https://doi.org/10.1140/epjp/s13360-025-06593-6
Regular Article
Verifying hierarchical thermal quantum nonlocalities of two coupled dipole–dipole electrons inside a pair quantum-dot molecules
1
Department of Mathematical Sciences, Princess Nourah bint Abdulrahman University, 11671, Riyadh, Saudi Arabia
2
Department of Physics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, 11942, Al-Kharj, Saudi Arabia
3
Department of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, 11942, Al-Kharj, Saudi Arabia
4
Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Received:
18
February
2025
Accepted:
25
June
2025
Published online:
11
July
2025
This study investigates hierarchical thermal quantum nonlocalities—entanglement, EPR steering, and Bell nonlocality—in two dipole–dipole-coupled electrons confined in spatially separated quantum-dot molecules. We analyze how dipole–dipole interaction, Coulomb interaction, energy detuning, and inter-dot tunneling influence the robustness of the thermal two-dot-molecule electrons nonlocalities. Results show that Coulomb interaction, energy detuning, and dipole–dipole coupling significantly enhance the robustness of two-electron nonlocalities against temperature-induced decay, effectively delaying the occurrence of the sudden death phenomenon to higher temperatures. In contrast, inter-dot tunneling accelerates this decay. Furthermore, we examine how the thermal behavior of two-electron Bell-type nonlocalities depends on energy detuning, inter-dot tunneling, Coulomb interaction strength, and dipole–dipole coupling. The findings confirm that thermal nonlocalities follow a hierarchical relationship and highlight the critical role of dipole–dipole interaction in sustaining strong thermal two-electron nonlocalities. Furthermore, it is found that increasing dipole–dipole interaction supports the dependence of these nonlocalities on energy detuning, tunneling, and Coulomb interaction coupling. These insights significantly advance our understanding of quantum critical properties in quantum-dot molecule models.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
