Dissipaton Equation Of Motion Approach To Non-linear Coupling Bath Effects

Quantum dissipation is one of the most important topic in quantum statistical me-chanics,thus plays a very important role in many fields of modern science.Feynman-Shannon influences functional path integral is a rigorous theory of quantum dissipation,but only for Gaussian environments.It also has the issue of numerical difficulty due to large computation cost.Hierarchical equation of motion(HEOM)formalism can be constructed based on the path integral theory,followed by consecutive time derivatives.HEOM formalism exploits certain sum-over-poles decomposition on spectral function of bath,via a series of auxiliary density operators.It has a great improvement in per-formance of numerical efficiency and application to various dynamics,comparing to path integral formalism.However,this method has similar disadvantages as the path integration method,as it only applies to Gaussian environments.Under strong system-bath interaction,non-Markov dissipation,and time-correlated external driving,the bath does not necessarily satisfy Gaussian statistics.Nonlinear coupling must be taken into account.Another drawback of HEOM formalism is lacks of physical meanings of its dynamical variable.Hybridizing dynamics of both system and coupled bath can hardly be simulated under HEOM.To overcome these challenges,we recently developed dissipation equation of mo-tion(DEOM)formalisim,which describes not only the reduced system,but also collec-tive bath dynamics.Those purely mathematical auxiliary density operators in HEOM were replaced by dissipaton density operators,which has a clear physical description on baths.The underlying dissipaton algebra,especially the generalized Wick's theorem can deal with quadratic-bath-couplimg case.In chapter 1,we introduced the background of quantum dissipation theories.Since most of then require Gaussian bath to work,we make a comment on Gaussian environ-ments.In chapter 2,the extended DEOM theory will be presented.According to fluctuation-dissipation theorem,the influence of bath coupling can be characterized by spectral density.Apply a sum-over-poles decomposition,we can describe the influ-ence of environment with a finite number of statistically independent quasi-particles,the dissipatons.The dissipaton algebra consists of the general diffusion equation and the generalized Wick's theorems.Universal DEOM evaluations will be implemented in DEOM-space,which maps density operator and Liouvillian to their correspondences form in DEOM-space.At last,we revisit the DEOM dynamics via standard HEOM construction,which de facto validates the dissipaton algebra.In chapter 3,our main purpose is to validate the dissipaton algebra with nonlinear coupling bath environments.Two approaches are applied.One is extended Fokker-Planck quantum master equation(FP-QME),which will be constructed based on its DEOM correspondence.The extended FP-QME is carried out with a straightforward algorithm.The underlying equivalence between these dynalics formalisms thus vali-date the nonlinear coupling dissipaton algebra.Another approach is extended Zusman equation,which is derived via a totally different way.We proved that the DEOM for-malism,if it started with the same setup,constitutes the dynamical resolutions to the extended Zusman equation.Therefore,as the algebraic construction is concerned,we would have also confirled the DEOM formulations.In chapter 4,the nonlinear coupling bath descriptors were concerned.The charac-terizations of nonlinear bath couplings should be physically well supported.To that end,we proposed a linear-displacement-mapping(LDM)scheme,together with the refercnce-bath invariance requirement.The quadratic coupling strength does influence the linear descriptor,but not the other way around.In chapter 5,we presented semiclassical(SC)approach for quantum dissipative dynamics,based on HEOM formalism,which is a highly numerically efficient method.The dynamical components in SC-HEOM are then different order phasespace moments of not only the primary reduced system density operator but also the auxiliary density operators of HEOM.Although this algorithm is based on HEOM fonnalism,it is pos-sible to apply to DEOM formnalism,due to equivalence between two approach.In chapter 6,we conclude this thesis,and discuss about the future work and appli-cations.