Bosonic bath

In this section we will consider the case of a multi-level electronic system in interaction with a bosonic bath [288,289], We will use unitary transformation techniques to deal with the problem, but will only focus on the low-bias transport, so that strong non-equilibrium effects can be disregarded. Our interest is to explore how the qualitative low-energy properties of the electronic system are modified by the interaction with the bosonic bath. We will see that the existence of a continuum of vibrational excitations (up to some cut-off frequency) dramatically changes the analytic properties of the electronic Green function and may lead in some limiting cases to a qualitative modification of the low-energy electronic spectrum. As a result, the I-V characteristics at low bias may display metallic behavior (finite current) even if the isolated electronic system does exhibit a band gap. The model to be discussed below... 

E) = t P E) is the crucial contribution to the GF since it contains the influence of the bosonic bath. Note that gT(E) includes the transversal hopping t to all orders, the leading one being t. ... 

The first summand has the same form as Landauer s expression for the current with an effective transmission function t(E) = Tt[G1TrC /r]. However, the reader should keep in mind that the GFs appearing in this expression do contain the full dressing by the bosonic bath and hence, t(E) does not describe elastic transport. The remaining terms contain explicitly contributions from the bath. It can be shown after some transformations that the leading term is proportional to (tjJ2 so that within a perturbative approach in t and at low bias it can be approximately neglected. We therefore remian with the exression / = yf / dETr(fh E) - /r(F)) t(E) to obtain the current. 

To describe the effect of the environment one usually needs to determine the bath correlation function C(t). Let us start discussing this function for a bosonic bath where the subscript Ph indicates a bath of phonons. Using the numerical decomposition of the spectral density Eq. (2) together with the theorem of residues one obtains the complex bath correlation functions... 

Similar to the case of the bosonic bath, one may also define correlation functions in the case of fermionic reservoirs. In contrast to the former case, two different correlation functions will be introduced since there are also two different parts of the system-reservoir interaction one creates and one annihilates an electron in the wire. The correlation functions are given by... 

After discussing the properties of the environments, let us now turn to the dynamics of a system in a dissipative environment, i.e. a system coupled to a bosonic bath. The case of fermionic reservoirs will be treated in the next section. 

As for the case of a bosonic bath, the Hamiltonian describing the molecular junction is separated into the relevant system Hs(t), describing the wire, the field-matter interaction Hp(t) and reservoirs Hr modeling the leads... 

As for the case of a bosonic bath, the starting point here is the TL approach and a TL QME based on a second-order perturbation theory in the molecule-lead coupling was developed for the reduced density matrix p(t) of the molecule [38,65]... 

Using a numerical decomposition of the spectral density which describes the coupling of the system to the environment allows one to develop TL and TNL non-Markovian QMEs. Using the hierarchical approach the results can be extended from second-order perturbation theory to higher orders to be able to study the convergence properties of the different approaches. As shown in the example for bosonic baths, the TL formalism shows numerically almost converged results. Actually, this numerical finding has been analytically proven... 

Let us now return to the two model Hamiltonians introduced in Section 12.2, and drop from now on the subscripts S and SB from the coupling operators. Using the polaron transformation we can describe both models (12.1), (12.2) and (12.4)-(12.6) in a similar language, where the difference enters in the form ofthe coupling to the boson bath... 

Equations (12.28) and (12.29) describe different spin-boson models that are commonly used to describe the dynamics of a two-level system interacting with a boson bath. Two comments are in order ... 

The coupling to the boson bath can change this in a dramatic way because initial levels of the combined spin-boson system are coupled to a continuum of other levels. Indeed Fig. 12.1 can be redrawn in order to display this feature, as seen in Fig. 12.4. Two continuous manifolds of states are seen, seating on level 1 and 2, that encompass the states l,v ) = 1) na Iv ) and 2,v) = 2) Ha Pa) with zero-order energies TTpv and 2.v, respectively, where... 

This implies that the effective spin-bath/system coupling is large, and the noise correlation gets enhanced. Thus, quantum dissipation is more facilitated as the temperature is lowered. As T- 0, the noise correlation of the spin-bath (Equation 9.33) tends to that of the bosonic bath. 

A major focus of the theory of quantum dissipation in a spin-bath is the conspicuous thermal behavior of the reservoir. Our analysis clearly shows that, at temperatures close to zero, a spin-bath behaves almost in the same way as a bosonic bath, implying a universality in the nature of bath as TWO. At higher temperatures (below saturation temperature), the system-bath coupling tends to diminish, which is reflected in the emergence of coherence in the dynamics, and the behavior of a spin-bath differs significantly from that of a bosonic bath. In what follows, we consider two specific examples to illustrate these aspects. 

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