Spin boson

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

This model, called the spin-boson Hamiltonian, is probably the only fully manageable problem of this kind (with the possible exception of some very artificial problems) with a transparent solution. 

The solution of the spin-boson problem with arbitrary coupling has been discussed in detail by Leggett et al. [1987]. The displacement of the equilibrium positions of the bath oscillators in the transition results in the effective renormalization of the tunneling matrix element by the bath overlap integral... 

When the potential V Q) is symmetric or its asymmetry is smaller than the level spacing (Oq, then at low temperature (T cuo) only the lowest energy doublet is occupied, and the total energy spectrum can be truncated to that of a TLS. If V Q) is coupled to the vibrations whose frequencies are less than coq and co, it can be described by the spin-boson Hamiltonian... 

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, the exchange reactions of proton transfer in condensed media [Borgis et al. 1989 Suarez and Silbey 1991a Borgis and Hynes 1991 Morillo et al. 1989 Morillo and Cukier 1990]. 

The analytic results for the spin-boson Hamiltonian with fluctuating tunneling matrix element (5.67) are investigated in detail by Suarez and Silbey [1991a]. Here we discuss only the situation when the qi vibration is quantum, i.e., (o P P 1. When the bath is classical, cojP, j 1, the rate... 

Leggett et al. [1987] have set forth a rigorous scheme that reduces a symmetric (or nearly symmetric) double well, coupled linearly to phonons, to the spin-boson problem, if the temperature is low enough. However, in the case of nonlinear coupling (which is necessary to introduce in order to describe the promoting vibrations), no such scheme is known, and the use of the spin-boson Hamiltonian together with (3.67) relies rather on intuition, and is not always Justifiable. 

Thus the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model (5.55), (5.67). As an illustration, three-dimensional trajectories for various interrelations between symmetric (Ws) and antisymmetric (oja) vibration frequencies, and odo are shown in fig. 33. 

It is to be emphasized that, despite the formal similarity, the physical problems are different. Moreover, in general, diabatic coupling is not small, unlike the tunneling matrix element, and this circumstance does not allow one to apply the noninteracting blip approximation. So even having been formulated in the standard spin-boson form, the problem still remains rather sophisticated. In particular, it is difficult to explore the intermediate region between nonadiabatic and adiabatic transition. 

Skinner, J. L. and Trommsdorf, H. P. Proton transfer in benzoic acid crystals A chemical spin-boson problem. Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments, J.Chem.Phys., 89 (1988), 897-907... 

In cases where both the system under consideration and the observable to be calculated have an obvious classical analog (e.g., the translational-energy distribution after a scattering event), a classical description is a rather straightforward matter. It is less clear, however, how to incorporate discrete quantum-mechanical DoF that do not possess an obvious classical counterpart into a classical theory. For example, consider the well-known spin-boson problem—that is, an electronic two-state system (the spin) coupled to one or many vibrational DoF (the bosons) [5]. Exhibiting nonadiabatic transitions between discrete quantum states, the problem apparently defies a straightforward classical treatment. 

Although the systems discussed so far exhibit fairly complex vibrational and diabatic electronic relaxation dynamics, their adiabatic population dynamics is relatively simple and thus is well-suited for a SH description. To provide a challenge for the SH method, we next consider various spin-boson-type models, which may give rise to a quite comphcated adiabatic population dynamics. 

In order to get a first impression on the performance of the QC Liouville approach, it is instructive to start with a simple one-mode spin-boson model, that is. Model IVa [205]. In what follows, the QCL calculations used the first-order Trotter scheme (61) with a time step 8r = 0.05 fs. If not noted otherwise, we have employed the momentum-jump approximation (59) and the initial number of random walkers employed was N = 50 000. 

Although the classical mapping formulation yields the correct quantum-mechanical level density in the special case of a one-mode spin-boson model, the classical approximation deteriorates for mulhdimensional problems, since the classical oscillators may transfer their ZPE. As a hrst example. Fig. 21a compares Nc E) as obtained for Model I in the limiting cases y = 0 and 1 (thin solid lines) to the exact quantum-mechanical density N E) (thick line). The classical level density is seen to be either much higher (for y = 1) or much lower (for y = 0) than the quantum result. Since the integral level density can be... 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

DoF by introducing the variables A = A2 = Atot — N, Q = Qi Q [89]. Assuming furthermore that Atot = (A tot) = 1, we obtain the classical spin-boson Hamiltonian... 

S2 conical intersection in pyrazine as well as several spin-boson models, we discuss advantages and problems of this semiclassical method. 

The semiclassical mapping approach outlined above, as well as the equivalent formulation that is obtained by requantizing the classical electron-analog model of Meyer and Miller [112], has been successfully applied to various examples of nonadiabatic dynamics including bound-state dynamics of several spin-boson-type electron-transfer models with up to three vibrational modes [99, 100], a series of scattering-type test problems [112, 118, 120], a model for laser-driven... 

A variation of the small polaron problem is the spin-boson Hamiltonian, which also belongs to the 2-level limit and is now known to have very rich... 

An alternative but related approach has been taken by Silbey and Suarez in their study of hydrogen hopping in solids. Instead of a Marcus model they used the spin-boson Harrriltonian with a turmeling sphtting that has the form Eq. (30). The envirorrment as described in the spin-boson Hamiltorrian has not only slow dyrramics (as in the Marcus model), but fast modes as well. 

Q vibration is not directly coupled to the bath of harmonic oscillators. This assumption is similar to the approach employed by Silbey and Suarez who used a tunneling splitting that depends on the oscillating transfer distance Q in their spin-boson Hamiltonian. Borgis and Hynes, too, have made this assumption in the context of Marcus theory. 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

Indeed, consider the spin-boson model, with the Hamiltonian... 

Wilhelm, F. K., Kleff, S., and von Delft, J. (2004). The spin-boson model with a structured environment A comparison of approaches. Chem. Phys., 296 345. 

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, proton exchange reactions in condensed media [Borgis and Hynes, 1991 Borgis et al., 1989 Morillo et al., 1989 Morillo and Cukier, 1990 Suarez and Silbey, 1991], The early approaches to this model used a perturbative expansion for weak coupling [Silbey and Harris, 1983], Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term ftrz, where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the operators, daldt = i[H, ], have the form... 

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