Belarusian State University, Belarus

At present, attempts have been made to construct nanoscale structures that are stable against noise and possess states with controllable degeneracy. As such quantum structures, double quantum dots are proposed for use as artificial molecules. By controllably changing the relative positions of energy levels, such devices can function as quantum bits. A controllable change of a quantum bit state is described by diabatic terms. However, in the absence of state correlations, the occupation probability of a diabatic term of the two-level quantum system depends on an initial phase difference between the spinor components [1]. The problem of maintaining the phase difference of a quantum bit at a given value and, as a result, returning the quantum bit to its initial state could be solved by using strongly correlated systems such as topologically nontrivial Kitaev chains with deconfined Majorana fermions at their ends. Opposite to Cooper pairs, phase correlations in a topologically nontrivial Kitaev chain are protected by the topological charge conservation law. However, models of Majorana-type graphene logic gates have been absent so far. Graphene is a strongly correlated system; therefore, a double graphene quantum dot (dGQD) consisting of two electrically confined pn-graphene junctions can been considered as a candidate for the role of the quantum bit. There exist very simplified dGQD pseudo-Dirac Hamiltonians with two-dimensional (2D) radial step-like potential wells [2]. The lack of quasi-bound quasi-zero-energy states in a pseudo-Dirac-fermion model with a 2D radial step-like potential well [3] makes theoretical predictions questionable even in tight-binding calculations based on the pseudo-Dirac fermion dGQD model [4]. Thus, computation of the molecular orbitals of double GQDs whose states are collapsing and Klein-tunneling-induced is a complicated and still unsolved problem. In this report, we study the effects of pairing between graphene charge carriers in a Majorana-like-type couple of electrically confined pn-graphene junctions on the interplay between the single-GQD repulsion and the Majorana-like quantum exchange. A Majorana-like Hamiltonian of the double electrically confined graphene quantum dot has been constructed. The energy levels, $E^e_b$ and $E^e_a$, $E^e_a - E^e_b = \Delta > 0$, for the symmetric (with respect to coordinates) bonding and antisymmetric (with respect to coordinates) anti-bonding electron dGQD states, which are generated by the Majorana-like modes, have been found to be similar to molecular orbitals, as Fig. 1 shows. We choose $E^e_a = \Delta/2$ and $E^e_b = -\Delta/2$. Due to the graphene electron-hole symmetry, the symmetric bonding and antisymmetric anti-bonding hole dGQD states with energies $E^h_b$ and $E^h_a$, respectively, coincide with the antisymmetric anti-bonding and symmetric bonding electron dGQD states, respectively. Correspondingly, $E^h_b = E^e_a$ and $E^h_a = E^e_b$. Alternating p-doping of the right and left $n$--$p$--$n$-GQDs alternately changes the polarization vector $\vec P$ of the artificial molecule. In this way, one obtains a system that can be conveniently described in a valence electron and hole number configuration consisting of four relevant charge states: the anti-bonding electron state $(1_e,1_h)$, the bonding electron state $(1_e,1_e)$, the anti-bonding hole state $(1_h,1_e)$, and the bonding hole state $(1_h,1_h)$. Here the first and second occupation numbers refer to the first and second GQDs, respectively; the hole and electron occupation numbers are marked by indices $h$ and $e$, respectively. As a result, we propose to realize a graphene quantum bit based on Majorana-like modes. References [1] P. O. Kofman, S.N. Shevchenko, Fr. Nori. Tuning the initial phase to control the final state of a driven qubit //arXiv:2308.03571v2 [quant-ph] 31 Jan 2024. [2] X.-F. Zhou, Y.-C. Zhuang, M.-H. Zhang, H. Sheng, Q.-F. Sun, L. He. Relativistic artificial molecule of two coupled graphene quantum dots at tunable distances. Nature Communications. Vol. 15, p. 8786 (2024). [3] H. Grushevskaya, G. Krylov. Topologically tuned obliquity of Klein-tunnelling charged currents through graphene electrostatically-confined p–n junctions. Int. J. Nonlinear Phenomena in Complex Systems. Vol. 25, p. 21 (2022). [4] D. Moldovan, F. Peeters. pybinding v0.9.5: a Python package for tight-binding calculations. Zenodo. https://doi.org/10.5281/ zenodo.4010216 (2020).