This thesis is concerned with the study of quantum systems coupled linearly to a continuous bath
of oscillators examples of which are the spin-boson model and the Nelson model. The main theme
throughout is to develop a better understanding of the properties of the bath of oscillators such
that more efficient representations of it can be made to facilitate the understanding of the system
dynamics. The main difficulty in simulating the system dynamics is that the bath of oscillators
composes an infinite number of degrees of freedom. In this thesis, we investigate the mathematical
properties of an approach in which the bath modes are written as a semi-infinite chain of nearest
neighbour interacting harmonic oscillators such that the efficient time dependent density-matrix
renormalisation group (t-DMRG) methods can be applied for simulation.
In the first section, we show how there are many different ways to represent the bath as semiinfinite
chains and prove that seemingly unrelated methods can all be achieved using the same
mathematical formalism. We show that in an iterative process the bath can be transformed into
a chain of oscillators with nearest neighbour interactions. This is achieved using the formalism
of orthogonal polynomials. This allows one to define a sequence of residual spectral densities at
each site along the chain. We show that this sequence of residual spectral densities is provided by
the so-called ”sequence of secondary measures”. We derive a systematic procedure to obtain the
spectral density of the residual bath in each step. We find that these residual spectral densities are
related to an old abstract problem in mathematics known as the ”secondary measures”. We solve
this problem from the field of orthogonal polynomials to give an explicit expression for the residual
spectral densities and go on to prove that these functions converge under very general conditions.
That is, the asymptotic part of the chain is universal, translation invariant with universal spectral
density. These results suggest efficient methods for handling the numerical treatment of the residual
bath.
In the second section, we take a different approach. Rather than studying the properties of
residual baths, we look at how system observables are affected by truncating the semi-infinite chain
of harmonic oscillators to a finite length chain. By developing locality bounds for the dynamics,
we derive an upper bound to the error introduced by such a truncation and show that for all finite
times it can be made arbitrarily small by including a sufficient number of harmonic oscillators
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before truncating. Furthermore, it is shown that the speed at which the system communicates
with different harmonic oscillators in the chain is proportional to the maximum frequency of the
environment but that this speed also depends on the particular version of the chain. These bounds
are given for when the dynamics are calculated in the Interaction picture and in the Schrodinger
picture.