Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators
Abstract
:1. Introduction
2. Is It Possible to Model an Open System in Terms of Single-Particle Pure States?
3. How Do We Select the Single-Particle Pure States Before and after the Collision?
3.1. Exact Solution in a Closed System
3.2. Approximate Solution with BCWF for an Open System
- At the initial time, , we consider an electron in the active region, with degree of freedom x with a central energy linked to zero photons wave function plus another electron far from the active region with degree of freedom y and energy linked to zero photons . At this initial time, the total energy involved in such a scenario is in the active region plus the energy outside.
- At the intermediate time, we consider that a spontaneous emission of a photon happens inside the active region. As seen in Figure 3, such an internal process ensures energy conservation. Therefore, the new photon inside the active region implies a change in energy there, , while the energy outside of the active region remains the same as before, . The total energy is the same as the initial one.
- At the final time t, we detect a photon at position y, far from the active region. Thus, the electron at y is now linked to one photon wave function , which implies an increment in the energy of far from the active region, . The conservation of the total energy implies that the same amount of energy is eliminated in the active region when the photon leaves, . The electron in the active region will have a new energy linked to the zero photons wave function . As we have seen in Figure 5, under such new energy conditions in the active region, such an electron will no longer be able to generate spontaneous emissions inside the RTD. Thus, the Rabi oscillations seen in Figure 3 for a closed system will not be present when we assume that the photon leaves the cavity.
4. Implementation of the Transition from Pre- to Post-Selected BCWF
4.1. Model A: Change in the Central Energy
4.2. Model B: Change in Central Momentum
5. Numerical Results
5.1. Collisions in Flat Potentials
5.2. Collisions in Arbitrary Potentials
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
BCWF | Bohmian Conditional Wave Function |
BITLLES | Bohmian Interacting Transport in non-equiLibrium eLEctronic Structures |
RTD | Resonant Tunneling Diode |
References
- Breuer, H.P.; Petruccione, F. Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2002; pp. 5–23. [Google Scholar]
- Klimeck, G. Single and multiband modeling of quantum electron transport through layered semiconductor devices. J. Appl. Phys. 1997, 81, 7845. [Google Scholar]
- Klimeck, G.; Ahmed, S.S.; Bae, H.; Kharche, N.; Clark, S.; Haley, B.; Lee, S.; Naumov, M.; Ryu, H.; Saied, F.; et al. Atomistic Simulation of Realistically Sized Nanodevices Using NEMO 3-D Part I: Models and Benchmarks. IEEE Trans. Electron Devices 2007, 54, 2079–2089. [Google Scholar] [CrossRef]
- Schmidt, A.; Cheng, B.; daLuz, M. Green function approach for general quantum graphs. J. Phys. A Math. Gen. 2003, 36, 42. [Google Scholar] [CrossRef]
- Rossi, F. Theory of Semiconductor Quantum Devices; The Density-Matrix Approach; Springer: Berlin, Germany, 2010; pp. 89–130. [Google Scholar]
- Iotti, C.; Ciancio, E.; Rossi, F. Quantum transport theory for semiconductor nanostructures: A density-matrix formulation. Phys. Rev. B 2005, 72, 125347. [Google Scholar] [CrossRef] [Green Version]
- Wigner, E.P. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 1932, 40, 749–759. [Google Scholar] [CrossRef]
- Frensley, W. Wigner-Function Model of Resonant-Tunneling Semiconductor Device. Phys. Rev. B 1987, 36, 1570–1580. [Google Scholar] [CrossRef]
- Weinbub, J.; Ferry, D.K. Recent advances in Wigner function approaches. Appl. Phys. Rev. 2018, 5, 041104. [Google Scholar] [CrossRef] [Green Version]
- Querlioz, D.; Nguyen, H.-N.; Saint-Martin, J.; Bournel, A.; Galdin-Retailleau, S.; Dollfus, P. Wigner-Boltzmann Monte Carlo approach to nanodevice simulation: From quantum to semiclassical transport. J. Comp. Electron. 2009, 8, 324–335. [Google Scholar] [CrossRef]
- Nedjalkov, M.; Querlioz, D.; Dollfus, P.; Kosina, H. Wigner Function Approach. In Nano-Electronic Devices; Springer: New York, NY, USA, 2011; pp. 289–358. [Google Scholar]
- Fan, Z.; Garcia, J.; Cummings, A.; Barrios-Vargas, J.; Panhans, M.; Harju, A.; Ortmann, F.; Roche, S. Linear scaling quantum transport methodologies. Phys. Rep. 2021, 903, 1–69. [Google Scholar] [CrossRef]
- Vyas, P.; Van de Put, M.; Fischetti, M. Master-Equation Study of Quantum Transport in Realistic Semiconductor Devices Including Electron-Phonon and Surface-Roughness Scattering. Phys. Rev. Appl. 2020, 13, 014067. [Google Scholar] [CrossRef]
- Fischetti, M. Theory of electron transport in small semiconductor devices using the Pauli master equation. J. Appl. Phys. 1998, 83, 270. [Google Scholar] [CrossRef]
- Kramer, T.; Kreisbeck, C.; Krueckl, V. Wave packet approach to transport in mesoscopic systems. Phys. Scr. 2010, 82, 038101. [Google Scholar] [CrossRef] [Green Version]
- Bracher, C.; Delos, J.; Kanellopoulos, V.; Kleber, M.; Kramer, T. The photoelectric effect in external fields. Phys. Lett. A 2005, 347, 62–66. [Google Scholar] [CrossRef] [Green Version]
- Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. Phys. Rev. 1952, 85, 166. [Google Scholar] [CrossRef]
- Bohmian Interacting Transport in Non-equiLibrium eLEctronic Structures. Available online: europe.uab.es/bitlles (accessed on 29 March 2021).
- Vacchini, B.; Smirne, A.; Laine, E.; Piilo, J.; Breuer, H. Markovianity and non-Markovianity in quantum and classical systems. New J. Phys. 2011, 13, 093004. [Google Scholar] [CrossRef]
- Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys. 1976, 48, 119–130. [Google Scholar] [CrossRef]
- Ferialdi, L. Exact Closed Master Equation for Gaussian Non-Markovian Dynamics. Phys. Rev. Lett. 2016, 116, 120402. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Vega, I.; Alonso, D. Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys. 2017, 89, 015001. [Google Scholar] [CrossRef] [Green Version]
- Ghirardi, G.C.; Rimini, A.; Webber, T. Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 1986, 34, 470. [Google Scholar] [CrossRef]
- Bassi, A.; Lochan, K.; Satin, S.; Singh, T.P.; Ulbricht, H. Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys. 2013, 85, 471. [Google Scholar] [CrossRef]
- Strunz, W.T.; Diósi, L.; Gisin, N. Open System Dynamics with Non-Markovian Quantum Trajectories. Phys. Rev. Lett. 1999, 82, 1801. [Google Scholar] [CrossRef] [Green Version]
- Strunz, W.T. The Brownian motion stochastic Schrödinger equation. Chem. Phys. 2001, 268, 237. [Google Scholar] [CrossRef]
- Ferialdi, L.; Bassi, A. Exact Solution for a Non-Markovian Dissipative Quantum Dynamics. Phys. Rev Lett. 2012, 108, 170404. [Google Scholar] [CrossRef] [Green Version]
- Gambetta, J.; Wiseman, H.M. Non-Markovian stochastic Schrödinger equations: Generalization to real-valued noise using quantum-measurement theory. Phys. Rev. A 2002, 66, 012108. [Google Scholar] [CrossRef] [Green Version]
- Gambetta, J.; Wiseman, H.M. The interpretation of non-Markovian stochastic Schrödinger equations as a hidden-variable theory. Phys. Rev. A 2003, 68, 062104. [Google Scholar] [CrossRef] [Green Version]
- Diósi, L.; Ferialdi, L. General Non-Markovian Structure of Gaussian Master and Stochastic Schrödinger Equations. Phys. Rev. Lett. 2014, 113, 200403. [Google Scholar] [CrossRef] [Green Version]
- Oriols, X. Quantum-Trajectory Approach to Time-Dependent Transport in Mesoscopic Systems with Electron-Electron Interactions. Phys. Rev. Lett. 2007, 98, 066803. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Dürr, D.; Teufel, S. Bohmian Mechanics: The Physics and Mathematics of Quantum Theory; Springer: Berlin, Germany, 2009. [Google Scholar]
- Oriols, X.; Mompart, J. Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology, 2nd ed.; Jenny Stanford Publishing: Singapore, 2019. [Google Scholar]
- Colomés, E.; Zhan, Z.; Marian, D.; Oriols, X. Quantum dissipation with conditional wave functions: Application to the realistic simulation of nanoscale electron devices. Phys. Rev. B 2017, 96, 075135. [Google Scholar] [CrossRef] [Green Version]
- Albareda, G.; López, H.; Cartoixà, X.; Suñé, J.; Oriols, X. Time-dependent boundary conditions with lead-sample Coulomb correlations: Application to classical and quantum nanoscale electron device simulators. Phys. Rev. B 2010, 82, 085301. [Google Scholar] [CrossRef]
- Marian, D.; Colomés, E.; Oriols, X. Quantum noise from a Bohmian perspective: Fundamental understanding and practical computation in electron devices. J. Phys. Condens. Matter 2015, 27, 245302. [Google Scholar] [CrossRef] [Green Version]
- Albareda, G.; Traversa, F.L.; Benali, A.; Oriols, X. Computation of Quantum Electrical Currents throught The Ramo–Shockley–Pellegrini Theorem with Trajectories. Fluctuation Noise Lett. 2012, 11, 1242008. [Google Scholar] [CrossRef]
- Jacoboni, C.; Reggiani, L. The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev. Mod. Phys. 1983, 55, 645. [Google Scholar] [CrossRef]
- Tzemos, A.C.; Contopoulos, G.; Efthymiopoulos, C. Bohmian trajectories in an entangled two-qubit system. Phys. Scr. 2019, 94, 105218. [Google Scholar] [CrossRef] [Green Version]
- Tzemos, A.C.; Contopoulos, G. Chaos and ergodicity in an entangled two-qubit Bohmian system. Phys. Scr. 2020, 99, 065225. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Villani, M.; Albareda, G.; Destefani, C.; Cartoixà, X.; Oriols, X. Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators. Entropy 2021, 23, 408. https://doi.org/10.3390/e23040408
Villani M, Albareda G, Destefani C, Cartoixà X, Oriols X. Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators. Entropy. 2021; 23(4):408. https://doi.org/10.3390/e23040408
Chicago/Turabian StyleVillani, Matteo, Guillermo Albareda, Carlos Destefani, Xavier Cartoixà, and Xavier Oriols. 2021. "Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators" Entropy 23, no. 4: 408. https://doi.org/10.3390/e23040408
APA StyleVillani, M., Albareda, G., Destefani, C., Cartoixà, X., & Oriols, X. (2021). Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators. Entropy, 23(4), 408. https://doi.org/10.3390/e23040408