Dissipative Quantum Methods

The dynamical methods used for treating atom-surface dynamics should be able to include the effect of changing the surface temperature. Thus the average energy transfer (the energy accommodation) should depend upon the surface temperature. Experimentally one finds an energy accommodation 

The detailed-balance corrected quantum-classical method described in section 8.2 is capable of giving this surface temperature dependence [62]. More rigorous methods involve the solution of the Liouville-von Neumann equation [159]. However, this appears to be close to impossible for systems involving the number of atoms and processes of interest for molecule-surface scattering [160]. At least this is so unless approximations are introduced, for instance, combining classical and quantum mechanical treatments or self-consistent field approximations [161]. 

An interesting alternative is the semigroup theory of Lindblad, Davies, and Kossakowski [162,165,163]. This and similar theories have been used in molecule surface scattering by Kosloff [164] and Jackson [166]. 

In order to incorporate the surface temperature dependence we need to introduce an ensemble average over the states of the solid. Thus, introducing the state vector 1 n(0)y we obtain the density operator for the total system 

The semigroup theory results in a quantum mechanical Markovian master equation for the evolution of the density operator p. Ceijan and Kosloff [164] introduced the dissipative Liouville operator [165], which in the Heisenberg representation yields the equations of motion for an operator O as 

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Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

A rather general method of the calculation of the tunneling taking account of the dissipation was given in Ref. 82. The cases of rather strong dissipation were considered in Refs. 81 and 82, where it was assumed that a thermodynamical equilibrium in the initial potential well exists. The case of extremely weak friction has been considered using the equations for the density matrix in Ref. 83. A quantum analogue of the Focker-Planck equation for the adiabatic and nonadiabatic processes in condensed media was obtained in Refs. 105 and 106. 

Results for two types of model systems are shown here, each at the two different inverse temperatures of P = 1 and P = 8. For each model system, the approximate correlation functions were compared with an exact quantum correlation function obtained by numerical solution of the Schrodinger equation on a grid and with classical MD. As noted earlier, testing the CMD method against exact results for simple one-dimensional non-dissipative systems is problematical, but the results are still useful to help us to better imderstand the limitations of the method imder certain circumstances. 

During the past few decades, various theoretical models have been developed to explain the physical properties and to find key parameters for the prediction of the system behaviors. Recent technological trends focus toward integration of subsystem models in various scales, which entails examining the nanophysical properties, subsystem size, and scale-specified numerical analysis methods on system level performance. Multi-scale modeling components including quantum mechanical (i.e., density functional theory (DFT) and ab initio simulation), atom-istic/molecular (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational... 

Consider, now, how the energy is dissipated. In the absence of B, A may lose its energy either as fluorescence emission or in some non-radiative process such as interaction with the solvent. On the convention used by Forster and Weller the rate coefficients or probabilities for these two processes are denoted by and sec S respectively. The lifetime Tq of the exdted spedes is then ( f+nquantum yield < >o of the fluorescence process is f/( f+nthird method of energy dissipation from A is by reaction with B for this a pseudo-first order rate coeffident k2C sec may be assigned. The lifetime of A (x) is now given by... 

In the case of s linearly damped oscillator, the transformation of the Heisenberg picture into the Schrodinger picture by the method applied in classical quantum theory is impossible because the operator has a time-dependent part due to the dissipative process. Thus, a new way must be found to construct the wave equation of the oscillator. Kostin introduced a supplementary dissipation potential into his wave equation and constructed this dissipation potential by an assumption that the energy eigenvalues of the oscillator decay exponentially over time [39]. In Kostin s version of the wave equation, the operators are time independent, but the dissipation potential is nonlinear with respect to the wave function. In our theory, it is assumed that the abstract wave equation of the linearly damped oscillator has the form... 

Jin J, Zheng X, Yan YJ (2008) Exact dynamics of dissipative electronic systems and quantum transport hierarchical equations of motion approach. J Chem Phys 128 234703 Mathews J, Walker R (1970) Mathematical methods of physics. Benjamin, New York Croy A, Saalmann U (2009) Partial fraction decomposition of the Fermi function. Phys Rev B 80 073102. doi 10.1103/PhysRevB.80.073102. http //link.aps.org/doi/10.1103/ PhysRevB.80.073102... 

It is obvious that in the real physical situations we are not able to avoid dissipation processes. For dissipative systems, we cannot take an external excitation too weak (the parameter e cannot be too small) since the field interacting with the nonlinear oscillator could be completely damped and hence, our model could become completely unrealistic. Moreover, the dissipation in the system leads to a mixture of the quantum states instead of their coherent superpositions. Therefore, we should determine the influence of the damping processes on the systems discussed here. To investigate such processes we can utilize various methods. For instance, the quantum jumps simulations [38] and quantum state diffusion method [39] can be used. Description of these two methods can be found in Ref. 40, where they were discussed and compared. Another way to investigate the damping processes is to apply the approach based on the density matrix formalism. Here, we shall concentrate on this method [12,41,42]. 

In order to improve the model further we are currently taking quantum effects in the lattice into account, i.e. treating the CH units not classically but on quantum mechanical basis. To this end we use an ansatz state similar to Davydov s so-called ID,> state [96] developed for the description of solitons in proteins. However, there vibrations are coupled to lattice phonons, while in tPA fermions (electrons) are coupled to the lattice phonons. The results of this study will be the subject of a forthcoming paper. Further we want to improve the description of the electrons by going to semiempirical all valence electron methods or even to density functional theories. Further we introduce temperature effects into the theory which can be done with the help of a Langevin equation (random force and dissipation terms) or by a thermal population of the lattice phonons. Starting then the simulations with an optimized soliton geometry in the center of the chain (equilibrium position) one can study the soliton mobility as function of temperature. Further in the same way the mobility of polarons can be... 

It is impossible to predict the complete dissipation factor (tan 8) curves of polymers as functions of temperature and frequency without detailed consideration of relaxation times. At present, the a priori estimation of relaxation times requires detailed computer-intensive calculations, such as force field or quantum mechanical methods to estimate rotation barriers, and molecular dynamics simulations. A much less ambitious goal was therefore pursued. A simple variable was sought, for use in an order of magnitude estimate of the "lossiness" of a polymer at room temperature over the most important frequency range for typical applications. 

A different theory of local control has been derived from the viewpoint of global optimization, applied to finite time intervals [58-60]. This approach can also be applied within a classical context, and local control fields from classical dynamics have been used in quantum problems [61]. In parallel, Rabitz and coworkers developed a method termed tracking control, in which Ehrenfest s equations [26] for an observable is used to derive an explicit expression for the electric field that forces the system dynamics to reproduce a predefined temporal evolution of the control observable [62, 63]. In its original form, however, this method can lead to singularities in the fields, a problem circumvented by several extensions to this basic idea [64-68]. Within the context of ground-state vibration, a procedure similar to tracking control has been proposed in Ref. 69. In addition to the examples already mentioned, the different local control schemes have found many applications in molecular physics, like population control [55], wavepacket control [53, 54, 56], control within a dissipative environment [59, 70], and selective vibrational excitation or dissociation [64, 71]. Further examples include isomerization control [58, 60, 72], control of predissociation [73], or enantiomer control [74, 75]. 

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