Wave-function approach to Master equations for quantum transport and measurement

Abstract

This paper presents a comprehensive review of the wave-function approach for derivation of the numberresolved Master equations, used for description of transport and measurement in mesoscopic systems. The review contains important amendments, clarifying subtle points in derivation of the Master equations and their validity. This completes the earlier works on the subject. It is demonstrated that the derivation does not assume weak coupling with the environment and reservoirs, but needs only high bias condition. This condition is very essential for validity of the Markovian Master equations, widely used for a phenomenological description of different physical processes.

Keywords

mesoscopic systems / quantum transport / Master equation / continuous measurement

References

[1] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge: Cambridge University Press, 1995

[2] Y. Imry, Introduction to Mesoscopic Physics, Oxford: Oxford University Press, 1997

[3] Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to Nanoscience, Cambridge: Cambridge University Press, 2009

[4] F. R. Waugh, M. J. Berry, D. J. Mar, R. M. Westervelt, K. L. Campman, and A. C. Gossard, Single-electron charging in double and triple quantum dots with tunable coupling, Phys. Rev. Lett. 75(4), 705 (1995)

[5] L. I. Glazman and K. A. Matveev, Coulomb correlations in the tunneling through resonance centers, Pis’ma Zh. Eksp. Teor. Fiz. 48(7), 403 (1988) [JETP Lett. 48(7), 445 (1988)]

[6] D. V. Averin and A. N. Korotkov, Influence of discrete energy spectrum on correlated single-electron tunneling via a mezoscopically small metal granule, Zh. Eksp. Teor. Fiz. 97(5), 1661 (1990) [Sov. Phys.-JETP 70, 937 (1990)]

[7] C. W. J. Beenakker, Theory of Coulomb-blockade oscillations in the conductance of a quantum dot, Phys. Rev. B 44(4), 1646 (1991)

[8] S. A. Gurvitz, H. J. Lipkin, and Ya. S. Prager, The Pauli principle and quantum transport, Mod. Phys. Lett. B 8(21–22), 1377 (1994)

[9] J. H. Davies, S. Hershfield, P. Hyldgaard, and J. W. Wilkins, Current and rate equation for resonant tunneling, Phys. Rev. B 47(8), 4603 (1993)

[10] Yu. V. Nazarov, Quantum interference, tunnel junctions and resonant tunneling interferometer, Physica B 189(1–4), 57 (1993)

[11] S. A. Gurvitz, H. J. Lipkin, and Ya. S. Prager, Interference effects in resonant tunneling and the Pauli principle, Phys. Lett. A 212(1–2), 91 (1996)

[12] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applications, New York: Wiley, 1992

[13] S. A. Gurvitz and Ya. S. Prager, Microscopic derivation of rate equations for quantum transport, Phys. Rev. B 53(23), 15932 (1996)

[14] S. Kohler, J. Lehmann, and P. Hänggi, Driven quantum transport on the nanoscale, Phys. Rep. 406(6), 379 (2005)

[15] V. May and O. Kühn, Optical field control of charge transmission through a molecular wire. I. Generalized master equation description, Phys. Rev. B 77(11), 115439 (2008)

[16] M. W. Y. Tu, and W. M. Zhang, Non-Markovian decoherence theory for a double-dot charge qubit, Phys. Rev. B 78(23), 235311 (2008)

[17] X. Q. Li, Number-resolved master equation approach to quantum measurement and quantum transport, Front. Phys. 11(4), 110307 (2016), and references therein

[18] G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48(2), 119 (1976)

[19] S. A. Gurvitz, Measurements with a noninvasive detector and dephasing mechanism, Phys. Rev. B 56, 15215 (1997)

[20] S. A. Gurvitz, Rate equations for quantum transport in multidot systems, Phys. Rev. 57(11), 6602 (1998)

[21] B. Elattari, and S. A. Gurvitz, Influence of measurement on the lifetime and the linewidth of unstable systems, Phys. Rev. A 62(3), 032102 (2000)

[22] L. Xu, Y. Cao, X. Q. Li, Y. J. Yan, and S. Gurvitz, Quantum transfer through a non-Markovian environment under frequent measurements and Zeno effect, Phys. Rev. A 90(2), 022108 (2014)

[23] S. Gurvitz, Does the measurement take place when nobody observes it? Fortschr. Phys. (in press), arXiv: 1605.05553

[24] T. H. Stoof, and Yu. V. Nazarov, Time-dependent resonant tunneling via two discrete states, Phys. Rev. B 53(3), 1050 (1996)

[25] A. Wolf, G. De Chiara, E. Kajari, E. Lutz, and G. Morigi, Entangling two distant oscillators with a quantum reservoir, Europhys. Lett. 95(6), 60008 (2011)

[26] J. Ping, Y. Ye, L. Xu, X. Q. Li, Y. J. Yan, and S. Gurvitz, Undetectable quantum transfer through a continuum, Phys. Lett. A 377(9), 676 (2013)

[27] Y. Ye, Yu. Cao, X. Q. Li, and S. Gurvitz, Decoherence and the retrieval of lost information, Phys. Rev. B 84(24), 245311 (2011)

[28] S. A. Gurvitz, Lapse of transmission phase and electron molecules in quantum dots, Phys. Rev. B 77, 201302(R) (2008)

[29] S. A. Gurvitz and D. Mozyrsky, Quantum mechanical approach to decoherence and relaxation generated by fluctuating environment, Phys. Rev. B 77(7), 075325 (2008)

[30] S. A. Gurvitz, Quantum interference in resonant tunneling and single spin measurements, IEEE Trans. Nano Technol. 4(1), 45 (2005)

[31] S. A. Gurvitz, D. Mozyrsky, and G. P. Berman, Coherent effects in magnetotransport through Zeeman-split levels, Phys. Rev. B 72(20), 205341 (2005)

[32] M. Field, C. G. Smith, M. Pepper, D. A. Ritchie, J. E. F. Frost, G. A. C. Jones, and D. G. Hasko, Measurements of Coulomb blockade with a noninvasive voltage probe, Phys. Rev. Lett. 70(9), 1311 (1993)

[33] E. Buks, R. Shuster, M. Heiblum, D. Mahalu, and V. Umansky, Dephasing in electron interference by a “which-path” detector, Nature 391, 871 (1998)

[34] B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18(4), 756 (1977)

[35] C. Presilla, R. Onofrio, and U. Tambini, Measurement quantum mechanics and experiments on quantum Zeno effect, Ann. Phys. 248(1), 95 (1996)

[36] A. G. Kofman and G. Kurizki, Quantum Zeno effect on atomic excitation decay in resonators, Phys. Rev. A 54(5), R3750 (1996)

[37] A. G. Kofman and G. Kurizki, Acceleration of quantum decay processes by frequent observations, Nature 405(6786), 546 (2000)

[38] S. Gurvitz, Single-electron approach for time-dependent electron transport, Phys. Scr. T165, 014013 (2015)

[39] S. Gurvitz, A. Aharony, and O. Entin-Wohlman, Temporal evolution of resonant transmission under telegraph noise, Phys. Rev. 94(7), 075437 (2016)

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg