Dissipation equation of motion approach to open quantum systems

Abstract

This paper presents a comprehensive account of the dissipaton-equation-of-motion (DEOM) theory for open quantum systems. This newly developed theory treats not only the quantum dissipative systems of primary interest, but also the hybrid environment dynamics that are also experimentally measurable. Despite the fact that DEOM recovers the celebrated hierarchical-equations-of-motion (HEOM) formalism, these two approaches have some fundamental differences. To show these differences, we also scrutinize the HEOM construction via its root at the influence functional path integral formalism. We conclude that many unique features of DEOM are beyond the reach of the HEOM framework. The new DEOM approach renders a statistical quasi-particle picture to account for the environment, which can be either bosonic or fermionic. The review covers the DEOM construction, the physical meanings of dynamical variables, the underlying theorems and dissipaton algebra, and recent numerical advancements for efficient DEOM evaluations of various problems. We also address the issue of high-order many-dissipaton truncations with respect to the invariance principle of quantum mechanics of Schrödinger versus Heisenberg prescriptions. DEOM serves as a universal tool for characterizing of stationary and dynamic properties of system-and-bath interferences, as highlighted with its real-time evaluation of both linear and nonlinear current noise spectra of nonequilibrium electronic transport.

Keywords

quantum dissipation / quantum transport / entangled system-and-bath dynamics

References

[1] A. G. Redfield, The theory of relaxation processes, Adv. Magn. Reson. 1, 1 (1965)

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

[3] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17(5), 821 (1976)

[4] Y. J. Yan, Quantum Fokker-Planck theory in a non-Gaussian–Markovian medium, Phys. Rev. A 58(4), 2721 (1998)

[5] R. X. Xu and Y. J. Yan, Theory of open quantum systems, J. Chem. Phys. 116(21), 9196 (2002)

[6] Y. J. Yan and R. X. Xu, Quantum mechanics of dissipative systems, Annu. Rev. Phys. Chem. 56(1), 187 (2005)

[7] R. P. Feynman and F. L. Jr Vernon, The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. 24, 118 (1963)

[8] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th Ed., Singapore: World Scientific, 2009

[9] U. Weiss, Quantum Dissipative Systems, 3rd Ed., Series in Modern Condensed Matter Physics, Vol. 13, Singapore: World Scientific, 2008

[10] J. S. Shao, Decoupling quantum dissipation interaction via stochastic fields, J. Chem. Phys. 120(11), 5053 (2004)

[11] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao, Hierarchical approach based on stochastic decoupling to dissipative systems, Chem. Phys. Lett. 395(4–6), 216 (2004)

[12] Y. Tanimura, Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath, Phys. Rev. A 41(12), 6676 (1990)

[13] Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems, J. Phys. Soc. Jpn. 75(8), 082001 (2006)

[14] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan, Exact quantum master equation via the calculus on path integrals, J. Chem. Phys. 122(4), 041103 (2005)

[15] R. X. Xu and Y. J. Yan, Dynamics of quantum dissipation systems interacting with bosonic canonical bath: Hierarchical equations of motion approach, Phys. Rev. E 75(3), 031107 (2007)

[16] J. J. Ding, J. Xu, J. Hu, R. X. Xu, and Y. J. Yan, Optimized hierarchical equations of motion theory for Drude dissipation and efficient implementation to nonlinear spectroscopies, J. Chem. Phys. 135(16), 164107 (2011)

[17] J. J. Ding, R. X. Xu, and Y. J. Yan, Optimizing hierarchical equations of motion for quantum dissipation and quantifying quantum bath effects on quantum transfer mechanisms, J. Chem. Phys. 136(22), 224103 (2012)

[18] J. S. Jin, X. Zheng, and Y. J. Yan, Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach, J. Chem. Phys. 128(23), 234703 (2008)

[19] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan, Electron transfer dynamics: Zusman equation versus exact theory, J. Chem. Phys. 130(16), 164518 (2009)

[20] K. B. Zhu, R. X. Xu, H. Y. Zhang, J. Hu, and Y. J. Yan, Hierarchical dynamics of correlated system-environment coherence and optical spectroscopy, J. Phys. Chem. B 115(18), 5678 (2011)

[21] Y. J. Yan, Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach, J. Chem. Phys. 140(5), 054105 (2014)

[22] H. D. Zhang, R. X. Xu, X. Zheng, and Y. J. Yan, Nonperturbative spin-boson and spin-spin dynamics and nonlinear Fano interferences: A unified dissipaton theory based study, J. Chem. Phys. 142(2), 024112 (2015)

[23] J. S. Jin, S. K. Wang, X. Zheng, and Y. J. Yan, Current noise spectra and mechanisms with dissipaton equation of motion theory, J. Chem. Phys. 142(23), 234108 (2015)

[24] S. Mukamel, The Principles of Nonlinear Optical Spectroscopy, New York: Oxford University Press, 1995

[25] Y. J. Yan and S. Mukamel, Electronic dephasing, vibrational relaxation, and solvent friction in molecular nonlinear optical lineshapes, J. Chem. Phys. 89(8), 5160 (1988)

[26] A. O. Caldeira and A. J. Leggett, Quantum tunnelling in a dissipative system, Ann. Phys. 1983, 149: 374 [Erratum: 153, 445 (1984)]

[27] A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121(3), 587 (1983)

[28] J. Hu, R. X. Xu, and Y. J. Yan, Padé spectrum decomposition of Fermi function and Bose function, J. Chem. Phys. 133(10), 101106 (2010)

[29] J. Hu, M. Luo, F. Jiang, R. X. Xu, and Y. J. Yan, Padé spectrum decompositions of quantum distribution functions and optimal hierarchial equations of motion construction for quantum open systems, J. Chem. Phys. 134(24), 244106 (2011)

[30] R. X. Xu, B. L. Tian, J. Xu, Q. Shi, and Y. J. Yan, Hierarchical quantum master equation with semiclassical Drude dissipation, J. Chem. Phys. 131(21), 214111 (2009)

[31] B. L. Tian, J. J. Ding, R. X. Xu, and Y. J. Yan, Biexponential theory of Drude dissipation via hierarchical quantum master equation, J. Chem. Phys. 133(11), 114112 (2010)

[32] H. D. Zhang and Y. J. Yan, Onsets of hierarchy truncation and self-consistent Born approximation with quantum mechanics prescriptions invariance, J. Chem. Phys. 143(21), 214112 (2015)

[33] X. Zheng, R. X. Xu, J. Xu, J. S. Jin, J. Hu, and Y. J. Yan, Hierarchical equations of motion for quantum dissipation and quantum transport, Prog. Chem. 2012, 24(06): 1129, http://www.progchem.ac.cn/EN/abstract/abstract10858.shtml

[34] P. Cui, X. Q. Li, J. S. Shao, and Y. J. Yan, Quantum transport from the perspective of quantum open systems, Phys. Lett. A 357(6), 449 (2006)

[35] J. S. Jin, J. Li, Y. Liu, X. Q. Li, and Y. J. Yan, Improved master equation approach to quantum transport: From Born to self-consistent Born approximation, J. Chem. Phys. 140(24), 244111 (2014)

[36] D. Hou, S. K. Wang, R. L. Wang, L. Z. Ye, R. X. Xu, X. Zheng, and Y. J. Yan, Improving the efficiency of hierarchical equations of motion approach and application to coherent dynamics in Aharonov–Bohm interferometers, J. Chem. Phys. 142(10), 104112 (2015)

[37] Y. Tanimura and P. G. Wolynes, Quantum and classical Fokker-Planck equations for a Guassian–Markovian noise bath, Phys. Rev. A 43(8), 4131 (1991)

[38] X. Q. Li and Y. J. Yan, Quantum master equation scheme of time-dependent density functional theory to time-dependent transport in nanoelectronic devices, Phys. Rev. B 75(7), 075114 (2007)

[39] Y. Tanimura, Real-time and imaginary-time quantum hierarchal Fokker–Planck equations, J. Chem. Phys. 142(14), 144110 (2015)

[40] L. H. Ryder, Quantum Field Theory, 2nd Ed., Cambridge: Cambridge University Press, 1996

[41] H. D. Zhang, J. Xu, R. X. Xu, and Y. J. Yan, Modified Zusman qquation for quantum solvation dynamics and rate processes, in: Reaction Rate Constant Computations: Theories and Applications, London: edited by K.-L. Han and T.-S. Chu, pp. 319–336, Ch. 13, RSC Theoretical and Computational Chemistry Series No.6, 2014, http://dx.doi.org/10.1039/9781849737753-00319

[42] X. Zheng, J. S. Jin, and Y. J. Yan, Dynamic electronic response of a quantum dot driven by time-dependent voltage, J. Chem. Phys. 129(18), 184112 (2008)

[43] X. Zheng, J. S. Jin, and Y. J. Yan, Dynamic Coulomb blockade in single-lead quantum dots, New J. Phys. 10(9), 093016 (2008)

[44] X. Zheng, J. Y. Luo, J. S. Jin, and Y. J. Yan, Complex non-Markovian effect on time-dependent quantum transport, J. Chem. Phys. 130(12), 124508 (2009)

[45] F. Jiang, J. S. Jin, S. K. Wang, and Y. J. Yan, Inelastic electron transport through mesoscopic systems: Heating versus cooling and sequential tunneling versus cotunneling processes, Phys. Rev. B 85(24), 245427 (2012)

[46] S. K. Wang, X. Zheng, J. S. Jin, and Y. J. Yan, Hierarchical Liouville-space approach to nonequilibrium dynamic properties of quantum impurity systems, Phys. Rev. B 88(3), 035129 (2013)

[47] X. Zheng, Y. J. Yan, and M. Di Ventra, Kondo memory in driven strongly correlated quantum dots, Phys. Rev. Lett. 111(8), 086601 (2013)

[48] L. Z. Ye, D. Hou, R. L. Wang, D. W. Cao, X. Zheng, and Y. J. Yan, Thermopower of few-electron quantum dots with Kondo correlations, Phys. Rev. B 90(16), 165116 (2014)

[49] Z. H. Li, N. H. Tong, X. Zheng, D. Hou, J. H. Wei, J. Hu, and Y. J. Yan, Hierarchical Liouville-space approach for accurate and universal characterization of quantum impurity systems, Phys. Rev. Lett. 109(26), 266403 (2012)

[50] D. Hou, R. Wang, X. Zheng, N. H. Tong, J. H. Wei, and Y. J. Yan, Hierarchical equations of motion for impurity solver in dynamical mean-field theory, Phys. Rev. B 90(4), 045141 (2014)

[51] T. Ozaki, Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations, Phys. Rev. B 75(3), 035123 (2007)

[52] P. W. Anderson, Localized magnetic states in metals, Phys. Rev. 124(1), 41 (1961)

[53] Y. Meir, N. S. Wingreen, and P. A. Lee, Low-temperature transport through a quantum dot: The Anderson model out of equilibrium, Phys. Rev. Lett. 70(17), 2601 (1993)

[54] J. Xu, R. X. Xu, D. Abramavicius, H. D. Zhang, and Y. J. Yan, Advancing hierarchical equations of motion for efficient evaluation of coherent two-dimensional spectroscopy, Chin. J. Chem. Phys. 24(5), 497 (2011)

[55] J. Xu, H. D. Zhang, R. X. Xu, and Y. J. Yan, Correlated driving and dissipation in two-dimensional spectroscopy, J. Chem. Phys. 138(2), 024106 (2013)

[56] R. W. Freund and N. M. Nachtigal, QMR: A quasiminimal residual method for non-Hermitian linear systems, Numer. Math. 60(1), 315 (1991)

[57] R. W. Freund, A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput. 14(2), 470 (1993)

[58] G. Stefanucci, Bound states in ab initio approaches to quantum transport: A time-dependent formulation, Phys. Rev. B 75(19), 195115 (2007)

[59] Y. Mo, R. X. Xu, P. Cui, and Y. J. Yan, Correlation and response functions with non-Markovian dissipation: A reduced Liouville-space theory, J. Chem. Phys. 122(8), 084115 (2005)

[60] Y. X. Cheng, W. J. Hou, Y. D. Wang, Z. H. Li, J. H. Wei, and Y. J. Yan, Time-dependent transport through quantum-impurity systems with Kondo resonance, New J. Phys. 17(3), 033009 (2015)

[61] J. Rammer and H. Smith, Quantum field-theoretical methods in transport theory of metals, Rev. Mod. Phys. 58(2), 323 (1986)

[62] H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, 2nd Ed., Berlin: Springer, 2007

[63] A. Croy and U. Saalmann, Propagation scheme for nonequilibrium dynamics of electron transport in nanoscale devices, Phys. Rev. B 80(24), 245311 (2009)

[64] J. S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, Nonequilibrium Green’s function method for quantum thermal transport, Front. Phys. 9(6), 673 (2014)

[65] W. Ji, H. Q. Xu, and H. Guo, Quantum description of transport phenomena: Recent progress, Front. Phys. 9(6), 671 (2014)

[66] K. G. Wilson, The renormalization group: Critical phenomena and Kondo problem, Rev. Mod. Phys. 47(4), 773 (1975)

[67] R. Bulla, T. A. Costi, and T. Pruschke, Numerical renormalization group method for quantum impurity systems, Rev. Mod. Phys. 80(2), 395 (2008)

RIGHTS & PERMISSIONS

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg