17] and (ii) superconducting transmons supported by a mature cryogenic control stack [18]. Silicon benefits from 300 mm fabrication and the prospect of dense 2D arrays with purely electrostatic control, which motivates our focus on electrostatically defined single-electron devices. Major system-level constraints remain error-correction overheads and the wiring/thermal budget at the mK stage, which strongly favor low-power, high-density solutions. In silicon quantum dots, progress in isotopic purification, gate-oxide quality, and cryo-CMOS control has reduced charge noise and improved uniformity, enabling reproducible exchange couplings across small arrays [17]. Against this backdrop, electrostatically defined single-electron devices offer a fabrication-aligned path to native two-qubit primitives (SWAP/$\sqrt{\mathrm{SWAP}}$) realized with detuning and barrier pulses, making them natural candidates for scalable, all-electrostatic architectures.

As summarized in Figure 1, platform roadmaps consistently indicate that silicon [19,20] and superconducting approaches currently offer the steepest scaling trajectories and meet DiVincenco criteria [21]. In silicon, the decisive levers are 300 mm manufacturing, isotopically purified ²⁸Si with long coherence, and the possibility of co-integrating cryo-CMOS control within a tight thermal budget. Superconducting circuits, by contrast, benefit from a mature microwave toolchain and rapid gate speeds yet face wiring density and footprint constraints at the mK stage. These system-level considerations motivate a device-centric look at CMOS-compatible, electrostatically defined qubits.

Figure 1. Indicative scalability of selected qubit architectures. Stylized comparison of estimated qubits-per-die across superconducting, photonic, ion-trap and silicon spin platforms (based on public roadmaps from Diraq company https://diraq.com/).

Figure 1. Indicative scalability of selected qubit architectures. Stylized comparison of estimated qubits-per-die across superconducting, photonic, ion-trap and silicon spin platforms (based on public roadmaps from Diraq company https://diraq.com/).

Figure 2. Representative implementations of silicon quantum dots and arrays (gate-defined MOS/SiGe, nanowires/holes, crossbar concepts, and 2D lattices): (a–c) Few electron single, double, and triple QDs. (d) 8-site 1D QD array. (e) 3 × 3 QD array. (f,g) SiMOS single and DQD devices. (h) Donor device fabricated using STM lithography. (i) Single-layer etch defined 1 × 6 QD array in Si/SiGe. (j) 1 × 9 QD array fabricated using overlapping Al gates on Si/SiGe. (k) Enhancement mode Ge/GeSi structure. Holes are confined in (k), while the remaining devices isolate electrons. After Burkard et al., Rev. Mod. Phys. 95, 025003 (2023) [22].

Figure 2. Representative implementations of silicon quantum dots and arrays (gate-defined MOS/SiGe, nanowires/holes, crossbar concepts, and 2D lattices): (a–c) Few electron single, double, and triple QDs. (d) 8-site 1D QD array. (e) 3 × 3 QD array. (f,g) SiMOS single and DQD devices. (h) Donor device fabricated using STM lithography. (i) Single-layer etch defined 1 × 6 QD array in Si/SiGe. (j) 1 × 9 QD array fabricated using overlapping Al gates on Si/SiGe. (k) Enhancement mode Ge/GeSi structure. Holes are confined in (k), while the remaining devices isolate electrons. After Burkard et al., Rev. Mod. Phys. 95, 025003 (2023) [22].

Zooming in, Figure 2 maps the concrete device landscape: gate-defined MOS and Si/SiGe quantum dots, hole-spin nanowires (Ge/Si), as well as crossbar and 2D-array layouts that enable capacitive and exchange coupling without permanent magnets. Two control knobs dominate across these realizations—detuning $\Delta$ (on-site energies) and barrier control (tunnel coupling $t_c$)—which together set the exchange interaction $J(\Delta ,t_c)$ and hence the entangling dynamics. This architecture aligns with standard CMOS stacks and provides native access to SWAP/$\sqrt{\mathrm{SWAP}}$ primitives via time-shaped detuning and barrier pulses, paving the way for the tunable electrostatic SWAP operation analyzed next (Figure 3).

Following [23], we use a spinless tight-binding model in which on-site detuning $\Delta$ and barrier-controlled tunnel coupling $t_c$ are the only electrostatic knobs. Coulomb-driven charge anticorrelation enforces single occupancy and supplies the resource for entanglement while keeping control fully capacitive—no micromagnets—thus aligning the device stack with CMOS (gate metals/oxide, mK operation, sub-10 $\mathsf{\mu }\mathrm{W}$/channel cryo-CMOS). Our numerical simulations reproduce the analytic gate-time conditions and indicate tolerance to moderate dispersion in $(\Delta ,t_c)$ along short chains, supporting the scalable SWAP primitive depicted in Figure 3. Various milestones to be achieved by position-based qubits are specified by Table 1.

Table 1. Selected milestones relevant to CMOS-compatible qubits to be achieved in the near future.

Milestone	Brief Note
Cryo-CMOS control < 10 $\mathsf{\mu }\mathrm{W}$/channel	mK operation with negligible coherence penalty.
Single-qubit fidelity $99.9\%$	Demonstrated on 300 mm lines; CMOS-compatible.
Two-qubit fidelity > 99%	Silicon entangling gates beyond $99\%$.
Benchmarking consortia	Toward standardized metrics (e.g., DARPA/QBI).

Figure 3. Basic logic elements of electrostatic position-based qubits in single-electron regime: (a)—single qubit, (b)—tunable SWAP in a CMOS-compatible DQD (double-quantum-dot) network with analytical treatment and numerical corroboration according to Pomorski [23].

Figure 3. Basic logic elements of electrostatic position-based qubits in single-electron regime: (a)—single qubit, (b)—tunable SWAP in a CMOS-compatible DQD (double-quantum-dot) network with analytical treatment and numerical corroboration according to Pomorski [23].

The purpose of this work is to describe position-based semiconductor qubits arranged in a chain of electrostatically controlled quantum dots using single electrons and neglecting spin, which is discussed in the preceding sections. In particular this work is the extension of previous work [23] treating the interaction of two single-electron semiconductor qubits in a more systematic way, while previous work assumed that only two position-based qubits are place in parallel at a certain distance. We also treat qubit interaction as under the influence of an external radio-frequency field. Furthermore, we provide fundamental derivation of the tight-binding model for the case of the perturbative electrostatic interaction of two qubits when they are placed at different semiconductors that are separated. We also consider a scenario in which two electrostatically interacting qubits with initially four energy levels acquire more energy levels, which violates the simplistic tight-binding model, which is important as it indicates that two interacting qubits break down after some interaction time as in Figure 4.

Figure 4. Stages of thermalization of quantum state and evolution of occupancy of energy levels with time, represented by time arrow moving from left to right and then from top to bottom and with each lower line representing the proceeding time steps. Initially weakly interacting particles A and B are placed at two different single-electron lines, with the initial non-entangled state being the tensor product of two qubits’ quantum states and each of them having the occupancy of two energy levels. The system of two qubits starts to occupy an increasing number of energy levels, and breakdown of the tight-binding model is observed. In the meantime the emergence of quantum entanglement is observed from the initial non-entangled state as by very left bottom diagram.

2. Introduction to the Tight-Binding Model and Its Applicability in Solid-State Physics

Before using a tight-binding Hamiltonian as the main analytical tool, the manuscript needs to define the physical regime in which that approximation is meaningful. This is especially important because the work later discusses the breakdown of a reduced four-state description. The motivation of this section is therefore methodological as well as physical: to distinguish a localized-orbital, weak-overlap description appropriate for quantum-dot charge states from regimes in which delocalization, disorder, strong correlations, lattice dynamics, or self-consistent screening become essential. The section evaluates the approximation through its basis assumptions. Electrons are represented by localized atomic-like or Wannier-like states, and kinetic motion is introduced through hopping matrix elements. The discussion then tests this picture against representative failure regimes, including Mott transitions, Anderson localization, Peierls or charge-density-wave transitions, magnetic or Slater transitions, and band-overlap transitions. Table 2 summarizes why a clean, non-interacting, nearest-neighbor tight-binding model is insufficient in those cases and points toward Hubbard-type interactions, disorder models, dynamical mean-field theory, DFT+U, GW, or other first-principles approaches when the simplified basis ceases to be predictive.

Table 2. Applicability of tight-binding model.

Transition Type	Justification Behind Validity of Tight-Binding Model
Mott transition	A partially filled band should be metallic in non-interacting tight binding, but strong electron–electron repulsion can split it into Hubbard bands and produce an insulator. This requires interactions such as the Hubbard U, not just hopping t.
Anderson localization	Disorder can localize electronic wavefunctions through quantum interference. A clean periodic tight-binding model cannot capture this; disorder and localization theory are needed.
Peierls or charge-density-wave transition	The insulating state is caused by lattice distortion or electron-phonon coupling. Tight binding can mimic the final distorted band structure, but does not predict the instability unless lattice dynamics are included.
Slater or magnetic transition	Magnetic ordering can open a gap. A nonmagnetic tight-binding model cannot describe the transition without spin-dependent interactions or exchange fields.
Band-overlap transition	In semi-metals or pressure-driven systems, the transition depends sensitively on orbital hybridization, screening, structural changes, and self-consistent charge redistribution. Simple fixed hopping parameters are usually insufficient.

A central point is that delocalized eigenstates do not automatically invalidate tight binding. A periodic tight-binding model can itself produce Bloch-like states. The limitation arises when the chosen localized basis no longer gives a controlled representation of the relevant low-energy physics, for example, when orbital overlap is large, long-range hopping is important, screening changes self-consistently, or many-body interactions determine the phase. This distinction is directly relevant to the quantum-dot problem: the model is most defensible when the electron remains largely confined to a small set of dot-localized states and when the main device physics can be captured by on-site energies, tunneling, and Coulomb terms. The later observation of additional populated eigenstates is therefore a physical signal that the original truncation must be reconsidered.

The tight-binding model is adopted in this work as a controlled low-energy approximation for localized single-electron quantum-dot states, not as a universal description of solid-state electrons. Its validity depends on the persistence of a small localized basis and on limited occupation of higher orbital states. This section therefore provides the conceptual criterion used later to interpret when the model succeeds and when a beyond-tight-binding treatment is required.

The tight-binding approximation assumes that electrons’ wavefunctions can be described as linear combinations of localized atomic-like orbitals, with relatively small overlap between neighboring sites. It works best for narrow bands, localized d or f orbitals, molecular crystals, semiconductors with directional covalent bonds, and lattice models where hopping between sites is the dominant kinetic process. However, it becomes less reliable when the electronic states are strongly delocalized, as in many simple metals, where the conduction electrons resemble nearly-free Bloch waves rather than electrons bound near particular atoms.

A key point is that tight binding does not fail merely because the final eigenstates are delocalized. Even in tight binding, the eigenstates of a perfect crystal are Bloch states spread over the lattice. The failure occurs when the basis assumption becomes poor: the atomic orbitals overlap strongly, the band width is large, hopping is not limited to nearest neighbors, and the electron wavefunction is better represented by plane waves or extended Wannier functions than by localized orbitals. In such cases, a simple tight-binding Hamiltonian with fermionic electron creation/annihilation operators ${\widehat{c}}_k^{†}\left({\widehat{c}}_k\right)$

$$\begin{array}{c}\hfill \widehat{H}=\sum _k{ϵ}_k{\widehat{c}}_k^{†}{\widehat{c}}_k+\sum _{k,s}{t}_{k\to s}{\widehat{c}}_s^{†}{\widehat{c}}_k\end{array}$$

(1)

may fit part of a band structure, but it no longer gives a transparent or predictive microscopic description.

For metals, especially simple s- and p-band metals, the nearly-free-electron model or first-principles band-structure methods are often more appropriate. Tight binding can still be used as an effective model if many hopping terms, several orbitals, and carefully constructed Wannier functions are included, but then it becomes a fitted representation rather than a simple physical approximation.

Metal–insulator transitions are especially problematic for the simplest tight-binding approximation because the transition is often controlled by physics that is not contained in a single-particle hopping model. A periodic non-interacting tight-binding model can describe band formation and band gaps, but it cannot generally describe the critical mechanisms responsible for real metal–insulator transitions. Important failure cases of the tight-binding model are summarized by Table 2.

For a Mott transition, the essential parameter is often the ratio ${\displaystyle \frac{W}{U}}$, where U is the on-site Coulomb repulsion and W is the electronic bandwidth. A simple tight-binding model describes W but not the correlation physics produced by U. Therefore, it predicts a metal for a half-filled band where the real material may be an insulator. This is the classic failure behind Mott insulators and strongly correlated oxides. The Hubbard model was introduced precisely to include local electron correlations in narrow bands, beyond ordinary tight binding [24].

For Anderson localization [25], even a tight-binding lattice can be used but only after adding random site energies or random hopping amplitudes. The critical phenomenon is not ordinary band formation but the disappearance of diffusion due to disorder-induced localization of the wavefunction. Anderson’s original work showed that sufficiently strong randomness can prevent diffusion in a lattice system.

A broader review by Imada, Fujimori, and Tokura [26,27] emphasizes that metal–insulator transitions can involve correlations, disorder, lattice effects, magnetism, and orbital physics, often producing resistivity changes over many orders of magnitude. This complexity is far beyond the predictive scope of a simple non-interacting tight-binding approximation.

Thus, the tight-binding approximation should be used with caution near metallic and metal–insulator transition regimes. It is reliable when localized orbitals, weak overlap, and fixed hopping parameters are physically justified. It becomes unreliable when electrons are strongly delocalized, when screening and long-range Coulomb effects are important, when disorder controls transport, or when electron–electron and electron–lattice interactions determine whether the system is metallic or insulating. In these cases, more advanced approaches are required, such as Hubbard-type models, dynamical mean-field theory, DFT+U, GW, disorder-based localization theory, or first-principles calculations with self-consistent structural and electronic degrees of freedom. In the whole manuscript we use $q^2={\displaystyle \frac{e^2}{4\pi {ϵ}_0}}$; e is electron charge $-1.60217663\times {10}^{-19}C$ (Coulombs), and ${ϵ}_0=8.854\times {10}^{-12}{\displaystyle \frac{F}{m}}$ is the dielectric constant of vacuum.

3. Description of a Position-Based Qubit in the Tight-Binding Model

The goal of this section is to define the elementary qubit used throughout the manuscript: one electron shared between two neighboring quantum dots, with logical information encoded in position. A mathematically explicit single-qubit description is required before Coulomb-coupled two-qubit gates can be analyzed. The section therefore establishes how experimentally meaningful controls—local electrostatic energies and interdot tunneling—enter the qubit Hamiltonian and how those parameters determine coherent state evolution. A two-state tight-binding Hamiltonian is written in the localized position basis using the on-site energies $E_{p1}$ and $E_{p2}$ and the generally complex tunneling amplitude $t_{s12}$. The same system is also represented in its instantaneous energy eigenbasis. Analytical expressions are introduced for the eigenenergies and eigenstates, followed by the Schrödinger time-evolution equation, the unitary propagator, and the density-matrix representation. A Pauli-matrix decomposition is then used to connect the Hamiltonian to the standard two-level formalism. Special limits—vanishing hopping, dominant hopping, and asymmetric on-site energies—are considered to show how the propagator depends on the physical control parameters.

This construction identifies the two quantities that play the role of primary control axes in the position-basis qubit: energy detuning biases localization, while the hopping term produces coherent transfer between the dots. The model makes the relation between voltage control and qubit dynamics transparent because both quantities can, in principle, be modified electrostatically. The density-matrix formulation is important for the later two-qubit discussion because it provides the natural route to occupation probabilities, coherences, and entanglement measures. The analytical propagators are most useful when the effective two-state basis remains valid; if additional orbital states become appreciably populated, $E_p$ and $t_s$ should be regarded as effective parameters of a truncated model rather than complete microscopic descriptors.

Section 3 supplies the single-qubit building block for the entire manuscript. It defines the localized basis, energy basis, Hamiltonian parameters, unitary dynamics, and density-matrix language that are reused for coupled devices. In physical terms, it shows how a double quantum dot can function as a voltage-controlled position qubit whose coherent evolution is governed by detuning and tunneling.

We are motivated by the methodology given in work [28]. We refer to the physical situation from Figure 5 as given by [29,30,31,32,33], and we consider a position-based qubit in the tight-binding model [33] Hamiltonian ${\widehat{H}}_P$ or in energy bases ${\widehat{H}}_E$ as given by

$$\begin{array}{c}\hfill {\widehat{H}}_P\left(t\right)={\left(\begin{array}{cc}{E}_{p1}\left(t\right)& t_{s12}\left(t\right)\\ t_{s12}^{†}\left(t\right)& E_{p2}\left(t\right)\end{array}\right)}_{[x=(x_1,x_2)]}\to \\ \hfill {\widehat{H}}_E\left(t\right)={\left(E_1\left(t\right)\right|E_1{⟩}_t⟨E_1{|}_t+E_2\left(t\right)|E_2⟩⟨E_2|)}_{[E=(E_1,E_2)]}.\end{array}$$

(2)

The Hamiltonian $\widehat{H}\left(t\right)$ eigenenergies $E_1\left(t\right)$ and $E_2\left(t\right)$ with $E_2\left(t\right)>E_1\left(t\right)$ are given as

$$\begin{array}{c}\hfill E_1\left(t\right)=\left(-\sqrt{{\displaystyle \frac{{(E_{p1}\left(t\right)-E_{p2}\left(t\right))}^2}{4}}+{\left|t_{s12}\left(t\right)\right|}^2}+{\displaystyle \frac{E_{p1}\left(t\right)+E_{p2}\left(t\right)}{2}}\right),\\ \hfill E_2\left(t\right)=\left(+\sqrt{{\displaystyle \frac{{(E_{p1}\left(t\right)-E_{p2}\left(t\right))}^2}{4}}+{\left|t_{s12}\left(t\right)\right|}^2}+{\displaystyle \frac{E_{p1}\left(t\right)+E_{p2}\left(t\right)}{2}}\right),\end{array}$$

(3)

and energy eigenstates $|E_1\left(t\right)⟩$ and $|E_2\left(t\right)⟩$ have the following form

$$\begin{array}{c}\hfill |E_1,t⟩=\left(\begin{array}{c}{\displaystyle \frac{(E_{p2}\left(t\right)-E_{p1}\left(t\right))+\sqrt{\frac{{(E_{p2}\left(t\right)-E_{p1}\left(t\right))}^2}{2}+{\left|t_{s12}\left(t\right)\right|}^2}}{-i{t}_{sr}\left(t\right)+t_{si}\left(t\right)}}\\ -1\end{array}\right),\\ \hfill |E_2,t⟩=\left(\begin{array}{c}{\displaystyle \frac{-(E_{p2}\left(t\right)-E_{p1}\left(t\right))+\sqrt{\frac{{(E_{p2}\left(t\right)-E_{p1}\left(t\right))}^2}{2}+{\left|t_{s12}\left(t\right)\right|}^2}}{t_{sr}\left(t\right)-i{t}_{si}\left(t\right)}}\\ 1\end{array}\right).\end{array}$$

(4)

This Hamiltonian gives a description of two coupled quantum wells as depicted in Figure 5. In such a situation we have real-valued functions $E_{p1}\left(t\right)$, $E_{p2}\left(t\right)$ and complex-valued functions $t_{s12}\left(t\right)=t_s\left(t\right)=t_{sr}\left(t\right)+i{t}_{si}\left(t\right)$ and $t_{s21}\left(t\right)=t_{s12}^{*}\left(t\right)$, which is equivalent to the knowledge of four real-valued time-dependent continuous or discontinuous functions $E_{p1}\left(t\right)$, $E_{p1}\left(2\right)$, $t_{sr}\left(t\right)$ and $t_{si}\left(t\right)$. The quantum state is a superposition of state localized at nodes 1 and 2 and therefore is given as

$$|\psi {⟩}_{\left[x\right]}=\alpha \left(t\right)|1,0{⟩}_x+\beta \left(t\right)|0,1{⟩}_x=\alpha \left(t\right)\left(\begin{array}{c}1\\ 0\end{array}\right)+\beta \left(t\right)\left(\begin{array}{c}0\\ 1\end{array}\right),$$

(5)

where ${\left|\alpha \left(t\right)\right|}^2$ (${\left|\beta \left(t\right)\right|}^2$) is the probability of finding a particle at node 1 (2), which brings ${\left|\alpha \left(t\right)\right|}^2+{\left|\beta \left(t\right)\right|}^2=1$ and obviously $⟨1,0{|}_x||1,0{⟩}_x=1=⟨0,1{|}_x||0,1{⟩}_x$ and $⟨1,0{|}_x||0,1{⟩}_x=0=⟨0,1{|}_x||1,0{⟩}_x$. In the Schrödinger formalism, states $|1,0{⟩}_x$ and $|0,1{⟩}_x$ are Wannier functions that are parameterized by position x. We work in tight-binding approximation and quantum state evolution with time as given by

$$i\mathit{\hslash }{\displaystyle \frac{d}{dt}}|\psi \left(t\right)⟩=\widehat{H}\left(t\right)|\psi \left(t\right)⟩=E\left(t\right)|\psi \left(t\right)⟩.$$

(6)

The last equation has an analytic solution

$$|\psi \left(t\right)⟩=e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}|\psi \left(t_0\right)⟩=e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}\left(\begin{array}{c}\alpha \left(0\right)\\ \beta \left(0\right)\end{array}\right)$$

(7)

and in quantum density-matrix theory we obtain

$$\begin{array}{c}\hfill \widehat{\rho }\left(t\right)={\widehat{\rho }}^{†}\left(t\right)=|\psi \left(t\right)⟩⟨\psi \left(t\right)|=\\ \hfill =\widehat{U}(t,t_0)\widehat{\rho }\left(t_0\right)\widehat{U}{(t,t_0)}^{-1}=\\ \hfill =e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}\left(\right|\psi \left(t_0\right)⟩⟨\psi \left(t_0\right)\left|\right)e^{-{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}=\\ \hfill =e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}\left(\left(\begin{array}{c}\alpha \left(0\right)\\ \beta \left(0\right)\end{array}\right)\left(\begin{array}{cc}{\alpha }^{*}\left(0\right)& {\beta }^{*}\left(0\right)\end{array}\right)\right)e^{-{\displaystyle \frac{{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}{i\mathit{\hslash }}}}\\ \hfill =\widehat{U}(t,t_0)\left(\begin{array}{cc}{\left|\alpha \left(0\right)\right|}^2& \alpha \left(0\right){\beta }^{*}\left(0\right)\\ \beta \left(0\right)\alpha {\left(0\right)}^{*}& {\left|\beta \left(0\right)\right|}^2\end{array}\right)\widehat{U}{(t,t_0)}^{†}.\end{array}$$

Pauli Decomposition Applied to Tight-Binding Hamiltonian

Any Hermitian $2\times 2$ matrix admits the expansion $A={\sum }_{k=0}^3{c}_k{\sigma }_k$ with $c_0=\frac{1}{2}\mathrm{Tr}A$ and $\overrightarrow{c}=\frac{1}{2}\mathrm{Tr}\left(A\overrightarrow{\sigma }\right)$. For composite systems, $A\in {\mathbb{C}}^{2^N\times 2^N}$ can be written as $A={\sum }_{\mathbf{k}}{c}_{\mathbf{k}}{\sigma }_{k_1}\otimes \cdots \otimes {\sigma }_{k_N}$, where $\mathbf{k}=(k_1,\dots ,k_N)$ and $k_i\in \{0,1,2,3\}$.

Using the above property for matrix of size 2×2 we obtain $e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}=\widehat{U}(t,t_0),$ and assuming $E_{p1}\left(t\right)=E_{p2}\left(t\right)=E_p\left(t\right)$ and we are given matrix $e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}=$

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{e}^{{\displaystyle \frac{-i{\int }_{t_0}^t{E}_p\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}}ch\left({\displaystyle \frac{\sqrt{-{\int }_{t_0}^t\left(\right|t_s\left(t^{\prime }\right){|}^2)d{t}^{\prime }}}{\mathit{\hslash }}}\right)& {\displaystyle \frac{e^{\frac{-i{\int }_{t_0}^t{E}_p\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}\left({\int }_{t_0}^t\left(t_s^{*}\left(t^{\prime }\right)\right)d{t}^{\prime }\right)sh\left(\frac{\sqrt{-{\int }_{t_0}^t|t_s\left(t^{\prime }\right){|}^2)}}{\mathit{\hslash }}\right)}{\sqrt{-{\int }_{t_0}^t\left({(t_{si}{\left(t^{\prime }\right)}^2+t_{sr}\left(t^{\prime }\right))}^2\right)d{t}^{\prime }}}}\\ {\displaystyle \frac{e^{\frac{-i{\int }_{t_0}^t{E}_p\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}\left({\int }_{t_0}^t(-t_s\left(t^{\prime }\right))d{t}^{\prime }\right)sh\left(\frac{\sqrt{-{\int }_{t_0}^t{\left|t_s\left(t^{\prime }\right)\right|}^{2}d{t}^{\prime }}}{\mathit{\hslash }}\right)}{\sqrt{-{\int }_{t_0}^t\left({(t_{si}{\left(t^{\prime }\right)}^2+t_{sr}\left(t^{\prime }\right))}^2\right)d{t}^{\prime }}}}& e^{{\displaystyle \frac{-i{\int }_{t_0}^t{E}_p\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}}ch\left({\displaystyle \frac{\sqrt{-{\int }_{t_0}^t\left(\right|t_s\left(t^{\prime }\right){|}^2)d{t}^{\prime }}}{\mathit{\hslash }}}\right)\end{array}\right),\end{array}$$

where $sh$(.) and $ch$(.) are sinh and cosh hyperbolic functions, where $|t_s{\left(t\right)|}^2=|t_{sr}{\left(t\right)|}^2+{\left|t_{si}\left(t\right)\right|}^2$. This matrix is unitary, so ${\widehat{U}}^{†}(t,t_0)={\widehat{U}}^{-1}(t,t_0)$. At the very end we will also consider a more general case when $E_{p1}\left(t\right)\ne E_{p2}\left(t\right)$. At first let us consider the case of two localized states in the left and right quantum wells so there is no hopping, which implies $t_s=0$. In such a case the evolution matrix $\widehat{U}(t,t_0)$ is unitarian and has the following form

$$\begin{array}{c}\hfill \widehat{U}(t,t_0)=\\ \hfill e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}=\\ \hfill \left(\begin{array}{cc}{e}^{\frac{-i{\int }_{t_0}^t{E}_{p1}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}& 0\\ 0& e^{\frac{-i{\int }_{t_0}^t{E}_{p2}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}\end{array}\right)=\\ \hfill {\displaystyle \frac{(e^{\frac{-i{\int }_{t_0}^t{E}_{p1}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}+e^{\frac{-i{\int }_{t_0}^t{E}_{p2}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}})}{2}}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-\\ \hfill {\displaystyle \frac{(e^{\frac{-i{\int }_{t_0}^t{E}_{p1}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}-e^{\frac{-i{\int }_{t_0}^t{E}_{p2}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}})}{2}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\\ \hfill ={\displaystyle \frac{(e^{\frac{-i{\int }_{t_0}^t{E}_{p1}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}+e^{\frac{-i{\int }_{t_0}^t{E}_{p2}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}})}{2}}{\sigma }_0+\\ \hfill {\displaystyle \frac{(e^{\frac{-i{\int }_{t_0}^t{E}_{p1}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}}-e^{\frac{-i{\int }_{t_0}^t{E}_{p2}\left(t^{\prime }\right)d{t}^{\prime }}{\mathit{\hslash }}})}{2}}{\sigma }_3,\end{array}$$

(8)

which implies that the left and right quantum dots are two disconnected systems, each undergoing its own time evolution. However since one electron is distributed between those physical systems the measurement conducted on the left quantum dot immediately affects the right quantum dot. Another extreme example is the situation where hopping energy is considerably bigger than localization energy. In such a case we set $E_{p1}=E_{p2}=0$, and in the case of non-zero hopping terms we obtain $\widehat{U}(t,t_0)=$ given below

$$\begin{array}{c}\hfill \widehat{U}(t,t_0)=e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}=\\ \hfill \left(\begin{array}{cc}ch\left({\displaystyle \frac{\sqrt{-{\int }_{t_0}^t\left(\right|t_s\left(t^{\prime }\right){|}^2)d{t}^{\prime }}}{\mathit{\hslash }}}\right)& {\displaystyle \frac{\left({\int }_{t_0}^t\left(t_s^{*}\left(t^{\prime }\right)\right)d{t}^{\prime }\right)sh\left(\frac{\sqrt{-{\int }_{t_0}^t|t_s\left(t^{\prime }\right){|}^2)}}{\mathit{\hslash }}\right)}{\sqrt{-{\int }_{t_0}^t\left({(t_{si}{\left(t^{\prime }\right)}^2+t_{sr}\left(t^{\prime }\right))}^2\right)d{t}^{\prime }}}}\\ {\displaystyle \frac{\left({\int }_{t_0}^t(-t_s\left(t^{\prime }\right))d{t}^{\prime }\right)sh\left(\frac{\sqrt{-{\int }_{t_0}^t{\left|t_s\left(t^{\prime }\right)\right|}^{2}d{t}^{\prime }}}{\mathit{\hslash }}\right)}{\sqrt{-{\int }_{t_0}^t\left({(t_{si}{\left(t^{\prime }\right)}^2+t_{sr}\left(t^{\prime }\right))}^2\right)d{t}^{\prime }}}}& ch\left({\displaystyle \frac{\sqrt{-{\int }_{t_0}^t\left(\right|t_s\left(t^{\prime }\right){|}^2)d{t}^{\prime }}}{\mathit{\hslash }}}\right)\end{array}\right),\\ \hfill \end{array}$$

(9)

Now it is time to move to the most general situation of $E_{p1}\ne E_{p2}$, $t_{sr},t_{si}\ne 0$. We have four elements of the evolution matrix given as

$$\begin{array}{c}\hfill \widehat{U}(t,t_0)=e^{{\displaystyle \frac{1}{i\mathit{\hslash }}}{\int }_{t_0}^t\widehat{H}\left(t_1\right)d{t}_1}=\\ \hfill \left(\begin{array}{cc}U{(t,t_0)}_{1,1}& U{(t,t_0)}_{1,2}\\ U{(t,t_0)}_{2,1}=U{(t,t_0)}_{1,2}^{*}& U{(t,t_0)}_{2,2}\end{array}\right).\end{array}$$

$$\begin{array}{c}\hfill U{(t,t_0)}_{1,1}=\\ \hfill {\displaystyle \frac{\exp \left(-\frac{\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}+i\mathit{\hslash }{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)+E_{p2}\left(t^{\prime }\right))}{2{\mathit{\hslash }}^2}\right)}{2\mathit{\hslash }\left({\left({\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right))\right)}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}}\times \\ \hfill \times [-i\left({\int }_{t_0}^{t}d{t}^{\prime }{E}_{p1}\left(t^{\prime }\right)\right)\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}+\\ \hfill +\mathit{\hslash }\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)\times \\ \hfill e^{{\displaystyle \frac{\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}}{{\mathit{\hslash }}^2}}}+\\ \hfill +\left(\left({\left({\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right))\right)}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)\right)+\\ \hfill +i\left({\int }_{t_0}^{t}d{t}^{\prime }{E}_{p1}\left(t^{\prime }\right)\right)e^{{\displaystyle \frac{\sqrt{-h^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}}{{\mathit{\hslash }}^2}}}\times \\ \hfill \sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}\\ \hfill -i\left({\int }_{t_0}^{t}d{t}^{\prime }{E}_{p2}\left(t^{\prime }\right)\right)e^{{\displaystyle \frac{\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}}{{\mathit{\hslash }}^2}}}\times \\ \hfill \sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}+\\ \hfill +i\left({\int }_{t_0}^{t}d{t}^{\prime }{E}_{p2}\left(t^{\prime }\right)\right)\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}].\end{array}$$

(10)

$$\begin{array}{c}\hfill U{(t,t_0)}_{1,2}=\\ \hfill 2\mathit{\hslash }\left({\int }_{t_0}^{t}d{t}^{\prime }(t_{si}\left(t^{\prime }\right)-i{t}_{sr}\left(t^{\prime }\right))\right)e^{-{\displaystyle \frac{i{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)+E_{p2}\left(t^{\prime }\right))}{2\mathit{\hslash }}}}\times \\ \hfill \sinh \left({\displaystyle \frac{\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}}{2h^2}}\right)\times \\ \hfill {\displaystyle \frac{1}{\sqrt{-{\mathit{\hslash }}^2\left(|{\int }_{t_0}^{t}d{t}^{\prime }(E_{p1}\left(t^{\prime }\right)-E_{p2}\left(t^{\prime }\right)){|}^2+4\left(|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{si}\left(t^{\prime }\right){|}^2+{\left|{\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr}\left(t^{\prime }\right)\right|}^2\right)\right)}}}=\\ \hfill =U{(t,t_0)}_{2,1}^{*}.\end{array}$$

Figure 5. Scheme of electrostatically controlled qubit made from two neighboring quantum dots and referring to Figure 3 showing implementation of the single qubit.

Figure 5. Scheme of electrostatically controlled qubit made from two neighboring quantum dots and referring to Figure 3 showing implementation of the single qubit.

4. Description of Two-Qubit Interaction in the General Static Case

The main scientific step of the work is the transition from one position-based qubit to two electrostatically interacting qubits. In case of many interacting particles one can use mapping between quantum mechanics and statistical physics [34]. A two-qubit model is necessary to describe SWAP-like dynamics, Coulomb-driven correlation and anticorrelation, and the generation of entanglement. The motivation of this section is therefore to construct the smallest Hamiltonian that retains the relevant device ingredients while remaining sufficiently compact for analytical treatment. Two double quantum dots, labeled as upper and lower qubits, are represented in a four-state product basis. The Hamiltonian contains the local confinement energies of both qubits, their individual tunneling amplitudes, and four geometry-dependent Coulomb energies associated with the possible electron-position configurations. The matrix is also decomposed into the tensor products of Pauli operators. Analytical expressions for unitary evolution are developed for selected parameter restrictions, while eigenenergy spectra are studied as functions of distance and hopping. The symmetric Q-SWAP subsection specializes the model to equal on-site potentials and equal tunnel couplings, allowing explicit discussion of entangled eigenstates, degeneracy, correlation structure, and perturbative transitions between energy levels.

The physical content of the four-state model is the competition among three mechanisms: localization by electrostatic bias, coherent redistribution by tunneling, and configuration-dependent energy shifts produced by Coulomb repulsion. Coulomb repulsion alone favors particular charge configurations, but a quantum gate requires coherent superpositions and therefore depends essentially on the tunneling terms. The symmetric case provides transparent analytical insight, yet the general Hamiltonian is more important for realistic devices because interdot distances, local biases, and couplings are not perfectly identical. The manuscript also connects entanglement to measurable charge correlations and to the von Neumann entropy, which gives the analytical formalism a possible experimental interpretation through charge-state readout.

This section establishes the general two-qubit electrostatic Hamiltonian and shows how SWAP-related behavior emerges from the combined action of tunneling and Coulomb interaction. The symmetric solution serves as a useful reference point, while the more general parameterization supplies design variables for nonideal geometries. It also defines the correlation and entanglement quantities used in later sections to characterize gate operation.

We consider the most minimalist model of two electrostatically interacting position-based qubits, which are double quantum dots A (with nodes 1 and 2 and named U-upper qubit) and B (with nodes 1′ and 2′ and named L-lower qubit), with local confinement potentials as given in Figure 6 and Figure 7. By introducing notation $|1,0{⟩}_x=|1⟩,|0,1{⟩}_x=|2⟩,|1^{\prime },0^{\prime }{⟩}_x=|1^{\prime }⟩,|0^{\prime },1^{\prime }{⟩}_x=|1^{\prime }⟩$ the minimalistic Hamiltonian of the system of electrostatically interacting position-based qubits can be written as

$$\begin{array}{c}\hfill \widehat{H}=\left(t_{s21}\left(t\right)\right|2⟩⟨1|+t_{s12}\left(t\right)|1⟩⟨2\left|\right){\widehat{I}}_b)+({\widehat{I}}_a\left(t_{s{2}^{\prime }{1}^{\prime }}\left(t\right)\right|2^{\prime }⟩⟨1^{\prime }|+t_{s{1}^{\prime }{2}^{\prime }}\left(t\right)|2^{\prime }⟩⟨1^{\prime }\left|\right)+\\ \hfill +\left(E_{p1}\left(t\right)\right|1⟩⟨1|+E_{p2}\left(t\right)|2⟩⟨2\left|\right){\widehat{I}}_b+{\widehat{I}}_a\left(E_{p{1}^{\prime }}\left(t\right)\right|1^{\prime }⟩⟨1^{\prime }|+E_{p{2}^{\prime }}\left(t\right)|2^{\prime }⟩⟨2^{\prime }\left|\right)+\\ \hfill +{\displaystyle \frac{q^2}{d_{{11}^{\prime }}}}|1,1^{\prime }⟩⟨1,1^{\prime }|+{\displaystyle \frac{q^2}{d_{{22}^{\prime }}}}|2,2^{\prime }⟩⟨2,2^{\prime }|+{\displaystyle \frac{q^2}{d_{{12}^{\prime }}}}|1,2^{\prime }⟩⟨1,2^{\prime }|+{\displaystyle \frac{q^2}{d_{{21}^{\prime }}}}|2,1^{\prime }⟩⟨2,1^{\prime }|=\\ \hfill H_{kinetic1}+H_{pot1}+H_{kinetic2}+H_{pot2}+H_{A-B}\end{array}$$

(11)

described by parameters $E_{p1}\left(t\right)$, $E_{p2}\left(t\right)$, $E_{p{1}^{\prime }}\left(t\right)$, $E_{p{2}^{\prime }}\left(t\right)$, $t_{s12}\left(t\right)$, $t_{s{1}^{\prime }{2}^{\prime }}\left(t\right)$ and distances between nodes k and l’: $d_{{11}^{\prime }}$, $d_{{22}^{\prime }}$, $d_{{21}^{\prime }}$, $d_{{12}^{\prime }}$. In such a case the q-state of the system is given as

$$\begin{array}{c}\hfill |\psi {⟩}_t={\gamma }_1\left(t\right)|1,0{⟩}_U|1,0{⟩}_L+{\gamma }_2\left(t\right)|1,0{⟩}_U|0,1{⟩}_L+{\gamma }_3\left(t\right)|0,1{⟩}_U|1,0{⟩}_L+{\gamma }_4\left(t\right)|0,1{⟩}_U|0,1{⟩}_L,\end{array}$$

(12)

where the normalization condition gives $|{\gamma }_1{\left(t\right)|}^2+\dots {\left|{\gamma }_4\left(t\right)\right|}^2$. The probability of finding an electron in the upper system at node 1 is determined by the action of the projection ${\widehat{P}}_{1U}=⟨1,0{|}_U⟨1,0{|}_L+⟨1,0{|}_U⟨0,1{|}_L$ on q-state ${\widehat{P}}_{1U}|\psi ⟩$, so it gives probability amplitude $|{\gamma }_1\left(t\right)+{\gamma }_3{\left(t\right)|}^2$. On the other hand the probability of finding an electron from qubit A (U) at node 2 and an electron from qubit B(L) at node 1 is obtained by the projection ${\widehat{P}}_{2U,1L}=⟨0,1{|}_U⟨1,0{|}_L$ acting on the q-state giving $(⟨0,1{|}_U⟨1,0{|}_L)|\psi ⟩$ that gives probability amplitude $|{\gamma }_3{\left(t\right)|}^2$. Referring to Figure 6 we set distances between nodes as $d_{{11}^{\prime }}=d_{{22}^{\prime }}=d_1$, $d_{{12}^{\prime }}=d_{{21}^{\prime }}=\sqrt{{(a+b)}^2+d_1^2}$ and assume the Coulomb electrostatic energy to be of the form $E_c(k,l)={\displaystyle \frac{q^2}{d_{k{l}^{\prime }}}}$, and hence we obtain the matrix Hamiltonian given as $\widehat{H}\left(t\right)=$

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}{E}_{p1}\left(t\right)+E_{p{1}^{\prime }}\left(t\right)+{\displaystyle \frac{q^2}{d_1}}& t_{s{1}^{\prime }{2}^{\prime }}\left(t\right)& t_{s12}\left(t\right)& 0\\ t_{s{1}^{\prime }{2}^{\prime }}{\left(t\right)}^{*}& E_{p1}\left(t\right)+E_{p{2}^{\prime }}\left(t\right)+{\displaystyle \frac{q^2}{\sqrt{{\left(d1\right)}^2+{(b+a)}^2}}}& 0& t_{s12}\left(t\right)\\ t_{s12}^{*}\left(t\right)& 0& E_{p2}\left(t\right)+E_{p{1}^{\prime }}\left(t\right)+{\displaystyle \frac{q^2}{\sqrt{{\left(d1\right)}^2+{(b+a)}^2}}}& t_{s{1}^{\prime }{2}^{\prime }}\left(t\right)\\ 0& t_{s12}^{*}\left(t\right)& t_{s{1}^{\prime }{2}^{\prime }}{\left(t\right)}^{*}& E_{p2}\left(t\right)+E_{p{2}^{\prime }}\left(t\right)+{\displaystyle \frac{q^2}{d1}}\end{array}\right)\end{array}$$

We can introduce notation $E_{c1}={\displaystyle \frac{q^2}{d_1}}$ and $E_{c2}={\displaystyle \frac{q^2}{\sqrt{d_1^2+{(b+a)}^2}}}$. In the most general case of 2-qubit electrostatic interaction, one qubit has four different Coulomb terms on matrix diagonal $E_{c1}={\displaystyle \frac{q^2}{d_{{11}^{\prime }}}}$, $E_{c2}{\displaystyle \frac{q^2}{d_{{12}^{\prime }}}}$, $E_{c3}={\displaystyle \frac{q^2}{d_{{21}^{\prime }}}}$, $E_{c4}={\displaystyle \frac{q^2}{d_{{22}^{\prime }}}}$ and $|\psi ,t⟩=\widehat{U}(t,t_0)|\psi ,t_0⟩$. We introduce $q_1=E_{p1}\left(t\right)+E_{p{1}^{\prime }}\left(t\right)+E_{c{11}^{\prime }}$, $q_2=E_{p1}\left(t\right)+E_{p{2}^{\prime }}\left(t\right)+E_{c{12}^{\prime }}$, $q_3=E_{p2}\left(t\right)+E_{p{1}^{\prime }}\left(t\right)+E_{c{21}^{\prime }}$, $q_4=E_{p2}\left(t\right)+E_{p{2}^{\prime }}\left(t\right)+E_{c{22}^{\prime }}$ and in such a case by using Formula (8) one can decompose the two-particle Hamiltonian as

$$\begin{array}{c}\hfill \widehat{H}=[{\displaystyle \frac{(q_1+q_2+q_3+q_4)}{4}}{\sigma }_0\times {\sigma }_0+{\displaystyle \frac{(q_1-q_2+q_3-q_4)}{4}}{\sigma }_0\times {\sigma }_3+\\ \hfill {\displaystyle \frac{(q_1+q_2-q_3-q_4)}{4}}{\sigma }_3\times {\sigma }_0+{\displaystyle \frac{(q_1-q_2-q_3+q_4)}{4}}{\sigma }_3\times {\sigma }_3+\\ \hfill +t_{sr1}\left(t\right){\sigma }_0\times {\sigma }_1-t_{si1}\left(t\right){\sigma }_0\times {\sigma }_2+t_{sr2}\left(t\right){\sigma }_1\times {\sigma }_0\\ \hfill -t_{si2}\left(t\right){\sigma }_2\times {\sigma }_0.\end{array}$$

(13)

A very similar procedure is for the case of three or N interacting particles concerning the tensor product of three or N Pauli matrices. In order to simplify the representation of the unitary matrix describing the physical system of the evolution of the two particles with time it is helpful to define $Q_1\left(t\right)={\int }_{t_0}^t(E_{p1}\left(t^{\prime }\right)+E_{p{1}^{\prime }}\left(t^{\prime }\right)+E_{c{11}^{\prime }})d{t}^{\prime }$, $Q_2\left(t\right)={\int }_{t_0}^t(E_{p1}\left(t^{\prime }\right)+E_{p{2}^{\prime }}\left(t^{\prime }\right)+E_{c{12}^{\prime }})d{t}^{\prime }$, $Q_3\left(t\right)={\int }_{t_0}^t(E_{p2}\left(t^{\prime }\right)+E_{p{1}^{\prime }}\left(t^{\prime }\right)+E_{c{21}^{\prime }})d{t}^{\prime }$, $Q_4\left(t\right)={\int }_{t_0}^t(E_{p2}\left(t^{\prime }\right)+E_{p{2}^{\prime }}\left(t^{\prime }\right)+E_{c{22}^{\prime }})d{t}^{\prime }$ and $TR1\left(t\right)={\int }_{t_0}^{t}d{t}^{\prime }{t}_{s1r}\left(t^{\prime }\right)$, $TI1\left(t\right)={\int }_{t_0}^{t}d{t}^{\prime }{t}_{s1i}\left(t^{\prime }\right)$. We consider the situation where there is no hopping between q-wells $t_{s2}=0$, so the second particle is localized among two quantum wells, and the first particle can move freely among 2 q-wells. We obtain the following unitary matrix evolution with time with $\widehat{U}{(t,t_0)}_{1,2}=\widehat{U}{(t,t_0)}_{1,4}=0=\widehat{U}{(t,t_0)}_{2,3}={\widehat{U}}_{3,4}$ and

$$\begin{array}{c}\hfill \widehat{U}{(t,t_0)}_{1,1}={\displaystyle \frac{1}{2\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}}\times [Q_1\left(t\right)\left(-e^{i\mathit{\hslash }\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}\right)\\ \hfill +\left(\sqrt{|Q_1\left(t\right)-Q_3{\left(t\right)|}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}+Q_3\left(t\right)\right)\times \left(-e^{i\mathit{\hslash }\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}\right)+\\ \hfill \sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}+(Q_1\left(t\right)-Q_3\left(t\right)))e^{-{\displaystyle \frac{1}{2}}i\mathit{\hslash }\left(\sqrt{|{\int }_{t_0}^{t}d{t}^{\prime }(q_1\left(t^{\prime }\right)-q_3\left(t^{\prime }\right)){|}^2+4\left(t_{s1r}^2+t_{si1}^2\right)}+(Q_1\left(t\right)+Q_3\left(t\right))\right)}]\end{array}$$

(14)

$$\begin{array}{c}\hfill \widehat{U}{(t,t_0)}_{1,3}={\displaystyle \frac{2(T{I}_1\left(t\right)-iT{R}_1\left(t\right))e^{-\frac{1}{2}(Q_1\left(t\right)+Q_3\left(t\right))i\mathit{\hslash }}\sin \left(\frac{1}{2}\mathit{\hslash }\sqrt{|Q_1\left(t\right)-Q_3{\left(t\right)|}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}\right)}{\sqrt{|Q_1\left(t\right)-Q_3{\left(t\right)|}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}}},\end{array}$$

(15)

$$\begin{array}{c}\hfill \widehat{U}{(t,t_0)}_{2,2}=[e^{\left({\displaystyle \frac{1}{2}}i\mathit{\hslash }\left(\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}-(Q_2\left(t\right)+Q_4\left(t\right))\right)\right)}\times \\ \hfill \times {\displaystyle \frac{\left(\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}-Q_2\left(t\right)+Q_4\left(t\right)\right)}{2\sqrt{{(Q_2-Q_4)}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}}}\\ \hfill -e^{\left({\displaystyle \frac{1}{2}}i\mathit{\hslash }\left(-\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}-(Q_2\left(t\right)+Q_4\left(t\right))\right)\right)}\times \\ \hfill \times {\displaystyle \frac{\left(-\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}-Q_2\left(t\right)+Q_4\left(t\right)\right)}{2\sqrt{{(Q_2-Q_4)}^2+4(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2)}}},\end{array}$$

(16)

$$\begin{array}{c}\hfill \widehat{U}{(t,t_0)}_{3,3}={\displaystyle \frac{\exp \left(-\frac{1}{2}i\mathit{\hslash }\left(\sqrt{{(Q_1{\left(t\right)}^2-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}+Q_1\left(t\right)+Q_3\left(t\right)\right)\right)}{2\sqrt{{(Q_1{\left(t\right)}^2-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}}\times \\ \hfill [Q_1\left(t\right)\left(-1+e^{i\mathit{\hslash }\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}\right)+\\ \hfill \left(\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}-\mathrm{q}3\right)e^{i\mathit{\hslash }\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}+\\ \hfill +\sqrt{{(Q_1\left(t\right)-Q_3\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}+Q_3\left(t\right)],\end{array}$$

(17)

$$\begin{array}{c}\hfill \widehat{U}{(t,t_0)}_{4,4}={\displaystyle \frac{\exp \left(-\frac{1}{2}i\mathit{\hslash }\left(\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}+Q_2\left(t\right)+Q_4\left(t\right)\right)\right)}{2\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}}\times \\ \hfill \times [Q_2\left(t\right)\left(-1+e^{i\mathit{\hslash }\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}\right)+\\ \hfill +\left(\sqrt{{(Q_2\left(t\right)-Q_4{\left(t\right)}^2)}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}-Q_4\left(t\right)\right)e^{i\mathit{\hslash }\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}+\\ \hfill \sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}+Q_4\left(t\right)],\end{array}$$

$$\begin{array}{c}\hfill \widehat{U}{(t,t_0)}_{2,4}={\displaystyle \frac{2(T{I}_1\left(t\right)-iT{R}_1\left(t\right))e^{-\frac{1}{2}i\mathit{\hslash }(Q_2\left(t\right)+Q_4\left(t\right))}\sin \left(\frac{1}{2}\mathit{\hslash }\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}\right)}{\sqrt{{(Q_2\left(t\right)-Q_4\left(t\right))}^2+4\left(T{R}_1{\left(t\right)}^2+T{I}_1{\left(t\right)}^2\right)}}}.\end{array}$$

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Figure 6. Scheme of electrostatic quantum gate as two interacting qubits ([23,32,33,35,36]) with geometric parametrization of generalized electrostatic quantum gate.

Figure 6. Scheme of electrostatic quantum gate as two interacting qubits ([23,32,33,35,36]) with geometric parametrization of generalized electrostatic quantum gate.

The example of function dependence of eigenenergy spectra of two electrostatically interacting qubits on distance is given by Figure 8.

An important observation is that any element of matrix $\widehat{H}\left(t^{\prime }\right)$ for $t^{\prime }\in (t_0,t)$ denoted as $H_{k,l}\left(t^{\prime }\right)$ is transferred to element ${\widehat{U}}_{k,l}(t,t_0)=e^{{\displaystyle \frac{1}{\mathit{\hslash }i}}{\int }_{t_0}^{t}d{t}^{\prime }\left(H_{k,l}\left(t^{\prime }\right)\right)}$ of matrix $\widehat{U}(t,t_0)$. We can easily generalize the presented reasoning for the system of N electrostatically coupled electrons confined by some local potentials. However we need to know the position-dependent Hamiltonian eigenstate at the initial time $t_0$. In case $N>2$ finding such an eigenstate is the numerical problem since analytical solutions for roots of polynomials of one variable for a higher order than 4 do not exist. Using the numerical eigenstate at time instance $t_0$ we can compute the system quantum dynamics in an analytical way. This gives us a strong and relatively simple mathematical tool allowing full determination of the quantum dynamical state at any instance of time. The act of measurement on the position-based qubit is represented by the operator $P_{Left}=|1,0{⟩}_{E_1,E_2}⟨1,0{|}_{E_1,E_2}$ and $P_{Right}=|0,1{⟩}_{E_1,E_2}⟨0,1{|}_{E_1,E_2}$.

Figure 7. From (left) to (right): anticorrelation principle in the Q-SWAP gate; scheme of the renormalization procedure in a system of coupled qubits, following the procedure given in the work by K. Pomorski entitled “Analytic view on N-body interaction in electrostatic quantum gates and decoherence effects in the tight-binding model” [36]; illustration of the anticorrelation principle in a classical or quantum SWAP gate; correlation principle in a chain of coupled dots aligned along one line or along parallel lines; correlation principle for a system of qubits in perpendicular alignment. This last system is also equivalent to the generalized electrostatic quantum SWAP gate from Figure 6.

Figure 7. From (left) to (right): anticorrelation principle in the Q-SWAP gate; scheme of the renormalization procedure in a system of coupled qubits, following the procedure given in the work by K. Pomorski entitled “Analytic view on N-body interaction in electrostatic quantum gates and decoherence effects in the tight-binding model” [36]; illustration of the anticorrelation principle in a classical or quantum SWAP gate; correlation principle in a chain of coupled dots aligned along one line or along parallel lines; correlation principle for a system of qubits in perpendicular alignment. This last system is also equivalent to the generalized electrostatic quantum SWAP gate from Figure 6.

Figure 8. Case of metal–insulator-like transition, interpreted as a transition from delocalized to more localized electronic states caused by decreasing the hopping term value, as seen from the energy spectra of the quantum SWAP gate as a function of the distance d for the case of two electrostatically interacting qubits shown in Figure 6. No level crossings are observed, even after careful magnification of the plots.

Figure 8. Case of metal–insulator-like transition, interpreted as a transition from delocalized to more localized electronic states caused by decreasing the hopping term value, as seen from the energy spectra of the quantum SWAP gate as a function of the distance d for the case of two electrostatically interacting qubits shown in Figure 6. No level crossings are observed, even after careful magnification of the plots.

Simplified Picture of a Symmetric Q-SWAP Gate

Now we need to find a system of four eigenvalues and eigenstates (four orthogonal 4-dimensional vectors), so we are dealing with a matrix eigenvalue problem), a standard topic in linear algebra. Let us assume that two double-quantum-dot systems are symmetric and biased by the same voltages generating potential bottoms $V_s$ so we have $E_{p1}=E_{p2}=E_{p{1}^{\prime }}=E_{p{2}^{\prime }}=E_p=V_s$ and $t_{s12}=t_{s{1}^{\prime }{2}^{\prime }}=t_s$. Denoting $E_c(1,1^{\prime })=E_c(2,2^{\prime })=E_{c1}$ and $E_c(1,2^{\prime })=E_c(2,1^{\prime })=E_{c2}$ we obtain four orthogonal Hamiltonian eigenvectors

$$\begin{array}{c}\hfill |E_1⟩=\left(\begin{array}{c}-1\\ 0\\ 0\\ +1\end{array}\right)=\\ \hfill =-|1,0{⟩}_U|1,0{⟩}_L+|0,1{⟩}_U|0,1{⟩}_L\\ \hfill \ne \left(a_1\right|1,0{⟩}_U+a_2|0,1{⟩}_U)\left(a_3\right|1,0{⟩}_U+a_4|0,1{⟩}_U),\end{array}$$

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$$\begin{array}{c}\hfill |E_2⟩=\left(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right)=\\ \hfill =|1,0{⟩}_U|0,1{⟩}_L-|0,1{⟩}_U|1,0{⟩}_L\ne \left(a_1\right|1,0{⟩}_U+a_2|0,1{⟩}_U)\left(a_3\right|1,0{⟩}_U+a_4|0,1{⟩}_U).\end{array}$$

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We observe that the first two energetic states are degenerated, so the same quantum state corresponds to two different eigenenergies $E_1$ and $E_2$. This degeneracy is non-present if we come back to the Schrödinger picture and observe that localized energy and hopping terms for one particle are dependent on the presence of another particle, which leads to renormalization of wavevectors. The situation is depicted in Figure 7. The degeneracy of the eigenstates is lifted if we set $E_{p1}\left(\right|\psi \left(1^{\prime }\right){|}^2,|\psi \left(2^{\prime }\right){|}^2),E_{p2}\left(\right|\psi \left(1^{\prime }\right){|}^2,|\psi \left(1^{\prime }\right){|}^2)$, $E_{p{1}^{\prime }}{\left(\right|\psi \left(1\right)|}^2{,\left|\psi \left(2\right)\right|}^2),E_{p{2}^{\prime }}{\left(\right|\psi \left(1\right)|}^2{,\left|\psi \left(1\right)\right|}^2)$ and $t_{1\to 2}\left(\right|\psi \left(1^{\prime }\right){|}^2,|\psi \left(2^{\prime }\right){|}^2)$,

$t_{1\to 2}{\left(\right|\psi \left(1\right)|}^2{,\left|\psi \left(2\right)\right|}^2)$.

The same argument for another wavevector is given below.

$$\begin{array}{c}\hfill |E_{3\left(4\right)⟩}=\left(\begin{array}{c}1\\ \mp {\displaystyle \frac{4t_s}{\pm (-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\\ \mp {\displaystyle \frac{4t_s}{\pm (-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\\ 1\end{array}\right)=\\ \hfill =|1,0{⟩}_U|1,0{⟩}_L+|0,1{⟩}_U|0,1{⟩}_L+c\left(\right|1,0{⟩}_U|0,1{⟩}_L+|0,1{⟩}_U|1,0{⟩}_L)=\\ \hfill =\left(\right|1,0{⟩}_U+|0,1{⟩}_U)\left(\right|1,0{⟩}_L+|0,1{⟩}_L)+(c-1)\left(\right|1,0{⟩}_U|0,1{⟩}_L+|0,1{⟩}_U|1,0{⟩}_L)\\ \hfill \ne \left(a_1\right|1,0{⟩}_U+a_2|0,1{⟩}_U)\left(a_3\right|1,0{⟩}_U+a_4|0,1{⟩}_U),\end{array}$$

(21)

where c = $\mp {\displaystyle \frac{4t_s}{\pm (-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}$. The first two $|E_1⟩$ and $|E_2⟩$ energy eigenstates are always entangled, while $|E_3⟩$ and $|E_4⟩$ eigenenergies are only partially entangled if $\mp {\displaystyle \frac{4t_s}{\pm (-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\ne 1$. If $c=1=\mp {\displaystyle \frac{4t_s}{\pm (-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}$, the last two energy eigenstates are not entangled. The situation of c=1 takes place when $E_{c1}=E_{c2}$, when two qubits are infinitely far away and thus electrostatically decoupled. The situation of c=0 is interesting because it means that $|E_3⟩$ and $|E_4⟩$ are maximally entangled, and it occurs when $t_s=0$, so when two electrons are maximally localized in each of the qubits so there is no hopping between the left and right wells.

The obtained eigenenergy states correspond to four eigenenergies

$$\begin{array}{c}\hfill E_1=E_{c1}+2V_s,E_2=E_{c2}+2V_s,E_1>E_2\\ \hfill E_3={\displaystyle \frac{1}{2}}((E_{c1}+E_{c2})-\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}+4V_s)=\\ \hfill ={\displaystyle \frac{1}{2}}(\left(q^2({\displaystyle \frac{1}{d_1}}+{\displaystyle \frac{1}{\sqrt{d_1^2+{(a+b)}^2}}})\right)-\sqrt{{\left(q^2({\displaystyle \frac{1}{d_1}}-{\displaystyle \frac{1}{\sqrt{d_1^2+{(a+b)}^2}}})\right)}^2+16t_s^2}+4V_s),\\ \hfill E_4={\displaystyle \frac{1}{2}}((E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}+4V_s)=\\ \hfill {\displaystyle \frac{1}{2}}((q^2({\displaystyle \frac{1}{d_1}}+{\displaystyle \frac{1}{\sqrt{d_1^2+{(a+b)}^2}}})+\sqrt{{\left(q^2({\displaystyle \frac{1}{d_1}}-{\displaystyle \frac{1}{\sqrt{d_1^2+{(a+b)}^2}}})\right)}^2+16t_s^2}+4V_s),E_4>E_3.\end{array}$$

(22)

We also notice that the eigenenergy states $|E_1⟩$, $|E_2⟩$, $|E_3⟩$, and $|E_4⟩$ do not have classical counterparts since an upper electron exists at both positions 1 and 2 and a lower electron exists at both positions at the same time. We observe that when the distance between two systems of double quantum dots goes to infinity the energy difference between the quantum state corresponding to $|E_3⟩$ and $|E_4⟩$ goes to zero. This makes those two entangled states degenerated.

Four normalized eigenvectors of two interacting qubits in the SWAP Q-Gate configuration are of the following form

$$\begin{array}{c}\hfill |E_1{⟩}_n={\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{t_{sr1}-t_{sr2}}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}-E_{c1}+E_{c2}}\right)\right)}^2+2}}}\left(\begin{array}{c}-1,\\ -{\displaystyle \frac{2(t_{sr1}-t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}-E_{c1}+E_{c2}}},\\ {\displaystyle \frac{2(t_{sr1}-t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}-E_{c1}+E_{c2}}}\\ 1\end{array}\right)=\\ \hfill {\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{t_{sr1}-t_{sr2}}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}-E_{c1}+E_{c2}}\right)\right)}^2+2}}}|E_1⟩\end{array}$$

(23)

$$\begin{array}{c}\hfill |E_2{⟩}_n=-{\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{t_{sr1}-t_{sr2}}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}+E_{c1}-E_{c2}}\right)\right)}^2+2}}}\left(\begin{array}{c}-1\\ {\displaystyle \frac{2(t_{sr1}-t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}+E_{c1}-E_{c2}}}\\ -{\displaystyle \frac{2(t_{sr1}-t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}+E_{c1}-E_{c2}}}\\ ,1\end{array}\right)=\\ \hfill =-{\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{t_{sr1}-t_{sr2}}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}-t_{sr2})}^2}+E_{c1}-E_{c2}}\right)\right)}^2+2}}}|E_2⟩\end{array}$$

(24)

$$\begin{array}{c}\hfill |E_3{⟩}_n={\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{\mathrm{tsr}1+\mathrm{tsr}2}{\sqrt{{(\mathrm{Ec}1-\mathrm{Ec}2)}^2+4{(\mathrm{tsr}1+\mathrm{tsr}2)}^2}-\mathrm{Ec}1+\mathrm{Ec}2}\right)\right)}^2+2}}}\left(\begin{array}{c}1,\\ -{\displaystyle \frac{2(t_{sr1}+t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}+t_{sr2})}^2}-E_{c1}+E_{c2}}},\\ -{\displaystyle \frac{2(t_{sr1}+t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}+t_{sr2})}^2}-E_{c1}+E_{c2}}},\\ 1\end{array}\right)=\\ \hfill ={\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{\mathrm{tsr}1+\mathrm{tsr}2}{\sqrt{{(\mathrm{Ec}1-\mathrm{Ec}2)}^2+4{(\mathrm{tsr}1+\mathrm{tsr}2)}^2}-\mathrm{Ec}1+\mathrm{Ec}2}\right)\right)}^2+2}}}|E_3⟩\end{array}$$

(25)

$$\begin{array}{c}\hfill |E_4{⟩}_n={\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{t_{sr1}+t_{sr2}}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}+t_{sr2})}^2}-E_{c2}+E_{c1}}\right)\right)}^2+2}}}\left(\begin{array}{c}1,\\ {\displaystyle \frac{2(t_{sr1}+t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}+t_{sr2})}^2}+E_{c1}-E_{c2}}},\\ {\displaystyle \frac{2(t_{sr1}+t_{sr2})}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}+t_{sr2})}^2}+E_{c1}-E_{c2}}},\\ 1\end{array}\right)=\\ \hfill {\displaystyle \frac{1}{\sqrt{{\left(8\left(\frac{t_{sr1}+t_{sr2}}{\sqrt{{(E_{c1}-E_{c2})}^2+4{(t_{sr1}+t_{sr2})}^2}-E_{c2}+E_{c1}}\right)\right)}^2+2}}}|E_4⟩.\end{array}$$

(26)

We are obtaining simplifications after assuming $t_{sr1}\left(t\right)=t_{sr2}\left(t\right)$, so we obtain

$$|E_1{⟩}_n={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}-1\\ 0\\ 0\\ 1\end{array}\right),|E_2{⟩}_n={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right),$$

(27)

$$\begin{array}{c}\hfill |E_3{⟩}_n=\sqrt{{\displaystyle \frac{4t_s}{(E_{c2}-E_{c1})+8t_s-\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}}\left(\begin{array}{c}1\\ -{\displaystyle \frac{4t_s}{(-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\\ -{\displaystyle \frac{4t_s}{(-E_{c1}+E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\\ 1\end{array}\right),\end{array}$$

(28)

$$\begin{array}{c}\hfill |E_4{⟩}_n=\sqrt{{\displaystyle \frac{4t_s}{(E_{c1}-E_{c2})+8t_s-\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}}\left(\begin{array}{c}1\\ {\displaystyle \frac{4t_s}{(E_{c1}-E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\\ {\displaystyle \frac{4t_s}{(E_{c1}-E_{c2})+\sqrt{{(E_{c1}-E_{c2})}^2+16t_s^2}}}\\ 1\end{array}\right).\end{array}$$

(29)

It is worth mentioning that if we want to bring two electrostatic qubits to the entangled state we need to cool down (or heat up) the system of interacting qubits to the energy $E_1$ (or to energy $E_2$). Otherwise we might also wish to disentangle two electrostatically interacting qubits. In this way, one of the scenarios is to bring the quantum system either to energy $E_3$ or $E_4$ so only partial entanglement is achieved. The other scenario would be introducing the occupancy of different energetic levels so the net entanglement is reduced. One can use the entanglement witness in quantifying the existence of entanglement. One of the simplest forms of q-state entanglement measurement is von Neumann entanglement entropy as expressed by Formula (A139), which requires the knowledge of q-system density matrix with time. Such matrices can be obtained analytically for the case of two electrostatically interacting qubits.

It is interesting to note the dependence of eigenenergies on distance between interacting qubits in the general case as depicted in Figure 7. Now we are moving toward description the procedure of cooling down or heating up in the Q-SWAP gate. The procedure was discussed previously in the case of a single qubit. Now it is exercised in the case of 2-qubit electrostatic interaction. For the sake of simplicity we change the occupancy of the energy level $E_1$ and energy level level $E_2$ and keep the occupancy of other energy levels unchanged. We can write the $|E_2⟩⟨E_1|$ as   

$$\begin{array}{c}\hfill |E_2{⟩}_n⟨E_1{|}_n={\displaystyle \frac{1}{2}}\left(\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right)\left(\begin{array}{cccc}-1& 0& 0& 1\end{array}\right)=\left(\begin{array}{cccc}-1& 0& 0& +1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ +1& 0& 0& -1\end{array}\right),\\ \hfill |E_1{⟩}_n⟨E_2{|}_n={\displaystyle \frac{1}{2}}\left(\begin{array}{c}-1\\ 0\\ 0\\ 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& -1\end{array}\right)=\left(\begin{array}{cccc}-1& 0& 0& +1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ +1& 0& 0& -1\end{array}\right).\end{array}$$

(30)

We introduce $f_1$ and $f_2$ real-valued functions of small magnitude $f\left(t\right)=f_1\left(t\right)=f_2\left(t\right),\left(\right|f_1|,|f_2|<<(E_1,E_2))$, and we consider the following Hamiltonian with $H_0$ that has part of time-independent and part of time-dependent as

$$\begin{array}{c}\hfill \widehat{H}={\widehat{H}}_0+f_1\left(t\right)|E_2{⟩}_n⟨E_1{|}_n+f_2\left(t\right)|E_1{⟩}_n⟨E_2{|}_n=\\ \hfill =E_1|E_1⟩⟨E_1|+E_2|E_2⟩⟨E_2|+f_1\left(t\right)|E_2{⟩}_n⟨E_1{|}_n+f_2\left(t\right)|E_1{⟩}_n⟨E_2{|}_n=\end{array}$$

$$\begin{array}{c}\hfill =\left(\begin{array}{cccc}2E_p+{\displaystyle \frac{q^2}{d_1}}& t_s& t_s& 0\\ t_s^{*}& 2E_p+{\displaystyle \frac{q^2}{\sqrt{{\left(d1\right)}^2+{(b+a)}^2}}}& 0& t_s\\ t_s^{*}& 0& 2E_p+{\displaystyle \frac{q^2}{\sqrt{{\left(d1\right)}^2+{(b+a)}^2}}}& t_s\\ 0& t_s^{*}& t_s^{*}& 2E_p+{\displaystyle \frac{q^2}{d1}}\end{array}\right)\\ \hfill +{\displaystyle \frac{1}{2}}\left(f_1\left(\begin{array}{cccc}-1& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& -1\end{array}\right)+f_2\left(\begin{array}{cccc}-1& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& -1\end{array}\right)\right)=\\ \hfill =\left(\begin{array}{cccc}2E_p+{\displaystyle \frac{q^2}{d_1}}-f\left(t\right)& t_s& t_s& f\left(t\right)\\ t_s^{*}& 2E_p+{\displaystyle \frac{q^2}{\sqrt{{\left(d1\right)}^2+{(b+a)}^2}}}& 0& t_s\\ t_s^{*}& 0& 2E_p+{\displaystyle \frac{q^2}{\sqrt{{\left(d1\right)}^2+{(b+a)}^2}}}& t_s\\ f\left(t\right)& t_s^{*}& t_s^{*}& 2E_p+{\displaystyle \frac{q^2}{d1}}-f\left(t\right)\end{array}\right)=\\ \hfill =\widehat{H}{\left(t\right)}_{E_1<->E_2,Q-Swap}.\end{array}$$

(31)

Initially we establish the following parameters of the tight-binding model as $t_{s12}=t_{s{1}^{\prime }{2}^{\prime }}$. Changing $t_{s12}$ into $t_{s12}-{\displaystyle \frac{f\left(t\right)}{2}}$ and $t_{s{1}^{\prime }{2}^{\prime }}$ into $t_{s{1}^{\prime }{2}^{\prime }}+{\displaystyle \frac{f\left(t\right)}{2}}$ while keeping other parameters of the tight-binding model unchanged results in the heating up (cooling down) of the q-state of the SWAP gate, so the population of energy levels $E_1$ and $E_2$ is time-dependent, while populations of energy levels $E_3$ and $E_4$ are unchanged. Practically our results mean that we need to keep all our confinement potential bottoms constant while changing the barrier height between neighboring q-dots in each of the position-based qubits. In this way we establish the procedure of perturbative cooling (heating up) of the q-state. A non-perturbative approach is absolutely possible, but it requires full knowledge of time-dependent eigenstates and eigenenergies (solutions of eigenenergies of fourth-order polynomials are very lengthy in the general case), and therefore corresponding expressions are very lengthy. In similar fashion we can heat up or cool down two coupled single-electron lines [33] as in Figure 6 and Figure 7 or any other q-system with N interacting q-bodies that can be represented by the system of N-interacting position-based qubits.

5. Quasiclassical Approach Toward a Tunable SWAP Gate

The fully quantum Hamiltonian gives the most complete low-energy description, but it can obscure the intuitive electrostatic conditions that favor one logical configuration over another. This section introduces a quasiclassical viewpoint to obtain direct design rules for a tunable SWAP gate. The objective is to understand how gate geometry and localizing potentials determine whether correlated or anticorrelated charge configurations are energetically preferred before coherent quantum dynamics is added. The analysis suppresses or strongly reduces the kinetic contribution by considering the limit of small tunneling. The resulting energy is written as a function of the occupation probabilities of the two qubits, the local electrostatic potentials, the interdot distances, and a geometric angle $\alpha$. Symmetric and asymmetric configurations are compared by assigning energies $V_1$ and $V_2$ to different logical charge arrangements and deriving inequalities that determine the energetically favorable states. The subsection on the transition from an anticorrelated symmetric SWAP gate to a correlated gate uses this energy landscape to show how changing geometry and bias can move the system between operating regimes.

The value of the quasiclassical model is not that it replaces the quantum description but that it separates electrostatic design from coherent dynamics. It shows which charge configurations are selected by Coulomb-energy minimization and therefore identifies the regions in parameter space where a quantum tunneling term can act on an already favorable electrostatic landscape. This creates a useful bridge between lithographic geometry, gate voltages, and the qualitative form of the two-qubit interaction. At the same time, because kinetic energy and coherence are reduced in this treatment, the quasiclassical energy minima should be interpreted as design guidance rather than as a complete prediction of entangling-gate fidelity or coherent time evolution.

Section 5 provides a device-oriented tuning picture for the electrostatic SWAP architecture. By expressing the gate energy in terms of occupations, biases, distances, and relative geometry, it identifies conditions that favor correlated or anticorrelated configurations. These conditions complement the quantum model by translating the Hamiltonian into practical electrostatic design rules.

5.1. Quasiclassical Approach Toward a Symmetric SWAP Gate with the Same Localizing Potentials

We assume that one particle is in the field of another particle that is given by integro-differential equations as often used in quantum chemistry. In addition we assume that the kinetic energy of particles A and B (electrons in qubits A and B) is very small, so $t_{s12}\to 0$ and $t_{s{1}^{\prime }{2}^{\prime }}\to 0$; however, still $t_{s12}\ne 0$ and $t_{s{1}^{\prime }{2}^{\prime }}\ne 0$, so after a certain length of time particles have the chance to find a proper configuration. We assume that $p_{A1}$ is the probability of finding particle A at node 1 and that $p_{B{1}^{\prime }}$ is the probability of finding particle B at node 1′. We have an effective Hamiltonian omitting kinetic terms in the form

$$\begin{array}{c}\hfill H(p_{A1},p_{B{1}^{\prime }},E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})=p_{A1}{E}_{p_1}+(1-p_{A1})E_{p_2}+p_{B{1}^{\prime }}{E}_{p_{1^{\prime }}}+(1-p_{B{1}^{\prime }})E_{p_{2^{\prime }}}+\\ \hfill +p_{A1}{p}_{B{1}^{\prime }}{\displaystyle \frac{q^2}{d}}+(1-p_{A1})(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{d}}+p_{A1}(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}+p_{B{1}^{\prime }}(1-p_{A{1}^{\prime }}){\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}.\end{array}$$

(32)

The well-designed classical SWAP and Q-SWAP gates (inverting states 0 to 1 and 1 to 0) have the property that the function $H(p_{A1},p_{B{1}^{\prime }},E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})$ reaches its minima at $(p_{A1}=0,p_{B{1}^{\prime }}=1)$ and at $(p_{A1}=1,p_{B{1}^{\prime }}=0)$ and that $H(1,0,E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})=H(0,1,E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})$. Consequently we assume

• $H(1,1,E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})>H(0,1,E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})$ and
• $H(0,0,E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})>H(0,1,E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})$,
• which implies that logical states that are not allowed have higher energy. Further simplification can be done by setting, for example, $E_{p1}$ and $E_{p{1}^{\prime }}$ to a constant value (as $E_{p1}=cons{t}_1$ and $E_{p{1}^{\prime }}=cons{t}_{1^{\prime }}$), for example, 0 or 1 or any other fixed real number. It is underlined that we can choose the one among four possible combinations $(E_{p1},E_{p{1}^{\prime }})$, $(E_{p2},E_{p{2}^{\prime }})$, $(E_{p1},E_{p{2}^{\prime }})$, $(E_{p2},E_{p{1}^{\prime }})$ whose values need to be fixed. We set $q=1$, $d=1$, $a+b=0.2$, $E_{p2}=E_{p{2}^{\prime }}=1$. We obtain

$$\begin{array}{c}\hfill H(p_{A1},p_{B{1}^{\prime }},E_{p1},1,E_{p{1}^{\prime }},1)=p_{A1}{E}_{p_1}+(1-p_{A1})+p_{B{1}^{\prime }}{E}_{p_{1^{\prime }}}+(1-p_{B{1}^{\prime }})+\\ \hfill +p_{A1}{p}_{B{1}^{\prime }}+(1-p_{A1})(1-p_{B{1}^{\prime }})+p_{A1}(1-p_{B{1}^{\prime }}){\displaystyle \frac{1}{\sqrt{1.01}}}+p_{B{1}^{\prime }}(1-p_{A{1}^{\prime }}){\displaystyle \frac{1}{\sqrt{1.01}}}.\end{array}$$

(33)

Now we need to trace the numerical behavior of four functions

$$\begin{array}{c}\hfill H(1,0,E_{p1},1,E_{p{1}^{\prime }},1)=E_{p_1}+1+{\displaystyle \frac{1}{\sqrt{1.01}}}.\end{array}$$

(34)

$$\begin{array}{c}\hfill H(0,1,E_{p1},1,E_{p{1}^{\prime }},1)=+1+E_{p_{1^{\prime }}}+{\displaystyle \frac{1}{\sqrt{1.01}}}.\end{array}$$

(35)

$$\begin{array}{c}\hfill H(1,1,E_{p1},1,E_{p{1}^{\prime }},1)=E_{p_1}+E_{p_{1^{\prime }}}+1.\end{array}$$

(36)

$$\begin{array}{c}\hfill H(0,0,E_{p1},1,E_{p{1}^{\prime }},1)=+1+1+1=3.\end{array}$$

(37)

Since we impose $H(1,0,E_{p1},1,E_{p{1}^{\prime }},1)=H(0,1,E_{p1},1,E_{p{1}^{\prime }},1)$ we obtain $E_{p{1}^{\prime }}=E_{p1}$. Imposing $H(0,0,E_{p1},1,E_{p{1}^{\prime }},1)=H(1,1,E_{p1},1,E_{p{1}^{\prime }},1)$ we finally obtain $E_{p{1}^{\prime }}=E_{p1}=1$. We observe that $H(1,0,E_{p1},1,E_{p{1}^{\prime }},1)<H(0,0,E_{p1},1,E_{p{1}^{\prime }},1)$.

5.2. Transition from an Anticorrelated Symmetric SWAP Gate to a Correlated SWAP Gate

We have an effective Hamiltonian omitting kinetic terms for any angle $\alpha$ in the form

$$\begin{array}{c}\hfill H(p_{A1},p_{B{1}^{\prime }},E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})=p_{A1}{E}_{p_1}+(1-p_{A1})E_{p_2}+p_{B{1}^{\prime }}{E}_{p_{1^{\prime }}}+(1-p_{B{1}^{\prime }})E_{p_{2^{\prime }}}+\\ \hfill +p_{A1}{p}_{B{1}^{\prime }}{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}+(1-p_{A1})(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(a+b)}^{2}Sin{\left(\alpha \right)}^2}}}+\\ \hfill p_{A1}(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}+p_{B{1}^{\prime }}(1-p_{A{1}^{\prime }}){\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}.\end{array}$$

(38)

The previously considered case was for the angle $\alpha =-\Pi /2$ that corresponds to the symmetric anticorrelated SWAP gate as specified by Formula (32). Let us set $E_{p2}$ and $E_{p{2}^{\prime }}$ to be 1. We obtain the following $H(p_{A1},p_{B{1}^{\prime }},E_{p1},1,E_{p{1}^{\prime }},1)$ values as

$$\begin{array}{c}\hfill H(p_{A1}=0,p_{B{1}^{\prime }}=0,E_{p1},E_{p2}=1,E_{p{1}^{\prime }},E_{p{2}^{\prime }}=1)=1+1+\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(a+b)}^{2}Sin{\left(\alpha \right)}^2}}}=V_1,\end{array}$$

(39)

$$\begin{array}{c}\hfill H(p_{A1}=1,p_{B{1}^{\prime }}=1,E_{p1},E_{p2}=1,E_{p{1}^{\prime }},E_{p{2}^{\prime }}=1)=E_{p_1}+E_{p_{1^{\prime }}}+\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}=V_1,\end{array}$$

(40)

$$\begin{array}{c}\hfill H(p_{A1}=1,p_{B{1}^{\prime }}=0,E_{p1},E_{p2}=1,E_{p{1}^{\prime }},E_{p{2}^{\prime }}=1)=E_{p_1}+1+\\ \hfill {\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}=V_2,\end{array}$$

(41)

$$\begin{array}{c}\hfill H(p_{A1}=0,p_{B{1}^{\prime }}=1,E_{p1},E_{p2}=1,E_{p{1}^{\prime }},E_{p{2}^{\prime }}=1)=1+E_{p_{1^{\prime }}}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}=V_2,\end{array}$$

(42)

with condition $V_2<V_1$ that implies

$$\begin{array}{c}\hfill 1+1+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(a+b)}^{2}Sin{\left(\alpha \right)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}=E_{p_1}+E_{p_{1^{\prime }}},\end{array}$$

(43)

$$\begin{array}{c}\hfill +{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}=E_{p_1}-E_{p_{1^{\prime }}}\end{array}$$

(44)

and we obtain

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(a+b)}^{2}Sin{\left(\alpha \right)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}=E_{p1},\end{array}$$

(45)

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(a+b)}^{2}Sin{\left(\alpha \right)}^2}}}+{\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}\\ \hfill -{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}=E_{p{1}^{\prime }},\end{array}$$

(46)

We can check the designing condition $V_1>V_2$ that implies

$$\begin{array}{c}\hfill H(p_{A1}=1,p_{B{1}^{\prime }}=1,E_{p1},E_{p2}=1,E_{p{1}^{\prime }},E_{p{2}^{\prime }}=1)=E_{p_1}+E_{p_{1^{\prime }}}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}=V_1>,\\ \hfill H(p_{A1}=1,p_{B{1}^{\prime }}=0,E_{p1},E_{p2}=1,E_{p{1}^{\prime }},E_{p{2}^{\prime }}=1)=E_{p_1}+1+{\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right)))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}=V_2,\end{array}$$

(47)

and consequently we obtain

$$\begin{array}{c}\hfill E_{p_{1^{\prime }}}>+1+{\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}\end{array}$$

(48)

which brings the condition

$$\begin{array}{c}\hfill {\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(a+b)}^{2}Sin{\left(\alpha \right)}^2}}}>{\displaystyle \frac{q^2}{\sqrt{{(d+(a+b)Cos\left(\alpha \right)))}^2+{\left((1+Sin\left(\alpha \right))(a+b)\right)}^2}}}.\end{array}$$

(49)

that is equivalent to the condition $Sin{\left(\alpha \right)}^2<{(1+Sin\left(\alpha \right))}^2=1+Si{n}^2\left(\alpha \right)+2Sin\left(\alpha \right)$ or $0<Sin\left(\alpha \right)$ where $\alpha \in (0,\Pi )$. The presented considerations are visualized by Figure 9 and Figure 18, which describes the dependence of the phenomenological electrostatic Hamiltonian on logical values $p_{A1}$ and $p_{B1}$, which enables a construction fuzzy logic SWAP gate or quantum SWAP gate to be implemented in single-electron devices.

Figure 9. Transition from the classical/quantum SWAP gate into the ANTISWAP gate by changing the angle $\alpha$ from $-\Pi$ to $\Pi$. Minima in phenomenological Hamiltonian identifies the preferable logical state (states of ($p_{A1}=1,p_{B1}=0$) and ($p_{A1}=0,p_{B1}=1$) are preferred in the same way) into only one preferred state ($p_{A1}=1,p_{B1}=0$). Contour plots describing qubit–qubit Hamiltonian values in function of probabilities ($p_{A1}$, $p_{B1}$) of SWAP and ANTISWAP gates were magnified.

6. Electrostatic Quantum ANTISWAP Gate

This section explores a complementary operating regime in which the two charge qubits described in tight-binding model are driven toward correlated and later anticorrelated occupancy as depicted in Figure 10. In meantime the quasi-classical SWAP-gate Hamiltonian on the electrostatically controlled polarizing potentials is depicted in the figure. Consequently later the manuscript describes tight-binding regime known as an electrostatic ANTISWAP or repeater-like operation as depicted in figures. Its motivation is to show that the same coupled-dot hardware can support more than one logical response, depending on the chosen geometry and bias conditions. The section starts from the occupation-dependent electrostatic Hamiltonian and selects local potentials such that equal or related charge configurations become energetically favorable. The energies of the relevant logical states are grouped into $V_1$ and $V_2$, and algebraic conditions are derived for the ordering $V_2<V_1$. The analysis relates those inequalities to the interdot distance, the dot separation $a+b$, and the local confinement energies. This produces explicit voltage and geometry conditions under which the correlated operating point is preferred.

The important scientific point is the tunability of the interaction sign at the level of charge-state preference. By changing the electrostatic energy landscape, the device can favor anticorrelated states associated with the SWAP-like configuration or correlated states associated with the ANTISWAP/repeater-like configuration discussed here. This reinforces the manuscript’s broader claim that the logical behavior is encoded not in a single fixed coupling constant but in a combination of geometry, confinement energies, and tunneling. The section is most naturally interpreted as a model of controllable charge-state correlation within the restricted two-level description.

The ANTISWAP analysis demonstrates that electrostatic design can select a correlated operating regime in the same family of double-dot structures used for the SWAP gate. The derived inequalities provide a direct prescription for choosing local potentials and geometry so that the desired logical charge configurations have lower energy than competing states.

We can use the fact that two qubits can be in maximum correlated states (if $p_{A1}=1$ we have $p_{B1}=1-p_{A1}$). Due to repulsive electrostatic interaction we can build the electrostatic quantum ANTISWAP gate so it corresponds to the situation where one qubits is being copied by another qubit. It is underlined that the same considerations apply for the electron-hole system ($q^2=-1$) and hole-hole system ($q^2=1$).

Figure 10. Scheme of electrostatic ANTISWAP gate.

Figure 10. Scheme of electrostatic ANTISWAP gate.

We have

$$\begin{array}{c}\hfill H(p_{A1},p_{B{1}^{\prime }},E_{p1},E_{p2},E_{p{1}^{\prime }},E_{p{2}^{\prime }})=p_{A1}{E}_{p_1}+(1-p_{A1})E_{p_2}+p_{B{1}^{\prime }}{E}_{p_{1^{\prime }}}+(1-p_{B{1}^{\prime }})E_{p_{2^{\prime }}}+\\ \hfill +p_{A1}{p}_{B{1}^{\prime }}{\displaystyle \frac{q^2}{d+a+b}}+(1-p_{A1})(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{d+a+b}}+p_{A1}(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{d+2(a+b)}}+p_{B{1}^{\prime }}(1-p_{A{1}^{\prime }}){\displaystyle \frac{q^2}{d}}.\end{array}$$

(50)

We set $E_{p1}=1,E_{p{2}^{\prime }}=1$ and we have

$$\begin{array}{c}\hfill H(p_{A1},p_{B{1}^{\prime }},1,E_{p2},E_{p{1}^{\prime }},1)=p_{A1}+(1-p_{A1})E_{p_2}+p_{B{1}^{\prime }}{E}_{p_{1^{\prime }}}+(1-p_{B{1}^{\prime }})+\\ \hfill +p_{A1}{p}_{B{1}^{\prime }}{\displaystyle \frac{q^2}{d+a+b}}+(1-p_{A1})(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{d+a+b}}+p_{A1}(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{d+2(a+b)}}+p_{B{1}^{\prime }}(1-p_{A{1}^{\prime }}){\displaystyle \frac{q^2}{d}}=\\ \hfill =p_{A1}+(1-p_{A1})E_{p_2}+p_{B{1}^{\prime }}{E}_{p_{1^{\prime }}}+(1-p_{B{1}^{\prime }})+\\ \hfill +(2p_{A1}{p}_{B{1}^{\prime }}+1-(p_{A1}+p_{B{1}^{\prime }})){\displaystyle \frac{q^2}{d+a+b}}+p_{A1}(1-p_{B{1}^{\prime }}){\displaystyle \frac{q^2}{d+2(a+b)}}+p_{B{1}^{\prime }}(1-p_{A{1}^{\prime }}){\displaystyle \frac{q^2}{d}}.\end{array}$$

(51)

In order to obtain the quantum repeater we have $p_{A1}=p_{B{1}^{\prime }}$, and such a configuration is energetically favorable (minimizes Hamiltonian energy) so we have

$$\begin{array}{c}\hfill H(0,0,1,E_{p2},E_{p{1}^{\prime }},1)=H(1,1,1,E_{p2},E_{p{1}^{\prime }},1)=V_1,\\ \hfill H(1,0,1,E_{p2},E_{p{1}^{\prime }},1)=H(0,1,1,E_{p2},E_{p{1}^{\prime }},1)=V_2,V_2<V_1.\end{array}$$

(52)

We have the condition

$$\begin{array}{c}\hfill H(0,0,1,E_{p2},E_{p{1}^{\prime }},1)=E_{p_2}+1+{\displaystyle \frac{q^2}{d+a+b}}=V_1,\end{array}$$

(53)

$$\begin{array}{c}\hfill H(1,1,1,E_{p2},E_{p{1}^{\prime }},1)=1+E_{p_{1^{\prime }}}+(2+1-2){\displaystyle \frac{q^2}{d+a+b}}=1+E_{p_{1^{\prime }}}+{\displaystyle \frac{q^2}{d+a+b}}=V_1.\end{array}$$

(54)

and it implies $E_{p_{1^{\prime }}}=E_{p_2}=E_p$. We have

$$\begin{array}{c}\hfill H(1,0,1,E_{p2},E_{p{1}^{\prime }},1)=1+1+{\displaystyle \frac{q^2}{d+2(a+b)}}=V_2,\end{array}$$

(55)

$$\begin{array}{c}\hfill H(0,1,1,E_{p2},E_{p{1}^{\prime }},1)=+E_{p_2}+E_{p_{1^{\prime }}}+{\displaystyle \frac{q^2}{d}}=V_2.\end{array}$$

(56)

We have

$$\begin{array}{c}\hfill E_p=1+{\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d}})=E_{p_2}=E_{p_{1^{\prime }}}.\end{array}$$

(57)

and it implies

$$\begin{array}{c}\hfill V_2=2+({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d}})+{\displaystyle \frac{q^2}{d}}=2+{\displaystyle \frac{q^2}{d+2(a+b)}},\end{array}$$

(58)

$$\begin{array}{c}\hfill V_1=2+{\displaystyle \frac{q^2}{d+a+b}}+{\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d}}).\end{array}$$

(59)

where

$$\begin{array}{c}\hfill V_2-V_1=({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d+a+b}})+{\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d}})<0,\end{array}$$

(60)

since $({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d+a+b}})<0$ and ${\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{d+2(a+b)}}-{\displaystyle \frac{q^2}{d}})<0$, so condition $V_2-V_1<0$ is fulfilled.

In the next section, we present a fundamental derivation of the tight-binding model from the Schrödinger equation.

7. From Eigenenergy Qubits to Position-Based Qubits

A position-based qubit is convenient for describing a physical double quantum dot, whereas many quantum-control formalisms are written in an energy eigenbasis. This section is motivated by the need to connect these two representations without changing the underlying quantum state. Establishing that connection is essential for translating between abstract qubit amplitudes, localized charge probabilities, and experimentally tunable device parameters. The section introduces a rotation between the energy-basis Hamiltonian ${\widehat{H}}_E$ and the position-basis Hamiltonian ${\widehat{H}}_P$. A mixing angle is used to construct maximally localized Wannier-like left and right states from the two lowest-energy eigenstates. Matrix elements of the microscopic Schrödinger Hamiltonian are then projected into the localized basis to identify the effective on-site energies and tunneling terms. The development also treats asymmetric cases and uses the same mapping to express occupation probabilities and electrostatic gate energies in either representation.

This basis transformation clarifies that the energy and position descriptions are not competing physical models when they span the same two-dimensional subspace. They are different coordinate systems for the same truncated quantum dynamics. The position basis is especially useful for semiconductor devices because charge sensing and gate voltages are naturally local, while the energy basis is useful for spectral analysis and control protocols. The manuscript also emphasizes that the mapping relies on a well-defined two-level subspace and on localized Wannier functions. If higher levels become populated, a two-state rotation is no longer sufficient, and the effective parameters must be generalized.

Section 7 provides the formal bridge between abstract eigenenergy qubits and the localized charge qubits used in the proposed CMOS-compatible architecture. It shows how energy eigenstates, localized Wannier states, confinement energies, and hopping amplitudes can be transformed into one another within the two-level approximation.

The Hamiltonian of the tight-binding model describing single-electron devices (as presented by Fujisawa [29] or Pomorski [37]) uses the Hamiltonian of position-based qubit ${\widehat{H}}_P=\widehat{R}{\widehat{H}}_E{\widehat{R}}^{-1}$ with rotation matrix $\widehat{R}$, fictitious angle ${\mathrm{\Theta }}_s$ and eigenenergy Hamiltonian ${\widehat{H}}_E$. In the case of the two-energy-level system, this can be expressed in the following form:

$$\begin{array}{c}{\widehat{H}}_P=\left[\begin{array}{cc}{E}_{P1}& t_s\cos {\theta }_P+\mathrm{i}{t}_s\sin {\theta }_P\\ t_s\cos {\theta }_P-\mathrm{i}{t}_s\sin {\theta }_P& E_{P2}\end{array}\right]=\left[\begin{array}{cc}{E}_{P1}& t_s{e}^{+i{\theta }_P}\\ t_s{e}^{-i{\theta }_P}& E_{P2}\end{array}\right]=\widehat{R}\left[E_1\right|E_1⟩⟨E_1|+E_2|E_2⟩⟨E_2\left|\right]{\widehat{R}}^{-1}\hfill \\ =\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}{E}_1& 0\\ 0& E_2\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]=\widehat{R}{\widehat{H}}_E{\widehat{R}}^{-1}=\\ =\left[\begin{array}{cc}{E}_1\cos {\left({\mathrm{\Theta }}_s\right)}^2+E_2\sin {\left({\mathrm{\Theta }}_s\right)}^2& (E_2-E_1)\sin \left({\mathrm{\Theta }}_s\right)\cos \left({\mathrm{\Theta }}_s\right)\\ +(E_2-E_1)\sin \left({\mathrm{\Theta }}_s\right)\cos \left({\mathrm{\Theta }}_s\right)& E_2\cos {\left({\mathrm{\Theta }}_s\right)}^2+E_1\sin {\left({\mathrm{\Theta }}_s\right)}^2\end{array}\right]=\left[\begin{array}{cc}{E}_1+(E_2-E_1)\sin {\left({\mathrm{\Theta }}_s\right)}^2& {\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)\\ +{\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)& (E_2-E_1)\cos {\left({\mathrm{\Theta }}_s\right)}^2+E_1\end{array}\right]=\\ =\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}(E_2+E_1)& 0\\ 0& {\displaystyle \frac{1}{2}}(E_2+E_1)\end{array}\right]+\left[\begin{array}{cc}{E}_1\cos {\left({\mathrm{\Theta }}_s\right)}^2+E_2\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_1+E_2}{2}}& {\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)\\ +{\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)& E_1\sin {\left({\mathrm{\Theta }}_s\right)}^2+E_2\cos {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_1+E_2}{2}}\end{array}\right]=\\ \hfill {\displaystyle \frac{E_1+E_2}{2}}\widehat{I}+\left[\begin{array}{cc}{E}_1(1-\sin {\left({\mathrm{\Theta }}_s\right)}^2)+E_2\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_1+E_2}{2}}& {\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)\\ +{\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)& E_1\sin {\left({\mathrm{\Theta }}_s\right)}^2+E_2{(1-\sin \left({\mathrm{\Theta }}_s\right))}^2-{\displaystyle \frac{E_1+E_2}{2}}\end{array}\right].\end{array}$$

(61)

We continue elementary mathematical steps

$$\begin{array}{c}{\widehat{H}}_P=\left[\begin{array}{cc}{E}_1(1-\sin {\left({\mathrm{\Theta }}_s\right)}^2)+E_2\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_1+E_2}{2}}& {\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)\\ +{\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)& E_1\sin {\left({\mathrm{\Theta }}_s\right)}^2+E_2{(1-\sin \left({\mathrm{\Theta }}_s\right))}^2-{\displaystyle \frac{E_1+E_2}{2}}\end{array}\right]+{\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I}=\hfill \\ ={\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I}{\widehat{\sigma }}_0+{\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right){\widehat{\sigma }}_1+\left[(E_2-E_1)\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_2-E_1}{2}}\right]{\widehat{\sigma }}_3=\\ =\left[\begin{array}{cc}+[(E_2-E_1)\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_2-E_1}{2}}]& {\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)\\ +{\displaystyle \frac{1}{2}}(E_2-E_1)\sin \left(2{\mathrm{\Theta }}_s\right)& -[(E_2-E_1)\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{E_2-E_1}{2}}]\end{array}\right]+{\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I}=\\ =(E_2-E_1)\left[\begin{array}{cc}+[\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{1}{2}}]& {\displaystyle \frac{1}{2}}\sin \left(2{\mathrm{\Theta }}_s\right)\\ +{\displaystyle \frac{1}{2}}\sin \left(2{\mathrm{\Theta }}_s\right)& -[\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{1}{2}}]\end{array}\right]+{\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I}=\\ =(E_2-E_1)\left[\begin{array}{cc}+[\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{1}{2}}]& \pm \sin \left({\mathrm{\Theta }}_s\right)\sqrt{1-\sin {\left({\mathrm{\Theta }}_s\right)}^2}\\ \pm \sin \left({\mathrm{\Theta }}_s\right)\sqrt{1-\sin {\left({\mathrm{\Theta }}_s\right)}^2}& -[\sin {\left({\mathrm{\Theta }}_s\right)}^2-{\displaystyle \frac{1}{2}}]\end{array}\right]+{\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I}=\\ =(E_2-E_1)\sin {\left({\mathrm{\Theta }}_s\right)}^2\left[\begin{array}{cc}+[1-{\displaystyle \frac{1}{2\sin {\left({\mathrm{\Theta }}_s\right)}^2}}]& \pm \sqrt{{\displaystyle \frac{1}{\sin {\left({\mathrm{\Theta }}_s\right)}^2}}-1}\\ \pm \sqrt{{\displaystyle \frac{1}{\sin {\left({\mathrm{\Theta }}_s\right)}^2}}-1}& -[1-{\displaystyle \frac{1}{2\sin {\left({\mathrm{\Theta }}_s\right)}^2}}]\end{array}\right]+{\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I}=\\ ={\displaystyle \frac{(E_2-E_1)}{2}}{\displaystyle \frac{1}{w}}\left[\begin{array}{cc}+[1-w]& \pm \sqrt{2}\sqrt{w-{\displaystyle \frac{1}{2}}}\\ \pm \sqrt{2}\sqrt{w-{\displaystyle \frac{1}{2}}}& -[1-w]\end{array}\right]+{\displaystyle \frac{1}{2}}(E_2+E_1)\widehat{I},w={\displaystyle \frac{1}{2\sin {\left({\mathrm{\Theta }}_s\right)}^2}},\end{array}$$

where we use the property of maximization of the presence of the left Wannier wavefunction $w_L\left(x\right)=\cos \left({\mathrm{\Theta }}_s\right){\psi }_{E1}\left(x\right)+\sin \left({\mathrm{\Theta }}_s\right){\psi }_{E2}\left(x\right)$ in the region from $-\infty$ to $x_0$ encoded in minimization of functional ${\int }_{-\infty }^{x_0}{\left|w_L\left(x\right)\right|}^{2}dx=F\left[{\mathrm{\Theta }}_s\right]$, so ${\displaystyle \frac{d}{d{\mathrm{\Theta }}_s}}F\left[{\mathrm{\Theta }}_s\right]=0$ is achieved. Having the logical state zero assigned from $-\infty$ to $x_0$ (medium distance between two quantum dots) we obtain the formula for angle ${\mathrm{\Theta }}_s$ as given by [37] and the expression

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_s={\displaystyle \frac{1}{2}}\arctan \left[{\displaystyle \frac{{\int }_{-\infty }^{+x_0}\frac{1}{2}[{\psi }_{E1}\left(x\right){\psi }_{E2}^{*}\left(x\right)+{\psi }_{E1}^{*}\left(x\right){\psi }_{E2}\left(x\right)]dx}{{\int }_{-\infty }^{+x_0}dx\left[\right|{\psi }_{E2}{\left(x\right)|}^2-|{\psi }_{E1}\left(x\right){|}^2]}}\right].\end{array}$$

(62)

It is useful to parameterize ${\widehat{H}}_P$ by the introduced quantity $w={\displaystyle \frac{1}{2\sin {\left({\mathrm{\Theta }}_s\right)}^2}}$ and two eigenenergies $E_1$ and $E_2$. It is proper to underline that

$$\begin{array}{c}\hfill {\widehat{H}}_E=\left[\begin{array}{cc}[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V_{eff}\left(x\right)]& 0\\ 0& [-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V_{eff}\left(x\right)]\end{array}\right]=\left[\begin{array}{cc}{E}_1& 0\\ 0& E_2\end{array}\right]=E_1|E_1⟩⟨E_1|+E_2|E_2⟩⟨E_2|,\\ \hfill {\widehat{H}}_E|{\psi }_{E⟩}={\widehat{H}}_E\left[\begin{array}{c}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}{\psi }_{E1}\left(x\right)\\ \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}{\psi }_{E2}\left(x\right)\end{array}\right]=i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}{\psi }_{E1}\left(x\right)\\ \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}{\psi }_{E2}\left(x\right)\end{array}\right]=i\mathit{\hslash }{\displaystyle \frac{d}{dt}}|{\psi }_{E⟩}=\\ \hfill =\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}{\psi }_{E1}\left(x\right)\\ \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}{\psi }_{E2}\left(x\right)\end{array}\right],\\ \hfill |\psi {⟩}_P=\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{c}{\psi }_{E1}\left(x\right)\\ {\psi }_{E2}\left(x\right)\end{array}\right],|\psi {⟩}_P=\left[\begin{array}{c}{\psi }_{E1}\left(x\right)\\ {\psi }_{E2}\left(x\right)\end{array}\right]=\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right],\\ \hfill {\widehat{H}}_E|\psi {⟩}_E={\widehat{H}}_E\left[\begin{array}{c}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}{\psi }_{E1}\left(x\right)\\ \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}{\psi }_{E2}\left(x\right)\end{array}\right]={\widehat{H}}_E\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\end{array}$$

(63)

$$\begin{array}{c}\hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}[\sin \left({\mathrm{\Theta }}_s\right)[\sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}-\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]+\\ \hfill +\left[\begin{array}{cc}\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ \sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right].\end{array}$$

Consequently using the fact that ${\mathrm{\Theta }}_s$ is time-independent we obtain the following chain of equivalent expressions giving the transformation from the energy base to the position base

$$\begin{array}{c}\hfill {\widehat{H}}_E\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& -\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\times \\ \hfill \times \left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\\ \hfill =\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\times \\ \hfill \times i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& -\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{cc}\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ \sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right],\\ \hfill \left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]{\widehat{H}}_E\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\right]\times \\ \hfill \times \left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& -\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\\ \hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& -\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{cc}\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ \sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\\ \hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}\cos {\left({\mathrm{\Theta }}_s\right)}^2+\sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\sin {\left({\mathrm{\Theta }}_s\right)}^2& [\sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}-\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}]\sin \left({\mathrm{\Theta }}_s\right)\cos \left({\mathrm{\Theta }}_s\right)\\ \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}-\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}]\sin \left({\mathrm{\Theta }}_s\right)\cos \left({\mathrm{\Theta }}_s\right)& \sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}\sin {\left({\mathrm{\Theta }}_s\right)}^2+\sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\cos {\left({\mathrm{\Theta }}_s\right)}^2\end{array}\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\\ \hfill =i\mathit{\hslash }{\displaystyle \frac{d}{dt}}|\psi {⟩}_P={\widehat{H}}_P|\psi {⟩}_P=\\ \hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{c}{\psi }_{E1}\left(x\right)\\ {\psi }_{E2}\left(x\right)\end{array}\right],\\ \hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}{\psi }_p(x,t)=\left[\begin{array}{cc}1,& 1\end{array}\right]i\mathit{\hslash }{\displaystyle \frac{d}{dt}}|\psi {⟩}_P=\left[\begin{array}{cc}1,& 1\end{array}\right]{\widehat{H}}_P|\psi {⟩}_P=\\ \hfill =\left[\begin{array}{cc}1,& 1\end{array}\right]\left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]{\widehat{H}}_E\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\right]\times \\ \hfill \times \left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& -\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\right]\left[\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right]=\\ \hfill =\left[\begin{array}{cc}1,& 1\end{array}\right][\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ -\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\times \\ \hfill \times \left[\begin{array}{cc}{\int }_{-\infty }^{+\infty }d{x}_1{\psi }_{E1}^{*}\left(x_1\right)[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}_1^2}}+V(x_1,t)]{\psi }_{E1}\left(x_1\right)& {\int }_{-\infty }^{+\infty }d{x}_1{\psi }_{E1}\left(x_1\right)[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}_1^2}}+V(x_1,t)]{\psi }_{E2}^{*}\left(x_1\right)\\ {\int }_{-\infty }^{+\infty }d{x}_1{\psi }_{E2}\left(x_1\right)[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}_1^2}}+V(x_1,t)]{\psi }_{E1}^{*}\left(x_1\right)& {\int }_{-\infty }^{+\infty }d{x}_1{\psi }_{E2}^{*}\left(x_1\right)[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V(x,t)]{\psi }_{E2}\left(x_1\right)\end{array}\right]\times \\ \hfill \times \left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& -\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& +\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]]\times \\ \hfill \times \left[\left[\begin{array}{cc}+\cos \left({\mathrm{\Theta }}_s\right)& +\sin \left({\mathrm{\Theta }}_s\right)\\ +\sin \left({\mathrm{\Theta }}_s\right)& -\cos \left({\mathrm{\Theta }}_s\right)\end{array}\right]\left[\begin{array}{cc}\sqrt{p_{E1}\left(t\right)}{e}^{i{\gamma }_1\left(t\right)}& 0\\ 0& \sqrt{p_{E2}\left(t\right)}{e}^{i{\gamma }_2\left(t\right)}\end{array}\right]\right]\left[\begin{array}{c}{\psi }_{E1}\left(x\right)\\ {\psi }_{E2}\left(x\right)\end{array}\right],\end{array}$$

where we introduce the complex value function ${\psi }_P\left(x\right)$ expressed as a linear combination of maximal localized Wannier functions localized on the left or on the right side ($w_L\left(x\right)$ and $w_R\left(x\right)$) or as a linear combination of the two lowest-energy eigenstates ${\psi }_{E1}\left(x\right)$ and ${\psi }_{E2}\left(x\right)$ with effective potential $V(x,t)$ describing a position-based qubit in the single-electron regime as depicted in Figure 5 and Figure 6. We assume that this potential is only weakly time-dependent.

$$\begin{array}{c}\hfill {\widehat{H}}_p=\left(\begin{array}{cc}\sin \left(2{\mathrm{\Theta }}_s\right){\displaystyle \frac{E_{12}+E_{21}}{2}}+E_1{\cos }^2\left({\mathrm{\Theta }}_s\right)+E_2{\sin }^2\left({\mathrm{\Theta }}_s\right)& E_{12}{\cos }^2\left({\mathrm{\Theta }}_s\right)+\sin \left(2{\mathrm{\Theta }}_s\right){\displaystyle \frac{E_2-E_1}{2}}-E_{21}{\sin }^2\left({\mathrm{\Theta }}_s\right)\\ -E_{12}{\sin }^2\left({\mathrm{\Theta }}_s\right)+\sin \left(2{\mathrm{\Theta }}_s\right){\displaystyle \frac{E_2-E_1}{2}}+E_{21}{\cos }^2\left({\mathrm{\Theta }}_s\right)& -\sin \left(2{\mathrm{\Theta }}_s\right){\displaystyle \frac{E_{12}+E_{21}}{2}}+E_1{\sin }^2\left({\mathrm{\Theta }}_s\right)+E_2{\cos }^2\left({\mathrm{\Theta }}_s\right)\end{array}\right),\\ \hfill E_{21}\left(t\right)=E_{12}^{*}\left(t\right)={\int }_{-\infty }^{-\infty }d{x}_1{\psi }_{E2}^{*}(x_1,t)[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V(x_1,t)]{\psi }_{E2}(x_1,t),\\ \hfill E_{ss}\left(t\right)=E_s\left(t\right)={\int }_{-\infty }^{-\infty }d{x}_1{\psi }_{Es}^{*}(x_1,t)[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V(x_1,t)]{\psi }_{Es}(x_1,t).\end{array}$$

(64)

We observe that a given quantum system expressing single-electron qubits can be described by eigenenergy Hamiltonian $H_E$, and this picture can be transferred into position-based Hamiltonian $H_P$ or reversely. A detailed case with two symmetric conditions for two interacting qubits in effective tight-binding Hamiltonian $\widehat{H}$ with respect to diagonal terms is given in Appendix C, when

$$\begin{array}{c}\hfill \widehat{H}=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U& 0& 1\\ 1& 0& U_1& 1\\ 0& 1& 1& U_1\end{array}\right).\end{array}$$

(65)

Another cases of analytic treatment of two interacting qubits in asymmetric case is treated by Appendix D

$$\begin{array}{c}\hfill \widehat{H}=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U_1& 0& 1\\ 1& 0& U_1& 1\\ 0& 1& 1& U\end{array}\right),\widehat{H}=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U_1& 0& 1\\ 1& 0& U& 1\\ 0& 1& 1& U_1\end{array}\right).\end{array}$$

(66)

An example of introduction of correlation/anticorrelation (simultaneous position of electrons on the same/opposite site) function

$$\begin{array}{c}\hfill C={\displaystyle \frac{N_{1,1^{\prime }}+N_{2,2^{\prime }}-N_{1,2^{\prime }}-N_{2,1^{\prime }}}{N_{1,1^{\prime }}+N_{2,2^{\prime }}+N_{1,2^{\prime }}+N_{2,1^{\prime }}}}.\end{array}$$

(67)

for two interacting qubits is treated by Appendix E. With solutions of the tight-binding model for static electric fields coming from qubit-position biasing gates we can explore the cases with the presence of an EM field, which will be done in the next sections. At first we investigate the tight-binding model from the point of view of self-consistency using the Schrödinger formalism.

8. Higher-Order Effects and a Picture Beyond the Tight-Binding Model

The earlier sections deliberately use a reduced tight-binding description. Section 8 asks whether that truncation remains self-consistent when two qubits interact for finite time and when external electromagnetic driving or radiative processes are included. The motivation is therefore to identify physical effects that cannot be captured indefinitely by a four-state model and to determine when a more microscopic Schrödinger description becomes necessary. The manuscript treats the Coulomb interaction between two confined electrons as a perturbation to the dominant single-particle confinement and follows the resulting two-body wavefunction in time. The analysis compares the evolution predicted by the reduced tight-binding picture with a perturbative Schrödinger treatment. The subsection on entangled two-photon emission considers transitions between interacting states and discusses how gate-voltage patterns could be optimized to enhance correlated radiative channels. The thermalization subsection tracks the redistribution of state occupation and the growth of entanglement, with Figure 4 and Figure 11 used to illustrate the appearance of additional energetic components and the evolution from initially separable to more entangled states.

Figure 11. Partial correspondence between a four-resonator photonic network and two electrostatically interacting single-electron lines as given by [38] and also by [39]. Similarly [40] presents resonant microwave-mediated interactions between distant electron spins. One can also place the physical system [12] among pointed resonant cavities [41] that will have coherent charge oscillations in a silicon nanowire double quantum dot by gate-based radio-frequency dispersive sensing.

This is the key limitation analysis of the manuscript. The central observation is that a system initially represented by two levels per qubit can populate additional eigenstates as the interaction evolves. In that regime, fixed effective values of $E_p$ and $t_s$ cease to provide a complete description; their apparent time dependence reflects the fact that the reduced basis is absorbing physics from degrees of freedom that were omitted. The section therefore distinguishes a short-time, low-energy regime in which tight binding is useful from longer-time or more strongly driven conditions where a larger basis is required. The photon-emission discussion also broadens the role of the coupled-dot system from purely logical gating toward possible interfaces with electromagnetic degrees of freedom.

Section 8 shows that the tight-binding model is a useful first approximation but not a closed description at arbitrary time scales. The perturbative Schrödinger analysis provides evidence for higher-state occupation and increasing entanglement, which marks the onset of a beyond-tight-binding regime. This result is essential for interpreting all preceding analytical gate formulas as low-energy, finite-validity descriptions rather than universal dynamics.

One particle is moving in the electric field of another particle that corresponds to the case of the Hamiltonian with a more prominent time-independent part and small time-dependent part.

We give the Schrödinger equation for the system of two interacting particles in the regime of small particle–particle interaction subjected to time-dependent electric biasing of metallic gates and thus having the form

$$\begin{array}{c}\hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}|\psi ⟩=\widehat{H}|\psi ⟩=[-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m_a}}{\displaystyle \frac{d^2}{d{x}_a^2}}-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m_b}}{\displaystyle \frac{d^2}{d{x}_b^2}}+V_A(x_A,t)+V_A(x_B,t)+{\displaystyle \frac{e^2}{4\pi {ϵ}_0}}{\displaystyle \frac{1}{\sqrt{d^2+{(x_1-x_2)}^2}}}]|\psi ⟩,\\ \hfill |\psi ⟩={\gamma }_1\left(t\right)|E_{1a}⟩|E_{1b}⟩+{\gamma }_2\left(t\right)|E_{1a}⟩|E_{2b}⟩+{\gamma }_3\left(t\right)|E_{2a}⟩|E_{1b}⟩+{\gamma }_4\left(t\right)|E_{2a}⟩|E_{2b}⟩,\\ \hfill |E_{s,a}⟩|E_{k,b}⟩={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }d{x}_{a}d{x}_b{\psi }_{E_{s,a}}\left(x_{s,A}\right){\psi }_{E_{s,b}}\left(x_{k,b}\right)|x_{s,a}⟩|x_{k,b}⟩,\end{array}$$

(68)

where $V_A(x_A,t)$ and $V_A(x_B,t)$ are effective potential for confined electrons for qubits A and B, so they have two minima for each separated qubits A and B with the presence of mutual qubit–qubit interaction given by ${\displaystyle \frac{e^2}{4\pi {ϵ}_0}}{\displaystyle \frac{1}{\sqrt{d^2+{(x_1-x_2)}^2}}}$. At first we refer to the situation of a toy model of two qubits with the occupancy of two energy levels $(E_{1a},E_{2a}),(E_{1b},E_{2b})$ each as for systems A and B and with the situation of weak qubit electrostatic interaction.

We go beyond results given in [42,43] and we have the following cases under the assumption that the time-dependent component of the Hamiltonian is of perturbative character:

• $(E_{1a}\to E_{1a},E_{2a}\to E_{2a})$, $(E_{1b}\to E_{1b},E_{2b}\to E_{2b})$ with no photon emission or absorption in neither qubit;
• $(E_{1a}\to E_{1a},E_{2a}\to E_{2a})$, $(E_{1b}\to E_{2b},E_{2b}\to E_{1b})$ with possible photon emission or absorption in qubit B;
• $(E_{1a}\to E_{2a},E_{2a}\to E_{1a})$, $(E_{1b}\to E_{1b},E_{2b}\to E_{2b})$ with possible photon emission or absorption in qubit A;
• $(E_{1a}\to E_{2a},E_{2a}\to E_{1a})$, $(E_{1b}\to E_{2b},E_{2b}\to E_{1b})$ due to mutual qubit–qubit interaction (with the possibility of photon emission or absorption by each of qubits A and B);
• $(E_{2a}\to E_{1a},E_{1a}\to E_{1a})$, $(E_{2b}\to E_{1b},E_{1b}\to E_{1b})$ with simultaneous photon emission (as in the case of entangled photon emission) by qubits A and B;
• $(E_{1a}\to E_{2a},E_{2a}\to E_{2a})$, $(E_{1b}\to E_{2b},E_{2b}\to E_{2b})$ with simultaneous photon absorption (as in the case of entangled photon emission) by qubits A and B;
• $(E_{2a}\to E_{1a},E_{1a}\to E_{1a})$, $(E_{1b}\to E_{1b},E_{2b}\to E_{2b})$ with photon emission by qubit A;
• $(E_{1a}\to E_{1a},E_{2a}\to E_{2a})$, $(E_{2b}\to E_{1b},E_{1b}\to E_{1b})$ with photon emission by qubit B.

We can state that the identity operators can be written as $\widehat{I_a}=|E_{1a}⟩⟨E_{1a}|+|E_{2a}⟩⟨E_{2a}|$ and $\widehat{I_b}=|E_{1b}⟩⟨E_{1b}|+|E_{2b}⟩⟨E_{2b}|$. We omit possible phonon emission as described by [1]. The system Hamiltonian in spectral form can be written as

$$\begin{array}{c}\hfill \widehat{H}=\left[f_1\left(t\right)\right|E_{1a}⟩⟨E_{2a}|\times \widehat{I_b}+f_2\left(t\right)|E_{2a}⟩⟨E_{1a}|\times \widehat{I_b}]+\\ \hfill +[\widehat{I_a}\times g_1\left(t\right)|E_{1b}⟩⟨E_{2b}|+\widehat{I_a}\times g_2\left(t\right)|E_{2b}⟩⟨E_{1b}\left|\right]+\\ \hfill +\left[E_{1a}\left(t\right)\right|E_{1a}⟩⟨E_{1a}\left|\right]\times \widehat{I_b}+\left[E_{2a}\left(t\right)\right|E_{2a}⟩⟨E_{2a}\left|\right]\times \widehat{I_b}+\\ \hfill +\widehat{I_a}\times \left[E_{1b}\left(t\right)\right|E_{1b}⟩⟨E_{1b}\left|\right]+\widehat{I_a}\times \left[E_{2b}\left(t\right)\right|E_{2b}⟩⟨E_{2b}\left|\right]+\\ \hfill +\begin{array}{c}\hfill \left[h_1[E_{1a}\to E_{1a},E_{1b}\to E_{1b}]\left(t\right)\right|E_{1a}⟩⟨E_{1a}|\times |E_{1b}⟩⟨E_{1b}\left|\right][11,11]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_2[E_{1a}\to E_{1a},E_{2b}\to E_{2b}]\left(t\right)\right|E_{1a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{2b}\left|\right][11,22]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_3[E_{2a}\to E_{2a},E_{1b}\to E_{1b}]\left(t\right)\right|E_{2a}⟩⟨E_{2a}|\times |E_{1b}⟩⟨E_{1b}\left|\right][22,11]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_4[E_{2a}\to E_{2a},E_{2b}\to E_{2b}]\left(t\right)\right|E_{2a}⟩⟨E_{2a}|\times |E_{2b}⟩⟨E_{2b}\left|\right][22,22]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_5[E_{1a}\to E_{2a},E_{2b}\to E_{1b}]\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][12,21]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_6[E_{2a}\to E_{1a},E_{1b}\to E_{2b}]\left(t\right)\right|E_{1a}⟩⟨E_{2a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][21,12]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_7[E_{1a}\to E_{2a},E_{1b}\to E_{2b}]\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][12,12]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_8[E_{2a}\to E_{1a},E_{2b}\to E_{1b}]\left(t\right)\right|E_{1a}⟩⟨E_{2a}|\times |E_{1b}⟩⟨E_{2b}\left|\right][21,21]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_9[E_{1a}\to E_{1a},E_{1b}\to E_{2b}]\left(t\right)\right|E_{1a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][11,12]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{10}[E_{2a}\to E_{2a},E_{1b}\to E_{2b}]\left(t\right)\right|E_{2a}⟩⟨E_{2a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][22,12]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{11}[E_{1a}\to E_{2a},E_{1b}\to E_{1b}]\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{1b}⟩⟨E_{1b}\left|\right][12,11]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{12}[E_{1a}\to E_{2a},E_{2b}\to E_{2b}]\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{2b}\left|\right][12,22]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{13}[E_{1a}\to E_{1a},E_{2b}\to E_{1b}]\left(t\right)\right|E_{1a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][11,21]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{14}[E_{2a}\to E_{2a},E_{2b}\to E_{1b}]\left(t\right)\right|E_{2a}⟩⟨E_{2a}|\times |E_{2b}⟩⟨E_{1b}\left|\right][22,21]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{15}[E_{2a}\to E_{1a},E_{1b}\to E_{1b}]\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{1b}⟩⟨E_{1b}\left|\right][21,11]\end{array}+\\ \hfill +\begin{array}{c}\hfill \left[h_{16}[E_{2a}\to E_{1a},E_{2b}\to E_{2b}]\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{2b}\left|\right][21,22]\end{array}.\end{array}$$

(69)

We introduce the following color notation in regard to the Coulomb interaction term:

• Red when both two qubits are not changing energy;
• Blue when both two qubits de-excite;
• Orange when both qubits transfer into higher energy;
• Brown when both qubits transfer into higher energy;
• Pink when we have emission of photons by one qubit, while the second state is unchanged.

Single-particle energies in the two single-electron lines under the effect of small magnitude time-dependent potential V(t) with dominance of the time-independent part are defined as expectation values of the corresponding time–dependent Hamiltonians,

$$\begin{array}{c}\hfill \int d{x}_A{\psi }_s^{\left(A\right)*}(x_A,t)\left[-\frac{{\mathit{\hslash }}^2}{2m_A}{\mathsf{\partial }}_{x_A}^2+V_A(x_A,t)\right]{\psi }_s^{\left(A\right)}(x_A,t)=E(E_{A,s}\to E_{A,s})=E_A(s,s)=E_{A,s},\end{array}$$

(70)

$$\begin{array}{c}\hfill \int d{x}_B{\psi }_s^{\left(B\right)*}(x_B,t)\left[-\frac{{\mathit{\hslash }}^2}{2m_B}{\mathsf{\partial }}_{x_B}^2+V_B(x_B,t)\right]{\psi }_s^{\left(B\right)}(x_B,t)=E(E_{Bs}\to E_{B,s})=E_B(s,s)=E_{B,s},\end{array}$$

(71)

where ${\psi }_s^{\left(A\right)}(x_A,t),{\psi }_s^{\left(B\right)}(x_B,t)$ are eigenstates corresponding to $E_{A,s}$ or $E_{B,s}$ eigenvalues. We also use short–hand energy transition matrix elements within each line,

$$\begin{array}{c}\hfill \int d{x}_A{\psi }_s^{\left(A\right)*}(x_A,t)\left[-\frac{{\mathit{\hslash }}^2}{2m_A}{\mathsf{\partial }}_{x_A}^2+V_A(x_A,t)\right]{\psi }_r^{\left(A\right)}(x_A,t)=E(E_{A,s}\to E_{A,r})=E_A(s,r),\end{array}$$

(72)

$$\begin{array}{c}\hfill \int d{x}_B{\psi }_w^{\left(B\right)*}(x_B,t)\left[-\frac{{\mathit{\hslash }}^2}{2m_B}{\mathsf{\partial }}_{x_B}^2+V_B(x_B,t)\right]{\psi }_o^{\left(B\right)}(x_B,t)=E(E_{B,o}\to E_{B,w})=E_B(o,w).\end{array}$$

(73)

In the particular case of a two-level system for A and B we thus have transition amplitude from ground to excited state given by

$$\begin{array}{c}\hfill \int d{x}_A{\psi }_{E_A\left(2\right)}^{*}(x_A,t)\left(-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m_A}}{\displaystyle \frac{d^2}{d{x}_A^2}}+V_A(x_A,t)\right){\psi }_{E_A\left(1\right)}(x_A,t)=E(E_{A,1}\to E_{A,2})=E_A(1,2)=E_A{(2,1)}^{*},\end{array}$$

(74)

and

$$\begin{array}{c}\hfill \int d{x}_B{\psi }_{E_B\left(2\right)}^{*}(x_B,t)\left(-{\displaystyle \frac{{\mathit{\hslash }}^2}{2m_B}}{\displaystyle \frac{d^2}{d{x}_B^2}}+V_B(x_B,t)\right){\psi }_{E_B\left(1\right)}(x_B,t)=E(E_{B,1}\to E_{B,2})=E_B(1,2)=E_B{(2,1)}^{*}.\end{array}$$

(75)

We represent the two-body perturbative electrostatic coupling in the spectral (single-particle eigenstate) basis as

$$\begin{array}{c}\hfill \int d{x}_{A}d{x}_B{\psi }_k^{\left(A\right)*}\left(x_A\right){\psi }_s^{\left(B\right)*}\left(x_B\right)\left[V_{\mathrm{int}}(x_A,x_B)\right]{\psi }_r^{\left(A\right)}\left(x_A\right){\psi }_p^{\left(B\right)}\left(x_B\right)=E_{A-B}(E_{A,k}\to E_{A,s},E_{B,r}\to E_{B,p})=E_{A-B}(k,s,r,p).\end{array}$$

(76)

For the discretized operators we use the standard finite–difference stencil for the second derivative (a tridiagonal Toeplitz operator). As a result, computing ${\psi }^{\left(A\right)}$ and ${\psi }^{\left(B\right)}$ reduces to an eigenproblem determined by the gate-defined effective potential; algebraically this resembles emitter–field models (e.g., Jaynes–Cummings), though here the structure is constrained by device geometry.

This allows us to write a 2-qubit Hamiltonian system with a non-dominant time-dependent component of the Hamiltonian in the form

$$\begin{array}{c}\hfill \widehat{H}={\displaystyle \sum _{k=1}^{+\infty }}{\displaystyle \sum _{s=1}^{+\infty }}\left[{\int }_{-\infty }^{+\infty }d{x}_A{\psi }_k^{\left(A\right)*}(x_A,t)\left[-\frac{{\mathit{\hslash }}^2}{2m_A}{\mathsf{\partial }}_{x_A}^2+V_A(x_A,t)\right]{\psi }_s^{\left(A\right)}(x_A,t)\right][\left(\right|E_{A,k}⟩⟨E_{A,s}\left|\right)\otimes {\widehat{I}}_B]+\\ \hfill +{\displaystyle \sum _{r=1}^{+\infty }}{\displaystyle \sum _{w=1}^{+\infty }}\left[{\int }_{-\infty }^{+\infty }d{x}_B{\psi }_r^{\left(B\right)*}(x_B,t)\left[-\frac{{\mathit{\hslash }}^2}{2m_B}{\mathsf{\partial }}_{x_B}^2+V_B(x_B,t)\right]{\psi }_w^{\left(B\right)}(x_B,t)\right][{\widehat{I}}_A\otimes \left(\right|E_{B,w}⟩⟨E_{B,r}\left|\right)]+\\ \hfill +{\displaystyle \sum _{a=1}^{+\infty }}{\displaystyle \sum _{b=1}^{+\infty }}{\displaystyle \sum _{c=1}^{+\infty }}{\displaystyle \sum _{d=1}^{+\infty }}\left[\int d{x}_B{\psi }_a^{\left(A\right)*}(x_A,t){\psi }_c^{\left(B\right)*}(x_B,t)V_{A-B}(x_A,x_B,t){\psi }_a^{\left(A\right)}(x_A,t){\psi }_d^{\left(B\right)}(x_B,t)\right]\left(\right|E_{A,b}⟩⟨E_{A,a}\left|\right)\otimes \left(\right|E_{B,d}⟩⟨E_{B,c}\left|\right).\end{array}$$

(77)

Having a very general formula for two interacting position-based qubits in the perturbative limit, we can now analyze more exotic cases.

8.1. Entangled Two-Photon Emission from Two Interacting Position-Based Qubits

We consider physical situation from Figure 12 and we have the situation of the emission of each photon by each qubit that is accounted for by the mathematical term

$\begin{array}{c}\hfill \left[h_8[E_{2a}\to E_{1a},E_{2b}\to E_{1b}]\left(t\right)\right|E_{1a}⟩⟨E_{2a}|\times |E_{1b}⟩⟨E_{2b}\left|\right][21,21]\end{array}$ and describes the physical situation where the photon state is given as $|Photon-pair⟩=|E_{ph1,A}⟩|E_{ph1,B}⟩$, when $E_{ph1,A}=E_{2a}-E_{1a}$ and $E_{ph2,B}=E_{2b}-E_{1b}$.

Two–photon entanglement is conveniently expressed in the Bell basis,

$$\begin{array}{cc}\hfill |{\mathrm{\Phi }}^{\pm }⟩& =\frac{1}{\sqrt{2}}\left(|{\omega }_1^{\left(A\right)}⟩|{\omega }_1^{\left(B\right)}⟩\pm |{\omega }_2^{\left(A\right)}⟩|{\omega }_2^{\left(B\right)}⟩\right),\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill |{\mathrm{\Psi }}^{\pm }⟩& =\frac{1}{\sqrt{2}}\left(|{\omega }_1^{\left(A\right)}⟩|{\omega }_1^{\left(B\right)}⟩\pm |{\omega }_2^{\left(A\right)}⟩|{\omega }_2^{\left(B\right)}⟩\right),\hfill \end{array}$$

(79)

provided that each emitter supports at least a three–level ladder enabling distinct radiative channels and can be incorporated into the Lindblad formalism.

One can expect the effective system Hamiltonian to be as follows

$$\begin{array}{c}\hfill \widehat{H}=\left(s_1\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{1b}|+s_2\left(t\right)|E_{3a}⟩⟨E_{1a}|\times |E_{3b}⟩⟨E_{1b}\left|\right),\end{array}$$

(80)

Since the aforementioned Hamiltonian is non-Hermitian one can suggest the maximum intensification of two entangled photons takes place for Hamiltonian as

$$\begin{array}{c}\hfill \widehat{H}=\left(s_{1a}\left(t\right)\right|E_{2a}⟩⟨E_{1a}|\times |E_{2b}⟩⟨E_{1b}|+s_{1b}\left(t\right)|E_{1a}⟩⟨E_{2a}|\times |E_{1b}⟩⟨E_{2b}\left|\right)+\\ \hfill +\left(s_{2a}\left(t\right)\right|E_{3a}⟩⟨E_{1a}|\times |E_{3b}⟩⟨E_{1b}|+s_{2b}\left(t\right)|E_{1a}⟩⟨E_{3a}|\times |E_{1b}⟩⟨E_{3b}\left|\right),\end{array}$$

(81)

Therefore, coefficients $s_{1a}\left(t\right)$, $s_{1b}\left(t\right)$, $s_{2a}\left(t\right)$, $s_{2b}\left(t\right)$ are pronounced and others less pronounced. This is thus the optimization problem from the point of view of electrode geometries and the function of the polarization voltage pattern applied to six subsequent metallic gates.

Figure 12. Geometry of 2 interacting Wannier qubits to be parameterized for effective emission of 2 entangled photons.

Figure 12. Geometry of 2 interacting Wannier qubits to be parameterized for effective emission of 2 entangled photons.

By proper numerical experiments one can establish proper $V_A(x_A,t)$ and $V_B(x_A,t)$ structure that maximizes the emission of entangled photons among all emitted photons that can be later translated into voltage signal pattern $V_{1a}\left(t\right),V_{2a}\left(t\right),V_{3a}\left(t\right)$ and $V_{1b}\left(t\right),V_{2b}\left(t\right),V_{3b}\left(t\right)$ with respect to the physical situation from Figure 13. The results obtained by the two-body Schrödinger formalism support the validity of the tight-binding model to a certain degree. More optimal two-entangled-photon emission as basing on structure depicted in Figure 12 can be optimized with reference to Figure 9, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19.

Figure 13. Wavefunctions’ absolute value of interacting particles A and B placed at two different single-electron lines with an initial non-entangled state in the case of energy occupancy for each particle at time $t_0$ and after some time t. Oscillations are never fully periodic as the system heats up and each particle starts to occupy an increasing number of energy levels, which is directly the breakdown of the tight-binding model in longer time scales. One can refer to the next figure for more details.

Figure 13. Wavefunctions’ absolute value of interacting particles A and B placed at two different single-electron lines with an initial non-entangled state in the case of energy occupancy for each particle at time $t_0$ and after some time t. Oscillations are never fully periodic as the system heats up and each particle starts to occupy an increasing number of energy levels, which is directly the breakdown of the tight-binding model in longer time scales. One can refer to the next figure for more details.

Figure 14. Dependence of the quasi-classical SWAP-gate Hamiltonian on the electrostatically controlled polarizing potentials $E_{p1}$ and $E_{p{1}^{\prime }}$, calculated for fixed values $E_{p2}=1$ and $E_{p{2}^{\prime }}=1$.

Figure 14. Dependence of the quasi-classical SWAP-gate Hamiltonian on the electrostatically controlled polarizing potentials $E_{p1}$ and $E_{p{1}^{\prime }}$, calculated for fixed values $E_{p2}=1$ and $E_{p{2}^{\prime }}=1$.

Figure 15. Role of polarizing voltages in transition from symmetric classical and quantum SWAP gate in asymmetric SWAP gate (4 upper and bottom left) and in transition into ANTISWAP gate ((bottom right) plots).

Figure 15. Role of polarizing voltages in transition from symmetric classical and quantum SWAP gate in asymmetric SWAP gate (4 upper and bottom left) and in transition into ANTISWAP gate ((bottom right) plots).

Figure 16. Two main scenarios of dependence of eigenenergies of electrostatically interacting qubits on angles. The (left) plot corresponds to parameters (d = 1, a + b = 0.8, q = 1, $E_{p1}=1$, $E_{p2}=-1$, $E_{p{1}^{\prime }}=-3$, $E_{p{2}^{\prime }}=-2$, $t_{s12}=1$, $t_{s{1}^{\prime }{2}^{\prime }}=1$), and the (right) plot has the same parameters except ($E_{p1}=E_{p2}=E_{p{1}^{\prime }}=E_{p{2}^{\prime }}=1$, a + b = 0.1).

Figure 16. Two main scenarios of dependence of eigenenergies of electrostatically interacting qubits on angles. The (left) plot corresponds to parameters (d = 1, a + b = 0.8, q = 1, $E_{p1}=1$, $E_{p2}=-1$, $E_{p{1}^{\prime }}=-3$, $E_{p{2}^{\prime }}=-2$, $t_{s12}=1$, $t_{s{1}^{\prime }{2}^{\prime }}=1$), and the (right) plot has the same parameters except ($E_{p1}=E_{p2}=E_{p{1}^{\prime }}=E_{p{2}^{\prime }}=1$, a + b = 0.1).

Figure 17. Energy spectrum of generalized electrostatic quantum gate in dependence on angle for fixed $E_{p1}$, $E_{p2}$, $E_{p{1}^{\prime }}$, $E_{p{2}^{\prime }}$, $t_{s12}=t_{s{1}^{\prime }{2}^{\prime }}=0.1$, $q=1$, $a+b=0.8$. No crossing of energy levels was reported.

Figure 17. Energy spectrum of generalized electrostatic quantum gate in dependence on angle for fixed $E_{p1}$, $E_{p2}$, $E_{p{1}^{\prime }}$, $E_{p{2}^{\prime }}$, $t_{s12}=t_{s{1}^{\prime }{2}^{\prime }}=0.1$, $q=1$, $a+b=0.8$. No crossing of energy levels was reported.

Figure 18. Continuation of Figure 9: additional contour plots for the transition from the classical/quantum SWAP gate into the ANTISWAP gate as the angle $\alpha$ is varied.

Figure 18. Continuation of Figure 9: additional contour plots for the transition from the classical/quantum SWAP gate into the ANTISWAP gate as the angle $\alpha$ is varied.

8.2. Case of Thermalization of Systems of Two Position-Based Qubits and Failure of Tight-Binding Model

We can set

$$\begin{array}{c}\hfill {\psi }_{A,B}(x_A,x_B,t_0)=(e^{i\gamma 1A}\sqrt{p(E1,A)}{\psi }_{A,E1}(x_A,t_0)+\sqrt{p(E2,A)}{e}^{i\gamma 2A}{\psi }_{A,E2}(x_A,t_0)\times \\ \hfill \times (\sqrt{p(E1,B)}{e}^{i\gamma 1B}{\psi }_{B,E1}(x_B,t_0)+\sqrt{p(E2,B)}{\psi }_{B,E2}(x_B,t_0)e^{i\gamma 1B})\end{array}$$

(82)

which gives evolution of wavefunction modulus of each particle as in Figure 4 when the perturbative Coulomb interaction is turned on. It is interesting to observe how the non-entangled system is becoming more entangled with time as indicated by Figure 11. The dynamics of two electrons confined by two local potentials is determined for various cases both in analytical and in numerical way in the tight-binding model [44]. In this way, the system of two coupled electrostatic position-based qubits can be used for the implementation of the quantum SWAP or ANTISWAP gate. A broader picture is drawn by [29,44,45]. The obtained results have their meaning in designing the proper operation of the quantum gates implemented in chain of coupled semiconductor quantum dots that are electrostatically controlled. They open the path for implementation of a CMOS quantum computer that is only controlled by voltages applied to CMOS transistors with no need for usage of a magnetic field. It is an appealing alternative for the implementation of quantum electronics other than by the usage of Josephson junctions [9,46]. The best way of detecting the entanglement present between electrostatically interacting qubits is by experimentally measuring the correlation–anticorrelation function. Both formulas for von Neumann entanglement entropy and anticorrelation functions are given in this work in analytical form. In the next section we describe the elementary derivation of the tight-binding model from the Schrödinger formalism.

Figure 19. Continuation of Figure 17: additional dependence of the same generalized electrostatic quantum-gate spectrum on the geometric and distance parameters.

Figure 19. Continuation of Figure 17: additional dependence of the same generalized electrostatic quantum-gate spectrum on the geometric and distance parameters.

9. Fundamental Derivation of the Tight-Binding Model from the Schrödinger Equation

After demonstrating both the usefulness and the limitations of the reduced model, this section returns to the microscopic Schrödinger equation to justify how the tight-binding parameters arise. Its purpose is to place the position-basis Hamiltonian on a more fundamental footing and to explain how the effective parameters should be interpreted when two separated qubits interact perturbatively. The derivation begins with a two-level energy-eigenstate description and constructs localized Wannier-like states through a basis rotation. Matrix elements of the kinetic-plus-potential Schrödinger operator are projected onto those localized states to obtain the effective on-site energies and tunneling amplitudes. For two interacting single-electron lines, Coulomb terms are added perturbatively and encoded through geometry-dependent coefficients. The manuscript then compares occupation dynamics obtained in the effective tight-binding model with the perturbative two-body Schrödinger evolution and discusses how the procedure can be generalized to more qubits provided that a restricted low-energy subspace remains meaningful.

The derivation clarifies that $E_p$ and $t_s$ are not fundamental constants of the device. They are matrix elements of a microscopic Hamiltonian evaluated in a chosen localized basis. Consequently, changes in confinement, electric field, interaction strength, or basis composition can change their effective values. This interpretation is consistent with the breakdown discussion of Section 8: when interactions populate states outside the original two-level subspace, a time-independent two-parameter description can no longer represent the full wavefunction. The section therefore connects the analytical convenience of tight binding with the microscopic conditions required for its validity.

Section 9 supplies the formal justification for the tight-binding Hamiltonian used throughout the manuscript and validates it as a projected low-energy model of the Schrödinger problem. It also explains why the effective parameters can become state- and time-dependent when the truncated basis is no longer closed. This provides the conceptual bridge between the compact gate model and the beyond-tight-binding conclusions of the work.

Being motivated by [45,46] and starting from the eigenenergy qubit as given by IBM Q-Experience and many other technologies ($|{\gamma }_{E1}{\left(t\right)|}^2+{\left|{\gamma }_{E2}\left(t\right)\right|}^2=1$) we arrive at

$$\begin{array}{c}\hfill \widehat{H}{\left(t\right)}_E|\psi \left(t\right){⟩}_E={\left(\begin{array}{cc}{E}_{1\to 1}\left(t\right)& E_{2\to 1}\left(t\right)\\ E_{1\to 2}\left(t\right)& E_{2\to 2}\left(t\right)\end{array}\right)}_E{\left(\begin{array}{c}{\gamma }_{E1}\left(t\right)\\ {\gamma }_{E2}\left(t\right)\end{array}\right)}_E=\\ \hfill =i\mathit{\hslash }{\displaystyle \frac{d}{dt}}{\left(\begin{array}{c}{\gamma }_{E1}\left(t\right)\\ {\gamma }_{E2}\left(t\right)\end{array}\right)}_E=i\mathit{\hslash }{\displaystyle \frac{d}{dt}}{\left(\right|\psi \left(t\right)⟩)}_E=\\ \hfill =i\mathit{\hslash }{\displaystyle \frac{d}{dt}}\left({\gamma }_{E1}\left(t\right)\right|E_1\left(t\right)⟩+{\gamma }_{E2}\left(t\right)|E_2\left(t\right)⟩)=\\ \hfill ={\left(\begin{array}{cc}{E}_{1\to 1}\left(t\right)& E_{2\to 1}\left(t\right)\\ E_{1\to 2}\left(t\right)& E_{2\to 2}\left(t\right)\end{array}\right)}_E\left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& -\sin \left(\mathrm{\Theta }\right)\\ +\sin \left(\mathrm{\Theta }\right)& +\cos \left(\mathrm{\Theta }\right)\end{array}\right)\times \\ \hfill \times \left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& +\sin \left(\mathrm{\Theta }\right)\\ -\sin \left(\mathrm{\Theta }\right)& +\cos \left(\mathrm{\Theta }\right)\end{array}\right){\left(\begin{array}{c}{\gamma }_{E1}\left(t\right)\\ {\gamma }_{E2}\left(t\right)\end{array}\right)}_E=\\ \hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}[\left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& -\sin \left(\mathrm{\Theta }\right)\\ +\sin \left(\mathrm{\Theta }\right)& +\cos \left(\mathrm{\Theta }\right)\end{array}\right)\times \\ \hfill \times \left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& +\sin \left(\mathrm{\Theta }\right)\\ -\sin \left(\mathrm{\Theta }\right)& +\cos \left(\mathrm{\Theta }\right)\end{array}\right){\left(\begin{array}{c}{\gamma }_{E1}\left(t\right)\\ {\gamma }_{E2}\left(t\right)\end{array}\right)}_E]\end{array}$$

(83)

where we have $E_{s\to k}\left(t\right)=⟨E_k\left(t\right)\left|\widehat{H}\left(t\right)\right|E_s\left(t\right)⟩$ since we assume $\widehat{H}\left(t\right)=E_{1\to 1}\left(t\right)|E_1\left(t\right)⟩⟨E_1\left(t\right)|+E_{2\to 2}\left(t\right)|E_2\left(t\right)⟩⟨E_2\left(t\right)|+E_{1\to 2}\left(t\right)|E_2\left(t\right)⟩⟨E_1\left(t\right)|+E_{2\to 1}\left(t\right)|E_1\left(t\right)⟩⟨E_2\left(t\right)|$. Here $|{\gamma }_{E1}{\left(t\right)|}^2$ and $|{\gamma }_{E2}{\left(t\right)|}^2$ account for the probability of occupancy of $E_1$ and $E_2$ energetic states. Maintenance of quantum information is about keeping it encoded in the Bloch sphere by two parameters ($\mathrm{\Theta }\left(t\right)$, $\varphi \left(t\right)$) and given in reference to quantum state by $|\psi {⟩}_E={\gamma }_{E1}\left(t\right)|E_1\left(t\right)⟩+{\gamma }_{E2}\left(t\right)|E_2\left(t\right)⟩=e^{i{\varphi }_0\left(t\right)}\left(cos\left(\mathrm{\Theta }\left(t\right)\right)\right|E_1\left(t\right)⟩+e^{i\varphi \left(t\right)}sin\left(\mathrm{\Theta }\left(t\right)\right)|E_2\left(t\right)⟩)$. Due to the compact size of the semiconductor quantum dots as depicted in Figure 3, it is vital to express the position-based qubit by stating its Hamiltonian in the so-called tight-binding model, where the quantum state is the linear superposition of two Wannier wavefunctions denoting maximal localization on the left or right quantum dot ($|{\alpha }_q{\left(t\right)|}^2+{\left|{\beta }_q\left(t\right)\right|}^2=1$):

$$\begin{array}{c}\hfill \widehat{H}{\left(t\right)}_W|\psi \left(t\right){⟩}_W={\left(\begin{array}{cc}{E}_{p1}\left(t\right)& t_{s12}\left(t\right)\\ t_{s12}^{*}\left(t\right)& E_{p2}\left(t\right)\end{array}\right)}_W{\left(\begin{array}{c}{\alpha }_q\left(t\right)\\ {\beta }_q\left(t\right)\end{array}\right)}_W=\\ \hfill i\mathit{\hslash }{\displaystyle \frac{d}{dt}}{\left({\alpha }_q\left(t\right)\right|w_L\left(t\right)⟩+{\beta }_q\left(t\right)\left|w_R\left(t\right)⟩\right)}_W=i\mathit{\hslash }{\displaystyle \frac{d}{dt}}|\psi \left(t\right){⟩}_W,\end{array}$$

(84)

where $|w_L\left(t\right)⟩$ and $|w_R\left(t\right)⟩$ are maximum localized Wannier wavefunctions ($⟨w_L|w_R⟩=0$, $⟨w_L|w_L⟩=1=⟨w_R|w_R⟩$) on the left and right side respectively, while $|{\alpha }_q{\left(t\right)|}^2$ and $|{\beta }_q{\left(t\right)|}^2$ give the probabilities for a particle (single electron or hole) to stay on the left and right quantum dots. Encoding quantum information by electron or hole position brings compactification of quantum chips. The quantum state in Wannier representation $|\psi \left(t\right){⟩}_W$ can be mapped to quantum state in energetic representation given as $|\psi \left(t\right){⟩}_E$ or reversely, and those two ways of description are fully equivalent.

Following [35,37] it is useful to introduce parameter

$$\begin{array}{c}\hfill r={\displaystyle \frac{{\int }_{-\infty }^{0}dx({\psi }_{E1}\left(x\right){\psi }_{E2}^{†}\left(x\right)+{\psi }_{E1}{\left(x\right)}^{†}{\psi }_{E2}\left(x\right))}{{\int }_{-\infty }^{0}dx\left(\right|{\psi }_{E1}{\left(x\right)|}^2-|{\psi }_{E2}\left(x\right){|}^2)}}.\end{array}$$

(85)

The precondition for occurrence of the position-based qubit is occupancy for at least two energetic levels, so neither $|{\gamma }_{E1}{|}^2$ nor $|{\gamma }_{E2}{|}^2$ is zero. Finally in the case of static Hamiltonian from Equation (83) also given by [37] the tight-binding Hamiltonian [35] appears as

$$\begin{array}{c}\hfill \left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& \sin \left(\mathrm{\Theta }\right)\\ -\sin \left(\mathrm{\Theta }\right)& \cos \left(\mathrm{\Theta }\right)\end{array}\right){\left(\begin{array}{cc}{E}_1& 0\\ 0& E_2\end{array}\right)}_E\left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& -\sin \left(\mathrm{\Theta }\right)\\ -\sin \left(\mathrm{\Theta }\right)& +\cos \left(\mathrm{\Theta }\right)\end{array}\right)={\left(\begin{array}{cc}{E}_{p1}& t_{s21}\\ t_{s12}& E_{p2}\end{array}\right)}_W=\\ \hfill =E_{p1}|w_L⟩⟨w_L|+E_{p2}|w_R⟩⟨w_R|+t_{s21}|w_R⟩⟨w_L|+t_{s12}|w_L⟩⟨w_R|={\widehat{H}}_W=\\ \hfill \left(\begin{array}{cc}{E}_1+{\left|sin\left({\displaystyle \frac{1}{2}}ArcTan\left(r\right)\right)\right|}^2(E_2-E_1)& (E_2-E_1){\displaystyle \frac{1}{2}}sin\left(ArcTan\left(r\right)\right)\\ (E_2-E_1){\displaystyle \frac{1}{2}}sin\left(ArcTan\left(r\right)\right)& E_1+{\left|cos\left({\displaystyle \frac{1}{2}}ArcTan\left(r\right)\right)\right|}^2(E_2-E_1)\end{array}\right),\end{array}$$

(86)

We propose maximum localized orthonormal Wannier functions (so occupancy of $w_L$ is maximized from minus infinity to zero) of the form

$$\begin{array}{c}\hfill \mathrm{\Theta }={\displaystyle \frac{1}{2}}ArcTan\left(r\right),|\psi {⟩}_W={\left(\begin{array}{c}{w}_L\left(x\right)\\ w_R\left(x\right)\end{array}\right)}_W=\left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& \sin \left(\mathrm{\Theta }\right)\\ -\sin \left(\mathrm{\Theta }\right)& \cos \left(\mathrm{\Theta }\right)\end{array}\right){\left(\begin{array}{c}{\psi }_L\left(x\right)\\ {\psi }_R\left(x\right)\end{array}\right)}_E=\left(\begin{array}{cc}+\cos \left(\mathrm{\Theta }\right)& \sin \left(\mathrm{\Theta }\right)\\ -\sin \left(\mathrm{\Theta }\right)& \cos \left(\mathrm{\Theta }\right)\end{array}\right)|\psi {⟩}_E,\\ \hfill \left(\begin{array}{cc}+cos\left({\displaystyle \frac{1}{2}}ArcTan\left[{\displaystyle \frac{{\int }_{-\infty }^{0}dx({\psi }_{E1}\left(x\right){\psi }_{E2}^{†}\left(x\right)+{\psi }_{E1}{\left(x\right)}^{†}{\psi }_{E2}\left(x\right))}{{\int }_{-\infty }^{0}dx\left(\right|{\psi }_{E1}{\left(x\right)|}^2-|{\psi }_{E2}\left(x\right){|}^2)}}\right]\right)& sin\left({\displaystyle \frac{1}{2}}ArcTan\left[{\displaystyle \frac{{\int }_{-\infty }^{0}dx({\psi }_{E1}\left(x\right){\psi }_{E2}^{†}\left(x\right)+{\psi }_{E1}{\left(x\right)}^{†}{\psi }_{E2}\left(x\right))}{{\int }_{-\infty }^{0}dx\left(\right|{\psi }_{E1}{\left(x\right)|}^2-|{\psi }_{E2}\left(x\right){|}^2)}}\right]\right)\\ -sin\left({\displaystyle \frac{1}{2}}ArcTan\left[{\displaystyle \frac{{\int }_{-\infty }^{0}dx({\psi }_{E1}\left(x\right){\psi }_{E2}^{†}\left(x\right)+{\psi }_{E1}{\left(x\right)}^{†}{\psi }_{E2}\left(x\right))}{{\int }_{-\infty }^{0}dx\left(\right|{\psi }_{E1}{\left(x\right)|}^2-|{\psi }_{E2}\left(x\right){|}^2)}}\right]\right)& cos\left({\displaystyle \frac{1}{2}}ArcTan\left[{\displaystyle \frac{{\int }_{-\infty }^{0}dx({\psi }_{E1}\left(x\right){\psi }_{E2}^{†}\left(x\right)+{\psi }_{E1}{\left(x\right)}^{†}{\psi }_{E2}\left(x\right))}{{\int }_{-\infty }^{0}dx\left(\right|{\psi }_{E1}{\left(x\right)|}^2-|{\psi }_{E2}\left(x\right){|}^2)}}\right]\right)\end{array}\right){\left(\begin{array}{c}{\psi }_{E1}\left(x\right)\\ {\psi }_{E2}\left(x\right)\end{array}\right)}_E.\end{array}$$

(87)

Two Interacting Single-Electron Lines Described in the Tight-Binding Model and Validated by a Perturbative Schrödinger Model

We consider the system of two position-based qubits interacting perturbatively. Two electrons on two separated single-electron lines tend to be in anti-correlated positions due to repulsion force as depicted in Figure 3, Figure 6 and Figure 7. In such a way we encounter the situation of two electrons placed on two different left (right) quantum dots with probabilities $|{\xi }_1{\left(t\right)|}^2$ ($|{\xi }_4{\left(t\right)|}^2$) or the case left (right) occupancy of first dot and right (left) occupancy of second dot with probability $|{\xi }_2{\left(t\right)|}^2$ ($|{\xi }_3{\left(t\right)|}^2$) that specifies normalized quantum state ($|{\xi }_1{\left(t\right)|}^2+\dots +{\left|{\xi }_4\left(t\right)\right|}^2=1$) to be in the form as given by [37]:

$$\begin{array}{c}\hfill |\psi ⟩={\xi }_1\left(t\right)|0{⟩}_A|0{⟩}_B+{\xi }_2\left(t\right)|0{⟩}_A|1{⟩}_B+{\xi }_3\left(t\right)|1{⟩}_A|0{⟩}_B+{\xi }_4\left(t\right)|1{⟩}_A|1{⟩}_B.\end{array}$$

(88)

In the case of two single-electron lines we can write the effective Hamiltonian to be of the form

$$\begin{array}{c}\hfill {\widehat{H}}_{A-B}={\widehat{H}}_A\times {\widehat{I}}_B+{\widehat{I}}_A\times {\widehat{H}}_B+{\widehat{H}}_{Coulomb:A-B}=\\ \hfill =\left(E_{p1A}\right|w_{L,A}⟩⟨w_{L,A}|+E_{p2A}|w_{R,A}⟩⟨w_{R,A}|+t_{s12A}|w_{R,A}⟩⟨w_{L,A}|+t_{s21A}|w_{L,A}⟩⟨w_{R,A}\left|\right)\times {\widehat{I}}_B+\\ \hfill +{\widehat{I}}_A\times \left(E_{p1B}\right|w_{L,B}⟩⟨w_{L,B}|+E_{p2B}|w_{R,B}⟩⟨w_{R,B}|+t_{s12B}|w_{R,B}⟩⟨w_{L,B}|+t_{s21B}|w_{L,B}⟩⟨w_{R,B}\left|\right)+\\ \hfill +\left[q_1\right|w_{L,A}⟩|w_{L,B}⟩⟨w_{L,A}|⟨w_{L,B}|+q_2|w_{L,A}⟩|w_{L,B}⟩⟨w_{R,A}|⟨w_{R,B}|+q_3|w_{L,A}⟩|w_{L,B}⟩⟨w_{R,A}|⟨w_{R,B}|+\\ \hfill +q_4|w_{R,A}⟩|w_{R,B}⟩⟨w_{R,A}|⟨w_{R,B}|{]}_{Coulomb}\end{array}$$

(89)

Finally, we obtain validation of the tight-binding model in terms of the Schrödinger equation, so we end up with the following effective Hamiltonian.

$$\begin{array}{c}\hfill \widehat{H}=\\ \hfill \left(\begin{array}{cccc}{E}_{1a}cos{\left({\mathrm{\Theta }}_A\right)}^2+E_{2a}sin{\left({\mathrm{\Theta }}_A\right)}^2& (E_{2b}-E_{1b}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_B\right)& (E_{2a}-E_{1a}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_A\right)& 0\\ (E_{2b}-E_{1b}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_B\right)& E_{1a}cos{\left({\mathrm{\Theta }}_A\right)}^2+E_{2a}sin{\left({\mathrm{\Theta }}_A\right)}^2& 0& (E_{2a}-E_{1a}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_A\right)\\ (E_{2a}-E_{1a}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_A\right)& 0& E_{1a}sin{\left({\mathrm{\Theta }}_A\right)}^2+E_{2a}cos{\left({\mathrm{\Theta }}_A\right)}^2& (E_{2b}-E_{1b}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_B\right)\\ 0& (E_{2a}-E_{1a}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_A\right)& (E_{2b}-E_{1b}){\displaystyle \frac{1}{2}}sin\left(2{\mathrm{\Theta }}_B\right)& E_{1a}sin{\left({\mathrm{\Theta }}_A\right)}^2+E_{2a}cos{\left({\mathrm{\Theta }}_A\right)}^2\end{array}\right)+\\ \hfill \left(\begin{array}{cccc}{E}_{1b}cos{\left({\mathrm{\Theta }}_B\right)}^2+E_{2b}sin{\left({\mathrm{\Theta }}_B\right)}^2& 0& 0& 0\\ 0& E_{1b}sin{\left({\mathrm{\Theta }}_B\right)}^2+E_{2b}cos{\left({\mathrm{\Theta }}_B\right)}^2& 0& 0\\ 0& 0& E_{1b}cos{\left({\mathrm{\Theta }}_B\right)}^2+E_{2b}sin{\left({\mathrm{\Theta }}_B\right)}^2& 0\\ 0& 0& 0& E_{1b}sin{\left({\mathrm{\Theta }}_B\right)}^2+E_{2b}cos{\left({\mathrm{\Theta }}_B\right)}^2\end{array}\right)+\\ \hfill +\left(\begin{array}{cccc}{q}_1& 0& 0& 0\\ 0& q_2& 0& 0\\ 0& 0& q_3& 0\\ 0& 0& 0& q_4\end{array}\right)\end{array}$$

(90)

where four terms are responsible for two qubits being at a distance d mutual Coulomb interaction (as depicted in Figure 3.) and given as

$$\begin{array}{c}\hfill q_1={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d{x}_{A}d{x}_B{e}^2}{4\pi {ϵ}_0\sqrt{d^2+{(x_A-x_B)}^2}}}|cos\left({\mathrm{\Theta }}_A\right){\psi }_{E1}\left(x_A\right)+sin\left({\mathrm{\Theta }}_A\right){\psi }_{E2}\left(x_A\right){|}^2\ast {|cos\left({\mathrm{\Theta }}_B\right){\psi }_{E1}\left(x_B\right)+sin\left({\mathrm{\Theta }}_B\right){\psi }_{E2}\left(x_B\right)|}^2\end{array}$$

$$\begin{array}{c}\hfill q_2={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d{x}_{A}d{x}_B{e}^2}{4\pi {ϵ}_0\sqrt{d^2+{(x_A-x_B)}^2}}}|cos\left({\mathrm{\Theta }}_A\right){\psi }_{E1}\left(x_A\right)+sin\left({\mathrm{\Theta }}_A\right){\psi }_{E2}\left(x_A\right){|}^2\ast {| - sin\left({\mathrm{\Theta }}_B\right){\psi }_{E1}\left(x_B\right)+cos\left({\mathrm{\Theta }}_B\right){\psi }_{E2}\left(x_B\right)|}^2\end{array}$$

$$\begin{array}{c}\hfill q_3={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d{x}_{A}d{x}_B{e}^2}{4\pi {ϵ}_0\sqrt{d^2+{(x_A-x_B)}^2}}}|-sin\left({\mathrm{\Theta }}_A\right){\psi }_{E1}\left(x_A\right)+cos\left({\mathrm{\Theta }}_A\right){\psi }_{E2}\left(x_A\right){|}^2\ast {|cos\left({\mathrm{\Theta }}_B\right){\psi }_{E1}\left(x_B\right)+sin\left({\mathrm{\Theta }}_B\right){\psi }_{E2}\left(x_B\right)|}^2\end{array}$$

$$\begin{array}{c}\hfill q_4={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d{x}_{A}d{x}_B{e}^2}{4\pi {ϵ}_0\sqrt{d^2+{(x_A-x_B)}^2}}}|-sin\left({\mathrm{\Theta }}_A\right){\psi }_{E1}\left(x_A\right)+cos\left({\mathrm{\Theta }}_A\right){\psi }_{E2}\left(x_A\right){|}^2{| - sin\left({\mathrm{\Theta }}_B\right){\psi }_{E1}\left(x_B\right)+cos\left({\mathrm{\Theta }}_B\right){\psi }_{E2}\left(x_B\right)|}^2.\end{array}$$

In the first approximation we consider two position-based qubits as non-interacting and slightly perturbed by the electrostatic single-electron to single-electron interaction, which naturally leads to first approximation of system eigenstate as the tensor product of two Hilbert spaces belonging to each qubits in the form as given below:   

$$\begin{array}{c}\hfill |\psi {⟩}_{Energy,A-B}=\left(\begin{array}{c}{\gamma }_{E1A}\\ {\gamma }_{E2A}\end{array}\right)\times \left(\begin{array}{c}{\gamma }_{E1B}\\ {\gamma }_{E2B}\end{array}\right)=\left(\begin{array}{c}{\gamma }_{E1A}{\gamma }_{E1B}=cos\left({\xi }_A\right)e^{i{\varphi }_{E1A}}cos\left({\xi }_B\right)e^{i{\varphi }_{E1B}}\\ {\gamma }_{E1A}{\gamma }_{E2B}=cos\left({\xi }_A\right)e^{i{\varphi }_{E1A}}sin\left({\xi }_B\right)e^{i{\varphi }_{E2B}}\\ {\gamma }_{E2A}{\gamma }_{E1B}=sin\left({\xi }_A\right)e^{i{\varphi }_{E2A}}cos\left({\xi }_B\right)e^{i{\varphi }_{E1B}}\\ {\gamma }_{E2A}{\gamma }_{E2B}=sin\left({\xi }_A\right)e^{i{\varphi }_{E2A}}sin\left({\xi }_B\right)e^{i{\varphi }_{E2B}}\end{array}\right).\end{array}$$

(91)

The search for the system ground state naturally implies minimization of Coulomb energy as a criterion for the lowest energy state that can be expressed by proper self-alignment of electrons in an anticorrelated way and thus leading to minimization of Coulomb interaction energy. Coulomb energy is given as

$$\begin{array}{c}\hfill E_c({\xi }_A,{\xi }_B)=⟨\psi \left|{\widehat{H}}_c\right|\psi ⟩=\\ \hfill =q_{1}cos{\left({\xi }_A\right)}^{2}cos{\left({\xi }_B\right)}^2+q_{2}cos{\left({\xi }_A\right)}^{2}sin{\left({\xi }_B\right)}^2+q_{3}sin{\left({\xi }_A\right)}^{2}cos{\left({\xi }_B\right)}^2+q_{4}sin{\left({\xi }_A\right)}^{2}sin{\left({\xi }_B\right)}^2.\end{array}$$

(92)

Coulomb energy minimization condition brings the following constrains

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d{\xi }_A}}{E}_c({\xi }_A,{\xi }_B)=0,{\displaystyle \frac{d}{d{\xi }_B}}{E}_c({\xi }_A,{\xi }_B)=0,\end{array}$$

(93)

and hence we have

$$\begin{array}{c}\hfill 0={\displaystyle \frac{d}{d{\xi }_A}}{E}_c({\xi }_A,{\xi }_B)=2(-sin\left({\xi }_A\right)[q_{1}cos{\left({\xi }_B\right)}^2+q_{2}sin{\left({\xi }_B\right)}^2]+\\ \hfill +cos\left({\xi }_A\right)[q_{3}cos{\left({\xi }_B\right)}^2+q_{4}sin{\left({\xi }_B\right)}^2]),\end{array}$$

(94)

that leads to

$$\begin{array}{c}\hfill tan\left({\xi }_A\right)={\displaystyle \frac{[q_{3}cos{\left({\xi }_B\right)}^2+q_{4}sin{\left({\xi }_B\right)}^2]}{[q_{1}cos{\left({\xi }_B\right)}^2+q_{2}sin{\left({\xi }_B\right)}^2]}}={\displaystyle \frac{[q_3+q_{4}tan{\left({\xi }_B\right)}^2]}{[q_1+q_{2}tan{\left({\xi }_B\right)}^2]}}.\end{array}$$

(95)

Now we exercise the second constraint on energy minimization, which brings

$$\begin{array}{c}\hfill 0={\displaystyle \frac{d}{d{\xi }_B}}{E}_c({\xi }_A,{\xi }_B)=2(+cos{\left({\xi }_A\right)}^2[-q_{1}sin\left({\xi }_B\right)+q_{2}cos\left({\xi }_B\right)]+\\ \hfill +sin{\left({\xi }_A\right)}^2[-q_{3}sin\left({\xi }_B\right)+q_{4}cos\left({\xi }_B\right)]),\end{array}$$

(96)

and we obtain

$$\begin{array}{c}\hfill tan{\left({\xi }_A\right)}^2={\displaystyle \frac{[-q_{1}sin\left({\xi }_B\right)+q_{2}cos\left({\xi }_B\right)]}{[q_{3}sin\left({\xi }_B\right)-q_{4}cos\left({\xi }_B\right)]}}={\displaystyle \frac{{[q_{3}cos{\left({\xi }_B\right)}^2+q_{4}sin{\left({\xi }_B\right)}^2]}^2}{{[q_{1}cos{\left({\xi }_B\right)}^2+q_{2}sin{\left({\xi }_B\right)}^2]}^2}}.\end{array}$$

(97)

Under the assumption that qubit A is occupying two energy levels $E_{1A}$ and $E_{2A}$, while qubit B is occupying two energy levels $E_{1B}$ and $E_{2B}$, we have

$$\begin{array}{c}\hfill tan{\left({\xi }_A\right)}^2={\displaystyle \frac{[-q_{1}tan\left({\xi }_B\right)+q_2]}{[q_{3}tan\left({\xi }_B\right)-q_4]}}={\displaystyle \frac{{[q_3+q_{4}tan{\left({\xi }_B\right)}^2]}^2}{{[q_1+q_{2}tan{\left({\xi }_B\right)}^2]}^2}},\end{array}$$

(98)

$|{\gamma }_{E1A}{|}^2=cos{\left({\xi }_A\right)}^2=p_{E1A}$, $|{\gamma }_{E2A}{|}^2=cos{\left({\xi }_A\right)}^2=p_{E1A}$, $|{\gamma }_{E1B}{|}^2=cos{\left({\xi }_B\right)}^2=p_{E1B}$ and $|{\gamma }_{E2B}{|}^2=sin{\left({\xi }_B\right)}^2=p_{E2B}$. Geometry of the two electrostatically interacting structures is encoded in the $q_1$, $q_2$, $q_3$ and $q_4$ coefficients. We compute $(r_A$, $r_A$) in accordance with Formula (85) for qubits (A,B), and thus consequently we obtain the ${\mathrm{\Theta }}_A,{\mathrm{\Theta }}_B$ coefficients. By having values of $q_1$ …$q_4$ and consequently initial probability of occupancy eigenenergies in the case of separated but perturbatively interacting qubits $(E_{1A},E_{2A})$ and $(E_{1B},E_{2B})$ we can determine quantum dot occupancy during next time steps. Such a procedure can be attempted for N interacting qubits under condition of occupancy of two eigenenergy levels.

Qubits considered electrostatically interacting can be further subjected to Rabi oscillations by local microwave fields acting on qubits in their direct proximity, which might imply evolution of the system ground state with time.

$$\begin{array}{c}\hfill {\widehat{H}}_{A-B}={\widehat{H}}_A\times {\widehat{I}}_B+{\widehat{I}}_A\times {\widehat{H}}_B+{\widehat{H}}_{Coulomb:A-B}=\\ \hfill =\left(E_{p1A}\right|w_{L,A}⟩⟨w_{L,A}|+E_{p2A}|w_{R,A}⟩⟨w_{R,A}|+t_{s12A}|w_{R,A}⟩⟨w_{L,A}|+t_{s21A}|w_{L,A}⟩⟨w_{R,A}\left|\right)\times {\widehat{I}}_B+\\ \hfill +{\widehat{I}}_A\times \left(E_{p1B}\right|w_{L,B}⟩⟨w_{L,B}|+E_{p2B}|w_{R,B}⟩⟨w_{R,B}|+t_{s12B}|w_{R,B}⟩⟨w_{L,B}|+t_{s21B}|w_{L,B}⟩⟨w_{R,B}\left|\right)+\\ \hfill +\left[q_1\right|w_{L,A}⟩|w_{L,B}⟩⟨w_{L,A}|⟨w_{L,B}|+q_2|w_{L,A}⟩|w_{L,B}⟩⟨w_{R,A}|⟨w_{R,B}|+q_3|w_{L,A}⟩|w_{L,B}⟩⟨w_{R,A}|⟨w_{R,B}|+\\ \hfill +q_4|w_{R,A}⟩|w_{R,B}⟩⟨w_{R,A}|⟨w_{R,B}|{]}_{Coulomb}+\\ \hfill +\sum _{r,s,u,d}\left(\right|w_L,A⟩⟨w_L,A|+|w_R,A⟩⟨w_R,A\left|\right)\left(\right|w_L,B⟩⟨w_L,b|+|w_R,B⟩⟨w_R,B\left|\right)\\ \hfill g(t,r,s,u,d)\left(\right|E_{L,r}⟩|E_{R,s}⟩)(⟨E_{L,u}|⟨E_{R,d}\left|\right)\left(\right|w_L,A⟩⟨w_L,A|+|w_R,A⟩⟨w_R,A\left|\right)\left(\right|w_L,B⟩⟨w_L,b|+|w_R,B⟩⟨w_R,B\left|\right).\end{array}$$

(99)

It is quite straightforward to apply the presented scheme for the case of N position-based qubits implementing various quantum gates [36]. The obtained results have their significance in quantum communication as well [47].

10. Conclusions

The final section integrates the technological, analytical, and microscopic parts of the manuscript into a single assessment of the proposed electrostatic quantum-gate architecture. Its purpose is to identify what the model establishes, which remains conditional on the tight-binding approximation, and how the results can guide the design of CMOS-compatible single-electron quantum circuits. Across the work, a hierarchy of descriptions is used. The analysis begins with a two-state position qubit, extends it to a four-state Hamiltonian for two Coulomb-coupled double quantum dots, derives symmetric and asymmetric SWAP-related solutions, introduces quasiclassical energy landscapes for device tuning, maps between energy and position bases, and finally compares the reduced model with a perturbative two-body Schrödinger treatment. Gate behavior is characterized through charge-state probabilities, correlation/anticorrelation functions, and the von Neumann entropy. The device discussion connects these variables with electrostatic control, cryogenic CMOS compatibility, charge sensing, and numerical design workflows such as QTCAD [14,15].

The two QTCAD citations have distinct roles. Beaudoin et al. [14] used nonlinear Poisson–Schrödinger finite-element simulations with adaptive meshing to reproduce side-gate-activated corner-dot confinement in an industrial UTBB FD-SOI device and demonstrated robust solver convergence at 100 mK. Kriekouki et al. [15] then applied 3D quantum TCAD to a 28 nm FD-SOI device, comparing simulations with experiments to determine conditions for single-electron occupancy and Coulomb blockade. In the present article these works motivate the transition from compact analytical parameters to realistic device geometry; they are not used as evidence for the analytical derivations themselves.

The main result is that electrostatic two-qubit behavior can be organized around a small set of physically interpretable parameters: local confinement energies, tunnel couplings, Coulomb interaction energies, and geometry. Within the low-energy localized regime, these parameters provide analytical descriptions of SWAP, $\sqrt{\mathrm{SWAP}}$, ANTISWAP/repeater-like, correlated, and anticorrelated operating points. The quasiclassical treatment supplies intuitive energy-selection rules, whereas the fully quantum treatment shows how tunneling and coherent superposition convert electrostatic charge preferences into entanglement dynamics. The correlation/anticorrelation function and the von Neumann entropy provide complementary observables for following the transition from separable to entangled charge configurations.

A second major result is the explicit limitation of this picture. The perturbative Schrödinger analysis indicates that additional eigenstates can become populated with time, so the original two-level-per-qubit basis does not remain closed indefinitely. In this regime, the effective on-site energies $E_p$ and hopping amplitudes $t_s$ should be understood as low-energy matrix elements whose apparent time dependence can signal the influence of neglected states. The observation does not remove the usefulness of tight binding; rather, it defines the regime in which the compact analytical formulas can be used reliably and indicates when a larger Hilbert space, self-consistent Schrödinger–Poisson treatment, Hubbard-type extension, or another beyond-tight-binding method is required. The discussion of photonic tight-binding analogies [48] further emphasizes that similar mathematics can appear in different physical platforms while the underlying particle statistics and interaction mechanisms remain different.

Taken together, the manuscript provides a coherent analytical framework for voltage-controlled quantum logic in coupled semiconductor quantum dots. Its strongest contribution is the connection between device geometry, electrostatic bias, tunneling, charge correlation, and entanglement within a CMOS-oriented position-basis architecture. The derived formulas provide design rules for tuning correlated and anticorrelated operating regimes and identify charge observables that can be related to experimental readout. At the same time, the work demonstrates that the tight-binding approximation has a finite domain of validity: longer-time evolution, higher-state occupation, stronger driving, and many-body effects require a broader microscopic treatment. The proposed architecture is therefore best interpreted as a scalable low-energy gate framework that can serve as the analytical core of more complete numerical design tools, while experiments must determine the parameter window in which the two-level-per-qubit approximation remains quantitatively accurate. Preliminary examples of computer code incorporating tight-binding model and mathematical formulas for noise model in tight-binding model approach to single-electron semiconductor devices are included in Appendix A and Appendix B. Topological phase transitions as given by [49,50,51] can take place in superconducting-single electron semiconductor devices interfaced to Josephson junction can be grasped by tight-binding model [23]. Various experiments on single-electrons in nanostructures as given by [52,53,54,55,56,57] can be described by tight-binding model.

Appendix C.1. Case I: E_f (1, 1′) = E_f (1, 2′) = U, E_f (2, 1′) = E_f (2, 2′) = U₁

We postulate

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U& 0& 1\\ 1& 0& U_1& 1\\ 0& 1& 1& U_1\end{array}\right).\end{array}$$

(A2)

We obtain

$$\begin{array}{c}\hfill U=E_{p1}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}},\\ \hfill U=E_{p1}+E_{p{2}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}},\end{array}$$

(A3)

and it implies

$$\begin{array}{c}\hfill E_{p{2}^{\prime }}-E_{p{1}^{\prime }}=+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}},\end{array}$$

(A4)

We also have

$$\begin{array}{c}\hfill U_1=E_{p2}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}},\\ \hfill U_1=E_{p2}+E_{p{2}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}.\end{array}$$

(A5)

and we obtain

$$\begin{array}{c}\hfill E_{p{2}^{\prime }}-E_{p{1}^{\prime }}=+{\displaystyle \frac{q^2}{d}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}.\end{array}$$

(A6)

that implies condition

$$\begin{array}{c}\hfill +{\displaystyle \frac{q^2}{d}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}=\\ \hfill {\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}.\end{array}$$

(A7)

that is fulfilled for two angles $\alpha$ as in accordance to numerical solutions. Usually $a+b<<d$ since in most cases qubit size is small in comparison to the distance between qubits. Let us solve the problem with certain approximation by using Taylor expansion

$$\begin{array}{c}\hfill +{\displaystyle \frac{q^2}{d}}-{\displaystyle \frac{q^2}{d}}+{\displaystyle \frac{q^2}{d^2}}(\sqrt{d^2+2dCos\left(\alpha \right)(a+b)+{\left(Cos\left(\alpha \right)\right)}^2{(a+b)}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}-d)=\\ \hfill {\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}+\\ \hfill +{\displaystyle \frac{q^2}{d^2}}(\sqrt{{\left(Cos\left(\alpha \right)\right)}^2{(a+b)}^2+2(a+b)dCos\left(\alpha \right)+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2+{(a+b)}^2+d^2+2Sin\left(\alpha \right){(a+b)}^2}-d).\end{array}$$

(A8)

It implies

$$\begin{array}{c}\hfill +{\displaystyle \frac{1}{d}}+{\displaystyle \frac{1}{d^2}}\left(\sqrt{d^2+2dCos\left(\alpha \right)(a+b)+{(a+b)}^2}\right)=\\ \hfill {\displaystyle \frac{1}{\sqrt{d^2+{(a+b)}^2}}}+{\displaystyle \frac{1}{d^2}}\left(\sqrt{2(a+b)dCos\left(\alpha \right)+2{(a+b)}^2+d^2+2Sin\left(\alpha \right){(a+b)}^2}\right).\end{array}$$

(A9)

and hence we have

$$\begin{array}{c}\hfill +1+\left(\sqrt{1+2Cos\left(\alpha \right){\displaystyle \frac{(a+b)}{d}}+{\left({\displaystyle \frac{a+b}{d}}\right)}^2}\right)=\\ \hfill {\displaystyle \frac{d}{\sqrt{d^2+{(a+b)}^2}}}+\left(\sqrt{2{\displaystyle \frac{(a+b)}{d}}Cos\left(\alpha \right)+2{\left({\displaystyle \frac{a+b}{d}}\right)}^2+1+2Sin\left(\alpha \right){\left({\displaystyle \frac{a+b}{d}}\right)}^2}\right).\end{array}$$

(A10)

Using Taylor expansion for the square root function we obtain relation

$$\begin{array}{c}\hfill +1+1+{\displaystyle \frac{1}{2}}(2Cos\left(\alpha \right){\displaystyle \frac{(a+b)}{d}}+{\left({\displaystyle \frac{a+b}{d}}\right)}^2)=\\ \hfill {\displaystyle \frac{d}{\sqrt{d^2+{(a+b)}^2}}}+1+{\displaystyle \frac{1}{2}}(2{\displaystyle \frac{(a+b)}{d}}Cos\left(\alpha \right)+2{\left({\displaystyle \frac{a+b}{d}}\right)}^2+2Sin\left(\alpha \right){\left({\displaystyle \frac{a+b}{d}}\right)}^2).\end{array}$$

(A11)

that can be simplified into simple relation for the sinusoidal function of the form

$$\begin{array}{c}\hfill {\displaystyle \frac{d^2}{{(a+b)}^2}}[+1-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{a+b}{d}}\right)}^2-{\displaystyle \frac{d}{\sqrt{d^2+{(a+b)}^2}}}]=Sin\left(\alpha \right).\end{array}$$

(A12)

Two angles fulfill such a relation.

We can also solve Equation (A7) in a rigorous way (without approximation) equivalent to the following relation

$$\begin{array}{c}\hfill +{\displaystyle \frac{1}{d}}-{\displaystyle \frac{1}{d_1}}={\displaystyle \frac{1}{d_2}}-{\displaystyle \frac{1}{d_1+x}}\mathrm{or}{\displaystyle \frac{1}{d}}-{\displaystyle \frac{1}{d_2}}={\displaystyle \frac{1}{d_1}}-{\displaystyle \frac{1}{d_1+x}}={\displaystyle \frac{x}{d_1(d_1+x)}}.\end{array}$$

(A13)

and this implies

$$\begin{array}{c}\hfill d_1^2{\displaystyle \frac{d_2-d}{d_{2}d}}+x{d}_1{\displaystyle \frac{d_2-d}{d_{2}d}}=x\mathrm{or}+(d_1{\displaystyle \frac{d_2-d}{d_{2}d}}-1)x=-d_1^2{\displaystyle \frac{d_2-d}{d_{2}d}}.\end{array}$$

(A14)

and finally we have

$$\begin{array}{c}\hfill \left({\displaystyle \frac{d_1{d}_2-d{d}_1-d_{2}d}{d_{2}d}}\right)x=-d_1^2{\displaystyle \frac{d_2-d}{d_{2}d}},x=-d_1^2{\displaystyle \frac{d_2-d}{d_1{d}_2-d{d}_1-d_{2}d}}={\displaystyle \frac{-d_1^2{d}_2+d{d}_1^2}{d_1{d}_2-d{d}_1-d_{2}d}}.\end{array}$$

(A15)

We obtain

$${\displaystyle \frac{x+d_1}{d_1}}={\displaystyle \frac{-d_1^2{d}_2+d{d}_1^2+d_1(d_1{d}_2-d{d}_1-d_{2}d)}{(d_1{d}_2-d{d}_1-d_{2}d)d_1}}={\displaystyle \frac{-d{d}_1{d}_2}{(d_1{d}_2-d{d}_1-d_{2}d)d_1}}={\displaystyle \frac{-d{d}_2}{(d_1{d}_2-d{d}_1-d_{2}d)}}={\displaystyle \frac{d{d}_2}{d_1(-d_2+d)+d_{2}d}}$$

(A16)

This implies

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{x+d_1}{d_1}}\right)}^2={\displaystyle \frac{{\left(d{d}_2\right)}^2}{{[d_1(-d_2+d)+d_{2}d]}^2}}={\displaystyle \frac{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}=\\ \hfill ={\displaystyle \frac{{\left(d\sqrt{d^2+{(a+b)}^2}\right)}^2}{{[\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}(-\sqrt{d^2+{(a+b)}^2}+d)+\left(d\sqrt{d^2+{(a+b)}^2}\right)]}^2}}.\end{array}$$

(A17)

We recognize that we have three free parameters $E_{p1}\in R,E_{p2}\in R,E_{p{1}^{\prime }}\in R$ that predetermine $E_{p{2}^{\prime }}$ given as

$$\begin{array}{c}\hfill E_{p{2}^{\prime }}=E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}.\end{array}$$

(A18)

Using three free parameters $E_{p1}\in R,E_{p2}\in R,E_{p{1}^{\prime }}\in R$ we obtain

$$\begin{array}{c}\hfill U_1=2E_{p{1}^{\prime }}+{\displaystyle \frac{2q^2}{d}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}},\end{array}$$

(A19)

$$\begin{array}{c}\hfill U=E_{p1}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}.\end{array}$$

(A20)

We have four energy eigenvalues

$$E_1={\displaystyle \frac{1}{2}}\left(-\sqrt{{(U-U_1)}^2+4}+U+U_1)-2\right),$$

(A21)

$$E_2={\displaystyle \frac{1}{2}}\left(-\sqrt{{(U-U_1)}^2+4}+U+U_1)+2\right),$$

(A22)

$$E_3={\displaystyle \frac{1}{2}}\left(+\sqrt{{(U-U_1)}^2+4}+U+U_1)-2\right),$$

(A23)

$$E_4={\displaystyle \frac{1}{2}}\left(+\sqrt{{(U-U_1)}^2+4}+U+U_1)+2\right).$$

(A24)

and we obtain the following eigenstates

$$|E_1⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}\left(\sqrt{U^2-2U{U}_1+U_1^2+4}-U+U_1\right)\\ {\displaystyle \frac{1}{2}}\left(-\sqrt{U^2-2U{U}_1+U_1^2+4}+U-U_1\right)\\ -1\\ +1\end{array}\right),$$

(A25)

$$|E_2⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}\left(-\sqrt{U^2-2U{U}_1+U_1^2+4}+(U-U_1)\right)\\ {\displaystyle \frac{1}{2}}\left(-\sqrt{U^2-2U{U}_1+U_1^2+4}+(U-U_1)\right)\\ 1\\ 1\end{array}\right),$$

(A26)

$$|E_3⟩=\left(\begin{array}{c}-{\displaystyle \frac{1}{2}}\left(\sqrt{U^2-2U{U}_1+U_1^2+4}+(U-U_1)\right)\\ +{\displaystyle \frac{1}{2}}\left(\sqrt{U^2-2U{U}_1+U_1^2+4}+(U-U_1)\right)\\ -1\\ +1\end{array}\right),$$

(A27)

$$|E_4⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}\left(\sqrt{U^2-2U{U}_1+U_1^2+4}+U-U_1\right)\\ {\displaystyle \frac{1}{2}}\left(\sqrt{U^2-2U{U}_1+U_1^2+4}+U-U_1\right)\\ +1\\ +1\end{array}\right).$$

(A28)

Appendix C.2. Case II: E_f (1, 1′) = E_f (2, 2′) = U, E_f (1, 2′) = E_f (2, 1′) = U₁

We postulate

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U_1& 0& 1\\ 1& 0& U_1& 1\\ 0& 1& 1& U\end{array}\right),\end{array}$$

(A29)

that implies

$$\begin{array}{c}\hfill U=E_{p1}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}},\\ \hfill U=E_{p2}+E_{p{2}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}.\end{array}$$

(A30)

and consequently we obtain

$$\begin{array}{c}\hfill (E_{p1}-E_{p2})+(E_{p{1}^{\prime }}-E_{p{2}^{\prime }})={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}.\end{array}$$

(A31)

We also have

$$\begin{array}{c}\hfill U_1=E_{p1}+E_{p{2}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}},\\ \hfill U_1=E_{p2}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}},\end{array}$$

(A32)

and it implies

$$\begin{array}{c}\hfill -(E_{p1}-E_{p2})+(E_{p{1}^{\prime }}-E_{p{2}^{\prime }})={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}.\end{array}$$

(A33)

This results in

$$\begin{array}{c}\hfill (E_{p{1}^{\prime }}-E_{p{2}^{\prime }})={\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A34)

and

$$\begin{array}{c}\hfill (E_{p1}-E_{p2})={\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill -{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A35)

Therefore we have three open parameters $E_{p1}\in R$, $E_{p{1}^{\prime }}\in R$, and for any $\alpha \in (0,2\pi )$ we have predetermined conditions for $E_{p2}$ and $E_{p{2}^{\prime }}$ given as

$$\begin{array}{c}\hfill E_{p2}=E_{p1}-{\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill -{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A36)

and

$$\begin{array}{c}\hfill E_{p{2}^{\prime }}=E_{p{1}^{\prime }}-{\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A37)

and implies $U\in R$

$$\begin{array}{c}\hfill U=E_{p1}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}},\end{array}$$

(A38)

as well as

$$\begin{array}{c}\hfill U_1=E_{p2}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}}=2E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}}-{\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A39)

We have the following eigenvalues

$$\begin{array}{c}\hfill E_1=U,E_2=U_1,E_3={\displaystyle \frac{(U+U_1-\sqrt{{\left(4\right)}^2+{(U-U_1)}^2})}{2}},E_4={\displaystyle \frac{(U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2})}{2}}.\end{array}$$

(A40)

and eigenstates

$$\begin{array}{c}\hfill |E_1⟩=\left(\begin{array}{c}-1\\ 0\\ 0\\ 1\end{array}\right),\end{array}$$

(A41)

$$\begin{array}{c}\hfill |E_2⟩=\left(\begin{array}{c}0\\ -1\\ +1\\ 0\end{array}\right),\end{array}$$

(A42)

$$\begin{array}{c}\hfill |E_3⟩=\left(\begin{array}{c}1\\ {\displaystyle \frac{4}{-U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ {\displaystyle \frac{4}{-U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ 1\end{array}\right),\end{array}$$

(A43)

$$\begin{array}{c}\hfill |E_4⟩=\left(\begin{array}{c}1\\ {\displaystyle \frac{4}{U-U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ {\displaystyle \frac{4}{U-U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ 1\end{array}\right),\end{array}$$

(A44)

Appendix C.3. Case III: E_f (1, 1′) = E_f (2, 1′) = U, E_f (1, 2′) = E_f (2, 2′) = U₁

We postulate

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U_1& 0& 1\\ 1& 0& U& 1\\ 0& 1& 1& U_1\end{array}\right).\end{array}$$

(A45)

that implies $U={\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}+E_p\left(1\right)+E_p\left(1^{\prime }\right)$, $U={\displaystyle \frac{q^2}{d}}+E_{p2}+E_p\left(1^{\prime }\right)$ and we obtain

$$\begin{array}{c}\hfill E_{p2}=E_{p1}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}},\end{array}$$

(A46)

We also have

$$\begin{array}{c}\hfill U_1={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}+E_{p1}+E_{p{2}^{\prime }},\\ \hfill U_1={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}+E_{p2}+E_{p{2}^{\prime }},\end{array}$$

(A47)

which implies

$$\begin{array}{c}\hfill E_{p2}=E_{p1}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}.\end{array}$$

(A48)

and consequently we obtain the condition

$$\begin{array}{c}\hfill {\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}=\\ \hfill {\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\end{array}$$

(A49)

that is fulfilled for one angle $\alpha$. It gives us three controlling parameters $E_{p1}$, $E_{p{1}^{\prime }}$, $E_{p{2}^{\prime }}$ for given $(a+b),d$ that determine

$$\begin{array}{c}\hfill U_1={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}+E_{p1}+E_{p{2}^{\prime }},U={\displaystyle \frac{q^2}{d}}+E_{p2}+E_p\left(1^{\prime }\right).\end{array}$$

We obtain the following Hamiltonian eigenvalues and eigenstates

$$\begin{array}{c}\hfill E_1={\displaystyle \frac{-2+U+U_1-\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}},\\ \hfill E_2={\displaystyle \frac{+2+U+U_1-\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}},\\ \hfill E_3={\displaystyle \frac{-2+U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}},\\ \hfill E_4={\displaystyle \frac{+2+U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}}.\end{array}$$

(A50)

and

$$\begin{array}{c}\hfill |E_1⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}(-U+\sqrt{4+{(U-U_1)}^2}+U_1)\\ -1\\ {\displaystyle \frac{1}{2}}(U-\sqrt{4+{(U-U_1)}^2}-U_1)\\ 1\end{array}\right),\end{array}$$

(A51)

$$\begin{array}{c}\hfill |E_2⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}(U-\sqrt{4+{(U-U_1)}^2}-U_1)\\ 1\\ {\displaystyle \frac{1}{2}}(U-\sqrt{4+{(U-U_1)}^2}-U_1)\\ 1\end{array}\right),\end{array}$$

(A52)

$$\begin{array}{c}\hfill |E_3⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}(-U-\sqrt{4+{(U-U_1)}^2}+U_1)\\ -1\\ {\displaystyle \frac{1}{2}}(U+\sqrt{4+{(U-U_1)}^2}-U_1)\\ 1\end{array}\right),\end{array}$$

(A53)

$$\begin{array}{c}\hfill |E_4⟩=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}(-U-\sqrt{4+{(U-U_1)}^2}+U_1)\\ 1\\ {\displaystyle \frac{1}{2}}(U+\sqrt{4+{(U-U_1)}^2}-U_1)\\ 1\end{array}\right),\end{array}$$

(A54)

Appendix C.4. Case of Density Matrix in the Case of Two Interacting Particles in the Symmetric Case

We consider the simplifying matrix and highly symmetric matrix of the form

$$\begin{array}{c}\hfill \widehat{H}\left(t\right)=\\ \hfill \left(\begin{array}{cccc}2E_p\left(t\right)+{\displaystyle \frac{q^2}{d_1}}=q_{11}+q_{22}& t_{sr2}\left(t\right)& t_{sr1}\left(t\right)& 0\\ t_{sr2}\left(t\right)& 2E_p\left(t\right)+{\displaystyle \frac{q^2}{\sqrt{{\left(d_1\right)}^2+{(b+a)}^2}}}=q_{11}-q_{22}& 0& t_{sr1}\left(t\right)\\ t_{sr1}\left(t\right)& 0& 2E_p\left(t\right)+{\displaystyle \frac{q^2}{\sqrt{{\left(d_1\right)}^2+{(b+a)}^2}}}=q_{11}-q_{22}& t_{sr2}\left(t\right)\\ 0& t_{sr1}\left(t\right)& t_{sr2}\left(t\right)& 2E_p\left(t\right)+{\displaystyle \frac{q^2}{d1}}=q_{11}+q_{22}\end{array}\right)=\\ \hfill ={\widehat{\sigma }}_0\times {\widehat{\sigma }}_0{q}_{11}+{\widehat{\sigma }}_3\times {\widehat{\sigma }}_3{q}_{22}+t_{sr2}\left(t\right){\widehat{\sigma }}_0\times {\widehat{\sigma }}_3+t_{sr1}\left(t\right){\widehat{\sigma }}_3\times {\widehat{\sigma }}_0\end{array}$$

(A55)

that has only real-valued components $H_{k,l}$ with $q_{11}=E_p\left(t\right)+{\displaystyle \frac{E_{c1}+E_{c2}}{2}}=E_p\left(t\right)+{\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{d_1}}+{\displaystyle \frac{q^2}{\sqrt{{\left(d_1\right)}^2+{(b+a)}^2}}})$, $q_{22}={\displaystyle \frac{E_{c1}-E_{c2}}{2}}={\displaystyle \frac{1}{2}}({\displaystyle \frac{q^2}{d_1}}-{\displaystyle \frac{q^2}{\sqrt{{\left(d_1\right)}^2+{(b+a)}^2}}})$ and $Q_{11}\left(t\right)={\int }_{t_0}^{t}d{t}^{\prime }{q}_{11}\left(t^{\prime }\right)$, $Q_{22}\left(t\right)={\int }_{t_0}^{t}d{t}^{\prime }{q}_{22}\left(t^{\prime }\right)$, $TR1\left(t\right)={\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr1}\left(t^{\prime }\right)$, $TR2\left(t\right)={\int }_{t_0}^{t}d{t}^{\prime }{t}_{sr2}\left(t^{\prime }\right)$. We obtain the density matrix

$$\begin{array}{c}\hfill \widehat{U}\left(t\right)=\left(\begin{array}{cccc}{U}_{1,1}\left(t\right)& U_{1,2}\left(t\right)& U_{1,3}\left(t\right)& U_{1,4}\left(t\right)\\ U_{2,1}\left(t\right)& U_{2,2}\left(t\right)& U_{2,3}\left(t\right)& U_{2,4}\left(t\right)\\ U_{3,1}\left(t\right)& U_{3,2}\left(t\right)& U_{3,3}\left(t\right)& U_{3,4}\left(t\right)\\ U_{4,1}\left(t\right)& U_{4,2}\left(t\right)& U_{4,3}\left(t\right)& U_{4,4}\left(t\right)\end{array}\right),\widehat{\rho }\left(t\right)=\widehat{U}(t,t_0)\left(\begin{array}{cccc}{\rho }_{1,1}\left(t_0\right)& {\rho }_{1,2}\left(t_0\right)& {\rho }_{1,3}\left(t_0\right)& {\rho }_{1,4}\left(t_0\right)\\ {\rho }_{2,1}\left(t_0\right)& {\rho }_{2,2}\left(t_0\right)& {\rho }_{2,3}\left(t_0\right)& {\rho }_{2,4}\left(t_0\right)\\ {\rho }_{3,1}\left(t_0\right)& {\rho }_{3,2}\left(t_0\right)& {\rho }_{3,3}\left(t_0\right)& {\rho }_{3,4}\left(t_0\right)\\ {\rho }_{4,1}\left(t_0\right)& {\rho }_{4,2}\left(t_0\right)& {\rho }_{4,3}\left(t_0\right)& {\rho }_{4,4}\left(t_0\right)\end{array}\right){\widehat{U}}^{-1}(t,t_0)\end{array}$$

(A56)

with the following components of unitary matrix

$$\begin{array}{c}\hfill U_{1,1}\left(t\right)={\displaystyle \frac{e^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}}{2}}[-i{Q}_{22}\left(t\right)\times \\ \hfill \times \left({\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}\right)+\\ \hfill +\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)].\end{array}$$

(A57)

$$\begin{array}{c}\hfill U_{1,2}\left(t\right)={\displaystyle \frac{i{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}((TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}\\ \hfill -{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right))}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}},\end{array}$$

(A58)

$$\begin{array}{c}\hfill U_{1,3}\left(t\right)=-i{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}{\displaystyle \frac{[(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}\\ \hfill +{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{|TR1\left(t\right)+TR2\left(t\right)|}^2}\right)]}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}.\end{array}$$

(A59)

$$\begin{array}{c}\hfill U_{1,4}\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}\left[i{Q}_{22}\left(t\right)\right[{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}\\ \hfill -{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}]\\ \hfill -\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]\end{array}$$

(A60)

$$\begin{array}{c}\hfill U_{2,1}\left(t\right)=-{\displaystyle \frac{i}{2}}{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}{\displaystyle \frac{[(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}\\ \hfill -{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}\end{array}$$

(A61)

$$\begin{array}{c}\hfill U_{2,2}\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}\left[i{Q}_{22}\left(t\right)\right[{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill {\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}]+\\ \hfill +\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]\end{array}$$

(A62)

$$\begin{array}{c}\hfill U_{2,3}\left(t\right)=e^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}[{\displaystyle \frac{-Q_{22}\left(t\right)\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}+\\ \hfill +{\displaystyle \frac{i\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-i\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{2}}]\end{array}$$

(A63)

$$\begin{array}{c}\hfill U_{2,4}\left(t\right)=-{\displaystyle \frac{i{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}[(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill +{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)]}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}\end{array}$$

(A64)

$$\begin{array}{c}\hfill U_{3,1}\left(t\right)=-i{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}{\displaystyle \frac{[(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill +{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}\end{array}$$

(A65)

$$\begin{array}{c}\hfill U_{3,2}\left(t\right)=e^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}[{\displaystyle \frac{-Q_{22}\left(t\right)\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill {\displaystyle \frac{Q_{22}\left(t\right)\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}+\\ \hfill +{\displaystyle \frac{i\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-i\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]}{2}}\end{array}$$

(A66)

$$\begin{array}{c}\hfill U_{3,3}\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}\left[i{Q}_{22}\left(t\right)\right[{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}\\ \hfill +{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}]+\\ \hfill +\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]\end{array}$$

(A67)

$$\begin{array}{c}\hfill U_{3,4}\left(t\right)=(\sin \left(\mathit{\hslash }{Q}_{11}\left(t\right)\right)+i\cos \left(\mathit{\hslash }{Q}_{11}\left(t\right)\right)){\displaystyle \frac{[(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill -{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}\end{array}$$

(A68)

$$\begin{array}{c}\hfill U_{4,1}\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}\left[i{Q}_{22}\left(t\right)\right[{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}\\ \hfill -{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}]+\\ \hfill -\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]\end{array}$$

(A69)

$$\begin{array}{c}\hfill U_{4,2}\left(t\right)=-{\displaystyle \frac{i{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}[(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill +{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}\end{array}$$

(A70)

$$\begin{array}{c}\hfill U_{4,3}\left(t\right)=i{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}[{\displaystyle \frac{(TR1\left(t\right)-TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill -{\displaystyle \frac{(TR1\left(t\right)+TR2\left(t\right))\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{2\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}]\end{array}$$

(A71)

$$\begin{array}{c}\hfill U_{4,4}\left(t\right)={\displaystyle \frac{1}{2}}{e}^{-i\mathit{\hslash }{Q}_{11}\left(t\right)}[-i{Q}_{22}\left(t\right)[{\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}}+\\ \hfill {\displaystyle \frac{\sin \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)}{\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}}}]+\\ \hfill +\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\cos \left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)+TR2\left(t\right))}^2}\right)]\end{array}$$

(A72)

We set the quantum state to be $|\psi ,t_0⟩=|E_1⟩$ at time $t_0$ so it is maximally entangled and its density matrix is $\rho \left(t_0\right)=|\psi ,t_0⟩⟨\psi ,t_0|=|E_1⟩⟨E_1|={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}+1& 0& 0& -1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -1& 0& 0& 1\end{array}\right)$. Finally we obtain the following density matrix

$$\begin{array}{c}\hfill {\rho }_{1,1}\left(t\right)={\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+2{\left|Q_{22}\left(t\right)\right|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A73)

$$\begin{array}{c}\hfill {\rho }_{1,2}\left(t\right)={\displaystyle \frac{(TR1\left(t\right)-TR2\left(t\right))[-i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A74)

$$\begin{array}{c}\hfill {\rho }_{1,3}\left(t\right)=-(TR1\left(t\right)-TR2\left(t\right)){\displaystyle \frac{[-i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A75)

$$\begin{array}{c}\hfill {\rho }_{1,4}\left(t\right)=-{\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+2{\left|Q_{22}\left(t\right)\right|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A76)

$$\begin{array}{c}\hfill {\rho }_{2,1}\left(t\right)={\displaystyle \frac{(TR1\left(t\right)-TR2\left(t\right))(i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A77)

$$\begin{array}{c}\hfill {\rho }_{2,2}\left(t\right)={\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2{\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A78)

$$\begin{array}{c}\hfill {\rho }_{2,3}\left(t\right)=-{\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2{\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A79)

$$\begin{array}{c}\hfill {\rho }_{2,4}\left(t\right)=-{\displaystyle \frac{(TR1\left(t\right)-TR2\left(t\right))[i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A80)

$$\begin{array}{c}\hfill {\rho }_{3,1}\left(t\right)=-(TR1\left(t\right)-TR2\left(t\right)){\displaystyle \frac{[i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A81)

$$\begin{array}{c}\hfill {\rho }_{3,2}\left(t\right)=-{\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2{\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A82)

$$\begin{array}{c}\hfill {\rho }_{3,3}\left(t\right)={\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2{\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{2\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A83)

$$\begin{array}{c}\hfill {\rho }_{3,4}\left(t\right)={\displaystyle \frac{(TR1\left(t\right)-TR2\left(t\right))(i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill {\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A84)

$$\begin{array}{c}\hfill {\rho }_{4,1}\left(t\right)=-{\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+2{\left|Q_{22}\left(t\right)\right|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A85)

$$\begin{array}{c}\hfill {\rho }_{4,2}\left(t\right)=-(TR1\left(t\right)-TR2\left(t\right)){\displaystyle \frac{[-i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill +{\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A86)

$$\begin{array}{c}\hfill {\rho }_{4,3}\left(t\right)=(TR1\left(t\right)-TR2\left(t\right)){\displaystyle \frac{[-i\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\sin \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}+\\ \hfill {\displaystyle \frac{Q_{22}\left(t\right)\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-Q_{22}\left(t\right)]}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A87)

$$\begin{array}{c}\hfill {\rho }_{4,4}\left(t\right)={\displaystyle \frac{{(TR1\left(t\right)-TR2\left(t\right))}^2\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+2{\left|Q_{22}\left(t\right)\right|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}{4\left(|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right)}}\end{array}$$

(A88)

It transpires that ${\rho }^n\left(t\right)=\rho \left(t\right)$, so one deals with a pure quantum state. Now we are obtaining reduced matrices describing the state of particle B from a two-particle density matrix.

$$\begin{array}{c}\hfill {\rho }_B\left(t\right)=\left(\begin{array}{cc}{\rho }_{11}\left(t\right)+{\rho }_{22}\left(t\right)& {\rho }_{13}\left(t\right)+{\rho }_{24}\left(t\right)\\ {\rho }_{31}\left(t\right)+{\rho }_{42}\left(t\right)& {\rho }_{33}\left(t\right)+{\rho }_{44}\left(t\right)\end{array}\right)=\\ \hfill \left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}\\ {\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}& {\displaystyle \frac{1}{2}}\end{array}\right).\end{array}$$

(A89)

Consequently we can compute entanglement entropy. At first we evaluate

$$\begin{array}{c}\hfill Log\left({\rho }_B\left(t\right)\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),\end{array}$$

(A90)

$$\begin{array}{c}\hfill a={\displaystyle \frac{1}{2}}\left[\log \right[Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\\ \hfill +|Q_{22}{\left(t\right)|}^2+Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right))+{(TR1\left(t\right)-TR2\left(t\right))}^2]\\ \hfill -2\log \left[|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right]\\ \hfill +\log \left[\right[Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right))\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\\ \hfill |Q_{22}{\left(t\right)|}^2+Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))+{(TR1\left(t\right)-TR2\left(t\right))}^2]-\log \left(4\right)]\end{array}$$

$b=-{\tanh }^{-1}\left({\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))\left(\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-1\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}\right)=c$

$$\begin{array}{c}\hfill d={\displaystyle \frac{1}{2}}\left[\log \right[Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\\ \hfill |Q_{22}{\left(t\right)|}^2+Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right))+{(TR1\left(t\right)-TR2\left(t\right))}^2]\\ \hfill -2\log \left[|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right]]\\ \hfill +\log \left[\right]Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right))\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\\ \hfill |Q_{22}{\left(t\right)|}^2+Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))+{(TR1\left(t\right)-TR2\left(t\right))}^2]-\log \left(4\right)]\end{array}$$

(A91)

and we obtain the formula when we start from $TR1\left(t_0\right)=TR2\left(t_0\right)$ as

$$\begin{array}{c}\hfill S_B\left(t\right)=Tr\left[{\rho }_B\left(t\right)Log\left[{\rho }_B\left(t\right)\right]\right]=\\ \hfill =Tr[\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}\\ {\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}& {\displaystyle \frac{1}{2}}\end{array}\right)\times \\ \hfill Log\left[\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}\\ {\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}& {\displaystyle \frac{1}{2}}\end{array}\right)\right]]=\\ \hfill =-\log \left(4\right){\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\left[\log \right[Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)+\\ \hfill +|Q_{22}{\left(t\right)|}^2+Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right))+{(TR1\left(t\right)-TR2\left(t\right))}^2]+\\ \hfill +\log [Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right))\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)\\ \hfill +|Q_{22}{\left(t\right)|}^2+Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))+{(TR1\left(t\right)-TR2\left(t\right))}^2]\\ \hfill -2\log \left[|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2\right]\\ \hfill +{\displaystyle \frac{4Q_{22}\left(t\right)(TR2\left(t\right)-TR1\left(t\right)){\sin }^2\left(\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}\times \\ \hfill \times {\tanh }^{-1}\left({\displaystyle \frac{Q_{22}\left(t\right)(TR1\left(t\right)-TR2\left(t\right))\left(\cos \left(2\mathit{\hslash }\sqrt{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}\right)-1\right)}{|Q_{22}{\left(t\right)|}^2+{(TR1\left(t\right)-TR2\left(t\right))}^2}}\right)]\end{array}$$

(A92)

Appendix D.1. Correlation Function for Case II: E_f (1, 1′) = E_f (2, 1′) = U, E_f (1, 2′) = E_f (2, 2′) = U₁

We postulate

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U_1& 0& 1\\ 1& 0& U_1& 1\\ 0& 1& 1& U\end{array}\right),\end{array}$$

(A95)

that implies

$$\begin{array}{c}\hfill U=E_{p1}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}},\\ \hfill U=E_{p2}+E_{p{2}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}.\end{array}$$

(A96)

and consequently we obtain

$$\begin{array}{c}\hfill (E_{p1}-E_{p2})+(E_{p{1}^{\prime }}-E_{p{2}^{\prime }})={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}.\end{array}$$

(A97)

We also have

$$\begin{array}{c}\hfill U_1=E_{p1}+E_{p{2}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}},\\ \hfill U_1=E_{p2}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}},\end{array}$$

(A98)

and it implies

$$\begin{array}{c}\hfill -(E_{p1}-E_{p2})+(E_{p{1}^{\prime }}-E_{p{2}^{\prime }})={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}.\end{array}$$

(A99)

This results in

$$\begin{array}{c}\hfill (E_{p{1}^{\prime }}-E_{p{2}^{\prime }})={\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A100)

and

$$\begin{array}{c}\hfill (E_{p1}-E_{p2})={\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill -{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A101)

Therefore we have three open parameters $E_{p1}\in R$, $E_{p{1}^{\prime }}\in R$, and for any $\alpha \in (0,2\pi )$ we have predetermined conditions for $E_{p2}$ and $E_{p{2}^{\prime }}$ given as

$$\begin{array}{c}\hfill E_{p2}=E_{p1}-{\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill -{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A102)

and

$$\begin{array}{c}\hfill E_{p{2}^{\prime }}=E_{p{1}^{\prime }}-{\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A103)

and implies $U\in R$

$$\begin{array}{c}\hfill U=E_{p1}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}},\end{array}$$

(A104)

as well as

$$\begin{array}{c}\hfill U_1=E_{p2}+E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}}=2E_{p{1}^{\prime }}+{\displaystyle \frac{q^2}{d}}-{\displaystyle \frac{1}{2}}[{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{d}}\\ \hfill +{\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{\left(Sin\left(\alpha \right)\right)}^2{(a+b)}^2}}}-{\displaystyle \frac{q^2}{\sqrt{d^2+{(a+b)}^2}}}].\end{array}$$

(A105)

We have the following eigenvalues

$$\begin{array}{c}\hfill E_1=U,E_2=U_1,E_3={\displaystyle \frac{(U+U_1-\sqrt{{\left(4\right)}^2+{(U-U_1)}^2})}{2}},E_4={\displaystyle \frac{(U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2})}{2}}.\end{array}$$

(A106)

and eigenstates

$$\begin{array}{c}\hfill |E_1⟩=\left(\begin{array}{c}-1\\ 0\\ 0\\ 1\end{array}\right),\end{array}$$

(A107)

$$\begin{array}{c}\hfill |E_2⟩=\left(\begin{array}{c}0\\ -1\\ +1\\ 0\end{array}\right),\end{array}$$

(A108)

$$\begin{array}{c}\hfill |E_3⟩=\left(\begin{array}{c}1\\ {\displaystyle \frac{4}{-U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ {\displaystyle \frac{4}{-U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ 1\end{array}\right),\end{array}$$

(A109)

$$\begin{array}{c}\hfill |E_4⟩=\left(\begin{array}{c}1\\ {\displaystyle \frac{4}{U-U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ {\displaystyle \frac{4}{U-U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}}\\ 1\end{array}\right),\end{array}$$

(A110)

Appendix D.2. Correlation Function for Case III: E_f (1, 1′) = E_f (2, 1′) = U

$$\begin{array}{c}\hfill C(t,{\varphi }_{E_1},{\varphi }_{E_2},{\varphi }_{E_3},{\varphi }_{E_4},U,U_1,E_1(U,U_1),E_2(U,U_1),E_3(U,U_1),E_4(U,U_1))=\\ \hfill \left(2\right(c_{E1}{c}_{E2}\sqrt{{(U-U_1)}^2+4}(U-U_1)\cos (-E_{1n}t+E_{2n}t+{\varphi }_{E1}-{\varphi }_{E2})+\\ \hfill +c_{E4}(c_{E1}\sqrt{(U-U_1)\left(\sqrt{{(U-U_1)}^2+4}+U-U_1\right)+4}\sqrt{4-(U-U_1)\left(\sqrt{{(U-U_1)}^2+4}-U+U_1\right)}\times \\ \hfill \times \cos ((-E_1+E_4)t+{\varphi }_{E1}-{\varphi }_{E4})+\\ \hfill +c_{E3}\sqrt{{(U-U_1)}^2+4}(U_1-U)\cos ((-E_3+E_4)t+{\varphi }_{E3}-{\varphi }_{E4}))+\\ \hfill +c_{E2}{c}_{E3}\sqrt{(U-U_1)\left(\sqrt{{(U-U_1)}^2+4}+U-U_1\right)+4}\sqrt{4-(U-U_1)\left(\sqrt{{(U-U_1)}^2+4}-U+U_1\right)}\times \\ \hfill \times \cos ((-E_2+E_3)t+{\varphi }_{E2}-{\varphi }_{E3})\left)\right){\displaystyle \frac{1}{{(U-U_1)}^2+4}},\end{array}$$

(A111)

where

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccc}U& 1& 1& 0\\ 1& U_1& 0& 1\\ 1& 0& U& 1\\ 0& 1& 1& U_1\end{array}\right).\end{array}$$

(A112)

and

$$\begin{array}{c}\hfill U_1={\displaystyle \frac{q^2}{\sqrt{{(d+Cos\left(\alpha \right)(a+b))}^2+{(1+Sin\left(\alpha \right))}^2{(a+b)}^2}}}+E_{p1}+E_{p{2}^{\prime }},U={\displaystyle \frac{q^2}{d}}+E_{p2}+E_p\left(1^{\prime }\right),\end{array}$$

$$\begin{array}{c}\hfill E_1={\displaystyle \frac{-2+U+U_1-\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}},\\ \hfill E_2={\displaystyle \frac{+2+U+U_1-\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}},\\ \hfill E_3={\displaystyle \frac{-2+U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}},\\ \hfill E_4={\displaystyle \frac{+2+U+U_1+\sqrt{{\left(4\right)}^2+{(U-U_1)}^2}}{2}}.\end{array}$$

(A113)

Now we generalize the Hamiltonian matrix of the form

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccc}U& t_{s2}& t_{s1}& 0\\ t_{s2}& U_1& 0& t_{s1}\\ t_{s1}& 0& U& t_{s2}\\ 0& t_{s1}& t_{s2}& U_1\end{array}\right).\end{array}$$

(A114)

and energy eigenvalues dependent on $t_{s1}$, $t_{s2}$, U and $U_1$ are given as

$$\begin{array}{c}\hfill E_1={\displaystyle \frac{1}{2}}(-2t_{s1}+U+U_1-\sqrt{{(U-U_1)}^2+{\left(2t_{s2}\right)}^2}),\\ \hfill E_2={\displaystyle \frac{1}{2}}(+2t_{s1}+U+U_1-\sqrt{{(U-U_1)}^2+{\left(2t_{s2}\right)}^2}),\\ \hfill E_3={\displaystyle \frac{1}{2}}(-2t_{s1}+U+U_1+\sqrt{{(U-U_1)}^2+{\left(2t_{s2}\right)}^2}),\\ \hfill E_4={\displaystyle \frac{1}{2}}(+2t_{s1}+U+U_1+\sqrt{{(U-U_1)}^2+{\left(2t_{s2}\right)}^2}).\end{array}$$

(A115)

It is surprising to discover that the corresponding energy eigenstates are dependent only on the $t_{s2}$ hopping constant and are not dependent on the $t_{s1}$ hopping constant and are given as

$$\begin{array}{c}\hfill |E_1⟩=\left(\begin{array}{c}{\displaystyle \frac{\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ -\left({\displaystyle \frac{\left(2t_{s2}\right)}{\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}}}\right),\\ -\left({\displaystyle \frac{\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}}\right),\\ {\displaystyle \frac{\left(2t_{s2}\right)}{\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}}},\end{array}\right),\\ \hfill |E_2⟩=\left(\begin{array}{c}-{\displaystyle \frac{\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ +{\displaystyle \frac{2t_{s2}}{\sqrt{8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2}}},\\ -{\displaystyle \frac{\sqrt{8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ +{\displaystyle \frac{2t_{s2}}{\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}}},\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill |E_3⟩=\left(\begin{array}{c}-{\displaystyle \frac{\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ -{\displaystyle \frac{(\left(2t_{s2}\right)}{\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}}),\\ {\displaystyle \frac{\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ {\displaystyle \frac{2t_{s2}}{\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}}\end{array}\right),\end{array}$$

(A116)

$$\begin{array}{c}\hfill |E_4⟩=\left(\begin{array}{c}{\displaystyle \frac{\sqrt{[8t_{s2}^2+2t_{s2}^2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ {\displaystyle \frac{2t_{s2}}{\sqrt{[8t_{s2}^2+2t_{s2}^2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}},\\ {\displaystyle \frac{\sqrt{[8t_{s2}^2+2t_{s2}^2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}},\\ {\displaystyle \frac{2t_{s2}}{\sqrt{[8t_{s2}^2+2t_{s2}^2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}}}\end{array}\right).\end{array}$$

(A117)

The minimalist density matrix of two electrostatically interacting qubits A and B in the tight-binding model in the case of the time-independent Hamiltonian is given as

$$\begin{array}{c}\hfill {\rho }_{AB}\left(t\right)=|c_{E1}{|}^2|E_1⟩⟨E_1|+|c_{E2}{|}^2|E_2⟩⟨E_2|+|c_{E3}{|}^2|E_3⟩⟨E_3|+{\left|c_{E4}\right|}^2|E_4⟩⟨E_4|+\\ \hfill +c_{E1}{c}_{E2}\left(\right|E_1⟩⟨E_2|e^{i({\varphi }_{E1\left(t_0\right)}-{\varphi }_{E2\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_1-E_2)(t-t_0)}+|E_2⟩⟨E_1\left|\right)e^{i({\varphi }_{E2\left(t_0\right)}-{\varphi }_{E1\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_2-E_1)(t-t_0)}+\\ \hfill +c_{E1}{c}_{E3}\left(e^{i({\varphi }_{E1\left(t_0\right)}-{\varphi }_{E3\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_1-E_3)(t-t_0)}\right|E_1⟩⟨E_3|+e^{i({\varphi }_{E3\left(t_0\right)}-{\varphi }_{E1\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_3-E_1)(t-t_0)}|E_3⟩⟨E_1\left|\right)+\\ \hfill +c_{E1}{c}_{E4}\left(e^{i({\varphi }_{E1\left(t_0\right)}-{\varphi }_{E4\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_1-E_4)(t-t_0)}\right|E_1⟩⟨E_4|+e^{i({\varphi }_{E4\left(t_0\right)}-{\varphi }_{E1\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_4-E_1)(t-t_0)}|E_4⟩⟨E_1\left|\right)+\\ \hfill +c_{E2}{c}_{E3}\left(e^{i({\varphi }_{E2\left(t_0\right)}-{\varphi }_{E3\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_2-E_3)(t-t_0)}\right|E_2⟩⟨E_3|+e^{i({\varphi }_{E3\left(t_0\right)}-{\varphi }_{E2\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_3-E_2)(t-t_0)}|E_3⟩⟨E_2\left|\right)+\\ \hfill +c_{E2}{c}_{E4}\left(e^{i({\varphi }_{E2\left(t_0\right)}-{\varphi }_{E4\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_2-E_4)(t-t_0)}\right|E_2⟩⟨E_4|+e^{i({\varphi }_{E4\left(t_0\right)}-{\varphi }_{E2\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_4-E_2)(t-t_0)}|E_4⟩⟨E_2\left|\right)+\\ \hfill c_{E3}{c}_{E4}\left(e^{i({\varphi }_{E3\left(t_0\right)}-{\varphi }_{E4\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_3-E_4)(t-t_0)}\right|E_3⟩⟨E_4|+e^{i({\varphi }_{E4\left(t_0\right)}-{\varphi }_{E3\left(t_0\right)})}{e}^{{\displaystyle \frac{1}{i\mathit{\hslash }}}(E_4-E_3)(t-t_0)}|E_4⟩⟨E_3\left|\right),\end{array}$$

(A118)

where $|c_{E1}{|}^2$, $|c_{E2}{|}^2$, $|c_{E3}{|}^2$ and $|c_{E4}{|}^2$ are probabilities of occupancy of eigenenergies $E_1$, $E_2$, $E_3$ and $E_4$ and we obtain two-particle density-matrix diagonal elements in the detailed form given below. We have the probability of finding electron A at node 1 and electron B at node 1’ given by formula

$$\begin{array}{c}\hfill \rho (1,1)=p(1,1^{\prime },t)={\displaystyle \frac{1}{4}}[c_{E2}^2+c_{E3}^2+{\displaystyle \frac{\left(c_{E4}^2{t}_{s2}^2(2+2t_{s2}^2+{(U-U_1)}^2)\right)}{(4t_{s2}^2+{(U-U_1)}^2)}}]\\ \hfill +{\displaystyle \frac{(-(c_{E2}^2-c_{E3}^2-c_{E4}^2{t}_{s2}^2)(U-U_1)+c_{E1}^2(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1))}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E1}{c}_{E2}(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)Cos[{\varphi }_{E1}-{\varphi }_{E2}+(-E_1+E_2)t]\right)}{\left(2\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E1}{c}_{E3}\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E1}-{\varphi }_{E3}+(-E_1+E_3)t]\right)}{\left(8(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill +{\displaystyle \frac{\left(c_{E2}{c}_{E3}\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E2}-{\varphi }_{E3}+(-E_2+E_3)t]\right)}{\left(8(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill +{\displaystyle \frac{\left(c_{E1}{c}_{E4}\sqrt{\left[t_{s2}^2(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)\right]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E1}-{\varphi }_{E4}+(-E_1+E_4)t]\right)}{\left(4\sqrt{2}(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E2}{c}_{E4}\sqrt{\left[t_{s2}^2(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)\right]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E2}-{\varphi }_{E4}+(-E_2+E_4)t]\right)}{\left(4\sqrt{2}(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E3}{c}_{E4}\sqrt{\left[t_{s2}^2(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)\right]}\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}Cos[{\varphi }_{E3}-{\varphi }_{E4}+(-E_3+E_4)t]\right)}{\left(4\sqrt{2}(4t_{s2}^2+{(U-U_1)}^2)\right)}}\end{array}$$

and the probability of finding electron A at node 1 and electron B at node 2’ given by formula

$$\begin{array}{c}\hfill \rho (2,2)=p(1,2^{\prime },t)={\displaystyle \frac{\left(2c_{E4}^2\right)}{(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)}}+\\ \hfill +{\displaystyle \frac{\left(4c_{E3}^2{t}_{s2}^2\right)}{(8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)}}+{\displaystyle \frac{\left((c_{E1}^2+c_{E2}^2)t_{s2}^2\right)}{(4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1))}}\\ \hfill -{\displaystyle \frac{\left(2c_{E1}{c}_{E2}{t}_{s2}^{2}Cos[{\varphi }_{E1}-{\varphi }_{E2}+(-E_1+E_2)t]\right)}{(4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1))}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E1}{c}_{E3}{t}_{s2}^{2}Cos[{\varphi }_{E1}-{\varphi }_{E3}+(-E_1+E_3)t]\right)}{\left(\sqrt{[4t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)]}\right)}}\\ \hfill -{\displaystyle \frac{\left(2c_{E2}{c}_{E3}{t}_{s2}^{2}Cos[{\varphi }_{E2}-{\varphi }_{E3}+(-E_2+E_3)t]\right)}{\left(\sqrt{[4t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)]}\right)}}+\\ \hfill -{\displaystyle \frac{\left(2c_{E1}{c}_{E4}{t}_{s2}^{2}Cos[{\varphi }_{E1}-{\varphi }_{E4}+(-E_1+E_4)t]\right)}{\left(\sqrt{\left[t_{s2}^2(2+2t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1))\right]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)]}\right)}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E2}{c}_{E4}{t}_{s2}^{2}Cos[{\varphi }_{E2}-{\varphi }_{E4}+(-E_2+E_4)t]\right)}{\left(\sqrt{\left[t_{s2}^2(2+2t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1))\right]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)]}\right)}}+\\ \hfill -{\displaystyle \frac{\left(2c_{E3}{c}_{E4}{t}_{s2}^{2}Cos[{\varphi }_{E3}-{\varphi }_{E4}+(-E_3+E_4)t]\right)}{\left(\sqrt{\left[t_{s2}^2(2+2t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1))\right]}\sqrt{[4t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)]}\right)}}\end{array}$$

(A119)

and the probability of finding electron A at node 2 and electron B at node 1′ given as

$$\begin{array}{c}\hfill \rho (3,3)=p(2,1^{\prime },t)={\displaystyle \frac{1}{4}}(c_{E2}^2+c_{E3}^2+{\displaystyle \frac{\left(c_{E4}^2{t}_{s2}^2(2+2t_{s2}^2+{(U-U_1)}^2)\right)}{(4t_{s2}^2+{(U-U_1)}^2)}})+\\ \hfill +{\displaystyle \frac{(-(c_{E2}^2-c_{E3}^2-c_{E4}^2{t}_{s2}^2)(U-U_1)+c_{E1}^2(-U+\sqrt{[4ts{2}^2+{(U-U_1)}^2]}+U_1))}{\left(4\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}}+\\ \hfill +{\displaystyle \frac{\left(c_{E1}{c}_{E2}(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)Cos[{\varphi }_{E1}-{\varphi }_{E2}+(-E_1+E_2)t]\right)}{\left(2\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}\right)}}+\\ \hfill -{\displaystyle \frac{\left(c_{E1}{c}_{E3}\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E1}-{\varphi }_{E3}+(-E_1+E_3)t]\right)}{\left(8(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E2}{c}_{E3}\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E2}-{\varphi }_{E3}+(-E_2+E_3)t]\right)}{\left(8(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E1}{c}_{E4}\sqrt{\left[t_{s2}^2(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)\right]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E1}-{\varphi }_{E4}+(-E_1+E_4)t]\right)}{\left(4\sqrt{2}(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill -{\displaystyle \frac{\left(c_{E2}{c}_{E4}\sqrt{\left[t_{s2}^2(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)\right]}\sqrt{[8t_{s2}^2+2{(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)}^2]}Cos[{\varphi }_{E2}-{\varphi }_{E4}+(-E_2+E_4)t]\right)}{\left(4\sqrt{2}(4t_{s2}^2+{(U-U_1)}^2)\right)}}\\ \hfill +{\displaystyle \frac{\left(c_{E3}{c}_{E4}\sqrt{\left[t_{s2}^2(4+{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)\right]}\sqrt{[8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2]}Cos[{\varphi }_{E3}-{\varphi }_{E4}+(-E_3+E_4)t]\right)}{\left(4\sqrt{2}(4t_{s2}^2+{(U-U1)}^2)\right)}}\end{array}$$

and the probability of finding electron A at node 2 and electron B at node 2′ as

$$\begin{array}{c}\hfill \rho (4,4,t)=p(2,2^{\prime },t)={\displaystyle \frac{\left(2c_{E4}^2\right)}{(4+{(U+\sqrt{[4t_{s2}^2+{(U-U1)}^2]}-U_1)}^2)}}+\\ \hfill +{\displaystyle \frac{\left(4c_{E3}^2{t}_{s2}^2\right)}{(8t_{s2}^2+2{(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)}^2)}}+\\ \hfill +{\displaystyle \frac{\left((c_{E1}^2+c_{E2}^2)t_{s2}^2\right)}{(4t_{s2}^2-(U-U_1)(-U+\sqrt{4t_{s2}^2+{(U-U_1)}^2}+U_1))}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E1}{c}_{E2}{t}_{s2}^{2}Cos[{\varphi }_{E1}-{\varphi }_{E2}+(-E_1+E_2)t]\right)}{(4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1))}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E1}{c}_{E3}{t}_{s2}^{2}Cos[{\varphi }_{E1}-{\varphi }_{E3}+(-E_1+E_3)t]\right)}{\left(\sqrt{[4t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)]}\right)}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E2}{c}_{E3}{t}_{s2}^{2}Cos[{\varphi }_{E2}-{\varphi }_{E3}+(-E_2+E_3)t]\right)}{\left(\sqrt{[4t_{s2}^2+(U-U1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U1)}\right])}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E1}{c}_{E4}{t}_{s2}^{2}Cos[{\varphi }_{E1}-{\varphi }_{E4}+(-E_1+E_4)t]\right)}{\left(\sqrt{\left[t_{s2}^2(2+2t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1))\right]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4ts{2}^2+{(U-U_1)}^2]}+U1)]}\right)}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E2}{c}_{E4}{t}_{s2}^{2}Cos[{\varphi }_{E2}-{\varphi }_{E4}+(-E_2+E_4)t]\right)}{\left(\sqrt{\left[t_{s2}^2(2+2t_{s2}^2+(U-U_1)(U+\sqrt{[4ts{2}^2+{(U-U1)}^2]}-U_1))\right]}\sqrt{[4t_{s2}^2-(U-U_1)(-U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}+U_1)]}\right)}}+\\ \hfill +{\displaystyle \frac{\left(2c_{E3}{c}_{E4}{t}_{s2}^{2}Cos[{\varphi }_{E3}-{\varphi }_{E4}+(-E_3+E_4)t]\right)}{\left(\sqrt{\left[t_{s2}^2(2+2t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1))\right]}\sqrt{[4t_{s2}^2+(U-U_1)(U+\sqrt{[4t_{s2}^2+{(U-U_1)}^2]}-U_1)]}\right)}}.\end{array}$$

(A120)