Non-Markovian dissipation

Recently we have constructed a complete second-order QDT (CS-QDT), in which all excessive approximations, except that of weak system-bath interaction, are removed [38]. Besides two forms of CS-QDT corresponding to the memory-kernel COP [Eq. (1.2)] and the time-local POP [Eq. (1.3)] formulations, respectively, we have also constructed a novel CS-QDT that is particularly suitable for studying the effects of correlated non-Markovian dissipation and external time-dependent field driving. This paper constitutes a review of the three nonequivalent CS-QDT formulations [38] from both theoretical and numerical aspects. Concrete comparisons will be carried out in connection with the exact results for driven Brownian oscillator systems, so that sensible comments on various forms of CS-QDT can be reached. Note that QDT shall describe not only the evolution of p(t), but also the reduced thermal equilibrium system as p t oo) = peq(7 )-... 

To construct a CS-QDT formulation, one shall treat the effects of system-bath coupling H [Eq. (2.1c)] to the second order exactly for not only the reduced density operator p t) evolution, but also the initial canonical state of the total composite system, Pt( o —oo) = p (T), before the external field excitation. Various CS-QDT formulations differ at their partial resummation schemes for the higher order contributions. We have recently arrived at three forms of CS-QDT in terms of differential equations of motion [38]. Two of them are in principle equivalent to the conventional second-order COP [Eq. (1.2)] and POP [Eq. (1.3)] formulations (cf. AppendixB). For the sake of clarity, we shall present here only the unconventional one that may be particularly suitable for the numerical study of non-Markovian dissipation in the presence of external time-dependent fields. 

The conventional quantum master equation approach remains of great value for its transparent physical implications. This approach can be exact for harmonic system with arbitrary non-Markovian dissipation, but for anharmonic systems it is only valid in the Markovian limit, together with high... 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

If the phonon distribution of the model Eq. (8) spans a dense spectrum - as is generally the case for the extended systems under consideration, which are effectively infinite-dimensional - the dynamics induced by the Hamiltonian will eventually exhibit a dissipative character. However, the effective-mode construction demonstrates that the shortest time scales are fully determined by few effective modes, and by the coherent dynamics induced by these modes. The overall picture thus corresponds to a Brownian oscillator type dynamics, and is markedly non-Markovian [81,82],... 

Equations (5.7) were introduced so as to treat the non-Markovian process of Eq. (5.8) in the frame of the time-independent Fokker-Planck formalism. The equivalence has been shown to require that the fluctuation-dissipation relationship (5.10) holds the white noise limit can then be recovered by making t vanish for a fixed value of D. If we substitute Eq. (5.10) into Eqs. 

In principle, since the potential V is not linear, Eqs. (4.1) can be used to simulate the non-Gaussian non-Markovian behavior of the variables < > and V as well as the rototranslational phenomena. Note that this non-Gaussian, non-Markovian behavior depends on the presence of the virtual body and the nonlinear nature of the potential V in spite of the Markovian-Gaussian character of the fluctuation-dissipation process governing the stodiastic torques and the stochastic forces Note also... 

Chemical Reactions Driven by Bona Fide Non - Markovian Fluctuation-Dissipation Processes... 

A more reahstic and more general treatment would presumably lead to a set of equations like Eqs. (52), with the potential V(x) fluctuating as a consequence of couplings with nonreactive modes (see Section III). For the sake of simplicity, we study separately the two different aspects. While Section III was devoted to pointing out the role of multiphcative fluctuations (derived from nonlinear microscopic Liouvillians) in the presence of additive white noise, this subsection is focused on the effects of a non-Markovian fluctuation-dissipation process (with a time convolution term provided by a rigorous derivation from a hypothetical microscopic Liouvillian) in the presence of a time-independent external potential. 

We believe that the arguments above should convince the reader that the interesting phenomenon detected by Carmeli and Nitzan is another manifestation of the decoupling effect, well understood at least since 1976 (see ref. 86). The only physical systems, the dissipative properties of which are completely independent of whether or not an external field is present, are the purely ideal Markovian ones. Non-Markovian systems in the presence of a strong external field provoking them to exhibit fast oscUlations are characterized by field-dependent dissipation properties. These decoupling effects have also been found in the field of molecular dynamics in the liquid state studied via computer simulation (see Evans, Chapter V in this volume). 

NON-MARKOVIAN QUANTUM DISSIPATION IN THE PRESENCE OF EXTERNAL FIELDS... 

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

NON-MARKOVIAN QUANTUM BATH AND FLUCTUATION-DISSIPATION THEOREM... 

NON-MARKOVIAN QUANTUM DISSIPATION THEORY then assumes [cf. Eq. (4.1)]... 

Non-Markovian quantum dissipation in the presence of external fields 7... 

The key quantity in quantum dissipative dynamics is the reduced system density operator, ps(t) = trBPT(0> Ihe bath-subspace trace over the total composite density operator. It is worth mentioning here that the harmonic bath described above assumes rather Gaussian statistics for thermal bath influence. Realistic anharmonic environments usually do obey Gaussian statistics in the thermodynamic mean field limit. For general treatment of nonperturbative and non-Markovian quantum dissipation systems, HEOM formalism has now emerged as a standard theory. It is discussed in the next section. 

Note that the Markovian dissipative dynamical process is governed by a frequency -independent Il-dissipator in eqn (13.48) that also implies an 5-in-dependent /C-tensor here, while the Markovian kinetic rate process is governed by the constant rate matrix, Al(j) = iC(0). Equation (13.52) would indicate non-Markovian rates in general, even with Markovian dissipative dynamics. However, kinetic rates are physically concerned with post-coherence events, in which the coherence-to-coherence dynamics timescale, the magnitude of l ccl is short compared with the relevant of interest. Therefore, the kinetic rate matrix of eqn (13.52) in the kinetics regime is often of K s) K K 0) = - /Cpp -I- /Cpc cc cp, where /Cpp = 0 in the absence of level relaxations. 

Keywords Cope rearrangement Diels-Alder reaction Optimal control Dissipative dynamics Non-Markovian quantum master equation... 

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