System-bath coupling spin-boson Hamiltonian

When the system-bath coupling is linear in the bath coordinates, as in the spin-boson Hamiltonian, the physical interpretation is that the minimum position of each bath oscillator is shifted proportionately to the value of the system variable to which it is coupled. The small-polaron transformation redefines the Hamiltonian in terms of oscillators shifted adiabatically as a function of the system coordinate here the system coordinate is tr, so that the oscillators will be implicitly displaced equally but in opposite directions for each quantum state. Note that in the limit that the TLS coupling J vanishes, this transformation completely separates the system and bath. This makes it an effective transformation for cases of small coupling, and it has in fact been long and widely used in many types of physical problems, although typically in a nonvariational form [102]. Harris and Silbey showed that while simple enough to handle analytically, a variational small-polaron transformation contained the flexibility to treat the spin-boson problem effectively in most parameter regimes (see below) [45-47]. 

The Hamiltonian equations 15.1-15.4 is applicable to various processes characteristic of molecular systems, including the dynamics at conical intersections (Coin s) [1-4] and excitation energy transfer (EET) processes [5,6,8]. Its simplest realization corresponds to a single system operator, in which case the classical spin-boson Hamiltonian [12,18] is obtained, where the bath coordinates couple to the energy gap operator = a) (a — J3) (J31 of a two-level system (TLS). 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

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