Particle transport in a network of quantum harmonic oscillators
https://doi.org/10.17586/2220-8054-2020-11-2-145-152
Abstract
In this paper, we address the problem of a particle dynamics in a network of quantum harmonic oscillators by solving the stationary Schrodinger¨ equation on metric graphs in the presence of harmonic oscillator potential with bond-dependent frequency. Particle transport is analyzed by considering wave packet dynamics in such a system modeled in terms of quantum graph.
About the Authors
J. R. Yusupov
Turin Polytechnic University in Tashkent
Uzbekistan
Uzbekistan
17 Niyazov Str., 100095, Tashkent
Kh. Sh. Matyokubov
Urgench State University
Uzbekistan
Uzbekistan
14 H. Olimjon Str., 220100, Urgench
K. K. Sabirov
Tashkent University of Information Technologies
Uzbekistan
Uzbekistan
108 Amir Temur Str., 100200, Tashkent
References
1. Pauling L. The Diamagnetic Anisotropy of Aromatic Molecules, J. Chem. Phys., 1936, 4, P. 673.
2. Ruedenberg K. and Scherr C.W. Free-Electron Network Model for Conjugated Systems. I. Theory, J. Chem. Phys., 1953, 21 P. 1565.
3. Exner P. and Seba P. Free quantum motion on a branching graph. Rep. Math. Phys., 1989, 28(1) P. 26.
4. Kostrykin R. and Schrader R. Kirchhoff’s rule for quantum wires. J. Phys. A: Math. Gen., 1999, 32, P. 595.
5. Hul O., Bauch S., Pakonski P., Savytskyy N., Zyczkowski K. and Sirko L. Experimental simulation of quantum graphs by microwave networks, Phys. Rev. E, 2004, 69, P. 056205.
6. Kottos T. and Smilansky U. Periodic Orbit Theory and Spectral Statistics for Quantum Graphs. Ann.Phys., 1999, 274(1), P. 124.
7. Gnutzmann S. and Smilansky U. Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv.Phys., 2006, 55(5-6) P. 527–625.
8. Gnutzmann S., Keating J.P. and Piotet F. Eigenfunction statistics on quantum graphs. Ann.Phys., 2010, 325(12) P. 2595–2640.
9. Goldman N. and Gaspard P. Quantum graphs and the integer quantum Hall effect, Phys. Rev. B, 2008, 77, P. 024302.
10. Russo A., Barnes E., Economou S.E. Photonic graph state generation from quantum dots and color centers for quantum communications, Phys. Rev. B, 2018, 98, P. 085303.
11. Bolte J. and Harrison J. The spin contribution to the form factor of quantum graphs. J. Phys. A: Math. Gen., 2003, 36, P. L433.
12. Sabirov K., Yusupov J., Jumanazarov D., Matrasulov D. Bogoliubov de Gennes equation on metric graphs. Phys. Lett. A, 2018, 382, P. 2856.
13. Giacomelli G., Lepri S., Trono C. Optical networks as complex lasers, Phys. Rev. A, 2019, 99, P. 023841.
14. Yusupov J.R., Sabirov K.K., Ehrhardt M. and Matrasulov D.U. Transparent quantum graphs. Phys. Lett. A, 2019, 383, P. 2382.
15. Aripov M.M., Sabirov K.K., Yusupov J.R. Transparent vertex boundary conditions for quantum graphs: simplified approach. Nanosystems: physics, chemistry, mathematics, 2019, 10(5), P. 501–602.
16. Matrasulov D.U., Sabirov K.K. and Yusupov J.R. PT-symmetric quantum graphs. J. Phys. A: Math. Gen., 2019, 52, P. 155302.
17. Yusupov J., Dolgushev M., Blumen A. and Mulken O. Directed transport in quantum star graphs.¨ Quantum Inf Process, 2016, 15, P. 1765.
18. Yusupov J. Tunable wave packet transport in branched periodic lattices with time-dependent external field. Nanosystems: physics, chemistry, mathematics, 2017, 8(1), P. 42–47.
19. Naimark K. and Solomyak M. Eigenvalue estimates for the weighted Laplacian on metric trees. Proc. London Math. Soc., 2000, 80, P. 690.
20. Keating J.P. Fluctuation statistics for quantum star graphs. Quantum graphs and their applications. Contemp. Math., 2006, 415, P. 191–200.
21. Harrison J.M., Smilansky U. and Winn B. Quantum graphs where back-scattering is prohibited. J. Phys. A: Math. Gen., 2007, 40, P. 14181.
22. Exner P. and Lipovsky J. On the absence of absolutely continuous spectra for Schr´ odinger operators on radial tree graphs.¨ J. Math. Phys., 2010, 51, P. 122107.
23. Berkolaiko G., Kuchment P. Introduction to Quantum Graphs, Mathematical Surveys and Monographs, AMS, 2013.
24. Mugnolo D. Semigroup Methods for Evolution Equations on Networks. Springer-Verlag, Berlin, 2014.
25. Kurasov P., Ogik R. and Rauf A. On reflectionless equi-transmitting matrices. Opuscula Math., 2014, 34, P. 483.
26. P. Exner and H. Kovarik, Quantum waveguides. Springer, 2015.
27. Joly P., Kachanovska M. and Semin A. Wave propagation in fractal trees. Mathematical and Numerical Issues. hal-01801394, 2018.
28. Yusupov J.R., Sabirov K.K., Ehrhardt M. and Matrasulov D.U. Transparent nonlinear networks. Phys. Rev. E, 2019, 100, P. 032204.
29. Aquino N. The isotropic bounded oscillators. J. Phys. A: Math. Gen., 1997, 30(7) P. 2403.
30. Stevanovic L. and Sen K.D. Eigenspectrum properties of the confined 3D harmonic oscillator.´ J. Phys. B: At.Mol., 2008, 41 P. 225002.
31. Amore P. and Fernandez F.M. One-dimensional oscillator in a box.´ Eur. J. Phys., 2010, 31, P. 69.
32. Rice M.J., Gartstein Yu.N. Excitons and Interband Excitations in Conducting Polymers Based on Phenylene. Phys. Rev. Lett., 1994, 73, P. 2504.
33. Brazovskii S., Kirova N., Bishop A.R. Theory of electronic states and excitations in PPV. Opt. Mat., 1998, 9 P. 465–471.
34. Moses D., Wang J., Heeger A.J., Kirova N., Brazovski S. Electric field induced ionization of the exciton in poly(phenylene vinylene). Synth. Met., 2001, 119, P. 503.
35. Kirova N., Brazovskii S. Optical and electrooptical absorption in conducting polymers. Thin Solid Films, 2002, 403, P. 419.
36. Kirova N., Brazovskii S. Conjugated polymers at the verge of strongly correlated systems and 1d semiconductors, Synth. Met., 2004, 141, P. 139.
37. Kirova N., Brazovskii S. Electronic interactions and excitons in conducting polymers, Current Appl. Phys., 2004, 4 P. 473–478.
38. Kobrak M.N., Bittner E.R. A dynamic model for exciton self-trapping in conjugated polymers. I. Theory, J. Chem. Phys., 2000, 112 P. 5399.
Review
For citations:
Yusupov J.R., Matyokubov Kh.Sh., Sabirov K.K. Particle transport in a network of quantum harmonic oscillators. Nanosystems: Physics, Chemistry, Mathematics. 2020;11(2):145–152. https://doi.org/10.17586/2220-8054-2020-11-2-145-152
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