Abstract
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.
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References
R. E. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957)
H. J. Berendsen, Simulating the Physical World: Hierarchical Modeling from Quantum Mechanics to Fluid Dynamics (Cambridge University Press, Cambridge, 2007)
A. O. Caldeira, An Introduction to Macroscopic Quantum Phenomena and Quantum Dissipation (Cambridge University Press, Cambridge, 2014)
S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15, 1 (1943)
S. Dattagupta, Relaxation Phenomena in Condensed Matter Physics (Academic Press, Orlando, 2012)
B. J. Berne, G. Cicootti, and D. F. Coker, eds., Classical and Quantum Dynamics in Condensed Phase Simulations, Computer Simulation of Rare Events and the Dynamics of Classical and Quantum Condensed-Phase Systems (World Scientific, Singapore, 1998)
W. Ji, H. Xu, and H. Guo, Quantum description of transport phenomena: Recent progress, Front. Phys. 9, 671 (2014)
A. Einstein, Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen, Ann. Phys. 322, 549 (1905)
M. von Smoluchowski, Zur kinetischen theorie der Brownschen molekularbewegung und der suspensionen, Ann. Phys. 326, 756 (1906)
M. Scott, Applied Stochastic Processes in Science and Engineering (University of Waterloo, Waterloo, 2013)
C. Gardiner, Handbook of Stochastic Methods, 3rd ed. (Springer, Berlin, 2004)
N. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, Amsterdam, 2007)
J. B. Johnson, Thermal agitation of electricity in conductors, Phys. Rev. 32, 97 (1928)
H. Nyquist, Thermal agitation of electric charge in conductors, Phys. Rev. 32, 110 (1928)
P. Langevin, Sur la théorie du mouvement Brownien, C. R. Acad. Sci. Paris 146 (1908)
D. S. Lemons and A. Gythiel, Paul Langevin’s 1908 paper “on the theory of Brownian motion” [“sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530-533 (1908)], Am. J. Phys. 65, 1079 (1997)
A. D. Fokker, Die mittlere energie rotierender elektrischer dipole im strahlungsfeld, Ann. Phys. 348, 810 (1914)
M. Planck, An essay on statistical dynamics and its amplification in the quantum theory, Sitz. Ber. Preuß. Akad. Wiss. 325, 324 (1917)
A. Kolmogoroff, Über die analytischen methoden in der wahrscheinlichkeitsrechnung, Math. Ann. 104, 415 (1931)
H. Risken, Fokker-Planck Equation, Springer Series in Synergetics (Springer, Berlin, 1984)
G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36, 823 (1930)
H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7, 284 (1940)
P. Hänggi, P. Talkner, and M. Borkovec, Reactionrate theory: fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990)
R. Kubo, A stochastic theory of line shape, Adv. Chem. Phys. 15, 101 (1969)
H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev. 83, 34 (1951)
R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966)
S. Nakajima, On quantum theory of transport phenomena: Steady diffusion, Prog. Theor. Phys. 20, 948 (1958)
R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33, 1338 (1960)
G. W. Ford, J. T. Lewis, and R. F. O’Connell, Quantum Langevin equation, Phys. Rev. A 37, 4419 (1988)
M. C. Wang and G. E. Uhlenbeck, On the theory of the Brownian motion II, Rev. Mod. Phys. 17, 323 (1945)
H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)
A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative twostate system, Rev. Mod. Phys. 59, 1 (1987)
P. Hänggi and G. Ingold, Fundamental aspects of quantum Brownian motion, Chaos 15, 026105 (2005)
U. Weiss, Quantum Dissipative Systems, 3rd ed., Series in Modern Condensed Matter Physics, Vol. 13 (World Scientific, Singapore, 2008)
A. Caldeira and A. Leggett, Quantum tunnelling in a dissipative system, Ann. Phys. 149, 374 (1983)
R. Feynman and F. Vernon Jr., The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. 24, 118 (1963)
J. Cao, L. W. Ungar, and G. A. Voth, A novel method for simulating quantum dissipative systems, J. Chem. Phys. 104, 4189 (1996)
J. T. Stockburger and C. H. Mak, Dynamical simulation of current fluctuations in a dissipative twostate system, Phys. Rev. Lett. 80, 2657 (1998)
J. T. Stockburger and H. Grabert, Exact cnumber representation of non-markovian quantum dissipation, Phys. Rev. Lett. 88, 170407 (2002)
W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold, Non-Markovian dissipative semiclassical dynamics, Phys. Rev. Lett. 100, 230402 (2008)
L. Diósi and W. T. Strunz, The non-Markovian stochastic schrödinger equation for open systems, Phys. Lett. A 235, 569 (1997)
L. Diósi, N. Gisin, and W. T. Strunz, Non-Markovian quantum state diffusion, Phys. Rev. A 58, 1699 (1998)
W. T. Strunz, L. Diósi, and N. Gisin, Open system dynamics with non-Markovian quantum trajectories, Phys. Rev. Lett. 82, 1801 (1999)
W. T. Strunz, L. Diósi, N. Gisin, and T. Yu, Quantum trajectories for Brownian motion, Phys. Rev. Lett. 83, 4909 (1999)
T. Yu, Non-markovian quantum trajectories versus master equations: Finite-temperature heat bath, Phys. Rev. A 69, 062107 (2004)
X. Zhao, J. Jing, B. Corn, and T. Yu, Dynamics of interacting qubits coupled to a common bath: Non-markovian quantum-state-diffusion approach, Phys. Rev. A 84, 032101 (2011)
H. Breuer, Exact quantum jump approach to open systems in bosonic and spin baths, Phys. Rev. A 69, 022115 (2004)
E. Calzetta, A. Roura, and E. Verdaguer, Stochastic description for open quantum systems, Physica A 319, 188 (2003)
J. Shao, Decoupling quantum dissipation interaction via stochastic fields, J. Chem. Phys. 120, 5053 (2004)
J. T. Stockburger and H. Grabert, Non-Markovian quantum state diffusion, Chem. Phys. 268, 249 (2001)
M. Suzuki, Generalized trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to manybody problems, Commun. Math. Phys. 51, 183 (1976)
D. Gatarek and N. Gisin, Continuous quantum jumps and infinite£dimensional stochastic equations, J. Math. Phys. 32, 2152 (1991)
A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121, 587 (1983)
W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973)
W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649 (1954)
D. Finkelstein, On relations between commutators, Commun. Pure Appl. Math. 8, 245 (1955)
E. H. Wichmann, Note on the algebraic aspect of the integration of a system of ordinary linear differential equations, J. Math. Phys. 2, 876 (1961)
G. H. Weiss and A. A. Maradudin, The baker-hausdorff formula and a problem in crystal physics, J. Math. Phys. 3, 771 (1962)
A. Murua, The hopf algebra of rooted trees, free lie algebras, and lie series, Found. Comput. Math. 6, 387 (2006)
Y.-A. Yan and Y. Zhou, Hermitian non-Markovian stochastic master equations for quantum dissipative dynamics, Phys. Rev. A 92, 022121 (2015)
J. Shao, Rigorous representation and exact simulation of real gaussian stationary processes, Chem. Phys. 375, 378 (2010)
R. B. Davies and D. S. Harte, Tests for hurst effect, Biometrika 74, 95 (1987)
A. T. A. Wood and G. Chan, Simulation of stationary gaussian processes in [0; 1]d, J. Comp. Graph. Stat. 3, 409 (1994)
G. Chan and A. Wood, Algorithm AS 312 -An algo-rithm for simulating stationary gaussian random fields, App. Stat. 46, 171 (1997)
G. Chan and A. T. A. Wood, Simulation of stationary gaussian vector fields, Stat. Comp. 9, 265 (1999)
C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. issues of principle, Rev. Mod. Phys. 52, 341 (1980)
D. Mozyrsky and V. Privman, Measurement of a quantum system coupled to independent heatbath and pointer modes, Mod. Phys. Lett. B 14, 303 (2000)
J. Shao, M. Ge, and H. Cheng, Decoherence of quantum-nondemolition systems, Phys. Rev. E 53, 1243 (1996)
P. Schramm and H. Grabert, Effect of dissipation on squeezed quantum fluctuations, Phys. Rev. A 34, 4515 (1986)
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 2nd ed. (Springer-Verlag, Berlin, 1995)
V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, 3rd ed. (WILEY-VCH, Weinheim, 2010)
M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)
R. Schatten, Norm Ideals of Completely Continuous Operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 27. (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960)
H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the degree of non-markovian behavior of quantum processes in open systems, Phys. Rev. Lett. 103, 210401 (2009)
Á. Rivas, S. F. Huelga, and M. B. Plenio, Quantum non-Markovianity: characterization, quantification and detection, Rep. Prog. Phys. 77, 094001 (2014)
A. Brissaud and U. Frisch, Solving linear stochastic differential equations, J. Math. Phys. 15, 524 (1974)
V. I. Klyatskin, Dynamics of Stochastic Systems (Elsevier Science, Amsterdam, 2005)
M. Ban, S. Kitajima, and F. Shibata, Reduced dynamics and the master equation of open quantum systems, Phys. Lett. A 374, 2324 (2010)
E. Novikov, Functionals and the randomforce method in turbulence theory, Sov. Phys. JETP 20, 1290 (1965)
J. Cao, A phasespace study of Bloch-Redfield theory, J. Chem. Phys. 107, 3204 (1997)
C. Fleming, A. Roura, and B. Hu, Exact analytical solutions to the master equation of quantum Brownian motion for a general environment, Ann. Phys. 326, 1207 (2011)
H. Dekker, Quantization of the linearly damped harmonic oscillator, Phys. Rev. A 16, 2126 (1977)
H. Dekker, Classical and quantum mechanics of the damped harmonic oscillator, Phys. Rep. 80, 1 (1981)
F. Haake and R. Reibold, Strong damping and low-temperature anomalies for the harmonic oscillator, Phys. Rev. A 32, 2462 (1985)
H. Grabert, P. Schramm, and G.-L. Ingold, Quantum Brownian motion: The functional integral approach, Phys. Rep. 168, 115 (1988)
W. G. Unruh and W. H. Zurek, Reduction of a wave packet in quantum Brownian motion, Phys. Rev. D 40, 1071 (1989)
V. Ambegaokar, Dissipation and decoherence in a quantum oscillator, J. Stat. Phys. 125, 1183 (2006)
B. L. Hu, J. P. Paz, and Y. Zhang, Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise, Phys. Rev. D 45, 2843 (1992)
B. L. Hu, J. P. Paz, and Y. Zhang, Quantum Brownian motion in a general environment. II. nonlinear coupling and perturbative approach, Phys. Rev. D 47, 1576 (1993)
J. J. Halliwell and T. Yu, Alternative derivation of the Hu-Paz-Zhang master equation of quantum Brownian motion, Phys. Rev. D 53, 2012 (1996)
R. Karrlein and H. Grabert, Exact time evolution and master equations for the damped harmonic oscillator, Phys. Rev. E 55, 153 (1997)
G. W. Ford and R. F. O’Connell, Exact solution of the Hu-Paz-Zhang master equation, Phys. Rev. D 64, 105020 (2001)
E. Calzetta, A. Roura, and E. Verdaguer, Master equation for quantum Brownian motion derived by stochastic methods, Int. J. Theor. Phys. 40, 2317 (2001)
W. T. Strunz and T. Yu, Convolutionless non-markovian master equations and quantum trajectories: Brownian motion, Phys. Rev. A 69, 052115 (2004)
C. Chou, T. Yu, and B. L. Hu, Exact master equation and quantum decoherence of two coupled harmonic oscillators in a general environment, Phys. Rev. E 77, 011112 (2008)
C. Chou, B. Hu, and T. Yu, Quantum Brownian motion of a macroscopic object in a general environment, Physica A 387, 432 (2008)
R. Xu, B. Tian, J. Xu, and Y. Yan, Exact dynamics of driven Brownian oscillators, J. Chem. Phys. 130, 074107 (2009)
P. S. Riseborough, P. Hanggi, and U. Weiss, Exact results for a damped quantum-mechanical harmonic oscillator, Phys. Rev. A 31, 471 (1985)
S. Kohler, T. Dittrich, and P. Hänggi, Floquet-Markovian description of the parametrically driven, dissipative harmonic quantum oscillator, Phys. Rev. E 55, 300 (1997)
C. Zerbe and P. Hänggi, Brownian parametric quantum oscillator with dissipation, Phys. Rev. E 52, 1533 (1995)
H. Li, J. Shao, and S. Wang, Derivation of exact master equation with stochastic description: Dissipative harmonic oscillator, Phys. Rev. E 84, 051112 (2011)
J. T. Stockburger, Simulating spin-boson dynamics with stochastic Liouville-von Neumann equations, Chem. Phys. 296, 159 (2004)
C. Meier and D. J. Tannor, Non-Markovian evolution of the density operator in the presence of strong laser fields, J. Chem. Phys. 111, 3365 (1999)
C. Kreisbeck and T. Kramer, Longlived electronic coherence in dissipative exciton dynamics of light-harvesting complexes, J. Phys. Chem. Lett. 3, 2828 (2012)
V. Shapiro and V. Loginov, Formulae of differentiation and their use for solving stochastic equations, Physica A 91, 563 (1978)
Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly gaussian-markoffian noise bath, J. Phys. Soc. Japan 58, 101 (1989)
Y. Tanimura, Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath, Phys. Rev. A 41, 6676 (1990)
Y. Zhou, Y. Yan, and J. Shao, Stochastic simulation of quantum dissipative dynamics, Europhys. Lett. 72, 334 (2005)
Z. Tang, X. Ouyang, Z. Gong, H. Wang, and J. Wu, Extended hierarchy equation of motion for the spinboson model, J. Chem. Phys. 143, 224112 (2015)
J. Jin, X. Zheng, and Y. Yan, Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach, J. Chem. Phys. 128, 234703 (2008)
Q. Shi, L. Chen, G. Nan, R.-X. Xu, and Y. Yan, Efficient hierarchical liouville space propagator to quantum dissipative dynamics, J. Chem. Phys. 130, 084105 (2009)
J. Hu, R.-X. Xu, and Y. Yan, Padé spectrum decom-position of fermi function and bose function, J. Chem. Phys. 133, 101106 (2010)
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, 5678 (2011)
D. Alonso and I. de Vega, Hierarchy of equations of multipletime correlation functions, Phys. Rev. A 75, 052108 (2007)
M. Sarovar and M. D. Grace, Reduced equations of motion for quantum systems driven by diffusive markov processes, Phys. Rev. Lett. 109, 130401 (2012)
I. de Vega, On the structure of the master equation for a twolevel system coupled to a thermal bath, J. Phys. A 48, 145202 (2015)
Z. Zhou, M. Chen, T. Yu, and J. Q. You, Quantum Langevin approach for non-Markovian quantum dynamics of the spin-boson model, Phys. Rev. A 93, 022105 (2016)
A. Ishizaki and G. R. Fleming, Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature, Proc. Nat. Acad. Sci. 106, 17255 (2009)
Y.-A. Yan and O. Kühn, Laser control of dissipative two-exciton dynamics in molecular aggregates, New J. Phys. 14, 105004 (2012)
Y.-A. Yan and S. Cai, Exciton seebeck effect in molecular systems, J. Chem. Phys. 141, 054105 (2014)
Y. Yan, Exciton interference revealed by energy dependent exciton transfer rate for ring-structured molecular systems, J. Chem. Phys. 144, 024305 (2016)
L. Chen, R. Zheng, Q. Shi, and Y. Yan, Two-dimensional electronic spectra from the hierarchical equations of motion method: Application to model dimers, J. Chem. Phys. 132, 024505 (2010)
X. Zheng, Y. Yan, and M. Di Ventra, Kondo memory in driven strongly correlated quantum dots, Phys. Rev. Lett. 111, 086601 (2013)
S. Chakravarty and A. J. Leggett, Dynamics of the two-state system with Ohmic dissipation, Phys. Rev. Lett. 52, 5 (1984)
Y. Zhou and J. Shao, Solving the spin-boson model of strong dissipation with flexible random-deterministic scheme, J. Chem. Phys. 128, 034106 (2008)
F. Lesage and H. Saleur, Boundary interaction changing operators and dynamical correlations in quantum impurity problems, Phys. Rev. Lett. 80, 4370 (1998)
G. M. Whitesides, Reinventing chemistry, Angew. Chem. Int. Ed. 54, 3196 (2015)
H. Primas, Chemistry, Quantum Mechanics and Reductionism: Perspectives in Theoretical Chemistry, Lecture Notes in Chemistry (Springer, Berlin, 1983)
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Yan, YA., Shao, J. Stochastic description of quantum Brownian dynamics. Front. Phys. 11, 110309 (2016). https://doi.org/10.1007/s11467-016-0570-9
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DOI: https://doi.org/10.1007/s11467-016-0570-9